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PROJECT 
PHYSICS 




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PROJECI 
PHYSICS 




Directors of Harvard Project Physics 

F. James Rutherford 

Department of Science Education, New York University 

Gerald Hoiton 

Department of Physics, Harvard University 

Fletcher G. Watson 

Harvard Graduate School of Education 

Special Consultants on the 1981 edition of the Te^t: 
Professor Jay Blake, Hansard University 
Dr. Thomas von Foerster, American Institute of Physics 
Professor living Kaplan, Massachusetts Institute of Technology 

Editorial Deve/opmenf; William N. Moore, Roland Cormier, Lorraine Smith-Phelan 

Editorial Processing: Margaret M. Byrne, Regina Chilcoat, Holly Massey 

Art, Production, and Photo Resources: Vivian Fenster, Fred C. Pusterla, Robin M. 

Swenson, Annette Sessa, Beverly Silver, Anita Dickhuth, Dorina Virdo 
Product Manager: Laura Zuckerman 
Advisory Board: John I'aggart, Maurice E. Fey, Norman Hughes, David J. Miller, 

John W. Griffiths, William L. Paul 
Consultant: John Matejowsky 
Researchers: Pamela Floch, Gerard LeVan 



Acknowledgments begin on page 742 
Picture Credits begin on page 744 



Copyright © 1981, 1975, 1970 by Project Physics 

All Rights Reserved 

Printed in the United States of America 

ISBN D-D3-DSSim-E 

8 -032-98 
Project Physics is a registered tiademark. 



Science is an adventure of the whole human race to learn to live in 
and perhaps to love the universe in which they are. To be a part of it is 
to understand, to understand oneself, to begin to feel that there is a 
capacity within man far beyond what he felt he had, of an infinite e;cten- 
sion of human possibilities. . . . 

I propose that science be taught at whatever level, from the lowest to 
the highest, in the humanistic way. It should be taught with a certain 
historical understanding, with a certain philosophical understanding, 
with a social understanding and a human understanding in the sense of 
the biography, the nature of the people who made this construction, the 
triumphs, the trials, the tribulations. 

I. I. Rabi, Nobel Laureate in Physics 



Preface 



Back^ound 

The Project Physics Course is based on the ideas and research of a national 
curriculum development project that worked for eight years. 

Preliminary results led to major grants fi^om the U.S. Office of Education and the 
National Science Foundation. Invaluable additional financial support was also 
provided by the Ford Foundation, the Alfred P. Sloan Foundation, the Carnegie 
Corporation, and Haivard University. A large number of collaborators were brought 
together from all parts of the nation, and the group worked together intensively for 
over four years under the title Harvard Project Physics. The instructors serving as field 
consultants and the students in the trial classes u^ere also of vital importance to the 
success of Haivard Project Physics. As each successive experimental version of the 
course was developed, it was tried out in schools throughout the United States and 
Canada. The instiuctors and students reported their criticisms and suggestions to the 
staff in Cambridge. These reports became the basis for the next year's revision. The 
number of participating instrvictors during this period grew to over 100. Five thousand 
students participated in the last year of tiyout in a large-scale formal research 
program to evaluate the results achieved with the course materials. Thereafter, the 
trial materials were again rewritten. The final version has been revised twice for new 
editions. 

Aims 

From the beginning, Harvard Project Physics had three major goals in mind. These 
were to design a humanistically oriented physics course, to attract more students to 
the study of introductory physics, and to find out more about the factors that 
influence the learning of science. The last of these goals involved extensive 
educational research, and has been reported to the teaching profession in books imd 
journals. 

The challenge facing us was to design a humanistic course that would be useful 
and interesting to students with widely differing skills, backgrounds, and career plans. 
In practice, this meant designing a new couse that would have the following effects: 

1. To help students increase their knowledge of the physical world by 
concentrating on ideas that characterize physics as a science at its best, rather than 
concentrating on isolated bits of infomiation. 

2. To help students see physics as the wonderfully many-sided human activity that 
it really is. This meant presenting the subject in historical and cultural perspective, 

PREFACE iii 



and shov\ing that the ideas ot physics ha\'e a traditicjn as well as ways ol e\oluticjnaiy 
adaptation and change. 

3. To increase the opportunits' foi- t^ac'h student to ha\'e inimediatc'K' lenvaicling 
experiences in science even v\hile gaining the knowUnlge and skill that will be useful 
in the long run. 

4. To make it possible for insti\ictots to adapt the courst; to the wide range of 
interests and abilities of their students. 

o. To take into account the impoilance of the instiiictoi- in \hv. educational process, 
and the \'ast spectriim of teac^hing situations that pi(;\ail. 

Unhappily, it is not feasible to list in detail the contributions of each person who 
par'tiri[:)ated in some pari of Harvard Project Physics. Previous editions ha\'e included 
a partial list of the contributor's. VVe take partic'irlar' pleasuix? in acknowl(Klging the; 
assistance of I3i'. Andrew Ahlgr-en of the Unixersity of Minn(!S(jta. Dr-. /Xhlgren was 
invaluable because of his skill as a physics instructor-, his editorial talent, his xer'satility 
and energy, and above all, his commitment to the goals of Harvard Project Physics. 

VV(! would also especially like to thank Ms. Joan Laws, whose administiati\e skills, 
dep(Midabilit\', and thoughtfulness Cf)ntributed so much to our- work. Holt, Rinehart 
and Winston, Publishers, of New Yor-k, provided the coordination, editorial sirpport, 
and general backing necessary to the lar^ge under-taking of preparing the final version 
of all components of the Project Physics Course. Damon-Educational Division located 
in VV-estwood, Massachusetts, worked closely with us to improve the engineering 
design of the authorized laboratory apparatus and to see that it was properly 
integrated into the program. 

Since their last use in experimental for^m, all of the instructional materials have 
been more closely integrated and rewritten in final form. The course now consists of a 
lar'ge variety' of coorxlinated learning materials of which this textbook is only one. With 
the aid of these materials and the guidance of the insti'uctor-, with tin; student's own 
interest and effort, every student can look forward to a successful and worthwhile 
experience. 

In the years ahead, the learning materials of the Project Physics Course will be 
r-e\ised as often as is necessary' to r-emoxe r-emaining ambiguities, to clarify 
instructions, and to continue to make the materials more inter^esting and relexant to 
the students. We therefore urge all who use this course to send to us (in care of Holt, 
Rinehart and Winston, Publishers, 383 Madison Avenue, New York, New York 10017) 
an\ criticisms or" suggestions they may have. And now, welcome to the study of 
physics! 

F. James Rutherford 
Gerald Holton 
Fletcher G. Watson 



IV PIU:;iACE 



Contents 

Unit 1 • Concepts of Motion 

PROLOGUE 1 

Chapter 1 / The Lan^ag^e of Motion 8 

1.1 The motion of things 8 

1.2 A motion experiment that does not quite work 9 

1.3 A better experiment 11 

1.4 Leslie's swim and the meaning of average speed 14 

1.5 Graphing motion and finding the slope 17 

1.6 Time out for a warning 22 

1.7 Instantaneous speed 23 

1.8 Acceleration by comparison 28 

Chapter 2 / Free Fall: Galileo Describes Motion 36 

2.1 The Aristotelian theory of motion 36 

2.2 Galileo and his times 41 

2.3 Galileo's Two New Sciences 43 

2.4 Why study the motion of freely falling bodies? 46 

2.5 Galileo chooses a definition of uniform acceleration 47 

2.6 Galileo cannot test his hypothesis directly 49 

2.7 Looking for logical consequences of Galileo's hyjaothesis 50 

2.8 Galileo turns to an indirect test 53 

2.9 Doubts about Galileo's procedure 55 

2.10 Consequences of Galileo's work on motion 56 

Chapter 3 / The Birth of fh^^namics: Netvton Explains Motion 67 

3.1 "Explanation " and the laws of motion 67 

3.2 The Aristotelian explanation of motion 70 

3.3 Forces in equilibrium 71 

3.4 About vectors 74 

3.5 Newton's first law of motion 76 

3.6 The significance of the first law 79 

3.7 Newton's second law of motion 79 

3.8 Mass, weight, and free fall 85 

3.9 Newton's third law of motion 87 

3.10 Using Newton's laws of motion 90 

3.11 Nature's basic forces 92 

Chapter 4 / Understanding Motion 101 

4.1 A trip to the moon 101 

4.2 Projectile motion 103 

4.3 What is the path of a projectile? 106 

4.4 Moving fi^ames of reference: Galilean relativity 108 

4.5 Circular motion 110 

4.6 Centripetal acceleration and centripetal force 113 

4.7 The motion of earth satellites 117 

4.8 What about other motions? Retrospect and prospect 120 

EPILOGUE 126 

CONTENTS V 



Unit 2 • Motion in the Heavens 



PROLOGUE 128 

Chapter 5 / Where is the Earth? ITie Greeks' /Vnswer 133 

5.1 Motions of the sun and stars 133 

5.2 Motions of the moon 138 

5.3 The "wandering" stars 139 

5.4 Plato's problems 142 

5.5 rhe Greek idea of "explanation " 143 

5.6 The first earth-centered solution 144 

5.7 A sun-centered solution 146 

5.8 The geocentric system of Ptolemy 148 

5.9 Successes and limitations of the Ptolemaic^ model 152 

Chapter 6 / Does the Earth iVIove? The Work of Copernicus anel Ticho lUS 

6.1 rhe Copernican system 155 

6.2 New conclusions 159 

6.3 Arguments for the Copernican system 164 

6.4 Arguments against the Copernican system 166 

6.5 Historical consequences 169 

6.6 Tycho Brahe 170 

6.7 TVcho's observations 173 

6.8 Tycho s compromise system 175 

Chapter 7 / A New Universe Appears: The Work of Kepler and G«ilileo 180 

7.1 rhe abandonment of uniform circular motion 180 

7.2 Kepler's law of areas 183 

7.3 Kepler's law of elliptical orbits 185 

7.4 Kepler's law of periods 188 

7.5 The new concept of physical law 191 

7.6 Galileo and Kepler 192 

7.7 The telescopic evidence 193 

7.8 Galileo focuses the controversy 196 

7.9 Science and freedom 199 

Chapter 8 / The Vnhy of Earth and Skv: The Work of Nei«^on 205 

8.1 Xevvlon and seventeenth-cenluiy science 205 

8.2 Newton's Pr/nc/p/a 211 

8.3 The inverse-square law of planetaiy force 213 

8.4 Law of universal gravitation 216 

8.5 Neulon and hvpotheses 221 

8.6 The magnitude of planetary force 224 

8.7 Planetary motion and the gravitational constant 227 

8.8 The value of G and the actual masses of the planets 229 

8.9 Fuilher successes 231 

8.10 Some effects and limitations of Neuron's work 236 

EPILOGUE 241 



*1 CXJMblMS 



Unit 3 • The Triumph of Mechanics 



PROLOGUE 246 

Chapter 9 / Conservation of Mass and Momentum 249 

9.1 Conservation of mass 249 

9.2 Collisions 253 

9.3 Conservation of momentuni 255 

9.4 Momentuni and Newton's laws of motion 260 

9.5 Isolated systems 263 

9.6 Elastic collisions 264 

9.7 Leibniz and the conservation law 266 

Chapter 10 / Energy 273 

10.1 Work and kinetic energy 273 

10.2 Potential energy 277 

10.3 Conservation of mechanical energy 279 

10.4 Forces that do no work 281 

10.5 Heat as energy 284 

10.6 The steam engine and the Industrial Revolution 287 

10.7 The efficiency of engines 294 

10.8 Energy in biological systems 297 

10.9 Arriving at a general consen^ation law 305 

10.10 The laws of thermodynamics 309 

10.11 Faith in the laws of thermodynamics 312 

Chapter 11 / The Kinetic Theory of Gases 321 

11.1 An overview of the chapter 321 

11.2 A model for the gaseous state 323 

11.3 The speeds of molecules 326 

11.4 The sizes of molecules 329 

11.5 Predicting the behavior of gases from the kinetic theory 331 

11.6 The second law of themiodynamics and the dissipation of energy 338 

11.7 Maxwell's demon and the statistical view of the second law of 
thennodynamics 340 

11.8 Time's arrow and the recurrence paradox 343 

Chapter 12 / Waves 354 

12.1 What is a wave? 354 

12.2 Properties of waves 355 

12.3 Wave propagation 358 

12.4 Periodic waves 359 

12.5 When waves meet: the superposition principle 362 

12.6 A two-source interference pattern 363 

12.7 Standing waves 368 

12.8 Wave fronts and diffraction 371 

12.9 Reflection 375 

12.10 Refraction 377 

12.11 Sound waves 380 

EPILOGUE 390 



COIVTENTS vii 



Unit 4 • Lig^ht and Electromag^netism 

PROLOGUE 392 

Chapter 13 / Light 396 

13.1 Introduction: VA'hat is light? 396 

13.2 Propagation of light 397 

13.3 Reflection and refraction 402 

13.4 Interference and diffraction 404 

13.5 Color 406 

13.6 Why is the sky blue? 411 

13.7 Polarization 413 

13.8 The ether 415 

Chapter 14 / Eleetric and Ma^etic Fields 421 

14.1 Introduction 421 

14.2 The curious properties of lodestone and ambei-: Gilbeil's De 
Magnete 422 

14.3 Electric charges and electric forces 425 

14.4 Forces and fields 432 

14.5 The smallest charge 438 

14.6 The law of conservation of electric charge 439 

14.7 Electric currents 442 

14.8 Electric potential difference 443 

14.9 Electric potential difference and current 446 

14.10 Electric potential difference and power 447 

14.11 Currents act on magnets 450 

14.12 Currents act on currents 453 

14.13 Magnetic fields and moving charges 455 

Chapter 15 / Faraday and the Electrical A^e 464 

15.1 The problem: Getting energy' from one place to another 464 

15.2 Faraday's first electric motor 466 

15.3 The discov'eiy of electromagnetic induction 467 

15.4 Generating electricity by the use of magnetic fields: The dynamo 471 

15.5 The electric motor 474 

15.6 The electric light bulb 477 

15.7 AC versus dc, and the Niagara Fiills power plant 480 

15.8 Electricity and society 485 

15.9 Alternate energy sources 490 

15.10 The efficiency of an electric- power plant 491 

Chapter 16 / Electromagnetic Radiation 499 

16.1 Introduction 499 

16.2 Maxwell's fomiulation of the principles of electromagnetism 500 

16.3 The propagation of electromagnetic waves 503 

16.4 Hertz's experiments 506 

16.5 The electromagnetic spectrum 510 

16.6 What about the ether now? 517 

EPILOGUE 522 



VIU CO\TL\TS 



Unit 5 • Models of the Atom 



PROLOGUE 524 

Chapter 17 / A Siunmaiy of Some Ideas from Chemistry 532 

17.1 Elements, atoms, and compounds 532 

17.2 Electricity and chemistry 535 

17.3 The periodic table 536 

Chapter 18 / Electrons and Quanta 540 

18.1 The idea of atomic structure 540 

18.2 Cathode rays 542 

18.3 The measurement of the charge of the electron: Millikan's 
experiment 545 

18.4 The photoelectric effect 546 

18.5 Einstein's theoiy of the photoelectric effect 550 

18.6 X rays 555 

18.7 Electrons, quanta, and the atom 559 

Chapter 19 / The Rutherford-Bohr Model of the Atom 566 

19.1 Spectra of gases 566 

19.2 Regularities in the hydrogen spectrum 571 

19.3 Rutherford's nuclear model of the atom 573 

19.4 Nuclear charge and size 576 

19.5 The Bohr theory: the postulates 578 

19.6 The size of the hydrogen atom 580 

19.7 Other consequences of the Bohr model 581 

19.8 The Bohr theory: the spectral series of hydrogen 583 

19.9 Stationary states of atoms: the Franck-Hertz experiment 587 

19.10 The periodic table of the elements 589 

19.11 The inadequacy of the Bohr theory and the state of atomic theory in the 
early 1920's 595 

Chapter 20 / Some Ideas from Modern Physical Theories 600 

20.1 Some results of relativity theory 600 

20.2 Particle-like behavior of radiation 605 

20.3 Wave-like behavior of particles 608 

20.4 Mathematical versus visualizable atoms 611 

20.5 The uncertainty principle 613 

20.6 Probability interpretation 617 

EPILOGUE 625 



COIVTENTS ix 



Unh 6 • The Nucleus 

FKOLOGUE 627 

Chapter 21 / Raclioactiiiti' 629 

21.1 Becquerel s discoveiy 629 

21.2 OthcM- ladioactive elements are discovered 633 

21.3 The penetrating power of the radiation: a, (3, and 7 rays 638 

21.4 The charge and mass of a, p, and 7 rays 639 

21.5 The identity of a rays: Rutherford's "mousetrap" 641 

21.6 Radioactive transfomiations 642 

21.7 Radioactive decay series 644 

21.8 Decay rate and half-life 647 

Chapter 22 / Isotopes G55 

22.1 The concept of isotopes 655 

22.2 Transformation rules 657 

22.3 Direct exidence for isotopes of lead 658 

22.4 Positive rays 659 

22.5 Separating isotopes 661 

22.6 Summaiy of a useful notation for- nuclides; nuclear reactions 665 

22.7 The stable isotopes of the elements and their relative abundances 668 

22.8 Atomic masses 670 

Chapter 23 / I*robin^ the Nucleus 674 

23.1 The problem of the structure of the atomic nucleus 674 

23.2 The proton-electron hypothesis of nuclear structure 675 

23.3 The discovery of artificial transmutation 677 

23.4 The discoveiy of the neuti'on 680 

23.5 The proton-neution theoiy of the composition of atomic nuclei 683 

23.6 The neutrino 685 

23.7 The need for particle accelerators 687 

23.8 \uclear reactions 692 

23.9 Artificially induced radioactivity 696 

Chapter 24 / Nuclear Energy ; Nuclear Forces 702 

24.1 Conseivation of energ\' in nuclear reactions 702 

24.2 The energv' of nuclear binding 703 

24.3 Nuclear binding energy' and stability 705 

24.4 The mass-energy balance in nuclear reactions 707 

24.5 Nuclear fission: discoveiy 709 

24.6 Nuclear fission: controlling chain reactions 713 

24.7 Nuclear fission: large-scale energx' release and some of its 
consequences 716 

24.8 Nuclear fusion 725 

24.9 Fusion reactions in stars 727 

24.10 The strength of nuclear forces 729 

24.11 The liquid-drop nuclear model 731 

24.12 The shell model 733 

24.13 Biological and medical applications of nuclear physics 734 

KPII.OGUE 740 
^ co\ri:\rs 




CHAPTER 1 The Language of Motion 
CHAPTER 2 Free Fall: GalUeo Describes Motion 
CHAPTER 3 The Birth of Dhmamics: Newton 

Explains Motion 
CHAPTER 4 Understanding Motion 



y^DT 




It is January 1934 in the city of 
Paris. A husband and wife are at 
work in a university laboratory. They are exposing a piece of 
ordinary aluminum to a stream of tiny charged bits of matter 
called alpha particles. Stated so simply, this hardly sounds like 
an important e\/ent. But look more closely, for it is important 
indeed. Later you will look at the technical details, but for now 
they will not get in the way of the stoiy. 

The story is something of a family affair. The husband and wife 
are the French physicists Frederic Joliot and Irene Curie. The 
alpha particles they are using in their experiment are shooting 
from a piece of i"iaturally radioactive metal. This metal is 
polonium, first identified 36 years before by Irene's parents, 
Pierre and Marie Curie, the discoverers of radium. What Frederic 
and Irene have found is that when common aluminum is 
bombarded by alpha particles, it too becomes radioactive for a 
short time. 

This was a surprise. Until that moment, a familiar, everyday 
substance becoming artificially radioactive had never been 



UNIT 1 / PROLOGUE 




Physicist Enrico I'rnni tl30l-W54l. 



obsei'V'ocl. But physicists in the laboratoiA (uinnot force new 
phenomena on nature. They can only show nioie clearly what 
nature is like. Scientists know now that this t\'pe of ladioactix ity 
occurs quite often. It happens, for example, in stars and in the 
atmosphere when it is bombarded by cosmic rays. 

Though it made few, if any, newspaper headlines, the news 
was exciting to scientists and traveled rapidly. Enrico Fermi, a 
young physicist at the University of Rome, became intrigued b\' 
the possibility of repeating the experiment. But Fermi added an 
important alteration. The stoiy is told in the book /\f 0/77 .s in the 
Family, mitten by Enrico Fei'mi's wife, Laura. 

. . . hv (l(!cidi;ci he would tiy to product; aitificial radioadiv itv' 
with neutrons (instead of alpha particlesi. Having no electric 
charge, neutrons are neither attracted by electrons nor re|)(!ll(ui 
by nuclei; their path inside mattni- is much longer than that of 
alpha particles; theii- spet^d and en(Mg\' remain higher; iheii- 
chances of hitting a nucleus with full impact are much greater. 

Usually a physicist is guided by some theory in setting up an 
experiment. This time, no woikable theoiy had yet been 
developed. Only through actual experiment could one tell 
whether or not neutrons could trigger- artificial iadioacti\it\' in 
the target nuclei. Fermi, already an outstanding theor'(!ti("al 
physicist at age 33, decided to design some experiments that 
could settle the issue. Mis first task was to obtain instruments 
suitable for- detecting the j^ariicles emitted by radioacti\(^ 
materials. The best such laboratory instruments by far wer'e 
Geiger- counter^s. But in 1934, Geiger- counter's wer-e still r'elatively 
new anti not readilv axailable. Therefore, Feimi built his owii. 



All quotations in the I^rologue ai'c 
from Laur-a I'crnii, Atoms in the 
Fanjily: My Life With Enrico Fermi, 
L'iii\er\sity of (Ihicago F^i'ess, Chi- 
cago, 19i^4 (available; as a papeiliack 
book in the Phoenix Rooks series i. 
Fermi was one of the major phy- 
sicists of the twentieth centui-v. 



The counters were soon in operation detecting the radiation 
from radioactive materials. Fermi also needed a source of 
neutrons. This he made by en(-losing berAilium pow(i(M' and the 
radioactive gas radon in a glass tube. Alpha par tides from [hv 
radon, striking the beryllium, caused it to emit neutrons, which 
passed fr-eely thr'ough the glass tube. 

Now Enrico was ready for- the fiist experiments. Being a man of 
method, he did not star-t by bombar-ding substances at lanciom, 
but proceeded in or-der-, star-ting from the lightest element, 
hydrogen, and following the periodic tabk; orelemeiils. 
Hydrogen gave no results: vvIkmi he boml)ar(ic(l water witb 
neutrons, nothing happened. He tried lilhiuin next, but a^ain 
without luck. He went to beryllium, then to boion, to carbon, to 
nitrogen. None were a(;ti\'ated. Enrico wavered, discouraged, 
and was on the point of giving up iiis researclies, but liis 
stubbornness made him refuse to yield. He; would trv' one more 
element. That oxygen would not become radioactive he kn(;w 
already, for his first bombar'dment had been on water-. So be 
irradiat(;d fluoiine;. Huriab! \lv was lewai'detl. Fluorine was 



I Ml 



PK()i.()(;( b 



strongly activated, and so were other elements that came after 
fluorine in the periodic table. 

This field of investigation appeared so fruitful that Enrico not 
only enlisted the help of Emilio Segre and of Edoardo Amaldi 
but felt justified in sending a cable to Rasetti [a colleague then 
in Morocco), to inform him of the experiments and advise him 
to come home at once. A short while later a chemist, Oscar 
D'Agostino, joined the group, and systematic investigation was 
c'arried on at a fast pace. 

With the help of his co-woi'kers, Fermi pursued his 
experiments with high spirits, as Laura Fe mil's account shows: 

. . . Irradiated substances were tested for radioactivity with 
Geiger counters. The radiation emitted by the neutron source 
would have disturbed the measurements had it reached the 
counters. Therefore, the room where substances were irradiated 
and the room with the counters were at the two ends of a long 
corridor. 

Sometimes the radioactivity produced in an element was of 
short duration, and after less than a minute it could no longer 
be detected. Then haste was essential, and the time to cover 
the length of the corridor had to be reduc;ed l)y swift running. 
Amaldi and Feinii prided themselves on being the fastest 
runners, and theirs was the task of speeding short-lived 
substances fi^om one end of the corridor to the other. They 
always raced, and Enrico claims that he could run faster than 
Edoardo. . . . 

And then, one morning in October 1934, a fateful discoveiy was 
made. Two of Fermi's co-workers were irradiating a hollow 
cylinder of silver to make it artificially radioactive. They were 
using neutrons from a source placed at the center of the 
cylinder. They found that the ainount of radioactixaty induced in 
the silver depended on other objects that happened to be 
present in the loom! 

. . . The objects around the cylinder seemed to influence its 
activity. If the cylinder had been on a wooden table while being 
irradiated, its activity was greater than if it had been on a piece 
of metal. 

By now the whole group's interest had been aroused, and 
everybody was paiticipating in the work. They placed the 
neutron source outside the cylinder- and interposed objects 
between them. A plate of lead made the activity increase 
slightly. Lead is a heavy substance. "Let's try a light one next," 
Fermi said, "for instance, paraffin. " The most plentiful element 
in paraffiri is hydrogen. Vhe experiment with paraffin was 
performed on the morning of October- 22. 

They took a big block of par-affin, dug a cavit\' in it, pirt the 
neutron source inside the cavity, irradiated the silver- cylinder-, 
and brought it to a Cieiger counter- to measur-e its actixity. The 
counter- clicked madly. Ihe halls of the physics building 

UMIT 1 / PROLOGUE 



Follow the story rather than worry 
about the techniques of the exper- 
iment. 



n^uiro/i Sourer, 



J 

Silver c ylindtt 




P' 



'airSfin block 



3 



resounded with loud exclamations: "F"antasticl Incredihli!! iilaek 
Magic!" Paratlin incr(;as(!d tlie aitificially induced radioactivity 
of silver up to one lumdiiHl times. 

By the time Fei-nii cuinie hack tioni lunch, he had alreacK 
found a theory to account Ibi- the stianfi;(> action ol the paraffin. 



Because of Fermi's earlier experi- 
ments, they knew the water would 
not become artiticially radioactive. 
Ho\\e\(;r, th(;\' now reasontid that 
it would slow down ntuitrons and 
so allow sil\(M- to become more 
strongly ividioactixe. 



Paraffin c;ontains a great deal of hydrogen. Hydrogen nuclei ai-e 
protons, particles having the same mass as neutrons When the 
source is enclosed in a paraftin block, the neution.s hit the 
protons in the |)araffin bc^fore i-eaching the silver nucku. In the 
collision with a [iroton, a iKuitron loses part of its (Mi(Mg\', in the 
same manner as a billiard ball is slowed down when it hits a 
ball of its same siz(\ v\'h(Meas it loses little spiuul if it is i(!tlected 
off a mu('h heavier- ball, or- a solid wall. Before (^merging from 
the paraffin, a neutron will have collided with many protons in 
succession, and its velocity will be greatly reduced. This .s7ov\ 
neutron will have a mu(;h better chance of being captured by a 
silver nucleus than a fast one, much as a slow golf ball has a 
better chance of making a hole than one which zooms fast and 
may bypass it. 

if Enrico's explarnations wer"e correct, any other substance 
containing a large proportion of hydrogen should have the 
same effect as paraffin. "Let's try and see what a considerable 
quantity of water" do(!s to the silver- activity, tnrico said on the 
same afternoon. 

There was no better place to find a "considerable (luantity of 
water " than the goldfish fountain ... in the gar-den behind the 
laboratorA'. . . . 

In that foirntain the jjhvsicists had sailed (-eitain small toy 
boats that had suddenly invaded the Italian market. Each little 
craft bore a tiny candle on its deck. When the candles were 
lighted, the boats sp(;d and puffed on the water like r-eal 
motor-boats. They were delightful. And the yoirng men, who had 
never- been able to r-esist the charm of a new toy, had s|)ent 
nuK-h time watching them run in the fountain. 

It was natural that, when in need of a considerable amount 
of water-, Fermi and his friends should think of that foirntain. 
On that after-noon of October 22, they r-ushed their source of 
neutrons and their silver cylirnder to that fourntain, and they 
placed both under water. The goldfish, I am sur-e, r-etained their 
calm and dignitv, despite the neutr-on shower-, more than did 
the crowd oirtside. I he men's excitement was fed on the r-esirlts 
of this experiment. It confirmed Fer-mi's theory. Water also 
increased the artificial ladioactivitv of silver manv times. 



Fermi and his co-workers had learned that slowed-dov\ n 
neutrons can produce much stronger effects in making certain 
atoms radioactive than can fast neutrons. This discovers turned 
out to be a ciucial step toward further discoveries which, years 
later, led Fenni and others to the controlie^d pr-oduction of 
atomic ener-^v Ir-om uranium. 



LMI I I'KOLUGLt; 



Fermi and his associates did not give up in the face of 
discouraging results. They showed imagination in the invention 
of theories and experiments. They remained alert to the 
appearance of unexpected results and resourceful in using the 
material resources at hand. Moreover, they found joy in 
discovering something new and important. These traits are of 
value in pursuing scientific work no less than elsewhere in life. 

Scientists build on what has been found out and reported by 
other scientists in the past. Yet eveiy advance in science can 
raise new scientific questions. The work of science is not to 
produce some day a finished book that can be closed once and 
for all. Rather, it is to cany investigation and imagination on into 
fields whose importance and interest have not been realized. 

Some work in science depends upon painstaking observation 
and measurement. The results sometimes stimulate new ideas 
and sometimes reveal the need to change or even completely 
discard existing theories. Measurement itself, however, is usually 
guided by a theoiy. One does not gather data just for their own 
sake. 

All these characteristics are true of science as a whole and not 
of physics alone. This being a physics text, you may well ask, 
"Yes, but just what is physics?" The question is fair enough, yet 
there is no simple answer. Physics can be thought of as an 
organized body of tested ideas about the physical world. 
Information about this world is accumulating ever more rapidly. 
The great achievement of physics has been to find a fairly small 
number of basic principles which help to organize and to make 
sense of certain parts of this flood of information. 




The Fermi National Accelerator 
Laboratory is e}<:ploring the value of 
neutron irradiation in the treat- 
ment of cancer. A beam of protons 
from a linear accelerator is di- 
rected onto a beryllium target. 
Neutrons are produced as a result 
of this collision; these neutrons 
are used in the cancer therapv re- 
search. 



UNIT 1 / PROLOGUE 



yittse Up I 

Our Place in Space and Time 



Physics deals with those laws of the universe that 
apply everywhere, from the largest to the smallest. 






ORDER OF MAGNITUDE 


Distance to the furthest observed galaxy 


1 0^ meters 


Distance to the nearest galaxy 


10^ 


Distance to the nearest star 


10'^ 


Distance to the sun 


10" 


Diameter of the earth 


10^ 


One kilometer 


10^ 


Human height 


10° 


Finger breadth 


^o■' 


Paper thickness 


10-^ 


Large bacteria 


10^ 


Small virus 


10' 


Diameter of atom 


10'° 


Diameter of nucleus 


10" 




A globular star cluster. 

The estimated size of the universe now is of 
the order of 100 million, million, million, 
million times a person's height (person's 
height x 10.000.000.000.000,000,000. 
000.000). The diameter of the cluster is 
about ?0" m. 



Atomic sites in tungsten. 
The smallest known constituent units of the 
universe are less in size than a hundredth 
of a millionth of a millionth of a person's 
height (person's height x 0.000, 
000,000,000,01). The two bright spots in the 
middle are produced by atoms about 10'^ 
m apart 



I Ml 1 (;()\(;i;prs oi moi ion 



Physicists study phenomena in the extremes of time-space 
and in the whole region between the longest and shortest. 





ORDER OF MAGNITUDE 


Age of universe 


10'' seconds 


Precession of the earth's axis 


10'= 


Human life span 


10' 


One year 


10' 


One day 


10^ 


Light from sun to earth 


10^ 


Time between heartbeats 


10° 


One beat of fly's wings 


10-^ 


Duration of strobe flash 


10^ 


Short laser pulse 


10' 


Time for light to cross an atom 


1Q-.6 


Shortest-lived subatomic particles 


10^= 




Fossilized trilobites. 

These fossils are about 5x10^ years old. 
The total history of the universe has been 
traced back as far into the past as 300 million 
times the length of a human life (human life 
X 300.000.000). 



Particle tracks in a bubble chamber. 
Events have been recorded that last only a 
few millionths of a millionth of a millionth of 
a millionth of a person's heartbeat (person's 
heartbeat x 0.000.000.000.000,000, 
000.000,001). These tracks each took about 
10'^ sec to make. 



Physics started not with these intriguing extremes, but 
with the human-sized world: the world of horse-drawn 
chariots, of falling rain, and of flying arrows. It is with the 
physics of phenomena on that scale that this course 
begins. 



UNIT 1 / PROLOGUE 




The Language of Motion 




If 



1.1 The motion of things 

1.2 A motion experiment that does not quite ivork 

1.3 A better experiment 

1.4 Leslie's siiim and the meaning ofat'erage speed 

1.5 Graphing motion and finding the slope 

1.6 Time out for a warning 

1.7 Inst.intaneous speed 

1.8 Acceleration bv comparison 



l.l I The motion of things 

The world is filled with things in motion, things as small as dust 
and as large as galaxies, all continually moxing. Your book may 
seem to be lying quietly on the desk, but each of its atoms is 
constantly vibrating. The "still" air around you consists of 
molecules tumbling wild at various speeds, most of them moving 
as fast as rifle bullets. Light beams dart through the room, 
covering the distance from wall to wall in about a hundred- 
millionth of a second and making about 10 million vibiations 
during that time. The whole eai-th, our spaceship itself, is moxing 
at about 29 kilometers per second (km/secl around the sun. 

There is an old maxim: "To be ignorant of motion is to be 
ignorant of nature." So, from this swirling, wiiirling, \ibrating 
world of ours let us first choose just one moving object for our 
attention. Then let us describe its motion. 

Shall we start with a machine, such as a rocket or a car? 
Though made and controlled by humans, machines and their 



UIVIT 1 / CONCEPIS OF VIO IIOX 



parts mov^e in fast and complicated ways. We really ought to start 
with something simpler, something that our eyes can follow in 
detail. How about a bird in flight? Or a leaf falling from a tree? 

Surely, in all of nature there is no motion more ordinary than 
that of a leaf fluttering down from a branch. Can you describe 
how it falls or explain why it falls? You will quickly realize that, 
while the motion is veiy "natural," it is also very complicated. 
The leaf twists and turns, sails right and left, back and forth, as it 
floats down. Even a motion as ordinaiy as this may turn out, on 
closer examination, to be more complicated than the motion of 
machines. And even if you could describe the motion of a leaf in 
detail, what would you gain? No two leav^es fall in quite the same 
way. Therefore, each leaf seems to require its owai detailed 
description. This indi\aduality is typical of most events you see 
occurring in nature. 

And so we face a problem. We want to describe motion, but 
the motions we encounter under ordinary circumstances appear 
too complex. Also, from the separate obsen^ations we are not 
likely to find general conclusions that apply to all motions. What 
shall we do? The answer is that, at least for a while, we must go 
into the physics laboratoiy. The laboratory is the place to 
separate the simple ingredients that make up all complex natural 
phenomena and to make those phenomena more easily visible 
to our limited human senses. 




Study for "Dynamism of a Cyclist" 
(1913) by Umberto Boccioni. Cour- 
tesy Yale University Art Gallery. 



1«2 I A motion experiment that does not 
quite work 

Having abandoned the fall of a leaf as the way to start on the 
physics of motion, we might select a clearly simpler case: a 
billiard ball, hit squarely in the center and speeding easily across 
a tabletop in a straight line. An even simpler motion (simpler 
because there is no rolling) can be obtained. Place a disk of what 
is called "dry ice" on a smooth floor and give it a gentle push. 
(Take care not to touch the extremely cold disk with bare hands 
for more than a brief moment!) The disk will move slowly and 
with veiy little friction, supported on its own vapor. 

We did this in front of a camera to get a photograph that 
would record the action for easier measurement later. While the 
dry-ice disk was moving on the well-leveled surface, the shutter 
of the camera was kept open; the resulting time-exposure shows 
the path taken by the disk. 

What can you learn about the disk's motion by examining the 
photograph? As nearly as you can judge by placing a nailer on 
the photograph, the disk moved in a straight line. This is a very 
useful result, and you will see later that it is really quite 
surprising. It shows how simple a situation can be made in the 
laboratoiy; the kinds of motion vou ordinarily see are almost 



"Dry ice" is really frozen carbon 
dioxide, at -79°C. 



CHAPTER 1 / THE LANGUAGE OF MOTION 



9 



Uiboruloiy sflup. 



ne\'er that sim|jl(!. Bui did the disk nio\e steadiK , oi- did it slow 
down? I'l'oni this j)h()t()giaph, we really eannot tell. We must 
improve the experiment. Before we do so. however, we must 
decide just how we |)laii to measure the speed. 





Close-up of a diy-ice disk. 




ruin; c\f)osurc of llw disk in mo- 
tion. 



From time to time you v\aU be re- 
ferred to items in the Study Guide 
found at the end of each chapter. 
UsualJy the letters SG plus a num- 
ber will indicate this. See SG 1 on 
page 31 for more information on 
how to study for this course; and on 
the use of the Study Guide. 



it would he nice; to use soiiKUliing like an aulomohiie 
speedometer. A speedometer ean lell diieetK the s|jeed at which 
a car is mo\dng at any time. We can say, for example, that a car 
is moving at 100 kilometei"s jjer hour (km hri. This means that 
if the car continues t(j moxe with the same speed it had at the 
instant the speed reading was taken, the car would mo\ e a 
distance of 100 km in a time intenal of 1.0 hr. Or we could say 
that the car would mo\e 1.7 km in Vwi of an hour II minutel or 
10 km in '/lo of an hour. In fact, we could use any distance and 
time intervals foi- which the ratio of distance to time? is 100 km/ 
hr. 

Of course, an automobile speedometer cannot be hooked to a 
disk of diy ice, or to a bullet, or to many other objects. Howexor, 
there is a rathei* sinijjle vva\ lo measure speeds, at least in most 
cases that would interest us. 

Think of what you could do if the speedometer in your car 
were broken and voir still wanted to know xour- s|)eed as you 
moved along a turnpike. You could do one ot two things (the 
r-esirlt is the; same in eithcM' easel. Voir c^oirld count the number of 
kilometer- markcMs [massed in one hour- lor- some known fraction 
of an houi-i and find the aver'age speed by computing the ratio of 
kilometers and hoiris. Or- voir coirld determine the fraction of an 
hour- it takes to go Ir-om one kilometer- marker to the next (or- to 



10 



Li\l I 1 COXGLPiS OI MOilON 



another marker a known number of kilometers away) and again 
find the average speed as a ratio of kilometers to hours. 

Either method gives, of course, only the average speed for the 
inteival during which speed is measured. That is not the same as 
instantaneous speed. Instantaneous speed is the speed at any 
given instant as a speedometer might register it. But average 
speed is good enough for a start. After you understand average 
speed, you will see a simple way of finding instantaneous 
speeds. 

To find the average speed of an object, measure the distance it 
moves and the time it takes to move that distance. Then dixide 
the distance by the time. The speed is in kilometers per hour or 
meters per second, depending upon the units used to measure 
the distance and time. With this plan of attack, we can return to 
the experiment udth the diy-ice disk. Oui' task now is to tind the 
speed of the disk as it movies along its straight-line path. If we 
can do it for the disk, we can do it for many other objects as 
well. 

Note: There vvdll usually be one or more brief questions at the 
end of each section in the text. Question 1, below, is the first. 
The end-of-section questions are there for your use, to check on 
your own progress. Answer the questions before continuing to 
the next section. Check your answers to these end-of-section 
questions. Whenever you find you did not get the correct answer, 
study thiough the section again. Of course, if anything is still 
unclear after you have tried to study it on your own or together 
udth other students, then ask vour instructor. 



The speed of an object is, of course, 
how fast it moves from one place 
to another. A more formal way to 
say the same thing is: Speed is the 
time rate of change of position. The 
term "displacement" is often used 
to refer to the straight-line distance 
between the beginning and end 
points of the change in position of 
a moving object. We will use this 
term in connection with vectors in 
Chapter 3, and more often still with 
wave motion in Chapter 12. 



# 1. Why is it impossible to determine the speed of the diy-ice 
disk from the time-e}iposure photograph on page 10? 

1 .3 I A better experiment 

To find speed, you need to be able to measure both distance and 
time. Repeat the experiment udth the diy-ice disk. First place a 
meter stick (100 cm) on the table parallel to the expected path of 
the disk. This is the photograph obtained: 




CHAPTER 1 / THE LAMGUAGE OF MOTION 



11 



You now have a way of measuring th(! tiistan("e tra\ eled by the 
disk. But you still need a way to measure the lime it takes the 
disk to travel a given distance. 

Ihis can be done in xaiious ways, but theie^ is a fine tiick that 
you can Uy in the laboratory. The cam(;ia shutter" is again kejjt 
open and everything else is the same as betor-e, except thai the 
only sour'ce of light in the darkened room comers liom a 
sti'oboscopic lamp. This lam|) pr'oduc(!s bright Hashers of light at 
intervals which we can set as we please. Each pulse or- flash of 
light lasts foi' only about 10 millionths of a sec'ond 110 
microseconds). I'herefore, the mo\ing disk appears in a ser-ies of 
separate, sharp exposur-es, r-athei' than as a continuous blur. Ihe 
photograph below was made by using such a stroboscopic lamp 
flashing 10 times per second, after- the disk hatl been gently 
pushed as before. 



This special setup enables us to r-ecor-d accurately a series of 
positions of the mo\ ing object. The meter- stick helps us to 
measure the distance moved by the front edge of the disk 
between one light flash and the next. The time interAal between 
images is, of course, equal to the time inters al between 
stroboscopic lamp flashes [0.10 second (sec) in these photos]. 

You can now deter-mine the speed of the disk at the l)eginning 
and end of its photographed path. The front edge of tiie second 
clear image of the disk at the left is 19 cm from the zero niai k on 
the meter stick. The front edge of the third image fr-om th(^ left 
is at the 32-cm position. The distance traxeled during that time 
was the difference between those two positions, or 13 cm. Ihe 
cor-responding time interAal was 0.10 sec. Therefore, the speed 
dur'ing the first part of the obserAation must hixve been 13 
cm/0.10 sec, or 130 cm/sec. 

Now look at the two images of the disk far'thest to th(> right in 
the photograph. Here, too, th(^ distance triueled during 0.10 sec 
was 13 cm. Thus, the speed at the right end was 13 cm/0.10 sec, 
or 130 cm/sec. 

The disk's motion was not measurabK' slower at the right end 
than at the left end. Its speed was 130 cm/sec near the beginning 
of the path and 130 cm/sec near- the end of the path. Howexei-, 
this does not yd pr'oxe that the speed was (-onstant all th(^ way. 



12 



L\l I 1 / CONCEPTS OI VIOTIOX 



You might well suspect that it was, and you can easily check to 
find out. Since the time intervals between images are known to 
be equal, the speeds will be equal if the distance inteivals are 
equal to one another. Is the distance between images always 13 
cm? Did the speed stay constant, as far as you can tell from the 
measurement? 

When you think about this result, there is something really 
unusual in it. Cars, planes, and ships do not move in neat, 
straight lines with precisely constant speed even when using 
power. Yet this disk did it, coasting along on its own, without 
any propulsion to keep it moving. You might consider this a rare 
event which would not happen again. In any case, you should 
try the experiment. The equipment you will need includes 
cameras, strobe lamps lor mechanical strobes, which work just 
as well), and low-friction disks of one sort or anothei-. Repeat the 
experiment with different initial speeds. Then compare your 
results with those found above. 

You may have a serious reseivation about the experiment. You 
might ask: 'How do you know that the disk did not slow dowoi 
an amount too small to be detected by your measurements?" 
The answer is that we do not know. All measurements involve 
some uncertainty, though one which can usually be estimated. 
With a meter stick, you can measure distances reliably to the 
nearest 0.1 cm. If you had been able to measure to the nearest 
0.01 cm or 0.001 cm, you might have detected some slowing 
down. But if you again found no change in speed, you could still 
raise the same objection. There is no way out of this dilemma. 
We must simply acknowledge that no physical measurements are 
ever perfectly precise. The results of any set of measurements are 
acceptable within its own limits of precision, and you can leave 
open the question of whether or not measurements made with 
increased precision could reveal other results. 

Briefly rexaew the results of this experiment. You devised a way 
to measure the successive positions of a moving dry-ice disk at 
known time intei^^als. From this you calculated first the distance 
inteivals and then the speed between selected positions. You 
soon discovered that (within the limits of precision of the 
measurements) the speed did not change. Objects that move in 
such a manner are said to have uniform speed, or constant speed. 
You know now how to measure uniform speed. 

But, of course, actual motions are seldom uniform. What about 
the more usual case of nonuniform speed? That is our next 
concern. 



Uncertainty of measurement is 
taken up in detail in the Handbook, 
particularly in Experiment 3. 



Some practice problems dealing 
with constant speed are given in 
Study Guide 3 (a, b, c, and d). 



• 2. Suppose the circles on page 14 represent the successive 
positions of a moving disk as photographed stroboscopicallv. 
Did the object move with uniform speed? How do you know? 



CHAPTER 1 / THE LANGUAGE OF MOTIOIV 



13 



oo oo o o 

3. Give a general description of uniform speed, without 
referring to dry-ice disks and strobe photography or to any 
particular object or technique of measurement. 



% 




45 m/56.1 sec = 0.80 m/sec. That 
is the equhalent of 2.9 km hr. i\o 
great speed! A sailfish can do o\'er 
60 kni hr. Rut hiinians are land an- 
imals. E\en the fastest swimmers 
are only ahout twice as fast as Les- 
lie. For short distances, a person 
can run up to ahout 30 km,hr. The 
world's record for a 100-m run is 
a bit more than 10 sec; the runner 
thus had an axerage speed of 36 
kni/hr. 



1. ,4 I Leslie's siiim and the meaning of 
average speed 

Consider the situation at a swimming meet. At the end of each 
race, the name of the winner, the swimmer with the shortest 
time, is announced. But in any given race, for example, the 100- 
meter exent, eveiA' swimmer goes the same distance. Therefore, 
the swimmer with the shortest time is also the one having the 
highest average speed while covering the measured distance. The 
ratio of the distance traveled to the elapsed time is the measure 
of average speed. This relationship is expressed in the following 
equation: 

distance traveled 

average speed — — 

elapsed time of travel 

which, in fact, is the definition of average speed. What 
information does a knowledge of the average speed give you? 
You can answer this question hv studying a real example. 

Leslie is not the fastest woman freestyle swimmer in the world, 
but Olympic speed is not necessaiy for this purfoose. Leslie was 
timed while swimming two lengths of an old, indoor pool. The 
pool is 22.5 meters (m) long, which makes it a bit too short for 
Olympic events but good enough for many sports. It took Leslie 
56.1 sec to swim the two lengths (45 m). Thus, her average speed 
for the 45 m was 



45 m 
56.1 sec 



= 0.80 msec 



Did Leslie swim the two lengths of the pool at imiform (that is, 
constant) speed? If not, when was she swimming the fastest? 
What was her greatest speed? Did she take less time to swim out 
than to return? How fast was she moving when she was one- 
quarter or halfway down the pool? The answers to these 
questions are useful to know when training for a meet. 

So far you do not have a way to answer any of these questions, 
but soon you will. L'ntil then, the average speed over the two 
lengths (0.80 m/sec) is the best single value v ou can use to 
describe the whole of Leslie's swim. 

To compare Leslie's speed at different points in the swim, 
observe the times and distances covered as you did in 
experimenting with the diy-ice disk. For this puipose, the event 
was arranged as follows. 



14 



L'lMT 1 CONCLPIS OI MOIION 



Obseivers stationed at 4.6 m inteivals fiom the zero mark along 
the length of the pool started their stopwatches when the 
starting signal was given. Each obseiver had two watches. The 
observers stopped one watch as Leslie passed them going down 
the pool, and they stopped the other as she passed on hei- 
return trip. The data are tabulated ne.xt to the photograjjh of this 
experiment. 






From these data, you can, for example, determine Leslie s 
average speed for the first length (22.5 m) and for the last length 
separately. 

distance traveled 

average speed for the first length = j — \ 

elapseQ time 

22.5 m 



22.0 sec 



average speed for the last length 



= 1.02 m/sec 

distance traveled 
elapsed time 

22.5 m 
56.1 sec - 22.0 sec 



22.5 m 
34.1 sec 



= 0.66 m/sec 



It is now clear that Leslie did not swim with unifomi speed. 
She swam the first length much faster (1.02 m/sec) than the 
second length (0.66 m/sec). Notice that the overall average speed 
(0.80 m/sec) does not describe either lap by itself veiy well. 

In a moment we will continue our analysis of the data we have 
obtained for Leslie's swim. This analysis is important because the 



0.0m 


0.0 sec 


4.6 


2.5 


9.1 


5.5 


13.7 


II. 


18.3 


I6.0 


22.9 


22.0 


27.4- 


26.5 


32.0 


32.0 


36.7 


39.5 


41.2 


47.5 


45.7 


56.0 



CHAPTER 1 / THE LAMGUAGE OF MOTION 



15 



v^^ is pronounced "vee av" or "vee 
sub-av." Id is pronounced "delta 
dee." 



Practice problems on average speed 
can bi' found in Study Guide 4 (e, 
f, g. and h). Study (iuides 6, 8, and 
10 otfcr- somewhat more challeng- 
ing pr()i)lems. Some suggestions 
for axcrage speeds to measure are 
Iist(!d in Study Ciuidc; 9. 



concepts we are developing for this exeiyday type of motion will 
be needed later to discuss other motions, ranging from that of 
planets to that of atoms. 

The following shorthand notation will simplify the delinitif)n of 
avenige speed: 



average speed - 



distance; traxeled 



elapsed time 

A more concise statement that says exactly the same thing is 

Ad 



v.. = 



At 



In this equation, v^,^ is the symbol lor the average speed, Ad is the 
symbol for change in position, and A/ is the sxmbol foi- an 
elapsed inteival of time. The symbol A is the fourth letter in the 
Greek alphabet and is called delta. When A pi^ecedes another 
symbol, it means "the change in ... . ' Thus, Ad does not mean 'A 
multiplied by d." Rather, it means "the change in d" or "the 
distance interval." Likewise, At stands for "the change in /" or 
"the time inten^al." 

You can now go back to the data and compute Leslie's average 
speed for each 4.5-m interval, from beginning to end. This 
calculation is easily made, especially if you reorganize the data 
as showai in the table on page 19. The values of v.,^ calculated at 
4.5-m intervals for the first lap are entered in the right-hand 
column. (The values for the second lap ai'e left for you to 
complete.) 

Much more detail is emerging from the data. Looking at the 
speed column, you see that Leslie's speed was greatest, as 
expected, near the start. Hei- racing jump into the water gave her 
extra speed at the beginning. In the middle of her first length, 
she swam at a fairly steady rate, and she slowed dowTi coming 
into the turn. Use your own calculations to see what happened 
after the turn. 

Although \'ou ha\(; detemiined Leslie's speeds at xarious 
intervals along the path, you are still dealing with avcrnga 
speeds. The inten^als during which you determine the average 
speeds are smaller: 4.5 m rather than the entire 45 m. But you do 
not know the details of what happened within any of the 4.5-m 
inter\'als. Thus, you know that Leslie's average speed between the 
13.5- and 18-m marks was 0.9 m/sec. You do not know \et how 
to compute her speed at the veiy instant and point where she 
was, say, 16.2 m or 18 m from the start. Even so, the average 
speed for the 4.5-m intenal between the 13.5- and 18-m marks is 
pi^obably a better estimate of her speed as she went through the 
16.2-m mark than is the average speed for the whole 45 m, or for 
either 22.5-m length. We will come back to this problem of 
determining "sjjeed at a particular instant and point " in Sec 17 



16 



UNIT 1 / COXCEmS Ol iVIOl l()\ 



4. A hoy on his way to the store ahout 720 m from his home 
stopped twice to tie his shoes and once to watch an airplane. 
What was his average speed for the trip if he took 540 sec to 
reach the store? In your own words, define the concept 
"average speed. " 

5. Use the formula for average speed to determine how far 
down the slope a skier will travel in 15 sec if she is moving at 
a speed of 20 m/sec. 

6. If you have not already completed the table on page 19, do 
so now before going on to the ne^t section. 



\ ,5 \ Graphing motion and Uncling 
the slope 

You can look at the data from Leslie's swim another and more 
informative way by plotting them on a graph instead of just 
writing them down in a table. In the first graph on page 19, 
the distance and time values that were measured for Leslie's 
swim are shoum. (The circles around the points have been put in 
only to make the points show up more clearly.) Each point on 
the distance-time graph shows the distance Leslie covered up to 
that particular time. We know that Leslie was in the pool 
between our measured points as well, but we do not know 
precisely where. The usual way to show this is to connect the 
known points with some kind of line or cur\'e. 

In the second graph we have done just that in the simplest 
way, by drawing straight lines between points. Because we do 
not know that Leslie was really "on " those lines between the 
measured points, we have drawoi the line segments as broken 
lines, with dashes, instead of as solid lines. 

You can get a better approximation of Leslie's actual motion by 
drawing one continuous 'smooth ' cuiA^e through all the data 
points. One experimenter's idea of a good curve is shown in the 
last graph. On first sight, this graph may not look veiy different 
from the second graph, but a closer look will show differences in 
detail. For example, in the second graph there is a rather sharp 
kink at the point corresponding to 5.5 sec. This would imply that 
Leslie changed her speed abruptly at that moment. We have no 
reason to think this happened, and it does not show up on the 
third graph. This is one of the reasons for preferring the third 
graph. 

What can you see (or "read ") on the last graph? Notice that the 
line is steepest at the start. This is because there was a 
comparatively large change in position in the first few seconds. 
In other words, Leslie got off to a fast start. You can also 





CHAPTER 1 / THE LANGUAGE OF MOTIOM 



17 



If this concept is new to you or if 
you wish to review it, turn now to 
Study Guide 11 before continuing 
here. 



determine that from the table. After the third measured point, 
the giaph becomes less steep. The time taken to eo\er the same 
distance is longer; therefore, the speed is slower. Again this 
agrees wath what you sec in the table of data and computations. 
The steepness of the gnif)h line clearly indicates how fast Leslie 
was moving. The faster she swam, the steeper is the line on the 
graph. If you follow the line along, you see that Leslie slowed 
down coming into the turn, had a brief spurt just after the turn, 
and then slowed down steadily until the finish. 

Looked at in this way, a graph proxides you at a glance with a 
visual representation of motion which can be \eiy useful. But 
this kind of representation does not tell directly what the actual 
value of Leslie's speed was at an\' particular momcMit. For- this, 
you need a way of measuiing the stt^^pness of the giaph line. 
Here we can turn to mathematics for help, as we often shall. 

There is an old method in geometiy for soK ing just this 
problem. The steepness of a graph at any point is related to the 
change in vertical direction (Ayl and the change in horizontal 
direction lAy). By definition, the ratio of these fvvo changes (Ay/ 
A^l is the slope: 

slope = -f- 
A,v 

Slope is a widely used mathematical concept. It can be used to 
indicate the steepness of a line in any graph. In a distance-time 
graph, like the one for Leslie's swim, the position, or distance 
from the start, is usually plotted on the vertical axis id replaces y) 
and time on the horizontal axis it replaces ;<). Therefore, in such 
a graph, the slope of a straight line is given by 

slope - — - 
At 

This should remind you of the definition of average speed, v^^ = 
Ad/A/. In fact, v'_^^ is numericalK ecjual to the slope! In other 
words, the slope of any straight-line part of a graph of distance 
versus time gives a measure of the average speed of the object 
during that inteival. 

When you measure slope on a graph, you do basically the 
same thing that highway engineers do when they specih' the 
steepness of a road. They simply measure the rise in the load 
and divide that rise by the horizontal distanc'e one must go in 
order- to achieve the rise. The only difference between this and 
what we have done is that the highway engineers are concerned 
with an actual physical slope. Thirs, on a graph of their- data, the 
vertical axis and hoi-izontal iixis both show distance. We, on the 
other- hand, ar-e using the niathenvitical concept of sloj^e as a 
wav of e.xpr-essing distance measuicd against lime 



18 



L\n 1 covcLPTS oi \iurio\ 



O.Om 

4.6 

9.1 

13.7 
18.3 
22.9 
27.4 
32.0 
36.7 
41 .2 
45.7 



Ad 



O.O.ecj ^ 



2.5 

5.5 

ILO 

16. 

22. 

26.5 

32.0 

39.5 

47.5 

S6.0 



6rn 
} 4.6 
} 4.6 
J -f.6 
} 4.6 
4.6 
4.6 
} 4.6 
] 4.6 
} 4.6 



At 



"^t (-Vav) 



2.56«c 
3.0 
5.5 
5.0 

O.O 
4.5 
5.5 



1.5 
0.8 
0.9 
0.8 

etc. 



50 



^ 40- 
V 



p 
o 

C 

o 



30»- 



20- 



10- 



(finish) 



(turn) 



® 



© 



® 



© 



© 



ft 






3 
J+-. 



10 20 30 40 

time (seconds) 



50 



60 



50 


- 






- 


-(finish) 




J® 


'-> 40 


_ 






M 






X- 


i. 






JSr 


.^ 






/ 

^ 


^ 30 


_ 






Vw/ 






^ 


^ 

^ 


-(turn) 




/ 


2 20 


/ 


y 




"0 


/ 




^^ 




/■ 




^-^ ^ 


10 


/ 




3 C 




r 

/ 




^-^ lis, 


0' 


k \ — 




X,.j_._ L i 1 1 



50 



10 10 50 40 rO 60 

hme (seconds) 



-- (finish) 



^40|- 

\- 
%t 

^ 30 

c 
p 



10- 



ok- 



^ 



Jir- 



J^ 







(turn) 









^^ 



J^ 



10 



20 10 40 
VirrxQ (seconds) 



50 60 



The slope, as defined here, is not exactly the same as the 
steepness of the line on the graph paper. Suppose we had 
chosen a different scale for either the distance or time axis, 
making the graph twice as tall or twice as wide. Then the 
apparent steepness of the entire graph would be different. The 
slope, however, is measured by the ratio of the distance and time 
units. A Ad of 10 m in a Af of 5 sec gives a ratio of 2 m/sec, no 
matter how much space on the dravvang oi- graph paper is used 
to represent meters and seconds. 

But the graph is more than just a "pictuie" of the values in the 
table. You can now ask questions that cannot be answered 
directly from the original data. For example, what was Leslie's 
speed 10 sec aftei' the start when she was not even opposite one 



Above are shown four wavs of rep- 
resenting Leslie's swim: a table of 
data and calculations, a plot of the 
data points, broken straight-line 
segments that connect the points, 
and a smooth curxe that goes 
through the points. 



Tfie 4-sec value for / is just for con- 
venience; some other value could 
have been used. Or we could have 
chosen a value for Ac/ and then 
measured the corresponding A/. 



CHAPTER 1 / THE LANGUAGE OF MOTIOM 



19 



of the data takers? What was hei* speed as she crossed the 
35-m mark? You can ansvvei- questions like these hv finding the 
slope of a fairly straight jjortion of the graph line near the point 
of interest. Tu'o examples have been worked out on the graph 
shown below. For' each example, At was chosen to be a 4-sec 
interval from 2 sec befor^e the point in question to 2 sec after' it. 
Then the Ad for- that A/ was measur-ed. 



50 r 



__ (finish) 



40 



+• 

£ 

w 
u 

c 

D 



30- 



20 



-0.fo3m/sec, 



10 




= 0.75m/5ec,o+t= lO.Osec. 
4.0 5ec ~ jc 



20 30 40 

time (seconds) 



50 



60 



*oo 



300 



^ 200 



100 




I 2 3 

time (hours) 



400 



300 



3. 200 



100 



01 2345678 

time (hours) 



You can check the r-easonableness of using the graph in this 
way by compar'ing the results v\ ith the \ alues listed in the table 
on page 19. For example, the speed near- the 10-sec mark is 
found from the gr^aph to be about 3.0 m/4.0 sec = 0.75 m/sec. 
This result is somewhat less than the v alue of 0.8 m/sec given in 
the table for the average speed for the time internal between t = 
5.5 sec and t = 11 sec. This is just what you would expect, 
because you can see that the smooth-cur^e graph does become 
slightly less steep around the 10-sec point. If the smooth curve 
r^eally describes Leslie's swimming better than the dashed 
straight-line graph does, then you can get more infoi'iiiation from 
that last graph than you can get just by looking at the data 
themselves! 



7. Turn buck to pni^e 12 unci dmw :i dislHiicc-Uiuc gr.i/;/? /or 
the motion of the dry-ice disk. 

8. Which of the two graphs in the ninrgin. for two ditfcrcnl 
objects, has the greater slope? 



20 



iM 1 1 (;o\ci:prs (ji moiion 



Close Upl 



The Language of Motion 



t = 



B. t = 19 mm 



C t = 36 mm 



D. t = 63 mm 




Af = 19 mm 



At ^ 17 mm 



At = 27 mm 









A. t = 



B. t = 17 hr 



C. t = 50 hr 



These photographs show a stormy 
outburst of incandescent gas at the 
edge of the sun, a developmg chive 
plant, and a glacier. From these 
pictures and the time intervals given 
between pictures, you can determine 
the average speeds of: (1) the growth 
of the solar flare with respect to the 
sun's surface (radius of sun is about 
691.200 km), (2) the growth of one of 
the chive shoots with respect to the 
graph paper behind it (large sguares 
are 2.5 cm). (3) the moving glacier with 
respect to its "banks." 




At = 4 years 




CHAPTER 1 / THE LANGUAGE OF MOTION 



21 



30 



• 20 



3 



10 







I 


i 


f^\ 


1 1 

Ad 


J 







30 60 90 

time (mm) 



9. L's/ng f/?e /asf graph for her swim, where in the pool was 
Leslie swimming most rapidly? Where was she swimniing most 
slowly? 

10. From that graph, find Leslie's speed at the 43.0-m mark. 
From the table on page 19. calculate her average speed over 
the last 4.5 m. How do the two values compare? 

1.6. I Time out for a ivarning 

Graphs are useful, but they can also be misleading. You must 
always be aware of the limitations of any graph you use. The 
only actual data in a graph are the plotted [)oints. There is a 
limit to the precision with which the points can be plotted, and 
a limit to how precisely they can be read from the giaph. 

The placement of a line through a series of data points, as in 
the graph on page 19, depends on personal judgment and 
inteipretation. The process of estimating values between data 
points is called interpolation. That is essentially what you are 
doing when you draw a line betAveen data points. E\en more 
risky than inteipolation is extrapolation, where the graph line is 
extended to provide estimated points beyond the known data. 

The description of a high-altitude balloon experiment carried 
out in Lexington, Massachusetts, illustrates the danger of 
extrapolation. A cluster of helium-filled balloons carried cosmic- 
ray detectors high above the earth's surt'ace. From time to time, 
obsen'ers measured the altitude of the cluster. The graph on the 
right shows the data for the first hour and a half. After the first 
20 min, the balloons seem to be rising in a cluster with 
unchanging speed. The average speed can be calculated from the 
slope: speed of ascent = Ad/Af = 9,000 m/30 min - 300 m/min 
= 5 m/sec. 

Now, suppose you were asked how high the balloons would be 
at the veiy end of the experiment, which came at f = 500 min. 
You might be tempted to extrapolate, either by extending the 
graph or by computing from the speed. In either case, you would 
obtain about 500 min X 300 m/min - 150,000 m, which is over 
150 km high! Would you be right? (The point is that mathe- 
matical aids, including graphs, can be a splendid help, but onl\' 
within the limits set by physical realities.) 



sr, iH 



% 11. Look at the pictures of the solar flare on the preceding 
page. Estimate the height of the flare at (a) 42 min and (b) 72 
min. (Draw a graph if necessary^. I Explain which cstin^ate 
required extrapolation and which required interpolation. 
12. Which estimate fi^om the graph would you expect to be 
less accurate: Leslie's speed as she crossed the 30-m n^ark, or 
her speed at the end of an additional third lap.' 



22 



L\IT 1 / COrslCLPlS OF MOTION 



1«T Instantaneous speed 



Now let us summarize the chief lessons of this first chapter. In 
Sec. 1.5 you saw that distance-time graphs could be veiy helpful 
in describing motion. Near the end of the section, specific speeds 
at particular points along the path ("the 35-m mark") and at 
particular instants of time ("the instant 10 sec after the start") 
were mentioned briefly. You may have been bothered by these 
comments, since the only kind of speed you can actually 
measure is average speed. To find average speed you need a 
ratio of distance intervals and time inter\^als. A particular point 
on the path, however, does not have any inten^al. Nevertheless, it 
does make sense to speak about the speed at a point. The 
follovvang is a summary of the reasons for using 'speed ' in this 
way. 

You remember that the answer to the question (page 19) "What 
was Leslie's speed 10 sec after the start? ' was 0.75 m/sec. You 
obtained that answer by finding the slope of a small portion of 
the curve around the point P when t = 10 sec. That section of 
the curve is reproduced in the margin here. Notice that the part 
of the cuive used appears to be nearly a straight line. As the 
table under the graph shows, the value of the slope for each 
interx^al changes veiy little as the time inteival Af is decreased. 
Below At = 4 sec, you keep getting the same value for Ad/Ar. 
Correspondingly, the chosen segment of the line on which P sits 
is more and more a straight line. Now imagine that you 
continued to 700 m, where t — 10 sec, until the amount of cuive 
remaining became vanishingly small. Can you safely assume that 
the slope of that very small part of the cuive has the same value 
as the slope of the short straight-line portion of which it seems 
to be a part? It seems reasonable. In any case, it is up to you to 
define what you mean by your concepts. That is why we took the 
slope of the straight line from t — 8 sec to f = 12 sec and called 
it the speed at the midpoint, t = 10 sec. The correct term for 
this value is the instantaneous speed at the instant t = 10 sec. 

Prevaously, you estimated Leslie's instantaneous speed at a 
particular time by actually measuring the average speed over a 
4.0-sec interval. This method can be modified so that it can be 
used in many different contexts in the future. The instantaneous 
speed at a particular instant has the same value as the average 
speed, Ad/At, as long as tw^o conditions are met: First, the 
particular instant must, of course, be included in At. Second, the 
ratio, Ad/At must cover a small part of the curve, one that is as 
nearly as possible a straight-line segment. LJnder this condition, 
the ratio Ad/Af will not change noticeably when you compute 
it again over a still smaller time inten^al. 

A second example will help to explain the concept of 
instantaneous speed. In the oldest known study of its kind, the 







fO 


20 




time 


(sec) 


^t 


Ad 


"^t 


h.O sec 


4.6 m 


0.78 m/seC 


4.0 sec 


3.0 m 


D.75 m/SCC 


2.0 sec 


l.Sm 


0-75 rn/sec 



CHAPTER 1 / THE LANGUAGE OF JVIOTION 



23 



J 



■ / 



'■■f 



French scientist de Montiieillard periodically recorded the height 
of his son duiing the years 1759-1777. A graph of height \ersus 
age for his son was published and is shown at the lower left. 

From the graph, you can compute the average growth rate, or 
average speed of growth \\\J oxer the entire 18-year inteival or 
over any shorter time interval within that period. Suppose, 
however, you wanted to know how fast the boy was growing just 
as he reached his fifteenth l)iithday. The answer becomes 
evident if you enlarge the graph in the vicinity of the fifteenth 
year. His height at age 15 is indicated at point P, and the other 
letters indicate instants of time on either side of P. The boy's 
average growth rate over a 2-year inteival is given by the slope of 



T 



0.0C i or:: ) 



.'6 17 /6 




the line AB. Oxer a 1-year interval, this average growth rate is 
gixen by the slope CD. (See the thiid graph at the middle of this 
page.) The slope of a straight line drawn ft^om E to F gi\'es the 
a\'erage growth rate over 6 months, etc. The four lines, AB, CD, 
EF, and GH, are not exactly parallel to each other, and so their 
slopes are different. Howexer, the difference in slope gets smaller 
and smaller. It is large xvhen you compare AB and CD, less if you 
compare CD and EF, and still less between EF and GH. For 
intervals less than Af = 1 year, the line segments do become 
parallel to each other, as far as you can tell, and gradually merge 
into the curve. For- xery small internals, you am jind the slope by 
drawing a straight line tangent to this curve at P. This method 
inx'olx'es placing a Riler parallel to line GH at P and extending it 
on both sides. 



Z4 



LM T 1 / CO\CEI>^rS Ol MO HON 



The values of the slopes of the straight-line segments in the 
two right-hand graphs on page 24 have been computed for the 
corresponding time intervals. These values appear in the table in 
the margin at the right. Note that values of v^^ calculated for 
shorter and shorter time inteivals approach closer and closer to 
6.0 cm/yr. In fact, for any time inteival less than 2 months, v,^ will 
be 6.0 cm/yr within the limits of accuracy of measuring height. 
Thus, you can say that, on his fifteenth birthday, young de 
Montbeillard was grovvang at a rate of 6.0 cm/yr. At that instant in 
his life, t = 15.0 yr, this was his instantaneous growth rate. lYou 
might also express it as instantaneous speed of his head vvdth 
respect to his feet when he was lying still!) 

Average speed over a time interval A^ as mentioned earlier, is 
by definition the ratio of distance traveled to elapsed time. In 
svmbols, 



Line 

between 

points 



At 



Ad 



Growth rate 
Ad 



AB 


2yr 


19.0 


cm 


9.5 cm/year 


CD 


1 yr 


8.0 




8.0 


EF 


6 mo 


3.5 




7.0 


GH 


4 mo 


2.0 




6.0 


IJ 


2 mo 


1.0 




6.0 



If you have taken a course in cal- 
culus, you will recognize that we 
are discussing "limits," but in a 
simpler and less rigorous way. 



Ad 

^t 



You now have the definition of instantaneous speed at a giv^en 
instant t: It is the final, limiting value approached by the average 
speeds when you compute v^^, for a smaller and smaller range 
of time intervals that include the desired instant t. In almost all 
physical situations, such a limiting value can be accurately and 
quickly estimated by the graphical method used. 

From now on, the letter v (without the subscript J will be 
used to represent instantaneous speed defined in this way. You 
may wonder why the letter v instead of s was used for speed. 
The reason is that speed is closely related to velocity. The term 
"velocity" is used to mean speed in a specified direction (such 
as 50 km/hr to the north) and will later be represented by the 
symbol v. When direction is not specified and only the 
magnitude (50 km/hr) is of interest, remove the arrow and just 
use the letter v. This symbol represents only the magnitude of 
the velocity, that is, the "speed." This distinction between speed 
and velocity will be discussed in more detail in later sections. 
You also vvdll learn why, in physics, velocity is a more important 
concept than speed. 



SG 17 



• 13. Define instantaneous speed in words and in symbols. 

14. My average speed during a recent trip by car to Cleveland 
was 20 m/sec. Estimate my instantaneous speed for the time 
when I was (a) caught in citv traffic, (b) traveling on dry, open 
highway, and (cl filling up with gas. E}iplain the difference 
between average speed and instantaneous speed. 



See SG 19, 20, and 24 for problems 
that check your understanding of 
the chapter up to this point. 



CHAPTER 1 / THE LANGUAGE OF MOTION 



25 



Close Upl 

Photography, 1 609 to the Present 










/ rdfis i^ireet scene, 18od. A daguerreotype 
made by Louis Daguerre himself. 




2 American street scene, 1859 



1 . Note the lone figure in ttie otfierwise empty street. 
He was getting his shoes shined. The other 
pedestrians did not remain in one place long 
enough to have their images recorded. With 
exposure times several minutes long, the outlook for 
the possibility of portraiture was gloomy. 

2. However, by 1859, improvements in photographic 
emulsions and lenses made it possible not only to 
photograph a person at rest, but also to capture 

a bustling crowd of people, horses, and carriages. 
(With a magnifying glass you can see the slight blur 
of the jaywalker's legs.) 

3. Today, one can "stop" action with an ordinary 
camera. 

4. A new medium — the motion picture. In 1873 a 
group of California sports enthusiasts called in the 
photographer Eadweard Muybridge to settle the 
question, "Does a galloping horse ever have all four 
feet off the ground at once?" Five years later he 
answered the question with these photos. The five 




3. The moving car is seen in focus in the 
foreground, vjhile the stationary background 
appears blurred 



^^^Sg" ^^^Sff^ /^bSt^ 4'^HV^ ^^'^SBw 

^ST^^Tk £^ .37 ?7^ 



4 '.' , , s series. 1878 



za 



I'M r 1 / cx)\(;i:p'rs oi moiiox 



pictures were taken with five cameras lined up 
along the track. Each camera was triggered when 
the horse broke a string that tripped the shutter. The 
motion of the horse can be reconstructed by making 
a flip pad of the pictures. 

With the perfection of flexible film, only one 
camera was needed to take many pictures in rapid 
succession. By 1895, there were motion picture 
parlors throughout the United States. Twenty four 
frames each second were sufficient to give the 
viewer the illusion of motion. 

5. A light can be flashed successfully at a controlled 
rate, and a multiple exposure (similar to the strobe 
photos in this text) can be made. In this photo of 

a golfer, the light flashed 100 times each second. 

6. It took another 90 years after the time the 
crowded street was photographed before a bullet in 
flight could be "stopped." This remarkable picture 
was made by Harold Edgerton of h/IIT, using a 
brilliant electric spark which lasted for about 1 
millionth of a second. 

7. An interesting offshoot of motion pictures is the 
high-speed motion picture. In the frames of the milk 
drop series shown below, 1 ,000 pictures were taken 
each second (by Harold Edgerton). The film was 
whipped past the open camera shutter while the 
milk was illuminated with a flashing light (similar to 
the one used in photographing the golfer) 
synchronized with the film. When the film Is 
projected at the rate of 24 frames each second, 
action that took place in 1 sec is spread out over 42 
sec. Very high-speed photography was used to 
make several of the Project Physics film loops. 

It is clear that the eye alone cannot see the 
elegant details of this event. This is precisely why 
photography of various kinds is used in the 
laboratory. 



■ 




1 


^^H^BI^^BB^^I 


1 




nVw 


tKt 


^KS^^^ftv^r""-''^ 


mt 


Br 


.|^3 




ii 


M 


^i^i^H 



5. Stroboscopic photo of golfer's swing. 




6. Bullet cutting through a playing card. 




7. Action shown in high-speed film of milk 
drop. 



CHAPTER 1 / THE LANGUAGE OF MOTION 



27 



Unless noted otherwise, "rate of 
change^' will always m(;an "rate of 
change with respect to time. " 



1.8 I Acceleration hy comparison 

\ou can infei" troni the photograph at Ihe hottoiii ot this page of 
a basehall rolling on an incline that the ball vvas changing sj^eed 
(acceleratingi. Assuming the time between flashes to haxe been 
constant, the incieasing distance between the images of the ball 
give you this information. But how can \{)u tell how much 
acceleration the ball has.^ 

To answer this question, you ha\e to \v.nm the definition of 
acceleration. The definition itself is simple. 1 he real task is to 
learn how to use it in situations like the one below. For the time 
being, acceleration can be defined as rate ofchans,e of speed. 
Later, this definition v\ill have to be modified somewhat when 
you encounter motion in which change in direction becomes 
impoilant. For now, you are dealing only with straight-line 
motion. Therefore, you can equate the rate of change of speed 
v\ith acceleration. 

Some of the effects of acceleration are familiar to exciAone. It is 
acceleration, not speed, that you notice when an elevator 
suddenly starts up or slows down. The flutter in your stomach 
comes only during the speeding up and slowing down. It is not 
felt during most of the ride, when the elevator is moving at a 
steady speed. Likewise, the excitement of the I'oUer coaster and 
other rides at amusement parks results from their unexpected 
accelerations. Speed by itself does not cause these sensations. If 
it did, you would feel them during a smooth plane ride at 900 
km/hr, or during the continuous motion of the earth around the 
sun at 105,000 km/hr. 

Simply stated, speed is a relationship between fwo objects. 
One object is taken to be the leference object, while the other 
moves with respect to it. Some examples are the speed of the 
earth with respect to the sun, the speed of the swimmei' with 
respect to the pool edge, the speed of the top of the growing 
boy's head with respect to his feet. In a perfectly smooth-riding 
train, you could tell that you were moving at a high speed only 
by seeing the sceneiy speeding by. Vou would have just the same 
experience if the train were somehow fixed and the earth, rails, 
etc., were to speed by in the other direction. If you "lost the 
reference object" (by pulling down the shades, say), you could 
not tell whetheivou were moving or not. In contrast, vou "feel" 




'ZH 



I M I 1 COVCLPI S Ol MOnON 



accelerations. You do not need to look out the train window to 
realize that the engineer has suddenly started the train or has 
slammed on the brakes. You might be pushed against the seat, or 
the luggage might fly from the rack. 

All this suggests a profound physical difference between 
motion at constant speed and accelerated motion. It is best to 
learn about acceleration at first hand (in the laboratoiy and 
through the film loops). But the main ideas can be summarized 
here. For the moment, focus on the similarities between the 
concepts of speed and acceleration. For motion in a straight line: 



The rate of change of 
position is called speed. 



The rate of change of 
speed is called 
acceleration. 



This similarity of form is veiy helpful. It enables you to use what 
you have just learned about the concept of speed as a guide for 
using the concept of acceleration. For example, you have learned 
that the slope of a line of a distance-iime graph is a measure of 
instantaneous speed. Similarly, the slope of a speed-time graph is 
a measure of instantaneous acceleration. 

This section concludes with a list of sbc statements about 
motion along a straight line. The list has two purposes: (1) to 
help you review some of the main ideas about speed presented 
in this chapter, and (2) to present the corresponding ideas about 
acceleration. For this reason, each statement about speed is 
immediately followed by a parallel statement about acceleration. 

1. Speed is the rate of change of position. y4cce/eran'on is the 
rate of change of speed. 

2. Speed is expressed in units of distance/time. Acce/eraf /on is 
expressed in units of speed/time. 

3. Average speed over any time interval is the ratio of the 
change of position Ad to the time interval At: 

Ad 

"- ^ a7 

Average acceleration over any time inteival is the ratio of the 
change of speed Av to the time interval Af : 



a„ — 



Av 
Af 



4. Instantaneous speed is the value approached by the average 
speed as Af is made smaller and smaller. Instantaneous 
acceleration is the value approached by the average acceleration 
as Af is made smaller and smaller. 

5. On a d/sfance-time graph, the instantaneous speed at any 
instant is the slope of the straight line tangent to the cuiv/e at the 
point of interest. On a speed-time graph, the instantaneous 



For example, if an airplane changes 
its speed from 800 km/hr to 850 
km/hr in 10 min, its average accel- 
eration would be 



Av 
At 



850 km/hr - 800 km/hr 



10 min 



50 km/hr 



= 5 



10 min 
km/hr 



mm 



or 5 km/hr/min 



That is, its speed changed at a rate 
of 5 km/hr per minute which, ex- 
pressed in a consistent system of 
units, is about 0.08 km/min" or 300 
km/hr". (If the speed was decreas- 
ing, the value of the acceleration 
would be negative.) 



AC" 




' At 



AW{ 




'- > ■> 



At 



CHAPTER 1 / THE LANGUAGE OF MOTIOM 



29 



Constant speed and constant accel- 
eration are often called "uniform" 
speed and "uniform" acceleration. 
In the rest of this course!, we will 
use the terms interchani'cahlv. 



SG 21 provides an opportunity to 
work with distance-time and 
speed-time graphs and to see their 
relationship to one another. 



SG 22-25 are review problems for 
this chapter. Some of these will test 
how thoroughly you grasp the lan- 
guage used for describing straight- 
line motion. 



acceleration at any instant is the slope of the straight line 
tangent to the cune at the point of interest. See the giaphs in 
the margin on page 29. 

6. For the partieular case oi conslnnt speed, [hv. tiistance-time 
graph is a stiaight line. Therefore, the instantaneous speed has 
the same value at every point on the graph. Further, this value is 
eciuai to the axeiage speed coniputeHJ loi- the whole trip. For the 
partieular ease oi constant acceleration, the speed-time graph 
is a straight line. Therefore, the instantaneous acceleration has 
the same value at every point on the graph. Further, this xaiue is 
equal to the axerage acceleration computed for the whole trip. 
When speed is constant, its value can he found from any 
corresponding Ad and Af. When acceleration is constant, its 
valire can be Idirnd from any corresponding A\' and A/, i This is 
useful to remember" because constant acceleration is the kind of 
motion you will encounter most often in the following chapters.) 

Vou now have most of the tools needed to get into some real 
physics problems. The first such problem will inxoKe the 
acceler'ated motion of bodies caused by gra\'itational attraction. It 
was by studying the motion of falling obj(;cts that Cialileo, in the 
early 1600s, first shed light on the nature of accelerated motion. 
His work remains a wonderful example of how scientific theory, 
mathematics, and actual measur'ements can be combined to 
develop physical concepts. More than that, Galileo's work 
opened one of the earliest and most inipoitant battles of the 
scientific r^evolution. The specific ideas he introduced are even 
today fundamental to the science of mechanics, the stud\ of 
bodies in motion. 




15. What is the average acceleration of an airplane that goes 
from to 100 km/hr in 5 sec? 

16. What is your average acceleration if, while walking, you 
change your speed from 5 km/hr to 2 km hr in an inter\al of 
15 min? Is your answer affected by how your change of speed 
is distributed over the 15 min? 

17. Using the formula for average acceleration, determine how 
long it takes a bird starting from rest to reach a speed of 12 
m/sec if it accelerates uniformly at 4 m/sec~. 

18. If the bird in Ojiiestion 17 continued to accelerate at the 
same rate, how much speed would it add in another 6 sec? 



30 



L.MT 1 co\c;Li»rs oi .Mono.v 



study 
guide 



1. This book is probably different in many ways 
from textbooks you have had in other courses. 
Therefore, it might help to make some suggestions 
about how to use it. 

(a) If this is your own personal copy and you intend 
to retain it after you have completed the course, in 
short, if you are in a situation that permits you to 
mark freely in the book, do so. You will note that 
there are wide mai'gins to record questions or 
statements as they occur to you. Mark passages that 
you do not understand so that you can seek help 
from your instructor. 

(b) If you may not wiite in the textbook itself; try 
keeping a notebook keyed to the text chapters. In 
this study notebook, jot down the kinds of remarks, 
questions, and answers that you would otherwise 
write in the textbook as suggested above. Also, you 
ought to write down the questions raised by the 
other learning materials you wiU use, by the 
experiments you do, by demonstrations or other 
observations, and by discussions you may have with 
fellow students and others with whom you talk 
physics. Most students find such an informal 
notebook to be enormously useful when studving, or 
when seeking help from their instructors (or, for 
that matter, from advanced students, scientists they 
may know, or anyone else whose understanding of 
physics they trust). 

(c) Always try to answer the end-of-section rexaew 
questions yourself first, and then check your 
answers. If your answer agrees with the one in the 
book, it is a good sign that you understand the main 
ideas in that section (although it is true that you can 
sometimes get the right answer for the wrong 
reason). Also, sometimes there may be other answers 
as good as (or better than!) those given in the book. 

(d) There are many different kinds of items in the 
Study Guide at the end of each chapter. It is not 
intended that you should do eveiy item. Sometimes 
material is included in the Study Guide which may 
especially interest onlv some students. Notice also 



that there are several kinds of problems. Some are 
intended to give practice in the use of a particular 
concept, while others are designed to help you bring 
together several related concepts. Still other 
problems are intended to challenge those students 
who particularly like to work with numbers. 

(e) This text is only one of the learning matericds of 
the Project Physics course. The course includes 
several other materiids, such as film loops and 
filmstrips or transparencies. Use them if they are 
available to you. Be sure to familiarize yourself also 
with the Handbook for students, which describes 
outside activities and laboratory experiments. Each of 
these learning aids makes its oun contribution to 
an understanding of physics, and all have been 
designed to be used together. 

Note: The Project Physics learning materials 
particularly appropriate for Chapter 1 include: 

Experiments (in the Handbook) 

Naked Eye Astronomy 
Regularity and Time 
Variations in Data 
Measuring Uniform Motion 

2. Define, in words and symbols, the following 
terms: speed, uniform motion, average speed, slope, 
instantaneous speed, average acceleration. What does 
the symbol "A" mean? 

3. A goalie shoots a puck to his teammate 30 m 
away. If the puck took 1.5 sec to cover the distance, 
what was its average speed? 

4. Some practice problems: 



Situation 



Find 



Speed uniform, distance 
72 cm, time = 12 sec 



Speed 



b Speed uniform at 60 km/hr 



Distance traveled in 
20 min 



c Speed uniform at 36 m/min 



Time to move 9 m 



CHAPTER 1 / STUDY GUIDE 



31 



sg 



d d, = 
dj = 1 5 cm 
d, = 30 cm 



f, = 

fj = 5.0 sec 

U = ^0 sec 



Speed and position at 
8.0 sec 



e You drive 240 km in 6.0 hr 



Average speed 



f Same as e 



Speed and position 
after 3.0 hr 



g Average speed is 76 cm/sec, 
computed over a distance 
of 418 cm 



Time taken 



h Average speed is 44 m/sec, 
computed over a time 
interval of 0.20 sec 



Distance moved 



5. After the parachute opens, a sky diver falls v\ath 
a roughly uniform speed of 12 m/sec. How long docs 
it take her to fall 228 m? If she; ('ontinues to fall for 
another 25 sec, what is the total distance she lias 
fallen? 

6. What is your average speed in each of these 
cases? 

la) Vou run 100 ni at a speed of 5.0 m/sec and then 
you walk 100 m at a speed of l.O m/sec. 

(b) You run for 100 sec at a speed of 5.0 nx'sec and 
then you walk for 100 sec at a speed of 1.0 nVsec. 

7. A rabbit and a turtle are practicing for their 
race. The rabbit covers a 30-m course in 5 sec; the 
turtle coxers the same distance in 120 sec. If the race 
is run on a 96-m course;, l)y how many seconds will 
the ral)bit beat tiie turtle? 

8. A tsunami caused by an earthquake occurring 
near Alaska in 1940 consisted of sexeral huge wa\'es 
which were found to travel at the average speed of 
790 km/hr. The first of the waves reached Hawaii 4 
hr 34 min after the eai'thciuake occurred. From these 
data, calculate how far the origin of the tsunami was 
from Hawaii. 

9. Design and describe experiments to enable you 
to make estimates of the average speeds for some 
of the following objects in motion. 

(a) A baseball thrown from the outluild to home 
plate 

(b) the wiiKJ 



(c) a cloud 

Id) an ant walking 

(e) a camera shutter opening and closing 

(f) an eye blinking 

(g) a whisker growing 

10. Light and radio waves tra\'el through a vacuum 
in a straight line at a speed of nearly 3 x uf m/sec. 

(a) How long is a "light yeai'" I the distance light 
travels in a year)? 

(b) The nearest star, Alpha Ccntauri, is 4.0(S x lo"' m 
distant from (;arth. If this star po.ss(!sses planets on 
which highly int(;lligent beings li\e, how soon, at the 
earliest, t;ould we expect to receive a reply after 
sending them a radio or light signal strong enough to 
be received there? 

(c) Sound moves very quickly; it is hard to notice the 
time elapsed between when you see somebody say 
something and when you hear the sound. 

(1) Can you think of situations when you can tell 
that sound does not reach you virtually 
instantaneously? 

(2) Try to d(!sign an e\p(M'iment to measure the 
speed of sound. 

(3) The measurements made in many experiments 
indicate that under ordinary' circumstances the 
speed of sound in air is about 330 m/sec ( = 0.33 
km/sec). The speed of light is about 300,000 km/ 
sec. Suppose lightning strikes 1 km away. How 
long does it take before you see the flash? How 
long before you hear the thunder? 

(4) Can you use the known speeds of light and 
sound to find the distance to any lightning stroke? 

11. 'I\vo cyclists race with nearly uniform speed on 
a 500-m course. The blue bic\'cle crosses the finish 
line 20 sec ahead of the red bicycle. If the red bicycle 
inaintained an average spe(!d of 10 m sec, what was 
the average speed of tin; h\uv. bicycle? 

12. /\fter starting from rest, a cai- reaches a speed of 
30 m/sec in 5 sec. What is its average acceleration? 

If the car accelerates at thai rale for an additional 5 
sec, what is its final speed? 



S'Z 



(;ii\i'ii;h i sitdv (;i iuij 



sg 



13. The foUouing graph represents a jogger mo\dng 
with uniform speed who passes posts at 20 ni, 
30 m, and 50 m at 4 sec, 6 sec, and 10 sec, 
respectively. The jogger's average speed over the 




f i"me (sec) 



SO 



40 - 



9 30 



u 
c 

^20 



10- 











- 




/ 


/ E 
/ % 






F 


AX= 6 sec 


/ 


1 


Ay 
Ax 


30m 

5m/sec 

6sec 

1 1 J 



A 6 

time (s€c) 



o 



interval from 4 sec to 10 sec can be calculated as 
shouTi on the second graph. (Remember, on a 
distance-time graph, the slope of the line between 
two points is the average speed over that interval.) 

(a) Use the same method to find the jogger's average 
speed for the following intervals: (1) starting line (0 
sec) to 6 sec; (2) starting line to 10 sec; (3) 6 sec to 10 
sec; (4) 5 sec to 8 sec. 

(b) Your answers to (a) should indicate that no 
matter how small the inten'al you choose, you will 
always find the siune speed for a straight line on a 
distance-time graph. Why? 

(c) Now find the instantaneous speed, which is the 
speed over an extremely small interval. Without 
measuring the small Ay or A;t, what is the 
instantaneous speed at the 8-sec mark? How do you 
know? [Hint: see part (b).] 

14. The following graph shows the motion of a ball; 
use it to answer the qviestions below. 




• 5 



\o 



5 lO 

time (sec> 

(a) Without doing any calculations, determine in 
which section the ball was traveling fastest; slowest. 

(b) Ccilculate the average speeds over each interval 
(AB, BC, CD) and for the entire distance (AD). 

(c) Without doing any calculations, determine the 
instantaneous speed at point f in section CD. 

15. The graph on page 34 represents the motion of 
a jogger who speeds up as she runs. Notice that the 
graph is not a straight line. Average speed can be 
calculated from the graph using the same method as 
for a straight-line graph (see Question 13). However, 
to find the instantaneous speed at any point on the 
curved graph, you must draw a line tangent to the 
curve at that point. This tangent line has the same 
slope (speed) as the curve at the point where they 
touch. 



CHAPTER 1 / STUDY GUIDE 



33 



ss 



(a) What is the instantaneous speed at the 10-sec 
mark? At the 25-sec mark? 

(b) What is the average acceleration hc^lween t = 10 
sec and t = 25 sec? 



50|- 
45 - 

40 - 

56 - 

1 

20 '^ 

I 

15 t- 
10 - 

5 - 
O 



'2.? sec 



fc sec 



-I J. ,-L-.,i .11 

(O 15 20 i5 ?o ?.^ 



4o -<5 50 



16. W'orkl's 4UU-m swimming record in miniil(!s and 
seconds for men and women: 

U)2(i 4:.'j7.() Johnny W'cissmiiller 

5:53.2 Gertrude Ederle 

WMi 4:46.4 Syozo Makino 

5:28.5 HehMie Madison 

1946 4:46.4 1 1936 record uni)roken) 

5:00.1 H. Hveger 

1956 4:33.3 Hironoshin Furuhashi 

4:47.2 Lorraine; C'rap|) 

1966 4:11.1 Frank Weigand 

4:38.0 Miutha Handidl 

1976 3:15.9 Brian Goodell 

4:09.8 Fetra Thumer 

By aljout iiow many meters would Martha Handall 
have beaten Johnny WeissmuUer if tliey had been 
able to race each other? Could you prcuiict the 1986 
records for the 400-m race i)y extrapolating the 
graphs of world's records \s. dates up to the year 
1986? 

17. Using the graph on p. 20, find the instantaneous 
speeds v at severjil points (0, 10, 20, 30, 40. and 50 



sec, and near 0, or at other points of your choice) by 
finding the slopes of lines tangent to the curve at 
each of those points. Make a graph of i' \ersus /. 

18. Discuss the following quotation from Mark 
■I\vain s iJJ'r on the Missi.ssiiipi 1 18751 as an example 
of e\trapf)lation. "In IIk; space of One hundr(<d and 
se\'enty-si\ years the Fower Mississippi has 
shortened itself two hundred and forty-two miles. 
That is an a\ erage of a triflf! over one; mih; and a 
third p(M- y(;ai'. Thcrcdbn;. any calm pcM'son, who is 
not blind or idiotic, can sv.r that in tlu; old Colitic 
Silurian Period, just a million years ago next 
November, the Lower Mississippi Biver was upward 
of one; million three hundred thousand miles long, 
anti stuck out o\er the (iulf of Mexico like; a iishing 
rod. And by the same token any person can see that 
seven hundred and forty-tvvo years from now the 
Lower Mississippi Ki\'er will be only a mile and 
three-quarters long, and ("airo and \(n\ Orh^ans will 
ha\'e joined their streets together, and be plodding 
comfortably along under a single mayor and a 
mutuid board of aldermen. There is .something 
fascinating about s( ienc(!. (Jne gets such wholesale 
returns of conjecture; out of such trifling investment 
of fact." 

19. Careful analysis of a stroboscopic photograph of 
a moving object yielded information that was plotted 
on the graph below. 



.CO 




34 



cii\i»ri i{ 1 sii I)^ (;i iDi; 



sg 



(a) At what moment or interval was the speed 
greatest? What was the speed at that time? 

(b) At what moment or in which interval was the 
speed least? What was it at that time? 

(c) What was the speed at time t = 5.0 sec? 

(d) What was the speed at time t = 0.5 sec? 

(e) How far did the object move from time t = 7.0 
sec to f = 9.5 sec? 

20. A band of pioneers left St. Louis in their wagon 
train. They traveled at a good rate for the first 2.5 
weeks lAB), but slowed down as the initial excitement 
wore off (BO. Then they picked up speed for a week 
(CD) and really raced through dangerous country 
(DE). They rested at a watering hole (EFI and finally 
moved on (FG). 

(a) By observing the graph, determine which intenal 
was covered fastest; slowest. 

(b) According to the information given, which 
interval does not look as if it were drawn correctly? 
What should it look like? 

(c) Find the average speed over the whole trip. 

(d) V\/hat are the instantaneous speeds at points P 
and Q? 




time (weeks) 



ai. The data below show the instantaneous speeds 
in a test run of a car starting from rest. Plot the 
speed-time grapli, then derive data from it and plot 
the acceleration-time graph. 

(a) What is the speed at t = 2.5 sec? 

(b) What is the maximum acceleration? 



Time 


Speed 


Time 


Speed 


(sec) 


(m/sec) 


(sec) 


(m/sec) 


0.0 


0.0 


6.0 


27.3 


1.0 


6.3 


7.0 


29.5 


2.0 


11.6 


8.0 


31.3 


3.0 


16.5 


9.0 


33.1 


4.0 


20.5 


10.0 


34.9 


5.0 


24.1 







22. The electron beam in a typical T\/ set sweeps 
out a complete picture in 0.03 sec, and each picture 
is composed of 525 lines. If the uddth of the screen 
is 50 cm, what is the speed of that beam horizontally 
across the screen? 

23. Turn back to p. 28. At the bottom of this page 
there is a multiple-exposure photograph of a baseball 
rolling to the right. The time inten'al between 
successive flashes was 0.20 sec. The distance between 
marks on the meter stick was 1 cm. You might 
tabulate your measurements of the ball's progress 
between flashes and construct a distance-time 
graph. From the distance-time graph, determine the 
instantaneous speed at several instants and construct 
a speed-time graph. You can check your results by 
referring to the answer page at the end of this book. 

24. Suppose you must measure the instantaneous 
speed of a bullet as it leaves the barrel of a rifle. 
Elxplain how you might do this. 

25. Discuss the motion of the horse in the following 
series of photographs by Muybridge. The time 
inter\'al between exjjosures is 0.045 sec. 




CHAPTER 1 / STUDY GUIDE 



35 




-r^-:*'i^'l :; 






Free Fall 

Galileo Describes lUlotion 




A sketch of a medieval 
world-svstem. 



SG 1 



The Aristotelian theori' of motion 

Galileo and his times 

Galileo's Tit'o IVtnr Sciences 

^^lA\ stufh' the motion ol' freeh- falling bodies? 

Galileo chooses a definition of uniform acceleration 

Galileo cannot test his Inpothesis directh- 

Lookin^ for lo^ciil consequences of Galileo's hypothesis 

Galileo tiu'ns to an indirect test 

Doubts about Galileo's procedure 



2.1 
2J3 
2.3 
2.4 
2.5 
2.6 
2.7 
2.8 
2.9 
2.10 Consequences of G<dileo\s itork on motion 

2«1 I The Aristotelian theori' of motion 

In this chapter you will follow the de\elopnient ot an important 
piece of basic research: Galileo's study of freely falling bodies. 
The phenomenon of free fall is interesting in itself. But the 
emphasis will be on the way Galileo, one of the fiist modern 
scientists, presented his argument. His view of the world, way of 
thinking, use of mathematics, and reliance upon experimental 
te.xts set the style for modern science. These aspects of his work, 
therefore, are as important as the actual results of his 
inxestigation. 

I'o understand the natuie and impoitancc of Galileo s work, 
you must first examine the previous system of physical thought 



36 



I Ml 1 (;{)\(;i:i>rs oi moiiox 



which his ideas eventually replaced. Medieval physical science, 
as Galileo learned it at the University of Pisa, made a sharp 
distinction between objects on the earth and those in the sky. All 
terrestrial matter, matter on or near the earth, was believed to 
contain a mixture of four "elements": Earth, Water, Air, and Fire. 
These elements were not thought of as identical with the natural 
materials for which they were named. Ordinary water, for 
example, was thought to be a mixture of all four elements, but 
mostly the ideal element Water. Each of the four elements was 
thought to have a natural place in the terrestrial region. The 
highest place was allotted to Fire. Beneath Fire was Air, then 
Water, and finally, in the lowest position, Earth. Each was 
thought to seek its own place. Thus, Fire, if placed below its 
natural position, would tend to rise through Air. Similarly, Air 
would tend to rise through Water, whereas Earth would tend to 
fall through both Air and Water. The movement of any real object 
depended on its particular mixture of these four elements and 
on where it was in relation to the natural places of these 
elements. When water boiled, for example, the element Water 
would be joined by the element Fire, whose higher natural place 
would cause the mixture to rise as steam. A stone, on the other 
hand, was composed mainly of the element Earth. Therefore, a 
stone would fall when released and would pass through Fire, Air, 
and Water until it came to rest on the ground, its natural place. 
Medieval thinkers also believed that stars, planets, and other 
celestial (heavenly) bodies differed in composition and behavior 
from objects on or near the earth. Celestial bodies were believed 
to contain none of the four ordinaiy elements, but to consist 
solely of a fifth element, the quintessence. The difference in 
composition required a different physics. Thus, the natural 
motion of celestial objects was thought to be neither rising nor 
falling, but an endless revoking in circles around the center of 
the universe. That center was considered to be identical with the 
center of the earth. Heavenly bodies, although moving, were at 
all times in their natural places. In this way, heavenly bodies 



A good deal of common-sense ex- 
perience supports this natviral-place 
view. See SG 2. 



From quinta essentia, meaning fifth 
(quint) element [essence). In earlier 
Greek writings the term for it was 
aether (also written ether). 




This painting, entitled "School of 
Athens,' was done by Raphael in 
the beginning of the sixteenth cen- 
tury. It reflects a central aspect 
of the Renaissance, the rebirth of 
interest in classical Greek culture. 
The central figures are Plato (on 
the left, pointing to the heavens! 
and Aristotle (pointing to the 
ground). 



CHAPTER 2 I FREE FALL 



37 



differed from terrestrial objects, which displayed natural iiiolion 
only as they returned to their naliuai places fioni wliich tlic^y 
had been displaced. 

This theoiy, so widely held in Galileo's time, had (jriginatcHJ 
almost 2,000 years before, in the fouilh centiA I5 c it is stated 
clearly in the writings of the dreek philosopher .AristotUv iSee the 
time chart on the opposite jJagei This phxsical science, built on 
notions of cause, order, class, place, and purpose, seemed to fit 
well with many exeiyday obsenations. Moieo\ei-, these ideas 
about matter" and motion were part of an all-embracing unixersal 
scheme, or cosmology. In this cosmology, Aristotle sought to 
relate ideas which today are discussed separateK' under su("h 
headings as science, poetiy, politics, ethics, and theolog\'. 

Not very much is known of Aristotle's physical appearance or 
life. It is thought that he was born in 384 H c: in the Greek 
province of Macedonia. His father was the physician to the King 
of Macedonia, so Aristotle's early childhood was spent in an 
environment of court life. He studied in Athens with Plato and 
later returned to Macedonia to become the pri\ate tutor to 
Alexander the Great. In 335 B.C., Aristotle came back to Athens 
and founded the Lyceum, a school and center of research. 

After the decline of the ancient Greek civilization, Aristotle's 
waitings remained almost unknown in VVestei'n Europe for 1,500 
years. They were rediscovered in the thirteenth centuiy a.d. and 
soon began to shape the thinking of Christian scholars and 
theologians. Aristotle became such a dominant influence in the 
late Middle Ages that he was refeired to simply as "The 
Philosopher. " 

Unfortunately, Aristotle's physical theories had serious 
limitations. (This does not, of course, detract from his great 
achievements in other fields.) Accoixling to Aristotle, the fall of a 
heavy object toward the center of the eaith is an example of 
"natural " motion. He evidently thought that any object, after 
release, quickly reaches some final speed of fall which it 
maintains to the end of its path. What factors determine" th(! final 
speed of a falling object? It is a common obseivation that a rock 
falls faster than a leaf. Therefore, Aristotle reasoiK^l, weight is a 
factor that governs the speed of fall. This fitted in vxell with his 
idea that the cause of weight was the presence of the element 
Earth, whose natural motion was to the center of the earth. 
Thus, a heavier object, ha\ing a greater content of llaith, has a 
stronger tendency to fall to its natural place. In turn, this 

. stronger tendency creates a greater speed of falling. 

Aristotle: Rate of fall is proportional I'he same object falls more slowly in v\ater than in air, so 

to weight divided by resistance. Aristotle reasoned that the resistance of the medium must also 

affect motion. Other factor's, such as the color* or- teiii|)(Miitur(! of 
the falling object, also might change the rate of tall. Ikil Aristotle 
de( id(Hl that such influences could not be im[i()r1aiit lie 

38 UNIT 1 / COXCEPI S OF MOTION 



500 BC 



400 BC 384 BC 322 BC 

ARISTOTLE 



300 BC 



200 BC 




PTOLEMY I of Egypt 
PHILIP II H 



HANNIBAL 



ALEXANDER 



CONFUCIUS 
ANAXAGORAS 
ZENO OF ELEA 
PARMENIDESOFELEA ■ 

PROTAGORAS WM 
DEMOCRITUS 
SOCRATES _ 

FEsro 



, a. 



MENCIUS(MENGTZU) 
EPICURUS 

ZENOOFCmUM 

HSiJN TZU 




DIOGENES 



EUCLID 



P^RCHIMEDES 

ERASTOSTHENES 




AESCHYLUS 
I 
^ PINDA 




DEMOSTHENES 



PLAUTUS 



CHUANG TZU 



SOPHOCLES 

HERODOTUS 

EURIPIDES 

THUCYDIDES 

ARISTOPHANES 
I 
XENOPHON 





^PHIDIAS fli^^^^ SC0PAS1 

MYRONSHl 

POLYGNdTOS 

POLYCLITUS 
ZEUXIS 

TIMOTHEUS 




LYSIPPUS 

PRAXITELES 
MILLES^^^ , 



ARISTOXENUS 



John Philoponus: Rate of fall is pro- 
portional to weight minus resis- 
tance. 



SG 3 



Qjualitativc refers to quality — the 
sort of thing that happens. Ouan- 
titHtivc refers to quantity — the 
measurement or prediction of nu- 
mcrinil \alues. Tliis distinction w ill 
be made often in the course. 



concludeci tiial the rate ol fall must increase in jjroportion to the 
weight of the object and decrease in proportion to the resisting 
force of the medium. The actual late of fall in any [particular case 
would he found bv dixiding the weight by the resistance. 

Aristotle also discussed "violent" motion, that is, an\' motion of 
an object othei- than going fieely towaicl its 'natuial place." Such 
motion, he argued, must always be caused by nforcc, and the 
speed of the motion must increase as the force increases. When 
the force is remoxed, the motion must stop. This theoiy agrees 
with common experience, for example, in pushing a chair oi" a 
table across the floor-. It does not woik ciuite so well for objects 
thrown thiough the air, since they keep moving for a v\'hile even 
aftei' you have stopped exerting a force on them. To explain this 
kind of motion, Aristotle pi^oposed that the aii' itself somehow 
exerts a force that keeps the object mo\ ing. 

Later scientists suggested certain changes in Aristotle s theoiy 
of motion. For example, in the fifth centuiy A.u. John Philoponus 
of Alexandria argued that the s|)eed of an object in natural 
motion should be found by subtractinu, the resistance of the 
medium fiom the weight of the object. (Aristotle, you recall, 
recommended dividing by the resistant'e.i Philoponus claimed 
that his expeiimental work supported his theoiy, though he did 
not report the details. He simply said that he dropped two 
weights, one twice as hea\y as the other, and obsent^d that the 
heavy one did not reach the ground in half the time taken by the 
light one. 

There were still other difficulties with Aristotle's theoiy of 
motion. However, the knowledge that his teachings had faults 
did little to lessen their influence in the universities of France 
and Italy during the fifteenth and sixteenth centuries and during 
Galileo's lifetime. Aristotle's theoiy of motion did, after all, fit 
much of ordinaiy experience in a general, if qualitative, way. 
Besides, the study of motion through space was of great interest 
to only a few scholars, just as it hail been only a ven' small part 
of Aristotle's own work. 

I\vo other influences stood in the way of major changes in the 
theoiy of motion. First, Aristotle beliexed that mathematics was 
of littl(! \ aku! in describing terrestrial jjhenomena. Setxjnd, he put 
great emphasis upon direct, qualitative observation as the basis 
for forming theories. Simple qualitative obsenation was veiy 
useful in Aristotle's biological studies. But as it turned out, real 
progress in physics began only when scientists recognized the 
value of mathematical prediction and detailed measurement. 

A number of scholars in the fifteenth and sixteenth centuries 
took part in this change to a new way of approaching science. 
But of all these, Galileo was by far the best known and most 
successful. He showed how to describ(> mathcmaticalK the 
motions of simi^le, ordinaiy objects, such as falling stomas and 



411 



UNIT 1 / CO\CEI>'rS OF MOTION 



balls rolling on an incline. Galileo's work paved the way for othei' 
scholars to describe and explain the motion of eveiything from 
pebbles to planets. It also began the intellectual revolution that 
led to what is now considered modern science. 



# 1. Which of the following properties do you believe might 
affect the rate of fall of an object: color, shape, size, or weight':' 
How could vou determine if your answers are correct? 

2. Describe two ways in which, according to the Aristotelian 
view, terrestrial and celestial bodies differ from each other. 

3. Which of these statements would be accepted in the 
fifteenth and si;<teenth centuries by persons who believed in 
the Aristotelian system of thought? 

(a) Ideas of motion should fit in with poetry, politics, theology, 
and other aspects of human thought and activity. 
(bj Heaxy objects fall faster than light ones. 

(c) E}icept for motion toward their natural location, objects 
will not move unless acted on violently by a force. 

(d) Mathematics and precise measurement are especially 
important in developing a useful theory of motion. 



2»2 I Galileo and his times 

Galileo Galilei was born in Pisa in 1564, the year of 
Michelangelo's death and Shakespeaie's birth. Galileo was the 
son of a noble family from Florence, and he acquired his father's 
active interest in poetiy, music, and the classics. His scientific 
inventiveness also began to show itself early. For example, as a 
young medical student, he constiTicted a simple pendulum-t>q3e 
timing device for the accurate measurement of pulse rates. 

After reading the classical Greek philosopher-scientists Euclid 
and Archimedes, Galileo changed his interest from medicine to 
physical science. He quickly became known for his unusual 
scientific ability. At the age of 26, he was appointed Professor of 
Mathematics at Pisa. There he showed an independence of spirit 
unmellowed by tact or patience. Soon after his appointment, he 
began to challenge the opinions of older professors, many of 
whom became his enemies. He left Pisa before his term was 
completed, appai'ently forced out by financial difficulties and by 
his enraged opponents. Later, at Padua in the Republic of Venice, 
Galileo began his work in astronomy. He supported the belief 
that the earth moves around the sun. This belief brought him 
additional enemies, but it also brought him immoital fame. That 
part of his work will be covered in L'nit 2. 






AUSTIfIA 






h/UM&ftRY 









mitf 



CdWSfCA 



Ki^&DOIfi OF 
SARDINIA 




KIN'^&OIA OF SICILY 



Italv about 1600. 



CHAPTER 2 / FREE FALL 



41 



1500 



1564 



1600 

I 

GALILEO 



1642 



1700 



IVAN THE TERRIBLE of Russia 
I 
ELIZABETH I of England 



E i 

i i 



CROI\^WELL 



LOUIS XIV of Franci 



HENRY VIM of England 

LUCREZIA BORGIA 



POPE URBAN VIII 
JMIARY STUART 



CARDINAL RICHELIEU 




COPERNICUS 
JEAN FERNEL 

^ANDREASVESALIUS 

I 

AMBROISE PARE 
I 
TYCHO BRAHE 



FRANCIS BACON 

KEPLER 

WILLIAM HARVEY 
I 
RENE DESCARTES 




NEWTON 
I 
GOTTFRIED LEIBNITZ 



GIORDANO BRUNO 




BLAISE PASCAL 
I 
ROBERT BOYLE 
I 
CHRISTIAN HUYGENS 




MACHIAVELLI 

I 
ERASIMUS 

MARTIN LUTHE 

1 

ST. IGNATIUS OF LOYOLA 
JOHN CALVIN 



THOMAS HOBBES 




SPINOZA 
I 
JOHN LOCKE 



1 

BEN JOHNSON 
I 
.M0NTAIGNE 
I 
CERVANTES 

EDMUND SPENSER 

WILLIAM SHAKESPEARE 



RABELAIS 




JOHN MILTON 

MOLIERE 




MICHELANGELO 
TITIAN 




PIETER BREUGHEL 
EL GRECO 



BERNINI 

I 

VELASQUEZ 

REMBRANDT VAN RUN 



RUBENS 



1 

PALESTRINA 
I 
ORLANDO Dl LASSO 

GIOVANNI GABRIELI 

MONTEVERDI 



HENRY PURCELL 



A generous offer of the Grand Duke drew Galileo back to his 
native province of luscanv in 1610. He became Couit 
Mathematician and Philosopher, a title which he chose himself. 
From then until his death at 78, despite illness, family troubles, 
occasional poverty, and quarrels with his enemies, Galileo 
continued his research, teaching, and writing. 



2*3 I Galileo's Ttvo JVcw Sciences 

Mechanics is the study of the behavdoi- of mattei' under the 
influence of forces. Galileo's early writings on this subject follow 
the standard medieval theories of physics, although he was 
aware of some of the shortcomings of those theories. Duiing his 
mature years his chief interest was in astronomy. However, his 
important astronomical book Dialogue on the Two Great World 
Systems (16321 was condemned by the hiquisition. Forbidden to 
teach the "new" astronomy, Galileo decided to concentrate again 
on mechanics. This woik led to his book Discourses and 
Mathematical Demonstrations Concening l\vo New Sciences 
Pertaining to Mechanics and Local \4otion (1638), usually referied 
to as Two New Sciences. This book signaled the beginning of the 
end of the medieval theoiy of mechanics and of the entire 
Aristotelian cosmology. 

Galileo was old, sick, and nearly blind at the time he u^rote 
Two New Sciences. Yet, as in all his writings, his style is lively 
and delightful. As he had in the l\vo Great World Systems, he 
presented his ideas in the form of a conversation among three 
speakers: Simplicio competently represents the Aristotelian vdew; 
Salviati presents the new views of Galileo; and Sagredo is a man 
of good will and open mind, eagei' to learn. Eventually, of course, 
Salviati leads his companions to Galileo's views. Listen to 
Galileo's three speakers as they discuss the problem of free fall: 

Salviati: I greatly doubt tliat /Xiistotle ever tested by experiment 
whether it is true that two stones, one weighing ten times as 
muc:h as the other, if allowed to fall at the same instant from a 
height of, say, 100 cubits, would so differ in speed that when 
the heavier had reached the ground, the other would not have 
fallen more than 10 culiits. |A "(-libit" is about 50 cm.] 

Simplicio: His language would indicate tliat he had tried the 
experiment, because he says: We see the heavier; now the woixl 
see shows that he had made the experiment. 

Sagredo: But, I, Simplicio, who hiwe made the test can assure 
you that a cannon ball weighing one or two hundred pounds, 
or even more, will not reach the ground by as much as a span 
I hand-breadth) ahead of a musket ball weighing only half a 
pound, provided both are diopped from a height of 200 cubits. 




Frontispiece of the book Dialogue 
on Two Great World Systems 
(163ZI. 



D I S C O R S I 

DIMOSTRAZIONI 

MATEMATICHE, 
intorno a due nuoue fcicnz^e 

Attcncnti alia 
MeCANICA & i MOVIMENTI LOCALI, 

delSignor 
GALILEO GALILEI LINCEO, 

Filolotoc M.itcniaticoprimariodcISercnilTimo 
Grand Ducadi Tofcana. 

Conx'na Apf endue ielcentro di gnmti i' iUumSoltii ■ 




IN L E I D A, 
AjiprcflTo gli Elfcvirii. .m. d. c. ,\x.vvni. 

Title page of Discourses and Math- 
ematical Demonstrations Concern- 
ing two New Sciences Pertaining 
to Mechanics and Local Motion 

I1G3SI. 



CHAPTER 2 / FREE FALL 



43 



SG 4 



Here, perhaps, one iniglit expeet a detailed repoit on an 
experiment done by Galileo or one of his students. Instead, 
Galileo uses a "thought exptMinient, " that is, an analysis ot what 
would happen in an imaginaiy experiment, to cast doubt on 
Aristotle's theoiv of motion: 



delGaliieo. if 

ytcm Ktafi firthtt it men , U fofn ica Jel Vtrui tjfuluH menic 
f'f'if uti "> reUiictc it mule , aca vien Jtihmu , mii per dire 
quel the fer 4ii»e»liirt felrehhtr rifpondere ifiitgU mlichi , teiii 
meeliojifer^A^fiitinle eeaeUdj U dtmeUnziene d' AriiteleU^mi 
Per (befifttrehhe uider eenire i i^li ejfuali di (fuell" , aef^ende^li 
tmendut. £ qteale elfrim^ie ^rmdemenle dmhiu, the AnSle- 
teU ntitf^ertmenteffe mei tjumlejii vere , (he due fitlre vnt fi» 
peue dtit alire ditti velie Ufcitte nel medefime initenie teder 
Jiim eliezziyV. ^r. di tenle hreciit fuffer lelmenle differenii 
welle ler veleiiu , i)>e tit erriiit dtSi mtf^ier in terre C ellre flirt- 
mtffe Hen htuert meeneefrfe died hreccit^ 

Simp. Si vede fare diUe fie ferele,! h' ei meflri tthiuerle/he- 
rimeuiele,fercheeidiie: le^giameilfiii greue: her tjuel vederji 
ettennt thxnemefiiilt tefperien':^. 

Sa-.'r. hliieS Simp, the n'he/iiie U prene,vi efpcure, rhe in^i 
telle eterlielterie » the prfttenie , du^ente , e Anteftii lihbre^ nun 
eniitiberedt vntelmo foUmenle^errtuoin lerrt detlepetle dun 
mefiheiie , the ne fefi vne mez,ze , venende entt deli' ellez z,e di 
dueente hrettie, 

Salu. Mifint.' elire ejfitrien'!^ien hreue ,e etntludenle dime- 
fireziene fejpeme ehiiremenie preuere nen effer lere.ihevn me- 
tile pill greHef mueui piii veletemente duo eliro men grene , in- 
tendende di mohili dell" iHe(fe meter le i (^ in fumme di ijuelli de i 
quell perle Jriileiele. Pert diiemi S. Simpfe vei emmetlete,(he 
dl liefthedane tcrpo ^rtue tedenlefie vne de nelure determinete 
veUeile \fiehe I iiirefcer^lieli ,edjminuirglieUnonfi pejjefe non 
(en vfirgli vitiUnze, i eppergli ijuelihe impedimenle. 

Simp. Sen fpuidutilaTe, the riilejfo mobile neW isleffomez- 
£# hehhie vne (letutte , e de nelure dttermtmte velettti, le t^ue- 
U ni,n fr fji ptffe eetrefcere ft nen ten nuouo impiio tonferin. 
^Im , e eiminuirglielefilue the eon quelthc imptdimenle the In ri- 
lerdi. 

Silu. ^iiend* dumqutnei heuejfme due mehili, le neiureli 



A page from the original Italian 
edition o/Two New Sciences, 
showing statements that are trans- 
lated in this te;<.t. 



SaKiati: But, even without tuither ex[5eiiriK'iit, it is po.s.siblc to 
prove cleai'ly, by means of a short and concliisi\(^ argument, 
that a heavier body does not move more rapidly than a lighter- 
one provided both bodies ar-e of the same material and in short 
such as those mentioned bv Aristotle. But tell me, Simplicio, 
whether- you admit that each falling bodv ac-cjuires a d(!finite 
speed fixed by naturae, a velocity which cannot be incr-eased or- 
diminished except by the use of violence or- r-esistance? 

Simplicio: There can be no doubt but that one and the same 
body moving in a single medium has a fixed velocity which is 
determined by nature and which cannot be incr-(?ased except by 
the addition of impetus or- diminished except by some 
r-esistance which retai-ds it. 

Salviati: If then we take two bodies whose natirral speeds are 
different, it is clear that on uniting the two, the more rapid one 
will be par-tly r-etarded by the slower, and the slower- will be 
somewhat hastened by the swifter-. Do you not agree with me 
in this opinion? 

Simplicio: You ar-e unquestionably right. 

Salviati: But if this is true, and if a lar-ge stone moves with a 
speed of say, eight, while a smaller- moves with a speed of four-, 
then when they are united, the system will move with a speed 
less than eight; but the two stones when tied together make 
a stone larger than that which before moved with a speed of 
eight. Hence the heavier- body moves with less s|jeed than the 
lighter one; an effect which is contrary to your- sirpposition. 
Thus you see how, from your assumption that the heavier' body 
moves more rapidly than the lighter- one, I infer- that the heavier- 
bodv moves more slowly. 



SG 



Simplicio: I am all at sea. 
compr-ehension. . . . 



. This is, indeed, quite beyond my 



Simplicio r'eti^eats in confusion when Salviati shows that the 
Aristotelian theory of fall contradicts itself. But while Simplicio 
cannot refute Galileo's logic, his own eyes tell him that a heavy 
object does fall faster than a light object: 

Simplicio: V'our- discirssion is reallv admirable; vet I do not find 
it easy to believe that a birdshot falls as swiftiv as a cannon ball. 

Salviati: Why not say a grain of sand as rapidiv as a giindslone? 
But, Simplicio, I trnjst you will not follow the example of many 
other's who divert the discussion fr-om its main intent and 
hasten upon some statement of mine that lacks a haiisbi-cadth 



44 



L'MT 1 / co\(:i:pts cji mo nov 



of the truth, and under this hair hide the fault of another that 
is as big as a ship's cable. Aristotle says that "an iron biiU of one 
hundred pounds falling from a height of 100 cubits reaches the 
ground before a one-pound ball has fallen a single cubit." I say 
that they arrive at the same time. You tind, on making the 
experiment, that the larger outstrips the smaller by two 
fingerbreadths. . . . Now you would not hide behind these two 
fingers the 99 cubits of Aristotle, nor would you mention my 
small error and at the same time pass over in silence his veiy 
large one. 

This is a clear statement of an important principle: Even in 
careful observation of a common natui^al event, a very minor 
effect may distract the obseiver's attention. As a i^esult, a much 
more important regularity niay be overlooked. Different bodies 
falling in air from the same height, it is true, may not reach the 
ground at exactly the same time. However, the important point is 
not that the times of arrival are slightly different, but that they 
are very nearly the samel Galileo regarded the failure of the 
bodies to arrive at exactly the same time as a minor effect which 
could be explained with a better understanding of motion in free 
fall. He himself correctly attributed the observed results to 
differences in the effect of air resistance on bodies of different 
size and weight. A few years after his death, the invention of the 
vacuum pump allowed others to show that Galileo was right. \n 
one experiment, for example, a feather and a heavy gold coin 
were di^opped from the same height at the same time inside a 
container pumped almost empty of air. With the effect of air 
resistance eliminated, the different bodies fell at the same rate 
and struck the bottom of the container at the same instant. Long 
after Galileo, scientists learned how to express the laws of air 
resistance in mathematical form. With this knowledge, one can 
understand exactly why and by how much a light object vvdll fall 
more slowly than a heavier one. 

Learning what to ignore has been almost as important in the 
growth of science as learning what to take into account. In the 
case of falling bodies, Galileo's explanation depended on his 
being able to imagine how an object would fall if there were no 
air resistance. His explanation seems simple today, when we 
know about vacuum pumps. But in Galileo's time it was difficult 
to accept. For most people, as for Aristotle, common sense said 
that air resistance is always present in nature. Thus, a feather 
and a coin could never fall at the same rate. Why talk about 
motions in a vacuum, when a vacuum could not be shovvai to 
exist? Physics, said Aristotle and his followers, should deal with 
the world all around us that we can readily obseive. It should 
not bother with imaginaiy situations which might never be seen 
or which, like the vacuum, were considered impossible. 

Aristotle's physics had dominated Europe since the thirteenth 
centuiT. To manv scientists, it seemed to offer the most 




A stroboscopic photograph of two 
freely falling balls of unequal 
weight. The picture shows the last 
part of the total path. The balls had 
been released simultaneously. The 
time interval between images is 
0.03 sec. 



The phrase "ft^ee fall' as now used 
in physics generally refers to fall 
when the only force acting is grav- 
ity, that is, when air friction is neg- 
ligible. 



One of the arguments against the 
existence of a vacuum was deduced 
from Aristotle's theory as follows: 
If the final speed of faU is propor- 
tional to the weight divided by the 
resistance, then, since the resis- 
tance in an assumed vacuum would 
be zero, the final speed of fall of all 
bodies must be infinite in a vac- 
uum. But such a result was re- 
garded as absurd, so the assump- 
tion of a vacuum was believed to 
have Ijcen showai to be impossible. 



CHAPTER 2 / FREE FALL 



45 






' ^^^\ 



%' 



jH^ 



reasonable method for describing natni ai phenomena. 1 o 
overthrow such a finnly established doctrine required much 
more than writing reasonable arguments. It e\en letiuircxi more 
than clear experimental pioot, such as dicjpping hea\A and light 
objects fiom a tall building. (Galileo is often said to have done 
this from th(; top of the Leaning Tower of Pisa, but probably did 
not.) It demanded Galileos unusual coml)ination of mathematical 
talent, experimental skill, literaiy stv'le, and tireless campaigning 
to disciedit Aristotle's theoiies and thus to begin the era of 
modern physics. 



Portrait of Galileo by Ottavio Leoni, 
a contemporary of Galileo. 



4. If a nail and a toothpick are dropped at the same time from 
the same height, they do not reach the ground at eyactly the 
same instant. (Try it with these or similar objects.) How would 
Aristotelian theory csplain this? What was Galileo's 
esplanation/ 

5. A paper bag containing a rock is dropped from a window. 
Using Aristotle s theory, eyplain whv the bag aiui the rock 
together fall slower than the rock would fall by itself, i'se the 
same theory to e.xplain why the two together fall faster than 
the rock alone. 



"Aristotelian cosmology" refers to 
the whole interlocking set of ideas 
about tbe structure of tlie pliysical 
iini\erse and th(! Ijebaxior of all tbe 
objects in it. This was t)rie(ly men- 
tioned in Sec. 2.1. Other aspects of 
it will be presented in I nit 2. 



In fact, more than mere "superti- 
cial obsenations" iiad been made 
long before (ialileo set to work. For 
example, \icoIas Oresme and oth- 
ers at the Uni\'ersity of Paris had l)y 
1330 discovered a distance-time re- 
lationship similar to that which 
(ialileo was to announce for falling 
bodies in two \cu Sciences. Some 
of their reasoning is discussed in 
SCi 7. It is, howe\'er, questional)ie 
how much of this prior work influ- 
enced (ialileo in d(!tail, rather tlian 
just in spirit. 



2.4 Uliv studi' the motion of freelv 
falling bodies? 

In Galileo's attack on the Aristotelian cosmologx , lew details were 
actually new. Howexer, Galileo's approach and his findings 
together provided the first woi'kable presentation of the science 
of motion. Galileo realized that understanding free-fall motion 
is the key to understanding all obsenable motions of all bodies 
in nature. To know which was the ke\' phenomenon to study was 
a gift of genius. But in many ways Galileo simply worked as do 
scientists in general. His approach to the problem of motion 
makes a good "case" to follow as an introduc^tion to stiategies of 
incjuiiy that are still used in science. 

Sexeral reasons for studying in detail Galileo's attack on the 
problem of free fall ha\'e been mentioned. Galileo himself 
recognized another reason: The study of motion which he 
proposed was only the starting phase of a mightv field of 
discoxeiy. 

My purpose is to set forth a very new science dealing with a 
\eiA' ancient subject. Ther'e is, in nature, per'haps nothing okler 
than motion, concerning vvhit'b tiie hooks written h\' 
pliilosophers are neither- few nor' small ; nevertheless, I have 
ciiscovered some properti(\s of it that arc worth know ing that 
have not hitherto bv,o,w either- oiisiM-vcd oi' denionstiated. Some 



46 



UNIT 1 / CONCEPTS OF MOTION 



supeiiicial observations have been made, as tor instance, that 
the natural motion of a heavo' falling bodv is continuously 
accelerated; but to just what extent this acceleration occurs has 
not yet been announced. . . . 

Other facts, not few in number or less worth knowing 1 have 
succeeded in proving; and, what I consider more important, 
there have been opened up to this vast and most excellent 
science, of which my work is merely the beginning, ways and 
means by which other minds more acute than mine vvdll 
explore its remote corners. 



2»S I Galileo chooses a definition of uniform 
acceleration 



Two New Sciences deals directly with the motion of freely falling 
bodies. In studying the following paragraphs from this work, you 
must be alert to Galileo s overall plan. First he discusses the 
mathematics of a possible, simple type of motion. (This motion is 
now called uniform acceleration or constant acceleration.) Then 
he proposes that heaw bodies actually fall v\dth just this kind of 
motion. Next, on the basis of this pioposal, he makes certain 
predictions about balls rolling down an incline. Finally, he shows 
that experiments bear out these predictions. 

The first part of Galileos presentation is a thorough discussion 
of motion with uniform speed, similar to the discussion in 
Chapter 1. This leads to the second part, where Salviati, one of 
Galileo's characters, says: 

When, therefore, I obseive a stone initially at rest falling fi^om 
an elevated position and continually acquiring new increments 
of speed, why should I not believe that such increases take 
place in a manner which is exceedingly simple and rather 
obvious to eveiybody? If now we examine the matter caiefully 
we find no addition or increment more simple than that which 
repeats itself always in the same manner. This we readily 
understand when we consider the intimate relationship 
between time and motion; for just as uniformity of motion is 
defined by and conceived through equal times and equal 
spaces (thus we call a motion uniform when equal distances 
are traversed during equal time-intervals), so also we may, in a 
similar manner, through equal time-intervals, conceive 
additions of speed as taking place without (-omplication. . . . 

Hence the definition of motion which we are about to 
discuss may be stated as follows: 

A motion is said to be unifonnaly accelerated when, starting 
from rest, it acquires during equal tinie-inten'als, equal 
increments of speed. 

Sagredo then questions whether Galileo's arbitrary definition of 
acceleration actually corresponds to the way objects fall. Is 



It will help you to have a plan 
clearly in mind as you progress 
through the I'est of this chapter. As 
you study each succeeding section, 
ask yourself whether Galileo is 
presenting a definition 
stating an assumption (or 
hyi3othesis) 

deducing predictions from his 
hypothesis 

experimentally testing the pre- 
dictions 



This is sometimes known as the 
rule of j3ai\simony: Unless forced to 
do otherwise, assume the simplest 
possible hv]30thesis to explain nat- 
ural events. 



We can rephi'ase Galileo's defini- 
tions, using the symbols from 
Chapter 1, as follows: for uniform 
motion, the r^atio i\d/At is constant; 
for unifoi'mly accelerated motion, 
the ratio Av/Af is constant. These 
definitions are equivalent to those 
given in Sees. 1.3 and 1.8. Other 
ways of describing uniform accel- 
eration are discussed in SG 8 and 
9. 



CHAPTER 2 / FREE FALL 



47 



Here Sahiati refers to the Aristote- 
lian assumption that air propels an 
object moving through it I see Sec. 
2.1). 



acceleration, as defined, really useful in desciibing their obseiAed 
change ot motion? Sagredo wonders about a turther point, so tar 
not laised by Galileo: 

From these considerations perhaps we can oi)tain an answer to 
a question that has been argued b\' philospheis, namely, whal 
is the cmise of the natural motion of hea\y bodies. . . . 

But Salviati, the spokesman of Galileo, rejects the ancient 
tendency to investigate phenomena by looking first for their 
causes. As we would today, Sahiati says it is pointless to ask 
about the cause of any motion until an accurate description of it 
exists: 

Sal\dati: The present does not seem [o be tlie proper lime to 
investigate the cause of the acceleration of natural motion 
concerning which various opinions have been expressed by 
philosophers, some explaining it by attraction to the center, 
others b\' repulsion b(M\\een the \'eiy small parts of the body, 
while still others attribute it to a certain stress in the 
surrounding medium which closes in behind the falling body 
and drives it from one of its positions to another. Now, all these 
fantasies, and others, too, ought to be examined; but it is not 
reiilly worth while. At present it is the puipose of our Author 
merely to investigate and to demonstrate some of the properties 
of accelerated motion, whatever the cause of this acceleration 
may be. 

Galileo has now introduced two distinct piopositions: 111 
"uniform" acceleration means that equal increases of speed Av 
occur in equal time intervals At; and (2) things actually fcdl that 
way. 

First look more closely at Galileo's proposed definition. Is this 
the only possible way of defining uniform acceleration? Not at 
all! Galileo says that at one time he thought it would be more 
useful to define uniform acceleration in terms of speed increase 
in proportion to distance traveled Ad, rather than to time Af. 
Notice that both definitions met (ialileo's re(|uirement of 
simplicity. (In fact, both delinilions had been discussed since 
eaily in the fourteenth centuiy.) Furtheimoi^, both definitions 
seem to match the common sense idea of acceleiation. To say 
that a body is "accelerating" seems to imply "the farther it goes, 
the faster it goes " as well as "the longer time it goes, the faster 
it goes." How should we choose between these two ways of 
putting it? Which definition wall be more useftil in describing 
nature? 

This is where expeiimentation becomes important. Galileo 
chose to define unifoi'm ac(;eleration as the motion in \\ hich the 
change in speed Av is proportional to elapsed time It. He then 
demonstrated that his definition matches the real behavior of 
mcning bodi(;s, in laboiatoix' situations as well as in ordinaix 



48 



L\i I 1 co\c:fc;prs oi moiiov 



"un-arranged" experience. As you will see later, he made the 
right choice. But he was not able to prove his case by direct or 
obvious means, as you will also see. 



• 6. Describe uniform speed without referring to dry-ice disks 
and strobe photography or to any particular object or 
technique of measurement. 

7. Express Galileo's definition of uniformly accelerated motion 
in words and in the form of an equation. 

8. Use Galileo's definition of acceleration to calculate the 
acceleration of a car that speeds up from 22 m/sec to 32 m/ 
sec in 5 sec. At this same acceleration, how much more time 
would the car take to reach 40 m/sec? 

9. What two conditions did Galileo want his definition of 
uniform acceleration to meet? 

2*G I Galileo cannot test his hypothesis directly 

After Galileo defined uniform acceleration in terms that matched 
the way he believed freely falling objects behaved, his next task 
was to show that his definition actually was useful for describing 
observed motions. 

Suppose you drop a heavy object from several different 
heights, for example, from udndows on different floors of a 

building. You want to check whether the final speed increases in 

proportion to the time the object falls; that is, you want to know The symbol « means "directly pro- 

whether Av ^ At or, in other words, whether Av/Af is constant. In portional to/' or "changes with. " 

each trial you must observe the time of fall and the speed just 

before the object strikes the ground. This presents a problem. 

Even with modern instruments, it would be very difficult to make 

a direct measurement of the speed reached by an object just 

before striking the ground. Furthemiore, the entire time intervals 

of fall (less than 3 sec from the top of a 10-story building) is 

shorter than Galileo could have measured accurately udth the 

clocks available to him. So a direct test of whether Av/Af is 

constant was not possible for Galileo. 



• 10. Which of these statements accurately explains why Galileo 
could not test directly whether or not the final speed reached 
by a freely falling object is proportional to the time of fall? 

(a) His definition was wrong. 

(b) He could not measure the speed attained by an object just 



before it hit the ground. SG 10 

CHAPTER 2 I FREE FALL 49 



(c) He could not incusurc limes iiccurutcly ciiouu^h. 

(d) He could not measure distances accurately enough. 

(e) Esperinientation was not f)rin\itU'd in Italy. 



More generally, the average speed 
would be 

V + V 



Galileo used a geometrical argu- 
ment. /Vlgebra was not used until 
more tlian 100 vears later. 



2*7 I Looking for logical consequences of 
Galileo's hipothesis 

Galileo's inability to make direct measurements to test his 
hypothesis that Av/Af is constant in free fall did not stop him. He 
turned to mathematics to derive from this hspothesis some other 
i^lationship that could be checked by measurements with 
equipment available to him. You will see that in a few steps he 
came much closer to a relationship he c:ould use to check his 
hypothesis. 

Large distances and large time intervals are, of coui-se, easier to 
measure than the veiy small \ alues of Ac/ and It needed to find 
the final speed just before a falling body hits. So Galileo tried to 
determine, by reasoning, how total distance of fall would 
increase with total time of fall if objects did fall with unifomi 
acceleration. You already know how to find the total distance 
from total time for motion at constant speed. Now you can derive 
a new equation that relates total distance of fall to total time of 
fall for motion at constant acceleration. In doing so, you will not 
follow Galileo's own calculations exactly, but the results will be 
the same. First, recall the definition of average speed as the 
distance traveled Ad dixided by the elapsed time Af: 



V,. = 



Ad 

^t 



From this general definition you can compute the average speed 
from measurement of Ad and A/, whethei- Ad and A/ are small 
or large. You can rewrite the equation as 



Ad = 



X At 



This equation, really a definition of v_^^, is always tnje. For the 
special case of motion at a constant speed v, v^^ — v, and 
therefore, Ad = v x At. Suppose the \alue of v is known, for 
example, w hen a car is driven with a steady reading of 96 km/hr 
on the speedometer. Then you can use this equation to figure 
out how far (Ad) the car would go in any given time interval {At}. 
But in the case of unifomily accelerated motion the speed is 
continually changing, rheiefore, what xalue can you use for v^^? 
The answer involves just a bit of algebra and some reasonable 
assumptions. Galileo reasoned (as others had before him) that 
for any quantit\' that changes unifonnly, the average value is just 
half\vay between the beginning value and the final value. For 
uniformly accelerated motion starling ftom r-est, v.,,-,.,, = 0. Thus, 



50 



I Ml 1 CONCLPIS ()l MOnON 



the rule tells you that the average speed is halfway between 
and Vp^^,; that is, v^^ = ^/^v^,,^,. If this reasoning is correct, it follows 
that 



1 

Ad = -v« , X Af 
2 



final 



for uniformly accelerated motion starting from rest. 

This relation could not be tested directly either, because the 

equation still contains a speed factor. What is needed is an SG 11, 12 

equation relating total distance and total time, without any need 
to measure speed. 

Now look at Galileo's definition of uniform acceleration: a — 
Av/Af. You can rewrite this relationship in the form Av = a x Af. 
The value of Av is just v^^^^, -v.^^^^.^^■, and v^,,,^^, = for motion that 
begins from rest. Therefore, you can write 

Av = a X Af 

V'Hnal - ^.nmal = 3 X Af 

v«„ai -^ a X Af 

Now you can substitute this expression for v^^^, into the 
equation for Ad above. Thus, //the motion starts from rest and if 
it is uniformly accelerated (and //the average rule is correct; as 
you have assumed), you can write 



Ad 


1 

~ 2 


'''^Hnal 


X Af 






1 
~ 2 


(a X 


Af) X 


Af 




Ad 


1 

~ 2 


a(Af)' 





Or, regrouping terms, 



This is the kind of relation Galileo was seeking. It relates total 
distance Ad to total time Af, without involving any speed term. 
Before finishing, though, you can simplify the symbols in the 
equation to make it easier to use. If you measure distance and 
time from the position and the instant that the motion starts, 
then dj,^j,j^ = and f^^^,;^ = 0. Thus, the intervals Ad and Af have 
the values given by d^^^, and ffj,,^,. You then can write the equation 
above more simply as 

1 2 

t^final = - affinal SG 13 

Remember that this is a very specialized equation. It gives the 
total distance fallen as a function of total time of fall, but it does 
so only if the motion starts from rest (v,,^;,^^, = 0) , if the 
acceleration is uniform (a = constant], and if time and distance 
are measured from the start (f , , =0 and d.^„. = 0). 

CHAPTER 2 / FREE FALL 51 



Galileo reached the same conclusion, though he did not use 
algebraic forms to express it. Since you are dealing only with the 
special situation in which acceleration a is constant, the quantity 
Vza is constant also. Therefore, you can write the conclusion in 
the form of a proportion: In uniform acceleration from rest, the 
distance traveled is proportional to the square of the time 
elapsed, or 



^ t: 



For example; if a uniformly accelerating car starting fi^om rest 
moves 10 m in the first second, in twice the time it would move 
four times as far, or 40 m during the first two seconds. In the 
first three seconds it would move nine times as far, or 90 m. 

Another way to express this relation is to say that the ratio 
^finai to f^,,,, has a constant value, that is. 



Because we will use the expression 
'^nna/^inai rnaHV times, it is simpler to 
write it as d/t\ It is understood that 
d and t mean total distance and 
time inter\'al of motion, starting 
from rest. 



Physics texts must be read with 
pencil in hand. Go oxer each step 
in this section, starting with the 
definition of average speed. Make 
a list of each simplifying assump- 
tion and each new definition used 
in the text. 



= constant 



This equation is a logical result of Galileo's original proposal for 
defining uniform acceleration. This result can be expressed as 
follows: If an object accelerates uniformly from rest the ratio d/t^ 
should be constant. Conversely, any motion for which this ratio 
of d and r is constant for different distances and their 
corresponding times is a case of uniform acceleration as defined 
by Galileo. 

Of course, you still must test the hypothesis that freely falling 
bodies actually do exhibit just such motion. You know that you 
cannot test directly whether Av/Af has a constant value. But a 
constant value of Av/At means there will be a constant ratio of 



to f^ ,. The values for total time and distance of fall d^ , 

final final 



are 



easier to measure than are the values for short intervals Ad and 
At needed to find Av. However, even measuring the total time of 
fall presented a difficult task in Galileo's time. So, instead of a 
direct test of his hypothesis, Galileo went one step further and 
deduced a clever indirect test. 



52 



% 11. Why was it more reasonable for Galileo to use the 

1 ■. 
equation d = - aV for testing his hypothesis thai} to use a = 

Av/At.^ 

12. If you simply combined the two equations A = \'At and Av 
= aAt, it looks as if you might get the results Ad = aAt~. What 
is wrong with doing this':* 

I \i I 1 (:()\(;i;pis oi vioi lov 



2 ,3 I Galileo turns to an indirect test 

Realizing that direct measurements involving a rapidly and freely 
falling body would not be accurate, Galileo decided to test an 
object that w^as moving less rapidly. He proposed a new 
hypothesis: If a freely falling body has constant acceleration, then 
a perfectly round ball rolling down a perfectly smooth inclined 
plane will also have a constant, though smaller, acceleration. 
Thus, Galileo claimed that if d/t^ is constant for a body falling 
freely from rest, this ratio will also be constant, although smaller, 
for a ball rolling from rest down a straight inclined plane. 

Here is how Salviati described Galileo's own experimental test 
in Two New Sciences: 



Note the careful description of the 
experimental apparatus. Today an 
experimenter would add to his or 
her verbal description any detailed 
drawings, schematic layouts, or 
photographs needed to make it 
possible for other competent sci- 
entists to duplicate the experiment. 
Experiment 1.5 in the Handbook 
is very similar to Galileo's test. 



A piece of wooden moulding or scantling, about 12 cubits long, 
half a cubit wide, and three finger-breadths thick, was taken; 
on its edge was cut a channel a little more than one finger in 
breadth; having made this groove very straight, smooth, and 
polished, and having lined it with parchment, also as smooth 
and polished as possible, we rolled along it a bard, smooth, and 
very round bronze ball. Having placed this board in a sloping 
position, by lifting one end some one or two cubits above the 
other, we rolled the ball, as I was just saying, along the 
channel, noting, in a manner presently to be described, the 
time required to make the descent. We repeated this 
experiment more than once in order to measure the time with 
an accuracy such that the deviation between two observations 
never exceeded one-tenth of a pulse beat. Having performed 
this operation and having assured ourselves of its reliability, we 
now rolled the ball only one-quarter of the length of the 
channel; and having measured the time of its descent, we 
found it precisely one-half of the former. Next we tried other 
distances, comparing the time for the whole length with that 
for the half, or with that for two-thirds, or three-fourths, or 
indeed for any ft'action; in sucb experiments, repeated a full 



This picture, painted in 1841 by G. 
Bezzuoli, attempts to reconstruct 
an experiment Galileo is alleged 
to have made during his time as 
lecturer at Pisa. Off to the left and 
right are men of ill will: the blase 
Prince Giovanni de Medici (Galileo 
had shown a dredging-machine in- 
vented by the prince to be unus- 
able) and Galileo's scientific oppo- 
nents. These were leading men of 
the universities; thev are shown 
here bending over a book of Aris- 
totle, in which it is written in black 
and white that bodies of unequal 
weight fall with different speeds. 
Galileo, the tallest figure left of 
center in the picture, is sur- 
rounded by a group of students 
and followers. 




CHAPTER 2 / FREE FALL 



53 





< 



X 




Spheres rolling down planes of in- 
creasingly steep inclination. For 
each angle, the acceleration has its 
own constant value. At 90° the in- 
clined plane situation looks almost 
like free fall, except that the ball 
would still be rolling. Galileo as- 
sumed that the difference between 
free fall and "rolling fall" is not 
important. (In most real situations, 
the ball would slide, not roll, down 
the really steep inclines.) 



hundred times, we always lound that the spaces traversed were 
to each other as the squares of the times, and this was true for 
all inclinations of the . . . channel along which we rolled the 
ball. . . . 

Galileo has packed a great deal of information into these lines. 
He describes his pro(;ediires and apparatus clearly enough to 
allow others to repeat the experiment foi" themselves if they wish. 
He indicates that consistent measurements can be made. Finally, 
he restates the two chief experimental results which he believes 
support his free-fall hypothesis. Examine the results carefully. 

(a) First, Galileo found that when a ball rolled down the 
incline, the ratio of the distance covered to the square of the 
cori-esponding time was always the same. Foi' example, if d^, d^, 
and d, represent distances measured from the same starting 
point on the incline and f,, t.^, and r, represent the times taken to 
roll down these distances, then 






In general, for each angle of incline, the value of dJt^ was 
constant. Galileo did not present his experimental data in the full 
detail which since has become the custom. However, others have 
repeated his experiment and have obtained results which 
parallel his. (See data in SG 16.) You can perform this experiment 
yourself with the help of one or two other students. 

(b) Galileo's second experimental finding relates to what 
happened when the angle of inclination of the plane was 
changed. Whenever the angle changed, the ratio d/t^ took on a 
new value, although for any one angle it remained constant 
regardless of distance of roll. Galileo confirmed this by repeating 
the experiment "a full hundred times " for each of the many 
different angles. After finding that the ratio d/r was constant for 
each angle at which t could be measured conveniently, Galileo 
was willing to extrapolate. He concluded that the ratio d/f is a 
constant even for steeper angles, wheie the ball moves too fast 
for accurate measurement of t. Now, finally, Galileo was ready to 
solve the problem that had started the whole argument: He 
reasoned that when the angle of inclination became 90°, the ball 
would move straight down as Si freely falling object. By his 
reasoning, d/t^ would still be constant even in that extreme case, 
although he couldn't say what the numerical value was. 

Galileo already had deduced that a constant value of d/v was 
characteristic of uniform acceleration. By extrapolation, he could 
conclude at last that free fall was uniformlv accelerated motion. 



• 13. U\ testing his hypothesis that free-fall niotion is unifonnly 
accelerated, Galileo made the unproved assumption that 



I MI 1 CONCLPIS Ol M()ll()\ 



(choose one or more): 

(a) d/t" is constant. 

(h) the acceleration has the same value for all angles of 

inclination of the plane. 

(c) the results for small angles of inclination can be 
extrapolated to large angles. 

(d) the speed of the ball is constant as it rolls. 

(e) the acceleration of the rolling ball is constant if the 
acceleration in free fall is constant, though the value of the two 
constants is not the same. 

14. Which of the following statements best summarizes the 
work of Galileo on free fall when air friction is not a factor? 
(Be prepared to defend your choice.) Galileo: 

(a) proved that all objects fall at exactly the same speed 
regardless of their weight. 

(b) proved that for any freely falling object the ratio d/f is 
constant for any distance of fall. 

(c) proved that an object rolling down a smooth incline 
accelerates in the same way, although more slowly than, the 
same object falling jreely. 

(d) supported indirectly his assertion that the speed of an 
object falling freely from rest is proportional to the elapsed 
time. 

(e) made it clear that until a vacuum could be produced, it 
would not be possible to settle the free fall question once and 
for all. 

2*9 I Doubts about Galileo's procedure 

This whole process of reasoning and experiment appears long 
and involved on first reading, and you may have some doubts 
concerning it. For example, was Galileo's measurement of time 
precise enough to establish the constancy of d/t^ even for a 
slowly rolling object? In his book, Galileo tries to reassure 
possible critics by providing a detailed description of his 
experimental arrangement: 

For the measurement of time, we employed a large vessel of 
water placed in an elevated position; to the bottom of this 
vessel was soldered a pipe of small diameter gi\ang a thin jet of 
water, which we collected in a small cup during the time of 
each descent, whether for the whole length of the channel or 
for a part of its length; the water thus collected was weighed on 
a very accurate balance; the differences and ratios of these 
weights gave us the differences and ratios of the time intervals, 
and this with such accuracy that, although the operation was 
repeated many, many times, there was no appreciable 
discrepancy in the results. 



Galileo's technique for measuring 
time is discussed in the next sec- 
tion. 



For problems that uill check and 
extend your understanding of uni- 
form acceleration, see SG 14 tlirough 
21. 




Earlv water clock. 



CHAPTER 2 / FREE FALL 



55 



The water clock described by Galileo was not invented by him. 
Indeed, water clocks existed in China as early as the sixth 
century B.C. and probably were used in Babylonia and India even 
earlier. In the early sixteenth century a good water* clock was the 
most accurate instrument available for measuring shoii time 
intervals. It remained so until shortly after- (ialileo's death, when 
the work of Christian Huygens and others led to practical 
pendulum clocks. When better clocks became available, Galileo's 
results on inclined-plane motion were confirmed. 

Another' reason for- questioning Galileo's results involved the 
gr eat difference between fi^ee-fall and rolling motion on a slight 
incline. Galileo does not repor-t what angles he used in his 
experiment. However, as you may have fournd out from doing a 
similar- experiment, the angles must be kept ratlier- small. As the 
angle increases, the speed of the ball soon becomes so great that 
it is difficult to measur-e the times involved. I'he largest usable 
angle i-eported in a recent repetition of Galileo's experiment was 
only 6°. It is not likely that Galileo worked with much larger 
angles. This means that his extrapolation to free fall 190° incline) 
was bold. A cautious person, or- one not already convinced of 
Galileo's argument, might well doubt it. 

There is still another- reason for questioning Galileo's results. 
As the angle of incline is increased, there comes a point where 
the ball starts to slide as well as roll. This change in behavior- 
could mean that the motion is very different at large angles. 
Galileo does not discuss these cases. If he had been able to use 
frictionless blocks that slid down the plane instead of rolling, he 
would have found that for sliding motion the ratio d/r is also a 
constant, although having a different numerical value than for 
r-olling at the same angle. 



SG 22 • 15. Which of the following statements could be regarded as 
major reasons for doubting the value of Galileo's procedure? 

(a) His measurement of time was not accurate enough. 

(b) He used too large an angle of inclination in his experiment. 

(c) It is not clear that his results apply when the ball can slide 
as well as roll. 

(d) In Galileo's experiment the ball was rolling, and therefore 
he could not extrapolate to the case of free fall where the ball 
did not roll. 

(e) d/t'' was not constant for a sliding object. 

2.1.0 I Consequences of Galileo's n^ork on 
motion 

Galileo seems to have understood that one cannot get the correct 
numerical valire for- the acceleration ol n bodv in free fall sinipK 



56 UNIT 1 / COiXCEPTS OF MOTION 



by extrapolating through increasingly large angles of incline. He 
did not attempt to calculate a numerical value for the 
acceleration of freely falling bodies. For his purposes it was 
enough that he could support the hypothesis that the 
acceleration is constant for any given body, whether rolling or 
falling. This is the first of Galileo's findings, and it has been fully 
borne out by all following tests. 

Second, spheres of different weights allowed to roll down an 
inclined plane set at a given angle have the same acceleration. 
We do not know how much experimental evidence Galileo 
himself had for this conclusion, but it agrees with his 
observations for fi^eely falling objects. It also agrees with the 
"thought experiment " by which he argued that bodies of 
different weights fall at the same rate (aside fi'om the effects of 
air resistance). His results clearly contradicted what one would 
have expected on the basis of Aristotle's theory of motion. 

Third, Galileo developed a mathematical theory of accelerated 
motion from which other predictions about motion could be 
derived. Just one example is mentioned here: it will be veiy 
useful in Unit 3. Recall that Galileo chose to define acceleration 
as the rate at which the speed changes with time. He then found 
by experiment that falling bodies actually do experience equal 
changes of speed in equal times, and not in equal distances. Still, 
the idea of something changing by equal amounts in equal 
distances has an appealing simplicity. You might ask if there is 
not some quantity that does change in that way during uniform 
acceleration. In fact, there is. If follows without any new 
assumptions that, during uniform acceleration from rest, the 
square of the speed changes by equal amounts in equal 
distances. There is a mathematical equation that expresses this 
result: If v.^.^.^ — and a = constant, then 



Scientists now know by measure- 
ment that the magnitude of the ac- 
celeration of gravity, symbol a,, is 
about 9.8 m/sec per sec at the 
earth's surface. The Project Physics 
Handbook contains five different 
experiments for finding a value of 
a^. (For many problems, the ap- 
proximate value 10 m/sec/sec is sat- 
isfactoi'v.) 



SG 23 



v!. 



2ad« 



In other words: If an object moves from rest with uniform 
acceleration, the square of its speed at any point is equal to 
twice the product of its acceleration and the distance it has 
moved. (You vvdll see the importance of this relation in Unit 3.) 

These results of Galileo's work were most important to the 
development of physics. But they could scarcely have brought 
about a revolution in science by themselves. No sensible scholar 
in the seventeenth centuiy would have given up a belief in 
Aristotelian cosmology only because some of its predictions had 
been disproved. Still, Galileos work on free-fall motion helped 
to prepare the way for a new kind of physics, and indeed a new 
cosmology, by planting the seeds of doubt about the basic 
assumptions of Aristotelian science. For example, when it was 
recognized that all bodies fall with equal acceleration if air 



You can derive this equation. (See 
SG 24.) 



SG 25, 26 



CHAPTER 2 / FREE FALL 



57 



friction is minor, the whole Aristotelian explanation of falling 
motion iSec. 2.1) broke tlovvn. 

The most disputed scientific problem during Galileo's lifetime 
was not in mechanics but in astronomy. A central question in 
cosmology was wheth(;i- the earth or the sun was the cent(;r of 
the universe. Galileo supported the view that the earth and other 
planets revolved around the sun, a \iew entirely contrary to 
Aristotelian cosmolog\'. But to support such a \ iew recjuir ed a 
physical theory of why and how the earth itself moved. Galileo's 
work on fi'ee fall and other' motions turned out to be just what 
was needed to begin constructing such a theory. His work did 
not have its full effect, however, until it had been combined with 
the studies of for^ces and motion by the English scientist Isaac 
Newton. But as Newlon acknowledged, (Jalileo was the pioneer. 
(In the next chapter, you will consider Newton's work on force 
and motion. In Chapter- 8, you will see its application to the 
motions in the heavens as well as the revolution it caused in 
science.) 

Galileo's work on motion introduced a new and important 
method of doing scientilic research, ibis method is as effective 
today as when Galileo demonstrated it. The basis of this 
procedur^e is a cycle, repeated as often as necessary, entirely oi' 
in pari, until a satisfactory theory has emerged. I'he cycle 
roughly follows this foriii: general obser^^ation — > hvpothesis —> 
mathematical analysis or- deduction from hvpothesis — » 
experimental test of deduction — * re\asion of hypothesis in light 
of test, and so forth. 

While the mathematical steps are detemiined mainly by 'cold 

logic, " this is not so for the other parts of the process. A variety' 

SG 27 of paths of thought can lead to a hypothesis in the fir^t place. 

A new hypothesis might come from an inspired hunch based on 
general knowledge of the experimental facts. Or- it might come 
from a desire for mathematically simple statements, or from 
modi^'ing a previous hvpothesis that failed. Moreoxer, there ar-e 
no general rules about exactly how well experimental data must 
agree with predictions based on theory. In some areas of science, 
a theory is expected to be accurate to better- than 0.001%. In 
other- ar eas, or at an early stage of any new work, one might be 
delighted with an erixjr of only 50% . Finally, note that while 
exper-iment has an impor-tant place in this process, it is not the 
only or even the main element. On the contrary, expeiiments are 
worthwhile only in combination with the other steps in the 
process. 

The general cycle of observation, hvpothesis, deduction, test, 
re\asion, etc., so skillfully demonstrated by Galileo in the 
seventeenth centuiy, commonly appears in the work of scientists 
today. Though ther-e is no such thing as the scientific method, 
some for-m of this cycle is almost always present in scientific 



58 I Ml 1 (X)\(;i;i»rs oi mo i ion 



research. It is used not out of respect for Galileo as a towering 
figure in the histoiy of science, but because it works so well so 
much of the time. What is too frequently underplayed is the 
sheer creativity that enters into each of these phases. There are 
no fixed rules for doing any one of them or for how to move 
from one to the next. 

Galileo himself was aware of the value of both the results and 
the methods of his pioneering work. He concluded his treatment 
of accelerated motion by putting the following words into the 
mouths of the characters in his book: 

Salviati: . . . we may say the door is now opened, for the first 
time, to a new method ft'aught with numerous and wonderful 
results which in future years will command the attention of 
other minds. 

Sagredo: I really believe that . . . the jorinciples which are set 
forth in this little treatise will, when taken up by speculative 
minds, lead to another more remarkable result; and it is to be 
believed that it will be so on account of the nobility of the 
subject, which is superior to any other in nature. 



16. Which one of the following was not a result of Galileo's 
work on motion? 

(a) Determination of the correct numerical value of the 
acceleration in free fall, obtained by extrapolating the results 
for larger and larger angles of inclination. 

(b) If an object starts from rest and moves with uniform 
acceleration a through a distance d, then the square of its 
speed will be proportional to d. 

(c) Bodies rolling on a smooth inclined plane are uniformly 
accelerated. 



Many details of physics, mathemat- 
ics, and history have appeared in 
this chapter. For a review of the 
most important ideas, see SG 28-32. 



CHAPTER 2 / FREE FALL 



59 



study 
guide 



1. Note that at the beginning of each chapter in 
this book there is a list of the section titles. This is a 
sort of road map you can refer to from time to time 
as you study the; chapter. It is important, especially 
in a chapter such as this one, to know how the part 
you are studying relates to w^hat preceded it and to 
have some idea of where it is leading. For this same 
reason , you will find it very helpful at first to skim 
through the entire chapter, reading it rapidly and not 
stopping to puzzle out parts that you do not quickly 
understand. Then you should return to the beginning 
of the chapter and work your way through it 
carefully, section by section. Rememijer also to use 
the end-of-section questions to check your progress. 

The Project Physics learning materials particularly 
appropriate for Chapter 2 include: 

Experiments 

A Seventeenth-Century Experiment 

Twentieth Century Version of Galileo's Experiment 

Measuring the Acceleration of Gravity, a 

Film Loops 

Acceleration Caused by Gravity. Method I 
Acceleration Caused by Gravity. Method II 

2. Aristotle's theoiy of motion seems to be 
supported to a great extent by common sense 
experience. For example, water bubbles up through 
earth at springs. When sufficient fire is added to 
water by heating it , the resulting mixture of elements 
(what we call steam) rises through the air. Can you 
think of other examples? 

3. Compare Aristotle's hypothesis about falling rate 
(weight divided by resistance) with Philoponus' 
(weight minus resistance) for some extreme cases: a 
very heaw body v\ith no resistance, a very light body 
v\ith great resistance. Do the two hypotheses suggest 
very different results? 

4. Consider Aristotle's statement "A given weight 
moves [falls] a given distance in a given time; a 
weight which is as great and more moxes the same 
distances in h;ss time, the times Ijcung in in\'erse 



proportion to the weights. For instance, if one 
weight is twice another, it will take half as long over 
a given movement." (Dp ('ado) 

Indicate what Simplicio and Sal\iati each would 
predict for the falling motion in these cases: 

(a) A 1-kg rock falls from a cliff and, while dropping, 
breaks into two equal pieces. 

(b) A 5-kg rock is dropped at Uie same time as a 4.5- 
kg piece of the same ty|De of rock. 

(c) A hundred 4. 5-kg pieces of rock, falling from a 
height, drop into a draw-string sack which closes, 
pulls loose, and falls. 

5. Tie two objects of greatly different weight (like a 
book and a pencil) together with a piece of string 
(see below). Drop the combination with different 
orientations of objects. Watch the string. In a few 
sentences summarize your results. 

6. (a) A bicyclist starting from rest accelerated 
uniformly at 2m/sec" for 6 sec. Vlliat distance did he 
cover in that time? Calculate the average speed for 
tliat time (B sec) by finding the average of the initial 
speed and fined speed. What distance would the 
bicyclist cover in 6 sec at this a\'erage speed? This 
problem illustrates the Merton theorem (see 
question 7). 

(b) Only in the special case of uniform acceleration 
does a simple arithmetic average of speeds give the 
correct average speed. Elxplain why this is so. 

7. A good deal of work on the topic of motion 
preceded tliat of Galileo. In the period 1280-1340, 



'^- f 



<-.^ 



^ 



l> 




60 



CILXPILK 2 SrLU\ Gl IIJL 



sg 



mathematicians at Merton College, Oxford, carefully 
considered different quantities that change with the 
passage of time. One result that had profound 
influence was a general theorem known as the 
Merton theorem or mean speed rule. 

This theorem might be restated in our language 
and applied to uniform acceleration as follows: The 
distance an object goes during some time while its 
speed is changing uniformly is the same distance it 
would go if it went at the average speed the whole 
time. 

(a) First show that the total distance traveled at a 
constant speed can be expressed as the area under 
the graph line on a speed-time graph. ("Area" must 
be found in speed units x time units.) 

(b) Assume that this area represents the total 
distance even when the speed is not constant. Draw 
a speed-time graph for uniformly increasing speed 
and shade in the area under the graph line. 

(c) Prove the Merton theorem by showing that the 
area is equal to the area under a constant-speed line 
at the average speed. 

8. According to Galileo, uniform acceleration 
means equal Av's in equal Af s. Which of the 
following are other ways of expressing the same 
idea? 

(a) Av is proportional to Af 

(b) Av/Af = constant 

(c) the speed-time graph is a straight line 

(d) V is proportional to t 

9. In his discussion of uniformly accelerated 
motion, Galileo introduced another relationship that 
can also be put to experimental test. GalUeo found 
that "the distances traversed by a body falling from 
rest during successive intervals of equal times will be 
in the ratios of the odd integers, 1:3:5:7. . . ." 

Show that this experimentally testable result is in 
accord with our definitions for v ^ and a for 
uniformly accelerated motion. (Hint: One way is to 
proceed as follows. For equal time intervals {At), the 
final speed reached is successively aAf, 2aAf, .... 
During each of these time intervals, the average 

1 1 

speeds are - (aAt), - (3a Af), . . . , and the 

2 2 



1 1 

corresponding distances covered are - (aAf) • {At), — 

^ 2 

(3aAf) • {At), 

{Note: You can also deduce this result from a 
speed-time graph. Since the distance traversed is 
just speed times time, that is, Ad = vAt, the area of 
any slice of the v versus t graph that has a height 
of V and a width At is just the distance traversed 
during At. Using this idea, show that the distances, 
Ad, traversed during equal Af's obey Galileo's rule, 
quoted above.) 

10. Using whatever modern equipment you wish, 
describe how you could find an accurate value for 
the speed of a falling object just before it strikes the 
ground. 



11. Show that the expression 



+ V,. 



is equivalent to the Merton theorem discussed in 

SG 7. 

12. For any quantity that changes uniformly , the 
average is the sum of the initial and final values 
divided by two. lYy it out for any quantity you may 
choose. For example: What is the average age in a 
group of five people having individually the ages of 
15, 16, 17, 18, and 19 years? What is your average 
earning power over 5 years if it grows steadily from 
$8,000 per year at the start to $12,000 per year at the 
end? 

13. Lt. Col. John L. Stapp achieved a speed of 284 
m/sec in an experimental rocket sled at the 
HoUoman Air Base Development Center, Alamogordo, 
New Mexico. Running on rails and propeUed by nine 
rockets, the sled reached its top speed within 5 sec. 
Stapp survived a maximum acceleration of 22 g's in 
slowing to rest during a time interval of 1.5 sec. (One 
g is an acceleration equal in magnitude to that due 
to gravity; 22 g's means 22 x a .) 

(a) Find the average acceleration in reaching 
maximum speed. 

(b) How far did the sled travel before attaining 
maximum speed? 

(c) Find the average acceleration while stopping. 



CHAPTER 2 / STUDY GUIDE 



61 



sg 



14. Indicate whether the follovxing statements are 
true or false when applied to the strobe photo below 
(you may assume that th(! strobe was flashing at a 
constant rate): 

(a) The speed of the ball is greater at the bottom 
than at the top. 

(b) This could be a freely falling object. (Make 
measurements on photograph.) 

(c) This could be a ball thrown straight upward. 

(d) If (b) is true, the speed increases with time 
because of the acceleration due to gravity. 

(e) If (c) is true, the speed decreases with time 
because of the effect of gravity; this effect could still 
be called acceleration due to gra\aty. 




15. rlic pliotograph above shows a ball falling next 
to a vertical meter stick. The time interxal betwecni 
strolje flashes was 0.035 sec. Use this information 
to make graphs of d versus t and \ \(!rsiis /, and find 
the a(-releration of the ball. (IMotc: The bottom of the 
ball, just on release, was next to the zero point of 
the meter stick. During the first few flashes, the 



images of the falling ball may ha\ e been so 
superposed as to be difficult to icsolxc. Ikit lor the 
purjioses of this problem. \\(! can iicgh^ct the (earliest 
part ol the fall, say. to 10 cm.) 

16. rh(! photograph in the figure; on page; (i;j is ol a 
l)all thrown upward. 1 In; accel(;ration due to gra\ily 
increases the speed of the ball as it desceiuls fioni 
its highest point (like any free-falling object I if air 
friction is negligible. Rut the accehM-ation due to 
gra\ity. which do(!s not change;, acts also while the 
IkiII is still on its way up. and for that portion of the 
path causes the ball to slow down as it ascends to 
the top point, C. 

VVlien there is both up and down motion, it will 
help to adopt a sign convcMition, an arbitral' but 
consistent set of rules, similar to designating the 
height of a place with respect to sea le\el. To identify 
distances measured above the point of initial release, 
give them positive values; for example, the distance 
at B or at L), measured from the release le\(;l. is 
about +60 cm and +37 cm, respectively. If 
measured below the release level, give them negative 
values; for example, E is at -23 cm. /\lso. assign a 
positive value to the speed of an object on its way up 
to the top (about + 3 m/sec at A and a negative value 
to a speed a body has on the way down after 
reaching the top (about — 2 m/sec at D and -6 m/ 
sec at El. 

(a) Fill in the table with + and — signs. 



AT POSITION 



SIGN GIVEN TO 

VALUE OF 

d V 



(b) Show that it follows from this con\ention iind 
from the definition of a = A\ Af that the \alue or 
sign given to the acceleration due to gra\it\' is 
negative, and for both parts of the path. 

(c) What would the sign of acceleration due to 
gra\ity be in each case if we had chosen the + and 
- sign conventions just the other way, that is, 
associating — with up, + with down? 



liZ 



(;ii\i»ri;i{ 2 snm (;i ini; 



8S 




Stroboscopic photoi^raph of a ball 
thrown into the air. 

17. Draw a set of points (as they would appear in a 
strobe photo) to show the successive positions of an 
object that by our convention in SG 16 had a positive 
acceleration, that is, "upward." Can you think of any 
way to produce such an event physically? 

18. Memorizing equations will not save you from 
having to think your way through a problem. You 
must decide if, when, and how to use equations. This 
means analyzing the problem to make certain you 
understand what information is given and what is to 
be found. Test yourself on the following problem. 
Assume that the acceleration due to gravity is equal 
to 10 m/sec/sec. 

Problem: A stone is dropped from rest from the 
top of a high cliff. 

(a) How far has it fallen after 1 sec? 

(b) What is the stone's speed after 1 sec of fall? 



(c) How far does the stone fall during the second 
second (that is, from the end of the first second to 
the end of the second second)? 

19. From the definition for a, show it follows 
directly that v^^^^ = ^,n,<,ai + ^^ ^'^^ motion with 
constant acceleration. Using this relation and the 
sign convention in SG 16, answer the questions 
below. (Assume a = 10 m/sec/sec.) /\n object is 
thrown upward with an initial speed of 20 m/sec. 

(a) What is its speed after 1.0 sec? 

(b) How far did it go in this first second? 

(c) How long did the object take to reach its 
maximum height? 

(d) How high is this maximum height? 

(e) When it descends, what is its final speed as it 
passes the throwing point? 

If you have no trouble with this, you may wish to try 
problems SG 20 and 21. 

20. A batter hits a pop fly that fravels straight 
upward. The ball leaves the bat with an initial speed 
of 40 m/sec. (Assume a =10 m/sec/sec.) 

(a) What is the speed of the ball at the end of 2 sec? 

(b) What is its speed at the end of 6 sec? 

(c) When does the ball reach its highest point? 

(d) How high is this highest point? 

(e) What is the speed of the ball at the end of 10 sec? 
(Graph this series of speeds.) 

(f) What is its speed just before it is caught by the 
catcher? 

ai. A ball starts up an inclined plane with a speed of 
4 m/sec, and comes to a halt after 2 sec. 

(a) What acceleration does the ball experience? 

(b) What is the average speed of the ball during this 
interval? 

(c) What is the ball's speed after 1 sec? 

(d) How far up the slope will the ball travel? 

(e) What will be the speed of the ball 3 sec after 
starting up the slope? 



CHAPTER 2 / STUDY GUIDE 



63 



ss 



(f) what is the total time for a round trip to the top 
and back to the start? 

22. As Director of Research in your class, you 
receive the following research proposals from 
physics students wishing to improve upon Galileo's 
free-fall experiment. Would you recommend support 
for any of them? If you reject a proposal, you should 
make it clear why you do so. 

(a) "Historians believe that Galileo never dropped 
objects from the Leaning Tower of Pisa. But such an 
experiment is more direct and more fun than 
inclined plane experiments, and of course, now that 
accurate stopwatches are available, it can be carried 
out much better than in Galileo's time. The 
experiment involves dropping, one by one, different 
size spheres made of copper, steel, and glass from 
the top of the Leaning Tower and finding how long it 
takes each one to reach the ground. Knowing d (the 
height of the tower I and time of fall t, I will 
substitute in the equation d = Vzat to see if the 
acceleration a has the same value for each sphere." 

(b) "An iron shot will be dropped from the roof of a 
4-story building. As the shot falls, it passes a window 
at each story. At each window there will be a student 
who starts a stopwatch upon hearing a signal that 
the shot has been released, and stops the watch as 
the shot passes the window. Also, each student 
records the speed of the shot as it passes. From 
these data, each student will compute the ratio v/t. I 
expect that all four students will obtain the same 
numerical value of the ratio." 

(c) "Galileo's inclined planes dilute motion all right, 
but the trouble is that there is no reason to suppose 
that a ball rolling down a board is beha\ing like a 
ball fiilling straight dowoiward. A better way to 
accomplish this is to use light, fluffy, cotton balls. 
These will not fall as rapidly as metal spheres, and 
therefore it would be possible to measure the time 
of the fall / for different distances. The ratio d/r 
could be determined for different distances to see if 
it remained constant. The compactness of the cotton 
ball could then be changed to see if a different value 
was obtained for the ratio." 

23. A student on the planet Arret in another solar 
system dropped an object in order to determine the 
acceleration due to gravity at that place. Hk; 



following data are recorded (in local unitsi 



TIME 


DISTANCE 


TIME 


DISTANCE 


(in surgs) 


(in welfs) 


(in surgs) 


(in welfs) 


0.0 


0.0 


2.2 


10.41 


0.5 


0.54 


2.4 


12.39 


1.0 


2.15 


2.6 


14.54 


1.5 


4.84 


2.8 


16.86 


2.0 


8.60 


3.0 


19.33 



(a) What is the acceleration due to gravity on the 
planet Arret, expressed in welfs/surg^? 

(b) A visitor from earth finds that one welf is equal 
to about 6.33 cm and that one surg is equiviilent to 
0.167 sec. What would this tell us about Arret? 

24. (a) Derive the relation v^ = 2ad from the 

1 
equations d =-af and v = at. What special 

conditions must be satisfied for the relation to be 
true? 

(b) Show that if a ball is thrown straight upward 
with an initial speed v, it will rise to a height 



h = 



2a 



25. Sometimes it is helpful to have a special 
equation relating certain variables. For example, for 
constant acceleration a, the final speed v, is related to 
initial speed v and distance traveled d by 

vf = vf + 2ad 

Try to derive this equation from some other 
equations you are familiar with. 

26. Use a graph like the one sketched below and the 
idea that the area under the graph line in a 
speed-time graph gives a value for the distance 
traveled, to derive the equation 

d = vt + Vzaf 



64 



CHAPI ER 2 / sum (aiDL 



sg 



27. List the steps by which Galileo progressed from 
his first definition of uniformly accelerated motion to 
his final confirmation that this definition is useful 

in describing the motion of a freely falling body. 
Identify each step as a hypothesis, deduction, 
observation, computation, etc. What limitations and 
idealizations appear in the argument? 

28. In these first two chapters we have been 
concerned with motion in a straight line. We have 
dealt with distance, time, speed, and acceleration, 
and with the relationships among them. 
Surprisingly, most of the results of our discussion 
can be summarized in the following three equations. 



Ad 



Av 



d = Vzaf 



Because these three equations are so useful, they are 
worth remembering. 

(a) State each of the three equations in words, and 
state explicitly any limitations on when they apply. 

(b) Make up a simple problem to demonstrate the 
use of each equation. (For example: How long will it 
take a jet plane to travel 3,200 km if it averages 1,000 
km/hr?) 

(c) Work out the solutions just to be sure the 
problems can be solved. 

29. What is wrong with the following common 
statements? "The Aristotelians did not observe 
natvire. They took their knowledge out of old books 
which were mostly wrong. Galileo showed it was 
wrong to trust authority in science. He did 
experiments and showed everyone direcdy that the 
old ideas on free fall motion were in error. He 
thereby started science and also gave us the scientific 
method." 

30. (a) What is the acceleration of a car that 
accelerates uniformly from 5 m/sec to 30 m/sec in 10 
sec? 

(b) How tall is a building if it takes an object 9.0 sec 
to hit the ground after falling from the roof? 

(c) A block slides down an inclined plane with a 
constant acceleration of 2 m/sec'. How long will it 
take the block to slide 20 m? How fast will the block 
be moving at the end of that time? 



(d) A particle with a velocity of 8 m/sec north starts 
accelerating. It accelerates uniformly at 5 m/sec"^ 
north for 10 sec. How far does the particle travel in 
10 sec? What is the speed of the particle after 10 sec? 

(e) What is the final speed of a model train that 
accelerates uniformly from rest at 2 m/sec' for a 
distance of 4 m? 

(f) A particle moving with a speed of 6 m/sec enters 
a region 2 m long where it is uniformly accelerated 
at 1 m/sec^. What is the speed of the particle at the 
end of that region? 

(g) What is the acceleration of a motor boat that 
accelerates uniformly from 5 m/sec to 55 m/sec over 
a distance of 100 m? 

(h) A ball is dropped from the roof of a 125-m 
building. At the same time, a second ball is thrown 
straight up to collide with the first ball. What is the 
initial speed of the second ball if the balls collide 45 
m from the ground 4 sec after they were released? 
{Hint: Some of this information is unnecessary.) 

31. (a) Using the methods you learned in Chapter 1, 
calculate the average speed of the object represented 
by the graph shown below in sections AB and CD. 

(b) Using the information from (a), calculate the 
average speed and average acceleration in section BC. 

(c) Discuss your results for (a) and (b). 



30 r 




time (sec) 

32. You have probably noticed that uniform motion 
is represented by a straight line on a distance 1/N 
time graph while accelerated motion is represented 
by a curved line. 



CHAPTER 2 / STUDY GUIDE 



65 



ss 



(a) Describe the motion represented by each of the 
graphs below. 

(b) Identify the direction of motion in each graph. 



t 
(A) 



t 
(B) 



t 



t 



66 



(;n\pn:i{ 2 s n i)\ (;i ini 














The Birth of Bynsmiics 

niemitoii EHplains Motion 



3.1 ''Explanation" and the laivs of motion 

3.2 The Aristotelian explanation of motion 

3.3 Forces in equilibrium 

3.4 About vectors 

3.5 Neuton's first law of motion 

3.6 The si^ificance of the first law 

3.7 NcTiton's second law of motion 

3.8 Mass, Yvei^t, and free fall 

3.9 Nemton's third law of motion 

3.10 Usin^ Newton's laivs of motion 

3.11 Nature's basic forces 



(gt^^PTl^ 





3.1 I "Explanation" and the laws of motion 

Kinematics is the study of how objects move, but not of why they 
move. Galileo investigated many topics in kinematics with 
insight, originality, and energy. The most valuable part of that 
work dealt uath special types of motion, such as free fall. In a 
clear and consistent way, he showed how to describe the motion 
of objects with the aid of mathematical ideas. 

Galileo had written that "the present does not seem to be the 
proper time to investigate the cause of the acceleration of natural 

motion " When Isaac New4on began his studies of motion in 

the second half of the seventeenth century, that statement was 
no longer appropriate. Indeed, because Galileo had been so 



SG 1 



CHAPTER 3 / THE BIRTH OF DYMAMICS 



67 



Soiiu! kiiinmatics roncrpts: posi- 
tion, time, speed, acceleration. 
Some dynamics concepts: mass, 
force, momentum (Ch.9l, energy 
(Ch. 10). 



In (Chapter 4 we will take up motion 
along cuncd paths. 



\'ewtons First Law: Every object 
continues in its state of rest or of 
uniff)rm motion in a straight line 
unless acted upon by an luihal- 
anced force. 

Newton s Second Law: I'he accel- 
eration of an object is directly pro- 
potional to, and in the same direc- 
tion as, the unbalanc(;d force acting 
on it, and in\ersely proportional to 
the mass of the object. 

\'ewton's Third Law: Vo e\erv ac- 
tion there is always opposed an 
equal reaction; or, mutuiU actions 
of two bodies upon each other are 
always equal and in opposite direc- 
tions. 



effective in describing motion, \e\\1on could turn his attention 
to dynamics. Dynamics is the study of why an object moves the 
way it does, for example, why it starts to mo\'e instead of 
remaining at rest, why it speeds up or moves on a cuned path, 
and why it comes to a stop. 

How does dynamics differ from kinematics? As mentioned 
above, kinematics treats the desciiption of motion, while 
dynamics treats the causes of motion, tracing these causes back 
to the play of forces. Each, of course, depends on the other in 
order to describe a motion in a way that will make its 
explanation as simple as possible. Conversely, given an idea for 
an explanation, that idea can be used to suggest better methods 
of description. 

The study of kinematics in Chapters 1 and 2 revealed that an 
object may: (a) remain at rest, (b) move uniformly in a straight 
line, (cl speed up during straight-line motion, (di slow down 
during straight-line motion. Because the last two situations are 
examples of acceleration, the list could actually be reduced to: la) 
rest, lb) unifomi motion, and (c) acceleration. 

Rest, unifomi motion, and acceleration are therefore the 
phenomena to explain. The word "explain" must be used with 
care. To the physicist, an event is "explained" when it is shown 
to be a logicid consequence of a law the physicist has reason to 
believe is ti\ie. In other words, a physicist with faith in a general 
law "explains" an exent by showing that it is consistent lin 
agreement) with the law. An infinite number of separate, 
different-looking events occur constantly all around you and 
within you. In a sense, the physicist's job is to show how each of 
these events results necessarily from certain general i\iles that 
describe the way the world operates. This approach to 
"explanation" is made possible by the fact that the number of 
general laws of physics is surprisingly small. Ihis chapter will 
discuss three such laws. Together with the mathematical 
schemes of Chapters 1 and 2 for describing motion, they will 
enable you to understand practically all motions that you can 
easily observe. Adding one more law, the law of unixersal 
gravitation (Unit 2i, you can explain the motions of stars, planets, 
comets, and satellites. In fact, throughout physics one sees again 
and again that nature has a manelous simplicity. 

l"o explain rest, unifomi motion, and acceleration of any 
object, you must be able to answer such questions as these: Why 
does a vase placed on a table remain stationary? If a diy-ice disk 
resting on a smooth, level surface is given a brief push, why does 
it move with unifoim speed in a straight line? Why does it 
neither slow down quickly nor cun^e to the right or left? These 
and almost all other specific questions about motion can be 
answered either directly or indirectly from Isaac i\ew1ons three 
general "Laws of Motion." These laws appear in his famous book, 



68 



IMI 1 COVCLPIS Ol MOnON 



Philosophise Naturalis Principia Mathematica (Mathematical 
Principles of Natural Philosophy, 1687), usually referred to simply 
as the Principia. They remain among the most basic laws in 
physics today. 

This chapter will examine Newton's three laws of motion one 
by one. If your Latin is fairly good, try to translate them from the 
original, reproduced below. A modern, English version of 
Newton's text of these laws appears in the margin on page 68. 

Before taking up Newton's ideas, it is helpful to see how other 
scientists of Newton's time, or earlier, might have answered 
questions about motion. One reason for doing this now is that 
many people who have not studied physics still tend to think a 
bit like pre-Newtonians! 



Pages 12 and 13 of the original 
(Latin) edition of Newton's Philoso- 
phiae Naturalis Principia Mathe- 
matica (The Mathematical Princi- 
ples of Natural Philosophy). These 
pages contain the three laws of 
motion and the parallelogram rule 
for the addition of forces (see Sees. 
3.3 and 3.4). 



[ u ] 



A X I O M A T A 

siv£ 
LEGES MOTUS 



Corpus mniie pcrfcvc!\iyc in jiMii [no ^jiin-Jt^-urli icl moz'cndi itinfur- 
iiiitcr in dirciiiim, nifi jii.itcniu ■ii'iiihns imfreff's (og^iir Jijttuit 
ilium iiiHlare. 

PRojcfrilij pcrkvcrjnr in motibiisfiiis nifi quarcniis aicfiftcn- 
ria acris rcr.iul.uuui Sc vi j;ra\it.ui^ imptlluntur dcoifiim. 
Trtxiuia, niHh p.iircs tolivicmlo pcrpctiio rctraliuiit fcfe 
a monlui- ficnlmci i-.on cdiat lor.iii nili i-;uarcmis ab acri- ic- 
tarJatm. M • i ;,i jutcm I'lancraitiiii Jk Conictariimcoipora mo 
ru^ luoi iv projuiHuj; & ciicularci ill fpatiis minus rcllftcntibiis 
facto; conllrvant tliutiu--. 

I. ex. II. . 

hlii1.Ttioticni »i0!nr pioforlionnlem tffeiii inotna iwprelft, d,'- fierife- 
cn>'d):iii Imfjiii recimii ijiut 'vts ilU imprimitiir. 

Sm ,.!;]i unotuni (picmvis gtncrcr, cKipia dupliini, tiiplatri- 
p!i.!iiL'ir.i-!,i!":- iTm. l":m!l& fcinil, fivcyia'Jarim&.- fiirctffivtijn- 
prilli kiLiir. I [ lii( iiinir ijror.i.ini in caniiciii Ic-niiKT pl.ijjiam 
cunn; L" mi.'t :n di tuinii.itui , li torpusantta movibarur, nio- 
t'JU!'- .■ ', t;;,i'|.: .ir.ii.iJiiitiir, \(.l mnnaiio Uibdiltitui, vd obf>. 
f]i',o t.l.]k]L,'_ ,, 'iiLitui; Ix Limi (.o U.vunduiii utriufipdctCTniinatio- 
fK.m.iiinpoi.if.ii. Lex. Hi. 



> 



I '3 3 
Lex. lU. 



A^ioni contrariam femferdf- cdjHJem effc rcuBtoneni : fi've corporum 

dmrrHm aSlio^ics m Ic mutiw jemptr tfjt; .t^Uiiles ^ in partes conlm- 
rui dnigi. 

Qiiicquid prcmit vtl nalik alttrimi, rantundcmab coprcmitur 
vcl traliiciir. Siijui^ lapidcm digito picnnt, piitnirui & luijiis 
digitus a lapide. Sieijimslapidem luni allegatuui rrahir, rclrahc- 
tiucti.iin S^ c'i)in:saqualiftTinl.ipidtni:namluriiiirrini|.dirttimis 
codeni nl.i\.ir.di li loiiatu iirgebit Kquuin vtilii' l.ipidcm, ac l.i- 
piJini Mifu c\]mim, raiitunuj, iiiipc'Ji<.t pionrifiimitinius quaii- 
tiiiii proiiiovct progulTuin .ilrciuis. Si coipiiv aliciiiod inccrpus 
.iliud iiiipingms, niotumtjusvi lua qiuMnuiJoLunq: niut.ntjit, i- 
dciiiquoqiit viciiTIm inniotti propiio c.mJcm niut.itioncm m pai- 
rcm uiiitr.iii.uii \i altciius ( ob .vqu.ilil.ircm picnioiii- nnirui-J 
fubibit. Hi .K'tmnibu' iqu^ili' tii lit irut.itionc^ nun Mlocitatum 
M inoniniii, !^ lulicct m coipoiilniMum .ilinndciinpiditi^ .^jMu- 
tatioiKs cnini vclocitatum, in tontraiia> itidcni partes iafl-ar,quia 
motiis .Kjii.ilitcr iiiurantur, Unit corpi)ribiii iiciprocc propottio- 
ii.ik-. 

Corol. I. 

Corpus iiirihiis ronjiin^lis di.jroihiLm p.ir.iUcloirjnimi eudem tempore 
dcjiribcrc^ ,jUo Ltlerd fcpjrjiis. 

Si coipu; datotcmpoio, \iibla Af, 
fcirtciir ab /I ail i>, 5c vi Tola ,V, ab 
A ad C, compleatiir parallclogiani- 
muin AliPC, &: vi iitiaq, fm tin id 
ctidcm reinpon- ab A ad /). N.iin 
qiionfain vii N ajjir ffrunduni Imcam 

AC ipfi B D paialliliiii, hvt vi, niliil nnirabir vtlocifatcm accc- 
dcndi ad lincain illani !> D a \i altera geiiitaiii. Aecedet iyiciu- 
c^ipiis eodem tempore ad line.iiii H P l7Ce \i, N imprimanir,"/;\o 
lion, atq, adeo in fine illiiis tenipons rqniietur aliciibi iii !inea 




1. A baseball is thrown straight upward. Which of these 
questions about the baseball's motion are kinematic and which 
dynamic? 

(a) How high will the ball go before coming to a stop and 
starting downward? 

(b) How long will it take to reach that highest point? 

(c) What would be the effect of throwing it twice as hard? 

(d) Which takes longer, the trip up or the trip down? 

(e) Why does the acceleration remain the same whether the 
ball is moving up or down? 



CHAPTER 3 / THE BIRTH OF DYNAMICS 



69 



3.2 I The Aristotelian explanation 
of motion 




Keeping an object in motion at u/i/- 
form speed. 



SG 2 



The idea of force played a central role in the dynamics of 
Aristotle 20 centuries before N'eulon. You will recall from 
Chapter 2 that in Aristotle's physics there weve two types of 
motion: "natural" motion and "violent motion. For example, a 
falling stone was thought to be in "natural " motion (towards its 
natural place i. On the other hand, a stone being steadily lifted 
was thought to be in "\iolent ' motion (away from its natural 
placel. To maintain this uniform \iolent motion, a force had to be 
continuously applied. Anyone lifting a large stone is very much 
aware of this force while straining to hoist the stone higher. 

The Aristotelian ideas agreed with many common-sense 
obseirations. But there were also difficulties. Take as a specific 
example an arrow shot into the air. It cannot be in \iolent 
motion without a moxer, or something pushing it. Aristotelian 
physics required that the arrow be constantly propelled by a 
force. If this propelling force were removed, the arrow should 
immediately stop its flight and fall directly to the ground in 
"natural" motion. 

But, of course, an arrow does not fall to the ground as soon as 
it loses diiect contact with the bowstring. What, then, is the force 
that piopels the arrow? Here, the Aristotelians offered a clever 
suggestion: The motion of the arrow through the air is 
maintained by the air itself! As the arrow starts to mo\'e, the air 
in front of it is pushed aside. More air rushes in to fill the space 
being vacated by the arrow. This iiish of air- around the arr'ow 
keeps it in flight. 

Other ideas to explain motion were developed before the mid- 
seventeenth Centura'. But in every case, a force was considered 
necessary to sustain uniform motion. The explanation of unifor-m 
motion depended on finding the force, and that was not always 
easy. There were also other problems. For example, a falling 
acor'n or stone does not move with unifomi speed. It accelerates. 
How is acceleration explained? Some Aristotelians thought that 
the speeding up of a falling object was connected with its 
approaching arrival at its natural place, the earth. In other words, 
a falling object was thought to be like a tir-ed horse that starts to 
gallop as it nears the barii. Others claimed that when an object 
falls, the weight of the air- above it incr-eases, pushing it harder. 
Meanwhile, the column of air below it decreases^ thus offering 
less resistance to its fall. 

When a falling object finally reaches the ground, as close to 
the center of the earth as it can get, it stops. And there, in its 
"natual place," it remains. Rest, being regarded as the natural 
state of objects on earth, required no firrther explanation. The 
thr'ee phenomena of rest, uniform motion, and acceleration 



70 



I \H 1 CO.NCliPTS OF MOI l()\ 



could thus be explained more or less reasonably by an 
Aristotelian. Now examine the Newtonian explanation of the 
same phenomena. The key to this approach is a clear 
understanding of the concept of force. 



# 2. According to Aristotle^ what is necessary to maintain 
uniform motion? 

3. Give an Aristotelian explanation of a dry-ice disk's uniform 
motion across a tabletop. 



3 .3 I Forces in equilibrium 

The common-sense idea of force is closely linked with muscular 
activity. You know that a sustained effort is required to lift and 
support a heavy stone. When you push a lawn mower, row a 
boat, split a log, or knead bread dough, your muscles indicate 
that you are applying a force to some object. Force and motion 
and muscular activity are naturally associated in our minds. 
When you think of changing the shape of an object, or moving it, 
or changing its motion, you automatically think of the muscular 
sensation of applying a force to the object. You wdll see that 
many, but not all, of your everyday, common-sense ideas about 
force are useful in physics. 

You know, without having to think about it, that forces can 
make things move. Forces can also hold things still. The cable 
supporting the main span of the Golden Gate Bridge is under the 
influence of mighty forces, yet it remains at rest. Apparently, 
more is required to start motion than just any application of 
forces. 

Of course, this is not suiprising. You have probably seen 
children quarreling over a toy. If each child pulls with 
determination in the opposite direction, the toy may go nowhere. 
On the other hand, the tide of battle may shift if one of the 
children suddenly makes an extra effort or if two children 
cooperate and pull side by side against a third. 

Likevvase, in a tug-of-war between two teams, large foices are 
exerted on each side, but the rope remains at rest. We might say 
that the forces "balance" or "cancel." A physicist would say that 
the rope was in equilibrium. That is, the sum of all forces applied 
to one side of the rope is just as great, though acting in the 
opposite direction, as the sum of forces applied to the other side. 
The physicist might also say the net force on the rope is zero. 
Thus, a body in equilibrium cannot start to move. It starts to 
move only when a new, "unbalanced" force is added, destroying 
the equilibrium. 





CHAPTER 3 / THE BIRTH OF DYNAMICS 



71 



This force is called tiie net force, 
because it is the sum of all the 
forces in one direction minus the 
sum of all the forces in the oppo- 
site direction. 



Ca) 

Jorce Fg force r, 

<r ^ 

teom 2 team t 

(b) 



Cc) 



F. 



K'-O 



There are se\'eral ways of express- 
ing the idea of unbalanc-ed force: 
net force, resultnnt force, total force, 
vector sum of forces. All mean the 
same thing. 



In all these examples, both the magnitude (size or amoimtl of 
the forces and their directions are important. The eftect of a 
force depends on the direction in vvhic-h it is a[)plied. You can 
represent both the sizes and directions of foi-ces in a sketch by 
using arrows. The direction in which an arrow points represents 
the direction in which the force acts. The length of the arrow 
represents how large the force is. For example, the force exerted 
by a 10-kg bag of potatoes is shown by an arrow twice as long 
as that for a 5-kg bag. 

If you know separately each of the forces applied to any object 
at rest, you can predict whether the object will remain at rest. 
It is as simple as this: An object acted on by forces is in 
equilibrium and remains at lest only if the arrows representing 
the forces all total zero. 

How do you "total" arrows? This can be done by means of a 
simple technique. Take the tug-of-war as an example. Call the 
force applied by the team pulling to the right F,. (The arrow over 
the F indicates that you are dealing with a quantity for* which 
direction is important.) The force applied to the rope by the 
second team you can call F,. Figure (a) in the margin shows the 
two arrows corresponding to the two forces, each applied to the 
same rope, but in opposite directions. Assume that these forces, 
F, and F^, were accurately and separately measured. For 
example, you might let each team in turn pull on a spring 
balance as hard as it can. You can then draw the arrows for F, 
and F^ carefully to a chosen scale, such as 1 cm = 100 N. Thus, 
a 200-N force in either direction would be represented by an 
arrow 2 cm in length. Next, take the arrows F, and F^ and draw 
them again in the correct directions and to the chosen scale. 
This time, howevei", put them "head to tail" as in Figure (b). 
Thus, you might dravv^ F, first, and then draw F^ with the tail of 
F, starting from the head of F,. (Since they would, of course, 
oveilap in this example, they are drawn slightly apart in Figure 
(b) to show them both clearly.) The technique is this: If the head 
end of the second arrow falls exactly on the tail end of the first, 
then you know that the effects of F, and F^ balance each other. 
The two forces, equally large and acting in opposite directions, 
total zero. If they did not, the excess of one force over the other 
would be the net force, and the rope would accelerate instead 
of being at rest. 

To be sure, this was an obvious case. But the "head-to-tail" 
method, using drawings, also works in cases that are not as 
simple. For example, apply the same procedure to a boat that is 
secured by three ropes attached to different moorings. Suppose 
F,, in this case, is a force of 24 N, F, is 22 N, and F., is 19 N, each 
in the direction shown in the sketch on p. 73. (A good scale for 
the magnitude of the forces here is 0.1 cm = 1 N of force.) Is the 
boat in ecjuilibrium when it is acted on by the foices'.^ \'es, if the 



7Z 



IM I 1 COXCKPTS Ol VIOTIOX 



forces add up to zero. With iT-iler and protractor you can draw 
the arrows to scale and in exactly the right directions. Adding Fj, 
F,, and F3 head to tail, as in the diagram to the right of the 
picture, you see that the head of the last arrow falls on the tail of 
the first. The sum of the forces, or the net force, is indeed zero. 
The forces are said to cancel, or to be balanced. Therefore, the 
object (the boat) is in equilibrium. This method tells when an 
object is in equilibrium, no matter how many different forces are 
acting on it. 






^--.^0 



We are defining equilibrium with- 
out worrying about whether the 
object will rotate. For e^cample, the 
sum of the forces on the plank in 
the diagram is zero, but it is ob- 
vious that the plank will rotate. 




SG5 




p . p ♦ r> =■ o 



ii.i.i_iLiiJil- 



We can now summarize our understanding of the state of rest 
as follows: If an object remains at rest, the sum of all forces 
acting on it must be zero. Rest is an example of the condition of 
equilibrium, the state in which all forces on the object are 
balanced. 

An interesting case of equilibrium, different fi^om the tug-of- 
war, is the last part of the fall of a skydiver. At the beginning, just 
after the jump, the person is in just the sort of free-fall, 
accelerated motion discussed in Chapter 2. But the force of air 
friction on the skydiver increases with speed. Eventually, the 
upward fHctional force becomes large enough to cancel the 
downward force of gravity (which is the force you experience as 
your weight). Under these conditions, the skydiver is in 
equilibrium, going at a constant (terminal) speed, kept from 
accelerating by friction with the air going by. The net force is 
zero, just as it is when you are lying still in bed. 

The speed at which this equilibrium occurs for a falling person 
without a parachute is veiy high. When the parachute is opened, 
its large area adds greatly to the friction, and therefore 
equilibrium is established at a much smaller terminal speed. 



-^" ■^Jf.f^^^ 




CHAPTER 3 / THE BIRTH OF DY\AIVIICS 



73 



4. A vase is standing, at rest on a table. What forces would you 
say are acting on the vase? Show how each force acts (to some 
scale) by means of an arrow. Can you show that the sum of the 
forces is zero? 

5. In which of these cases do the forces cancel? 




if 



6. What is 3 plus 3? What is 3 north plus 3 south? What is 4 
plus 2? What is 4 up plus 2 down? What is 4 in plus 2 in? 

7. If an object esperiences forces of 5 N right and 3 \' left, is it 
in equilibrium? What is the definition of equilibrium? Does an 
object in equilibrium have to be at rest? 

3«4r About vectors 




The method for representing forces by arrows really can predict 
whether the forces cancel and will leave the object in 
equilibrium. It will also show whether any net force is left over, 
causing the object to accelerate. You can demonstrate for 
yourself the reliability of the addition rule by doing a few 
experiments. For example, you could attach three spring scales 
to a ring. Have three persons pull on the scales with forces that 
just balance, keeping the ring at rest. While they are pulling, read 
the magnitudes of the forces on the scales and mark the 
directions of the pulls. Then make a sketch with arrows 
representing the forces, using a convenient scale, and see 
whether they total zero. Many different experiments of this kind 
ought all to show a net force equal to zero. 

It is not obvious that forces should behave like anT)ws. But 
arrows drawn on paper happen to be useful for calculating how 
forces add. (If they were not, we simply would look for other 
svnnbols that did work.) Forces belong in a class of concepts 
called vector quantities, or just vectors for short. Some 
characteristics of vectors are easy to represent by arrows. In 
particular, vector quantities have magnitude, which can be 
represented by the length of an arrow drawn to scale. They also 
have direction, which can be showii by the direction of an arrow. 
By experiment, we find that vectors can be added in such a way 
that the total effect of two or more can be represented b\' the 
head-to-tail addition of arrows. This total effect is called the 
vector resultant, or vector sum. 



74 



IMF 1 COVCKPiS ()l MOi l()\ 



In the example of the tug-of-war, you determined the total 
effect of equally large, opposing forces. If two forces act in the 
same direction, the resultant force is found in much the same 
way, as shown below. 





If two forces act as some angle to each other, the same type of 
sketch is still useful. For example, suppose two forces of equal 
magnitude are applied to an object at rest but free to move. One 
force is directed due east and the other due north. The object 
will accelerate in the northeast direction, the direction of the 
resultant force. (See sketch in the margin.) The magnitude of the 
acceleration is proportional to the magnitude of the resultant 
force, shown by the length of the arrow representing the 
resultant. 

The same adding procedure works for forces of any magnitude 
and acting at any angle to each other. Suppose one force is 
directed due east and a somewhat larger force is directed 
northeast. The resultant vector sum can be found as shown 
below. 



Any vector quantity is indicated by 
a letter with an arrow over it, for 
example, F, a, or v. 






To summarize, a vector quantity has both direction and 
magnitude. Vectors can be added by constructing a head-to-tail 
arrangement of vector arrows (graphical method) or by an 
equivalent technique known as the parallelogram method, which 
is briefly explained in the marginal note at the right. (Vectors also 
have other properties which you will study if you take further 
physics courses.) By this definition, many important concepts in 
physics are vectors, for example, displacement, velocity, and 
acceleration. Some other physical concepts, including volume, 
distance, and speed, do not require a direction, and so are not 
vectors. Such quantities are called scalar quantities. When you 
add 10 liters (L) of water to 10 L of water, the result is always 20 
L; direction has nothing to do vWth this result. Similarly, the tenn 
speed has no directional meaning; it is simply the magnitude of 
the velocity vector. Speed is shown by the length of the vector 
arrow, wdthout regard to its direction. By contrast, suppose you 



You can use equally well a graphi- 
cal construction called the "paral- 
lelogram method." It looks differ- 
ent from the "head-to-tail" method, 
but is really exactly the same. In the 
parallelogram construction, the 
vectors to be added are represented 
by arrows joined tail-to-tail instead 
of head-to-tail, and the resultant is 
obtained by completing the diago- 
nal of the parallelogram. (See SG 6.1 




CHAPTER 3 / THE BIRTH OF DYIVAMICS 



75 



add two forces of 10 X each. The lesiiltant force may be 
anwvhere between and 20 i\', depending on the direction of the 
two indixidual forces. 

In Sec. 1.8, acceleration was defined as the rate of change of 
speed. That was onl\ partly correct because it was incomplete. 
We must also consider- changes in the direction of motion as 
well. Acceleration is best defined as the rate of change of velocity, 
where velocitv is a vector- ha\ing both niiignitude and direction. 
In symbols this definition may be written 

- - ^ 
^" ~ At 

where Av is the change in xelocity. Velocity' can change in two 
ways: by changing its magnitude (speed) and by changing its 
direction. In other' words, an object is acceler'ating when it 
speeds up, slows down, or changes dir^ection. This definition and 
its important uses will be explor^ed mor'e fully in later- sections. 



Because constant \elocity means 
both constant speed and constant 
direction, we can write Mewlon's 
first law more concisely: 

V = constant 

if and only if 

F , = 

net 

This statement includes the condi- 
tion of rest, since r'est is a speciiil 
case of unchanging velocity, the 
case where v = 0. 



# 8. Classijy each of the following as vectors or scalars: (al 2 m 
up; (b) volume; (c) 4 sec; (d) 3 m/sec west; ((e) 2 m/sec'; (f) 
velocity. 
9. List three properties of vector quantities. 
10. How does the new definition of acceleration given above 
differ from the one used in Chapter 1? 

3.S I Neivton's first laiv of motion 

You probably were surprised when you first watched a moving 
dry-ice disk or some other nearly frictionless objec't. Remember 
how smoothly it glided along after the slightest .slio\e? How it 
showed no sign of slowing douTi or speeding up? From your 
everyday experience, you automatically think that some force is 
constantly needed to keep an object mo\ ing. But the disk does 
not act according to common-sense Aristotelian expectations. It 
is always surpr-ising to see this for the first time. 

In fact, the disk is beha\ing quite naturally. If the for^ces of 
friction were absent,, a gentle push would send tables and chairs 
gliding across the floor like drA'-ice disks. Neulon's fir-st law 
directly challenges the Aristotelian idea of what is "natur-al." It 
declares that the state of rest and the state of uniforrn, 
unaccelerated motion in a straight line ar-e equalK' natural. Only 
the existence of some force, friction for- example, keeps a moving 
object from moving /brever! Newton's first law of motion can be 
stated in modem language as follows: 

E\'ety object continues in its state of rest or of uniform 
rectilinear istraigbt-line) motion unless acted upon b\' an 



76 



I MI 1 CONCLI'IS Ol .MOnUV 



unbalanced force. Conversely, if an object is at rest or in 
uniform rectilinear motion, the unbalanced force acting upon it 
mvist be zero. 

In order to understand the motion of an object, you must take 
into account all the forces acting on it. If all forces (including 
friction) are in balance, the body will be moving at constant v. 

Although Newton was the first to express this idea as a general 
law, Galileo had made similar statements 50 years before. Of 
course, neither Galileo nor Newton had dry-ice disks or similar 
devices. Therefore, they were unable to observe motion in which 
friction had been reduced so greatly. Instead, Galileo devised a 
thought experiment in which he imagined the friction to be zero. 

This thought experiment w^as based on an actual observation. 
If a pendulum bob on the end of a string is pulled back and 
released from rest, it wall swing through an arc and rise to very 
nearly its starting height. Indeed, as Galileo showed, the 
pendulum bob will rise almost to its starting level even if a peg is 
used to change the path as shown in the illustration below. 



d- 




A 



/ 



O 




SG 8 



From this observation Galileo went on to his thought 
experiment. He predicted that a ball released from a height on a 
frictionless ramp would roll up to the same height on a similar 
facing ramp. Consider the diagram at the top of the next page. As 
the ramp on the right is changed fi^om position (a) to (b) and 
then to (c), the ball must roll farther in each case to reach its 
original height. It slows down moie gradually as the angle of the 
incline decreases. If the second ramp is exactly level, as shown 
in (d), the ball can never reach its original height. Therefore, 
Galileo believed, the ball on this frictionless surface would roll on 
in a straight line and at an unchanged speed forever. This could 
be taken to mean the same as Newton's first law. Indeed, some 
historians of science do give credit to Galileo for having come up 
with this law first. Other historians, however, point out that 
Galileo thought of the "rolling on forever " as "staying at a 
constant height above the earth." He did not think of it as 
"moving in a straight line through space." 

This tendency of objects to maintain their state of rest or of 
unifomi motion is sometimes called the principle of inertia. 



Inside the laboratory there is no 
detectable difference between a 
straight (horizontal) line and a con- 
stant height above the earth. But on 
a larger scale, GalUeos eternal roll- 
ing would become motion in a cir- 
cle around the earth. Newton made 
clear what is really important: In 
the absence of the earth's gravita- 
tional puU or other external forces, 
the bfill's undisturbed path would 
extend straight out into space. 



CHAPTER 3 / THE BIRTH OF DYNAMICS 




Newton's first law is therefore sometimes referred to as the law of 
inertia. Inertia is a property of all objects. Material bodies have, 
so to speak, a stubborn streak concerning their state of motion. 
Once in motion, they continue to move with unchanging velocitv' 
(unchanging speed and direction) unless acted on by some 
unbalancx^l external force. If at rest, they i-emain at rest. This 
tendency is what m^ikes seat belts so necessaiy when a car stops 
very suddenly. It also explains why a car may not follow an icy 
road around a turn, but tiaxcl a straightei' path into a field or 
fence. Ihe greater the inertia of an object, the greater its 
resistance to change in its state of motion. Therefore, the greater 
is the force needed to produce a change in the state of its 
motion. For example, it is more difficult to start a train or a ship 
and to bring it up to speed than to keep it going once it is 
mox'ing at the desired speed. (In the absence of frictiorn, it would 
keep moving without any applied force at all.) For- the same 
reason it is also difficult to bring it to a stop, and passengers and 
cai^go keep going for-war'd if the xehicle is suddenK' br'aked. 

Newton's first law says that if an object is mo\ing with a 
constant speed in a straight line, the forces acting on it must be 
balanced; that is, the object is in ecjuilibrium. Does this mean 
that in Nevvlonian physics the state of rest and the state of 
unifonn motion are equivalent? It does indeed! If a body is in 
equilibrium, v = constant. Whether' the \alue of this constant is 
zer o or- not depends in any case on the frame of reference foi- 
measuring the magnitude of v. You can say whether a body is at 
rest or- is moxing with constant v lar-ger than zei'o orily by 
reference to some other- body. 

Take, for example, a tug-of-war. The two teams are sitting on 
the deck of a bar-ge that is dr-ifting with unifor-m \elocit\' down a 
lazy river. An observer- on the same barge and one on the shore 
repor^t on the incident. Each observes from a particular frame of 
r-eference. The observer on the bar'ge repor^ts that the forces on 
the rope are balanced and that it is at rest. The observer on the 
shore reports that the forces on the rope are balanced and that il 
is in unifor-m motion. Which observer- is right? They are both 
light; Newton's fiist law of motion applies to both observations. 
Whether a body is at rest or in unifbr-m motion depends on 
which fr'arrie of r^efer-encx^ is used to obseiAe the event. In both 
(-as(^s, the for-ces on the objec-t inx'ohed are balanc(Hi. 



78 



IM I 1 COXCIIPTS Ol MOI l()\ 



11. How would Aristotle have explained the fact that a bicyclist 
must keep pedaling in order to move with uniform speed? How 
would Newton explain the same fact? If Aristotle's e^cplanation 
is "wrong, " why do you think you study it? 

12. What is the net force on the body in each of the four cases 
sketched in the margin of page 80? 

13. What may have been a difference between Nen^on's 
concept of inertia and Galileo's? 



3*6 I The significance of the first law 

Newton's laws invohe many deep philosophical concepts. (See 
SG 7.) However, the laws are easy to use, and you can see the 
importance of Newton's first law without going into any complex 
ideas. For convenience, here is a list of the important insights the 
first law provides. 

1. It represents a break with Aristotelian physics: The 'natural" 
motion, the motion that needs no further explanation, is not a 
return to a position of repose at an appropriate place. It is^ 
rather, any motion that takes place with a uniform velocity. 

2. It presents the idea of inertia, that is, of the basic tendency 
of all objects to maintain their state of rest or uniform motion. 

3. It says that, ft^om the point of view of physics, a state of rest 
is equivalent to a state of unifomi motion at any speed in a 
straight line. There is nothing "absolute " or specially 
distinguished about any one of the states, uniform motion or 
rest. This raises the need to specil\' a "frame of reference " for 
describing motion, since an object that is stationarv' with respect 
to one obseiA'er, or frame of reference, can be in motion v\ith 
respect to another. 

4. It, like the other physical law^s, is a universal law, claiming to 
be valid for objects anvwhere in the universe. That is, the same 
law applies on the earth, on the moon, throughout the galaxy, 
and beyond, and the same law applies to the motions of atoms, 
magnets, tennis balls, stars, and every other thing. (A rather 
grand claim, but, as far as we can tell, a valid one.) 

5. The first law describes the behavior of objects when no net 
force acts on them. Thus, it sets the stage for the question: 
Exactly what happens when an unbalanced force does act on an 
object? 



Of course , the idea of inertia does 
not explain why bodies resist change 
in their state of motion. It is simply 
a term that helps us to talk about 
this basic, e.xperimentaUy observed 
fact of nature. (See SG 10.1 



The correct reference frame to use 
in our physics turns out to be any 
reference frame that is at rest or in 
uniform rectilinear motion with re- 
spect to the stars. Therefore, the 
rotating earth is, strictly speaking, 
not allowable as a Newtonian ref- 
erence frame; but for most pur- 
poses the earth rotates so little dur- 
ing an experiment that the rotation 
can be neglected. (See SG 11.) 



3.7 Newton's second laiv of motion 



SG 12 



In Sec. 3.1, we stated that a theoiy of dynamics must account for 
rest, unifomi motion, and acceleration. So far, we have met two 



CHAPTER 3 / THE BIRTH OF DYNAMICS 



79 




Feather foiling at nearly 
constant- speed 




Kite held suspended in the wlr>d 



DRAG 




PUSH 




Athlete running against the 

wind 




of these three objectives: the explanation of rest and of uniform 
motion. In terms of the first law, the states of rest and iiniibini 
motion are equivalent. 

Neuron's second law provides an answer to the question: What 
happens when an unbalanced force acts on an obje(-t.^ Since we 
think of the force as causing the resulting motion, this law is the 
fundamental law of dvnamics. 

In qualitatixe terms, the second lav\' of motion says little more 
than this: The necessary and sufficient cause for a deviation fixDm 
"natural" motion (that is, motion with a constant or zero velocitv) 
is that a nonzero net force act on the object. However, the law 
goes further and provides a simple quantitative relation between 
the change in the state of motion and the net force. In order to 
be as clear as possible about the meaning of the law, we will first 
consider a situation in which different forces act on the same 
object; we will then consider a situation in which the same force 
acts on different objects; finally, we will combine these results 
into a general relationship. 

Force and acceleration: The acceleration of an object is directly 
proportional to, and in the same direction as, the net force acting 
on the object. Note that force and acceleration are both vector 
quantities. Since acceleration is the rate at which velocity 
changes, the force is proportional to the change in the velocity; 
the faster or the greater the change, the larger the force must be. 
If a stands for- the acceleration of the object and F ,^., stands for 
the net force on it, the relationship is 

I'his relationship is equivalent to the statement above if it is also 
understood that when vectors are proportional, they must point 
in the same direction as well as have proportional miignitudes. 
To say that tvvo quantities are proportional means that if one 
quantity is doubled lor multiplied by any numberl, the other- 
(juantity is also doubled lor mirltiplied by the same number). 
1 bus, for example, if a certain force produces a certain 
acceleration, Uvice the force Ion the same object) v\'ill produce 
twice as great an acceleration in the same direction. In symbols, 
for the same object, 

if F„,., will cause a, then 

2 F ,^., will cause 2a 

'/2F„g,will cause Vza 

5.2 F„^, wall cause 5.2a 

;cF„^,,will cause ^a 

It is easy to perform a rough experiment to test this law . Place 
a dry-ice disk or other nearly frictionless object on a flat table, 
attach a spring balance, and pull with a steady for'ce so that it 
accelerates continuously, i he pull registered In the balance is 



HO 



IM 1 1 (;()\(;i;prs oi moi ion 



the only unbalanced force, so it is the net force on the object. 
You can determine the acceleration by measuring the time to go 
a fixed distance la oc y, t'}. Repetitions of this experiment should 
show Fj^p, ^ a, to within experimental error; at least, they have 
shown it whenever such an experiment was done in the past. 

Mass and acceleration: The acceleration of an object is inversely 
proportional to the mass of the object; that is, the larger the mass 
of an object, the smaller will be its acceleration if a given net 
force is applied to it. The mass of an object is therefore what 

determines how large a force is required to change its motion; in 

other words, the mass of an object is a measure of its inertia. It SG 13 

is sometimes called inertial mass, to emphasize that it measures 

inertia. Mass is a scalar quantity (it has no direction), and it does What does it mean to say that mass 
not affect the direction of the acceleration. is a scalar quantity? 

The relation between acceleration and mass can be ua itten in 

symbols. Let a stand for the acceleration (the magnitude of the 

acceleration vector a) and m stand for mass. Then, See Sec. 3.4. 

1 
a a — 

m 

as long as the same net force is acting. Notice that an object with 
twice (or three times . . .) the mass of another will experience one 
half (or one third . . .) the acceleration if subjected to the same 
net force; that is, for the same net force. 



if 


m 


experiences 


a, then 




2m 


experiences 


la 
2 




Im 


experiences 


3a 




3 








^m 


experiences 


1 
-a 



This law can also be demonstrated to hold by experiment. Can 
you suggest a way to do that? 

The general relationship. Acceleration (for constant m) is 
proportional to the net force, and (for constant F,^^.,) it is 
proportional to 1/m, the reciprocal of the mass. It follovv^s that, in 
general, acceleration is proportional to the product of the net 
force and the reciprocal of the mass: 

-, -. 1 

a oc F ■ — 

""' m 

Are there any other quantities (other than net force and mass) 
on which the acceleration depends? Newton proposed that the 
answer is no. Only the net force on the object and the mass of 
the object being accelerated affect the acceleration. All 
experience since then suggests he was right. 



CHAPTER 3 / THE BIRTH OF DYNAMICS 81 



Since there are no otlier- factors to be (-onsidered, we can make 
the proportionality into an equiility; that is, ue can write 



m 



= 1. 



Actually, from a « ^'„,,/"i 't follows 

that a = k (F_,./m), where k is a (;on- 3 = 

stant. But by choosing tiie units for 

a, F, and m pvopv.rW, we ha\(! set k ^he same relationship can, ot course, also l)c written as the 

famous equation 

In both these equations we have again written F, ., to emphasize 
that it is the net for'ce that cl(!t(Mniin(!s the; acce^leration. 

This relationship is probably tlu; most basic; e(}uation in 
mechanics and, therefore, in physics. Without symbols we can 
state it as follows. Newton's second hnv: The net force on an 
object is numerically equal to, and in the same direction as, the 
acceleration of the object multiplied by its mass. It does not 
matter" whethtM' the; forcx^s that act are magnetic, gia\ itational, 
simple pushes and pirlls, or- any comljination; whetiier- the 
masses are those of electrons, atoms, stars, or cars; whether the 
acceleration is large or- small, in this direction oi' that. The law- 
applies universally. 

Measuring mass and force. We have already mentioned a 
method of mcasirring for'ce and have used it in talking about 
Newton's first law. The method is based on the fact that the 
extension of a spr ing (as long as it is not stretched or bent out of 
shape) is propor^tional to the for'ce. Thcr^efore, you can use a 
spring balance to measure forces. A force that is twice as big as 
another" will stretch the same spring twice as far". 1 he spring 
balance must be c^aliljrated but that can be d(jn(> aft(M" the units 
of mass have been detined. 

Measuring mass lor inertia) is quite different fr"om measuring 
forx;e. When you think of measuring mass, you might first think 
of weighing it. But if you take apart a typical scale (the kitchen or" 
bathroom variety, for example) you will find that it is usually just 
a spr"ing balance. They measur"e a force and not the inertia of an 
object. The force they measure, called weight, is the graxitational 
force exerted on the object by the earth. If you stand on your 
bathroom scale on the moon, it will show a much smaller weight, 
but you will have just the same mass, or" inertia. 

You will need to calibrate the spring balance, that is, to decide 
how mu(ii stretch of the spring corresponds to one unit of for'ce. 
In modern practice, this decision is made after- deciding how to 
measure masses. 

A reminder": When we sp(Mk of the mass of a IkkIv, we mean 
its iner-tia, not its weight. The difference will be discussed in 
more detail in Sec. 3.8 and Chapter 8. For now you can get a 
quick feeling for" the difference berv\-c;en the two In thinking of an 
expcM-imc;nt in a spacc^craft that is mo\ ing witb constant \cic)cit\ . 



SG 15 



SG 18 



H'Z 



I Ml 1 t:o\(:iiPrs oi vionox 



far from the earth or other planets. Suppose you are trying to 
move a puck on a table inside the spacecraft, as in the 
experiment in the photograph on p. 11. You find that the push 
needed to bring the puck up to a given speed on its table is just 
the same as on earth. The mass of an object is a measure of its 
inertia, its resistance to changes in motion; and that is the same 
for that object everywhere . The weight, on the other hand, 
depends on its location, for example, on how close the object is 
to a planet that exerts the gravitational pull called weight. When 
the spacecitift reaches outer space, the weight of an object 
becomes negligible, but its mass or inertia remains the same. 

This fact suggests that the wise thing to do is to choose the 
unit of mass first, and let the unit of force follow later. The 
simplest way to define a unit of mass is to choose some 
convenient object as the universal standard of mass, and 
compare the mass of all other objects with that one. What is 
selected to seiA^e as the standard object is arbitrary. In 
Renaissance England the standard used was a grain of barley 
("from the middle of the ear"). The original metric commission in 
France in 1799 proposed the mass of a cubic centimeter of water 
as a standard. Today, for scientific puiposes, the standard mass 
is a cylinder of platinum-iridium alloy kept at the International 
Bureau of Weights and Measures near Paris. The mass of this 
cylinder is defined to be 1 kilogram I kg), or 1,000 grams (g). 
Accurately made copies of this cylinder are used in various 
standards laboratories throughout the world to calibrate 
precision equipment. Further copies are made from these for 
distribution. 

The same international agreements that have established the 
kilogram as the unit of mass also established units of length and 
time. The meter (m) was originally defined in terms of the 
circumference of the earth, but modem measurement techniques 
make it more precise to define the meter in tenns of the 
wavelength of light, generated in a specific way. The second of 
time (often abbreviated sec, the official svmbol is s) was also 
originally defined with respect to the earth las a certain fraction 
of the year), but it, too, is now more precisely defined in temis 
of light waves emitted by a specific group of atoms. The meter 
and second together determine the units of speed Im/sec) and 
acceleration (m/sec^). 

With these units, you can now go back and calibrate the spring 
balances used for measuring force. The unit of force, 1 newton 
(N), is defined to be the force required to give an acceleration of 1 
m/sec" to a mass of 1 kg. Because of Neuron s second law (F,,^, 
= ma], 

1 N = 1 kg X 1 m/sec' = 1 kg m/sec"" 

Imagine now, step by step, how you would calibrate a spring 
balance. To begin, take a 1-kg standard object. Put it on a 




The standard kilogram at the U.S. 
Bureau of Standards. 



The mass of 1 cm' of water is 1 
gram (g) (approximately). The mass 
of 1 liter (L) of water is just about 
1,000 g, or 1 kg. 



SG 16 



1 m/sec' is an acceleration of one 
meter per second per second. For 
comparison, note that the acceler- 
ation in free fidl on earth is about 
10 m/sec\ The sec' means that di- 
vision by time units occurs twice. 



SG 17, 18 



In this equation we use only the 
magnitudes; the direction is not 
part of the definition of the unit of 
force. 



CHAPTER 3 / THE BIRl H OF DYMAMICS 



83 



frictionless horizontal surface. Attac^h the spring halance, and 
pull hoiizontally. Ihe net force on the object, supplied by the 
spring balance, is the net force. Pull steadily to get an 
acceleration of 1 msec". Mark the place to which the pointer of 
the stretched balance pointed as "1 i\.' Repeat the procedure 
vvdth an acceleration of 2 m/sec", to get 2 N, etc. Of course, once 
this is done, other spring balances can be more easily calibrated, 
for example, by hooking a pair together, one of which is 

calibrated, and pulling. (Complete this "thought experiment.") 

SI stands for Systeme Internation- The kilogram, together with the meter and second, are the 

3/e. fundamental units of the "mks' system of measurement; together 

with units for light and electricity, the mks units form the 
International System of units ISH. Other systems of units are 

possible. But since the ratios between related units aie more 

SG 19-22 convenient to use in a decimal system, all scientific and 

technical work, and most industiial work, is now done with SI 
units in most countries, including the United States. 



P 



# 14. What combination of the three fimdamental units is 
represented by the newton, the unit of force? 

15. A net force of 10 N gives an object a constant acceleration 
of 4 m/sec'. What is the mass of the object? 

16. True or false? Newton's second law holds only when 
frictional forces are absent. 

17. A 2-kg object being pulled across thejloor with a speed of 
10 m/sec is suddenly released and slides to rest in 5 sec. What 
is the magnitude of the frictional force producing this 
deceleration? 

18. Newton's second law, a = F/m, claims that the 
acceleration of an object depends on three things. What are 
they? 

19. Complete the table below which lists some accelerations 
resulting from applying equal forces to objects of different 
mass. 



MASS 


ACCELERATION 


1 kg 


30 m/sec^ 


2 kg 


15 m/sec^ 


3 kg 




1 5 kg 




0.5 kg 




45 kg 






3 msec^ 




75 m sec^ 



t*4 LM r 1 COXCEPl S OF VIOIIOV 



3«8 I Mass, iveight, and free fall 

You will now examine some more details concerning the veiy 
important topic of mass and weight and how they are related. 
The idea of force in physics includes much more than muscular 
pushes and pulls. Whenever you obsewe an acceleration, you 
know that there is a force acting. Forces need not be 
"mechanical" (exerted by contact only). They can also result fi'om 
gravitational, electric, magnetic, or other actions. Newton's laws 
hold true for all forces. 

The force of gravity acts between objects even without direct 
contact. Such objects may be separated by only a few meters of 
air, as is the case with the earth and a falling stone. Or they may 
be separated by many kilometers of empty space, as are artificial 
satellites and the earth. 

The symbol F^ is used for gravitational force. The magnitude of 
the gravitational pull F^ is roughly the same anywhere on the 
surface of the earth for a given object. When we wish to be veiy 
precise, we must take into account the facts that the earth is not 
exactly spherical and that there are irregularities in the makeup 
of the earth's ciust. These factors cause slight differences (up to 
0.5%) in the gravitational force on the same object at different 
places on the earth. An object having a mass of 1 kg will 
experience a gravitational force of 9.812 N in London, but only 
9.796 N in Denver, Colorado. Geologists make use of these 
variations in locating oil and othei- mineral deposits. 

The term weight is sometimes used in eveiyday conversation 
as if it meant the same thing as mass. This is quite wi^ong, of 
course. In physics, the weight of an object is defined as the 
magnitude of the gravitational force acting on the body. Your 
weight is the downward force the planet exerts on you whether 
you stand or sit, fly or fall, orbit the earth in a space vehicle, or 
merely stand on a scale to "weigh ' yourself. Only in interstellar 
space, far from planets, would you truly have no weight. 

Think for a moment about what a scale does. The spring in it 
compresses until it exerts on you an upward force strong 
enough to hold you up. So what the scale registers is really the 
force with which it pushes up on your feet. When you and the 
scale stand still and are not accelerating, the scale must be 
pushing up on your feet wdth a force equal in magnitude to your 
weight. That is why you are in equilibrium. The sum of the 
forces on you is zero. 

Now imagine for a moment a ridiculous but instructive 
thought experiment. As you stand on the scale, the floor (which, 
while sagging slightly, has been pushing up on the scale) 
suddenly gives way. You and the scale drop into a deep well in 
free fall. At every instant, your fall speed and the scale's fall 
speed will be equal, since you fall with the same acceleration. 




Alice falling down the rabbit hole, 
by Willy Pogany (19291. 



In a few physics books, weight is 
defined as the force needed to sup- 
port an object. In that case, you 
would be "weightless " if you were 
falling freely. We avoid this usage. 



CHAPTER 3 / THE BIRTH OF DYNAMICS 



85 




ConsidcM' SG 16 iigain. 



Some books 
stead of a . 



use the sxanbol 



g in- 



Vour feet now toiicli the scale only l)arel\ lif at alh. \ou look at 
the dial and see that the scale registers zero. This does not mean 
\oLi iia\e lost \'our weight: that could onl\ hapjjen if the earth 
suddenly disapjjeared or if yon were snckientK renuned to deep 
space. No, F^ still acts on you as hefore, accelerating you 
downward. But since the scale is acc(!lerating with you, you are 
no longer pushing dovxn on it, nor is it pushing up on voir 

You can experience firsthand the difference between the 
properties of WfMght and mass h\ h(jlding a hook. First, la\ the 
hook on yuuv hand; \'ou feel th(^ weight of the hook acting dowii. 
Next, grasp the hook and shake; it hack and forth. Wm still feel 
the weight downward, hut xou also feel how hard it is to 
acceleiate the hook bac'k and forth. This resistance to 
acceleration is the book's iiKMtia. \ou couki Cancer [hv 
sensation of tht> hook's \\riu,ht by hanging the book on a sti'ing, 
but the sensation of its inertia as \ ou shake it would remain the 
same. Ibis is only a crude dcnnonstration. More elaborate 
experiments would shov\', howexer, that weight can (change 
without any change of mass. Thus, when an astronaut on the 
moon's suiface uses a large (uuiiera, the camera is much easier 
to hold than on earth. In terms of tbt; moon's gra\it\', the 
camera's \veis,ht is an\y '/(; of its weight on earth. But its mass or 
inertia is not less, so it is as hard to swing the camer-a around 
suddenly into a new position on the moon as it is on earth. 

You c;an nov\- understand more clearK' the rt^sirlts of Galileo's 
experiment on falling objects. Galileo showed that any given 
object (at a gixen localitx i falls with unifor-m acceleration, a„. 
\\ hat is responsible tor- this uniform aci^eleration.^ Since the 
object is in free fall, the onl\ force acting on it is F,, due to the 
eaith's grax'ity. Nev\1on's second law allows us to relate this foi'ce 
to the acceleration a^ of the object. Applying the eqiraticjn F,,^., = 
nin to this case, where F^^,, = F^ and a and a^, we can write 

We can, of course, r'ewrite this ec|uation as 



From Newton's second law, \'ou can now see why the 
acceleration of a body in free fall is ccjnstant. Vhc reason is that, 
for- an object of gi\en mass ni, the graxitational force F^ oxer 
normal distances of fall is nearly constant. 

Cialileo, howexer-, did more than claim that exerA' object falls 
xvith cnnstnnt acceleration. Ik; k)irnd that at an\ one place all 
objects fall xxith the same uniform acc(;l(Mation. ScicMitists noxv 
knoxv that at the earth's surface this acceleration has [Uv \alue of 
[).H m/sec". Regardless of the mass m or xveight F^, all i)odi(;s in 
free fall (in the same locality) have the same acceleration a^. 



HH 



I MI 1 (;()\c;i;i»is oi motion 



Does this agree with the relation a^ = F^Jm? It does so only if 
F^ is directly proportional to mass m for eveiy object. In other 
words, if m is doubled, F^ must double; if m is tripled, F must 
triple. This is an important result indeed. Weight and mass are 
entirely different concepts. Weight is the gra\dtational force on an 
object (thus, weight is a vector). Mass is a measure of the 
resistance of an object to a change in its motion, a measure of 
inertia (thus, mass is a scalar). Yet, you have seen that different 
objects fall freely vvdth the same acceleration in any given locality. 
Thus, in the same locality, the magnitudes of these two quite 
different quantities are proportional. 



• 20. What quality of an object is measured by weight; what 
quality is measured by mass':^ Using both words and symbols, 
define weight and mass. 

21. The force pulling domi on a hammer is more than 20 
times the force pulling down on a nail. Why, then, do a 
hammer and a nail fall with nearly equal acceleration^ 

22. If a 30 -N force is applied to an object whose mass is 3 kg, 
what is the resulting acceleration^ If the same force is applied 
to the object on the moon, where the object's weight is one 
si}:th of its weight on earth, what is the acceleration':^ What is 
the acceleration in deep space where the object's weight is 
zero? 

23. An astronaut has left the earth and is orbiting the earth 
with a space vehicle. The acceleration due to gravity at that 
distance is half its value on the surface of the earth. Which of 
the following are true? 

(a) The astronaut's weight is zero. 

(b) The astronaut's mass is zero. 

(c) The astronaut s weight is half its original value. 

(d) The astronaut's mass is half its original value. 

(e) The astronaut's weight remains the same. 
(fj The astronaut's mass remains the same. 

3 .9 I Newton's third laiv of motion 

In his first law, Newton described the behavior of objects when 
they are in a state of equilibrium, that is, when the net force 
acting on them is zero. His second law explained how their 
motion changes when the net force is not zero. Newton's third 
law added a new and surprising insight into forces. 

Consider this problem: In a 100-m dash, an athlete goes from 
lest to nearly top speed in less than a second. We could measure 




Wiliuii Ihidolph at ttie start of the 
200-m sprint in which she set an 
Olympic record of 23.2 sec. 



CHAPTER 3 / THE BIRTH OF DYIVAMICS 



87 



The runner is, to be sure, pushing 
against the ground, but thai is a 
force acting un the ground. 



the runner's mass before the dash, and we could use high-speed 
photography to measure the initial acceleration. With mass and 
acceleration known, we could use F,,,., = ma to find the force 
acting on the injnner during the initial acceleration. But where 
does the force come from? It must have something to do with 
the iTjnner herself. Is it possible for her to exert a force on 
herself as a whole? Can she, for example, ever lift herself by her 
ov\Ti bootstraps? 

Newton's third law of motion helps explain just such puzzling 
situations. First, what does the third law claim? In \e\\1on's 
words: 



SG 27 



In the collision between the tennis 
ball and the racket, the force the 
ball exerts on the racket is equal 
and opposite to the force the 
racket everts on the ball. Both the 
racket and the ball are deformed 
bv the forces acting on them (see 
diagram). 




To every action there is always opposed an equal reaction: or, 
mutual actions of tvvo bodies upon each other are always equal 
and directed to contrary parts. 

This is a word-for-word translation fiom the Principia. In 
modern usage, however, we would use force where Nev\^on used 
the Latin word for action. So we could rewrite this passage as 
follows: If one object exerts a force on another, then the second 
also exerts a force on the first; these forces are equal in 
magnitude and opposite in direction. 

The most startling idea in this statement is that forces always 
exist in mirror- twin pairs and act on two different objects. 
Indeed, the idea of a single force acting without another force 
acting somewhere else is without any meaning whatsoever. 

Now apply this idea to the athlete. You now see that her act of 
pushing with her feet back against the ground (call it the action) 
also involves a push of the ground forward on her (call it the 
reaction). It is this reaction that propels her forward. In this and 
all other cases, it really makes no difference which force you call 
the action and which the reaction, because they occur at exactly 
the same time. The action does not "cause " the reaction. If the 
earth could not "push back " on her feet, the athlete could not 
push on the earth in the first place. Instead, she would slide 
around as orn slippery ice. Action and reaction coexist. You 
cannot have one without the other. Most important, the two 
forces are not acting on the same body. In a way, they are like 
debt and credit. One is impossible without the other; they ar^e 
equally large but of opposite sign, and the\' happen to two 
different objects. 

OrT this point Newton wrote: "VVhatexer dr^aws or' pr'esses 
another- is as much drawn or' pr'essed by that other-. If \'ou pi"ess 
a stone with your finger, the finger is also pressed by the stone." 
This statement suggests that for^ces always arise as a result of 
nurtual actions ("interactions "i between objects. If object A 
pushes or pulls on B, then at the same time object B pushes or 
}3ulls with precisely equal for^ce on A. These pair-ed pulls and 
pushes are always ecjual in magnitude, opjiosite in clireH'tioii <»/((/ 



88 



LMl 1 / CONCLPIS Ol .MOl IU\ 



on two different objects. Using the efficient shorthand of algebra 
to express this idea, whenever bodies A and B interact, 



F„, = 



This equation clearly sums up Newton's third law. A modern 
way to express it is as follows: Whenever two bodies interact, the 
forces thev e^iert on each other are equal in magnitude and 
opposite in direction. 

Every day you see hundreds of examples of this law at work. A 
boat is propelled by the water that pushes forward on the oar 
while the oar pushes back on the water. A car is set in motion by 
the push of the ground on the tires as they push back on the 
ground; when fiiction is not sufficient, the push on the tires 
cannot start the car forward. While accelerating a bullet foiAvard, 
a rifle experiences a recoil, or "kick. " A balloon shoots foiTvard 
while the air spurts out from it in the opposite direction. Many 
such effects are not easily observed. For example, when an apple 
falls, pulled down by its attraction to the earth, i.e., by its weight, 
the earth, in turn, accelerates upward slightly, pulled up by the 
attraction of the earth to the apple. 

Note what the third law does not say. The third law speaks of 
forces, not of the effects these forces produce. Thus, in the last 
example, the earth accelerates upward as the apple falls dovvTi. 
The force on each is equally large. But the accelerations 
produced by the forces are quite different. The mass of the earth 
is enormous, and so the earth's upward acceleration is far too 
small to notice. The third law also does not describe how the 
push or pull is applied, whether by contact or by magnetic 
action or by electrical action. Nor does the law require that the 
force be either an attraction or repulsion. The third law in fact 
does not depend on any particular kind of force. It applies 
equally to resting objects and to moving objects, to accelerating 
objects as well as to objects in uniform motion. It applies 
whether or not there is fiiction present. This universal nature of 
the third law makes it extremely valuable in physics. 





Force on ba I) due 
to racket 



For'c$ on racket 
due -to bull . 



foxi.e- on car'h 



force on mooh. 



The force on the moon owing to 
the earth is equal and opposite to 
the force on the earth owing to 
the moon. 



Z4. Two objects are next to one another; one object everts a 

force of 3 N to the right on the other object. Describe the 

three qualities of the second force that is immediately present 

according to Newton s third law. 

25. Identify the forces that act according to Newton's third law 

when a horse accelerates. When a swimmer moves at constant 

speed. 



CHAPTER 3 / THE BIRTH OF DVMAMICS 



89 



26. A piece of fishing line breaks if the force e^ierted on it is 
greater than 500 A'. Will the line break if two people at 
opposite ends of the line pull on it, each with a force of 300 N? 

27. State Newton's three laws of motion as clearly as vou can 
in your own words. 



3.1.0 I Using Neiiton's laivs of motion 

Each of Newton's three laws of motion has heen discussed in 
some detail. The first law emphasizes the modern point of view 
in the study of motion. It states that what requires explanation is 
not motion itself, but change of motion. The first law stresses that 
one must account for why an object speeds up or slows down 
or changes direction. The second law asserts that the rate of 
change of velocity of an object is related to both the mass of the 
object and the net force applied to it. In fact, the very meanings 
of force and mass are shown by the second law to be closely 

related to each other. The third law describes a relationship 

SG 29-31, 34 between interacting objects. 

Despite their individual importance, Newton's three laws are 
most powerful when they are used together. The mechanics 
based on Newton's laws was very successful. Indeed, until the 
late nineteenth century it seemed that the entire univer'se must 
be understood as "matter in motion. " Below are two specific 
examples that illustrate the use of these laws. 

E^cannple 1 

On September 12, 1966, a dramatic experiment based on 
Newton's second law was carried out high over the earth. From a 
previous space flight, a spent piece of an Agena r'ocket was 
quietly floating in its orbit ar^ound the Earth. In this experiment, 
the mass of the Agena piece was detemiined by acceler ating it 
with a push from a Gemini spacecraft. After the Gemini 
spacecraft made contact with the Agena rocket case, the 
thrusters on the Gemini were fired for 7.0 sec. These thrtisters 
wer-e set to give an average thrusting force of 890 N. The change 
in velocity of the spacecraft and Agena was found to be 0.93 m/ 
sec. The mass of the Gemini spacecr^aft was known to be about 
3,400 kg. The question to be answered was: What is the mass of 
the Agena? 

(Actually, the mass of the Agena was known ahead of time, 
from its constrtiction. But the purpose of the experiment was to 
develop a method for- finding the unknown mass of a foreign 
satellite in orbit. I 

In this case, a known force of magnitude 890 N was a( ting on 
two objects in contact, with a total mass of m, ,,,, where 

90 I Ml 1 ClONCLP^rS OF .MOI ION 



'^total ^Gemini """ ^Agena 



= 3,400 kg + m 



Agena 



The magnitude of the average acceleration produced by the 
thrust is found as follows: 









a 


0.93 nVsec 
7.0 sec 

— 0.13 nVsec 


Newton's 


second law 


gives 


the relation 




F - 


-- "l.o.al X 


a 



or 

^ = (^Agena + 3,400 kg) X R 

Solving for m^,„, gives 

F 890 N 

"^Afiena = " " 3,400 kg = ; - 3,400 kg 

^^""'^ a ° 0.13 m/sec- ^ 

= 6,900 kg - 3,400 kg 

= 3,500 kg 

The actual mass of the orbiting portion of the Agena, as 
previously determined, was about 3,660 kg. The method for 
finding the mass by pushing the Agena while in orbit therefore 
gave a result that was accurate to vvdthin 5% . This accuracy was 
well within the margin of error expected in making this 
measurement. 

Example 2 

Imagine taking a ride on an elevator: (A) At first it is at rest on SG 16 

the ground floor; (B) it accelerates upward unifonnly at 2 m/sec^ 
for a few seconds; then (C) it continues to go up at a constant 
speed of 5 m/sec. Suppose a 100-kg man (whose weight would 
therefore be about 1,000 N) is standing in the elevator. With what 
force is the elevator floor pushing up on him during (A), (B), 
and (O? 

Parts (A) and (C) are the same in terms of dynamics. Since the 
man is not accelerating, the net force on him must be zero. So 
the floor must be pushing up on him just as hard as gravity is 
pulling him dowoi. The gravitational force on him (his weight) is 
1,000 N. So the floor must be exerting an upward force of 
1,000 N. 

CHAPTER 3 / THE BIRTH OF DYMAMICS 91 



f „„. = o 



I '-y 



^net = 



(A) 



(B) 



(C) 



(AJ At rest (Bl Accelerating upward 
(C) Rising at constant speed 



In Part IBI, since the man is accelerating upward, there must 
be a net force upward on him. The unbalanced force is 



^ne, = ^^up 



= 100 kg X 2 m/sec' 
= 200 N 



SG 32 is an elaboration of a similar Clearly, the floor must be pushing up on him with a force 200 N 
example. For a difficult, worked- greater than what is required just to balanc:e his weight. 



out example, see SG 33. 



Therefore, the total force upward on him is 1,200 N. 



• 28. What force must the runner's legs produce in order to 
accelerate a 70-kg runner at 3 m/sec'? (Ignore air resistance or 
other effects.) 

29. An object with a mass of 3 kg is pulled down with a force 
of 29.4 N. What is the acceleration? What is the weight of the 
object? 

3*1.1. Nature's basic forces 



The study of Newton's laws of motion has increased your 
understanding of objects at rest, moxing uniformly, and 
accelerating. However, you have learned much more in the 
process. Newton's first law emphasized the importance of ftames 
of reference. In fact, an undeistaiiding of the relationship 



92 



L Mi 1 CONCLFIS Ol MO 1 ION 



between decriptions of the same event seen from different frames 
of reference was the necessary first step toward the theory of 
relativity. 

Newton's second law shows the fundamental importance of 
the concept of fo'^ce. It sayS; in effect, "When you observe 
acceleration, find the force!" This is how scientists were first 
made aware of gravitational force as an explanation of Galileo's 
kinematics. They discovered that, at a given place, a^ is constant 
for all objects. And since a = F^m by Newton's second law, the 
magnitude of F^ is always proportional to m. 

But this is only a partial solution. Why is F, proportional to m 
for all bodies at a given place? How does F, change for a given 
body as it is moved to places more distant ft^om the earth? Is 
there a "force law" connecting F^, m, and distance? As Unit 2 will 
show, there is indeed. Knowing that force law, you can 
understand all gravitational interactions among objects. 

Gravitational attraction is not the only basic force by which 
objects interact. However, there appear to be very few such basic 
forces. In fact, physicists now believe that eveiything obserx'ed 
in nature results from only four basic types of interactions. In 
terms of present understanding, all ev^ents of nature, from those 
among subnuclear particles to those among vast galaxies, involve 
one or more of only these few types of forces. There is, of course, 
nothing sacred about the number four. New discoveries or 
insights into present theories might increase or reduce the 
number. For example, two lor more) of the basic forces might 
someday be seen as arising from an even more basic force. 

The first of the four interactions is the graxatational force. This 
force becomes important only on a relatively large scale, when 
tremendous numbers of atoms of matter are involved. Between 
indi\ddual atoms, gravdtational force is extremely weak. But it is 
this v^ery weak force that literally holds the universe together. The 
second interaction involves electric and magnetic processes and 
is most important on the atomic and molecular scale. It is chiefly 
the electromagnetic force that holds together objects in the size 
range between an atom and a mountain. 

Scientists know the force laws governing gravitational and 
electromagnetic interactions. Therefore, these interactions are 
fairly well "understood. " Considerably less is known about the 
tvvo remaining basic interactions. They are the subject of much 
research today. The third interaction (the so-called 'strong " 
interaction) somehow holds the particles of the nucleus together. 
The fourth interaction (the so-called 'weak" interaction) governs 
certain reactions among subnuclear particles. 

There are, of course, other names for forces, but each of these 
forces belongs to one of the basic tv^Des. One of the most 
common is the "ftictional " force. Friction is thought to be an 
electrical interaction; that is, the atoms on the surfaces of the 



Einstein spent most of the latter 
hcdf of his life seeking a theory that 
would express gravitational and 
electromagnetic effects in a unified 
way. A satisfactory "unified field 
theory ' is stUl being sought. 

Recently, however, some suc- 
cesses have come in an entirely dif- 
ferent direction. A theory that con- 
siders electromagnetism and weak 
interactions to be aspects of the 
same fundamental force (in the 
way that electricity and magnetism 
are different aspects of the same 
force) has led to some interesting 
experimental predictions that have 
been verified. The details must stiU 
be worked out. 



CHAPTER 3 / THE BIRTH OF DWAMICS 



93 



"The Starr}' Night, " by Vincent Van 
Gogh. The intuitive feeling that all 
of nature's phenomena are interre- 
lated on a grand scale is shared 
bv scientists as well as artists. 



objects sliding or rubbing against each other are believed to 
interact electrically. 

You will encounter these ideas again. The gravitational force is 
covered in Unit 2, the electrical and magnetic forces in Units 4 
and 5, and the forces between nucleai- paiticles in Unit 6. In all 
of these cases, all objects subjected to a force behave in 
agreement with Newton's laws of motion, no matter what kind of 
force is involved. 

The knowledge that there are so few basic interactions is both 
surprising and encour^aging. It is surprising because, at first 
glance, the events all around us seem so varied and complex. It 
is encouraging because it makes the elusive goal of 
understanding the events of nature look more attainable. 




94 



UMiT 1 / coxcEPis oi ivio rio\ 



stnily 
guide 



1. The Project Physics learning materials 
particularly appropriate for Chapter 3 include the 
following: 

Experiments 

Newton's Second Law 
Mass and Weight 

Activities 

Checker Snapping 

Beaker and hammer 

Pulls and Jerks 

Experiencing Newton's Second Law 

Make One of These Accelerometers 

Film Loops 

Vector Addition: Velocity of a Boat 

2. The Aristotelian explanation of motion should 
not be dismissed lightly. Great intellects of the 
Benaissance, such as Leonardo da Vinci, who among 
other things designed devices for launching 
projectiles, did not challenge such explanations. One 
reason for the longevity of these ideas is that they 
are so closely aligned with our common-sense ideas. 

In what ways do your common-sense notions of 
motion agree with the Aristotelian ones? 

3. (a) Explain mechanics, dynamics, and 
kinematics. 

(b) Classify the followdng values as m, t, d, v, a, or/. 
Indicate whether they are scalars or vectors. 

(1) 5 m 

(2) 2 kg • m/sec' up 

(3) 10 m/sec 

(4) -8N 

(5) 5 kg 

(6) 5 m west 

(7) + 10 sec 



(8) 6 m/sec' left 

(9) 4 m/sec down 

4. A man walked 3 blocks north, 4 blocks east, 5 
blocks south, 1 block west, and 2 blocks north. 

(a) Where did he end up? 

(b) How far did he walk? 

(c) Which part, (a) or (b), is a vector problem? Which 
is a scalar problem? 

5. Three ants are struggling with a breadcrumb. 
One ant pulls toward the east v\ith a force of 8 units. 
Another pulls toward the north with a force of 6 
units, and the third puUs in a direction 30° south of 
west with a force of 12 units. 

(a) Using the "head-to-tail" construction of arrows, 
find whether the forces balance, or whether there is 
a net (unbalanced) force on the crumb. 

(b) If there is a net force, you can find its direction 
and magnitude by measuring the line drawn from 
the tail of the first arrow to the head of the last 
arrow. What is its magnitvide and direction? 

6. Show why the pcirallelogram method of adding 
arrows is geometrically equivalent to the head-to- 
tail method. 

7. A parachutist whose weight is 750 N falls with 
uniform motion. What is the size and direction of 
the force of air resistance? How do you know? 

8. There are many familiar situations in which the 
net force on a body is zero, and yet the body moves 
with a constant velocity. One example of such 
"dynamic equilibrium" is an automobile traveling at 
constant speed on a straight road. The force the road 
exerts on the tires is just balanced by the force of 
air friction. If the gas pedal is depressed further, the 
tires will push against the road harder, and the road 
will push against the tires harder. The car vvtII 
accelerate forward until the air friction builds up 
enough to balance the greater drive force. 

Give another example of a body moving with 
constant velocity under balanced forces. Specify the 
source of each force on the body and, as in the 
automobile example, explain how these forces could 
be changed to affect the body's motion. 



CHAPTER 3 / STUDY GUIDE 



95 



so 



9. Aristotle thought that objects in any kind of 
motion and objects at rest represented two different 
dynamical situations and had to bo explained 
separately. X'ewlon claimed tiiat oijjects in 
"equilibrium," moving or not, represented one 
dynamical situation. 

(a) What is equilibrium? 

(b) What are two possible states of motion for 
objects in equilibrium? 

10. (a) Vou exert a force on a box, but it does not 
move. How would you explain this? How might an 
Aristotelian explain it? 

(b) Suppose now that you exert a greater force and 
the box moves. Explain this from your (Newtonian) 
point of view and from an Aristotelian point of view. 

(c) You stop pushing on the box, and it quickly 
comes to rest. Explain this from both the Newlonian 
and the AiMstotelian points of view. 

11. (a) Assume that the floor of a bus could be 
made perfectly horizontc'U and perfectly smooth. A 
dry-ice puck is placed on the lloor and given a small 
push. Predict the way in which the puck would 
move. How would this motion differ if the whole bus 
were moving uniformly during the experiment? How 
would it differ if the whole bus were accelerating 
along a straight line? If the puck were seen to move 
in a curved path along the floor, how would you 
explain this? From these results, can you argue that 
uniform motion and rest reidly represent the same 
dynamical situation and are different from 
accelerated motion? 

(b) A man gently starts a dry-ice puck in motion 
while both are on a rotating platform. What will he 
report to be the motion he obsen'es as the platform 
keeps rotating? How will he explain what he sees if 
he believes he can use Nevvlon's first law to 
understand observations made in a rotating 
reference frame? Will he be right or v\Tong? 

12. (a) In terms of Newton's first law, explain: 

(1) why people in a moving car lurch forward 
when the car suddenly slows dovvai. 

(2) what happens to the passengers of a car that 
makes a shaip, quick turn. 



(b) When a coin is put on a phonograph turntable 
and the motor is started, does the coin fly off when 
the turntable reaches a certain speed? Why doesn't it 
fly off sooner? 

13. In an ac^tual experiment on applying the same 
force to different masses, how would you know it 
was the "same force"? 

14. (a) What three things does Newton's second law 
tell you about acceleration? 

(b) The acceleration of an object is 10 msec" north. 
In a second experiment, the force is divided in half 
and turned to the east, and the mass is reduced to 
one third. What is the object's acceleration now? 

15. Several proportionalities can be combined into 
an equation only if care is taken with the units in 
which the factors are exjiressed. When we vvTote Ad 
= V X Af in Chapter 1, we chose meters as units 
for d, seconds as units for t, and then made sure 
that the equation came out right by using meters per 
second as units for v. In other words, we let the 
equation define the unit for v. If we; had already 
chosen some other units for v, s£iy miles per hour, 
then we would have had to write instead something 
like 

Ad = /c X vAr 

where k is a constant factor that matches up the 
units of d, t, and v. What value would k have if d 
were measured in miles, t in seconds, and i' in miles 
per hour? 

Writing a = F^^Jm before we have defined units of 
F and m is not the very best mathematictd 
procedure. To be perfectly correct in expressing 
Newton's law, we would have had to write 

P 

a = k X ^^ 
m 

where k is a universally constant factor that woidd 
match up whatever units we choose for a, F, and m. 
The SI units have been chosen so that k = \. 

16. You can confirm the results of Example 2 in 
Sec. 3.10 by taking a bathroom scale into an (^levator. 
By how much does your registered "weight " seem 

to be increased when the elevator starts to go up 
(accelerates upward I? What happcms while it slows to 



Ub 



(;HAi>ri:i{ ;{ sri l)^ (;i ini; 



sg 



a stop? What happens when it goes up or down at 
constant speed? Does your weight recdly change? If 
not, why does the scale show what it does? 

How would all these measurements differ if this 
elevator were in a space vehicle in interstellar space? 

17. Ask your instructor for a simple spring balance, 
and examine hovA' it works. Then describe as a 
thought experiment how you could calibrate the 
spring balance in force units. What practical 
difficulties would you expect if you actually tried to 
do the experiment? 

IS. Hooke's law says that the force exerted by a 
stretched or compressed spring is directly 
proportional to the amount of the compression or 
extension. As Robert Hooke put it in announcing his 
discover}': 

. . . the power of any spring is in tlie same 
proportion with the tension thereof: that is, if one 
power stretch or bend it one space, two will bend 
it twO; three will bend it three, and so forward. 
Now as the theor\' is very short, so the way of 
trying it is ver\' easie. 

You can probably think immediately of how to test 
this law using springs and weights. 

(a) Tr\' designing such an experiment; then after 
checking with your instructor, carry it out. What 
limitations do you find to Hooke's law? 

(b) How could you use Hooke's law to simplify the 
calibration procedure asked for in SG 17? 



19. Complete this table: 



NET FORCE 

(N) 



MASS 

(kg) 



RESULTING 
ACCELERATION 

(m/secf) 



a 


1.0 down 


1.0 


1.0 down 


b 


24.0 west 


2.0 


12.0 west 


c 




3.0 


8.0 out 


d 




74.0 


0.2 left 


e 




0.0066 


130.0 north 


f 


72.0 in 




8.0 in 


g 


3.6 right 




12.0 right 


h 


1.3 up 




6.4 up 


1 


30.0 east 


10.0 




J 


0.5 left 


0.20 




k 


120.0 down 


48.0 





30. A rocket-sled has a mass of 4,440 kg and is 
propelled by a solid-propellent rocket motor of 
890,000-N thrust which burns for 3.9 sec. 

(a) What is the sled's average acceleration and 
maximum speed? 

(b) This sled has a maximum acceleration of 30 g ( = 
30 a). How can that be, considering the data given? 

(c) If the sled travels a distance of 1,530 m while 
attaining a top speed of 860 m/sec, what is its 
average acceleration? How did it attain that high a 
speed? 




21. Describe in detail the steps you would take in an 
idealized e.xjDeriment to determine the unknov\Ti 
mass m of a certain object (in kilograms) if you were 
given notliing but a frictionless horizontal plane, a 
1-kg standard, an uncalibrated spring balance, a 
meter stick, and a stopwatch. 

Z2. A block is dragged with constant velocity along a 
rough , horizontal tabletop by means of a spring 
balance horizontally attached to the block. The 
balance shows a reading of 0.40 N at this and any 
other constant velocity'. This means that the 
retarding fiictional force between block and table is 
0.40 N and is not dependent on speed. 

Now the block is pulled harder and given a 
constant acceleration of 0.85 m/sec'; the balance is 
found to read 2.1 N. Compute the mass of the block. 

23. We have claimed that any body in free fall is 
"weightless" because any weight-measuring device 
falling with it would read zero. This is not an entirely 
satisfactoiy explanation, because you feel a definite 
sensation during free fall that is exacth' the same 



CHAPTER 3 / STUDY GUIDE 



97 



Si 



sensation you would feel if you were truly without 
weight, for example, deep in space far from any star 
or planet. iVoii feel the same sensation on jumping 
off a roof or a di\ing board, or when somt^one pulls 
a chair out from under you.) Can you explain why 
your insides react in the same w£iy to lack of \\'eight 
and to free fall? 

24. (a) A replica of the standard kilogram is 
constructed in Paris and then sent to the \ationaI 
Bureau of Standards near Washington, D.C. 
Assuming that this secondary' standard is not 
damaged in transit . what is 

(1) its mass in Washington? 

(2) its weight in I'aris and in Washington? (in Paris, 
a, = 9.81 m/sec"; in Washington, a, = 9.80 m/sec~.l 

(b) What is the change in your own weight as you 
go from Paris to Washington? 

25. You have probably seen signs that niisleadingly 
convert people's weights from pounds to kilograms. 

(a) Since the pound is a unit of force, why are these 
signs wrong? Explain why they are only valid on 
earth land then only approximately!. 

(b) Calculate your mass in kilograms and your 
weight in newtons. 

(c) How much force is needed to accelerate you Im/ 
sec^? How many kilograms can you lift? How many 
newtons of force must you apply to do this? 

26. Why is it often said that astronauts in orbit 
around a planet or satellite iire "weightless"? 

27. Quite apart from pushing dov\Ti on the ground 
owing to a runner's own weiglit, the sole of a 
runner's shoe pushes on the earth in a horizontal 
direction and the earth pushes with an eqiuil and 
opposite force on the sole of the shoe. This latter 
force has an accelerating effect on the runner, liut 
what does the force acting on the earth do to the 
earth? From N'evx'ton's second law, we would 
conclude that such an unbalanced force would 
accelerate the earth. The mass of the earth is xen' 
great, however, so the acceleration caused by the 
runner is very small. A reasonable value for the 
average acceleration of a runner is 5m/sec" , and a 
reasonable value for the runner's mass would be (SO 



kg. The mass of the earth is approximated 60 x lO'^ 
kg. 

(a) What acceleration of the earth would the runner 
cause? 

(b) If the acceleration lasts for 2 sec, what speed will 
the runner have reached? 

(c) What speed wiU the earth have reached? 

28. A boy of mass 70 kg and a girl of mass 40 kg are 
on ice skates holding opposite ends of a 10-m rope. 
The boy pulls on tlu; rojje toward hims(;lf with a 
force of 80 \. Assuming that there is \ irtually no 
friction between the skates and the ice surface, what 
is the girl's acceleration? According to Mewlon's third 
law, what is the force on the box? What is his 
acceleration? 

29. In terms of X'ewlon's third law, assess the 
following statements: 

(a) You are standing perfectly still on the ground; 
therefore you and the earth exert equal and opposite 
forces on each other. 

(b) The reason that a propeller airplane cannot fl\' 
abo\e the atmosphere is that there is no air to push 
one way while the plane goes the other. 

(cl Object A rests on object B. The mass of object A 
is 100 times as great as that of object B, but even so, 
the force A exerts on B is no greater than the force 
of B on A. 

30. (Consider a tractor pulling a heavy log in a 
straight line. On the basis of \ewtons third law, one 
might argue that the log pulls back on the tractor 
just as strongly as the tractor pulls the log. But why, 
then, does the trat'tor move? (Make a large drawing 
of the tractor, rop(!, log, and the earth, and (Miter 

all the forc(;s acting on each.i 

31. (Consider the system consisting of a 1.0-kg ball 
and the earth. The ball is dropped from a short 
distance above the ground and falls freely. .Assuming 
that the mass of the earth is 0.0 x lO" kg. 

(a) make a vector diagram illustrating the important 
forces acting on each member of the system. 

(b) calculate the acceleration of the earth in this 
interaction. 



98 



CJIAPIhli A Sll I3\ (il IDi; 



sg 



(c) find the ratio of the magnitude of the ball's 
acceleration to that of the earth's acceleration iajaj. 

(d) make a vector diagram as in (a) but shovAing the 
situation when the ball has come to rest after hitting 
the ground. 

32. (a) A 75-kg person stands in an elevator. What 
force does the floor exert on the person when the 
elevator 

(1) starts moving upward with an acceleration of 
1.5 m/sec"? 

(2) moves upward with a constant speed of 2.0 m/ 
sec? 

(3) starts accelerating downward at 1.5 m/sec"? 

(b) If the person were standing on a bathroom 
(spring! scale during the ride, what readings would 
the scale have under conditions (1), (21, and (3) above? 

(c) It is sometimes said that the "apparent weight " 
changes when the elevator accelerates. What could 
this mean? Does the weight really change? 

33. Useful hints for solving problems about the 
motion of an object and the forces acting on it: 

(a) Make a light sketch of the physical situation. 

lb) In heax/y line, indicate the liinits of the pai'ticular 
object you are interested in, and draw all the forces 
acting on that object. (For each force acting on it, 
it wiU be exerting an opposite force on something 
else, but you can ignore these forces.) 

(c) Find the vector sum of all these forces, for 
example, by graphical construction. 

(d) Using Newton's second law, set this sum, F^^, 
equal to ma. 

(e) Solve the equation for the unknown quantity. 

(f) Put in the numerical values you know, and 
calculate the answer. 

Example: 

A ketchup bottle whose mass is 1.0 kg rests on a 
table. If the friction force between the table and the 
bottle is a constant 3 N, what horizontcil pull is 
required to accelerate the bottle from rest to a speed 
of 6 m/sec in 2 sec? 

First, sketch the situation: 

Second, draw in arrows to represent all the forces 




'■/: 




1 




'^ 5 


■^^w 







acting on the object of interest. There will be the 
horizontal pull F, the friction F^, the gravitational 
pull F (the bottle's weight), and the upward force F, 
exerted by the table. (There is, of course, also a force 
acting down on the table, but you are interested only 
in the forces acting on the bottle.) 

Next, draw the arrows alone. In this sketch all the 
forces can be considered to be acting on the center 
of mass of the object. 



A > 



Because the bottle is not accelerating up or down, 
you know there is no net force up or down. 
Therefore, F, must just balance F,. The net force 
acting on the bottle is just the vector sum of F and 
F^. Using the usual tip-to-tail addition: 



Ff 



'? 



As the last arrow diagram shows, the horizontal pull 
must be greater than the force required for 
acceleration by an amount equal to the friction. You 
already know F^. You can find F, , from Newton's 
second law if you know the mass and acceleration of 
the bottle, since F ^^, = ma. The net force required 



CHAPTER 3 / STUDY GUIDE 



99 



ss 



to accelerate the bottle is found from Mevvton's 
second law: 

I'he mass m is ^i\cn as 1.0 k^. i'lic; acceleration 
iinoKcd in going from rest to 6.0 nVsec in 2 sec is 

Ai' 6.0 m sec 

a = — - = 3.0 m/sec" 

Af 2 sec 

So the net force required is 

F . = 1.0 kg X 3.0 m/sec^ 

= 3.0 kg m/sec' 

= 3.0 N 

If you consider toward the right to be the positi\ e 
direction, F^ , is 3.0 W and h], which is directed to 
the left, is -3.0 \. 

F , = F + F, 

net p J 

3.0 \' = F + (-3.0N) 

I' 

F = 3.0 N + 3.0 N 

p 

F = 6.0 i\ 

p 

If you prefer not to use + and - signs, you can 
work directly from your final diagram and use only 
the magnitudes of the forces: 

' ■> 
^^ 
from which the magnitude of/-', is obviously 6.0 \. 



34. (a) Two forces act on an object of 5 kg mass. 
One force is 20 N right and the other is 5 N left. How 
far mil the object move in 10 sec? 

(b) A box weighing 500 i\ can be moved across the 
floor in uniform motion by a force of 200 X. If the 
force is suddenly increased to 1,200 \, what will be 
tin; spe(!d of the box afteu- 20 s(u ? 

(c) What is the net force on an object of 4 kg mass if 
its speed is changed from 40 m/sec to 80 m/sec in 

10 sec? 

(d) A 40-N force acts on an object. This force 
overcomes a li)-\ frictional force and accelerates the 
object so that its speed increases by 55 m/sec in 1 1 
sec. What is the mass of the object? 

(e) A 5-kg object experiences an 80-i\ force on the 
earth's surface. What is the acceleration of the 
object? If the weight of the object is reduced by one 
half, what is its acceleration gi\'en the same 80-\ 
force? 

(f) 'I\vo wooden blocks resting on a table top ha\e a 
coiled spring b(;tween them. When the spring is 
released, it exerts a 40-\ force on each block. If one 
block is three times as massive as the other block 
(whose mass is 3 kg I how long will it take the blocks 
to mo\'e 200 m apiirt? I The frictional force is 
estimated at 4 N.I 

(g) A 6-kg block is pulled along the floor whose 
frictional force is estimated at 3 \. Forces of 18 i\ 
right and 15 N left are exerted on the block 
simultiineously. What is the acceleration of the 
block? 



lOO 



CM API hu 3 sum (a inij 




Understanilins Motion 

4.1 A trip to the moon 

4J3 Projectile motion 

4.3 Vlliat is the path of a projectile? 

4.4 Moving frames of reference: Galilean relatiiity 

4.5 Circular motion 

4.6 Centripetal acceleration and centripetal force 

4.7 The motion of earth satellites 

4.8 nliat about other motions? Retrospect and prospect 



4r*l. I A trip to the moon 

Imagine a Saturn rocket taking off from its launching pad at the 
Kennedy Space Flight Center on Cape Canaveral. It climbs above 
the earth, passing through the atmosphere and beyond. 
Successive stages of the rocket shut off, finally leaving a capsule 
hurtling through the near- vacuum of space. Approximately 65 
hr after takeoff, the capsule reaches its destination 384,000 km 
away. It circles the moon and descends to its target, the center 
of the lunar crater Copernicus. 

The complexity of such a voyage is enormous. To direct and 
guide the flight, a great number and variety of factors must be 
taken into account. The atmospheric drag in the early part of the 
flight depends upon the rocket's speed and altitude. The engine 
thrust changes wdth time. The gravitational pulls of the sun, the 
earth, and the moon change as the capsule changes its position 
relative to them. The rocket's mass changes as it burns fuel. 
Moreover, it is launched from a spinning earth, which in turn is 




= -^ ^ -- 



In his science-fiction novels of 
more than a hundred years ago, 
the French author Jules Verne 
(1828-1905) launched three space- 
men to the moon by means of a 
gigantic charge fised in a steel pipe 
deep in the earth. 



SG 1 



CHAPTER 4 / UNDERSTANDING MOTION 



lOl 




circling the sun. Meanwhile, the moon is moving around the 
eailh at a speed of about 1,000 m/sec relative to the eai'th. 

Yet, like almost any complex motion, the flight can be broken 
down into small portions, each of which is relatively simple. 
What you have learned in earlier chapters uill be useful in this 
task. 

In simplified form, the earth-moon trip can be divided into 
eight pails or steps: 

Step 1. The rocket accelerates vertically upward from the surface 
of the earth. The force acting on the rocket is not really 
constant, and the mass of the rocket decreases as the 
fuel burns. The value of the acceleration at any instant 
can be computed by using Newton's second law. The 
value is given by the ratio of net force (thnjst minus 
weight) at that instant to the mass at that instant. 

Step 2. The rocket, still accelerating, follows a curved path as it is 
"injected" into an orbit about the earth. 

Step 3. In an orbit 185 km above the earth's surface, the capsule 
moves in a nearly circular arc. Its speed is constant at 
7,769 m/sec. 

Step 4. The rocket engines are fired again, increasing the 

capsule's speed so that it follows a much less cuived 
path into space. (The minimum speed necessary to 
escape the earth completely is 11,027 m/sec.) 




\ 



Step 5. In the flight between the ear-th and moon, occasional 

short bursts from the capsule's rockets keep it precisely 
on course. Between these correction thrusts, the capsule 



102 



LM 1 1 COVCEPTS OF AlOllUN 



moves under the influence of the gravitational forces of 
earth, moon, and sun. You know from Newton's first law 
that the capsule would move wdth constant velocity if it 
were not for these forces. 

Step 6. On nearing the moon, the rocket engines are fired again 
to give the capsule the correct velocity to "inject " into a 
circular orbit around the moon. 

Step 7. The capsule moves with a constant speed in a nearly 
circular path 80 km above the moon's surface. 

Step 8. The rocket engines are fired into the direction of motion, 
to reduce the speed. The capsule then accelerates 
downward as it falls toward the surface of the moon. It 
follows an arcing path toward a landing at the chosen 
site. (Just before impact, the rocket engines fire a final 
time to reduce speed of fall and prevent a hard landing.) 




Motion along a straight line (as in Steps 1 and 5) is easy 
enough to describe. However, it is useful to analyze in greater 
detail other parts of this trip. Motion in a circular arc, as in Steps 
3 and 7, and projectile motion, as in Step 8, are two important 
cases. 

How can you go about making this analysis? Followdng the 
example of Galileo and Newton, you can learn about motions 
beyond your reach, even on the moon or in the farthest parts of 
the universe, by studying motions near at hand. If physics is the 
same everywhere, the path of a lunar capsule moving as in Step 
8 can be understood by studying a bullet fired ft^om a horizontal 
rifle. 



SG 2 



4r.2 I Prcijectile motion 

Imagine an experiment in which a rifle is mounted on a tower 
with its barrel parallel to the ground. The ground over which the 
bullet will travel is level for a great distance. At the instant a 
bullet leaves the rifle, an identical buUet is dropped from the 
height of the rifle's barrel. This second bullet has no horizontal 
motion relative to the ground; it goes only straight down. Which 
bullet will reach the ground first? 



CHAPTER 4 / UNDERSTANDING MOTION 



103 




■V ;lv :<i^»- 7.'. . .',v-^;-.. < -•-5.. /■ • 







6rv> -< 




You do not need to know anv'thing about the speed of the 
bullet or the height of the tower in order to answer this question. 
Consider first the motion of the second bullet, the one that is 
dropped. As a freely falling object, it accelerates toward the 
ground with constant acceleration. As it falls, the elapsed time 
inteival Af and the corresponding downward displacement Ay 
are related by the equation 

Ay = la^Af^ 

where a^ is the acceleration due to gravity' at that location. 
Following usual practice, you can now drop the A S3anbols, but 
keep in mind that y and t stand for displacement and time 
interval, respectively. So you can write the last equation as 



-ar 




Nov\' consider the bullet that is fired horizontally from the rifle. 
When the gun fires, the bullet is driven by the force of expanding 
gases. It accelerates very rapidly until it reaches the muzzle of 
the rifle. On reaching the muzzle, the gases escape and no longer 
push the bullet. At that moment, however, the bullet has a great 
horizontal speed, v^. The air slows the bullet slightly, but you can 
ignore that fact and imagine an ideal case with no air friction. 
As long as air ftiction is ignored, there is no horizontal force 
acting on the projectile. Therefore, the horizontal speed will 
remain constant. From the instant the bullet leaves the muzzle, 
its horizontal motion is described by the following equation 
involving the horizontal displacement A^c: 



or again dropping the As, 



A^: = V Af 



?c = vj 



These equations describe the forward part of the motion of the 
bullet. There is, however, another part that becomes more and 
more important as t increases. From the moment the bullet 
leaves the gun, it falls toward the earth while it moves forward, 



104 



IM I 1 COXCLPIS Ol ,M()ri()\ 



like any other unsupported body. Can you use the same 
equation to describe its fall that you used to describe the fall of 
the dropped bullet? How will falling affect the bullet's horizontal 
motion? These questions raise a more fundamental one which 
goes beyond just the behavior of the bullets in this experiment. 
Is the vertical motion of an object affected by its horizontal 
motion or vice versa? 

To answer these questions, you could carry out a real 
experiment similar to the thought experiment. A special 
laboratory device which fires a ball horizontally and at the same 
moment releases a second ball to fall freely can be used. Set up 
the apparatus so that both balls are the same height above a 
level floor and release them at exactly the same time. Although 
the motion of the balls may be too rapid to follow with the eye, 
your ears will tell you that they do in fact reach the floor at the 
same time. This result suggests that the vertical motion of the 
projected ball is unaffected by its horizontal motion. 

In the margin is a stroboscopic photograph of this experiment. 
Equally spaced horizontal lines aid the examination of the two 
motions. Look first at the ball on the left, the one that was 
released without any horizontal motion. You can see that it 
accelerates because it moves a greater distance between 
successive flashes of the strobe's light. Careful measurement of 
the photograph shows that the acceleration is constant, within 
the uncertainty of the measurements. 

Now, compare the vertical positions of the second ball, fired 
horizontally, wdth the vertical positions of the ball that is falling 
freely. The horizontal lines show that the distances of fall are the 
same for corresponding time inteivals. The two balls obey the 
same law for motion in a vertical direction. That is, at every 
instant they both have the same constant acceleration a^, the 
same downward velocity, and the same vertical displacement. 
Therefore, the experiment supports the idea that the vertical 
motion is the same whether or not the ball has a horizontal 
motion also. The horizontal motion does not affect the vertical 
motion. 

You can also use the strobe photo to see if the vertical motion 
of the projectile affects its horizontal velocity. Do this by 
measuring the horizontal distance between images. You vvdll find 
that the horizontal distances are practically equal. Since the time 
intervals between images are equal, you can conclude that the 
horizontal velocity v^ is constant. Therefore, the vertical motion 
does not affect the horizontal motion. 

The experiment shows that the vertical and horizontal parts, 
or components, of the motion are independent of each other. 
This experiment can be repeated from different heights and with 
different horizontal velocities, but the results lead to the same 
conclusion. 








9r "■* 

^'< ' "" 

In one uf the most famous allegori- 
cal frontispieces of Renaissance 
science, from Nova Scientia (1537) 
by Nicola Tartaglia, Euclid greets 
students at the outer gate of the 
circle of knowledge. The fired can- 
non and mortar show the trajecto- 
ries defined by Tartaglia. Plato and 
Aristotle are shown in the inner 
circle. 




The two balls in this stroboscopic 
photograph were released simulta- 
neously. The one on the left was 
simply dropped from a rest posi- 
tion; the one on the right was given 
an initial velocity in the horizontal 
direction. 



CHAPTER 4 / UNDERSTANDING MOTION 



105 



SG 3 



SG 4 



The independence of horizontal and \ ertical motions has 
important consequences. For example, it allows you to piedict 
the displacement and the velocity of a projectile at any time 
during its flight. You need merely to consider the horizontal and 
vertical aspects of the motion separately, and then add the 
results by the vector method. You can predict the magnitude of 
the components of displacement ;c and y and the components of 
velocity' v^ and v at any instant by applying the appropriate 
equations. For the horizontal component of motion, the 
equations are 



constant 



and 



and for the vertical component of motion, 




\ 



v^ 



1. tlow do you know that it is correct to simply break down 
complicated motion into separate vertical and horizontal 
components in the case described in Sec. 4.2? 

2. If a body falls from rest with acceleration a^, with what 
acceleration will it fall if it also has an initial horizontal 
speed v^? 




4.3 I llliat is the path of a prcijectile? 

it is easy to see that a thrown object, such as a lock, follows a 
curved path. But it is not as easy to see just what kind of cuia e it 
traces. Arcs of circles, ellipses, parabolas, hvperbolas, and 
cycloids (to name only a few geometric figures i all pio\ide likely 
looking cuived paths. 

Eai'ly scientists gained better" knowledge about the path of a 
pr'ojectile when they applied mathematics to the problem. This 



106 



I'M I 1 COXCliPi S OF MO ri()\ 



was done by deriving the equation that expresses the shape of 
the path. Only a few steps are involved. First list equations you 
already know for a projectile launched horizontally: 



and 



vt 



V = -aj- 



You could plot the shape of the path, or trajectory as it is often 
called in physics, if you had an equation that gave the value of y 
for each value of ,y. You could find the fall distance y for any 
horizontal distance ,v by combining these two equations in a way 
that eliminates the time variable. Solving the equation ;c = vj for 
t gives 

t = — 

Because t means the same in both equations, you can substitute 
^/v for t in the equation for y: 



y 



T^r 



and thus, 



y = -3« 



This last equation is a specialized equation of the kind that 
need not be memorized. It contains two variables of interest, ,v 
and y. It also contains three constant quantities: the number V-i, 
the unifomi acceleration of free fall a^, and the horizontal speed 
v^, which is constant for any one flight, from launching to the 
end. The vertical distance y that the projectile falls is thus a 
constant times the square of the horizontal displacement ^. In 
other words, the two quantities y and }C are proportional: y ^ }C. 

Thus, there is a fairly simple relationship between x and y for 
the trajectory. For example, when the projectile goes twice as far 
horizontally from the launching point, it drops vertically four 
times as far. 

The mathematical curve represented by this relationship 
between ;c and y is called a parabola. Galileo deduced the 
parabolic shape of trajectories by an argument similar to the one 
used here. (Even projectiles not launched horizontally, as in the 
photographs on page 106, have parabolic trajectories.) This 
discovery greatly simplified the study of projectile motion, 
because the geometiy of the parabola had been established 
centuries earlier by Greek mathematicians. 





« 


6 


f < 


• 


^ 


o . 


^ 


/ 


1 


^^^""^ 


^-— ^^' 




I 

rv 









































Drawing of a parabolic trajectory 
from Galileo 's Two New Sciences. 



CHAPTER 4 / UNDERSTANDING MOTION 



107 



SG 5-10 



Here is a clue to one of the important strategies in modern 
science. Whenever possihle, scientists ^^\press the leatiii-es of a 
phenomenon quantitatively and put the relations between them 
into equation form. Then the rules of mathemati('s can be used 
to shift and suljstitute terms, opening the way to unexpected 
insights. 






SJ 

^<^ 



m 



trrrst 






M, 



.ff. 

The critics of Galileo claimed that 
if the earth moved, a dropped 
stone would be left behind and 
land hevond the foot of the tower. 



m 



m 



p3 






Galileo argued that the falling 
stone continued to share the mo- 
tion of the earth, so that an ob- 
ser\'er on earth could not tell 
whether or not the earth moved by 
watching the stone. 



• 3. Rewrite the steps in Sec. 4.3 yourself, defining all the 
variables and e;<plaining each step and each equation used. 

4. Which of the conditions below must hold in order for the 
relationship y ^ ^' to describe the path of a projectile? (a) a^ is 
a constant (b) a^ depends on t (c) a^ is straight down (d) v^ 
depends on t (e) air friction is negligible 

5. How far, vertically and horizontally, will a projectile travel 
in 10 sec if it is launched with an initial horizontal speed of 
4 m/sec? (For simplicity, use a_, as approximately 10 m/sec^.) 

4.4 I Moving frames of reference: 
Galilean relativity 

Galileo's work on projectiles leads to thinking about reference 
frames. As you will see in Unit 2, Galileo strongly supporled the 
idea that the proper reference frame for discussing motions in 
our planetary system is one fixed to the sun, not the earth. From 
that point of view, the earth both revolves around the sun and 
rotates on its oun axis. For many scientists of Galileo's time, this 
idea was impossible to accept, and they thought they could 
prove their case. If the earth rotated, they said, a stone dropped 
fi'om a tower would not land directly at the tower's base. For if 
the earth rotates once a day, the tower would move onward 
hundreds of meters for every second the stone is falling. The 
stone would be left behind while falling through the air and so 
would land far away from the base of the tower. But this is not 
what happens. As nearly as one can tell, the stone lands directly 
below the point of release. Therefore, many of Galileo s critics 
believed that the tower and the earth could not possibly be in 
motion. 

To answer these arguments, Galileo used the same example to 
support his own view. During the time of fall, Galileo said, the 
tower and the ground supporting it move fonvaixi together with 
the same unifomi velocity. While the stone is held at the top of 
the tower, it has the same horizontal velocity as the tower. 
Releasing the stone allows it to gain veitical speed. But by the 
principle of independence of v^ and v; discussed in Sec. 4.2, this 
vertical component does not diminish any horizontal speed the 
stone had on being released. In other words, the falling stone 
behaves like any other projectile. The horizontal and vcilical 



108 



LMT 1 c(),\c;i;prs oi ,\ionc).\ 



components of its motion are independent of each other. Since 
the stone and tower continue to have the same v^ throughout, 
the stone will not be left behind as it falls. Therefore, no matter 
what the speed of the earth, the stone will land at the foot of the 
tower. So the fact that falling stones are not left behind does not 
prove that the earth is standing still. 

Similarly, Galileo said, an object released from a crow's nest at 
the top of a ship's mast lands at the foot of the mast, whether 
the ship is standing still or moving with constant velocity in calm 
water. This was actually tested by experiment in 1642 (and is also 
the subject of three Project Physics film loops). Many everyday 
observations support this view. For example, when you drop or 
throw a book in a bus or train or plane that is moving with 
constant velocity, you see the book move just as it would if the 
vehicle were standing still. Similarly, if the udnd is small enough 
not to interfere, an object projected vertically upward fi om inside 
an open truck moving at constant velocity udll fall back into the 
truck. A person in the truck sees the same thing happen whether 
the truck is moving at constant velocity or standing still. 

These and other observations lead to a valuable generalization: 
If Newton's laws hold in any one laboratory, then they will hold 
equally well in any other laboratoiy (or "reference frame ") 
moving at constant velocity with respect to the first. This 
generalization is called the Galilean relativity principle. It holds 
true for all "classical " mechanical phenomena, that is, 
phenomena where the relative velocities are in the range from 
almost negligible up to millions of kilometers per hour. 

Even if the laws of mechanics are the same for all reference 
frames moving wdth constant velocity vvdth respect to each other, 
a problem still arises. Namely, there is no way to find the speed 
of one's own reference frame from any mechanical experiment 
done within that frame. Nor can one pick out any one reference 
frame as the 'true " frame, the one that is, say, at absolute rest. 
Thus, there can be no such thing as the "absolute" velocity of a 
body. All measured velocities are only relative. 

What about observations of phenomena outside of one's own 
frame of reference? Certainly some outside phenomena appear 
differently to observers in different reference frames. For 
example, the observed velocity of an airplane will have a value 
when measured from the earth different from that when 
measured from a moving ship. Other quantities, such as mass, 
acceleration, and time interval, have the same values when 
measured from different reference frames moving with constant 
velocity with respect to one another. Moreover, certain 
relationships among such measurements will be the same for 
these different reference frames. Newton's laws of motion are 
examples of such "invariant " relationships, and so are all the 
laws of mechanics that follow from them. 



At high speeds, air drag will affect 
the results considerably. The situ- 
ation is still distinguishable from a 
car at rest, but in a high wind! 



CHAPTER 4 / UNDERSTANDIIVG MOTION 



109 



when rclath'o speeds become u no- 
ticeable fraction of the s|}eed of 
light (approximately 300, 000 km seel, 
som(! {l(n iations from this simph; 
relatix it\ i)rin(ipl(' begin to appeal'. 
We will consider soriu; of them in 
Unit 'i. 



SG 11-13 



Notice that the relativaty principle, even in this restricted, 
classical foim, does not say "everything is relative. ' On the 
contrary, it asks you to look for i elationships that do not change 
when vou transfer vour attention from one frame to another. 



• 6. If the laws of mechanics are found to be the same in two 
reference frames, what must he true of the motions of those 
frames? 

7. An outfielder running at constant speed under a falling ball 
sees the ball falling straight down to her. What is the path of 
the ball as seen by someone in the stands? Explain how this 
common experience supports (a) Galilean relativity and (b) the 
breakdown of projectile motion into independent horizontal 
and vertical parts. 



In discussing circular motion it is 
useful to keep clearly in mind a dis- 
tinction between revolution and ro- 
tation. We define these terms dif- 
ferently: Re\'olution is the act of 
traxeling along a circular or ellip- 
tical path; rotation is the act of 
spinning rather than traveling. A 
point on th(' rim of a phonograph 
turntable travels a long way; it is 
revolving about the axis of the 
turntable. But the turntable as a 
unit does not move from place to 
place; it merely rotates. In some 
situations both processes occur si- 
multan(!ously; for examph;, the 
earth rotates about its own axis, 
while it also rexohcs lin a nearly 
circular path) around the sun. 



4r.5 I Circular motion 

A pixjjectile launched horizontally from a tall tower strikes the 
earth at a point determined by three factors. These factors are 
the horizontal speed of the projectile, the height of the tower, 
and the acceleration due to the force of gravity. As the 
projectile's launch speed is increased, it strikes the earth at 
points farther and farther from the tower's base. It is then no 
longer true that the trajectory is a simple parabola, because the 
force of gravity does not pull on the projectile in the same 
direction throughout the path. Eventually, then, you would have 
to consider a fourth factor: The earth is not flat but cuned. If the 
launch speed were increased even more, the projectile would 
strike the earth at points even farther from the tower, and at last 
it would iTJsh around the earth in a nearly circular orbit. (See the 
quotation from Neuron, page 112.) At the launching speed that 
puts the projectile into orbit, the fall of the projectile away from 
straight-line forward motion is just matched by the cuivature of 
the earth's surface. Therefore, the projectile stays at a constant 
distance above the surface. 

What horizontal launch speed is required to put an object into 
a circular orbit about the earth or the moon.'' Vou will be able to 
answer this question quite easily after you have learned about 
circular motion. 

The simplest kind of circailar motion is uniform circular 
motion, that is, motion in a circle at constant speed. If you are in 
a car that goes around a perfectly circular track so that at eveiy 
instant the speedometer reading is 64 km/hr, you are in unifomi 
circular motion. This is not the case if the track is of any shape 
other than circular or if your speed changes at an\' point. 



110 



L!,M T 1 co\c:L;i»rs oi mo nox 



How can you find out if an object in circular motion is moving 
at constant speed? You can apply the same test used in deciding 
whether or not an object traveling in a straight line does so with 
constant speed. That is, measure the instantaneous speed at 
many different moments and see w^hether the values are the 
same. If the speed is constant, you can describe the circular 
motion of any object by means of two numbers: the radius R of 
the circle and the speed v along the path. For regularly repeated 
circular motion, you can use a quantity more easily measured 
than speed: either the time required by an object to make one 
complete revolution, or the number of revolutions the object 
completes in a unit of time. The time required for an object to 
complete one revolution in a circular path is called the period (T) 
of the motion. The number of revolutions completed by the same 
object in a unit time interval is called the ^equencv if) of the 
motion. 

An an example, these terms are used to describe a car moving 
vWth uniform speed on a circular track. Suppose the car takes 
20 sec to make one lap around the track. Thus, T - 20 sec. On 
the other hand, the car makes three laps in 1 min. Thus,/ = 3 
revolutions per minute, or/ = 1/20 revolution per second. The 
relationship between frequency and period (when the same time 
unit is used) is / = 1/T. If the period of the car is 20 sec/rev, then 
the frequency is given by 

11 1 

f = = — rev/sec = — Hz 

20 sec/rev 20 20 

All units are a matter of convenience. Radius may be expressed 
in terms of centimeters, kilometers, or any other distance unit. 
Period may be expressed in seconds, minutes, years, or any other 
time unit. Frequency may be expressed as "per second," "per 
minute," "per year, " and so on. The most widely used units of 
radius, period, and frequency in scientific work are, respectively, 
meter, second, and hertz (or per second). 



TABLE 4.1 COMPARISON OF THE FREQUENCY AND PERIOD FOR VARIOUS 
KINDS OF CIRCULAR MOTION. 



PHENOMENON 


Period (T) 


Frequency (f) 


Electron in circular accelerator 


10' 


sec 


10' Hz (per sec) 


Ultra-centrifuge 


0.00033 


1 sec 


3000 Hz 


Hoover Dam turbine 


0.33 


sec 


3 Hz 


Rotation of earth 


24 


hr 


0.0007 per min 


Moon around the earth 


27 


days 


0.0015 per hour 


Earth around the sun 


365 


days 


0.0027 per day 



Note: always use t for time, T for 
period, and in general stick to svm- 
bols as given. Other\\asfe, you may 
confuse tlieir meanings. 



1/20 revolution per second can be 
written 0.05 rev/sec, or more briefly, 
just 0.05 sec'. In this last expres- 
sion the symbol sec ' stands for 
1/sec, or "per second." The unit 
sec ' is called hertz, svmbol Hz. 



SG 14 



The term "revolutions" is not as- 
signed any units because it is a 
pure number, a count. There is no 
need for a standard as there is for 
distance, mass, and time. So, the 
unit for freqviency is usually given 
without "rev." This looks strange, 
but one gets used to it. It is not veiy 
important, because it is merely a 
matter of terminologx', not a fact of 
physics. 



"Note the difference between units. 



If you know the frequency of revolution / and the radius R of 
the path, you can compute the speed v of any object in unifomi 



CHAPTER 4 / UNDERSTANDING MOTION 



111 



". . . the greatejr the velocity . . . with 
which [a stone] is projected, the 
farther it goes before it falls to the 
earth. We may therefore suppose 
the velocity to be so increased, that 
it would describe an arc of 1, 2, 5, 
10, 100. 1000 miles before it ar- 
rived at the earth, till at last, ex- 
ceeding the limits of the earth, it 
should pass into space without 
touching it." — Newton, System ol 
the World. 




circular motion without difficulty. 1 he distance traveled in one 
revolution is simply the perimeter of the circular path, that is, 
ZtiH. The time for one revolution is by definition the period T. 
For uniform motion along any path, it is always tn.ie that 



speed = 



distance traveled 
time elapsed 



By substitution, 



V = 



2tiR 
T 



To express this equation for circular motion in terms of the 
frequency/ rewrite it as 



Now, since by definition 



= 2ttR X - 
T 



f = 



112 



IMI 



(;()\(;i;prs oi moiiox 



you can write 



= 27Tfl X / 



If the body is in uniform circular motion, the speed computed 
by this equation is both the instantaneous speed and the average 
speed. If the motion is not unifomi, the formula gives only the 
average speed. The instantaneous speed for any point on the 
circle can be determined if you find Ad/Af from measurements of 
very small portions of the path. 

How can you best use the last equation above? You can, for 
example, calculate the speed of the tip of a helicopter rotor blade 
in its motion around the central shaft. On one model, the main 
rotor has a diameter of 7.50 m and a frequency of 480 
revolutions/min under standard conditions. Thus, R = 3.75 m, 
and so 



V = ZTTRf 



V = 2 (3.14) (3.75) I I nVsec 

. 60 

V = 189 m/sec 



480 



or about 680 km/hr. 



8. If a phonograph turntable is running at 33.3 revolutions 
per minute, 

(a) what is its period (in minutes)? 

(b) what is its period (in seconds)? 

(c) what is its frequency in hertz? 

9. What is the period of the minute hand of an ordinary 
clock? If the hand is 3.0 cm long, what is the speed of the tip 
of the minute hand? 

10. The terms frequency and period can also be used for any 
other regular, repetitive phenomenon. For e^iample, if your 
heart beats 80 times per minute, what are the frequency and 
period for your pulse? 



4,6 I Centripetal acceleration and centripetal 
force 



The adjective "centripetcU" means 
literally "moving, or directed, to- 
ward the center." 



Assume that a stone on the end of a string is whirling about udth 
unifomi circular motion in a horizontal plane. The speed of the 
stone is constant. The velocity, however, is always changing. 
Velocity is a vector quantity which includes both speed and 
direction. Up to this point, you have dealt with accelerations in 
which only the speed was changing. In uniforai circular motion 
the speed of the revolving object remains the same, while the 



In uniform circular motion , the in- 
stantaneous velocity and the cen- 
tripetal force at any instant of time 
are pei-pendicular, one being along 
the tangent, the other along the ra- 
dius. So instantaneous velocity and 
the acceleration are also always at 
right angles. 



CHAPTER 4 / UNDERSTANDING MOTION 



113 




a and F are parallel, but v is per- 
pendicular to a and F . Note that 
usually one should not draw differ 
ent kinds of vector quantities on 
the same drawina. 



direction of motion changes continually. The top figure in the 
margin shows the whirling stone at three successive moments in 
its revolution. At any instant, the direction of the velocity vector 
is tangent to the cumng path. Notice that the stone's speed, 
represented by the leitgth of the velocity ariovv, does not change. 
But its direction does change from moment to moment. Since 
acceleration is defined as a change in velocity per unit time, the 
stone is, in fact, accelerating. 

To produce an acceleration, a net force is needed. In the case 
of the whirling stone, a force is exerted on the stone by the 
string. If you ignore the weight of the stone and air resistance, 
this force is the net force. If the string were suddenly cut, the 
stone would go flying off with the velocity it had at the instant 
that the string was cut. Its path would stai1 off on a tangent to 
the circular path. But as long as the string holds, the stone is 
forced to move in the circular path. 

rhe direction of this force which is holding the stone in its 
circular path is along the string. Thus, the force vector always 
points toward the center of rotation. This kind of force, which is 
always directed toward the center of rotation, is called 
centripetal force. 

From Newton's second law, you know that net force and 
corresponding acceleration are in the same direction. Iherefore, 
the acceleration vector is also directed toward the center. This 
acceleration is called centripetal acceleration and has the symbol 
a^. Any object moving along a circular path has a centripetal 
acceleration. 

You know now the direction of centripetal acceleration. What 
is its magnitude? An expression for a^ can be derived from the 
definition of acceleration a^ = Av/Af . The details for- such a 
derix'ation are given on the next page. The result shows that a,, 
depends on v and H. In fact, the magnitude of a^ is given by 



v^ 



a. = 




You can verify this relationship with a numerical example. As 
sketched in the diagram on page 116, a car goes around a 
circular cuive of radius R = 100 m at a uniform speed of v = 20 
m/sec. What is the car's centripetal acceleration a, toward the 
center of curvature? By the equation deiived on the next page, 



a., = 



v^ 
R 

(20 m/sec I 
100 m 



114 



LMT 1 / co\t:L;i»rs oi \iorio\ 



Close Upl 



Derivation of the Equation o^ 

Assume that a stone on the end of a string is 
moving uniformly In a circle of radius R. You can 
find the relationship between a^, v, and R by treating 
a small part of the circular path as a combination 
of tangential motion and acceleration toward the 
center. To follow the circular path, the stone must 
accelerate toward the center through a distance h 
in the same time that it would move through a tan- 
gential distance d. The stone, with speed v, would 
travel a tangential distance d given by d = vAt. In 
the same time At, the stone, with acceleration a , 

' ' c' 

would travel toward the center through a distance 
h given by h = -a^Af. (You can use this last equa- 
tion because a\ t = 0, the stone's velocity toward 
the center is zero.) 

You can apply the Pythagorean theorem to the 
triangle in the figure below. 

R^ + d' = {R + hf 

= R^ + 2Rh + h^ 

When you subtract R^ from each side of the equa- 
tion, you are left with 

d' = 2Rh + h' 

You can simplify this expression by making an ap- 
proximation. Since h is very small compared to R, 
ff will be very small compared to Rh. And since At 
must be vanishingly small to get the instantaneous 
acceleration, h^ will become vanishingly small com- 
pared to Rh. So you can neglect if and write 

d^ = 2Rli 






K 

Also, d = vAt and /7 = -a^Af, so you can substitute 
for d^ and for in accordingly. Thus, 



or 



{vAtf = 2R ■ '-a^iAtr 



v'iAtf = Ra^(AtY 
v' = Ra^ 



a„ = 



R 



The approximation becomes better and better as 
At becomes smaller and smaller. In other words, v^/ 
R gives the magnitude of the instantaneous centri- 
petal acceleration for a body moving on a circular 
arc of radius R. For uniform circular motion, v^/R 
gives the magnitude of the centripetal acceleration 
at every point of the path. (Of course, it does not 
have to be a stone on a string. It can be a small 
particle on the rim of a rotating wheel, or a house 
on the rotating earth, or a coin sitting on a rotating 
phonograph disk, or a car in a curve on the road, 
or an electron in its path through a magnetic field.) 

The relationship between a^, v, and R was dis- 
covered by Christian Huygens and was published 
by him in 1673. Newton, however, must have known 
it in 1 666, but he did not publish his proof until 1 687, 
in the Pnncipia. 

We can substitute the relation v = 2TTR/f or v 
= 2ttR/T (derived in Chapter 3 for uniform circular 
motion) into the equation for a : 



V 



a„ = 



R 



{27:Rf Y 
R 



2ttR 



R 



/ P- 



= 4 Tr'Rf ' 



4'n'R 



These two resulting expressions for a^ are entirely 
equivalent. 



CHAPTER 4 / UNDERSTANDING MOTION 



115 




SG 15 
SG 16 



A 



2Dm 



/ 
/ 
/ 

/\00ru 





_ 400 mVsec^ 
100 ni 

= 4.0 m/sec^ (about 0.4 g) 

Does this make sense? You can check the result by going back 
to the basic vector definition of acceleration: a^^ = Av/Af. You will 
need a scale drawing of the car's velocity vector at two instants 
a short time A/ apart. Then, you can measure the change in 
velocity Av between these points, and divide the magnitude of Av 
by Af to get a,^ over the interval. 

Consider a time interval of Af = 1 sec. Since the car is moving 
at 20 m/sec, its position will change 20 m during Af. Two 
positions P and P', separated by 20 m, are marked in Diagram A. 

Now, draw aiiows representing velocity vectors. If you choose 
a scale of 1 cm = 10 m/sec, the velocity vector for the car will 
be represented by an arrow 2 cm long. These are drav\Ti at P and 
P' in Diagram B. 

Now, put these two arrows together tail to tail as in Diagram C. 
It is easy to see what the change in the velocity vector has been 
during Af. Notice that if you had drawai Av halfway between P 
and P', it would point directly toward the center of the curve. So 
the average acceleration between P and P' is indeed directed 
centripetally. The Av arrow in the diagram is 0.40 cm long, so it 
represents a velocity change of 4.0 m/sec. This change occurred 
during Ar = 1 sec, so the rate of change is 4.0 m/sec'^. This is the 
same value found by using a^ = v^/fil 

The relation a^. = \riR agrees completely with the mechanics 
developed in Unit 1. You can show this by doing some 
experiments to measure the centripetal force required to keep an 
object moving in a circle. Return to the example of the car. If its 
mass were 1,000 kg, you could find the (centripetal force acting 
on it as follows: 

F^ — m X a,, 

= 1000 kg X 4.0 m/sec'' 

= 4000 kg m/sec^ 

= 4000 N 

This force would be directed toward the center- of curvatur-e of 
the road. That is, it would always be sideways to the direction 
the car is moving. This for^ce is exerted on the tires by the road. 
If the road is wet or icy and cannot exert the force of 4,000 N 
sideways on the tires, the centripetal acceleration will be less 
than 4.0 m/sec". I'hen the car will follow a less curbed path as 
sketched in Diagram D. In situations where the cars path is less 
curved than the road, you would say the car "left the road." Of 
course, it might be just as appropriate to say the road loft the 
car. 



lie 



LM 1 1 CO.VCLPIS Ol MOnON 



11. Is a body accelerating when it 

(a) moves with constant speed? 

(b) moves in a circle with constant radius? 

(c) moves with constant velocity? 

12. A car of mass m going at speed v enters a curve of radius 
R. What is the force required to keep the car curving with the 
road? 

13. A rock of mass 1 kg is swung in a circle with a frequency 
ofl revolution/sec on a string of length 0.5 m. What is the 
magnitude of the force that the string e\erts on the rock? 
What is the magnitude and direction of the force on the string 
that must be present according to Newton's third law? If the 
string were cut, what kind of path would the rock follow? How 
fast would it move? 



SG 17-20 



4*T The motion of earth satellites 



Nature and technology provide many examples of objects in 
uniform circular motion. The wheel has been a main 
characteristic of civilization, first appearing on crude carts and 
later forming essential parts of complex machines. The historical 
importance of rotary motion in the development of modem 
technology has been described by the historian V. Gordon Childe 
in The History of Technology: 

Rotating machines for performing repetitive operations, driven 
by water, by thenncd power, or by electrical energy, were the 
most decisive factors of the industrial revolution, and, from the 
first steamship till the invention of the jet plane, it is the 
application of rotary motion to transport that has 
revolutionized communications. The use of rotary machines, as 
of any other human tools, has been cumulative and progressive. 
The inventors of the eighteenth and nineteenth centuries were 
merely extending the applications of rotaiy motion that had 
been devised in previous generations, reaching back thousands 
of years into the prehistoric past. . . . 

As you will see in Unit 2, there is another rotational motion that 
has concerned scientists throughout recorded history. This 
motion is the orbiting of planets around the sun and of the 
moon around the earth. 

The kinematics and d3^amics for any unifoiTn circular motion 
are the same. Therefore, you can apply what you have learned 
so far to the motion of artificial earth satellites in circular (or 
nearly circular) paths. A typical illustration is Alouette I, Canada's 
first satellite, which was launched into a nearly circular orbit. 



SG25 




CHAPTER 4 / UNDERSTANDIIVG MOTION 



117 



ALOUETTEII 




The same equation iv = 27r/J/7') can 
be used to find the speed of any sat- 
ellite in nearly or fiilK' cii'cular or- 
bit, for example, that of our moon. 
The a\erag(; distance from the cen- 
ter of the moon is approximately 
3.82 X 10 km. and the moon takes 
an a\erag(! of 27 days. 7 hr. 43 min 
to complete one re\'olution around 
the earth with respect to the fixed 
stars. I'hus 

_ 2-ITI3.82 X 10) km 

3.93 X 10' min 
= 61.1 km/min 
or about 3,666 km/hr. 




An artist s impression of the Space 
Shuttle retrieving n sntcllile from 
orbit. When operational, the Shut- 
tle Orhiter will function as the 
world s first reusable space cargo 
plane. 



SG 29 
SG 30 



Tracking stations located in many places around the world 
maintain a record of anv satellite's position in the sky. Fixam the 
position data, the satellite's period of revolution and its distance 
aho\'e the eai-th at any time are found. By means of such 
tracking, scientists know that Alouette I moves at an average 
height of 1,010 km ahove sea level. It takes 105.4 min to complete 
one revolution. 

You can now quickly calculate the orhital speed and the 
centripetal acceleration of Alouette I. rhe relation v = ZirB/T 
gives the speed of any ohject moving uniformly in a circle if you 
know its period T anci its distance B from the center of its path, 
hi this case, the center of the path is the center of the earth. So, 
adding 1,010 km to the earth's radius of 6,380 km, you get R = 
7,390 km. Thus, 

_ ZttH 
^ ~ T 

_ 2tt X 7390 km 
105.4 min 

46,432 km 



6324 sec 



= 7.34 km/sec 



To calculate the centripetal acceleration of Alouette I, you can 
use this value of v along with the relationship a^ = \^/B. Thus, 



a = — 
B 



_ (7.34 km/sec)^ 
7390 km 

= 0.0073 km/sec^ 

= 7.3 m/sec^ 

(You could just as well have used the values of B and T directly 
in the relationship a,, = 4'n~B/T'.) 

What is the origin of the force that gives rise to this 
acceleration? You will not find a good argument for the answer 
until Chapter 8, but you surely know already that it is due to the 
earth's attraction. Exidently the centripetal acceleration a, of the 
satellite is just the gravitational acceleration a, at that height. 
(Note that a^ at this height has a value about 25% less than a^ 
very near the earth's surface.) 

Earlier we asked the question, 'What speed is required for an 
object to stay in a circular orbit about the earth? You can 
answer this question now for an orbit 1,010 km above the earth's 



118 



I M I I CONCUPIS ()l MOilON 



surface. To get a general answer, you need to know how the 
acceleration due to gravaty changes with distance. Chapter 8 will 
come back to the problem of injection speeds for orbits. 

The same kind of analysis applies to an orbit around the 
moon. For example, consider the first manned orbit of the moon. 
The mission control group wanted to put the capsule into a 
circular orbit 110 km above the lunar surface. They knew (from 
other arguments) that the acceleration due to the moon's gravity 
at that height would be a^ = 1.43 m/sec". What direction and 
speed would they have had to give the capsule to inject it into 
lunar orbit? 

The direction problem is fairly easy to solve. To stay at a 
constant height above the surface, the capsule would have to be 
moving horizontally at the instant the orbit correction was 
completed. So injection would have to occur just when the 
capsule was moving on a tangent, at a height of 110 km, as 
shown in the sketch in the margin. What speed (relative to the 
moon, of course) would the capsule have to be given? The 
circular orbit has a radius 110 km greater than the radius of the 
moon, which is 1,740 km. So /? = 1,740 km + 110 km = 1,850 
km = 1.85 X 10*^ m. The centripetal acceleration is just the 
acceleration caused by gravity, namely, 1.43 m/sec^, so 

a„ = a„ 



v" 



\r = fia„ 



= V(1.85 X lO^m) X 1.43 m/sec' 



= V2.65 X 10' mVsec' 




Chariot, by the sculptor Alberto 
Giacometti, 1950. 




= 1.63 X 10 m/sec 

The necessary speed for an orbit at 110 km above the surface is 
1,630 m/sec. Knowing the capsule's speed, ground control could 
calculate the speed changes needed to reach 1,630 m/sec. 
Knowing the thrust force of the engines and the mass of the 
capsule, they could calculate the time of thrust required to make 
this speed change. 



SG 31-33 



Because the injection thrust is not 
applied instantaneously, the details 
are actually more difficult to cal- 
culate. 



• 14. What information was necessary to calculate the speed for 
an orbit 110 km above the moon's surface? 



CHAPTER 4 / UIVDERSTANDIIVG MOTION 



119 



15. If you know the period and speed of a satellite, you also 
know the acceleration of gravity at the height of the satellite. 
Why? 

16. What is the magnitude of the force that holds a 500-kg 
satellite in orbit if the satellite circles the earth every 3 hr at a 
height of 3,400 km above the surface? 



TABLE 4.2 SOME INFORMATION ON A FEW EARLY ARTIFICIAL SATELLITES. 



Name 



Launch Date Weight (kg) Period (min) 



Height (km) 
Perigee-Apogee 



Remarks 
(including purpose) 



Sputnik 1 
1957 (USSR) 



Explorer 7 
1958 (USA) 



Lunik 3 
1959 (USSR) 

Vostok 1 
1961 (USSR) 

Midas 3 

1961 (USA) 

Telestar 1 

1962 (USA) 



Alouette 1 

1962 

(USA-Canada) 

Luna 4 
1963 (USSR) 

Vostok 6 
1963 (USSR) 

Syncom 2 
1963 (USA) 



Oct. 4, 1957 



Jan. 31, 1958 



July 10, 1962 



Sept. 29, 1962 



July 26, 1963 



83.4 



14.0 



Oct. 4, 1959 433 

Apr. 12, 1961 4,613 

July 12, 1961 1,590 



77 



145.3 



Apr. 2, 1963 1,423 
June 16, 1963 4,600 



39 



96.2 229-947 First earth satellite. Internal 

temperature, pressure inside 
satellite. 

114.8 301-2,532 Cosmic rays, 

micrometeorites, internal and 
shell temperatures, discovery 
of first Van Allen belts. 



22,300 


48,300-468,510 


Transmitted photographs 
of far side of moon. 


89.34 


175-303 


First manned orbital flight 
(Major Yuri Gagarin; one 
orbit). 


161.5 


3,428-3,466 


Almost circular orbit. 


157.8 


955-5,640 


Successful transmission 
across the Atlantic: 
telephony, phototelegraphy, 
and television. 


105.4 


998-1,030 


Joint project between NASA 
and Canadian Defense 
Research Board; 
measurement in ionosphere. 


42,000 


90,160-700,350 


Passed 8,500 km from moon 
very large orbit. 


88.34 


171-216 


First orbital flight by a 
woman (Valentina 
Tereshkova; 48 orbits). 


1,460.4 


35,721-35,729 


Successfully placed in near- 



synchronous orbit (stays 
above same spot on earth) 



4r.8 I What about other motions? Retrospect and 
prospect 

So far we have described straight-line motion, projectile motion, 
and unifoiTn circular motion. We have considered onlv examples 
where the acceleration was constant (in magnitude if not in 
direction) or very nearly constant. There is another basic kind of 
motion that is equally common and important in physics. In this 
kind of motion the acceleration is ahvaxs changing. .\ common 



rzo 



LiMT 1 / co\c:L;i»'rs oi .vio rio\ 



example is seen in playground swings or in vibrating guitar 
strings. Such back and forth motion, or oscillation, about a center 
position occurs when there is a force always directed toward the 
center position. When a guitar string is pulled aside, for example, 
a force arises which tends to restore the string to its undisturbed 
center position. If it is pulled to the other side, a similar 
restoring force arises in the opposite direction. 

In very common types of such motion, the restoring force is 
proportioned, or neariy proportional, to how far the object is 
displaced. This is true for the guitar string, if the displacements 
are not too large. Pulling the string aside 2 mm produces twice 
the restoring force that pulling it aside 1 mm does. Oscillation 
with a restoring force proportional to the displacement is called 
simple harmonic motion. The mathematics for describing simple 
harmonic motion is relatively simple, and many phenomena, 
from pendulum motion to the vibration of atoms, have aspects 
that are very close to simple harmonic motion. Consequently, the 
analysis of simple harmonic motion is used very uddely in 
physics. The Project Physics Handbook describes a variety of 
activities you can do to become familiar with oscillations and 
their description. 

The dynamics discussed in this chapter will cover most 
motions of interest. It provides a good start toward 
understanding apparently very complicated motions such as 
water ripples on a pond, a person nanning, the swaying of a tall 
building or bridge in the vvdnd, a small particle zig-zagging 
through still air, an amoeba seen under a microscope, or a high- 
speed nuclear particle moving in the field of a magnet. The 
methods developed in this unit give you the means for dealing 
with any kind of motion whatsoever, on earth or anywhere in the 
universe. 

When you considered the forces needed to produce motion, 
Newton's laws supplied the answers. Later, other motions, 
ranging from motion of the planets to the motion of an alpha 
particle passing near a nucleus inside an atom, will be discussed. 
You wdll continue to find in Newton's laws the tools for 
determining the magnitude and direction of the forces acting in 
each case. 

If you know the magnitude and direction of the forces acting 
on an object, you can determine what its change in motion will 
be. If you also know the present position, velocity, and mass of 
an object, you can reconstruct how it moved in the past and 
predict how it will move in the future under these forces. Thus, 
Newion's laws provide a neariy unlimited view of forces and 
motion. It is not surprising that Newtonian mechanics became a 
model for many other sciences. They seemed to provide a 
method for understanding all motions, no matter how 
mysterious the motions appeared to be. 





SG 34 
SG 35 







'''■'!>iii(iiluuiJia«BaBi 

£?■■«■■■ ■ « B ■ ••»! 

" lift!.?!*' • W" * "^ ** ?* "^ 

HUH Umii Ml M»>«>f 




CHAPTER 4 / UNDERSTANDING MOTION 



121 



study 
guide 



1. The Project Physics learning materials 
particularly appropriate for Chapter 4 include the 
following: 

Experiments 

Curves of IVajectories 
Prediction of IVajectories 
Centripetal Force 
Centripetal Force on a Turntable 

Activities 

Projectile Motion Demonstration 

Speed of a Stream of Water 

Photographing a Waterdrop Parabola 

Ballistic Cart Projectiles 

Motion in a Rotating Reference Frame 

Penny and Coat Hanger 

Measuring Unknown Frequencies 

Film Loops 

A Matter of Relative Motion 

Galilean Relativity: Ball Dropped from Mast of Ship 

Galilean Relativity: Projectile Fired V'ertically 

Analysis of Hurdle Race. 1 

Analysis of Hurdle Race. II 

2. The thrust developed by a Saturn Apollo rocket 
is 7370,()()0 N and its mass is 540,000 kg. What is the 
acceleration of the vehicle relative to the earth's 
surface at lift-off? How long would it take for the 
vehicle to rise 50 m? 

The acceleration of the vehicle increases greatly 
with time (it is 47 m/sec^ at first stage burnout) even 
though the thrust force does not increase 
appreciably. Explain why the acceleration increases. 

3. A person points a gun barrel directly at a bottle 
on a distant rock. Will the bullet follow the line of 
sight along the barrel? If the bottle falls off the rock 
at the very instant of firing, will it then be hit by the 
bullet? Explain. 

4. It is helpful to consider the vertical and 
horizontal motions of a projectile separately. You 
can do the same for almost any motion, uniform or 
accelerated. 



(a) What are the pc and y displacements of a particle 
that travels uniformly v\ith a velocity v^ = 4 m/sec 
and a velocity v; = 3 m/sec after 2 sec of travel? Find 
the displacements after 5 sec and 10 sec. 

(b) Add the two displacement vectors pc and y that 
you found in part (a) to determine the distance of 
the particle from the origin after 2 sec, 5 sec, and 10 
sec. 

(c) Use your results from part (b) to find the velocity 
of the particle along its path. Can you relate this 
velocity to the velocity components v and v ? 

5. If you like algebra, try this general proof: If a 
body is launched with speed v at some angle other 
than 0°, it will initially have both a horizontal speed 
V and a vertical speed v . The equation for its 
horizontal displacement is ;»: = vt, as before. But the 
equation for its vertical displacement has an 

additional term: y = vf + —af. Show that the 

2 « 

trajectory is still parabolic in shape. 

6. A lunch pail is accidently kicked off a steel beam 
on a skyscraper under construction. Suppose the 
initial horizontal speed v^ is 1.0 m/sec. Where is the 
pail (displacement!, and what is its speed and 
direction (velocity) 0.5 sec after launching? 

7. (a) A ball is thrown from a roof with a 
horizontal speed of 10 m/sec toward a building 25 m 
away. How long will it take the ball to hit the 
building? 

(b) How far will the ball have fallen when it hits the 
building? 

(c) If the roof ft'om which the ball is tlirown is 20 m 
above the ground, what is the minimum speed with 
which the ball must be throv\ii to hit the building? 

8. A shingle slides down a roof having a 30° pitch 
and falls off with a velocity of 2 m/sec. How long will 
it take to hit the ground 45 m below? Why can you 
ignore the horizontal velocity of the shingle in 
calculating the answer? 

9. A projectile is launched with a horizontal speed 
of 8 m/sec. 

(a) How much time will the projectile take to hit the 
ground 80 m below? 



rz'z 



ciiyXFiLK 4 s run\ (;i idl 



S5 



(b) How does this time change if the horizontal 
speed is doubled? 

(c) What is the vertical speed of the projectile when 
it hits the ground? 

(d) What is the horizontal speed of the projectile 
when it hits the ground? 

10. In Galileo's drawing on page 107, the distances 
be, cd, de, etc., are equal. What is the relationship 
among the distances bo, og, gl, and In? 

11. You are inside a van that is moving with a 
constant velocity. You drop a ball. 

(a) What would be the ball's path relative to the van? 

(b) Sketch its path relative to a person driving past 
the van at a high uniform speed. 

(c) Sketch its path relative to a person standing on 
the road. 

You are inside a moving van that is accelerating 
uniformly in a straight line. When the van is traveling 
at 10 km/hr (and still accelerating) you drop a ball 
from near the roof of the van onto the floor. 

(d) What would be the ball's path relative to the van? 

(e) Sketch its path relative to a person driving past 
the van at a high uniform speed. 

(f) Sketch its path relative to a person standing on 
the road. 

12. Two persons watch the same object move. One 
says it accelerates straight dovmward, but the other 
claims it falls along a curved path. Describe 
conditions under which each observation would be 
correct. 

13. An airplane has a gun that fires bullets straight 
ahead at the speed of 1 ,000 km/hr while the plane 
is stationary on the ground. The plane takes off and 
flies due east at 1 ,000 km/hr. Which of the following 
describes what the pOot of the plane will see? In 
defending your answers, refer to the GalUean 
relativity principle. 

(a) When fired direcfly ahead, the bullets moved 
eastward at a speed of 2,000 km/hr. 

(b) When fired in the opposite direction, the bullets 
dropped vertically downward. 



(c) When fired vertically downward, the bullets 
moved eastward at 1,000 km/hr while they fell. 

Specify the frames of reference from which (a), (b), 
and (c) are the correct observations. 

14. Many commercial record turntables are 
designed to rotate at frequencies of 16% rpm (called 
transcription speed), 33'/3 rpm (long playing), 45 rpm 
(pop singles), and 78 rpm (old fashioned). What is 
the period corresponding to each of these 
frequencies? 

15. Passengers on the right side of the car in a left 
turn have the sensation of being "thrown against the 
door." Elxplain what actually happens to the 
passengers in terms of force and acceleration. 

16. The tires of the turning car in the example on 
page 116 were being pushed sideways by the road 
with a total force of 4 kN. Of course the tires would 
be pushing on the road with a force of 4 kN also. 

(a) What happens if the road is covered with loose 
sand or gravel? 

(b) How would softer (lower pressure) tires help? 

(c) How would banking the road (that is, tilting the 
surface toward the center of the curve) help? (Hint: 
Consider the extreme case of banking in the bobsled 
photo on p. 114.) 

17. Using a full sheet of paper, make and complete 
a table like the one below. 



Name of 
Concept 


Symbol 


Definition 


Example 






Length of a 
path between 
any two points, 
as measured 
along the path. 










Straight-line 
distance and 
direction from 
Detroit to 
Chicago. 




V 






Instanta- 
neous 
speed 









CHAPTER 4 / STUDY GUIDE 



123 



so 









An airplane 
flying west at 
640 km hr at 
constant 
altitude. 






Time rate of 
change of 
velocity. 






^9 






Centripetal 
accelera- 
tion 














The drive shaft 
of some 
automobiles 
turns 600 rpm 
in low gear. 






The time it 
takes to make 
one complete 
revolution. 





18. Our sun is located at a point in our galaxy about 
30,000 light years (1 light year = 9.46 x lo" kml 
from the galactic center. It is thought to be revolving 
around the center at a linear speed of approximately 
250 km/sec. 

(a) What is the sun's centripetal acceleration with 
respect to the center of the galaxy? 

(b) The sun's mass can be taken to be 1.98 x 10^" kg. 
What centripetal force is required to keep the sun 
moving in a circular orbit about the galactic center? 

(c) Compare the centripetal force in (b) with that 
necessaiy to keep the earth in orbit about the sun. 
(The earth's mass is 5.98 x lO"^ kg, and its average 
distance from the sun is 1.495 x lO" km. I 

19. The hammer thrower in the photograph on 
page 121 is exerting a large centripetal force to keep 
the hammer moving fast in a circle and applies it 

to the hammer through a connecting wire. The mass 
of the hammer is 7.27 kg. 

(a) Estimate \hv. radius of the circle and th(; pcM-iod. 
Calculate a rough \alue for the amount of fbi'ce 
required just to keep the hammer moving in a circle. 

(b) What other components are there to the total 
force e\(M-ted on the hammer? 



20. Contrast rectilinear motion, projectile motion, 
and uniform circular motion Ijy 

(a) defining each. 

(b) giving examples. 

(c) describing the relation between velocity and 
acceleration in each case. 

21. An object of mass 2 kg is svvTing in a circle of 
radius 2 m, taking 2 sec to complete the circle. What 
is the speed of the object? What is the size and 
direction of the acceleration keeping the object 
moving in a circle? What is the size and direction of 
the force? 

22. A force of 5 M is required to hold a stone of 
mass 2.5 kg in a circle of 5-m radius. What must be 
the speed of uniform circular motion? 

23. An object 3 m away from the center of its 
uniform circular motion moves with an acceleration 
of 10 m/sec" toward the center. How long does the 
object take to complete one circle? Why is it not 
necessaiy to know the mass of the object in order to 
find the answer? 

24. A fan blade takes 0.1 sec to go around. What is 
its frequency? Wliat is the acceleration of a point 0.5 
m from the center? 

25. These questions refer to Table 4.2 on page 120. 

(a) Which satellite has the most nearly circular orbit? 

(b) Which satellite has the most eccentric orbit? How 
did you arrive at your answer? 

(c) Which has the longest period? 

(d) How does the position of Syncom 2 relative to a 
point on eiirth change over one day? 

26. A satellite is put into an orbit 400 km above the 
surface of the earth (6,800 km from the center of 
the earth) where the acceleration owing to gravity is 
8.7 m/sec\ What is the satellite's speed? Based on 
the orbit, are there any restrictions on the mass of 
the satellite? 

27. A satellite with a mass of 500 kg requires 380 
min to circle the earth in an orbit 18,000 km from 
the center of the earth. What is the magnitude of the 
force holding the satellite in orbit? 

28. A satellite with a mass of 500 kg orbits the moon 
(whose gravity is one sixth that of earth's) at a 
distance of 18,000 km from \\\v. center of the moon. 



1:24 



ciL\i»ri:it 4 s ruD^ (;i idl 



ss 



what is the magnitude of the force acting on the 
satellite? How long would it take to circle the moon? 

39. If the earth had no atmosphere, what would be 
the period of a satellite skimming just above the 
earth's surface? What would its speed be? 

30. Elxplain why it is impossible to have an earth 
satellite orbit the earth in 80 min. Does this mean 
that it is impossible for any object to go around the 
earth in less than 80 min? 

31. What was the period of the "110-km " Apollo 8 
lunar orbit? 

32. Knowing a near the moon's surface and the 
orbital speed in an orbit near the moon's surface, we 
can now work an example of Part 8 of the 
earth-moon trip described in Sec. 4.1. The Apollo 8 
capsule was orbiting about 100 km above the surface. 
The value of a near the moon's surface is about 1.5 
m/sec~. 

If the rocket's retro-enj^ines are fired, it will slow 
down. Consider the situation in which the rockets 
fire long enough to reduce the capsule's horizontal 
speed to 100 m/sec. 

(a) About how long will the fall to the moon's 
surface take? 

(b) About how far will it have moved horizontally 
during the fall? 

(c) About how far in advance of the landing target 
might the "braking" maneuver be performed? 



33. Assume that a capsule is approaching the moon 
along the right trajectory, so that it uill be moving 
tangent to the desired orbit. Given the speed v^ 
necessary for orbit and the current speed v, how 
long should the engine with thrust F fire to give the 
capsule of mass m the right speed? 

34. The intention of the first four chapters has been 
to describe "simple" motions and to progress to the 
description of more "complex" motions. Classify 
each of the following examples as "simplest motion," 
"more complex," or "very complex." Be prepared 

to defend your choices and state any assumptions 
you made. 

(a) helicopter landing 

(b) "human cannon ball" in flight 

(c) car going from 50 km/hr to a complete stop 

(d) tree growing 

(e) child riding a Ferris wheel 

(f) rock dropped 3 km 

(g) person standing on a moving escalator 
(h) climber ascending Mt. Everest 

(i) person walking 

(j) leaf falling from a tree 

35. Write a short essay on the physics involved in 
the motions shoun in the picture below, using the 
ideas on motion from Unit 1. 



■ .n^^Li^ 



/ 




V 




CHAPTER 4 / STUDY GUIDE 



125 



:i 







This unit dealt v\ith the 
fundamental concepts of 
motion. We started by analyzing very simple kinds of motion. 
\fter learning the "ABC's" of physics, we expected to be iible to 
turn our attention to some of the more complex features of the 
world. To what extent were our expectations fulfilled? 

We did find that a lelatixely few basic concepts ga\e us a fairly 
solid understanding of motion. We could describe many motions 
of objects by using the concepts of distance, displacement, time, 
speed, velocity, and acceleration. To these concepts we added 
force and mass and the relationships expressed in Newton's 
three laws of motion. With this knowledge, we found we could 
describe most observed motion in an effective way. The 
surprising thing is that these concepts of motion, which were 
developed in very restricted circumstances, can be so widely 
applied. For example, our discussion of motion in the laboratory 
centered around the use of sliding dr\'-ice disks and steel balls 
rolling down inclined planes. These are not objects ordinarily 
found moving around in the everyday "natural" world. Yet we 
found that the ideas obtained from those specialized 
experiments led to an understanding of objects falling near the 
earth's surface, of projectiles, and of objects moving in circular 
paths. We star-ted by analyzing the motion of a disk of dry ice 
moving across a smooth surface. We ended up analyzing the 
motion of a space capsule as it circles the moon and descends 
to its surface. 

We have made quite a lot of progress in analyzing complex 
motions. On the other harnd, we cannot be certain that we have 
her-e all the tools needed to understand all the phenomena that 
interest us. In Unit 3, we will add to our stock of fundamental 
concepts a few additional ones, particularly those of momentum, 
work, and ener'gy. They will help us when we turn our* attention 
fr om interactions involving a relatively few objects of easily 
measured size, to interactions involving countless numbers of 
submicroscopic objects such as molecules and atoms. 

In this unit, we have dealt mainly with concepts that ow e their 
greatest debts to Galileo, Newion, and their- followers. If space 
had permitted, we should also have included the contributions 
of Rene Descartes and the Dutch scientist Christian Huygens. 
The mathematician and philosopher A. N. Whitehead, in Science 
and the Modern World, has summarized the role of these four 
men and the importance of the concepts we have been studying 
as follows: 

This subject of the formation of the three laws of motion and of 
the law of gravitation (which we sliall take up in Unit 2] 
deserves critical attention. The whole development of thought 
occupied exactly two generations. It commenced with Galileo 
and ended with Nevvlon's Principia: and \ev\lon was born in 



126 



LMT 1 / LPILCJGl E 



the year that Galileo died. Also the lives of Descartes and 
Huygens fall within the period occupied by these great terminal 
figures. The issue of the combined labours of these four men 
has some right to be considered as the greatest single 
intellectual success which mankind has achieved. 

The laws of motion that Whitehead mentions were the subject 
of this unit. They u^ere important most of all because they 
suddenly allowed a new understanding of celestial motion. For 
at least 20 centuries people had been trying to reduce the 
complex motions of the stars, sun, moon, and planets to an 
orderly system. The genius of Galileo and Newton lay in their 
studying the nature of motion as it occurs on earth, and then 
assuming that the same laws would apply to objects in the 
heavens beyond human reach. 

Unit 2 is an account of the centuries of preparation that paved 
the way for the great success of this idea. We will trace the line 
of thought, starting with the formulation of the problem of 
planetary motion by the ancient Greeks. We will continue 
through the woi^k of Copernicus, Tycho Brahe, Kepler, and 
Galileo to provide a planetary model and the laws of planetary 
motion. Finally, we vvdll discover Newton's magnificent synthesis 
of terrestrial and celestial physics through his law of universal 
gravitation. 



UNIT 1 / EPILOGUE 127 



umnr 





CHAPTER 5 Where Is the Earth? ITie Greeks' Answers 

CHAPTER 6 Does the Earth Move? The Work of 

Copernieus and Tycho 

CHAPTER 7 A New Universe Appears: The Work of 

Kepler and GalUeo 

CHAPTER 8 The Un% of Earth and Skv: The Work of 

Newton 



Astronomy, the oldest science, 
deals with objects now known 
to lie vast distances from the earth. To early observers, the sun, 
moon, planets, and stars did not seem to be so far away. Yet 
always, even today, the majesty of celestial events has fired our 
imagination and curiosity. The ancients noted the gieat variety of 
objects visible in the sky, the regularit\' of their motions, the 
strangely slow changes in their position and brightness. This 
whole mysterious pattern of motions required some reason, 
some cause, some explanation. 

To the eye, the stars and planets appear as very well-defined 
pinpoints of light. Phey are easy to observe and follow precisely, 
unlike most other natur^ally occurring phenomena. The sky 
therefore pro\aded the natural "laboratory" for beginning a 
science based on the abilitv to abstract, measure, and simpli^. 



128 



UMIT 2 / PROLOGUE 



Astronomical events not only affected the imagination of the 
ancients, but also had a practical effect on everyday life. The 
working day began when the sun rose, and it ended when the 
sun set. Before electric lighting, human activity was dominated 
by the presence or absence of daylight and the sun's wannth, 
which changed season by season. 

Of all time units commonly used, "one day " is probably the 
most basic and surely the most ancient. For counting longer 
intervals, a "moon " or month was an obvious unit. Over the 
centuries, clocks were devised to subdivide days into smaller 
units, and calendars were invented to record the passage of days 
into years. 

When the early nomadic tribes settled down to live in villages 
some 10,000 years ago, they became dependent upon agriculture 
for their food. They needed a calendar for planning their plowing 
and sovvdng. Throughout recorded histoiy, most of the world's 
population has been involved in agriculture and so has 
depended on a calendar. If seeds were planted too early, they 
might rot in the ground, or the young shoots might be killed by a 
frost. If they were planted too late, the crops would not ripen 
before winter came. Therefore, a knowledge of the best times for 
planting and haivesting was important for sundval. Because 
religious festivals were often related to the seasons, the job of 
making and improving the calendar often fell to priests. Such 
improvements required observation of the sun, planets, and stars. 
The first astronomers, therefore, were probably priests. 

Practical needs and imagination acted together to give 
astronomy an early importance. Many of the great buildings of 
ancient times were constructed and situated with a clear 
awareness of astronomy. The great pyramids of Egypt, tombs of 
the Pharaohs, have sides that run due north-south and 
east-west. The awesome circles of giant stones at Stonehenge in 
England appear to have been arranged about 2000 B.C. to permit 
accurate observations of the positions of the sun and moon. The 
Mayans and the Incas in America as well as the ancient 
civilizations of India and China put enormous effort into 
buildings from which they could measure changes in the 
positions of the sun, moon, and planets. At least as early as 1000 
B.C. the Babylonians and Egyptians had developed considerable 
ability in timekeeping. Their recorded observations are still being 
unearthed. 

Thus, for thousands of years, the motions of the heavenly 
bodies were carefully observed and recorded. In all science, no 
other field has had such a long accumulation of data as 
astronomy. 

But our debt is greatest to the Greeks, who began trying to 
deal in a new way udth what they saw. The Greeks recognized 
the contrast between the apparently haphazard and short-lived 







The Aztec calendar, carved over 
100 years before our calendar was 
adopted, divides the year into 18 
months of 20 days each. 



Even in modern times people who 
live much in the open use the sun 
by day and the stars by night as a 
clock. True south can be deter- 
mined from the position of the sun, 
at local noon. The Pole Star gives a 
bearing on true north after dark. 




I he positions of Jupiter from I'SZ 
B c to 60 B.C. are recorded on this 
section of Babylonian clav tablet, 
now in the British Museum. 



UNIT 2 / PROLOGUE 



129 




Stonchcnge, England, was appar 
ently a prehistoric obser\'atory. 



motions of objects on earth and the unending cycles of the 
heavens. About 600 b c: they began to ask new questions: How 
can we explain these cyclic exents in the sky in a simple way? 
What order and sense can we make of them? The Greeks' 
answers, discussed in Chapter 5, had an important effect on 
science. For example, the writings of Aristotle labout 330 B.C.) 
became widely studied and accepted in Western Europe after 
1200 A.n , and they were important factors in the scientific 
revolution that followed. 

After the conquests of Alexander the Great, the center of Greek 
thought and science shifted to Eg\pt. At the new citA of 
Alexandria, founded in 332 B.c:., a great museum similar to a 
modern research institute was created. It flourished for many 
centuries. But as Greek cixilization gradualK' declined, the 
practical-minded Romans captured Egxpt, and interest in science 
died out. In 640 A.D., Alexandria was captur-ed by the Muslims 
as they swept along the souther n shor^e of the MediterTanean Sea 
and moved northward through Spain to the Pvr-enees. Along the 
way they seized and preserved many libraries of Greek 
documents, some of which were later tr^anslated into Arabic and 
carefully studied. During the tbilowing c'enturii^s, Muslim 
scientists made new and better observations of the heavens. 
However, they made no major- changes in the explanations or 
theories of the (ireeks. 



130 



L!MT 2 PKOLOGIE 



In Western Europe during this period, the works of tlie Greeks 
were largely forgotten. Eventually Europeans rediscovered them 
through Arabic translations found in Spain after the Muslims 
were forced out. By 1130 A.D., complete manuscripts of at least 
one of Aristotle's books were known in Italy and France. After the 
founding of the University of Bologna in the late twelfth century, 
and of the University of Paris around 1200, many other wi itings 
of Aristotle were acquired. Scholars studied these writings both 
in Paris and at the new English universities, Oxford and 
Cambridge. 

During the next century, the Dominican monk Thomas 
Aquinas blended major elements of Greek thought and Christian 
theology into a single philosophy. His work was widely studied 
and accepted in Western Europe for several centuries. In 
achieving this synthesis, Aquinas accepted the physics and 
astronomy of Aristotle. Because the science was blended with 
theology, any questioning of the science seemed also to question 
the theology. Thus, for a time there was little effective criticism 
of Aristotelian science. 

The Renaissance movement, which spread out across Europe 
from Italy, brought new art and music. It also brought new ideas 
about the universe and humanity's place in it. Curiosity and a 
questioning attitude became acceptable, even prized. Scholars 
acquired a new confidence in their ability to learn about the 
world. Among those whose work introduced the new age were 
Columbus and Vasco da Gama, Gutenberg and da Vinci, 
Michelangelo and Raphael, Erasmus and Vesalius, Luther, Calvin, 
and Henry VIII. (The chart in Chapter 6 shows their life spans.) 
Within this emerging Renaissance culture lived Niklas 
Koppernigk, later called Copernicus, vv^hose reexamination of 
astronomical theories is discussed in Chapter 6. 

Further improvements in astronomical theory were made in 
the seventeenth century by Kepler, mainly through mathematical 
reasoning, and by Galileo, through his observations and writings. 
These contributions are discussed in Chapter 7. Chapter 8 deals 
with Newton's work in the second half of the seventeenth 
century. Neuron's genius extended ideas about motion on earth 
to explain motion in the heavens, a magnificent synthesis of 
terrestrial and celestial dynamics. These men, and others like 
them in other sciences such as anatomy and physiology, literally 
changed the world. The results they obtained and the ways in 
which they went about their work had effects so far-reaching 
that we generally refer to their work as the Scientific Revolution. 

Great scientific advances can, and often do, affect ideas outside 
science. For example, Newton's work helped to create a new 
feeling of self-confidence. It seemed possible to understand all 
things in the heavens and on the earth. This great change in 
attitude was a major characteristic of the eighteenth century. 



In the twelfth century, the Muslim 
scholar Ibn Rashd had attempted a 
similar union of Aristotelianism and 
Islam. 



UNIT 2 / PROLOGUE 



131 





ON 




THE 01 


K.IN (IF 


SPECIES 


BY MKANS OK NATURAL' SELEaiOS, 


PiiESKIlVATIIlN OK FAVOUICEI) IIACES 


IX THE STRUGGLE 




KDH LIKE. 




Dv 


JHARLES riARWIN 


M.A., 




L or an^jiKHCs i^'iiiiio ii. 
LONDON: 


M. •. BKAQLC^ VOTUK 


JOHN Ml 


RIIAY. A1-UEMARI,E STIiKEl | 




IHS'J. 






,^..^u^r„^ 





which has heen called the Age of Reason. To a degree, what we 
think today and how we njn our affair's are still affected by 
scientific discoveries made centuries ago. 

Broad changes in thought de\'eloped at the start of the 
Renaissance and grev\' for neariy a centu^\^ from the work of 
Copernicus to that of Newton. In a sense, this era of invention 
resembles the sweeping changes that have occurred during the 
past hundred years. This recent period might extend from the 
publication of Darwin's Origin of Species in 1859 to the first 
controlled release of nuclear energy' in 1942. Within this inteival 
lived such great scientists as Mendel and Pasteur, Planck and 
Einstein, Rutherford and Fermi. The ideas they and others 
introduced into science have become increasingly important. 
These scientific ideas are just as much a part of our time as the 
ideas and works of people such as Roosevelt and Ghandi, Martin 
Luther King and Pope John XXIII, Marx and Lenin, Freud and 
Dewey, Picasso and Stravinsky, Shaw and Joyce. If we understand 
how science influenced the people of past centuries, we can 
better understand how science influences our thought and lives 
today. This is clearly one of the basic aims of this course. 

The material treated in this unit is historical as well as 
scientific. Even today, the historv of science remains of great 
importance to anyone interested in understanding science. The 
reasons for presenting the science of astronomy in its historical 
framework include the follov\ang: The results that were finally 
obtained still hold tiue and rank among the oldest ideas used 
every day in scientific work. The characteristics of all scientific 
work are cleariv visible. We can see the role of assumptions, of 
experiment and observations, of mathematical theory. We can 
note the social mechanisms for cooperation, teaching, and 
disputing. And we can appreciate the possibilitv of ha\ing one's 
scientific findings become par't of the accepted knowledge of the 
time. 

There is an interesting conflict between the rival theories used 
to explain the same set of astr-onomical observations. This 
conflict is typical of all such disputes down to our day. Thus, it 
helps us to see clearly what standar-ds may be used to judge one 
theory against another. 

This subject matter includes the main reasons for the rise of 
science as we understand it now. The story of the Scientific 
Revolution and its many effects outside science itself is as 
necessary to understanding our current age of science as is the 
story of the American Re\'olution to an understanding of America 
today. 



132 



IMF 2 PK()LO(;i b 




llVbere Is the Earth? 

The Greeks' Answers 



5.1 Motions of the sun and stars 

5.2 Motions of the moon 

5.3 The 'Svandering" stars 

5.4 Plato's problem 

5.5 The Greek idea of "explanation" 

5.6 The first earth-centered solution 

5.7 A sun-centered solution 

5.8 The geocentric system of Ptolemy 

5.9 Successes and limitations of the Ptolemaic model 



S.l I Motions of the sun and stars 

The facts of eveiyday astronomy, the celestial events themselves, 
are the same now as in the times of the ancient Greeks. You can 
observe with your unaided eyes most of what these early 
scientists saw and recorded. You can discover some of the long- 
known cycles and rh34hms: the seasonal changes of the suns 
height at noon, the monthly phases of the moon, and the 
glorious spectacle of the slowly turning night sky. If you wash 
only to forecast eclipses, planetary positions, and the seasons, 
you could, like the Babylonians and Egyptians, focus your 
attention on recording the details of the cycles and rhythms. 
Suppose, however, like the Greeks, you wish to explain these 
cycles. Then you must also use your data and imagination to 



SG 1 



The motions of these bodies, essen- 
tially the same as they were thou- 
sands of years ago, are not difficult 
to observe. You should make a 
point of doing so. The Handbook 
has many suggestions for observing 
the sky, both with the naked eye 
and with a smaU telescope. 



CHAPTER 5 / WHERE IS THE EARTH? 



133 




The midr^ight sun, photographed at 
5-min inten'als over the Ross Sea 
in Antarctica. 



This description is for observers in 
the northern hemisphere. For ob- 
ser\'ers south of the equator, ex- 
change "north" and "south." 



constiTJct some sort of simple model or theory with which you 
can predict the obsen^ed variations. Before you explore sexeral 
theories pioposed in the past, re\ievv the major obser\'iitions 
which the theories tried to explain: the motions of the sun, 
moon, planets, and stars. 

The most basic celestial cycle as seen from the earth is, of 
course, that of day and night. Each day tiie sun rises aboxe the 
local horizon on the eastern side of the sky and sets on the 
western side. The sun follows an arc across the sky, as is 
sketched in part (ai of the diagram at the top of the next page. At 
noon, halfway between sunrise and sunset, the sun is highest 
above the horizon. Exeiy day, it follows a similar path from 
sunrise to sunset, hideed all the objects in the sky show this 
pattern of daily motion. They all rise in the east, reach a high 
point, and drop lower in the west. iHowexer, some stars nexer 
actually sink below the horizon. I 

As the seasons change, so do the details of the sun s path 
across the sky. In the northern hemisphere duiing winter, the 
sun rises and sets moie to the south. Its altitude at noon is 
lower, and so its run across the sky lasts for a shorter period of 
time. In summer' the sun rises and sets more toward the north. 
Its height at noon is greater, and its track across the sky lasts a 
longer time. The whole cycle takes a little less than 365 '/» days. 
In the southern hemisphere the pattern is similar, but is 
displaced bv' half a year. 



134 



LJ,\IT 2 / iVlO TIO\ I.N THE HEAVliIVS 



This year-long cycle north and south is the basis for the 
seasonal or "solar" year. Apparently, the ancient Egyptians once 
thought that the year had 360 days, but they later added five 
feast days to have a year of 365 days. This longer year agreed 
better with their observations of the seasons. Now^ we know that 
the solar year is 365.24220 days long. The decimal fraction 
0.24220 raises a problem for the calendar maker, who works with 
whole days. If you used a calendar of just 365 days, after four 
years New Year's Day would come early by one day. In a century, 
you would be in error by almost a month. In a few centuries, the 
date called Januaiy 1 would come in the summertime! In ancient 
times, extra days or even vv^hole months were inserted from time 
to time to keep a calendar of 365 days in fair agreement with the 
seasons. 

Such a makeshift calendar is, however, hardly satisfactory. In 
45 B.C., Julius Caesar decreed a new 365-day calendar (the Julian 
calendar) with one extra whole day (a "leap day") inserted each 
fourth year. Over many years, the average would therefore be 
365 V4 days per year. This calendar was used for centuries, during 
which the small difference between '/t (0.25) and 0.24220 added 
up to several days. Finally, in 1582 a.d. Pope Gregory announced 
a new calendar (the Gregorian calendar). This calendar had only 
97 leap days in 400 years, and the new approximation has lasted 
satisfactorily to this day wdthout revision. 

You may have noticed that a few stars are bright and many are 
faint. The brighter stars may seem to be larger, but if you look 
at them through binoculars, they still appear as points of light. 
Some bright stars show colors, but most appear whitish. People 
have grouped many of the brighter stars into patterns, called 
constellations. Examples of constellations include the familiar Big 
Dipper and Orion. 

You may have also noticed a particular pattern of stars 
overhead and then several hours later noticed it low in the west. 
What was happening? More detailed observation, for example, 
by taking a time-exposure photograph, would show that the 
entire bowl of stars had moved from east to west. New stars had 
risen in the east, and others had set in the west. As seen from 
the northern hemisphere, during the night the stars appear to 
move counterclockwdse around a point in the sky called the 
north celestial pole. This stationary' point is near the fairly bright 
star Polaris (see the photograph at the top of page 136). 

Some star patterns, such as Orion (the Hunter) and Cygnus 
(the Swan, also called the Nor^thern Cr^oss), wer^e described and 
named thousands of yearns ago. Since the star patterns described 
by the ancients still fit, we can conclude that the star positions 
change very little, if at all, over the centuries. This constancy of 
relative positions has led to the terTn "fixed stars. ' 

Thus, we observe in the heavens unchanging relationships over 
the centuries and smooth, orderly motions each day. But, 









-«)A 






rjjiiiiii 



(a) Path of the sun through the sky 
for one day of summer and one 
day of winter. 

(b) Noon altitude of the sun as seen 
from St. Louis, Missouri, through- 
out the year. 

SG 2 
SG3 

To reduce the number of the leap 
days from 100 to 97 in 400 years, 
century years not divisible by 400 
were omitted as leap years. Thus, 
the year 1900 was not a leap year, 
but the year 2000 will be a leap 
year. 

SG4 



SG5 



A very easy but precise way to time 
the motions of the stars is ex- 
plained in the Handbook. 



CHAPTER 5 / WHERE IS THE EARTH? 



135 





A combination trail and star photo- 
graph of the constellation Orion. 
The camera shutter was opened 
for several hours while the stars 
moved across the sky (leaving trails 
on the photographic plate). Then 
the camera was closed for a few 
minutes and reopened while the 
camera was moved to follow the 
stars. 

Flight: 

Time-exposure showing star trails 
around the north celestial pole. 
The diagonal line was caused by 
the rapid passage of an artificial 
earth satellite. 

You can use a protractor to de- 
termine the duration of the expo- 
sure; the stars appear to move 
about 15° per hour. 



although the daily rising and setting cycles of the sun and stars 
are similar, they are not identical. Unlike the sun's path, the 
paths of the stars do not vary in altitude from season to season. 
Also, stars do not have quite the same rhythm of rising and 
setting as the sun, but go a little faster. Some constellations seen 
high in the sky soon after sunset appear low in the west at the 
same time several weeks later. As measured by sun-time, the 
stars set about 4 minutes earlier each da\^ 

Thus far, the positions and motions of the sun and stars have 
been described in relation to the observer's horizon. But different 
obser\'ers have different horizons. Therefore, the horizon cannot 
be used as a frame of reference from which all observers will see 
the same positions and motions in the sky. However, the fixed 
stars pro\ide a ft ame of reference which is the same for all 
obsen^ers. The positions of these stars relative to one another do 
not change as the observer moves over the earth. Also, their daily 



• • • 



URSA MAJOR 



URSA MINOR 



• • 



CASSIOPEIA 



CYGNUS 



LYRA 



»-. 



ORIW 






136 



UNIT 2 / MOTION IN 1 HE HEAV ENS 



motions are simple circles with almost no changes during a year 
or through the years. For this reason, positions in the heavens 
are usually described in terms of a frame of reference defined by 
the stars. 

A description of the sun's motion must include the daily 
crossing of the sky, the daily difference in rising and setting 
times, and the seasonal change in noon altitude. You have 
already seen that, as measured by sun-time, each star sets about 
4 minutes earlier each day than it did the previous day; it goes 
ahead of the sun toward the west. You can just as well say that, 
measured by star-time, the sun sets about 4 minutes later each 
day; that is, the sun appears gradually to slip behind the daily 
east-to-west motion of the stars. In other words, the sun moves 
very slowly eastward against a background of "fixed " stars. 

The difference in noon altitude of the sun during the year 
corresponds to a drift of the sun's path north and south on the 
background of stars. In the first diagram below, the middle 
portion of the sky is represented by a band around the earth. 
The sun's yearly path against this background of stars is 
represented by the dark line. If you cut and flatten out this band, 
as showm in the second and third diagrams, you get a chart of 
the sun's path during the year. (The 0° line is the celestial 
equator, an imaginary line in the sky directly above the earth's 
equator.) The sun's path against the background of the stars is 
called the ecliptic. Its drift north and south of the celestial 
equator is about 23.5°. You also need to define one point on the 
ecliptic so you can locate the sun or other celestial objects along 
it. For centuries this point has been the place where the sun 
crosses the equator from south to north on about March 21. This 
point is called the vernal (spring) equina^. It is the zero point 
from which positions among the stars usually are measured. 

Thus, there are three apparent motions of the sun: (1) its daily 
westward motion across the sky, (2) its yearly drift eastward 
among the stars, (3J its yearly cycle of north-south drift in noon 
altitude. These cyclic events can be described by using a simple 
model to represent them. 





The differences between the two 
frames of reference (the horizon 
and the fixed stars) are the basis for 
establishing a position on the earth, 
as in navigation. 




In New Me}iico, a construction of 
stone slabs directs beams of sun- 
light onto spiral carvings on a cliff 
face. The light forms changing pat- 
terns throughout the year and also 
marks the solstices and equinoxes. 
This unique astronomical marker 
makes use of the changing height 
of the midday sun throughout the 
year; it was apparently built by the 
Anasazi Indians. The above photo 
shows the light pattern at 11:13 
on the morning of summer sol- 
stice. 





^ 


--^^^^^5t/e 


oelestial 


equator 


Wevnsl e>c|uin 


ox 


"~^~^--^____ 


^____-^-^^^ 







t^arch 2.1 



June 21 



Sept 25 



Dec 21 



March 21 



• 1. If you told time by the stars, would the sun set earlier or 
later each day? 

2. For what practical purposes were calendars needed? 

3. What are the observed motions of the sun during one year? 



These end-of-section questions are 
intended to help you check your 
understanding before going on to 
the next section. 



CHAPTER 5 / WHERE IS THE EARTH? 



137 



S»2 \ Motions of the moon 

The moon shares the general east-to-west daily motion ot the 
sun and stare. But the moon slips eastward against the 
background of the stars faster than the sun does. Each night the 
moon lises nearly 1 hour later. When the moon rises in the east 
at sunset (opposite the sun in the sky', it is a bright, full disc (full 
moon). Each day after that, it rises later and appears less round. 
Finally, it wanes to a thin crescent low in the sky at davvii. After 
about 14 days, the moon is passing near the sun in the sky and 
rising with it. During this time (new mooni, you cannot see the 
moon at all. After the new moon, you first see the moon as a thin 
crescent low in the western sky at sunset. As the moon moves 
rapidly eastward from the sun, its crescent fattens to a half disc 
at first quarter. Within another week, it reaches full moon again. 
After each full moon the cycle repeats itself. 




The moon as it looks 26 days after 
new moon (left); 1 7 days after new 
moon (middle]; and 3 days after 
new moon (right). 



As early as 380 B.C., the Greek philosopher Plato recognized 
that the phases of the moon could be explained by thinking of 
the moon as a globe reflecting sunlight and moving around the 
earth in about 29 days. Because the moon appears so big and 
moves so rapidly compared to the stars, people in earl\' times 
thought that it must be quite close to the eaith. 

Ihe moons path around the sky is close to the yearly path of 
the sun; that is, the moon is always near the ecliptic. But the 
moon's path tilts a bit with respect to the sun's path. If it did 
not, the moon would come e.xactK in front of the sun at e\'ery 
new moon, causing an eclipse of the sun. It would be exactly 
opposite the sun at e\'er\' full moon, moxing into the earth's 
shadow and causing an e(li|)se of the moon. 



138 



UMT 2 iVIOTIOX l\ niL HE \\ L.NS 



The motions of the moon have been studied with great care 
for centuries, partly because of interest in predicting eclipses. 
These motions are veiy complicated. The precise prediction of 
the moon's position is an exacting test for any theory of motion 
in the heavens. 



SG 6 



# 4. Draw a rough diagram to show the relative positions of the 
sun, earth, and moon during each of the moon s four phases. 

5. Why do eclipses not occur each month? 

6. (a) What are the observed motions of the moon during a 
month? 

(b)) How do these motions change during the year? 

(c) What do you think is the origin of the word "month"? Look 

it up in the dictionary. 

S.3 I The ''ivandering" stars 

Without a telescope you can see, in addition to the sun and 
moon, five rather bright objects that move among the stars. These 
are the "wanderers, " or planets: Mercury, Venus, Mars, Jupiter, 
and Saturn. With the aid of telescopes, three more planets have 
been discovered: Uranus, Neptune, and Pluto. None of these 
three planets were known until nearly a century after- the time of 
Isaac Newton. Like the sun and moon, all the planets rise daily 
in the east and set in the west. Also like the sun and moon, the 
planets generally move slowly eastward among the starts. But 
they have another^ r emarkable and puzzling motion of their own. 
At certain times, each planet stops moving eastward among the 
stars and for some months loops back westward. This westward 
or "uTong-way " motion is called retrograde motion. The 
retrograde loops made by Mercury, Mar s, and Saturn during 1 
year are plotted on page 140. 



PL^n&t 



E^rt-h 



The paths of all the planets are close to the sun's path among 
the stars — the ecliptic. Mercury and Venus are always fairly near 
the sun. The greatest angular distance east or west of the sun 
is 28° for Mercury and 48° for Venus. The westward, or 
retrograde, motions of Mercury and Venus begin after the planets 
are farthest east of the sun and visible in the evening sky. The 
other planets may have any position relative to the sun. Their 
westward retrograde motion occurs around the time when they 
are opposite the sun (highest in the sky at midnight). 



When a planet is observed directly 
opposite the sun, the planet is said 
to be in opposition. Retrograde 
motions of Mars, Jupiter, and Sat- 
urn are observed about the time 
they are in opposition. 



CHAPTER 5 / WHERE IS THE EARTH? 



139 



«5* 



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Mars 



The retrograde motions of Mer- 
cury (marked at 5-day inten^als), 
Mars (at 10-day intervals), and Sat- 
urn (at 20-day intervals) in 1963, 
plotted on a star chart. The dotted 
line is the annual path of the sun, 
called the ecliptic. 



Saturn 




yph 



2'/* 



-20 
310' 



140 



UMT 2 / MOTION l\ IHE HEA\ EXS 



Sun 





The ma;<.imum angles from the sun 
at which Mercury and Venus can 
be observed. Both planets can, at 
timesi be observed at sunset or 
at sunrise. Mercury is never more 
than 28° from the sun, and Venus is 
never more than 48° from the sun. 



Earth 



The planets change considerably in brightness. When Venus 
first appears in the evening sky as the "evening star," it is only 
fairly bright. During the folloudng 4-5 months, it moves farther 
eastward from the sun. Gradually it becomes so bright that it 
often can be seen in daytime if the air is clear. A few weeks later, 
Venus moves westward toward the sun. It fades rapidly, passes 
the sun, and soon reappears in the morning sky before sunrise 
as the "morning star." Then it goes through the same pattern, 
but in the opposite order: bright at first, then gradually fading. 
Mercury follows much the same pattern. But because Mercuiy is 
seen only near the sun (that is, only during twilight), its changes 
are difficult to observe. 

Mars, Jupiter, and Saturn are brightest about the time that they 
are in retrograde motion and opposite the sun. Yet over many 
years their maximum brightness differs. The change is most 
obvious for Mars; the planet is brightest when it is opposite the 
sun during August or September. 

The sun, moon, and planets generally slip behind as the 
celestial sphere goes around the earth each day, and thus they 
appear to move eastward among the stars. Also, the moon and 
planets (except Pluto) are always found vvdthin a band, called the 
Zodiac, only 8° udde on either side of the sun's path. 

These, then, are some of the main observations of celestial 
phenomena. All of them were known to the ancients. In their day 
as in ours, the puzzling regularities and variations seemed to cry 
out for some explanation. 



SG 7 



• 7. In what part of the sky must you look to see the planets 
Mercury and Venus? 

8. In what part of the sky would you look to see a planet that 
is in opposition to the sun? 

9. When do Mercury and Venus show retrograde motion? 



CHAPTER 5 / WHERE IS THE EARTH? 



141 



10. when do Mars, Jupiter, and Saturn show retrograde 
motion':' 

11. Can Mars, Jupiter, and Saturn appear any place in the sky? 

12. What are the obserx^ed niotions of the planets during the 
year? 



S.4r I Plato's problem 



Several centuries later, a more ma- 
ture Islamic culture led to extensive 
study and scholarly commentan,' 
on the remains of Greek thought. 
Several centuries later still, a more 
mature Clhristian culture used the 
ideas preserved by the Muslims to 
evolve early parts of modern sci- 
ence. 



In the fourth century B.C., Greek philosophers asked new 
questions: How can we explain the cycles of changes obseived in 
the sky? What model can consistently and accurately account 
for the obseived motions? Plato sought a theory to explain what 
u'as seen or, as he phrased it, "to save the appeai^ances." The 
Greeks were among the first people to desire clear, rational 
explanations for natural events. Their attitude was an important 
step toward science as we know it today. 

How did the Greeks begin their explanation of celestial 
motion? What were their assumptions? 

Any answer's to these questions must be partly guesswor'k. 
Many scholar's oxer' the centuries ha\e dexoted themselves to the 
study of Greek thought. But the documents on which our 
knowledge ©f the Greeks is based ar'e mostly copies of copies and 
translations of tr'anslations. Many error's and omissions occur. 
In some cases, all we have are reports from later writer's on what 
certain philosophers did or said. These accounts may be 
distorted or incomplete. The historians s task is diflrcult. Most of 
the original Greek writings were on papyr-us or cloth scrolls 
which have decayed through the ages. Wars, plundering, and 
burning have also destroyed many important documents. 
Especialh' tragic was the burning of the famous library of 
Alexandria in Egypt, which contained sexeral hundred thousand 
documents. (It was bur'ned three times: in part by Caesar 's troops 
in 47 B.C., in the fourth century a.d. by Christians, and about tS40 
A.D. by early Muslims when they overran the country.! The 
general pictur-e of Greek culture is fairly clear, but many 
interesting details are missing. 

The approach to celestial motion taken by the Greeks and their 
intellectual followers for many centuries was outlined by Plato 
in the fourth century B.C. He defined the problem to his students 
in terms of order and status. The stars, Plato said, represent 
eternal, divine, unc;hanging beings. They move at a unilbrm 
speed around the earth in the most regular and perfect of all 
paths, an endless circle. But the sun, moon, arid planets wander 
across the sky by complex paths, including e\en letrogiade 
motions. Yet, being heavenly bodies, surely they too are really 
moving in a way that suits their high status. Their motions, if not 
in a single perfect circle, must be in some combination of peifect 



142 



LMT 2 MO TIO\ I\ IHti HLLW L\S 



circles. What combinations of circular motions at uniform speed 
could account for these strange variations? 

Notice that the problem deals only with the changing apparent 
positions of the sun, moon, and planets. The planets appear to 
be only points of light moving against the background of stars. 
From tw^o obsen/ations at different times, an observer can obtain 
a rate of motion — a value of so many degrees per day. The 
problem then is to invent a "mechanism," some combination of 
motions, that reproduces the obseiA^ed angular motions and 
leads to accurate predictions. The ancient astronomers had no 
data for the distance of the planets from the earth. All they had 
were directions, dates, and rates of angular motion. They did 
know that the changes in brightness of the planets were related 
to their positions with respect to the sun. But these changes in 
brightness were not included in Plato's problem. 

Plato and many other Greek philosophers assumed that there 
were only a few basic "elements." Mixed together, these few 
elements gave rise to the great variety of materials obsen^ed in 
the world. (See Unit 1, Chapter 2.) Perfection could only exist in 
the heavens, which were separate from the earth, and were the 
home of the gods. Just as motions in the heavens must be 
eternal and perfect, the unchanging heavenly objects could not 
contain elements normally found on or near the earth. Therefore, 
they were supposed to consist of a changeless fifth element (or 
quintessence) . 

Plato's problem in explaining the motion of planets remained 
the most important problem in astronomy for nearly 2,000 years. 
In later chapters, you wall explore the different interpretations 
developed by Kepler, Galileo, and Newton. But in order to 
appreciate these efforts, you must first examine the solutions 
offered by the Greeks to Plato's problem. For their time, these 
solutions were useful, intelligent, and, indeed, beautiful. 



# 13. What was Plato's problem of planetary motion? 

14. Why is our knowledge of Greek science incomplete? 

15. Why did the Greeks feel that they should use only uniform 
circular motions to explain celestial phenomena? 

5.5 I The Greek idea of "explanation" 

Plato's statement of this historic problem of planetary motion 
illustrates these major contributions of the Greek philosophers. 
With slight changes, these concepts are still basic to an 
understanding of the nature of physical theories: 

1. A theory should be based on simple ideas. Plato regarded it 
not merely as simple, but also as self-evident, that heavenly 



CHAPTER 5 / WHERE IS THE EARTH? 143 




21 ■^° 



The annual north-south (seasonal) 
motion of the sun was explained 
by having the sun on a sphere 
whose a^is was tilted 23.5° from 
the a^is of the eternal sphere of 
the stars. 



bodies must move uniiomily along circular paths. Only in recent 
centuries have scientists learned that such common-sense beliefs 
may be misleading. While unproved assumptions are made at the 
outset, they must be examined closely and never accepted 
without reservation. As you will see often in this course, it has 
been very difficult to identify hidden assumptions in science. Yet 
in many cases, when the assumptions were identified and 
questioned, entirely new theories followed. 

2. Physical theory must agree with the measured results of 
observation oi phenomena, such as the motions of the planets. 
The purpose of a theory is to discover the uniformity of behavior, 
the hidden simplicity underlying apparent irregularities. For 
organizing obseivations, the language of numbers and geometry 
has become useful. Plato stressed the ftmdamental role of 
numerical data only in his astronomy, while Aristotle largely 
avoided detailed measurements. This was unfortunate because, 
as reported in the Prologue, Aristotle greatly influenced later 
scholars. His arguments, which gave little attention to the idea of 
measurement of change as a tool of knowledge, were adopted 
centuries afterward by such important philosophers as Thomas 
Aquinas. 

3. To "explain" complex phenomena means to develop or 
invent a physical model, or a geometrical or other mathematical 
construction. This model must reproduce the same features as 
the phenomena to be explained. For example, v\ith a model of 
interlocking spheres, a point on one of the spheres must have 
the same motions as the planet which the point represents. 

^•6 I The first earth-centered solution 

The Greeks observed that the earth was large, solid, and 
peiTnanent. Meanwhile the heavens seemed to be populated by 
small, remote objects that were continually in motion. What was 
more natural than to conclude that the big, heaw earth was the 
steady, unmoving center of the universe? Such an earth-centered 
viewpoint is called geocentric. From this viewpoint the daily 
motion of the stars could be explained easily: The stars were 
attached to, or were holes in, a large, dark, spherical shell 
surrounding the earth. They were all at the same distance from 
the earth. Daily, this celestial sphere turned once on an axis 
through the earth. As a result, all the stars fixed on it moved in 
circular paths around the pole of rotation. Thus, a simple model 
of a rotating celestial sphere and a stationary earth could explain 
the daily motions of the stars. 

The three observ^ed motions of the sun required a somewhat 
more complex model. To explain the sun's motion with respect 
to the stars, a separate, imisible shell was imagined. This shell 
was fixed to the celestial sphere and shared its daily motion. But 



144 



UNIT 2 I MOTION IN THE HEAVENS 



it also had a slow, opposite motion of its own, amounting to one 
360° cycle per year. The yearly north-south motion of the sun 
was accounted for by tilting the axis of its sphere. This 
adjustment matched the 23.5° tilt of the sun's path from the axis 
of the dome of stars. 

The motions of the visible planets (Mercury, Venus, Mars, 
Jupiter, and Saturn) were more difficult to explain. These planets 
share generally the daily motion of the stars, but they also have 
peculiar motions of their own. Saturn moves most slowly among 
the stars, revolving once in 30 years. Therefore, its sphere was 
assumed to be largest and closest to the stars. Inside the sphere 
of Saturn were spheres cariying the faster-moving Jupiter (12 
years) and Mars (687 days). Since they all require more than a 
year for a complete trip among the stars, these three planets 
were believed to lie beyond the sphere of the sun. Venus, 
Mercury^ and the moon were placed between the sun and the 
earth. The fast-moving moon was assumed to reflect sunlight and 
to be closest to the earth. 

This imaginary system of transparent shells or spheres 
provided a rough model for explaining the general motions of 
heavenly objects. By choosing the sizes of the spheres and their 
rates and direction of motions, one could roughly match the 
model with the obseivations. If additional obseivations revealed 
other cyclic variations, more spheres could be added to adjust 
the model. (An interesting description of this general system of 
cosmology appears in the Divine Comedy, written by the poet 
Dante about 1300 a.d. This was shortly after Aristotle's wintings 
became known in Europe.) 

You may feel that Greek science was bad science because it 
was different from our own or because it was less accurate. You 
should understand from your study of this chapter that such a 
conclusion is not justified. The Greeks were just beginning the 
development of scientific theories. Naturally, they made 
assumptions that appear odd or inaccurate to us today. Their 
science was not 'bad science, " but in many ways it was a 
different kind of science from ours. And our science is not the 
last word, either. You must realize that to scientists 2,000 years 
from now our efforts may seem clumsy and strange. 

Even today's scientific theory does not and cannot claim to 
account for every detail of every specific situation. Scientific 
concepts are general ideas which treat only selected aspects of 
observations. Thev do not cover the whole mass of raw data and 
raw experience that e^cists in the universe. Also, each period in 
history puts its own limits on the range of human imagination. 
As you learned in Unit 1, important general concepts such as 
force and acceleration were invented specifically to help organize 
observations. Such concepts are human inventions. 

The history of science contains many cases in which certain 
factors overlooked by one researcher later turn out to be very 



Dt X.// COS UOGK.ytr H. 

La Figure & nombrc 

dcs Sphacs. 




Dcs Ccrclcs dc la Sphere. Chap. 1 1 I 
Quelle chofe eft la Sphere. 

Quelle choft eft 1 exieu dc la Sphere, 

Li«,« dc U Sphm(com, Jicl D^odocliMJ <Ji U diamtnt J»i fof- 



u ditnt fn^erficc rondt,4U mitlieu 
Ui lim<s'j>*i enfi'icprorratfies 



A geocentric cosmological scheme. 
The earth is fi^ed at the center of 
concentric rotating spheres. The 
sphere of the moon (lune) sepa- 
rates the terrestrial region (com- 
posed of concentric shells of the 
four elements Earth, Water, Air, 
and Fire) from the celestial region. 
In the latter are the concentric 
spheres carrying Mercury, Venus, 
Sun, Mars, Jupiter, Saturn, and the 
stars. To simplify the diagram, only 
one sphere is shown for each 
planet. (From the DeGolyer copy of 
Petrus Apianus Cosmographia, 
1551.) 



CHAPTER 5 / WHERE IS THE EARTH? 



145 



Sun 




As the earth passes a planet in its 
orbit around the sun, the planet 
appears to move backwards in the 
sh,'. The arrows show the sight 
lines toward the planet for the dif- 
ferent numbered positions of the 
earth. The lower-numbered circles 
symbolize the resulting apparent 
positions of the planet when seen 
against the background of the dis- 
tant stars. 



important. But how would better systems for making predictions 
be developed without first trials? Theories are impro\ed through 
tests and revisions, and sometimes are completely replaced by 
better ones. 



• 16. What is a geocentric system? How does it account for the 
motions of the sun? 
17. Describe the first solution to Plato's problem. 

^•T I A sun-centered solution 

For nearly 2,000 years after Plato and Aristotle, the basic 
geocentric model was generally accepted, though scholai"s 
debated certain details. But a veiy different model, based on 
different assumptions, had been pro[)os(xl in the third ccmtury 
B.C. The astronomer Aristarchus, peihaps influenced by the 
earlier writings of Heracleides, offered this new model. 
Aristarchus suggested that a simpler explanation of heavenly 
motion would place the light-giving sun at the center, with the 
earth, planets, and stars all revolving around it. A sun-centered 
system is called heliocentric. 

Aristarchus proposed that the celestial sphere is motionless 
and that the earth rotates once daily on an axis of its own. He 
believed that this assumption could explain all the daily motions 
observed in the sky. In this heliocentric system, the apparent tilt 
of the paths of the sun, moon, and all the planets results from 
the tilt of the earth's own axis. The vearK' changes in the sky, 
including retrograde motions of planets, are explained by 
assuming that the earth and the planets revolve around the sun. 
In this model, the motion previously assigned to the sun around 
the earth is assigned to the earth moving around the sun. Also, 
the earth becomes just one among several planets. The planets 
are not the homes of gods, but are now considered to be bodies 
rather like the earth. 

The diagram in the margin shows how such a system can 
explain the retrograde motions of Mais, Jupitei', and Saturn. ,An 
outer planet and the earth are assumed to be moving around the 
sun in circular orbits. The outer planet moves more slowly than 
the earth. As a result, when the earth is diiectlv between the sun 
and the planet, the earth moves lapidlv past the planet. To us 
the planet appears for a time to be moving backward in 
retrograde motion aci'oss the skv. 

The interlocking sphei-es are no longer- needed. The 
heliocentric hypothesis has one further advantage. It explains the 
obserA'ation that the planets are bi ighter- dur ing retrograde 
motion, since at tliat tinre the [)lanets are nearer to [he earth. 



146 



U.MT 2 I MOTIOM Ii\ THE HEA\Ti,\S 



Even so, the proposal by Aristarchus was severely criticized for 
three basic reasons. The idea of a moving earth contradicted the 
philosophical doctrines that the earth is different from the 
celestial bodies and that its natural place is at the center of the 
universe. In fact, his contemporaries considered Aristarchus 
impious for even suggesting that the earth moved. Also, this new 
picture of the solar system contradicted common sense and 
everyday obseivations: The earth certainly seemed to be at rest 
rather than rushing through space. 

Another criticism was that expected motions of the stars were 
not observed. If the earth moved in an orbit around the sun, it 
would also move back and forth under the fixed stars. As shown 
in the sketch in the margin, the angle from the vertical at which 
any star is seen would be different for various points in the 
earth's annual path. This shift is called paralla^. This difference 
was not obsei-ved by the Greek astronomers. This awkward fact 
could be explained in two ways: either (1) the earth does not go 
around the sun and there is no shift, or (2) the earth does go 
around the sun but the stars are so far away that the shift is too 
small to observe. As the Greeks realized, for the shift to be too 
small to detect, the stars must be enomiously far away. 

Today, the annual shift of the stars can be observed with 
telescopes; thus, Aristarchus' model is in fact useful. The shift is 
so small that even with telescopes it was not measured until 
1838. The largest annual shift is an angle of only 1/100 of the 
smallest angle observable by the human eye. The shift exists, but 
we can sympathize with the Greeks who rejected the heliocentric 
theory partly because they could not observe the required shift. 
Only Aristarchus imagined that the stars might be as immensely 
distant as we now know them to be. 

Finally, Aristar chus was criticized because he did not develop 
his system in detail or use it to predict planetary positions. His 
work seems to have been purely qualitative, a general scheme of 
how things might be. 

The geocentric and heliocentr ic systems offer ed two different 
ways of explaining the same observations. The heliocentric 
proposal required such a drastic change in people's image of the 
universe that Aristarchus' hypothesis had little influence on 
Greek thought. Fortunately, his arguments were recorded and 
handed down. Eighteen centuries later, they gained new life in 
the thoughts of Copernicus. Ideas are not bound by space or 
time. 




&«r-t.h 



// the earth goes around the sun, 
then the direction in which we have 
to look for a star should change 
during the year. A shift in the rela- 
tive observed positions of objects 
that is caused by a displacement of 
the observer is called a parallax. 
The greatest observed parallax of a 
star caused by the earth's annual 
motion around the sun is about 
1/2400°. This is explained by the 
fact that the distance to this near- 
est star is not just hundreds of mil- 
lions of kilometers but 40 million 
million kilometers. 



• 18. What two new assumptions were made by Aristarchus? 
What simplification resulted? 

19. How can the heliocentric model proposed by Aristarchus 
explain retrograde motion? 



CHAPTER 5 / WHERE IS THE EARTH? 



147 



20. What change predicted by Aristarchus' theory was not 
obscrx'cd by the Greeks':' 

21. Why was Aristarchus considered impious? Why was his 
system neglected? 

5.8 I The geocentric system of Ptolemy 

Disregarding the heliocentric model suggested by Aristarchus, 
the Greeks continued to develop their geocentric system. As 
noted, the first solutions in terms of interiocking spheres lacked 
accuracy. During the 500 years after Plato and Aristotle, 
astronomers began to seek more accurate predictions. To fit the 
obseived data, a complex mathematical theory was required for 
each planet. 

Several Greek astronomers made imporiant contributions, 
which climaxed about 150 A.u. in the geocentric theoiy of 
Claudius Ptolemy of Alexandria. Ptolemy's book on the motions 

of heavenly objects is a masterpiece of analysis. 

The Arabic title given to Ptolemy's Ptolemy wanted a system that would predict accurately the 

book, the Almagest, means "the positions of each planet. The type of system and the motions he 
°^^^ ^ ' accepted were based on the assumptions of Aristotle. In the 

preface of his Almagest, Ptolemy defined the problem and stated 
his assumptions as follows: 

. . . we wish to find the evident and certain appearances from 
the observations of the ancients and our own, and applving the 
consequences of these conceptions by means of geometrical 
demonstrations. 

And so, in general, vvc have to state, that tiie heavens are 
spherical and move spherically; that the earth, in tigure, is 
sensibly spherical . . .; in position, lies right in the middle of the 
heavens, like a geometrical center; in magnitude and distance, 
[the earth) lias the ratio of a point uith respect to the sphere of 
the fixed stars, liaving itself no local motion at all. 

Ptolemy then argued that each of these assumptions was 
necessary and fit all the observations. The str^ength of his belief is 
illustrated by his statement "... it is once for all clear from the 
very appearances that the earth is in the middle of the world 
and all weights move towards it." Notice that he supported his 
interpretation of astr^onomical observations by citing the physics 
of falling bodies. Later, Ptolemy applied this mixture of 
astronomy and physics to the earth itself and to its place in the 
scheme. In doing so, Ptolemy tried to disprove Aristarchus' idea 
that the earth might rotate and revolve: 

Now some people, although they lia\e nothing to oppose to 
these arguments, agree on something, as they think, moi-e 
plausible. And it seems to them there is nothing af^ainst their 

148 UMT 2 / MO IKJ.X l\ THE HEAVENS 



supposing, for instance, tlie heavens immobile and the earth as 
turning on the same axis [as the stars] from west to east very 
nearly one revolution a day. . . . 

But it has escaped their notice that, indeed, as far as the 
appearances of the stars are concerned, nothin ' would perhaps 
keep things ft cm being in accordance with this ;impler 
conjecture, but that in the light of what happens around us in 
the air such a notion would seem altogether absurd. 

Ptolemy believed that if the earth rotated, it would not pull its 
blanket of air around with it. As a result, all clouds would fly 
past toward the west. All birds and other things in the air also 
would be carried away to the west. Even if the earth did drag the 
air along with it, objects in the air would still tend to be left 
behind by the earth and air together. 

The paragraphs quoted above contain a main theme of Unit 2. 
Ptolemy recognized that the two systems were equally successful 
in describing motion, that is, in terms of kinematics. He preferred 
the geocentric theory because it fit better the causes of motion, 
that is, the dynamics, as accepted at the time. Much later, when 
Newton developed a completely different dynamics, the choice 
fell the other way. 

Ptolemy developed veiy clever and rather accurate procedures 
for predicting the positions of each planet on a geocentric 
model. He went far beyond the scheme of the earlier Greeks, 
constnjcting a model out of circles and three other geometrical 
devices. Each device provided for variations in .e rate of angular 
motion as seen from the earth. In order to appi ;ciate Ptolemy's 
solution, examine one of the very small variations he was 
attempting to explain. 

The sun's yearly 360° path across the background of stars can 
be divided into four 90° parts. If the sun is at the zero point on 
March 21, it will be 90° farther east on June 21, 90° farther still on 
September 23, another 90° farther on December 22, and back at 
the starting point on March 21, one whole year later. If the sun 
moves uniformly on a circle around the earth, the times between 
these dates ought to be equal. But, as you will find by consulting 
a calendar, they are not equal. The sun takes a few days longer 
to move 90° in spring or summer than it does in fall or vvdnter. So 
any simple circular system based on motion vvdth constant speed 
will not work for the sun. 

The three devices that Ptolemy used to improve geocentric 
theory were the eccentric, the epicycle, and the equant. 

Agreeing with Plato, astronomers had held previously that a 
celestial object must move at a unifomi angular rate and at a 
constant distance from the center of the earth. Ptolemy, too, 
believed that the earth was at the center of the universe. But he 
did not insist that it stood at the geometrical centers of all the 
perfect circles. He proposed that the center C of a circle could be 



SG 8 



PUan^t. 




An eccentric 



PLanet. 




An epicycle 



CHAPTER 5 / WHERE IS THE EARTH? 



149 



off-center from the earth, in an eccentric position. Thus, motion 
that was really uniform around the center C would not ap|3ear to 
be uniform when observed from the earth. An eccentric oibit of 
the sun would therefore account for the seasonal \ariation 
obseived in the sun's rate of motion. 

The eccentric can also account for small variations in the rate 
of motion of planets. Howexer, it cannot describe such drastic 
changes as retrograde motion of the planets. To account for 
retrograde motion, Ptolemy used another device, the epicycle (see 
the figure on page 1491. The planet is considered to be moxing at 
a uniform rate on the small epicvle. The center of the epicvcle 
moves at a uniforai rate on a large circle, called the deferent, 
around the earth. 



Retrograde motion created by a 
simple epicycle machine: (a) Stro- 
boscopic photograph of epicyclic 
motion. The flashes were made 
at equal time internals. Note that 
the motion is slowest in the loop. 
(b) Loop seen from near its plane. 




SG 9 



If a planet's speed on the epicycle is greater than the speed of 
the epicycle on the large circle, the ]Dlanet as seen from above 
the system appears to move through loops. When obsened from 
a location near the center of the system, these loops look like the 
retrograde motions actually obseived for planets. The 
photographs above show two \iews of the motions pioduced b\' 
a simple mechanical model, an "epicycle machine." A small light 
takes the place of the planet. The photo on the left was taken 
fi'om 'abo\'e," like the diagram on page 149. Ihe photo on the 
right was taken "on edge," almost in the plane of the motion. 
Thus, the loop looks much as it w ould if \iewed from near the 
center. 

With epicycles it was not loo difficult to produce a system that 
had all the main features of obsened planetary motion. Ptolem\' s 
system included a unique pattern for the epicycles foi- the outer 
planets. All had the same period, exactly 1 year! Moreover, the 
positions of the outer planets on their epicx'cles alwaxs matched 
the position of the sun relatixe to the earth. See the sketches ip. 
151) for this matching of epicyles to the relative motion of sun 
and earth. Fourteen centuries later, this peculiar feature became 
a ke\' point of concern to Copernicus. 



150 



UMT 2 / MO riO\ IN THE HEAVENS 




Simplified representation of the 
Ptolemaic system. The scale of the 
upper drawing, which shows the 
planets between the earth and the 
sun, is eight times that of the lower 
drawing, which shows the planets 
that are beyond the sun. The 
planets' epicycles are shown along 
one straight line to emphasize the 
relative sizes of the epicycles. 



Mars plotted at 4-day intervals on 
three consecutive oppositions. 
Note the different sizes and shapes 
of the retrograde curves. 






CHAPTER 5 / WHERE IS THE EARTH? 



151 



Ptolemy did not picture the plane- 
tary motions as those of an inter- 
locking machine in which each 
planet determined the motion of 
the next. Because there was no in- 
formation about the distances of 
the planets, Ptolemy adopted an old 
order of distances from the ejirth: 
stars being the most remote, then 
Saturn, Jupiter, Mars, the sun, Ve- 
nus, MercuPk', and the moon. The 
orbits were usually shown nested 
inside one another so that their epi- 
cycles did not overlap. 




An equant. C is the center of the 
circle. The planet P moves at a uni- 
form rate around the off-center 
point C. The earth is also off cen- 
ter. 

SG 10 

Astronomical observations were all 
obsenations of angles. A small loop 
in the sky could be a small loop 
fairly near, or a lai'ger loop much 
farther away. 



SG 11 



So far, the system of epicycles and deferents "works" well 
enough. It explains not only letrograde motion, but also the 
greater brightness of the planets when they are in retrograde 
motion. Then a planet is closest to the eai'th, and so appears 
brightest. This is an unexpected bonus, since the model was not 
designed to explain the brightness change. 

Even with combinations of eccentiics and epicycles, Ptolemy 
could not fit the motions of the five planets exactly. For example, 
as you see in the three figures on page 151, the retrograde 
motion of Mars is not always of the same angular size or 
duration. To allow for such difficulties, Ptolemy used a third 
geometrical device, called the equant. The equant is a variation 
of the eccentric with the uniform motion about an off-center 
point C. 

^•9 I Successes and limitations of the Ptolemaic 
model 

Ptolemy's model always used a uniform rate of angular motion 
around some center. To that extent, it stayed close to the 
assumptions of Plato. But, to fit the observations, Ptolemy was 
willing to displace the centers of motion from the center of the 
earth as much as necessary. By combining eccentrics, epicycles, 
and equants, he described the positions of each planet 
separately. For each planet, Ptolemy found a combination of 
motions that predicted its observed positions over long periods 
of time. These predictions were accurate to within about 2° 
Ir'oughH' four diameters of the moonj. This accuracy was a gr-eat 
improvement over earlier systems. 

Ptolemy's model was quite successful, especially in its 
unexpected explanation of varying brightness. Such success 
might be taken as proof that objects in the sky actually move on 
epicycles and equants around off-center points. Ptolemy did not 
believe he was pr oxiding an actual physical model of the 
universe. He created a mathematical model, like equations, for 
computing positions. 

The Ptolemaic model was a series of mathematical dexices 
meant to match and predict the motion of each planet 
separ'ately. His geometrical analyses xvere like complicated 
equations of motion for- each individual planet. But in the 
following centuries most scholat^s, including the poet Dante, 
accepted the model as r^eal. They actuallx' beliexed that the 
planets mox'ed on transparent, inxisible spheres. Also, the}' felt 
that somehow the motion of all these separ ate spheres should be 
related. IrT Ptolemy's original work, each planet xvas independent 
of the other s. 

Ptolemy proposed his model of the planetary system in 150 
AD. Although it is noxv discarded, it xvas used for- about 1,500 
year's, iher'e xver-e good reasons lor- this long acceptance. 



1S2 



UMT 2 / MOTION IIVI THE HEAMiNS 



1. It predicted fairly accurately the positions of the sun, moon, 
and planets. 

2. It explained why the fixed stars do not show an annual shift 
(parallax) when obsen^ed with the naked eye. 

3. It agreed in most details with philosophies developed by the 
early Greeks, including the ideas of "natural motion" and 
"natural place. ' 

4. It had common-sense appeal to all who saw the sun, moon, 
planets, and stars moving around them. 

5. It agreed with the comforting assumption that we live on an 
unmoxdng earth at the center of the universe. 

6. Later, it fitted into Thomas Aquinas' widely accepted 
synthesis of Christian belief and Aristotelian physics. 

Yet Ptolemy's system eventually was displaced by a 
heliocentric one. Why did this occur? What advantages did the 
new theory have over the old? From this historic argument about 
competing theories, w^hat can you learn about the relative value 
of rival theories in science today? These are some of the 
questions to consider in the next chapter. 



SG 12 



Study 
guide 



1. The Project Physics learning materials 
particularly appropriate for Chapter 5 include: 

Experiments 

Naked-Eye Astronomy 

Size of the Earth 

The Distance to the Moon 

Height of Piton — A Mountain on the Moon 

Retrograde Motion 

Activities 

Making Angular Measurements 
Epicycles and Retrograde Motion 
Celestial Sphere Model 
How Long is a Sidereal Day? 
Scale Model of the Solar System 
Build a Sundial 
Plot an Analemma 
Stonehenge 



Moon Crater Names 
Literature 

Film Strip 

Retrograde Motion of Mars 

Film Loops 

Retrograde Motion — Geocentric Model 

2. How could you use the shadow cast by a vertical 
stick on horizontal ground to find 

(a) the local noon? 

(b) which day was June 21st? 

(c) the length of a solar year? 

3. What is the difference between 365.24220 days 
and 365 '/i days (a) in seconds (bl in percent? 

4. (a) List the observations of the motions of 
heavenly bodies that you might make which would 
also have been possible in ancient Greek times. 

(b) For each observation, list some reasons why the 
Greeks thought these motions were important. 



CHAPTER 5 / STUDY GUIDE 



153 



sg 



5. Which of the apparent motions of the stars 
could be exjjlained by a flat eiU'th and stars fixed to a 
bowl that rotated around it? 

6. Describe the obsened motion of the moon 
during one month, using drawings. (Use your own 
observations if possible. i 

7. Mercur\' and Venus show retrograde motion 
after they ha\ e been farthest east of the sun and 
visible in the evening sky. Then they quickly move 
ahead westward toward the sun, pass it, and 
reappear in the morning sky. During this motion 
they are moving westward relative to the stars, as is 
show II l)v the plot of Merruiy on page 140. Describe 
the rest of the cyclic motion of Mercur\' and Venus. 

8. Center a protractor on point C in the top 
diagram on page 149 and measure the number of 
degrecis in the four quadrants. Consider each 1° 
around C as one day. Make a table of the days 
needed for the planet to move through the four arcs 
as seen from the earth. 

9. (a) How many degrees of terrestrial longitude 
does the sun move each hour? 

(b) What rough value for the diameter of the earth 
can you obtain from the following information: 

(1) Washington, D.C., and San Francisco have 
about the same latitude. How can one easily test 
this? 

(2) A nonstop jet plane, going up wind at a ground 
speed of 800 km/hr from Washington, 13. C., to San 
Francisco, takes 5 hr to get there. 



(3) When it is just sunset in Washington, D.C., a 
person th(!r(! turns on a 'l\' sv.\ to watch a l)a.sebcdl 
game that is just beginning in San I'raiK isco. The 
game goes into extra innings. /Vlter 3 hr the 
announcer notes that the last out occurred just as 
the sun set. 

10. In Ptolemy's theory of the planetarv' motions 
there were, as in all theories, a numl)er of 
assumptions. Which of tiie follovxing did Ptolemy 
assume? 

(a) The vault of stfirs is spherical in form. 

(b) The earth has no motions. 

(c) The e£irth is spherical. 

(d) The earth is at the center of the sphere of stars. 

(e) The size of the earth is extremely small 
compared to the distance to the stars. 

(f) Uniform angular motion along circles (even if 
measured from an off-center point i is the only 
proper behavior for celestial objects. 

11. As far as the Greeks were concerned, and indeed 
as far as we are concerned, a reasonable argument 
can be made for either the geocentric or the 
heliocentric theorv' of the universe. 

(a) In what ways were both ideas successful? 

(b) In terms of Greek science, what are some 
advantages and disadvantages of each system? 

(c) What were the major contributions of Ptolemy? 

12. Why was astronomy the first successful science, 
rather than, for example, meteorology or zoology? 



154 



CHAPILK 5 / S rLU\ GLIUE 




Does the Earth Moue? 

The Work of Copernicus and Tyoho 



6.1 The Copernican svstem 

6.2 Xen' conclusions 

6.3 Ar^ments for the Copernican system 

6.4 Ar^ments against the Copernican system 

6.5 Historical consequences 

6.6 Tycho Brahe 

6.7 Tycho's observations 

6.8 Tycho's compromise system 



6.1. I The Copernican system 

Nicolaus Copernicus (1473-1543) was a young student in Poland 
when America was discovered by Europeans. An outstanding 
astronomer and mathematician, Copernicus was also a talented 
and respected churchman, jurist, administrator, diplomat, 
physician, and economist. During his studies in Italy he read the 
writings of Greek and other early philosophers and astronomers. 
As Canon of the Cathedral of Frauenberg he was busy with civdc 
and church affairs and also worked on calendar refomi. It is said 
that on the day of his death in 1543, he saw the first copy of his 
great book, on which he had worked most of his life. It was this 
book which opened a whole new vision of the universe. 
Copernicus titled his book De Revolutionibus Orbium 
Coelestium, or On the Revolutions of the Heavenly Spheres. This 




Nicolas Copernicus (1473-1543). (In 
Polish his name was Koppernigk, 
but, in keeping with the scholarly 
tradition of the age, he gave it the 
Latin form Copernicus.) 

SG 1 



CHAPTER 6 / DOES THE EARTH MOVE? 



155 



I 

1350 



1400 



1450 



I 
1473 ' ^1^^ 1543 

COPERNICUS 



I 

1550 



I 
1600 



-5 E 



JMCHARD Ij 



LORENZO DE MEDICI 

ISABELLA OF CASTILE 
1 
FERDINAND of Aragon | 



EIZABETH I of England 

1 

HyAN THE TERRIBLE ol Russia 

HENRY IV of France ^ 



RICHARD III HENRY VIII of England) 



I 

MONTEZUMA of Mexico 



MARY STUART 



PHINCb Hk'MWfflirWSvlgator 

igl^STO PHER COLUMBU 




TYCHO BRAHE 

I 

GIORDANO BR( 

JOHANNES KEPLEHfc^ 




JEAN FERNEL ^ 

I 
ANDREAS VESALIUSj 

1 , 
illMBROISE PARE 
I 

WILLIAM QILBEf 




ITHOMAS A KEMPU 

I 

"TOhn huss^h 



JOAN ot ARC 



MARTIN LUTHER 
PAVONAR&LA 
ERASMU 
JA^CHIAVELLI 

THOMAS MORE ^gf^ 
IGNATIUS LOYOLA 




PETRARCH 

1 
GEOFFREY CHAUCER 

i 
JEAN FROISSART 



RABELAIS 



FRANCOIS VILLOl^ 



SHAKESPEARE > 



CERVANTI 
EDMUND iSPENSER 



OONATELL 



IFRA FILLIPO LIPPI 




pXfimRO bOTTICELLI 1H |tfl^T6^ETT0 



li^EONARDO DA VINCI 



PIETER BRUEGMEf 



JflJ^yg^A^j^g^^ 



ROGIER VAN DER WEYDEN 



RAPHAEL 



EL GRECO 




JOSQUIN DEPRES 



WILLIAM BYRD 



GUILLAUME DUFAY 
■^UNSTABLE 




PALESTRINI 

I 

fANDREA GABRIELLI 



title suggests the early Greek notions of the spheres. Copernicus 
was indeed concerned with the old problem of Plato: how to 
construct a planetary system by combinations of the fewest 
possible unifoiTH circular motions. He began his study to rid the 
Ptolemaic system of the equants, which seemed contrary to 
Plato's assumptions. In his words, taken from a short summary 
written about 1512, 

. . . the planetaiy theories of Ptolemy and most other 
astronomers, although consistent with the numerical data, 
seemed likewise to present no small difficulty. For these 
theories were not adequate unless certain equants were also 
conceived; it then appeared that a planet moved with uniform 
velocity neither on its deferent nor about the center of its 
epicycle. Hence a system of this sort seemed neither sufficiently 
absolute nor sufficiently pleasing to the mind. 

Having become aware of these defects, I often considered 
whether there could perhaps be found a more reasonable 
arrangement of circles, from which every apparent inequality 
would be derived and in which everything would move 
uniformly about its proper center. 

In De Revolutionibus he wrote: 



rF-T'3': 



Copernicus' diagram of his helio- 
centric system (from his manu- 
script o/De Revolutionibus, 1543). 
This simplified representation 
omits the many small epicycles ac- 
tually used in the system. 



We must however confess that these luovements [of the sun, 
moon, and planets] are circular or are composed of many 
circular movements, in that they maintain these irregularities in 
accordance with a constant law and with fixed periodic returns, 
and that could not take place, if they were not circular. For it 
is only the circle which can bring back what is past and over 
with. . . . 

I found first in Cicero that Nicetas thought that the Earth 
moved. And afterwards I found in Plutarch that there were 
some others of the same opinion. . . . Therefore I also . . . began 
to meditate upon the mobility of the Earth. And idtbough the 
opinion seemed absurd, nevertheless, because I knew that 
others before me bad been granted the liberty of constructing 
whatever circles they pleased in order to demonstrate astral 
phenomena, I thought that I too would be readily permitted to 
test whether or not, by the laying down that the Earth had 
some movements, demonstrations less shaky than those of my 
predecessors could be found for the revolutions of the celestial 

spheres I finally discovered by the help of long and 

numerous observations that if the movements of the other 
wandering stars are correlated with the circular movement of 
the Earth, and if the movements are computed in accordance 
with the revolution of each planet, not only do all their 
phenomena follow from that but also this correlation binds 
together so closely the order and magnitudes of all the planets 
and of their' spheres or orbital circles and the heavens 
themselves that nothing can be shifted around in any part of 
them without disr-upting the remaining parts and the universe 
as a whole. 



The "wandering stars" are the 



planets. 



CHAPTER 6 / DOES THE EARTH MOVE? 



157 



SG 2 




In his final work, the result of nearly 40 years of study, 
Copernicus proposed a system of more than 30 eccentrics and 
epicycles. These would, he said, "suffice to explain the entire 
structure of the universe and the entire ballet of the planets. " 
Like Ptolemy's Almagest, De Rcvoliitionibus uses long geometrical 
analyses and is difficult to lead. Comparison of the two hooks 
strongly suggests that Copernicus thought he was producing an 
improved xersion of the Almagest. He used many of Ptolemy's 
obsenations plus some more recent ones. Yet Copernicus' 
system differed from Ptolemy's in several fundamental ways. 
Above all, Copernicus adopted a sun-centered system which in 
general matched that of Aristarchus. 

Like all scientists, Copernicus made a number of assumptions 
in his system. In his own words (using more modern temis in 
several places!, his assumptions were: 

L There is no precise, geometrical center of all tlic (-clestial 
circles or spheres. 

2. The center of the eailh is not the center of the uni\'erse, but 
only of gra\itation and of the lunar sphere. 

3. All the spheres revolve about the sun . . . and thei'efore the 
sun has a central location in the universe. 

4. The distance from the earth to the sun is \ eiy small in 
comparison with the distance to the stars. 

5. Whatever motion appears in the sky ai'ises not from any 
motion of the sky, but from the earth's motion. The earth 
together with its water and air performs a complete rotation on 
its fixed poles in a daily motion, while the sky remains 
unchanged. 

6. What appears to us as motions of the sun arise not from its 
motion but from the motion of [he eaith and . . . we ie\ol\'e about 
the sun like any other planet. The earth has, then, nunc than 
one motion. 

7. The apparent retrograde motion of the planets arises not 
from their motion but from the earth's. The motions of the earth 
alone, therefore, are enough to explain man\' apparent motions 
in the sky. 

Compare this list with the assumptions of Ptolemy, given in 
Chapter 5. You will see close similarities and important 
differences. Notice that Copernicus proposed that the eailh 
rotates daily. As Aristarchus and others had realized, this 
rotation would explain all the daily risings and settings seen in 
the sky. Copernicus also proposed, as had Aristarchus, that the 
sun was stationary and stood at the center of the universe. The 
eaiih and other planets each mo\(;d about a different central 
point near the sun. 

The figure at the left shows the main spheres earning the 
planets around the sun I sol I. Copernicus' text explains the basic 
features of his system: 



158 



UNIT 2 / MOTION IN THE HEAVENS 



The ideas here stated are difficult, even almost impossible, to 
accept; they are quite contrary to popular notions. Yet uath the 
help of God, we will make even^hing as clear as day in what 
follows, at least for those who are not ignorant of 
mathematics. . . . 

The first and highest of all the spheres is the sphere of the 
fixed stars. It encloses all the other spheres and is itself self- 
contained; it is immobile; it is certainly the portion of the 
universe with reference to which the movement and positions 
of all the other heavenly bodies must be considered. If some 
people are yet of the opinion that this sphere moves, we are of 
contrary mind; and after deducing the motion of the earth, we 
shall show why we so conclude. Saturn, first of the planets, 
which accomplishes its revolution in thirty years, is nearest to 
the first sphere. Jupiter, making its revolution in twelve years, is 
next. Then comes Mars, revolving once in two years. The fourth 
place in the series is occupied by the sphere which contains 
the earth and the sphere of the moon, and which performs an 
annual revolution. The fifth place is that of Venus, revohdng in 
nine months. Finally, the sixth place is occupied by Mercuiy, 
revolving in eighty days. ... In the midst of all, the sun reposes, 
unmoving. 

Already you can see an advantage in Copernicus' system that 
makes it "pleasing to the mind." The rates of rotation for the 

heavenly spheres increase progressiv^ely, fi^om the motionless 

sphere of stars to speedy Mercury. SG 3 



# 1. What reasons did Copernicus give for rejecting the use of 
equants? 

2. In the following list of propositions, mark with a P those 
made by Ptolemy and with a C those made by Copernicus. 
(a) The earth is spherical. 

(bj The earth can be thought of as a point in reference to the 
distance to the stars. 

(c) The heavens rotate daily around the earth. 

(d) The earth has one or more motions. 

(e) Heavenly motions are circular. 

(f) The observed retrograde motion of the planets results fi^om 
the earth's motion around the sun. 



6.2 I Neir conclusions 

A new w^ay of looking at old obseivations la new theory) can 
suggest quite new kinds of obseivations to make, or new uses for 
old data. Copernicus used his movdng-eaith model to obtain two 
important results which were not possible with the Ptolemaic 

CHAPTER 6 / DOES THE EARTH MOVE? 159 



Close Upl 



The Periods of Revolution of the Plonets 



The problem is to find the rate at which a planet 
moves around the sun by using observations made 
from the earth, which is itself moving around the 
sun. Say, for example, that a planet closer to the 
sun than the earth is goes around the sun at the 
frequency (rate) of 1 Va cycles per year. The earth 
moves around the sun also, in the same direction, 
at the rate of 1 cycle per year. Because the earth 
follows along behind the planet, the planet's motion 
around the sun. as seen from the earth, appears 
to be at a rate less than 1 Va cycles per year. In fact, 
as the diagrams below suggest, the planet's ap- 
parent rate of motion around the sun equals the 
difference between the planet's rate and the earth's 
rate: VA cycle per year minus 1 cycle per year, or 
Va cycle per year. In general, if an inner planet 
moves around the sun at frequency f^ and the earth 
moves around the sun with frequency f^, then the 
planet's apparent rate of motion, f^^, as seen from 
the earth, is given by f = f - f . 

^ •' pe p e 

A similar argument holds for planets farther from 
the sun than the earth is. (See Diagram B.) Since 
these "outer planets" revolve about the sun more 
slowly than the earth does, the earth repeatedly 
leaves the planets behind. Consequently, for the 
outer planets, the sign in the equation for f^^ is re- 
versed: f = f + f . 

pe p e 

The apparent frequency f^^ represents what is 
actually observed. Since f^ is by definition 1 cycle 
per year, either equation is easily solved for the 
unknown actual rate f^: 
For inner planets: f^ = 1 cycle/yr + f^^ 
For outer planets: f = 1 cycle yr - f 



Earth 




(A) A planet that is inside the Eanh's orbit 
and moves 1 'A revolutions around the sun 
in a year would, as seen from the earth, ap- 
pear to have made only a 'A cycle relative 
to the sun. 




(B) A planet that is outside the Earth's orbit 
and moves only % revolution around the sun 
in a year would, as seen from the earth, ap- 
pear to make about 7% revolutions relative 
to the sun. 



TABLE 6.1 





Number 


Apparent 


Apparent 


Frequency f^ 


Period 




of Years 


Number of 


Frequency 


Around Sun 


Around 




of Ob- 


Cycles with 


Ke in 


in Cycles 


Sun 




serva- 


Respect to 


Cycles per 


per Year 


(TK> 




tion 


Sun During 


Year 




in Years 




(t) 


t (n) 


(n,t) 






Mercury 


46 


145 


3.15 


4.15 


0.241 


Venus 


8 


5 


0.625 


1.625 


0.614 


Mars 


79 


37 


0.468 


0.532 


1.88 


Jupiter 


71 


65 


0.915 


0.085 


11.8 


Saturn 


59 


57 


0.966 


0.034 


29.4 





160 



I IM I 2 iVlO HON l\ IHE HEAVENS 



theory. Copernicus was able to calculate: (a) the period of motion 
of each planet around the sun, and (b) the sizes of each planet's 
orbit compared to the size of the earth's orbit. These 
calculations, for the first time, gave a scale for the dimensions of 
the planetary system, based on observations. 

To calculate the periods of the planets around the sun, 
Copernicus used observations recorded over many centuries. The 
method of calculation is similar to the "chase problem" of how 
often the hands of a clock pass one another. The details of this 
calculation are shouai on page 160. In Table 6.Z below, 
Copernicus' results are compared with accepted values. 

TABLE 6.2 



Planet 



Copernicus' Value 



Modern Value 



Mercury 


0.241 y (88 d) 


Venus 


0.614 y (224 d 


Mars 


1.88 y (687 d) 


Jupiter 


11.8y 


Saturn 


29.5 y 



87.97 d 

224.70 d 

686.98 d 

11.86y 

29.46 y 



SG4, 7 



Copernicus was also able, for the first time in histoiy, to derive 
relative distances between the planets and the sun. Remember 
that the Ptolemaic system had no distance scale. It proxaded only 
a way of predicting the planets' angular motions and positions. 

Ptolemy's system described the motions of the sun and five 
planets in terms of 1-yr epicycles on deferent circles. It gave only 
the relative sizes of epicycle and deferent circle, and gave them 
separately for each planet. Copernicus, on the other hand, 
described all these features of planetary motion in temis of the 
motion of the earth's yearly revolution around the sun. (The 
details of hou^ this can be done are given on pages 162 and 163.) 
Thus, it became possible to compare the radii of the planets' 
orbits with the radius of the earth's orbit. Because all distances 
were compared to it, the average sun-earth distance is called 
one astronomical unit, abbreviated 1 AU. 

Table 6.3 below compares Copernicus' values for the orbital 
radii (deferent circles only, the radii of the epicycles being 
relatively smcdl) vvdth the currently accepted values for the 
average distances to the sun. 

TABLE 6.3 





Radii of Planetary Orbits 




Planet 


Copernicus' 


Values 


Modern Value 


Mercury 


0.38 AU 




0.39 AU 


Venus 


0.72 




0.72 


Earth 


1.00 




1.00 


Mars 


1.52 




1.52 


Jupiter 


5.2 




5.20 


Saturn 


9.2 




9.54 




In the Ptolemaic system, onlv the 
relative sizes of epicycle and defer- 
ent were specified. Their sizes 
could be changed at will, as long as 
they kept the same proportions. 



CHAPTER 6 / DOES THE EARTH MOVE? 



161 



Hiose up I 

Changing Frame of Reference 
from the Earth to the Sun 



Sun 




Earth 



The change of viewpoint from Ptolemy's 
system to Copemicus" involved what today 
would be called a shift in frame of refer- 
ence. The apparent motion previously at- 
tributed to the deferent circles and epicy- 
cles was attributed by Copernicus to the 
earth's orbit and the planets' orbits around 
the sun. 

For example, consider the motion of 
Venus. In Ptolemy's earth-centered sys- 
tem, the center of Venus' epicycle was 
locked to the motion of the sun, as shown 
in the top diagram at the left. The size of 
Venus' deferent circle was thought to be 
smaller than the sun's. The epicycle was 
thought to be entirely between the earth 
and the sun. However, the observed mo- 
tions to be explained by the system re- 
quired only a certain relative size of epi- 
cycle and deferent. The deferent could be 
changed to any size, as long as the epi- 
cycle was changed proportionally. 

The first step toward a sun-centered 
system was taken by moving the center of 
Venus' 1-yr deferent out to the sun. Venus' 
epicycle was enlarged proportionally, as 
shown in the middle diagram at the left. 
Now the planet moved about the sun, while 
the sun moved about the earth. Tycho 
Brahe later proposed such a system with 
all visible planets moving about the moving 
sun. 

Copernicus went further. He accounted 
for the relative motion of the earth and sun 
by considering the earth to be moving 
around the sun, instead of the sun moving 
about the earth. In the Copernican system, 
Venus' enlarged epicycle became its orbit 
around the sun. Also, the sun's deferent 
was replaced by the earth's orbit around 
the sun. See the bottom diagram at the 
left. All three systems, Ptolemy's, Coper- 
nicus', and Tycho's, explain the same ob- 
servations. 



For the outer planets the argu- 
ment was similar, but the roles of 
epicycle and deferent circle were re- 
versed. For the outer planets In the 
Ptolemaic model, the epicycles in- 
stead of the deferent circles had 1- 
yr periods and moved in parallel with 
the sun in its orbit. The sizes of the 
deferents were chosen so that the 
epicycle of each planet would just 
miss the epicycles of the planets 
next nearest and next farthest from 
the sun. (This was a beautiful ex- 
ample of a simplifying assumption. 
It filled the space with no overlap and 
no gaps.) This system is repre- 
sented in the top diagram at the 
right; the planets are shown in the 
unlikely condition of having their epi- 
cycle centers along a single line. 

The first step in shifting to a sun- 
centered view for these planets in- 
volves adjusting the sizes of the de- 
ferent circles, keeping the epicycles 
in proportion. Eventually, the 1-yr 
epicycles are the same size as the 
sun's 1-yr orbit. See the middle dia- 
gram at the right. Next, the sun's 
apparent yearly motion around the 
earth is explained just as well by 
having the earth revolve around the 
sun. Also, the same earth orbit would 
explain the retrograde loops asso- 
ciated with the outer planets' matched 
1-yr epicycles. So all the matched 
epicycles of the outer planets and 
the sun's orbit are replaced by the 
single device of the earth's orbit 
around the sun. This shift is shown 
in the bottom diagram at the right. 
The deferent circles of the outer 
planets become their orbits around 
the sun. 




SAt-urn 




Jujoi-t-er 



Sat-urn 



Notice that Copernicus now had one system which lelated the 
size of each planet's orhit to the sizes ot all the other planets' 
orbits. Contrast this to Ptolemy's solutions, which were 
completely in(lep(>nclent foi- each planet. .\o wonder Copernicus 
said that 'nothing can l)e shitted around in any part of them 

without disrupting the remaining parts and the universe as a 

SG 8 whole. " 



• 3. What new kinds of results did Copernicus obtain with a 
moving-earth model which were not possible with a geocentric 
model for the planetary system? 



6.3 I Ar^ments for the Copernican system 

Copernicus knew that to many his work would seem absurd, 
"nay, almost contrarv' to ordinary human understanding, " so he 
tried in several ways to meet the old arguments against a mo\ing 
earth. 

1. Copernicus argued that his assumptions agreed with 
religious doctrine at least as well as Ptolemy's. Copeiiiicirs' book 
had many sections on the faults of the Ptolemaic system (most 
of which had been known for centuries). To Copernicus, as to 
many scholars, complex events wei'e merely symbols of God's 
thinking. lo find order and symmetry in them was an act of 
piety, for order and symmetry w^ere proofs of God's existence. As 
a church official, Copernicus would have been stunned to think 
that, in Galileo's time, his theory would contribute to the confiict 
between religious doctrine and science. 

2. Copernicus carefully calculated relatixe r-adii and speeds of 
the circular motions in his system. Fr^om these data, tables of 
planetaiy motion could be made. Actually, the theories of 
Ptolemy and Copernicus wer^e about ecjually accurate in 
predicting planetary positions. Both theories often diftered from 
the observed positions by as much as 2° (about four diameters 
of the moon I. 

3. Copernicus tried to answer sever'al other objections. Most of 
them had been raised against Aristarchus' heliocentric system 
nearly 19 centuries earlier. One ar'gument h{>ld that a rapidly 
rotating earth would sur^ely fly apart. Copernicus replied, 'Why 
does the defender of the geocentric theory not fear the same fate 
for" his rotating celestial spher'c — so mirch faster becairse so 
much larger?" It was argued that birds and clouds in the sky 
would be left behind by the earth's rotation and revolution. 
Copernicus answered this objection In- indicating that the 
atmospher'c is dr'agged along with \Uv cailh. I o \hv lack of 

164 UMT 2 / MOTIOX Ii\ THE HEAVTINS 



observed annual shift for the fixed stars, he could only give the 
same kind of answer that Aristarchus had proposed: 

. . . though the distance from the sun to the earth appears very 
large as compared with the size of the spheres of some planets, 
yet compared with the dimensions of the sphere of the fixed 
stars, it is as nothing. 

4. Copernicus claimed that the greatest advantage of his 
scheme was its simple description of the general motions of the 
planets. There certainly is a basic overall simplicity to his system. 
Yet for precise calculations, because Copernicus would not use 
equants, he needed more small motions than did Ptolemy to 
explain the observations. A diagram from Copernicus' manuscript 
shows more detail (page 166). 

5. Copernicus pointed out that the simplicity of his system was 
not merely convenient, but also beautiful and "pleasing to the 
mind. " The pleasure which scientists find in the simplicity of 
their models is one of the most powerful experiences in science. 
Far from being a "cold," merely logical exercise, scientific work 

is full of such recognitions of harmony and beauty. Another sign 
of beauty that Copernicus saw in his system was the central 
place given to the sun, the biggest, brightest object in the 
heavens and the giver of light, warmth, and life. As Copernicus 
himself put it: 

In the midst of all, the sun reposes, unmoving. Who, indeed, in 
this most beautiful temple would place the light-giver in any 
other part than whence it can illumine all other parts? So we 
find underlying this ordination an admirable symmetry in the 
Universe and a clear bond of the harmony in the motion and 
magnitude of the spheres, such as can be discovered in no 
other wdse. 



Look again at SG 2. 



4. Which of these arguments did Copernicus use in favor of his 
system? 

(a) It was obvious to ordinary common sense. 

(b) It was consistent with Christian beliefs. 

(c) It was much more accurate in predicting planet positions. 

(d) Its simplicity made it beautiful. 

(e) The stars showed an annual shift in position due to the 
earth's motion around the sun. 

5. What were the largest differences between observed 
planetary positions and those predicted by Ptolemy? by 
Copernicus? 

6. Did the Copernican system allow simple calculations of 
where the planets should be seen? 



CHAPTER 6 / DOES THE EARTH MOVE? 



165 




This drawing in Copernicus rnnnu- 
script ofDe Revolutionibus shows 
details of some epicycles in his 
model. 



6.4 I Arguments against the Copernican svstem 



Copernicus' hopes for acceptance of his theory were not quickly 
fulfilled. More than 100 years passed hefore the heliocentric 
system was generally accepted even by astronomers. E\'en then, 
the acceptance came on the basis of arguments quite different 
from those of Copernicus. In the meantime, the theon' and its 
few defenders met powerful opposition. Most of the criticisms 
were the same as those used by Ptolemy against Aristarchus. 

1. Apart from its apparent simplicity, the Copeinican sxstem 
had no clear scientijic advantages over the geocentric theoiy. No 
known observation was explained by one system and not by the 
other. Copernicus had a different \iev\point. But he had no new 
types of obseivations, no experimental data that could not be 
explained by the old theory. Furthermore, the accuracy of his 
predictions of planetaiy positions was little better than that of 
Ptolemy's. As P'rancis Bacon wrote in the early seventeenth 
century: "Now it is easy to see that both they who think the 
eai1h revolves and they who hold the old constiTiction are about 
equally and indifferently supported by the phenomena. " 

Basically, the rival systems differed in their choice of a 
reference fiame for describing the observed motions. Copernicus 
himself stated the problem clearly: 



Ptolemy» too, had recognized the 
possibilitv' of aJternatixe frames of 
reference. (Reread the quotation on 
page 148 in C:hapter 5.i Most of Ptol- 
emy's followers did not share this 
insight. 



Although there are so many authorities for saying that the Earth 
rests in the centre of the world that people think the contraiy 
supposition . . . ridiculous; . . . if, liowever, we consider the thing 
attentively, we will see that the question has not yet been 
decided and accordingly is by no means to be scorned. For 
every apparent change in place occurs on account of the 
movement eithei' of the thing seen or of the spectator, or on 
account of the necessarily unequal moxement of both. For no 
movement is perceptible relatively to things moved equally in 
the same directions — I mean relatively to the thing seen and 
the spectator. Now it is from the Earth that tlie celestial circuit 
is beheld and [presented to our sight. 1 herefore, if some 
movement should belong to the Eaith ... it will appear, in the 
parts of the universe which are outside, as the same movement 
but in the opposite direction, as though the things outside were 
passing over. And the daily revolution ... is such a mo\ ement. 



Here Copernicus invites the reader to shift the frame of reference 
from the earth to a remote position overlooking the whole system 
with the sun at its center. As you may know liom personal 
experience, such a shift is not easy. We can sympathize with 
those who preferred to hold an earth-centered s\ stem for- 
descr'ibing what they saw. 



166 



L\IT 2 / MO HON 1\ 1 HE HLAX E\S 



Physicists now generally agree that any system of reference 
may in principle be used for describing phenomena. Some 
systems are easier and others more complex to use or think 
about. Copernicus and those who followed him felt that the 
heliocentric system was right in some absolute sense: that the 
sun was really fixed in space. The same claim was made for the 
earth by his opponents. The modern attitude is that the best 
frame of reference is the one that allows the simplest discussion 
of the problem being studied. You should not speak of reference 
systems as being right or wrong, but rather as being convenient 
or inconvenient. (To this day, navigators use a geocentric model 
for their calculations. See the page of a navigation book in the 
margin.) 

2. The lack of an observable annual shift for the fixed stars was 
contraiy to Copernicus' model. His only possible reply was 
unacceptable because it meant that the stars were at an 
enormous distance fi^om the earth. Naked-eye instiTiments 
allowed positions in the sky to be measured to a precision of 
about 0.10°. But for an annual shift to be less than 0.10°, the stars 
would have to be more than 1,000 times farther from the sun 
than the earth is! To us this is no shock, because we live in a 
society that accepts the idea of enormous extensions in space 
and in time. Even so, such distances strain the imagination. To 
the opponents of Copernicus, such distances were absurd. 
Indeed, even if an annual shift in star position had been 
observable, it might not have been accepted as unmistakable 
evidence against one and for the other theory. One can usually 
modify a theoiy more or less pleasingly to fit in a bothersome 
finding. 

The Copernican system demanded other conclusions that 
puzzled or threatened its critics. Copernicus determined the 
distances between the sun and the planetaiy orbits. Perhaps, 
then, the Copernican system was not just a mathematical model 
for predicting the positions of the planets! Perhaps Copernicus 
was describing a real system of planetary orbits in space (as he 
thought he was). This would be difficult to accept, for the 
described orbits were far apart. Ev^en the small epicycles which 
Copernicus still used to explain variations in planetary motions 
did not fill up the spaces between the planets. Then what did fill 
up these spaces? Because Aristotle had stated that "nature 
abhors a vacuum," it was agreed that something had to fill all 
that space. Even many of those who believed in Copernicus' 
system felt that space should contain something. Some of these 
scholars imagined various invisible fluids to fill up the emptiness. 
More recently, similar imaginaiy fluids were used in theories of 
chemistry and of heat, light, and electricity. 

3. No definite decision between the Ptolemaic and the 
Copernican theories could be made based on the astronomical 



ci:i.i':sTiAL 

OItSl<:iiVATI(»NS 

i;y PHILI1> KIsSAM, c, E. 

/■ro/.s,or oj Cvil E^sim-cri,,^. 

l'ri,uclon r,ir.rrs,ly 

I. Til*' I'riiiciplos iii>«>ii which 
t-t'l<'stijil < >h.s<Tviiti<>iis jir<" 
Kased. 

A. CONCEPTS. 

1. The Celestial .Siiliere. To simplify the 
computations necessary for tiie determinations 
of the direction of the meridian, of latitude, and 
of longitude or time, certain concepts of the 
heavens liavo been generally adopted. They are 
the following: 

a. The earth is stationary. 

b. The heavenly bodies have been jjrojected 
outward, along lines which extend from 
the center of the earth, to a sphere of 
infinite radius called the celestial sphere. 

The celestial sphere has the following char- 
acteristics: 

a. Its center is at the center of"the earth. 

1). Its equator is on the projection of the 
earth's equator. 

c. With respect to the earth, the celestial 
si)here rotates from east to west about 
a line which coincides with the earth's 
axis. Accordingly, the poles of the celes- 
tial sphere are at the l)rolongations of the 
earth's Jxiles. 

d. The speed of rotation of the celestial 
sphere is .'!r>0 59.15' i)cr L'-l hours. 

e. With the important exception of bodies-- 
in the solar system, which change position 
slowly, all heavenly bodies remain prac- 
tically fixed in their positions on the 
celestial sphere, never changing more than 
negligible amounts in 24 liours, and ac- 
cordingly are often called Jind .■ilara. 



Celestial navigation involves com- 
paring the apparent position of the 
sun (or star) with the "actual' posi- 
tion as given in a table called an 
"ephemer is.' Above is an excerpt 
from the introduction to the tables 
in the Solar Ephemeris. (Keuffel 
and Esser Co.) 

SG 9 
SG 10 

Galileo had this experience (see 
Chapter 7). 



CHAPTER 6 / DOES THE EARTH MOVE? 



167 



evidence. Therefore, attention was focused on the argument 
concerning the central, ininiovable position of the earth. Despite 
his efforts, Copernicus could not persuade most of his i-eaders 
that his heliocentric system reflected the mind of God as closely 
as did the geocentric system. 

4. The Copemican theory conflicted with the basic ideas of 
Aristotelian physics. This conflict is well described by H. 
Butterfield in Origins ofModcrn Science: 

... at least some of the economy of the Copernican system is 
rather an optical illusion of more recent centuries. We 
nowadays may say that it requires smaller effort to mo\'e the 
earth round upon its axis tlian to swing the v\holo univei-se in a 
twentv'-four hour revolution about the earth; but in the 
Aristotelian physics it required something colossal to shift the 
heavy and sluggish earth, whUe all the skies were made of a 
subtle substance that was supposed to have no weight, and 
they were comparatively easy to turn, since turning was 
concordant with their nature. Above all, if you grant Copernicus 
a certain advantage in respect of geometrical simplicity, the 
sacrifice that had to be made for the sake of this was 
tremendous. You lost the whole cosmology associated with 
Aiistotelianism — the whole intricately dovetailed sx'Stem in 
which the nobility of the various elements and the hierarchical 
arrangement of these had been so beautifully interlocked. In 
fact, you had to throw overboard the very framework of existing 
science, and it was here that Copernicus clearly failed to 
discover a satisfactory alternative. He provided a neater 
geometry of the heavens, but it was one which made nonsense 
of the reasons and explanations that had pre\dously been given 
to account for the movements in the sky. 

All religious faiths in Europe, including the new Protestants, 
opposed Copernicus. They used biblical quotations (for example, 
Joshua 10:12-13) to assert that the Divine Architect must have 
worked from a Ptolemaic blueprint. Indeed, Martin Luther called 
Copernicus "the fool who would overturn the whole science of 
astronomy." 

Eventually, in 1616, more storm clouds were raised by the case 
of Galileo. The Inquisition put De Revolutionibus on the Inde^c of 
forbidden books as "false and altogether opposed to Holy 
Scriptures." Some Jewish communities also prohibited the 
teaching of Copemicus's theory. It seems that humanity, 
believing itself central to God's plan, had to insist that the earth 
stood at the center of the physical imi\erse. 

The assumption that the earth was not the center of the 
universe was offensive enough. Exen worse, the Copernican 
system suggested that the other planets were similar to the 
earth. Thus, the concept of the distinctly different heavenly 
matter was threatened. What next? What if some rash person 
suggested that the sun and possibK' e\cn the stars wpi(> made of 



168 UNIT 2 / MOTIO.N I\ THE HE UTi.VS 



earthly materials? If other celestial bodies were similar to the 
earth, they might even be inhabited. And the inhabitants might 
be heathens, or beings as well-beloved by God as humans, 
possibly ev^en more beloved! Thus, the whole Copernican scheme 
led to profound philosophical questions which the Ptolemaic 
scheme avoided. 

In short, the sun-centered Copernican scheme was 
scientifically equivalent to the Ptolemaic scheme in explaining 
astronomical observations. But, philosophically, it seemed false, 
absurd, and dangerous. Most learned Europeans at that time 
recognized the Bible and the wiitings of Aristotle as their two 
supreme sources of authority. Both appeared to be challenged by 
the Copernican system. Although the fieedom of thought that 
marked the Renaissance was just beginning, the old image of the 
universe provided security and stability to many. Belief in a sun- 
centered rather than an earth-centered universe allowed a gain 
in simplicity; but it also seemed to contradict all common sense 
and observation. It required a revolution in philosophy, religion, 
and the physical science of the time. No wonder Copernicus had 
so few believers! 

Conflicts between accepted beliefs and the philosophical 
content of new scientific theories have occurred many times and 
are bound to occur again. During the last centuiy there were at 
least two such conflicts. Neither is completely resolved today. In 
biology, the theory of evolution based on Darwin's work has 
caused major philosophical and religious reactions. In physics, 
developing theories of atoms, relativity, and quantum mechanics 
have challenged long-held assumptions about the nature of the 
world and our knowledge of reality. Units 4, 5, and 6 touch upon 
these new theories. As the dispute between the Copernicans and 
the Ptolemaists illustrates, the assumptions which "common 

sense " defends so fiercely are often only the remains of an 

earlier, less complete scientific theory. SG ii 



• 7. Why were many people, such as Francis Bacon, undecided 
about the correctness of the Ptolemaic and Copernican 
systems? 

8. How did the astronomical argument become involved with 
religious beliefs? 

9. From a modern viewpoint, was the Ptolemaic or the 
Copernican system of reference more valid? 

6.0 I Historical consequences 

Eventually Copernicus' moving-earth model was accepted. But 
acceptance came veiy slowly. John Adams, who later became the 

CHAPTER 6 / DOES THE EARTH MOVE? 169 



second president of the I'nited States, wrote that he attended a 
lecture at HaiAard College in which the correctness of the 
Copernican viewpoint was debated on June 19, 1753! 

The Copernican model with moving earth and fixed sun 
opened a floodgate of new possibilities tor' anahsis and 
description. According to this model the planets could be 
thought of as real bodies mo\'ing along actiral or'bits. Xow Kepler 
and others could consider" these planetary paths in quite new 
ways. In science, the sweep of possibilities usually cannot be 

for-eseen by those who begin the r^exolution or by their- critics. 

SG 12 Today, Copernicus is honored not so much for* the details of 

his theory, but for his successful challenge of the prevailing 
world-picture. His theory became a major force in the intellectual 
r'evolution which shook humanit)' out of its self-center ed \'iew 
of the universe. As people gradually accepted the Copernican 
system, they also had to accept the \ievv that the earth was only 
one of sexeral planets cir'cling the sun. Thus, it became 
increasingly difficult to believe that all creation centered on 
human beings. At the same time, the new system stimulated a 
new self-r^eliance and curiosity about the world. 

Acceptance of a revolutionary idea based on (jirite new 
assumptions, such as Copemicais' shift of the frame of r^eference, 
is always slow. Sometimes compromise theories are pr'oposed 
as attempts to unite conflicting theories, to "split the difference." 
As you will see in later units, such compromises ar^e r'arely 
successful. But often they do stimulate new observations and 
concepts. In turn, these may lead to a xery useful development 
or restatement of the original rexolutionary theorA'. 

Such a restatement of the heliocentric theory came during the 
150 years after Copernicus. Many scientists provided observations 
and ideas. In Chapter^s 7 and 8, you uill see the major 
contributions made by Kepler, Galileo, and Isaac X'ewton. First 
we will consider the work of Tycho Biahe, who devoted his life 
to impr (ning the precision with whit^h planetary positions wer'e 
observed and to the working out of a compromise theory' of 
planetary motion. 



• 10. In terms of historical perspective, what were the greatest 
contributions of Copernicus to modern planetary theory? 

6.6 I Tvcho Brahe 

Tycho Brahe was born in 1546 of a noble, but not particular 1\' 
rich, Danish family. By the time he was 13 or 14, he had become 

intenseK' inter-ested in astronomy. Although he was studxing law, 
Tycho secr-etK' spent his allowance on astrononiical tablets and 

170 UiMT 2 I MOTIO.N I\ THE HEAVTiNS 



books. He read the Almagest and De Hevolutionibus. Soon he 
discovered that both Ptolemy and Copernicus had relied upon 
tables of planetary positions that were inaccurate. He concluded 
that astronomy needed new observ^ations of the highest possible 
precision gathered over many years. Only then could a 
satisfactory theory of planetary motion be created. 

Tycho's interest in studying the heavens vv^as increased by an 
exciting celestial event. Although the ancients had taught that 
the stars were unchanging, a "new star" appeared in the 
constellation Cassiopeia in 1572. It soon became as bright as 
Venus and could be seen even during the daytime. Then over 
several years it faded until it was no longer visible. To TVcho 
these changes in the stariy sky were astonishing. Evidently at 
least one assumption of the ancients was vvi ong. Perhaps other 
assumptions were wrong, too. 

After observing and waiting about the new star, Tycho traveled 
through northern Europe. He met many other astronomers and 
collected books. Apparently he was considering moving to 
Germany or Switzerland where he could easily meet other 
astronomers. To keep the young scientist in Denmark, King 
Frederick II offered him an entire small island and also the 
income from various farms. This income would allow Tycho to 
build an observatory on the island and to staff and maintain it. 
He accepted the offer, and in a few years Uraniborg ("Castle of 
the Heavens") was built. It was an impressive structure with four 
large observatories, a libraiy, a laboratoiy, shops, and living 
quarters for staff, students, and obseivers. There was even a 
complete printing plant. Tycho estimated that the obseiA^atory 
cost Frederick II more than a ton of gold. For its time, this 
magnificent laboratoiy was at least as important, complex, and 
expensive as some of today's great research centers. Uraniborg 
was a place where scientists, technicians, and students from 
many lands could gather to study astronomy. Here, a group 
effort under the leadership of an imaginative scientist was to 
advance the boundaries of knowledge in one science. 

In 1577, Tycho obseived a bright comet, a fuzzy object whose 
motion seemed irregular, unlike the orderly motions of the 
planets. To find the distance to the comet, Tycho compared its 
position as observed from Denmark with its positions as 
observed from elsewhere in Europe. Some of these obsei^ation 
points lay hundreds of kilometers apart. Yet, at any given time, 
all obseivers reported the comet as having the same position 
with respect to the stars. By contrast, the moon's position in the 
sky was measurably different w^hen obseived from places so far 
apart. Therefore, Tycho concluded, the comet must be at least 
several times farther away than the moon. 

This was an important conclusion. Up to that time, people had 
believed that comets were some sort of local event, like clouds 



Although there were precision 
sighting instruments, all observa- 
tions were with the naked eye. The 
telescope was not to be invented for 
another 50 years. 



CHAPTER 6 / DOES THE EARTH MOVE? 



171 



ttiose Up L 

Tycho Drohe 




At the top right is a plan of the observatory and gardens built 
for Tycho Brahe at Uraniborg, Denmark. 

The cross section of the observatory, above center, shows 
where most of the important instruments, including large 
models of the celestial spheres, were housed. 

The picture at the left shows the room containing Tycho's 
great quadrant. On the walls are pictures of some of his 
instruments. He is making an observation, aided by assist- 
ants. 

Above IS a portrait of Tycho, painted about 1597 



172 



U\IT 2 I MOTION IN THE HEAV15NS 




The bright comet of 1965. 



or lightning. Now comets had to be considered distant 
astronomical objects from the realm of eternal things beyond the 
moon. Stranger still, they seemed to move right through the 
ciystalline spheres that were still generally believed to cany the 
planets. Tycho's book on this comet was widely read and helped 
to weaken old beliefs about the nature of the heavens. 



SG 13 



• 11. What event stimulated Tycho's interest in astronomy? 

12. In what ways was Tycho's observatory like a modern 
research institute? 

13. Why were Tycho's conclusions about the comet of 1577 
important? 



6.7 I Tycho's observations 



Tycho's fame results from his lifelong devotion to making 
unusually accurate observations of the positions of the stars, sun, 
moon, and planets. These observ^ations were made before the 
telescope was invented. Over the centuries, many talented 
observers had recorded the positions of the celestial objects. But 
the accuracy of Tycho's w^ork was much greater than that of the 
best astronomers before him. How was Tycho Brahe able to do 
what no others had done before? 

Singleness of purpose certainly aided Tycho. He knew that 
highly precise obseivations must be made during many years. 
For this he needed improved instruments that would give 
consistent readings. Fortunately, he had the mechanical skill to 
devise such instruments. He also had the funds to pay for their 
construction and use. 

Tycho's first improvement on the astronomical instruments of 
the day was to make them larger. Most of the earlier instruments 
had been rather small, of a size that could be moved by one 
person. In comparison, TVcho's instruments were gigantic. For 
instance, one of his early devices for measuring the angular 
altitude of planets had a radius of about 1.8 m. This wooden 
instrument, shovni in the etching on page 174, was so large that 
it took several workers to set it into position. Tycho put his 



For a more modern example of this 
same problem of instrumentation, 
you may wish to read about the de- 
velopment and construction of the 
500-cm Hale telescope on Mt. Pal- 
omar. 



CHAPTER 6 / DOES THE EARTH MOVE? 



173 




Johannes Hevelius and his wife 
jointly using a quadrant in his oh- 
ser\'ator\' in Danzig (seventeenth 
century J. 




instruments on heaw, firm foimdations or attached them to a 
wall that ran exactly noith-south. By fixing the instruments so 
solidly» IVcho increased the reliability of the readings over long 
periods of time. Throughout his career Tycho cieated better 
sighting devices, more precise scales, and stronger support 
systems. He made dozens of other changes in design which 
increased the precision of the obsenations. 

Tycho did more than just devise better instruments for making 
his observations. He also determined and specified the actual 
limits of precision of each instiimient. He realized that merely 
making larger and larger instruments does not always result in 
greater precision. In fact, the very size of the instrument can 
cause errors, since the parts bend under their own weight. 
Tycho tried to make his instruments as large and strong as he 
could without introducing such errors. Furthermore, in modem 
style, he calibrated each instrument and determined its range 
of error. (Nowadays many commercial instrument makers supply 
a measurement report with scientific instruments designed for 
precision work. Such reports are usually in the form of a table of 
small corrections that have to be applied to the direct readings.! 

Like Ptolemy and the Muslim astronomers, Tycho knew that 
the light coming from any celestial bod\' was bent dov\'nvvard by 
the earth's atmosphere. He knew that this bending, oi- refraction, 
increased as the celestial object neared the horizon. To improve 
the precision of his observations, TVcho carefully determined the 
amount of refraction involved. Thus, each observation could be 
corrected for refraction effects. Such careful work was essential 
to the making of improved records. 

IVcho worked at Uraniborg from 1576 to 1597. After- the death 
of King Frederick II, the Danish government became less 
inter^ested in helping to pay the cost of Tycho's observatory. Yet 
lycho was unudlling to corisider- any reductions in the cost of 
his activities. Because he was promised support by Emperor 
Rudolph of Bohemia, Tvcho moved his recor ds and several 
instrximents to Prague. There, fortunately, he hir-ed as an 
assistant an able, imaginative young man named Johannes 
Kepler. After Tycho's death in 1601, Kepler- obtained all his 
recor'ds of observations of the motion of Mars. /\s Chapter 7 
reports, Kepler's analysis of lychos data solved many of the 
ancient problems of planetary motion. 



One of Tycho's sighting devices. 
Unfortunately lychos instruments 
were destroyed in 1619, during the 
Thirtv Years' War. 



14. What improvements did Tycho make in astronomical 
instruments? 

15. In what way did Tycho correct his ohsenations to provide 
records of hie,her acciiracv? 



174 



L'M 1 2 / MOriOiV L\ THL HEA\ EXS 



G*3 l\'cho's compromise svstem 

Tycho hoped that his observ^ations would provide a basis for a 
new theory of planetary motion, which he had outlined in an 
early book. He saw the simplicity of the Copernican system, in 
which the planets moved around the sun. Yet, because he 
observed no annual parallax of the stars, he could not accept an 
annual motion of the earth around the sun. In Tycho's system, 
all the planets except the earth moved around the sun. 
Meanw^hile, the sun moved around the stationaiy earth, as 
shown in the sketch in the margin. Thus, Tycho devised a 
compromise model which, as he said, included the best features 
of both the Ptolemaic and the Copernican systems. However, he 
did not live to publish quantitative details of his theory. 

The compromise Tychonic system was accepted by only a few 
people. Those who accepted the Ptolemaic model objected to 
having the planets movdng around the sun. Those who accepted 
the Copernican model objected to havdng the earth held 
stationary. So the argument continued. Many scholars clung to 
the seemingly self-evident position that the earth was stationary. 
Others accepted, at least partially, the strange, exciting proposals 
of Copernicus that the earth might rotate and revolve around the 
sun. The choice depended mainly on one's philosophy. 

All planetaiy theories up to that time had been developed only 
to provdde some system for predicting the positions of the 
planets fairly precisely. In the terms used in Unit 1, these would 
be called kinematic descriptions. The causes of the motions, now 
called dynamics, had not been considered in any detail. Aristotle 
had described angular motions of objects in the heavens as 
"natural. " Everyone, including Ptolemy, Copernicus, and Tycho, 
agreed. Celestial objects were still considered to be completely 
different from earthly materials and to behave in quite different 
ways. That a single theory of dynamics could describe both 
earthly and heavenly motions was a revolutionary idea yet to be 
proposed. 

As long as there was no explanation of the causes of motion, a 
basic problem remained unsolved. Were the orbits proposed for 
the planets in the various systems actual paths of real objects in 
space? Or were they only convenient imaginary devices for 
making computations? The status of the problem in the early 
seventeenth century was later described Well by the English poet 
John Milton in Paradise Lost: 

... He his fabric of the Heavens 
Hath left to their disputes, perhaps to move 
His laughter at their quaint opinions wide 
Hereafter, when they come to model Heaven 
And calculate the stars, how they will wield 
The mighty frame, how build, unbuild, contrive 



Observed 
I direction 
1 of light path 
; from star 



Light path 
-from star 



Earth's atmosphere 



Observer 



Refraction, or bending, of light 
from a star by the earth s atmos- 
phere. The amount of refraction 
shown in the figure is a great exag- 
geration of what actually occurs. 




Main spheres in Tycho Brahe's sys- 
tem of the universe. The earth was 
fixed and was at the center of the 
universe. The planets revolved 
around the sun, while the sun, in 
turn, revolved around the fixed 
earth. 



CHAPTER 6 / DOES THE EARTH MOVE? 



175 



Giose Up 




Observatories 



Observing instruments have changed dramati- 
cally in the years since Tycho's work at Uraniborg. 
Kitt Peak National Observatory is located in the 
mountains southwest of Tucson, Arizona. This cen- 
ter for ground-based optical astronomy in the north- 
ern hemisphere has the largest concentration of fa- 
cilities for stellar, solar, and planetary research in 
the world. The largest of Kitt Peak's 14 instruments 
is the 4-m Mayall telescope, which is the second 
largest reflecting telescope in the United States (the 
largest is the 5-m Hale telescope on Palomar Moun- 
tain). An almost identical 4-m telescope is located 
at the Cerro Tololo Inter-American Observatory in 
Chile. The location of this observatory in the Andes 
Mountains provides a clear view of the southern sky 
not visible from Kitt Peak and other northern hem- 
isphere observatories. Technical improvements over 





the 13 years necessary to design and build these 
two telescopes have led to a significant increase in 
the efficiency of the telescopes. 

A different type of telescope is the Multiple Mirror 
Telescope (MMT), also located in the Arizona des- 
ert. The MMT represents the first major innovation 
in telescope design in a century. Light is collected 
by six 1 .8-m mirrors and brought to a common focus 
(the alignment is corrected by lasers). Thus, the 
MMT has the light-gathering capacity of a conven- 
tional 4. 5-m telescope making it the third largest 
telescope in the world. The telescope is housed in 
a four-story rotating structure, rather than the con- 
ventional dome. Both the building and the telescope 
move in unison when in operation. 

The largest optical telescope in the world is the 
6-m instrument located in the mountains of the 
North Caucasus in the Soviet Union. 




Above left: The eye end of the ^)il- 
cm refractor at Lick Ohsenator'x' 
show ins, the automatic camera for 
direct photography. 



To save appearances, how gird the sphere 
With centric and eccentric scribbled o'er 
Cycle and epicycle, orb in orb. 

The eventual success of Newton's universal dynamics led to the 
belief that scientists were describing the "real world. " This belief 
was held confidently for about two centuries. Later chapters of 
this text deal with recent discoveries and theories which have 
lessened this confidence. Today, scientists and philosophers are 
much less certain that the common-sense notion of "reality " is 
very useful in science. 



• 16. In what ways did Tycho's system for planetary niotions 
resemble the Ptolemaic and the Copernican systems? 



SG 14 

This unit presents the first example 
of the highly successful trend of 
modern science toward synthesis, 
that is, not two or more kinds of 
science, but only one. For example, 
not a separate physics of energy in 
each branch, but one conservation 
law; not separate physics for opti- 
cal, heat, electric, magnetic phe- 
nomena, but one (Maxwell's); not 
two kinds of beings (animal and hu- 
man) but one (in Darwinian views); 
not space and time separately, but 
space-time; not mass and energy 
separately, but mass-energy, etc. 
To a point, at least, the great ad- 
vances of science are the results of 
such daring extensions of one set 
of ideas into new fields. The danger 
is the false extrapolation that sci- 
ence by itself can solve all prob- 
lems, including political, health, or 
educational, and "explain" all hu- 
man emotions. The majority of sci- 
entists do not believe this extrapo- 
lation, but many nonscientists falsely 
believe that all scientists do. 



Study 
guide 



1. The Project Physics learning materials 
particularly appropriate for Chapter 6 include the 
following: 



Experiments 

The Shape of the Earth's Orbit 
Using Lenses to Make a Telescope 

Activity 

Frames of Reference 

Film Loop 

Retrograde Motion: Heliocentric Model 



CHAPTER 6 / STUDY GUIDE 



177 



sg 



2. The first diagram on the nv.\\ pag(! shows 
numbered positions of the sun and Mars ion its 
epicycle) at equal time intervals in their motion 
around the earth, as described in the Ptolemaic 
system. You can easily redraw the relative positions 
to change from the earths frame of reference to the 
suns. Mark a sun-sized circle in the middle of a thin 
piece of paper; this will be a frame of reference 
centered on the sun. Place the circle over each 
successive position of the sun, and trace the 
corres])onding numbered |Josition of Mars and the 
position of the earth. I Be sure to keep the piece of 
paper straight.) When you have done this for all 15 
positions, you will have a diagram of the motions of 
Mars and the earth as seen in the sun's frame of 
reference. 



time scal(! is indicatc^d at lO-day inteiTals along the 
central line of the sun's position. 

(a) Can you explain why Mercuiy and \'enus appear 
to move from farthest east to farthest west more 
quickly thiin from farthest west to farthest east? 

(b) From this diagram, can you find a period for 
Mercury's apparent position in the sky relative to the 
sun? 

(c) Can you derive a period for Mercury's actual 
orbital motion around the sun? 

(d) What are the major sources of uncertainty in the 
results you derived? 

(e) Similarly, can you estimate the orbital period of 
Venus? 



3. What reasons did Copernicus give for believing 
that the sun is fixed at or near the center of the 
planetaiy system? 

4. Consider the short and long hands of a clock or 
watch. If, starting from 12:00 o'clock, you rode on 
the slow short hand, how many times in 12 hr would 
the long hand pass you? If you are not certain , 
slowly turn the hands of a clock or watch, and keep 
count. From this information, can you derive a 
relation by which you could conclude that the period 
of the long hand around the center is 1 hr? 

5. Copernicus' theory is considered valuable 
because it allows new predictions and conclusions. 
What new conclusions resulted from Copernicms' 
theory? Why do these conclusions make this theory 
"better" than previous theories? 

6. Section 6.4 states that Copernicus' heliocentric 
theory is scientifically equivalent in many ways to 
Ptolemy's geocentric theoiy and merely represents a 
change in the frame of reference from a fixed earth 
to a fixed sun. In what ways is the Copernican 
system more than just "a change in the frame; of 
reference"? 

7. The diagram at the upper right section of the 
next pag(; shows the motions of Mercury and X'enus 
east and west of the sun as seen from the earth. 'I'Ik; 



H. From the sequence of orbital radii from 
Mercur\' to Saturn, estimate what the orbital radius 
would Ije for a new planet if one were discovered. 
What is the basis for your estimate? 

9. The largest observed annual shift in star position 
is about 1/2400 of a degree. What is the distance (in 
iistronomical units I to this closest star? 

10. How might a Ptolemaic astronomer have 
inodified the geocentric system to account for 
observed stellar parallax? 

11. What conflicts between scientific theories and 
common sense do you know of today? 

12. How did the C'opernican system encourage the 
suspicion that there might be life on objects other 
than the earth? Is such a possibility seriously 
considered today? What important questions would 
such a possibility raise? 

13. How can you explain the obsened motion of 
Halley's comet during 1909-1910, as shown on the 
star chart on the next page? 

14. To what extent do you feel that the Copernican 
system, with its many motions in eccentrics and 
epicycles, revtuils r(!al paths in space, rather than 
provides only another way of computing planetary 
positions? 



178 



CH/XITLIK « / STUDY {iVlDE 



ss 




Apparent motion of Mars and the 
sun around the earth. 




Position of Venus and 
Mercury Relative +0 
Sun, \<^b(>'G>7 



Positions of Venus and Mercury 
relative to the sun. 



i ^ 



Sut"'^ posi+ion at 
10- do-, intervals 




• • 



J ♦ i» 



6h 
SOUTH 



24h 23h 



Observed motion of Hal ley's comet 
against the background of stars 
during 1909-1910. 



CHAPTER 6 / STUDY GUIDE 



179 





A Hemr Uniuerse 
Appears 

The Work of Kepler and Galileo 



7.1 The abandonment of uniform cireular motion 

7.2 Kepler's lai«' of areas 

7.3 Kepler's Ian' of elliptical orbits 

7.4 Kepler's law of periods 

7.5 The new- eoncept of phvsic<il law 

7.6 Galileo and Kepler 

7.7 The telescopic evidence 

7.8 Galileo focuses the controi'ersy 

7.9 Science and freedom 



SG 1 



7«1. I The abandonment of uniform cireular 
motion 

Kepler, who became Tycho's assistant, had the lifelong desire to 
perfect the heliocentric theory. He vdevved the hamionx' and 
simplicity of that theoiy with "incredible and raxishing delight." 
To Kepler, such patterns of geometric order and numericcil 
relation offered clues to God's mind. Kepler sought to unfold 
these patterns fui'ther through the heliocentric thcoiA'. In his Hrst 
major work, he attempted to explain the spacing of the planetary 
orbits as calculated by Copernicus (page 161 of C'hapter 6i. 
Kepler was searching for the I'easons wh\' there are just six 



180 



UNIT 2 / IVI0110\ l\ I HE HLA\ E\S 



visible planets (including the earth) and why they are spaced as 
they are. These are excellent scientific questions, but even today 
they are too difficult to answer. Kepler thought that the key lay 
in geometry. He began to wonder whether there was any relation 
between the six known planets and the five regular solids. A 
regular solid is a polyhedron whose faces all have equal sides 
and angles. From the time of the Greeks, it was known that there 
are just five regular geometrical solids. Kepler imagined a model 
in which these five regular solids nested one inside the other, 
somewhat like a set of mixing bowls. Between the five solids 
would be spaces for four planetary spheres. A fifth sphere could 
rest inside the whole nest and a sixth sphere could lie around 
the outside. Kepler then sought some sequence of the five solids 
that, just touching the spheres, would space the spheres at the 
same relative distance from the center as were the planetary 
orbits. Kepler said: 

I took the dimensions of the planetary orbits according to the 
astronomy of Copernicus, who makes the sun immobile in the 
center, and the earth movable both around the sun and upon 
its own axis; and I showed that the differences of their orbits 
corresponded to the five regular Pythagorean figures. . . . 

By trial and error Kepler found a way to arrange the solids so 
that the spheres fit wdthin about 5% of the actual planetary 
distances. We now know that this arrangement was entirely 
accidental. But to Kepler it explained both the spacings of the 
planets and the fact that there were just six. Also, it had the 
unity he expected between geometiy and scientific observations. 
Kepler's results, published in 1597, demonstrated his imagination 
and mathematical ability. Furthemiore, his work came to the 
attention of major scientists such as Galileo and Tycho. In 1600, 
Kepler was invited to become one of Tycho s assistants at his 
new observatory in Prague. 

There, Kepler was given the task of determining in precise 
detail the orbit of Mars. This unusually difficult problem had not 
been solved by Tycho and his other assistants. As it turned out, 
Kepler's study of the motion of Mars was only a starting point. 
From it, he went on to redirect the study of celestial motion. In 
the same way, Galileo used the motion of falling bodies to 
redirect the study of terrestrial motion. 

Kepler began by trying to fit the observed motions of Mars with 
motions of an eccentric circle and an equant. Like Copernicus, 
Kepler eliminated the need for the large epicycle by putting the 
sun motionless at the center and ha\ang the earth move around 
it (see page 157). But Kepler made an assumption which differed 
from that of Copernicus. Recall that Copernicus had rejected the 
equant as an improper type of motion, using small epicycles 
instead. Kepler used an equant, but refused to use even a single 
small epicycle. To Kepler the epicycle seemed 'unphysical. " He 




The five 'regular solids " (also 
called Pythagorean figures or Pla- 
tonic solids I, taken fiom Kepler's 
Harmonices Mundi (Harmony of 
the World). The cube is a regular 
solid with 6 square faces. The do- 
decahedron has 12 five-sided faces. 
The other three regular solids have 
faces that are equilateral triangles. 
The tetrahedron has 4 triangular 
faces, the octahedron has 8 trian- 
gular faces, and the icosahedron 
has 20 triangular faces. 



For Kepler, this geometric view was 
related to ideas of harmonv. 



CHAPTER 7 / A NEW UNIVERSE APPEARS 



181 




Kepler's model for c\[)Uiining the 
spacing of the planetary orbits by 
means of the regular geometrical 
solids. Notice that the planetary 
spheres were thick enough to in- 
clude the small epicycle used by 
Copernicus. 

SG 2 

In keeping with Aiistotelian phys- 
ics. Kepler beliexed that force was 
necessaiy to dri\ e the planets along 
their circles, not to hold them in 
circles. 

Fortunately, Kepler had made a 
major discovery earlier that was 
crnciid to his later work. He found 
that the orbits of the earth and 
other planets were in planes that 
passed through the sun. Ptolemy 
and Copernicus required special 
explanations for the motion of 
planets north and south of the 
ecliptic, but Kepler found that these 
motions were simply the result of 
the orbits lying in planes tilted to 
the plane of the earth s orbit. 



reasoned that the center of the epicycle was empty, and empty 
space could not exert any force on a planet. I bus, from the start, 
Kepler assumed that tiie orbits were real and that the motion 
resulted from pbvsic;al causes, namely, the action of forces on 
the planets. E\en bis beloved teacher, Maestlin, advised the 
young man to stick to geometrical models and astronomical 
obsei-xation and to a\oid physical assumptions. But Kepler 
stubbornly stuck to bis idea that the motions must l)e produced 
and explained by forces. When he finally published his results on 
Mars in bis book Astronomia Nova (New AslronomyJ, it was 
subtitled Celestial Physics. 

For a year and a iialf Kepler struggled to fit bis findings with 
TVcho's obsenations of Mars by xarious arrangements of an 
eccentric and an equant. When, aftei" 70 trials, success finally 
seemed near, be made a discouraging discovery. He could 
represent faii'ly well the motion of Mars in longitude least and 
west along the ecliptic), but be failed markedly with the latitude 
(north and south of the ecliptic). Even in longitude his very best 
predictions still differed by 8 min of arc from Tycho's obseiA'ed 
positions. 

Eight minutes of arc, about one-fourth of the moon's diameter, 
may not seem like much of a difference. Others might ba\e been 
tempted to explain it away as an obseiAational erioi\ But Kepler 
knew that Tycho's instruments and observations were rarely in 
error by even 2 min of ar'c. Those 8 min of ar c meant to Kepler 
that bis best system, using the old, accepted de\ices of eccentric 
and e(|uant, wcniid never- wor^k. In New Astronomy, Kepler- wrote: 

Since divine kindness granted us T\'cho Brahe, the most 
diligent obserAcr, by whose obser^ations an error of eight 
minutes in the case of Mar-s is brought to light in this Ptolemaic 
calculation, it is fitting that we recognize and honor this favor 
of God with gratitude of mind. Let us certainly work it out, so 
that we finally show the tr-ue form of the celestial motions (by 
supporting ourselves with these proofs of the fallacy of the 
suppositions assumed). 1 myself shall pr-epare this way for 
other's in the following chapter-s according to my small abilities. 
For if I thought that the eight minutes of longitude wer-e to be 
ignored, I would already have corrected the hxpothesis which 
he had made earlier in the book and which worked moderateK' 
well. But as it is, because they could not be ignor-ed, these eight 
minutes alone have pr-epared the way for reshapirig the whole 
of astronomy, and they ar-e the material which is made into a 
great pai't of this work. 



Kepler concluded that the orbit was not a circle and that ther-e 
was no point aroirnd whie^b the motion was unilorrn. Plato s idea 
of fitting perfect circles to the heavens had guided astronomers 
for 20 centuries. Now, Ke|)ler realized, this idea mirst be 
abandoned. Kepler- bad in bis bands the firn^st observations ever 



182 



U.MT 2 / iVIO TIO,\ I\ THL HEAVE.NS 



made, but now he had no theoiy by which they could be 
explained. He would have to start over facing two altogether new 
questions. First, what is the shape of the orbit followed by Mars? 
Second, how does the speed of the planet change as it moves 
along the orbit? 



1. When Kepler joined lycho Brake what task was he assigned? 

2. Why did Kepler reject the use of epicycles in his theory? 
Why is this reason important? 

3. Why did Kepler conclude that Plato s problem, to describe 
the motions of the planets by combinations of circular 
motions, could not be solved? 




The diagram depicts a nearly edge- 
on view of the orbital planes of 
earth and Mars, both intersecting 
at the sun. 



7*2i I Kepler's law of areas 



Kepler's problem was immense. To solve it would demand all of 
his imagination and skill. As the basis for his study, Kepler had 
Tycho's observed positions of Mars and the sun on certain dates. 
But these observations had been made from a moving earth 
whose orbit was not well knovvm. Kepler realized that he must 
first determine more accurately the shape of the earth's orbit. 
This would allow him to calculate the earth's location on the 
dates of the various observations of Mars. Then he might be able 
to use the observations to determine the shape and size of the 
orbit of Mars. Finally, to predict positions for Mars, he would 
need to discover how fast Mars moved along different parts of its 
orbit. 



MArs 





As you follow this brilliant analysis here, and particularly if you 
repeat some of his work in the laboratory, you will see the series 
of problems that Kepler solved. 

To find the earth's orbit he began by considering the moments 
when the sun, earth, and Mars lie almost in a straight line 
(Figure A). After 687 days, as Copernicus had found, Mars would 
return to the same place in its orbit (Figure B). Of course, the 
earth at that time would not be at the same place in its own 
orbit as when the first observation was made. As Figures B and C 



Ma.rs 



CHAPTER 7 / A IVEW UNIVERSE APPEARS 



183 



SG 3 



l{^' 




SG4 

Another way to express this rela- 
tionship for the nearest and far- 
thest positions would he to say the 
speeds were inversely proportional 
to the distance; but this rule does 
not generalize to any other points 
on the orbit. 



indicate, the directions to the sun and Mars, as seen from the 
earth against the fixed stars, would be known. The crossing point 
of the sight-lines to the sun and to Mars must be a point on the 
earth's orbit. Kepler worked with several groups of obseivations 
made 687 days apart lone Mars "year "). In the end, he 
determined fairly accurately the shape of the earth's orbit. 

The orbit Kepler found for the earth appeared to be almost a 
circle, with the sun a bit off center. Knov\ing now the shape of 
the earth's path and knowing also the recorded apparent 
position of the sun as seen from the earth for each day of the 
year, he could locate the position of the earth on its orbit and its 
speed along the orbit. Now he had the oibit and the timetable 
for the earth's motion. You may have made a similar plot in the 
experiment 'The Shape of the Earth's Orbit." 

Kepler's plot of the earth's motion revealed that the earth 
moves fastest when nearest the sun. Kepler wondered why this 
occurred. He thought that the sun might exert some force that 
drove the planets along their orbits. This concern with the 
physical cause of planetary motion marked the change in 
attitude toward motion in the heavens. 

The drawings at the left represent (with great exaggeration) the 
earth's motion for two parts of its orbit. The different positions 
on the orbit are separated by equal time inteivals. Between 
points A and B there is a relatively large distance, so the planet is 
moving rapidly. Between points C and D it moves more slowly. 
Kepler noticed, however, that the two areas swept over by a line 
from the sun to the planet are equal. It is believed that he 
actually calculated such areas only for the nearest and farthest 
positions of two planets, earth and Mars. Yet the beautiful 
simplicity of the relation led him to conclude that it was 
generally true for all parts of orbits. In its general form, the law of 
areas states: The line from the sun to the moving planet sweeps 
over areas that are proportional to the lime internals. Later, when 
Kepler found the exact shape of orbits, his law of areas became 
a powerful tool for predicting positions. 

You may be surprised that the first lule Kepler found for the 
motions of the planets dealt with the areas swept over by the 
line from the sun to the planet. Scientists had been considering 
circles, eccentric circles, epicycles, and equants. This was a quite 
unexpected property: The area swept over per unit time is the 
first property of the orbital motion to remain constant. (As you 
will see in Chapter 8, this major law of nature applies to all 
oibits in the solar system and also to double stars. i Besides being 
new and difl'erent, the law of areas drew attention to the central 
role of the sun. Thus, it strengthened Keplei's faith in the still 
widely neglected Copernican idea of a heliocentric system. 

As you will see, Kepler's other labors would have been of little 
use without this basic discoveiy. However, the njle does not give 



184 



UNIT 2 / MOTION IN THE HEAVENS 



any hint why such regularity exists. It merely describes the 
relative rates at which the earth and Mars (and, Kepler thought, 
any other planet) move at any point of their orbits. Kepler could 
not fit the rule to Mars by assuming a circular orbit, so he set out 
to find the shape of Mars' orbit. 



• 4. What observations did Kepler use to plot the earth's orbit? 

5. State Kepler's law of areas. 

6. Where in its orbit does a planet move the fastest? 

T«3 I Kepler's laii" of elliptical orbits 

Kepler knew the orbit and timetable of the earth. Now he could 
reverse his analysis and find the shape of Mars' orbit. Again he 
used obseivations separated by one Martian year. Because this 
interval is somewhat less than two earth years, the earth is at 
different positions in its orbit at the two times. Therefore, the 
two directions from the earth to Mars differ. Where they cross is 
a point on the orbit of Mars. From such pairs of observations 
Kepler fixed many points on the orbit of Mars. The diagrams 
below illustrate how two such points might be plotted. By 
drawing a curve through such points, Kepler obtained fairly 



\ Ma.r-» 





Ea.rth 



accurate values for the size and shape of Mars' orbit. Kepler saw 
at once that the orbit of Mars was not a circle around the sun. 
You will find the same result from the experiment, "The Orbit of 
Mars. " What sort of path was this? How could it be described 
most simply? As Kepler said, "The conclusion is quite simply 
that the planet's path is not a circle — it curves inward on both 
sides and outward again at opposite ends. Such a curve is called 
an oval. " But what kind of oval? 

Many different closed curves can be called ovals. Kepler 
thought for a time that the orbit was egg-shaped. However, this 
shape did not agree with his ideas of physical interaction 
between the sun and the planet. He concluded that there must 
be some better way to describe the orbit. For many months, 
Kepler struggled with the question. Finally, he realized that the 
orbit was a simple curve that had been studied in detail by the 




Kepler's law of areas. A planet 
moves along its orbit at a rate 
such that the line from the sun to 
the planet sweeps over areas which 
are proportional to the time inter- 
vals. The time taken to cover AB 
is the same as that for BC, CD, etc. 



SG 5 



CHAPTER 7 / A NEW UNIVERSE APPEARS 



185 



In this esperiment, the orbit of 
Mars was plotted from uwasure- 
ments made on pairs of sky photo- 
graphs taken 1 Martian year apart. 




a I 

An ellipse showing the long 
a^is a, the semiminor a;>iis b, and 
the two foci F, and F,. The shape of 
an ellipse is described by its eccen- 
tricity e, where e = c/a. 

S(. 6 

SG 10 



In the "Orbit of Mercury" Experi- 
ment, you can plot the shape of 
Mercury's very eccentric orbit ft'om 
observational data. See also SG 13. 

SG 13 




Gi^eks 2,000 years before. Ihe curve is called an ellipse. It is the 
shape you see when you view a circk; at a slant. 

Ellipses can differ greatly in shape. Ihey haxe many interesting 
properties. For example, you can draw an ellipse by looping a 
piece of string around two thumbtacks pinned to a drawing 
board at points F, and F^ as shown at the left. Full the loop taut 
with a pencil point (P) and run the pencil around the loop. Vou 
will have dr^awrr an ellipse, ilf the two thumb tacks had been 
together, on the same point, what curve would you ha\e drawn? 
What results do you get as you separate the two tacks more?) 

Each of the points F, and F, is called a focus of the ellipse, the 
greater the distance between F, and F^, the flatter, or more 
eccentric the ellipse becomes. As the distance between F, and F, 
shrinks to zer'o, the ellipse becomes more nearly circular'. A 
measure of the eccentricity of the ellipse is the ratio of the 
distance FjF, to the long axis. Since the distance between F, and 
F, is c and the length of the long iixis is a, the eccentricity, e, is 
defined by the equation e = c/a. 

The eccentricities are given for each of the ellipses shown in 
the mar^gin of the next page. You can see that a circle is the 
special case of an ellipse with e = 0. Also, note that the greatest 
possible eccentricity for an ellipse is e = 1.0 

TABLE 7.1 THE ECCENTRICITIES OF PLANETARY ORBITS 



Planet 



Orbital 
Eccentricity 



Notes 



Mercury 


0.206 


Venus 


0.007 


Earth 


0.017 


Mars 


0.093 


Jupiter 


0.048 


Saturn 


0.056 


Uranus 


0.047 


Neptune 


0.009 


Pluto 


0.249 



Too few observations for Kepler to study 

Nearly circular orbit 

Small eccentricity 

Largest eccentricity among planets Kepler could 

study 
Slow moving in the sky 
Slow moving in the sky 
Not discovered until 1781 
Not discovered until 1846 
Not discovered until 1930 



186 



UNIT 2 / MOTIOIV li\ THE HEAXlilNS 



Kepler's discovery that the orbit of Mars is an ellipse was 
remarkable enough in itself. But he also found that the sun is at 
one focus of the orbit. (The other focus is empty.) Kepler stated 
these results in his law of elliptical orbits: The planets move in 
orbits that are ellipses and have the sun at one focus. 

As Table 7.1 shows, Mars has the largest orbital eccentricity of 
any planet that Kepler could have studied. (The other planets 
for which there were enough data at the time were Venus, Earth, 
Jupiter, and Saturn.) Had he studied any planet other than Mars, 
he might never have noticed that the orbit was an ellipse! Even 
for Mars, the difference between the elliptical orbit and an off- 
center circle is quite small. No wonder Kepler later wrote: "Mars 
alone enables us to penetrate the secrets of astronomy which 
otherwise would remain forever hidden from us." 

Like Kepler, modern scientists believe that obseivations 
represent some aspects of reality more stable than the changing 
emotions of human beings. Like Plato and all scientists after him, 
scientists assume that nature is basically orderly and consistent. 
Therefore, it must be understandable in a simple way. This faith 
has led to great theoretical and technical gains. Kepler's work 
illustrates a basic scientific attitude: Wide varieties of phenomena 
are better understood when they can be summarized by a simple 
law, preferably expressed in mathematical form. 

After Kepler's initial joy at discovering the law of elliptical 
paths, he may have asked himself a question. Why are the 
planetary orbits elliptical rather than in some other geometrical 
shape? While Plato's desire for uniform circular motions is 
understandable, nature's insistence on the ellipse is suiprising. 

In fact, there was no satisfactory answer to this question for 
almost 80 years until, at last. Neuron showed that elliptical orbits 
were necessary r^esults of a more gener^al law of nature. You can 
accept Kepler's laws as rules that contain the observed facts 
about the motions of the planets. As empirical laws, they each 
summarize the data obtained by obserAdng the motion of any 
planet. The law of or^bits describes the paths of planets as 
ellipses around the sun. It gives all the possible positions each 
planet can have if the orbit's size and eccentricity are known. 
However, it does not tell when the planet will be at any particular 
position on its ellipse or how r^apidly it will be moving then. The 
law of areas, on the other hand, does not specify the shape of 
the orbit. But it does describe how the angular speed changes as 
the distance fr^om the sun changes. Cleariy, these two laws 
complement each other. Using them together, you can deterinine 
both the position and angular speed of a planet at any time, past 
or future. To do so, you need only to know the values for the 
size and eccentricity of the orbit and to know the position of the 
planet at any one time on its orbit. You can also find the earth's 
position for the same instant. Thus, you can calculate the 









Ellipses of dijferent eccentricities. 
(The pictures were made by photo- 
graphing a saucer at different an- 
gles.! 



CHAPTER 7 / A NEW UNIVERSE APPEARS 



187 



Empirical means based on obser- 
vation, not derived from theor\'. 



position of the planet as it would have been or uill be seen from 
the eai-th. 

The elegance and simplicity of Kepler's t\vo laws are 
impressive. Surely Ptolemy and Copernicus would haxe been 
amazed to see the problem of planetary motions solved in such 
shor-t statements. You must remember, however, that these laws 
wer-e distilled from Copernicus' idea of a moving ear1h and the 
great labors that went into Tycho's fine observations, as well as 
fr om the imagination and devotion of Kepler. 



Conic sections, as shown in the 
diagram, are figures produced by 
cutting a cone with a plane. The 
eccentricity of a figure is related to 
the angle of the cut. In addition 
to circles and ellipses, parabolas 
and hyperbolas are conic sections, 
with eccentricities greater than 
ellipses. Newton eventually showed 
that all of these shapes are possi- 
ble paths for a body moving under 
the gravitational attraction of the 
sun. 



7. If you noticed a man walking to the store to buy a 
newspaper at 8:00 A.M. for three days in a row, what empirical 
law might you propose-' What is an empirical law? If you later 
found that the man did not go to the store on Sunday, what 
conclusions could you make about empirical laws? 

8. What special feature of Mars' orbit made Kepler's study of it 
so fortunate? 

9. If the average distance and eccentricity of a planet's orbit 
are known, which of the following can be predicted from the 
law of areas alone? fi"om the law of elliptical orbits alone? 
Which require both? 

(a) all possible positions in the orbit 

(b) speed at any point in an orbit 

(c) position at any given time 





Ci'rcLe 





£LLi lose. 






R3.ra.b0La. 



HvjpcrboLaL 




T.4 I Kepler's laiv of periods 



As Einstein later put it: 'The Lord 
is subtle, but He is not malicious. ' 



Kepler published his fir^st two laws in 1609 in his book 
Astronomia Nova. But he was still dissatisfied. He had not yet 
found any relation among the motions of the differ^ent planets. 
Each planet seemed to have its ouii elliptical orbit and speed. 
There appear^ed to be no overall pattern relating all planets to 
one another. Kepler had begun his career by trying to explain 
the number of planets and their spacing. He was con\inced that 
the observed or^jits and speeds could not be accidental. There 
must he some regularity linking all the motions in the solar 
system. His conviction was so strong that he spent years 



188 



UNIT 2 I MOTIOIV IN THE HEA\^NS 



examining possible combinations of factors by trial and error. 
Surely one combination would reveal a third law, relating all the 
planetary orbits. His long, stubborn search illustrates a belief that 
has run through the whole history of science: Despite apparent 
difficulties in getting a quick solution, nature's laws are rationally 
understandable. This belief is still a source of inspiration in 
science, keeping up one's spirit in periods of seemingly fruitless 
labor. For Kepler it made endurable a life of poverty, illness, and 
other personal misfortunes. Finally, in 1619, he wrote 
triumphantly in his Harmony of the World: 

. . . after I had by unceasing toil through a long period of time, 
using the observations of Bi ahe, discovered the true relation 
. . . [It] overcame by storm the shadows of my mind, with such 
fullness of agreement between my seventeen years' labor on the 
observations of Brahe and this present study of mine that I at 
first believed that I was dreaming. . . . 

Kepler's law of periods, also called the harmonic law, relates 
the periods of planets to their average distances from the sun. 
The period is the time taken to go once completely around the 
orbit. The law states that the squares of the periods of the planets 
are proportional to the cubes of their average distances fi^om the 
sun. Calling the period T and the average distance R^^, this law 
can be expressed as 



T oc /i^ 



T- = kR' 



T- 



= k 



where A: is a constant. This relation applies to all the planets as 
well as to comets and other bodies in orbit around the sun. You 
can use it to find the period of any planet if you know its average 
distance from the sun, and xdce versa. 



For the earth, T is 1 year. The av- 
erage distance R^^ of the earth from 
the sun is one astronomical unit (1 
AU). So one way to express the 
value of the constant k is k = 1 
vearVAU'. 



SG 15-18 



TABLE 7.2 VERIFICATION OF KEPLER'S LAW OF PERIODS 



Using Copernicus' Values 



Using Modern Values 



Planet 



Period T 
(Years) 



Average ^ 

Distance X. 
RJAU) R3 



Period T 
(Years) 



Average 
Distance 
RJAU) 



Mercury 


0.241 


0.38 


1.06 


0.241 


0.387 


1.00 


Venus 


0.614 


0.72 


1.01 


0.615 


0.723 


1.00 


Mars 


1.881 


1.52 


1.01 


1.881 


1.523 


1.00 


Jupiter 


11.8 


5.2 


0.99 


11.862 


5.20 


1.00 


Saturn 


29.5 


9.2 


1.12 


29.458 


9.54 


1.00 



Kepler's three laws are so simple that their great power may 
be overlooked. Combined with his discovery that each planet 
moves in a plane passing through the sun, their value is greater 
still. They let us derive the past and future history of each planet 
if we know six quantities about that planet. Two of these 




The value of R^ for an ellipse is just 
half the major axis. 



CHAPTER 7 / A IVEW UNIVERSE APPEARS 



189 



The tables, named for IVcho's and 
Kepler's patron, Emperor Rudolph 
II, were called the Hudulphinc ta- 
bles. They were also important for 
a quite different reason. In them, 
Kepler pioneered in the use of log- 
arithms for making rapid calcula- 
tions and included a long section, 
practically a textbook, on the na- 
ture and use of logarithms (first de- 
scribed in 1614 l)y Xapier in Scot- 
land). Kepler's tables spread the 
use of this computational iiid, widely 
needed for nearly three centuries, 
until modern computing machines 
came into use. 



You know (see Question 11) that 
7^/fl^^ equals 1 for any object in or- 
bit around the sun. Therefore, if T 
— 5 years, then 



= 1 



R'^ = 25 

fl^ = approximately 3 AU 

Find R^^ for an object whose period 
is 4 vears. 



quantities are the size (long axis, a) and eccentricity (e) of the 
orbit. Three others are angles that relate the plane of the orbit to 
that of the earth's orbit. The si.xth quantity needed is the location 
of the planet in its orbit on any one date. These quantities are 
explained more fully in the Actixaties and Experiments listed in 
the Handbook for Chapters 7 and 8. 

In this manner, the past and future positions of each planet 
and each comet can be found. Kepler's system was vastly simpler 
and more precise than the multitude of geometrical devdces in 
the planetary theories of Ptolemy, Copernicus, and Tycho. With 
different assumptions and procedures Kepler had at last solved 
the problem which had occupied so many great scientists over 
the centuries. Although he abandoned the geometrical devices of 
Copernicus, Kepler did depend on the Copernican viewpoint of 
a sun-center ed universe. None of the eailh-centered models 
could have led to Kepler's three laws. 

In 1627, after many troubles v\ith his publishers and Tycho s 
heirs, Kepler published a set of astronomical tables. These tables 
combined TVcho's observations and the three laws in a way that 
permitted accur^ate calculations of planetar\' positions for any 
time, past or futurx\ These tables r^emained useful for a century, 
until telescopic observations of greater precision replaced 
I'ycho's data. 

Kepler s scientific interest went beyond the planetary problem. 
Like Tycho, who was fascinated by the new star of 1572, Kepler 
observed and wrote about new starts that appeared in 1600 and 
1604. His obserAations and comments added to the impact of 
Tycho s earlier- statement that changes did occur in the starr\' 
sky. 

As soon as Kepler learned of the development of the telescope, 
he spent most of a year studying how the images were formed. 
He published his findings in a book titled Dioptrice (16111, which 
became the standard work on optics for many years. Kepler 
wrote other important books on mathematical and astronomical 
problems. One u'as a popular and vxidely read descr iption of the 
Copernican system as modified by his own discoveries. This book 
added to the growing interest in and acceptance of the sun- 
centered svstem. 



10. State Kepler's law of periods. 

11. Use the periods and average radii of Jupiter's and Saturn s 
orbits to show that T^/Ri)^ is the same for both. What do all 
objects in orbit around the sun have in common':' 

12. What is the average orbital radius of a planetoid that 
orbits the sun every 5 years':' 

13. Why is it that none of the earth-centered models of the 
solar system could have led to Kepler's law of periods'^ 



190 



L'MIT 2 / MOTIOM IIV THE HEAVENS 



T.5 I The new concept of physical laiv 

One general feature of Kepler's lifelong work greatly affected the 
development of all the physical sciences. When Kepler began his 
studies^ he still accepted Plato's assumptions about the 
importance of geometric models. He also agreed with Aristotle's 
emphasis on "natural place " to explain motion. But later he 
came to concentrate on algebraic laws describing how planets 
moved. His successful statement of empirical laws in 
mathematical form helped to establish the equation as a normal 
fomi of stating physical laws. 

More than anyone before him, Kepler expected a theoiy to 
agree with precise and quantitative observation. From Tycho's 
observ^ations he learned to respect the power of precise 
measurement. Models and theories can be modified by human 
inventiveness, but good data endure regardless of changes in 
assumptions or viewpoints. 

Going beyond observation and mathematical description, 
Kepler attempted to explain motion in the heavens in terms of 
physical forces. In Kepler's system, the planets no longer moved 
by some divine nature or influence, or in "natural " circular 
motion caused by their spherical shapes. Rather, Kepler looked 
for physical laws, based on obseived phenomena and describing 
the whole universe in a detailed quantitative manner. In an early 
letter to Henvart (1605), he expressed his guiding thought: 

I am much occupied with the investigation of the physical 
causes. My aim in this is to show that the celestial machine is 
to be likened not to a divine organism but rather to a clockwork 
. . . insofar as nearly all the manifold movements are carried out 
by means of a single, quite simple magnetic force, as in the 
case of a clockwork, all motions are caused by a simple weight. 
Moreover, I show how this physical conception is to be 
presented through calculation and geometry. 

Kepler's likening of the celestial machine to a clockwork driven 
by a single force was like a look into the future of scientific 
thought. Kepler had read William Gilbert's work on magnetism, 
published a few years earlier. Now he could imagine magnetic 
forces from the sun driving the planets along their orbits. This 
was a reasonable and promising hypothesis. As Newton later 
showed (Chapter 8), the basic idea that a single kind of force 
controls the motions of all the planets was correct. The force is 
not magnetism, however, and does not keep the planets moving 
forward, but rather bends their paths into closed orbits. 

Even though Kepler did not understand correctly the nature of 
the forces responsible for celestial motion, his work illustrates 
an enonnous change in outlook that had begun more than two 
centuries earlier. Although Kepler still shared the ancient idea 



CHAPTER 7 / A NEW UNIVERSE APPEARS 191 



that each planet liad a "soul, " he lelused to base his explanation 
of planetaiy motion on this idea. Instead, he began to search for 
physical causes. Copernicus and lycho were willing to settle for 
geometrical models by which planetary positions could be 
predicted. Kepler was one of the first to seek dynamic causes for 
the motions. This new desire for physical explanations marked 
the beginning of a chief characteristic of modern physical 
science. 

Kepler's statement of empirical laws reminds us of Galileo's 
suggestion, made at about the same time. Galileo said that 
science should deal first with the how of motion in free fall and 
then with the why. A half-centuiy later, Nev\4on used the concept 
of gravitational force to tie together Kepler's three planetaiy laws 
with laws of terrestrial mechanics. This magnificent synthesis 
will be the subject of Chapter 8. 



14. In what ways did Kepler's work e^emplijy a "new" concept 
of physical law? 



T.6 I Galileo and Kepler 

One of the scientists with whom Kepler corresponded about 
scientific developments was Galileo. Kepler's main contributions 
to planetary theory were his empirical laws based on Tycho's 
observations. Galileo contributed to both theory and observation. 
As r^epor-ted in Chapters 2 and 3, Galileo based his theory of 
motion on observations of bodies moving on the earth's surface. 
His work in the new science of mechanics contradicted 
Aristotelian assumptions about ph^'sics and the nature of the 
heavens. Galileo's books and speeches triggered wide discussion 
about the differences or similarities of earth and heaven. Interest 
extended far outside of scientific circles. Some years after his visit 
to Galileo in 1638, the poet John Milton wrote: 



{Paradise Lost, Book V, line 574, . . . What if earth 

published 1667.) Be but the shadow of Heaven, and things therein 

Each to the other- like, more than on eaith is thought? 

Galileo challenged the ancient interpretations of experience. As 
stated earlier, he focused attention on new concepts: time and 
distance, velocity and acceleration, forces and matter. In contrast, 
the Aristotelians spoke of essences, final causes, and fixed 
geometric models. In Galileo's study of lalling bodies, he insisted 
on fitting the concepts to the observed facts. By seeking results 
that could be expressed in algebraic for-m, he paralleled the new 
stv'le being used by Kepler. 

192 UMT 2 I MOnO.X li\ THE HliAVEVS 



The sharp break between Galileo and most other scientists of 
the time arose from the kind of questions he asked. To his 
opponents, many of Galileo's problems seemed trivial. His 
procedures for studying the world also seemed peculiar. What 
was important about watching pendulums swing or rolling balls 
down inclines when deep philosophical problems needed 
solving? 

Although Kepler and Galileo lived at the same time, their lives 
were quite different. Kepler lived in near poverty and was driven 
from city to city by the religious wars of the time. Few people, 
other than a handful of friends and correspondents, knew of or 
cared about his studies and results. He wrote long, complex 
books that demanded expert knowledge from his readers. 

Galileo, on the other hand, wrote his essays and books in 
Italian. His language and style appealed to many readers who 
did not know scholarly Latin. Galileo was a master at publicizing 
his work. He wanted as many people as possible to know of his 
studies and to accept the Copernican theoiy. He wrote not only 
to small groups of scholars, but to the nobles and to civic and 
religious leaders. His arguments included humorous attacks on 
individuals or ideas. In return, Galileo's efforts to infoiTn and 
persuade on such a 'dangerous " topic as cosmological theory 
stirred up ridicule and even xaolence. Those who have a truly 
new point of view often must face such a reaction. 



In recent times, similar receptions 
were initially given to such artists 
as the painter Picasso, the sculptor 
Giacometti, and the composers 
Stravinsky and Schonberg. The same 
has often been true in most fields, 
whether literature, mathematics, 
economics, or politics. While great 
creative novelty is often attacked at 
the start, it does not follow that, 
conversely, everything that is at- 
tacked must be creative. 



15. Which of the following would you associate more with 
Galileo's work than with that of his predecessors: qualities and 
essences, popular language; concise mathematical e;>cpression; 
final causes? 



T.T I The telescopic evidence 

Like Kepler, Galileo was surrounded by scholars who believed 
the heavens were eternal and could not change. Galileo therefore 
took special interest in the sudden appearance in 1604 of a new 
star, one of those observed by Kepler. Where there had been 
nothing visible in the sky, there was now a brilliant star. Like 
Tycho and Kepler, Galileo realized that such events conflicted 
with the old idea that the stars could not change. This new star 
awakened in Galileo an interest in astronomy that lasted all his 
life. 

Four or 5 years later, Galileo learned that a Dutchman "had 
constructed a spy glass by means of which visible objects, 
though very distant from the eye of the observer, were distinctly 
seen as if nearby. ' Galileo (as he tells it) quickly worked out some 
of the optical principles involved. He then set to work to grind 




Th'o of Galileo's telescopes, dis- 
played at the Museum of Science 
in Florence. 



CHAPTER 7 / A NEW UNIVERSE APPEARS 



193 



Galileo meant that the area of the 
object was nearly 1,000 times 
greater. The area is proportionid to 
the square of the magnification lor 
"power") as we define it now. 




Two of Galileo s early drawings of 
the moon from Siderius Nuncius 
rrhe Starry Messengers 



the lenses and build su( h an insti\iment himself. Galileo's first 
telescope made objects appeal' three times closer- than when 
seen with the nciked eye. He reported on his third telescope in 
his book The Starry Messenger: 

Finally, sparing neither labor- nor expense, I succeeded in 
constrxicting for- myself so excellent an instrxinient that objects 
seen by means of it appeared nearly one thousand times larger 
and over thirty times closer than when regarded with oirr 
natural \ision. 

What would you do if you were handed "so excellent an 
instrument "? Like the scientists of Galileo's time, you probably 
would put it to practical uses. 'It would be superfluous, " Galileo 
agreed, 

... to enumerate the number- and importance of the advantages 
of such an instrument at sea as well as on land. But forsaking 
terr-estrial observations, 1 turned to celestial ones, and first I 
saw the moon trom as near- at hand as if it were scarceK' two 
terrestrial radii away. After that I observed often witli wondering 
delight both the planets and the fixed stars 

In a few shor't weeks in 1609 and 1610, Galileo used his 
telescope to make several major discoveines. First, he pointed his 
telescope at the moon. What he saw conxined him that 

. . . the surface of the moon is not smooth, uniform, and 
precisely spherical as a great number of philosophers believe it 
land other heavenly bodies) to be, but is uneven, rough, and 
fuU of caxities and prominences, being not unlike the face of 
the earth, relieved by chains of moirntains and deep \alleys. 

Galileo did not stop with that simple obser\'ation, so contrary 
to the Aristotelian idea of heaxenly perfection. He supported his 
conclusions with sever al kinds of evidence, including cai'eful 
measurement. For instance, he worked out a method for 
detei'mining the height of mountains on the moon from the 
length of their shadows. iHis value of about 6.4 km for' the height 
of some lunar mountains is not far fixjm modern results. For 
example, try the experiment, "The Height of Piton — A Mountain 
on the Moon, ' in the Handbook.} 

Next Galileo looked at the stars. To the naked eye, the Milky 
Way had seemed to be a continuous blotchy band of light. But 
through the telescope it was seen to consist of thousands of faint 
stars. Wherever Galileo pointed his telescope in the sky, he saw 
many moi^e stars than appeared to the unaided eye. This 
obser^•ation clashed with the old argument that the stains were 
created to help humans to see at night. By this argument, there 
should not be star\s inxisible to the naked eye. But Galileo foirnd 
thoirsands. 



194 



UNIT 2 / MO riOV l\ TfIL HE U E\S 



Galileo soon made another discoveiy which, in his opinion, 
"... deserves to be considered the most important of all — the 
disclosure of four Planets never seen from the creation of the 
world up to our time." He was referring to his discovery of four 
of the satellites which orbit Jupiter. Here before his eyes was a 
miniature solar system with its own center of revolution. Today, 
as to Galileo so long ago, it is a sharp thrill to see the moons of 
Jupiter through a telescope for the first time. Galileo's discoveiy 
strikingly contradicted the Aristotelian notion that the earth is 
the center of the universe and the chief center of revolution. 

The manner in which Galileo discovered Jupiter's "planets " is 
a tribute to his ability as an observer. On each clear night during 
this period he was discovering dozens if not hundreds of new 
stars never before seen. On the evening of January 7, 1610, he 
was looking in the vicinity of Jupiter. He noticed "... that beside 
the planet there were three starlets, small indeed, but veiy bright. 
Though I believe them to be among the hosts of fixed stars, they 
aroused my curiosity somewhat by appearing to lie in an exact 
straight line. . . . " (The sketches in which he recorded his 
observations are reproduced in the margin.) When Galileo looked 
again on the following night, the "starlets ' had changed position 
with reference to Jupiter. Each clear evening for weeks he 
obsewed and recorded their positions in drawings. Within days 
he had concluded that there were four "starlets " and that they 
were indeed satellites of Jupiter. Galileo continued obsemng 
until he could estimate their periods of revolution around 
Jupiter. 

Of all of Galileo's discoveries, that of the satellites of Jupiter 
caused the most stir. His book The Starry Messenger was an 
immediate success. Copies sold as fast as they could be printed. 
For Galileo, the result was a great demand for telescopes and 
great fame. 

Galileo continued to use his telescope with remarkable results. 
By projecting an image of the sun on a screen, he observed 
sunspots. This seemed to indicate that the sun, like the moon, 
was not perfect in the Aristotelian sense. It was disfigured rather 
than even and smooth. Galileo also noticed that the sunspots 
moved across the face of the sun in a regular pattern. He 
concluded that the sun rotated with a period of about 27 days. 

Galileo also found that Venus showed phases, just as the moon 
does (see photos on page 196). Therefore, Venus could not stay 
always between the earth and the sun, as Ptolemaic astronomers 
assumed. Rather, it must move completely around the sun as 
Copernicus and Tycho had believed. Saturn seemed to cany 
bulges around its equator, as indicated in the drawings on the 
next page. Galileo's telescopes were not strong enough to show 
that these were rings. (He called them "ears .") With his 
telescopes, Galileo collected an impressive array of new 




Telescopic photograph of Jupiter 
and its four bright satellites. This is 
appro,xinjately what Galileo saw 
and what you see through the sim- 
ple telescope described in the 
Handbook. 






Jo.i,.**^' 


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3.<f^.- 


0* » * 


3-'TV«^r' 


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lHv.r. 


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^ ♦►*«' 


.,0 - 


p. M-r^^l. 


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Ul <^,^c' 


* * . » 


M • J( O • 


K^.H.^J^- ♦ O • 


I7 7».»^' 


* ..0 » 


141^^ 


* ' » • 


\r- 


* . 


1<i (%^n'. 


• » » • 


17. c^Cr-n- 


• " - 


\f. 


♦ ' • • 


■Ul . >.».n,' 


* , • * _ 


it- 


» »■ Q * 


■\.r. 


« r ♦ 




»» 


'J»r'^>y -f 


..xrv' • • ♦ 


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tuu^; » . » 


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7 u^y^-^ , Q ,^ 


n- « « • 



These sketches of Galileo's are 
from the first edition of The Starry 
Messenger. 



Thirteen satellites of Jupiter have 
now been observed. 



CHAPTER 7 / A NEW UNIVERSE APPEARS 



19S 



SG 23 



information about the heavens, all ot which seeniod to contraclicU 
tiie basic assunijitions of the Ptolemaic world scheme. 



Photographs of Venus at various 
phtiscs with a constant magnifica- 
tion. 







Drawings of Saturn, made m the 
seventeenth centurv. 



? 16. Which commonly held beliefs of Gnlileo's contemporaries 
were contradicted by these obscrxations of Galileo? 

(a) A "new" star appeared in the heavens. 

(b) The moon has mountains. 

(c) There are many stars in dark areas of the sky. 

(d) Jupiter has several satellites. 

(e) The sun has spots and rotates. 

(f) Venus has phases like the moon. 

17. Could Galileo s observations of all phases of Venus support 
the heliocentric theory, the lychonic system, or Ptolemaic 
system? 

18. In what way did telescopic observation of the moon and 
sun weaken the earth-centered view of the universe? 

r.8 I Galileo focuses the controversy 

Galileo's observations supported his belief in the heliocentric 
Copernican system, but they were not the cause of his belief. His 
great work. Dialogue Concerning the 7\vo Chief World Systems 
11632), was based more on assumptions that seemed self-evident 
to him than on observations. Galileo recognized, as Ptolemy and 
Copernicus had, that the observed motions of planets do not 
prove either the heliocentric or the geocentric hypothesis right 
and the other one wrong. With proper adjustments of the 
systems, said CJalileo, "The same phenomena would result from 
either hypothesis." He accepted the earth's motion as real 
because the heliocentric system seemed simpler and more 
pleasing. 1 he sup|)oit foi' the heliocenti-ic view [ii-ovided by the 



196 



IMI I 2 / M(n lOIV IM THE HEAVENS 



"observed facts" was of course necessary, but the "facts" by 
themselves were not sufficient. Elsewhere in this course you will 
find other cases like this. Scientists quite often accept or reject 
an idea because of some strong belief or feeling that, at the time, 
cannot be proved decisively by experiment. 

In the Dialogue Concerning the Two Chief World Systems , 
Galileo presents his arguments in a systematic and lively way. 
Like his later book, Discourses Concerning Two New Sciences, 
mentioned in Chapter 2, it is in the form of a discussion between 
three learned men. Salvdati, the voice of Galileo, udns most of the 
arguments. His opponent is Simplicio, an Aristotelian who 
defends the Ptolemaic system. The third member, Sagredo, 
represents the objective and intelligent citizen not yet committed 
to either system. However, Sagredo usually accepts Galileo's 
arguments in the end. 

In Two Chief World Systems, Galileo's arguments for the 
Copernican system are mostly those given by Copernicus. Oddly 
enough, Galileo made no use of Kepler's laws, although Galileo's 
observations did proxdde new evidence for Kepler's laws. In 
studying Jupiter's four moons, Galileo found that the larger the 
orbit of the satellite, the longer was its period of revolution. 
Copernicus had noted that the periods of the planets increased 
with their average distances from the sun. Kepler's law of periods 
had stated this relation in detailed, quantitative form. Now 
Jupiter's satellite system showed a similar pattern, reinforcing 
the challenge to the old assumptions of Plato, Aristotle, and 
Ptolemy. 

Two Chief World Systems relies upon Copernican arguments, 
Galilean observations, and attacks on basic assumptions of the 
geocentric model. In response, Simplicio desperately tries to 
dismiss all of Galileo's arguments vvath a typical counter 
argument: 

. . . with respect to the power of the Mover, which is infinite, it 
is just as easy to move the universe as the earth, or for that 
matter a straw. 

To this argument, Galileo makes a very interesting reply. Notice 
how he quotes Aristotle against the Aristotelians: 

. . . what I have been saying was with regard not to the Mover, 
but only the movables . . . Giving our attention, then, to the 
movable bodies, and not questioning that it is a shorter and 
readier operation to move the earth than the universe, and 
paying attention to the many other simplifications and 
conveniences that follow from merely this one, it is much more 
probable that the diurnal motion belongs to the earth alone 
than to the rest of the universe excepting the earth. This is 
supported by a very true maxim of Aristotle's which teaches 
that ... "it is pointless to use many to accomplish what may be 
done with fewer." 



CHAPTER 7 / A NEW UNIVERSE APPEARS 197 



Some of the arguments brought 
forward against the new discover- 
ies sound silly to the modern mind. 
"One of his [Galileo's] opponents, 
who admitted that the surface of 
the moon looked rugged, main- 
tained that it was actually quite 
smooth and spherical as /Vristotle 
had said, reconciling the two ideas 
by saying that the moon was cov- 
ered with a smooth transparent 
materiiil through which mountains 
and craters inside it could be dis- 
cerned. Galileo, sarcastically ap- 
plauding the ingenuity' of this con- 
tribution, offered to accept it 
gladly — pro\ided that his opjjonent 
would do him the equal courtesy of 
allowing him then to assert that the 
moon was e\'en more rugged than 
he had thought before, its surface 
being covered with mountains and 
craters of this invisible substance 
ten times as high as any he had 
seen." [Quoted from Uiscoverics 
and Opinions of Galileo, by Still- 
man Drake. J 



Galileo thought his telescopic discoveries would soon 
demolish the assumptions that prevented uide ac(;eptance ol the 
Copernican theory. But people cannot believe what they are not 
ready to believe. The Aristotelians firmly believed that the 
heliocentric theory was obviously false and contrary to 
obsen'ation and common sense. The evidences pro\aded by the 
telescope could be distorted; after all, glass lenses change the 
path of light rays. Even if telescopes secMiied to work on ear'th, 
nobody could be sure they worked wIkmi pointed at the vastly 
distant stars. 

Most Aristotelians r eally could not even consider- the 
Copernican system as a possible theory. To do so would involve 
questioning too many of their own basic assumptions, as you 
saw in Chapter- 6. It is nearly humanly impossible to gi\e up all 
of one's common-sense ideas and find new bases for one's 
r-eligious and moral doctrines. The Aristotelians would haxe to 
admit that the earth is not at the center of creation. Then 
per haps the universe was not created especially for humanity. 
No wonder Galileo's arguments stinted up a storm of opposition! 

Galileo's observations intrigued many, but were unacceptable 
to Aristotelian scholars. Most of these critics had reasons one 
can respect. But a few were driven to positions that must have 
seemed silly even then. For example, the Flor-entine astronomer 
Francesco Sizzi argued in 1611 that there could not possibly be 
any satellites ar-ound Jupiter: 

There are seven windows in the head, two nostrils, two ears, 
two eyes and a mouth; so in the heavens ther-e ar-e two 
favorable stars, two unpropitious, two luminaries, and Mercury 
alone undecided and indifferent. From which and many other 
similar phenomena of nature such as the seven metals, etc., 
which it were tedious to enumerate, we gather that the number 
of planets is necessarily seven [including the sun and moon 

but excluding the earth] Besides, the Jews and other ancient 

nations, as well as modern Europeans, have adopted the 
division of the week into seven days, and have named them 
from the seven planets; now if we increase the number of 

planets, this whole system falls to the ground Moreover-, the 

satellites are invisible to the naked eye and therefore can have 
no influence on the earth, and therefore u'ould be useless, and 
therefore do not exist. 

A year after his discoveries, Galileo wrote to Kepler: 

You are the fii-st and almost the only pei-son who, even after a 
but cursor^' investigation, has . . . gi\en entire credit to my 
statements. . . . What do you say of the; leading philosophers 
hei-e to whom I haxe offered a thousand times of my own 
accord to show my studies, but who with the lazy obstinacy of 
a sequent who has eaten his fill haxe never consented to look 
at the jjlanets, oi- moon, or telescope? 



198 



LMT 2 / MOTION IN THL HLA\ DXS 



19. Did Galileo's telescopic observations cause him to believe 
in the Copernican viewpoint? 

ZO. What reasons did Galileo s opponents give for ignoring 
telescopic observations? 



T.9 I Science and freedom 

The political and personal tragedy that struck Galileo is 
described at length in many books. Only some of the major 
events are mentioned here. Galileo was warned in 1616 by the 
highest officials of the Roman Catholic Church to cease teaching 
the Copernican theory as true. It could be taught only as just 
one of several possible methods for computing the planetary 
motions. The Inquisitors held that the theory was contrary to 
Holy Scripture. At the same time, Copernicus' book was placed 
on the lnde}i of Forbidden Books "until corrected." As you saw 
before, Copernicus had used Aristotelian doctrine whenever 
possible to support his theory. Galileo had reached a new point 
of view. He urged that the heliocentric system be accepted on 
its merits alone. Although Galileo himself was a devoutly 
religious man, he deliberately ruled out questions of religious 
faith from scientific discussions. This was a fundamental break 
with the past. 

Cardinal Barberini, once a close friend of Galileo, was elected 
in 1623 to be Pope Urban VIII. Galileo talked with him about the 
decree against Copernican ideas. As a result of the discussion, 
Galileo considered it safe to write again on the topic. In 1632, 
having made some required changes, Galileo obtained consent to 
publish Two Chief World Systems. This book presented very 
persuasively the Ptolemaic and Copernican viewpoints and their 
relative merits. After its publication, his opponents argued that 
Galileo had tried to get around the warning of 1616. Furthermore, 
Galileo sometimes spoke and acted vvdthout tact. This fact, and 
the Inquisition's need to demonstrate its power over suspected 
heretics, combined to mark Galileo for punishment. 

Among the many factors in this complex story, it is important 
to remember that Galileo considered himself religiously faithful. 
In letters of 1613 and 1615, Galileo wrote that Gods mind 
contains all the natural laws. Consequently, the occasional 
glimpses of these laws that scientists might gain are direct 
revelations of God, just as true as those in the Bible: 'From the 
Divine Word, the Sacred Scripture and Nature did both alike 
proceed. . . . Nor does God less admirably discover himself to us 
in Nature's action than in the Scripture's sacred dictions. " These 
opinions are held by many people today, whether scientists or 
not. Few people think of them as conflicting vvath religion. In 



CRAPTER 7 / A NEW UNIVERSE APPEARS 199 



Pantheism refers to the idea that 
God is no more (and no less) than 
the forces and laws of nature. 



According to a well-known, but 
probably apocryphal story, at the 
end of these proceedings (ialileo 
muttered, ' E pur si miiovc' (but it 
does move). 




Over 200 years after his confine- 
ment in Home, opinions had 
changed so that Galileo was hon- 
ored as in the fresco "Galileo pre- 
senting his telescope to the Vene- 
tian Senate,' by Luigi Sabatelli 
(1772-1850). 



Galileo's time, such ideas were widely regarded as symptoms of 
pantheism. Pantheism was one of the religious "crimes " or 
heresies for which Galileo's contemporary, Giordano Bruno, was 
burned at the stake. The Inquisition was alamied by Galileo's 
seeming denial of the Bible as a certain source of knowledge 
about natural science. In reply, arrogant as Galileo often was, he 
quoted Cardinal Baronius: "The Holy Spirit intended to teach us 
how to go to hea\en, not how the heavens go " 

Though he was old and sick, Galileo was called to Rome and 
confined for a few months. The records of his trial are still partK' 
secret. It is known that he was tried, threatened with torture, 
and forced to make a formal confession for holding and teaching 
forbidden ideas. He was also forced to deny the Copernican 
theory. In return for his confessions and denial, Galileo was 
sentenced only to perpetual house arrest. Galileo's friends in 
Italy did not dare to defend him publicly. His book was placed 
on the Indcs- It remained there, along with that of Copernicus 
and one of Kepler's, until 1835. Thus, Galileo was used as a 
w^arning to all people that demands for spiritual conforniitv' also 
required intellectual confor-mity. 

Without intellectual freedom, science cannot flourish for long. 
Italy had given the worid many outstanding scholars. But for 2 
centuries after Galileo, Ital}' produced har'dly a single great 
scientist, while elsewhere in Europe many appeared. Today, 
scientists ar^e acutely aware of this famous part of the story of the 
development of planetary theories. Teachers and scientists in our 
time have had to face strong enemies of open-minded inquiry 
and of unr^estricted teaching. Today, as in Galileo's time, men 
and women who cr^eate or publicize new thoughts must be ready 
to stand up for them. There still are people who fear and wish 
to stamp out the open discussion of new ideas and new 
evidence. 

Plato knew that a government that wishes to control its people 
totiilly is threatened by new ideas. To prevent the spr-ead of such 
ideas, Plato recommended the now well-knowii treatment: 
reeducation, prison, or death. Not long ago, Soviet geneticists 
were required to discard well-established theories. They did so, 
not on the basis of new scientific evidence, but because pari\' 
"philosophers " accused them of conflicts with political doctrines. 
Similarly, the theory of relativity was banned from textbooks in 
Nazi Germany because Einstein's Jewish background was said to 
make his work worthless. Another example of intolerance was 
the prejudice that led to the "Monkey Trial " held in 1925 in 
Tennessee. At that trial, the teaching of Darwin's theory of 
biological evolution was attacked because it conflicted with 
certain types of biblical interpretation. 

On two points, one must be cautious not to romanticize the 
lessons of this episode. First, while a Galileo sometimes still may 



200 



UNIT 2 / MO'l ION IN THE HEAVENS 



be neglected or ridiculed, not eveiyone who feels neglected or 
ridiculed is for that reason a Galileo. The person may in fact be 
just wrong. Second, it has turned out that, at least for a time, 
science in some form can continue to live in the most hostile 
surroundings. When politicians decide what may be thought and 
what may not, science will suffer like eveiything else. But it will 
not necessarily be extinguished. Scientists can take comfort from 
the judgment of histoiy. Less than 50 years after Galileo's trial, 
Newton's great book, the Principia, appeared. Newton brilliantly 
united the work of Copernicus, Kepler, and Galileo with his own 
new statement of the principles of mechanics. Without Kepler 
and Galileo, there probably could have been no Newton. As it 
was, the work of these three, and of many others working in the 
same spirit, marked the triumphant beginning of modern 
science. Thus, the hard -won new laws of science and new \dews 
of humanity's place in the universe were established. What 
followed has been termed by historians The Age of 
Enlightenment. 



• 21. Which of the following appears to have contributed to 
Galileo's being tried by the Inquisition? 

(a) He did not believe in God. 

(b) He was arrogant. 

(c) He separated religious and scientific questions. 

(d) He v^ote in Italian. 



SG24 




Palomar Observatory houses the 
500-cm Hale reflecting telescope. It 
is located on Palomar Mountain 
in southern California. 



CHAPTER 7 / A NEW UNIVERSE APPEARS 



201 



study 
guide 



1. The Project Physics learning materials 
particuliirly appropriate for chapter 7 include the 
following: 

Experimente 

The Orhit of Mars 
Inclination of Mars' Orbit 
The Orbit of Mercury 

Actiiities 

Three-Dimensional Models of Two Orbits 

Demonstrating SateUite Orbits 

Galileo 

Conic-Section Models 

Challenging Problems: Finding Earth-Siin Distance 

Measuring Irregular Areas 

Film Loop 

Jupiter Satellite Orbit 

2. How large is an error of 8 min of arc in 
degrees? What fraction of the moons obsen'ed 
diameter does 8 min of arc represent? 

3. Summarize the steps Kepler used to determine 
the orbit of the earth. 

4. For the orbit positions nearest and farthest from 
the sun, a planet's speeds are inversely proportioned 
to the distances from the sun. What is the 
percentage change between the earth's slowest speed 
in July, when it is 1.02 AU from the sun, and its 
greatest speed in Januarv'. when it is 0.98 AL' from 
the sun? 

5. Summarize the steps Kepler used to determine 
the orbit of Mars. 

6. In any ellipse , the sum of the distances from the 
two foci to a point on the cur\'e equals the length 

of the major axis, or (F, P + F,P) — 2a. This property 
of ellipses allows us to draw them by using a loop 
of string around two tacks at the foci. What should 
the length of the looped string be? 

7. Draw an ellipse by looping a siring around two 
thumbtacks in a piece of paper and pulling th(' loop 



taut with a pencil, if the two tacks were on the same 
point, what kind of geometrical figure would \'ou 
draw? As the two points arv. separated more and 
more, what shapes do you draw? 

8. What is the eccentricity of an ellipse if the tw o 
points (foci) are 5 cm apart and the ellipse is 9 cm 
long at its widest part? 

9. (a) Draw an ellipse and label the following 
distances: a, c, perihelion, aphelion. 

(b) State the algebraic expression for c and a in 
terms of the perihelion and aphelion; state the 
expression for perihelion and aphelion in terms of a 
and c. 

(c) What is R,^ in terms of c and a? in terms of 
perihelion and aphelion? 

10. Kepler found that the sun is at one focus of the 
ellipse that describes the orbit of a planet. What is 
at the other focus? 

11. In describing orbits around the sun, the point 
nearest the sun is called the perihelion point, and the 
point farthest from the sun is called the aphelion 
point. The distances of these two points from the 
sun are called the perihelion distance and the 
aphelion distance, respectively. The terms perihelion 
and aphelion come from the Greek, in which helios 
is the sun, peri means near, and apo means away 
from. 

(a) List some other words in which the prefixes peri 
and apo or ap have similar meanings. 

(b) In describing earth satellite orbits, the terms 
apogee and perigee are often used. What do they 
mean? 

(c) What would such points for satellites orbiting the 

moon be called? 

p 




20Z 



CH^XiyiER 7 / STUDY Gl'IDL 



sg 



12. (a) An ellipse is 5 cm from one focus at its 

! farthest point and 2 cm from the same focus at its 
' nearest point. Find c, a, and the eccentricity of the 
ellipse. 

(b) An ellipse has an eccentricity of 0.5 and a = 10 
cm. What are the distances of nearest and farthest 
I approach from one of the foci? 

I (c) An ellipse is 5 cm from one of its foci at its 
nearest approach and has an eccentricity' of 0.8. 

. What is its greatest distance from that focus? Find c 
and a for the ellipse. 

13. For the planet Mercury, the perihelion distance 
(closest approach to the sun) has been found to be 
about 45.8 x lo'' km, and the aphelion distance 
(greatest distance from the sun I about 70.0 x lo'' 
km. What is the eccentricity of the orbit of Mercury? 

14. The eccentricity of Pluto's orbit is 0.254. What is 
the ratio of the minimum orbital speed to the 
maximum orbital speed of Pluto? 

15. Halley's comet has a period of 76 years, and its 
orbit has an eccentricity of 0.97. 

(a) What is its average distance from the sun? 

j' (b) What is its greatest distance from the sun? 

(' (c) What is its least distance from the sun? 
t 

II (d) How does its greatest speed compare udth its 

least speed? 

16. The mean distance of the planet Pluto from the 
sun is 39.6 AU. What is the orbital period of Pluto? 

17. Three major planets have been discovered since 
Kepler's time. Their orbital periods and mean 
distances from the sun are given in the table below. 
Determine whether Kepler's law of periods holds for 
these planets also. 

Average Eccen- 

Distance tricity 
Discovery Orbital From of 

Date Period Sun Orbit 



Uranus 1781 

Neptune 1846 

Pluto 1930 



84.013 yr 19.19 AU 0.047 
164.783 30.07 0.009 

248.420 39.52 0.249 



18. Considering the data available to him, do you 
think Kepler was justified in concluding that the 
ratio T/Rl is a constant? 



19. What is T^/Rl, for a satellite orbiting the earth if 
the average orbital radius is 18,000 km and the 
satellite orbits the earth every 380 min? 

ao. A satellite already in orbit around a planet is put 
into a new orbit whose radius is 4 times as large as 
the old radius. How many times longer is the new 
period than the old? 

21. Using the value of T^/fi", tliat you found in 
question 19, what is the average distance of a satellite 
from the center of the earth if its period is 28 days? 

22. Using the table of periods and orbital radii of 
earth satellites on p. 120 of Chapter 4, verify that 
Kepler's law of periods holds for these satellites. 

23. The chart on p. 204 is reproduced from the 
January, 1979, issue of Sky and Telescope. 

(a) Make a sketch of how Jupiter and its satellites 
appeared at one-week intervals, beginning with day 
"0." 

(b) Make measurements of the chart to find R^^ and 
T for each satellite. (For this problem, fi ^ can be to 
any convenient scale, such as centimeters on the 
diagram.) 

(c) Does Kepler's law of periods, T^/Rl,. = constant, 
hold for Jupiter's satellites? 

24. What are the current procedures by which the 
public is informed of new scientific theories? Do you 
think they are adequate? To what extent do news 
media emphasize clashes of points of view? Bring in 
some examples from news magazines. 

25. Kepler discovered his three laws because he 
believed that they should exist; that is, he believed 
that nature exhibits simple uniformity in motion. 

(a) Elxplain the hard work and courage that were 
necessary for Kepler to be successful. 

(b) Is it fair to say that Kepler was 'lucky" to have 
studied Mars, whose orbit is the most elliptical? 

(c) Discuss Kepler's reliance on the work of those 
who preceded him, particularly Copernicus and 
Brahe. 

(d) In what ways was Kepler independent enough 
not to rely completely on Copernicus and Brahe , but 
to go beyond the limits of their work? 



CHAPTER 7 / STUDY GUIDE 



203 



sg 



26. List at least three ways in which Kepler's 
approach to science differed from that of his 
predecessors. 



Z7. Kepler's laws are empirical laws. What is an 
empirical law? What are its limitations? Why are 
empirical laws important? 



Jupiter s satellites. The four cun- 
ing lines represent Jupiter's four 
bright (Galilean) satellites: (!) lo, III) 
Europa, (111) Ganymede, (I\0 Cal- 
listo. The location of the planet's 
disk is indicated by the pairs of 
vertical lines. If a moon is invisible 
because it is behind the disk (that 
is, occulted by Jupiter), the curxe is 
broken. For successive dates, the 
horizontal lines mark O^ Universal 
time, or 7 p.m. Eastern standard 
time (or 4 p.m. Pacific standard 
time) on the preceding date. Along 
the vertical scale, 0.16 cm is al- 
most 7 hours. In this chart, west is 
to the left, as if} an iinerting tele- 
scope for a northern hemisphere 
observer. At the bottom, 'd" is the 
point of disappearance of a satel- 
lite in the shadow of Jupiter; "r" is 
the point of reappearance. From 
the American Ephemeris and 
Nautical .\Jmanac. 




bATKl-Ll'l>«S UF JUl'lTEK, IHT'J 

CONKIGIIUATIONS OK SATELLITES I IV FOR NOVEMBER 

UNIVRRSAI. TIME 




PHASES OF THE ECLIPSES 



C^ 



204 



CH.\I»TL;R 7 / STL'D\ GUIDE 




The Unity of Earth anil Sky 

The Work of lUewton 



8.1 Neivton and seventeenth-centurr' science 

8.2 Neivton's Principia 

8.3 The inverse-square law of planetary force 

8.4 Law of universal ^aiitation 

8.5 Newton and hypotheses 

8.6 The ma^itude of planetary force 

8.7 Planetary motion and the ^avitational constant 

8.8 The value of G and the actual masses of the planets 

8.9 Further successes 

8.10 Some effects and limitations of Neuton's ivork 



8.1. I Neivton and seventeenth-century 
science 

Forty-five years passed between the death of Galileo in 1642 and 
the publication of Newton's Principia in 1687. In those years, 
major changes occurred in the social organization of scientific 
studies. The new philosophy of experimental science, applied 
with enthusiasm and imagination, produced a wealth of new 
results. Scholars began to work together and organize scientific 
societies in Italy, France, and England. One of the most famous, 
the Royal Society of London for Improving Natural Knowledge, 
was founded in 1662. Through these societies, scientific 
experimenters exchanged information, debated new ideas, 
argued against opponents of the new experimental activities, and 



SG 1 



CHAPTER 8 / THE UNITY OF EARTH AND SKY 



205 



Isaac Newton (1642-1727) 





^. 



T^'^-^4^// 



In referring to the time period be- 
tween 1500 and 1600, the forms 
"1500's" and "sixteenth century" 
are often used interchangeably, al- 
though the latter is preferable. 



published technical papers. Each society sought public support 
for its work and published studies in widely read scientific 
journals. Through the societies, scientific activities became well- 
defined, strong, and international. 

This development was part of the general cultural, political, 
and economic change occurring in the sixteen! Ii and 
seventeenth centuries. (See the time chart on page 207.1 Artisans 
and people of wealth and leisure became in\ol\(ul in sc ientific 
studies. Some sought to improve technological metliods and 
products. Others found the study of nature through experiment 
a new and exciting hobby. However, the availability of money and 
time, the growing interest in science, and the creation of 



206 



LMT 2 \IO HON I.\ THE HEAXTiNS 



1600 



1650 
1642 



1700 



1750 



NEWTON 



1727 





Newton entered Trinity College, 
Cambridge University, in 1661 at 
the age of 18. He was doing experi- 
ments and teaching while still a 
student. This early engraving shows 
the quiet student wearing a wig 
and heavy academic robes, as was 
customary. 



This drawing of the reflecting tele- 
scope he invented was done by 
Newton while he was still a student. 



organizations are not enough to explain the growing success of 
scientific studies. This rapid growlh also depended upon able 
scientists, vvell-tbiniulated piohlenis, and good experimental and 
mathematical tools. Some of the important scientists who lived 
between 1600 and 1750 are shown in the time chart for the Age 
of Newton. The list includes amateuis as well as unixersity 
professors. 

Many well-formulated problems appear in the writings of 
Galileo and Kepler. Theii- studies showed how useful 
mathematics could be when combined with experimental 
observation. Furthermore, their works raised exciting new 
questions. For example, what forces act on the planets and cause 
the paths actually observed? Why do objects fall as they do near 
the earth's surface? 

Good experimental and mathematical tools were being created. 
With mathematics being applied to physics, studies in each field 
stimulated development in the others. Similarly, the instiument 
maker and the scientist aided each other. 

Another factor of great importance was the rapid build-up of 
scientific knowledge itself. From the time of Galileo, scientists 
had reported repeatable experiments in books and journals. 
Theories could now be tested, modified, and applied. Each study 
built on those done previously. 

Newlon, who lived in this new scientific age, is the central 
person in this chapter. However, in science as in any other field. 







208 



UNIT 2 / MOTION IN THE HEAVENS 



many workers made useful contributions. The structure of 
science depends not only upon recognized geniuses, but also 
upon many lesser-known scientists. As Lord Rutherford, one of 
the founders of modern atomic theory, said: 

It is not in the nature of things for any one man to make a 
sudden violent discovery; science goes step by step, and every 
man depends upon the work of his predecessors. . . . Scientists 
are not dependent on the ideas of a single man, but on the 
combined wisdom of thousands. . . . 

To tell the story properly, each scientist's debt to others who 
worked previously and in the same age, and each scientist's 
influence upon future scientists should be traced. Within the 
space available, we can only briefly hint at these relationships. 

Isaac Neuron was bor^n on Christmas Day, 1642, in the small 
English vfllage of Woolsthorpe in Lincolnshire. He was a quiet 
farm boy. Like young Galileo, Newton loved to build mechanical 
gadgets and seemed to have a liking for mathematics. With 
financial help from an uncle, he went to Trinity CoUege of 
Cambridge University in 1661. There he enrofled in the study of 
mathematics and was a successful student. In 1665, the Black 
Plague swept through England. The college was closed, and 
Newton went home to Woolsthorpe. There, by the time he was 
24, he had made spectacular discoveries. In mathematics, he 
developed the binomial theorem and diff'erential calculus. In 
optics, he worked out a theory of colors. In mechanics, he 
already had formulated a clear concept of the first two laws of 
motion and the law of gravitational attraction. He also had 
discovered the equation for centripetal acceleration. However^ 
Newton did not announce this equation untU many years after 
Huygens' equivalent statement. 

This period at Woolsthorpe must have been the time of the 
famous and disputed fall of the apple. One version of the apple 
story appears in a biography of Newton written by his friend 
William Stukeley. In it we read that on a particular occasion 
Stukeley was having tea udth Newton. They were sitting under 
some apple trees in a garden, and Newton said that 

... he was just in the same situation, as when formerly, the 
notion of gravitation came into his mind. It was occasion 'd by 
the fall of an apple, as he sat in a contemplative mood. Why 
should that apple always descend perpendicularly to the 
ground, thought he to himself. Why should it not go sideways 
or upwards, but constantly to the earth's centre? 

The main emphasis in this story probably should be placed on 
the 'contemplative mood " and not on the apple. You have seen 
this pattern before: A great puzzle (here, that of the forces acting 
on planets) begins to be solved when a clear-thinking person 



CHAPTER 8 / THE UNITY OF EARTH AND SKY 309 



contemplates a familiar event (the fall of an object on eai1h). 
Where others had seen no relationship, Nevvlon did. Refening to 
the plague yeais, NevMon once wrote; 

I began to think of graxity extending to the oil) of the moon, 
and . . . from Kepler's rule (third law, law of periods) ... I 
deduced that the forces which kept the planets in their ort)s 
must be reciprocally as the squares of their distances from the 
centers about which they revoke: and thereby c-omijaied the 
force requisite to keep tlie moon in her orb with the foi'ce of 
gravity at the surface of the ecirth, and found them to answer 
pretty nearly. All this was in the two plague years of 1665 and 
1666, for in those days [at age 21 or 22] I was in the prime of 
my age for invention, and minded mathematics and philosophy 
more than at anv time since. 



PHILOSOPHI.t^ 

' N A T U R. A [. I S 

P R I N C I P I A 

MATHEMATICA 



Autorc JS. N Elf TON. Inn. CM. CmU. Sx. Miihifroi 
ProfclTorc Lncifum, i- Socirfaiis Rrgjib Sodili. 



IMPRIMATUR 

P E P \ S, Kig. Soc. P R it S E S. 
•J.U, 5. 1686. 



L ti D I N I, 

luff, Hcicljlil Kciit ac Typi« Jnfrfh SirtJUr. PioftJt apt 

l-lu,, B.l.l,o|«,la^. ^,w MDCXXXW II. 



Title page o/A'enton.s Piincipia 
Mathematica. Because the Hoyal 
Society sponsored the book, the ti- 
tle page includes the name of the 
Society's president, Samuel Pepys, 
famous for his diary, which de- 
scribes life during the seventeenth 
centurx. 



Soon after Newton's return to Cambridge, he was chosen to 
follow his former teacher as professor of mathematics. X'evvlon 
taught at the university and contributed papers to the Royal 
Societv'. At first, his contributions were mainly on optics. His 
Theory of Light and Colors, finally published in 1672, fired a long 
and bitter controversy wdth certain other scientists. Newton, a 
private and complex man, resoKed never to publish anything 
more. 

In 1684, Newton's devoted friend Halley, a noted astronomer, 
came to ask his advice. Halley was inxolved in a controversy with 
Christopher Wren and Robert Hooke about the force needed to 
cause a body to move along an ellipse in accord with Kepler's 
laws. This was one of the most debated and interesting scientific 
problems of the time. Halley was pleasantly suiprised to learn 
that Newton had already solved this problem I "and much other 
matter"!. Halley then persuaded his friend to publish these 
important studies. To encourage Newton, Hidley became 
responsible for all the costs of publication. Less than 2 years 
later, Newton had the Principia ready for the printer. Publication 
of the Principia in 1687 quickly established Newton as one of the 
greatest thinkers in histoiy. 

Several years afterward, Newlon appears to ha\'e had a nen'ous 
breakdown. He recovered, but from then until his death, 35 years 
later, Newton made no major scientific discoveries. He rounded 
out earlier studies on heat and optics and turned more and 
mor^e to writing on theologv'. During those years, he receixed 
many honors. In 1699, Newton was appointed Master of the Mint, 
partly because of his gr^eat knowledge of the chemistrv' of metals. 
In this position, he helped to reestablish the \alue of British 
coins, in which lead and copper had been introduced in place of 
silver and gold. In 1689 and 1701, Newlon represented 
Cambridge University in Parliament, and he was knighted in 1705 
by Queen Anne. He was president of the Royal Society from 1703 
until his death in 1727. Newton is buried in Westminster Abbe\ . 



210 



U\IT 2 / MOTIO\ IN THE HEAVENS 



• 1. List five characteristics of the society during Newton's 
lifetime that fostered scientific progress. 



8.2 I Newton's PtHncipia 

The original preface to Neuron's Principia, parts of which you 
have already studied, gives an outline of the book: 

Since the ancients (as we are told by Pappus) esteemed the 
science of mechanics of greatest importance in the investigation 
of natural things, and the moderns, rejecting substantial forms 
and occult qualities, liave endeavored to svibject the 
phenomena of nature to the laws of mathematics, I have in this 
treatise cultivated mathematics as far as it relates to philosophy 
[we would say physical science) . . . for the whole burden of 
philosophy seems to consist in this — from the phenomena of 
motions to investigate [induce] the forces of nature, and then 
from these forces to demonstrate [deduce] the other 
phenomena, and to this end the general propositions in the 
first and second Books are directed. In the third Book I give an 
example of this in the explication of the system of the World; 
for by the propositions mathematically demonstrated in the 
former Books, in the third I derive from the celestial 
phenomena the forces of gravity with which bodies tend to the 
sun and the several planets. Then from these forces, by other 
propositions which are also mathematical, I deduce the 
motions of the planets, the comets, the moon, and the sea 
[tides] 

The work begins with the definitions of mass, momentum, 
inertia, and force. Next come the three laws of motion and the 
principles of addition for forces and velocities (discussed in Unit 
1). Newton also included an equally important and remarkable 
passage on "Rules of Reasoning in Philosophy." The four iTiles, or 
assumptions, reflect Newton's profound faith in the unifomiity 
of all nature. Newton intended the i^ules to guide scientists in 
making hypotheses. He also wanted to make clear to the reader 
his own philosophical assumptions. These rules had their roots 
in ancient Greece and are still useful. The first has been called a 
piinciple of pai simony, the second and third, principles of unity. 
The fourth rule expresses a faith needed to use the process of 
logic. 

In a brief form, and using some modern language, Newton's 
four rules of reasoning are: 

1. "Nature does nothing ... in vain, and more is in vain when 
less will serve." Nature is essentially simple. Therefore, scientists 
ought not to introduce more hypotheses than are needed to 
explain obseived facts. This fundamental faith of all scientists 



These rules are stated by Newton at 
the beginning of Book III of the 
Principia. 



CHAPTER 8 / THE UNITY OF EARTH AND SKY 



211 



Notice that Newton's assumption 
denies the distinction between ter- 
restriiU and celestial matter. 



You should restate these rules in 
your ov\Ti words before going on to 
the next section. lA good topic for 
an essay would be whether New- 
ton's rules of reasoning are appli- 
cable outside of science.) 



had been also expressed in Galileo's "Nature . . . does not that by 
many things, which may be done by few. " Galileo in turn was 
reflecting an opinion of Aristotle. Thus, the belief in simplicity 
has a long histoiy. 

2. "Therefore to the same natural effects wc must, as far as 
possible, assign the same causes. As to respiration in a man and 
in a beast; the descent of stones in Europe and in America; . . . 
the reflection of light in the earth, and in the planets. ' 

3. Properties common to all bodies within reach of 
experiments are assumed (imtil proved othenvise) to apply to all 
bodies in general. For example, all physical objects known to 
experimenters had always been found to have mass. So, by this 
rule, Newton proposed that cvery^ object has mass, even those 
beyond our reach in the celestial region. 

4. In "experimental philosophy, ' hypotheses or generalizations 
based on experience should be accepted as "accurately or veiy 
nearly true, notwithstanding any contrary hvjDotheses that may 
be imagined. ' Scientists must accept such hvpotheses until they 
have additional exidence by which the hypotheses may be made 
more accurate or revised. 

The Principia is an extraordinaiy document. Its thici; main 
sections contain a wealth of mathematical and physical 
discoveries. Overshadowing eveiything else is the theoiy of 
universal gravitation, with the proofs and arguments leading to it. 
Newton used a form of argument patterned after that of Euclid. 
You may have encountered this tvpe of proof in studying 
geometiy. But the style of detailed geometrical steps used in the 
Principia is unfamiliar today. Therefore, many of the steps 
Newton used in his pioofs will be more understandable when 
restated in modern terms. 

The central idea of universal gr'a\dtation can be simply stated: 
Every object in the universe attracts every other object. Moreover, 
the amount of attr action depends in a simple way on the masses 
of the objects and the distance between them. 

This was Newton's great synthesis, boldly combining terrestrial 
laws of for^ce and motion with astronomical laws of motion. 
Gravitation is a universal forxe. It applies to the earth and apples, 
to the sun and planets, and to all other bodies (such as comets) 
moving in the solar- system. Heaven and earth wer'e united in one 
grand system dominated by the law of universal gravitation. The 
general astonishment and awe were reflected in the w(M'ds of the 
English poet Alexander" Pope: 



Nature and Naluie's laws ia\' hid in iiiglir: 
God said, Let Newton bel and all wa.s ligiit. 

The Principia, written in Latin, was lilled w ith long geometrical 
arguments and was diflicirlt to read. Ila|ipilv, sineral giftt^d 
writer's wr'ote summaries that allowed a wide* ( ircic of icadiMs to 



Zl'Z 



UNIT 2 / MOTION IN THE HEAVENS 



learn of Newton's arguments and conclusions. One of the most 
popular of these books was published in 1736 by the French 
philosopher and reformer Voltaire. 

Readers of these books must have been excited and perhaps 
puzzled by the new approach and assumptions. For 2,000 years, 
ft om the time of the ancient Greeks until w^ell after Copernicus, 
the ideas of natural place and natural motion had been used to 
explain the general position and movements of the planets. From 
the time of the Greeks, scholars had widely believed that the 
planets' orbits w^ere their "natural motion. " However, to Neuron 
the natural motion of a body was at a uniform rate along a 
straight line. Motion in a curve showed that a net force was 
continuously accelerating the planets away from their natural 
straight-line motion. Yet the force acting on the planets was 
entirely natural and acted between all bodies in heaven and on 
earth. Furthemiore, it w^as the same force that caused bodies on 
the earth to fall. What a reversal of the old assumptions about 
what was "natural "I 



2. Explain Newton's concept of the "whole burden of 
philosophy, " that is, the job of the scientist. 

3. In your own words, state Newton's four rules of reasoning 
and give an example of each. 

4. State, in your own words, the central idea of universal 
gravitation. 

5. How did Newton differ from Aristotle, who believed that the 
rules of motion on earth are different from the rules of motion 
in the heavens? 



8.3 I The inverse-square lan^ of planetary force 

Newton believed that the natural straight-line path of a planet 
was forced into a curve by the influence of the sun. He 
demonstrated that Kepler's law of areas could be true if, and 
only if, forces exerted on the planets were always directed 
toward a single point. (Details of his argument for this "central" 
force are given on the special pages entitled "Motion under a 
central force. ") New1:on also showed that the single point was the 
location of the sun. The law of areas is obeyed no matter what 
magnitude the force has, as long as the force is always directed 
to the same point. Newton still had to show that a central 
gravitational force would cause the exact relationship observed 
between orbital radius and period. How great was the 
gravitational force and how did it differ for different planets? 



CHAPTER 8 / THE UNITY OF EARTH AND SKY 213 



The combination of Kepler's laws with Newton's laws provides 
a fine example of the |)owei' of logical reasoning. Compare these 
laws: 

Newton's Laws Kepler's Laws 

1. A body continues in a state 1. The planets move in orbits 

of rest, or of uniform motion that are ellipses and have the 

in a straight line, unless acted sun at one focus, 

upon by a net force Haw of 2. The line from the sun to a 

inertia). planet sweeps oxer areas that 

Z. The net force acting on an are proportional to the time 

object is directly proportional intervals. 

to and in the same direction as 3. The squares of the periods 

the acceleration. of the planets are proportional 

3. To every action there is an to the cubes of their mean 

equal and opposite reaction. distances ft'om the sun iT^ = 

kHl). 

According to Newton's first law, a change in motion, either in 
direction or in magnitude Ispeedl, r'equires the action of a net 
force. According to Kepler , the planets move in orbits that are 
ellipses, that is, curved oi'bits. Thei'efore, a net for'ce must be 
acting to change their motion. Notice that this conclusion does 
not specify the type or direction of the net for ce. 

Combining Newlon's second law with the first two laws of 
Kepler clarifies the direction of the force. According to Newlon's 
second law, the net force is exerted in the direction of the 
observed acceleration. What is the dir^ection of the for^ce acting 
on the planets? Neu'ton employed the geometrical analysis 
described on pages 218-219, "Motion under* a central force. " 

Newton's analysis indicated that a body moving under a 
centr'al for-ce will, when viewed from the center* of the force, 
move according to Kepler's law of areas. Kepler's law of areas 
relates to the distance of the planets fr'om the sun. Ther^efore, 
Newton could conclude that the sun at one focus of each ellipse 
was the source of the central force acting on the planets. 

Newton then found that motion in an elliptical path would 
occur only when the central force was an inverse-square force, F 
oc 1/R^. Thus, only an inverse-square force exerted by the sun 
would r'esult in the observed ellipitcal orbits described by Kepler. 
Newton then pr^oved the argument by showing that such a for'ce 
law would also r-esult in Kepler's third law, the law of periods, 
T" = kR^. 

From this analysis, Newton concluded that one general law of 
universal gravitation applied to all bodies moving in the solar 
system. This is the centr'al argument of Newlon's gr-eat synthesis. 

Consider the motions of the sl\ then-known planets in terms 
of their centripetal acceleration toward the sun. By Newton's 
proof, mentioned above, this acceler^ation decr'eases inversely as 



ai4 UNIT 2 / MOTION IN THE HEAVTINS 



the square of the planets' average distances from the sun. The 
proof for circular orhits is very short. The expression for 
centripetal acceleration a^ of a body moving uniformly in a 
circular path, in temis of the radius H and the period T, is 

a = — — 

(This expression was derived in Chapter 4.) Kepler's law of 
periods stated a definite relation between the orbital periods of 
the planets and their average distances from the sun: 

— - = constant 

Using the symbol k for constant, 

T" = kR' 

av 

For circular orbits, fi,^ is just R. Substituting kR^ for T' in the 
centripetal force equation gives 

4'n'R 4iT' 
a = 



Since 4tt7/c is a constant, 



kR' kK 



1 
a ot — 7 
' R~ 



This conclusion follows necessarily from Kepler's law of periods 
and the definition of acceleration. If Newton's second law, F ^ 
a, holds for planets as well as for bodies on earth, then there 
must be a centripetal force F^ acting on a planet. Furthermore, 
this force must decrease in proportion to the square of the 
distance of the planet fiom the sun: 

1 

p cc 

Newton showed that the same result holds for all ellipses. 
Indeed, it holds for any object moving in an orbit around a 
center of force. [The possible orbital shapes are circle, ellipse, 

parabola, or hyperbola. These shapes are all conic sections (see 

page 188)]. Any such object is being acted upon by a centripetal j^^ Newton's time, four of Jupiter's 

force that varies inversely with the square of the distance from satellites and four of Saturn's sat- 

Ihe center of force. ellites had been observed. 

Newton had still more evidence from telescopic observ^ations of 

Jupiter's satellites and Saturn's satellites. The satellites of each SG 6 

planet obeyed Kepler's law of areas around the planet as a 
center. For Jupiter's satellites, Kepler's law of periods, T~/R^ — 
constant, held. But the value of the constant was different from 

CHAPTER 8 / THE UNITY OF EARTH AND SKY 215 



that for the planets around the sun. The law held also for 
Saturn's satellites, but vvdth still a different constant. Therefore, 
Jupiter's satellites were acted on by a central force directed 
toward Jupiter and decreasing with the square of the distance 
fi'om Jupiter. The same held true for Saturn's satellites and 
Saturn. These observed interactions of astronomical bodies 
supported Newton's proposed 1/R' central attractive force. 



• 6. What can be proved from the fact that the planets sweep 
out equal areas with respect to the sun in equal times? 

7. With what relationship can T"/R^^ = constant be combined 
to prove that the gravitational attraction varies as l/R'? 

8. What simplijying assumption was made in the derivation 
given in this section? 

9. Did Newton limit his own derivation by the same 
assumption? 

10. If two objects are moved twice as far away from one 
another, by how much is the gravitational force between them 
decreased? if they are moved three times as far? five times as 
far? By how much is the gravitational force increased if the 
objects are moved together to one-fourth their original 
separation? 

11. The moon is 60 times farther from the center of the earth 
than objects at the earth's surface. How much less is the 
gravitational attraction of the earth acting on the moon than 
on objects at its surface? E;<.press this value as a fraction of 9.8 
m/sec^. 

3 •4: I Laiv of universal gravitation 

Subject to further evidence, you can now accept the idea that a 
central force is holding the planets in their orbits. Furthermore, 
the strength of this central force changes inversely with the 
square of the distance from the sun. This strongly suggests that 
the sun is the source of the force but it does not necessarily 
require this conclusion. Newton's results so far describe the force 
in mathematical terms but do not provide any mechanism for 

its transmission. 

SG 7 The French philosopher Descartes (1596-16501 had proposed 

that all space was filled with a thin, invisible fluid. This fluid 
carried the planets around the sun in a huge whirlpool-like 
motion. This interesting idea was widely accepted at the time. 
However, Newton proved by a precise argument that this 
mechanism could not explain the details of planetary motion 
summarized in Kepler's laws. 

Z1& UNIT 2 / MOTION IN THE HEAVENS 



Kepler had made a different suggestion some years earlier. He 
proposed that some magnetic force reached out from the sun 
to keep the planets moving. Kepler was the first to regard the 
sun as the controlling mechanical agent behind planetary 
motion. But Kepler's magnetic model was inadequate. The 
problem remained: Was the sun actually the source of the force? 
If so, on what properties of the sun or planets did the amount 
of the force depend? 

As you read in Sec. 8.1, Newton had begun to think about 
planetary force while living at home during the Black Plague. 
There an idea came to him, perhaps when he saw an apple fall, 
and perhaps not. Newton's idea was that the planetary force was 
the same kind of force that caused objects near the earth's 
surface to fall. He first tested this idea on the earth's attraction 
for the moon. The data available to him fixed the distance 
between the center of the earth and the center of the moon at 
nearly 60 times the radius of the earth. Newton believed that the 
attractive force varies as 1/R^. Therefore, the gravitational 
acceleration the earth exerts on the moon should be only 1/(60)^ 
= 1/3,600 of that exerted upon objects at the earth's surface. 
Observations of falling bodies had long established gravitational 
acceleration at the earth's surface as about 9.80 m/sec/sec. 
Therefore, the moon should fall at 1/3,600 of that acceleration 
value: 9.80 m/sec' x (1/3,600) = 2.72 x 10"' m/sec'. 

Newton started from the knowledge that the orbital period of 
the moon was very nearly 27.33 days. The centripetal 
acceleration a,, of a body moving uniformly with period T in a 
circle of radius R is a^. = 47t"/J/7^. (This equation was developed 
in Sec. 4.6, Unit 1.) When you insert values for the known 
quantities R and T (in meters and seconds) and do the 
arithmetic, you find that the observed acceleration is 



To you, who have heard about grav- 
ity from your earliest years, this 
may not seem to have been a par- 
ticularly clever idea. But in New- 
ton's time, after centuries of believ- 
ing celestial events to be completely 
different from earthly events, it was 
the mental leap of a genius. Newton 
had already assumed the planets to 
be subject to the earth's laws of 
motion when he derived a 1/fl" 
force law using the formula for a^. 
It was a stm greater step to guess 
that the force on planets was not 
some special celestial force, but 
nothing other than the earth's grav- 
itational pull, which gave apples 
and everything else on earth their 
weight. 



a^ = 2.74 X 10 m/sec" 

This is in very good agreement with the value of 2.72 x 10"' 
m/sec^ predicted above. From the values available to Newton, 
which were close to these, he concluded that he had 

. . . compared the force requisite to keep the moon in her orbit 
with the force of gravity at the surface of the earth, and found 
them to answer pretty nearly. 

Therefore, the force by which the moon is retained in its 
orbit becomes, at the very surface of the earth, equal to the 
force of gravity which we observe in heavy bodies there. And, 
therefore, (by rules of reasoning 1 and 2) the force by which the 
moon is retained in its orbit is that very same force which we 
commonly call gravity. . . . 

This was really a triumph. The same gravity that brings apples 
down from trees also holds the moon in its orbit. This assertion 



CHAPTER 8 / THE UNITY OF EARTH AND SKY 



air 



Close Up I 

Motion under o Centrol Force 



How will a moving body respond to a central 
force? In order to follow Newton's analysis, remem- 
ber that the area of a triangle equals V2 base x 
altitude. Any of the three sides can be chosen as 
the base, and the altitude is the perpendicular dis- 
tance to the opposite corner. 

Suppose that a body passing some point P, was 
moving at uniform speed v along the straight line 
through PQ. (See Figure A below.) If it goes on with 
no force acting, then in equal intervals of time It it 
will continue to move equal distances, PQ, QR, RS, 
etc. 



(A^ 



How will its motion appear to an observer at some 
point O? Consider the triangles OPQ and OQR in 
Figure B below. 




So, strange as it may seem at first, Kepler's law 
of areas applies even to a body on which the net 
force has the value zero and which therefore is 
moving uniformly along a straight line. 

Suppose that the object discussed in Figure A, 
while passing through point Q, is briefly exposed to 
a force, such as a blow. If this force is directed 
toward point O, how will the object's motion change? 
(Refer to Figure D below.) 




The triangles have equal bases, PQ = QR = 
RS, and also equal altitudes, ON, for all three. 
Therefore, the triangles OPQ and OQR have equal 
areas. And therefore the line drawn from an ob- 
server at point to the body moving at a uniform 
speed in a straight line PQR will sweep over equal 
areas in eaual times. 




o (0) 

First, consider what happens if a body initially at 
rest at point Q were exposed to the same blow. The 
body would be accelerated during the blow toward 
O. It would then continue to move toward O at con- 
stant speed. After some definite time interval It, it 
would have moved a definite distance to a new point 
Q'. (See Figure E on the next page.) 



218 



Lli\IT 2 / MOTIOiX I\ THE HEAVENS 




Now, consider the effect of the blow on the object 
that was initially moving toward point R. The re- 
sultant motion is the combination of these two com- 
ponents, and the object moves to point R'. (See 
Figure F below.) 




Earlier we found that the areas of the triangles 
OPQ and OQR were equal. Is the area of the tri- 
angle OQR' the same? Both triangles OQR and 
OQR' have a common base, OQ. Also, the altitudes 
of both triangles are the perpendicular distance from 
line OQ to line RR'. (See Figure G.) Therefore, the 
areas of triangles OQR and OQR' are equal. 

If another blow directed toward O were applied 
at point R', the body would move to some point S", 




(G) 



as indicated in Figure H below. By a similar analysis 
you can find that the areas of triangles OR'S" and 
OR'S' are equal. Their areas also equal the area 
of triangle OPQ. 




o (H) 

In this geometrical argument we have always ap- 
plied the force toward the same point, O. A force 
always directed toward a single point is called a 
central force. (Notice that the proof has nothing to 
do with the magnitude of the force or with how it 
changes with distance from O.) Also, we have ap- 
plied the force at equal intervals At. If each time 
interval Af were made vanishingly small, the force 
would appear to be applied continuously. The ar- 
gument would still hold. We then have an important 
conclusion: If a body is acted upon by any central 
force, it will move in accordance with Kepler's law 
of areas. 



CHAPTER 8 / THE UNITY OF EARTH AND SKY 



219 



5un 



:w 



The sun, moon, and earth each pull 
on the other. The forces are in 
matched pairs, in agreement with 
Newton's third law of motion. As 
the moon moves through space, 
the gravitational attraction of the 
earth causes the moon to fall" 
toward the earth. The continuous 
combination of its straight-line in- 
ertial motion and its fall' pro- 
duces the curved orbit. 



is the first portion of what is knovvTi as {he law of iinixersal 
gravitation: Every object in the universe attracts every other object 
with a gravitational force. If this is so, there must be gravitational 
forces not only between a lock and the earth, but also between 
the earth and the moon, between Jupiter and its satellites, and 
between the sun and each of the planets. 

Newton did not stop at saying that a gravitational force exists 
between the planets and the sun. He further claimed that the 
force is exactly the right size to explain completely the motion of 
every planet. No other mechanism iwhirlpools of imisible fluids 
or magnetic forces! is needed. Gravitation, and gravitation alone, 
underiies the dynamics of the heavens. 

This concept is so commonplace that you might be in danger 
of passing it by without really understanding what Newton was 
claiming. First, he proposed a truly universal physical law. 
Following his rules of reasoning, Newton extended to the whole 
universe what he found tiue for its observable parts. He excluded 
no object in the universe ftom the effect of gravity. 

The idea that terrestrial laws and forces were the same as 
those that regulated the whole universe had stunning impact. 
Less than a centuiy before, it would have been dangerous even 
to suggest such a thing. Kepler and Galileo had laid the 
foundation for combining the physics of the heavens and earth. 
Newton carried this work to its conclusion. Today, Newton s 
extension of the mechanics of terrestrial objects to the motion of 
celestial bodies is called the Newtonian synthesis. 

Newton's claim that a planet's orbit is detemiined by the 
gravitational attraction between it and the sun had another 
effect. It moved science away from geometrical explanations and 
towards physical ones. Most philosophers and scientists before 
Newton had been occupied mainly with the question 'What are 
the motions? " Nev\ton asked instead "What force explains the 
motions?' In both the Ptolemaic and Copernican systems, the 
planets moved about points in space rather than about objects. 
The planets moved as they did because of their "nature " or 
geometrical shape, not because forces acted on them. Newton, 
on the other hand, spoke not of points, but of things, of objects, 
of physical bodies. Unless the gravitational attraction to the sun 
deflected them continuously from straight-line paths, the planets 
would fly out into the darkness of deep space. Thus, it was the 
physical sun that was important, not the point at which the sun 
happened to be located. 

Newton's synthesis centered on the idea of gravitational force. 
By calling it a force of gravity, Nevvt(Hi knew that he was not 
explaining why it existed. When you hold a stone above the 
surface of the earth and release it, it accelerates to the ground. 
The laws of motion tell you that there must be a force acting on 
the stone to accelerate it. You know the direction of the force. 



220 



UNIT 2 / MOTION IM THE HEAVENS 




/ /■ / / A drawing by which Descartes 

' '/ (1596-1650) illustrated his theory of 
space being filled with whirlpools 
jf matter that drive the planets 
along their orbits. 







You can find the magnitude of thie force by multiplying the mass 
of the stone by the acceleration. You know that this force is 
weight, or gravitational attraction to the earth. But why such an 
interaction between bodies exists remains a puzzle. It is still an 
important problem in physics today. 



12. What idea came to Newton while he was thinking about 
filling objects and the moon's acceleration? 

13. Kepler, too, believed that the sun e?certed forces on the 
planets. How did his view differ from Newton's? 

14. The central idea of Chapter 8 is the "Newtonian synthesis." 
What did Newton synthesize (bring together)? 



3»S I Neivton and hypotheses 

Newton's claim that there is a mutual force (gravitational 
interaction) between a planet and the sun raised a new question: 



CHAPTER 8 / THE UNITY OF EARTH AND SKY 



221 



How can a planet and the sun act upon each other at enormous 
distances without any \asible connections between them? On 
earth you can exert a force on an object by pushing it or pulling 
it. You are not surprised to see a cloud or a balloon drifting 
across the sky, even though nothing seems to be touching it. Air 
is inxisible, but you know that it is actually a material substance 
that you can feel when it moves. Falling objects and iron objects 
being attracted to a magnet are harder to explain, but the 
distances are small. However', the earth is over 144 million 
kilometers, and Saturn more than 1 billion kilometers, from the 
sun. How could there possibly be any physical contact between 
such distant objects? How can we account for such "action at 
a distance"? 

In Newton's time and for a long time afterwarxl, scholar's 
advanced suggestions for solving this problem. Most solutions 
involved imagining space to be filled with some invisible 
substance (an "ether") that transmitted for^ce. Newton himself 
piixately guessed that such an ether was involved. But he could 
find no way to test this belief. Therefor e, at least in public, he 
refused to speculate on possible mechanisms. As Neuron said in 
a famous passage which he added in the second edition of the 
Principia (17131: 

. . . Hitherto I have not been able to discover the cause of those 
properties of gia\ity from phenomena, and I frame no 
hypotheses; for whatever is not deduced from [he phenomena 
is to be called an hvpothesis; and hypotlieses, whether 
metaphysical or physical, whether of occult qualities or 
mechanical, have no place in experimental philosophy. . . . And 
to us it is enough that gravity does really exist, and acts 
according to tlie laws which we have explained, and 
abundantly serves to account for- all the motions of the celestial 
bodies, and of our sea. 

Newton is quoted at length here because one particular phrase 
is often taken out of context and misinterpreted. I'he original 
Latin reads: hypotheses nonfingo. This means "I ftame no 
hypotheses " or "I do not feign hypotheses." The sense is, "I do 
not make false hypotheses." Newton in fact made many 
hypotheses in his publications. Also, his letters to friends contain 
many speculations which he did not publish. So his stern denial 
of "framing" hypotheses must be properly interpreted. 

The fact is that there are two main kinds of hypotheses or 
assumptions. The most common hypothesis is a pr'oposal of 
some hidden mechanism to explain observations. For- example, 
you observe the moving hands of a watch. You might propose or 
imagine some arrangement of gears and springs that cairses the 
motion. This would be a hypothesis that is directly or indirectly 
testable, at least in principle, by reference to phenomena. The 
hypothesis about the watch, for example, can be tested h\ 



222 UIVIT 2 I MOTI(3IV I\ THE HEAVENS 



opening the watch or by making an X-ray photograph of it. In 
this context, consider an invisible fluid that transmitted 
gravitational force, the so-called "ether." Newton and others 
thought that certain direct tests might establish the presence of 
this substance. Many experimenters tried to "catch" the ether. 
A common approach involved pumping the air from a bottle. 
Then tests were made to see if any wind, pressure, or ftiction 
due to the ether remained to affect objects in the bottle. Nothing 
of this sort worked (nor has it since). So Newton wisely avoided 
making public any hypothesis for which he could not also 
propose a test. 

A quite different type of assumption is often made in 
published scientific work. It involves a hypothesis which 
everyone knows is not directly testable, but which still is 
necessary yusf to get started on one's work. An example is such a 
statement as "nature is simple" or any other of Newton's four 
rules of reasoning. Acceptance of either the heliocentric system 
or the geocentric system is another example. In choosing the 
heliocentric system, Copernicus, Kepler, and Galileo made the 
hypothesis that the sun is at the center of the universe. They 
knew that this hypothesis was not directly testable and that 
either system seemed to explain "the phenomena" equally well. 
Yet they adopted the point of view that seemed to them simpler, 
more convincing, and more "pleasing to the mind. ' It was this 
kind of hypothesis that Newton used vvdthout apology in his 
published work. Every scientist's work involves both kinds of 
hypothesis. The popular image of the scientist is of a person who 
uses only deliberate, logical, objective thoughts, and immediately 
tests them by definitive experiments. But, in fact, the working 
scientist feels quite tree to entertain any guess, imaginative 
speculation, or hunch, provable or not, that might be helpfiil. 
(Sometimes these hunches are dignified by the phrase "working 
hypotheses. " Without them there would be little progress!) Like 
Newton, however, most scientists today do not like to publish 
something that is still only an unproven hunch. 



# 15. Did Newton explain the gravitational attraction of all 
bodies? 

16. What was the popular type of explanation for "action at a 
distance "? Why did Newton not use this type of explanation? 

17. What are two main types of hypotheses used in science? 

18. Newton's claim to "frame no hypotheses" seems to refer to 
hypotheses that cannot be tested. Which of the following 
claims are not testable? 

(a) Plants need sunlight to grow, even on other planets. 




Stephen W. Hawking, an astrophysi- 
cist at Cambridge University, is 
considered by many scientists to 
be the equal of Newton and Ein- 
stein. 'Professor Hawking is search- 
ing for a quantum theory of gravi- 
tation by studying the phenomena 
associated with black holes. The 
achievement of such a theory 
would be the last step in the theo- 
retical unification of all the forces 
in the universe. 



CHAPTER 8 / THE UNITY OF EARTH AND SKY 



223 







The gravitational force on a planet 
owing to the sun s pull is equal and 
opposite to the gravitational force 
on the sun owing to the planet. 



(b) This bandage is guaranteed to be free from germs unless 
the package is opened. 

(c) Virtual particles e^ist for a time that is too short for them 
to affect anything. 

(d) Life exists in the distant galas^ies. 

(e) The earth really does not move, since you would feel the 
motion if it did. 

(f) Universal gravitation holds between every pair of objects in 
the universe. 

8.6 I The magnitude of planetari' force 

The general statement that gravitational forces exist universally 
must now be turned into a quantitative law. An expression is 
needed for both the magnitude and direction of the forces any 
two objects exert on e ich other. It was not enough for Newlon to 
asseit that a mutual gravitational attraction exists between the 
sun and Jupiter. To be convincing, he had to specify what 
quantitative factors determine the magnitudes of those mutual 
forces. He had to show how they could be measured, either 
directly or indirectly. 

The first problem was defining precisely the distance R. Should 
it, for example, be taken as the distance between the surface of 
the earth and the surface of the moon? For many astronomical 
problems, the sizes of the interacting bodies are extremely small 
compared to the distances between them. In such cases, 
the distance between the surfaces is practically the same as the 
distance between the centers. (For the earth and the moon, 
the distance between centers is only about 2% greater than the 
distance between surfaces.) Yet, some historians believe Newton's 
uncertainty' about a proper answer to this problem led him to 
drop the study for many years. 

Eventually, Neuron solved the problem. The gravitational force 
exerted by a spherical body is the same as if all its mass were 
concentrated at its center. The gravitational force exerted on a 
spherical body by another body is the same as would be exerted 
on it if all its mass were concentrated at its center. Therefore, the 
distance R in the law of gravitation is the distance between 
centers. 

This was a very important discovery. The gravitational 
attraction between spherical bodies can be considered as though 
their masses were concentrated at single points. Thus, in 
thought, the objects can be replaced by mass points. 

Newton's third law states that action equals reaction. If tiiis is 
universally true, the amount of force the sun exerts on a planet 
must exactly equal the amount of force the planet exerts on the 
sun. For such a veiy large mass and suc'h a relatively small mass, 



224 



UM T 2 / V10TI0\ IN THE HEAVEiXS 



this may seem contrary to common sense. But the equality is 
easy to prove. First, assume only that Newton's third law holds 
between small pieces of matter. For example, a 1-kg piece of 
Jupiter pulls on a 1-kg piece of the sun as much as it is pulled by 
it. Now consider the total attraction between Jupiter and the sun, 
whose mass is about 1,000 times greater than Jupiter's. As the 
figure in the right margin indicates, you can consider the sun as 
a globe containing 1,000 Jupiters. Define one unit of force as the 
force that two Jupiter-sized masses exert on each other when 
separated by the distance of Jupiter from the sun. Then Jupiter 
pulls on the sun (a globe of 1,000 Jupiters) with a total force of 
1,000 units. Each of the 1,000 parts of the sun cdso pulls on the 
planet Jupiter with 1 unit. Therefore, the total pull of the sun on 
Jupiter is also 1,000 units. Each part of the massive sun not only 
pulls on the planet, but is also pulled upon by the planet. The 
more mass there is to attract, the more there is to be attracted. 
(Although the mutual attractive forces are equal in magnitude, 
the resulting accelerations are not. Jupiter pulls on the sun as 
hard as the sun pulls on Jupiter, but the sun responds to the 
pull with only 1/1,000 of the acceleration, because its inertia is 
1,000 times Jupiter's.) 

Sec. 3.8 of Unit 1 explained why bodies of different mass fall 
wdth the same acceleration near the earth's surface. The greater 
the inertia of a body, the more strongly it is acted upon by 
gravity; that is, near the earth's surface, the gravitational force on 
a body is directly proportional to its mass. Like Nevvl^on, extend 
this earthly effect to all gravitation. You then can assume that the 
gravitational force exerted on a planet by the sun is proportional 
to the mass of the planet. Similarly, the gravitational force 
exerted on the sun by the planet is proportional to the mass of 
the sun. You have just seen that the forces the sun and planet 
exert on each other are equal in magnitude. It follows that the 
magnitude of the gravitational force is proportional to the mass 
of the sun and to the mass of the planet; that is, the gravitational 
attraction between two bodies is proportional to the product of 
their masses. If the mass of either body is tripled, the force is 
tripled. If the masses of both bodies are tripled, the force is 
increased by a factor of 9. Using the symbol F for the 
magnitude of the forces, F^,,, oc m^,,,,,,., m,,„. 

The conclusion is that the amount of attraction between the 
sun and a planet is proportional to the product of their masses. 
Earlier you saw that the attraction also depends on the square 
of the distance between the centers of the bodies. Combining 
these two proportionalities gives one force law, which now 
includes mass and distance: 



lOOO Jupiter-s 




Juipiter puds on \ooO 
pa-rts o'^ -the- sun 

\000 pArt^ of t^ie eon 
pull on Jupit-er 

tflttttlll 

O 

Jupite-r" 



SG 8 



SG 9 



m 



CHAPTER 8 / THE UNITY OF EARTH AND SKY 



225 



Such a propoi-tionaliU' can be mitten as an equation by 
intixjclucing a constant. iThe constant allows for the units of 
measurement used.) Using G for the proportionality constant, the 
law of planetary forces can be wiitten as 

P p. planet sun 

grav - ^2 

This equation asserts that the force between the sun and any 
planet depends only upon three factors. These factors are the 
masses of the sun and planet and the distance between them. 
The equation seems unbelievably simple when you remember 
how complex the observed planetary motions seemed. Yet every 
one of Kepler's empirical laws of planetarv motion agrees with 
this relation. In fact, you can even derive Kepler's empirical laws 
from this force law and Neuron's second law of motion. More 
important still, details of planetaiy motion not obtainable with 
Kepler's laws alone can be calculated using this force law. 

Newton's proposal that this simple equation describes 
completely the forces between the sun and planets was not the 
final step. He saw nothing to limit this mutual force to the sun 
and planets, or to the earth and apples. Rather, Newton insisted 
that an identical relation should apply universally. This relation 
would hold tnje for any tv^o bodies separated by a distance that 
is large compared to their dimensions. It would apply equally 
to two atoms or two stars. In short, Newton proposed a general 
law of universal gravitation: 

SG 10 F ^ G \ 

where m, and m_, are the masses of the bodies and Fi is the 
distance between their centers. The numerical constant G is 
called the constant of universal gravitation. Newton assumed it to 
be the same for all gravitational interactions, whether between 
two grains of sand, two members of a solar system, or two stars 
in different parts of the sky. As you udll see, the successes made 
possible by this simple relationship have been veiy great. In fact, 
scientists have come to assume that this equation applies 
everywhere and at all times, past, present, and future. 

Even before you gather more supporting exidence, the 
sweeping majesty of Neuron's theory should command your 
wonder and admiration. It also leads to the question of how 
such a bold universal theory can be pr'oved. Ther-e is no 
complete proof, of course, for that would mean examining every 
interaction between all bodies in the universe! But the greater 
the \'ar iet>' of single tests made, the greater will be the belief in 
the coriectness of the theory. 

226 UNIT 2 I MO TIOM li\ THE HEAVENS 



# 19. According to Newton's law of action and reaction, the 

earth should experience a force and accelerate toward a falling 
stone. 

(a) How does the force on the earth compare with the force on 
the stone? 

(b) How does the earth s acceleration compare with the 
stone's acceleration? 

20. Diagram A at the right represents two bodies of equal 
mass that e?cert gravitational forces of magnitude F on one 
another. What is the magnitude of the gravitational attractions 
in each of the other cases? 

21. A, B, C, and D are bodies with equal masses. How do the 
forces of attraction that A and B e^ert on each other compare 
with the forces that C and D e;<ert on each other? 

(a) F^ = 3 X F,,„ 

(b) F^ = 4 X F,„ 

(c) F^ - 9 X F,„ 

(d) ¥,, = 16 X F,„ 

22. Why is it a great simplification to use the distance between 
the centers of spherical objects in the formula for gravitational 
force? How is the use of the distance between centers 
justified? What does it mean to say that the law of gravitation 
is important because it is simple? 



(a)i 9^ -^ < ^ 



ih) •? 7% 



) 

j ._ _ 

(d)i Jv '/•, 



Ai m 



\ i ; 






I 

I 



8*7 Planetary motion and the 
^avitational constant 

Suppose that a planet of mass m^^ is moving along an orbit of 
radius R and period T. According to Newton's mechanics, there 
is a continual centripetal acceleration a^ = 4'n~R/T^. Therefore, 
there must be a continual force F^ 
gravity is the central force, then 

F = F 

grav c 



m^a^ = 4'^■^fimp/T^ If 



or 



m^/n^ 



4iT~Hm^ 



Simplifying this equation and rearranging some terms gives an 
expression for G: 



G - 



477" B 



m 



T 



CHAPTER 8 / THE UMTY OF EARTH AND SKY 



227 



This photograph, of the surface of 
the moon, shows some latter-dav 
evidence that the laws of mechan- 
ics for heavenly bodies are at least 
similar to those applying on earth. 
The trails of tixo huge boulders 
that rolled about 300 m down a lu- 
nar slope are shown. 




>i 



You know from Kepler that for the planets' motion around the 
sun, the ratio Fi^/l^ is a constant; 4it~ is a constant also. If the 
mass of the sun is assumed to be constant, then all factors on 
the right of the equation for G are constant. So G must be a 
constant for the gravitational effect of the sun on the planets. By 
similar reasoning, the value of G must be a constant for the effect 
of Jupiter on its moons. It must also be a constant for Saturn and 
its moons, for earth and its moon, and for an apple falling to the 
earth. But is it the same value of G for all these cases? 

It is impossible to prove that G is the same for the gravitational 
interaction oi all bodies. But by assuming that G is a universal 
constant, the relative masses of the sun and the planets can be 
obtained. 

Begin by again equating the centripetal force on the planets 
with the gravitational attraction to the sun. This time solve the 
equation for m^^^,^: 

F = F 

grav c 






477 flm^ 



m 



47T^ 

gt' 



Writing A:^^„, for the constant ratio 7 7fi^ gives 

4lT~ 



m - 



Gk 



By similar derivation, 

4tt' 



47r' 



•Jupitr 



Gk 



-' "1s.„.„n 



lup 



Gk 



4it' 



m. 



Gk 



228 



UMIT 2 / MOTION I\ THE HEAVENS 



Here k^^^,,^^, k^ 



and fcgg,^^ are the known values of the constant 



ratios T^/R^ for the satellites of Jupiter, Saturn, and the earth. 
To compare Jupiter's mass to the mass of the sun, simply 
divide the formula for mj^pj,^^ by the formula for m^^^: 



4tt' 



m 



Jupiter 



Gk 



Jupiter 



m 



m. 



477^ 

Gk~. 



or 



Jupiter 



m„ 



Jupiter 



Similarly, you can compare the masses of any tu^o planets if you 
know the values of T^/R^ for them both; that is, both must have 
satellites whose motion has been carefully observed. 

These comparisons are based on the assumption that G is a 
universal constant. Calculations based on this assumption have 
led to consistent results for a vvdde variety of astronomical data. 
One example is the successful orbiting and landing of a space 
vehicle on the moon. Results consistent with this assumption 
also appeared in difficult calculations of the small disturbing 
effects that the planets have on each other. There is still no way 
of proving G is the same everywhere and always. But it is a 
reasonable working assumption until evidence to the contrary 
appears. 

If the numerical value of G were known, the actual masses of 
the earth, Jupiter, Saturn, and the sun could be calculated. G is 
defined by the equation Fg,^^ = Gmjm//?^ . To find the value of G, 
you must know values for all the other variables; that is, you 
must measure the force F^^^^ between two measured masses m^ 
and m^, separated by a measured distance R. Newton knew this. 
In his time there were no instruments sensitive enough to 
measure the very tiny force expected between masses small 
enough for experimental use. 



Masses Compared to Earth 

Earth 1 

Saturn 95 

Jupiter 318 

Sun 333,000 



• 23. What information can be used to compare the masses of 
two planets? 

24. What additional information is necessary for calculation of 
the actual masses? 



8.8 I The value of G and the actual masses 
of the planets 

The masses of small solid objects can be found easily enough 
from their weights. Measuring the distance between solid objects 
of spherical shape presents no problem. But how can one 
measure the tiny mutual gravitational force between relatively 



Calculation of G from approximate 

experimental values: 

Mm 
F = G-^ 

grav ^2 



G = 



F„,. R' 



Mm 

do' N) (0.1 m)' 
(100 kg) (1 kg) 

10"'' X 10 



10' 



N = mVkg^ 



= 10""'Nm'/kg' 



CHAPTER 8 / THE UNITY OF EARTH AND SKY 



229 



N • mVkg'' can be expressed as 
mVkg • sec". 



SG 14-19 



Actual Masses (in 


units of 10'' kg) 


Sun 


1,980,000 


Mercury 


0.328 


Venus 


4.83 


Earth 


5.98 


Mars 


0.637 


Jupiter 


1,900 


Saturn 


567 


Uranus 


88.0 


Neptune 


103 


Pluto 


0.014 



SG 20-22 



small objects in a laboratory? (Remember that each object is also 
experiencing separately a huge gravitational for ce toward the 
tremendously massive earth.) 

This serious technical problem was eventirally solved by the 
English scientist, Henry Cavendish (1731-1810). For- measuring 
gravitational forces, he employed a torsion balance. In this 
device, the gr-a\itational attr-action between two pairs of lead 
spheres twisted a wire holding up one of the pairs. The twist of 
the wire could be calibrated from the twist produced by small 
known forces. A typical experiment might invohe a 100-kg sphere 
and a 1-kg sphere at a center-to-center distance of 0.1 m. The 
Insulting force would be about one-millionth of a newton 
(0.000001 N)! As the calculations in the margin show, these data 
lead to a value for- G of about lO'" (N • mVkg^J. This experiment 
has been steadily improved, and the accepted value of G is now 

G = 6.67 X 10" N • mVkg' 

Evidently, gravitation is a weak force that becomes important 
only when at least one of the masses is very great. The 
gravitational fot-ce on a 1-kg mass at the surface of the ear'th is 
9.8 N; if released, a 1-kg mass falls with an acceleration of 9.8 
m/sec^. Substituting 9.8 N for F^^^^and the radius of the earth for 
B, you can calculate the mass of the earth! (See SG 11.) 

Assume that the same value for G applies to all gravitational 
interaction. Now you can calculate the masses of the planets 
from the known values of T^/H^ for their satellites. Since Newton's 
time, satellites have been discovered around all of the outer 
planets. The values of these planets' masses, calculated from m 
= 4t:^/G X B^/T^, are given in the table in the margin. Venus and 
Mercury have no satellites. Their masses are found by analyzing 
the slight disturbing effects each has on other planets. Modem 
values for the actual masses of the planets are listed in the 
margin. Notice that the planets taken together make up not 
much more than 1/1000 of the mass of the solar system. By far, 
most of the mass is in the sun. For this reason, the sun 
dominates the motion of the planets, acting almost like an 
infinitely massive, fixed object. 

In light of Newton's third law, this picture must be modified a 
little. For every pull the sun exerts on a planet, the sun itself 
experiences an equally strong pull in the opposite direction. Of 
course, the very much greater mass of the sun keeps its 
acceleration to a correspondingly smaller value. But some slight 
acceleration does exist. Therefore, the sun cannot really be fixed 
in space even uithin the solar system, if we accept Newlonian 
dynamics. Rather-, it moves a little about the point that forms the 
common center of mass of the sun and each moving planet. This 
is true for every one of the nine planets. Since the planets rar-ely 
move all in one line, the sun's motion is a(-tuall\ a conibination 



230 



UNIT 2 / MOTION IN THE HEAVENS 



of nine small ellipses. Such motion might be important in a solar 
system in which the planets were very heavy compared to their 
sun. In our solar system, it is not large enough to be of interest 
for most purposes. 




Z5. Which of the quantities in the equation F^^^^ = Gm^ m/R" 
did Cavendish measure.^ 

26. Knowing a value for G, what other information can be used 
to find the mass of the earth? 

27. Knowing a value for G, what other information can be used 
to find the mass of Saturn? 

28. The mass of the sun is about 1,000 times the mass of 
Jupiter. How does the sun's acceleration due to Jupiter's 
attraction compare with Jupiter's acceleration due to the sun's 
attraction? 

8.9 I Further successes 

Newton did not stop udth the fairly direct demonstrations 
described so far. In the Principia, he showed that his law of 
universal gravitation could explain other complicated 
gravitational interactions. Among these were the tides of the sea 
and the peculiar drift of comets across the sky. 

The tides: Knowledge of the tides had been vital to navigators, 
traders, and explorers through the ages. The cause of the tides 
had remained a mystery despite the studies of such scientists as 
Galileo. However, by applying the law of gravitation, Neuron was 



Schematic diagram of the device 
used by Cavendish for determining 
the value of the gravitational con- 
stant G. Large lead balls of masses 
M| and M, were brought close to 
small lead balls of masses m^ and 
m,. The mutual gravitational at- 
traction between M^ and m^ and 
between M^ and m^ caused the ver- 
tical wire to be twisted by a meas- 
urable amount. 

Right: 

Cavendish s original drawing of his 
apparatus for determining the 
value ofG. To prevent disturbance 
from air currents, he enclosed it 
in a sealed case. Cavendish ob- 
served the deflection of the balance 
rod from outside with telescopes. 



SG 23 



CHAPTER 8 / THE UNITY OF EARTH AND SKY 



231 



Moon (^-j X 




Tidal forces. The earth-moon dis- 
tance indicated in the figure is 
greatly reduced because cfthe 
space limitations. 



able to explain the main features of the ocean tides. He found 
them to result from the attraction of the moon and the sun upon 
the waters of the earth. As you can calculate in SG 16, the 
moon's tide-raising force is greater than the sun s. Each day, two 
high tides normally occur. Also, twice each month, the moon, 
sun, and earth are in line with each other. At these times the 
tidal changes are greater than average. 

Two questions about tidal phenomena demand special 
attention. First, why do high tides occur on both sides of the 
earth, including the side away from the moon? Second, w^hy does 
high tide occur at a given location some hours after the moon 
is highest in the skv'? 

Newton knew that the gravitational attractions of the moon 
and sun accelerate the whole solid earth. These forces also 
accelerate the fluid water at the earth s surface. Neulon retdized 
that the tides result from the difference in acceleration of the 
earth £ind its waters. The moon's distance from the earth's center 
is 60 earth radii. On the side of the earth necirer the moon, the 
distance of the water from the moon is only 59 radii. On the side 
of the earth away from the moon, the water is 61 earth radii from 
the moon. The accelerations are shown in the figure at the left. 
On the side nearer the moon, the acceleration of the water 
toward the moon is greater than the acceleration of the etirth as 
a whole. The net effect is that the water is accelerated away from 
the earth. On the side of the earth away from the moon, the 
acceleration of the water toward the moon is less than that of 
the earth as a whole. The net result is that the earth is 
accelerated away from the water there. 

Perhaps you have watched the tides change at the seashore or 
examined tide tables. If so, you know that high tide occurs some 
hoars after the moon is highest in the sky. To understand this, 
ex'en qualitatively, you must remember that on the whole the 
ocefins cire not ver^' deep. The ocean waters mo\ing in from 
more distant parts of the oceans in response to the moon's 
attraction are slowed by friction with the ocean floors, especiailly 
in shallow water. Thus, the time of high tide is delayed. In any 
particular place, the amount of delay and the height of the tides 
depends greatly upon how easily the waters can flow. No general 
theory can account for all the pcirticulcir details of the tides. Most 
local predictions in the tide tables are based on empirical rtiles 
using the tidal patterns recorded in the past. 

Since there ar-e tides in the seas, you may wonder if the 
atmosphere and the earth itself undergo tides. They do. The 
earth is not completely rigid, but bends somewhat, like steel. The 
tide in the ear1h is about 30 cm high. The atmospheric tides are 
generally masked by other weather changes. However, at 
altitudes of about 160 km, satellites have recorded considerable 
rises anr) falls in the thin atmrjsphere 



'z:rz 



UM r Z I MOI IO\ l\ IHIi HtAVENS 



Comets: From earliest histon- through the Middle -Ages comets 
ha\'e been interpreted as omens of disaster. HaBe\ and Ne\^ton 
sho^^'ed them to be onh shin\ , cloud\ masses mo\ing around 
the sun according to Kepler s laA\'5 iust as planets do. The\ 
found that most comets are \lsible onh" A\±ien closer to the sun 
than the distance of Jupiter. Sev'eral A^en bright comets have 
orbits that take thexn \\^I1 inside the orbit of Mercun . Such 
comets pass u-ithin a fe\v million kilometers of the sun as the 
figure at the right indicates. Man\ orbits ha\'e eccentricities near 
1.0 and are almost parabolas: these comets ha\'e periods of 
thousands or ev-en millions of \'e.ars. Some other faint conaets 
hax-e periods of onh- 5-10 xTiars. 

L'nlike the planets all of \N±iose orbits lie nearh in a single 
plcine the planes of comet orbits tilt at all angles. Vet like all 
members of the solar s\'stem the\- obe^ all the la\^'5 of d\Tiamics 
including the law of unix'eTsal gra\itation. 

Edmund Halle\ 1656-1742 applied Newton s concepts of 
celestial motion to the motion of bright comets. .Among the 
comets he studied \%iiere those seen in 1531 1607 and 1682 
HaUey found the orbits for these coniets to be \'erv- nearh the 
same. He suspected that the\ might be one comet mD\ing in a 
closed orbit with a period of about 75 \'ears. He predicted that 
the comet A%T>uld return in about 1757, A^±lich it did although 
Halle) did not live to see it. Halley s comet appeared in 1S33 and 
1909 and is due to be near the sun and \isible again in 19S5— S6. 

With the f)ejiod of this bright comet kno\Mi its dates of 
appearance could be tracked bacJk in histon . .Ancient Indian 
Chinese, and Japanese documents record all ejq^ected 
appearances except one since 240 b c Almost no European 
records of this great comet e.vist. This is a sad commejit upon 
the lev'el of culture in Europ)e during the so-c^ed Dark .Ages. 
One of the few European records is the famous BaA-eux tapestn 
embroidered \\ith 72 scenes of the Norman Conquest of England 
in 1066. One scene show^ the comet OA^erhead ^^ilile King Harold 
of England and his court cowex below. .A maior triumph of 
Newtonian science was its e.\planation of comets. Now the\ were 
seen to be regular members of the solar sx'stemi instead of 
unpredictable, fearful ex'ents. 

The scope of the principle of: ~ / p^a\itation: Ne\^"tnn 

^plied the law of uni^ej^al gra^ . .j many other problems 

wiiich cannot be considej^d in detail haB, For example, he 
inxTistigated the causes of the some^^±lat irregular motion of the 
moon. He shoxN'ed that these motions are explained by the 
gra\itational forces acting on the moon. As the moon mo\«s 
around the earth the moon s distance from the sun changes 
continualh . This changes the resultant force of the earth and the 
sun on the oihitir^ moon. Ne^\Ton also shoA%^d that o^er 
chances ■■ :'-;- -■,.-■ > ;".^:\^:^ .^.-. .. :^: .-jiuse the e:i"":"' - , ' . 



L-org. 5 3-1) ■ 




: qflhr arDv: cf 
rmTo thf f.clipTir 
f-dm 



CHAPTER !i THE I MT\ OF EARTH ANT) SK\ 



i3.3 




A scene from the Bayeu}c tapestry, 
which was einbrnidered about 
1070. The bright comet of 106G can 
be seen at the top of the figure. 
This comet was later identified as 
Halley's comet. At the right, Har- 
old, pretender to the throne of 
England, is warned that the comet 
is an ill omen. Later that year, at 
the Battle of Hastings, Harold was 
defeated by William the Conqueror. 



perfect sphere. IThe earth's diameter at the equator is 43.2 km 
greater than the diameter through the poles.) On the problem of 
the moon's motion Newton commented that 'the calculation of 
this motion is difficult. " Even so, he obtained predicted xalues 
reasonably close to the observed values available at that time. He 
even predicted some details of the motion which had not been 
noticed before. 

Newton investigated the variations of gra\ity at different 
latitudes on the spinning and bulging earth. He noted differences 
in the rates at which pendulums svxang at different latitudes. 
From these data, he derived an approximate shape for the earth. 

In short, Newton created a whole new quantitative approach 
to the study of astronomical motion. Because some of his 
predicted variations had not been observed, improved 
instruments were built. These instruments improxed the old 
observations that had been fitted together under the grand 
theory. Many new theoretical problems also clamored for 
attention. For example, what were the predicted and observed 
influences among the planets themselves upon th(Mr motions? 
Although the planets are small compared to the sun and are \'eiy 
far apart, their interactions are obseiAable. As precise data 
accumulated, the Newlonian theoiA' permitted calculations about 
the past and future of the planetar\' system. For past or future 
intenals beyond some hundreds of billions of years, such 
extrapolations become too uncertain. But for shorter inteiAals, 
Newtonian theory says that the planetary system has been and 
will continue to be about as it is now. 



234 



U\IT 2 / MOTIOM IN THE HEAVENS 



Newton's greatness went beyond the scope and genius of his 
work in mechanics. It went beyond the originality and elegance 
of his proofs. It had another dimension: the astonishing detail 
in which he developed the full meaning of each of his ideas. Sure 
of his law of universal gravitation, Newton applied it successfully 
to a vast range of terrestrial and celestial problems. As a result, 
the theoiy became more and more widely accepted. Newton's 
theoiy has been the chief tool for solving all of the new problems 
concerning motion in the solar system. For example, the motion 
of every artificial satellite and space probe is calculated 
according to Newton's law of universal gravitation. You can well 
agree wdth the reply given to ground control as Apollo 8 returned 
from the first trip to the moon. Ground control : "Who's driving 
up there? " Captain of Apollo 8: "I think Isaac Newton is doing 
most of the driving right now. " 

Beyond the solar system: You have seen how Newton's laws 
explain motions and other physical events on the earth and in 
the solar system. Now consider a new and even broader 
question: Do Newton's laws also apply at greater distances, for 
example, among the stars? 

Over the years following publication of the Principia, several 
sets of obseiA/ations provided an answer to this important 
question. One observer was William Herschel, a British musician 
turned amateur astronomer. In the late 1700's, with the help of 
his sister Caroline, Herschel made a remarkable series of 
observations. Using homemade, high-quality telescopes, Herschel 
hoped to measure the parallax of stars due to the earth's motion 
around the sun. Occasionally he noticed that one star seemed 
quite close to another. Of course, this might mean only that two 
stars happened to lie in the same line of sight. But Herschel 
suspected that some of these pairs were actually double stars 
held together by their mutual gravitational attractions. He 
continued to observe the directions and distances from one star 
to the other in such pairs. In some cases, one star moved during 
a few years through a small arc of a curved path around the 
other. (The figure shows the motion of one of the two stars in a 
system.) Other astronomers gathered more infonnation about 
these double stars, far removed from the sun and planets. 
Eventually, it was clear that they move around each other 
according to Kepler's laws. Therefore, their motions also agree 
with Newton's law of universal gravitation. Using the same 
equation as that used for planets (see page 228), astronomers 
have Ccilculated the masses of these stars. They range from about 
0.1 to 50 times the sun's mass. 

A theory can never be completely proven. But theories become 
increasingly acceptable as they are found useful over a wider and 
wdder range of problems. No theory has stood this test better 
than Newton's theoiy of universal gravitation as applied to the 



Tiny variations from a 1/fi" centrip- 
etal acceleration of satellites in or- 
bit around the moon have led to a 
mapping of "mascons" on the 
moon. Mascons are unusueilly dense 
concentrations of mass under the 
surface. 




The motion over many years for 
one of the two components of a bi- 
nary star system. Each circle indi- 
cates the average of observations 
made over an entire vear. 



CHAPTER 8 / THE UNITY OF EARTH AlVD SKY 



235 



planetary system. It took nearly a century for physicists and 
astronomers to comprehend, verify, and extend Neulon's work 
on planetary motion. As late as the nineteenth centuiy, most of 
what had been accomplished in mechanics since Nev\lon's day 
was but a dexelopment or application of his work. 



• 29. How does the moon cause the water level to rise on both 
sides of the earth? 

30. In which of the followine, does the moon produce tides? (a) 
the seas (h) the atmosphere Id the solid earth 

31. Why is the calculation of the moon s motion so difficult? 

32. How are the orbits of comets different from the orbits of 
the planets? 

33. Do these differences affect the validity of Newton s law of 
universal gravitation for comets? 

8.10 I Some effects and limitations 
of Neiiton's w ork 

Today Newton and his system of mechanics are honored for 
many reasons. The Principia formed the basis for the 
development of much of our physics and technologv. Also, the 
success of Newton's approach made it the model for all the 
physical sciences for the next 2 centuries. 

Throughout Newton's work, you will find his basic belief that 
celestial phenomena can be explained by applving quantitative 
earthly laws. Newton felt that his laws had real ph\sical 
meaning, that they were not just mathematical con\eniences 
behind which unknowable laws lay hidden. The natural physical 
laws goxerning the uni\'erse could be known. The simple 
mathematical forms of the laws were e\idence of their reality. 

Newton combined the skills and approaches of both the 
experimental and the theoretical scientist. He invented pieces of 
equipment, such as the first reflecting telescope. He performed 
skillful experiments, especially in optics. Yet he also applied his 
great mathematical and logical powers to the creation of specific, 
testable predictions. 

Many of the concepts that Newton used came from earlier 
scientists and those of his owti time. Galileo and Descartes had 
contributed the first steps leading to a proper idea of ineitia, 
which became Newton's first law of motion. Kepler's planetary 
laws were central in Newton's consideration of planetary 
motions. Huygens, Hooke, and others clarified the concepts of 
force and acceleration, ideas that had been evoKing for centuries. 

In addition to his owii experiments, Newlon selected and used 



236 UiMT 2 I MOTIOiX IIV THE HEA\^IVS 



data from a large number of sources. Tycho Brahe was only one 
of several astronomers whose observations of the motion of the 
moon he used. When Newton could not complete his own 
measurements, he knew whom he could ask. 

Last, recall how completely and how fruitfully he used and 
expanded his own specific contributions. A good example is his 
theoiy of universal gravitation. In developing it, Newton used his 
laws of motion and his various mathematical inventions again 
and again. Yet Newton was modest about his achievements. He 
once said that if he had seen further than others "it was by 
standing upon the shoulders of Giants." 

Scientists recognize today that Newton's mechanics hold tn.ie 
only udthin a well-defined region of science. For example, the 
forces within each galaxy appear to be Newtonian. But this may 
not be true for forces acting between one galaxy and another. At 
the other end of the scale are atoms and subatomic particles. 
Entirely non-Newtonian concepts had to be developed to explain 
the observed motions of these particles. 

Even within the solar system, there are several small 
differences between the predictions and the observations. The 
most famous involves the angular motion of the axis of Mercury's 
orbit. This motion is greater than the value predicted from 
Neuron's laws by about 1/80° per century. What causes this 
difference? For a while, it was thought that gravitational force 
might not vary inversely exactly with the square of the distance. 
Perhaps, for example, the law is F^,.,^, = -^/^2oooooi 

Such difficulties should not be hastily assigned to soi e minor 
mathematical imperfection. The law of gravitation applies wath 
unquestionable accuracy to all other planetary motions. It may 
be that the basic assumptions in the theory make it too limited, 
as with the Ptolemaic system of epicycles. Many studies have 
shown that there is no way to modiiy the details of Newtonian 
mechanics to explain certain observations. Instead, these 
observations can be accounted for only by constructing new 
theories based on some very different assumptions. The 
predictions fr-om these theories are almost identical to those 
from Newton's laws for familiar phenomena. But they are also 
accurate in some extreme cases where the Newtonian 
predictions begin to show inaccuracies. Thus, Newtonian science 
is linked at one end with relativity theory, which is important for 
bodies with very great mass or moving at very high speeds. At 
the other end Newtonian science approaches quantum 
mechanics, which is important for particles of extremely small 
mass and size, for example, atoms, molecules, and nuclear 
particles. For a vast range of problems between these extremes, 
Newtonian theory gives accurate results and is far simpler to use. 
Moreover, it is in Newtonian mechanics that relativity theoiy and 
quantum mechanics have their roots. 



Newtonian mechanics refers to the 
science of the motion of bodies, 
based on Newton's work. It in- 
cludes his laws of motion and of 
gravitation as applied to a range of 
bodies from microscopic size to 
stars and incorporates develop- 
ments of mechanics for over two 
centuries after Newton's own work. 



SG 26 



SG 29, 30 



CHAPTER 8 / THE UNITY OF EARTH AND SKY 



237 



study 
guide 



1. The Project Physics learning materials 
particularly appropriate for Chapter 8 include: 

Experiments 

Stepwise /Xpproxiniation lo an Orbit 
Model of the Orbit of Halley's comet 

Activities 

Other Comet Orbits 

Forces on a Pendulum 

Trial of Copernicus 

Discovery of Neptune and Pluto 

Film Loops 

Jupiter Satellite Orbit 

Program Orbit 1 

Program Orbit II 

Central Forces: Iterated Blows 

Kepler's Laws 

Unusual Orbits 

Z. Complete the foUowing stateinents: 

(a) Nevvlon believed that the natural path of any 
moving object in the absence of forces is 



(b) Since Kepler's first law claims that planets travel 
along elliptical paths, Xewlon hypothesized a force 
that 

(c) Nevx'ton then discovered that if the motion of the 
planets follows Kepler's law of areas, this force must 
be 

(d) Nevx^on also disco\erefl that planets that obey 
Kepler's harmonic law of periods require; a force that 



3. (a) People have always known that ajiples fall to 
the ground. What particular thing about the fall of 
an apple led Newton to compare the apple's fiUl to 
the motion of the moon? 

(b) if the moon is 60 times farther from the center 
of the earth than an apple , what is the moon's 
acceleration toward the earth's center? 

(c) Use the formula for centrip(;tal accf^hn-ation to 



find the acceleration of the moon in its orbit. 
Compare this value to your answer in part (b). 

4. How would you answer the following question: 
What keeps the moon up? 

5. State the law of universal gravitation in words 
and svmbols, defining each sxTiibol. What is the 
direction of the force of gravity'? Give the value of C. 
Does this value ever change? 

6. In the table below are the periods and distances 
from Jupiter of the four large satellites, as measured 
by telescopic obsei'vations. Does Kepler's law of 
periods apply to the Jupiter system? 

Distance from Jupiter's Center 
Satellite Period (in terms of Jupiter's radius, r) 



1 


1.77 days 


6.04r 


II 


3.55 


9.62 


III 


7.15 


15.3 


IV 


16.7 


27.0 



7. Give some reasons why Descartes' theory of 
planetary motion inight ha\'e been "a useful idea." 

8. On p. 22.'> it was claimed that the dependence of 
the gravitational forcj; on the masses of both 
interacting bodies could hv, expressed as "'.,u„"i,,i^,„.,- 

(a) Using a diagram similar to that for Question 19 
on p. 227, show that this is correct. 

(b) To test alternatives to using the product, consider 
the possibilities that the force could depend upon 
the inasses in either of two ways: 

(1) total force depends on (fn^^,, +"^pia.iei '' °^ 

(2) total force depends on (m^^ym^i^,,^,). 

With these relationships, what would happen to the 
force if either mass were reduced to zero? Would 
there still hv. a force even though there; were only 
one mass left? Could you speak of a gi-a\itational 
force when there was no body to be accelerated? 

9. Use the values for the mass and size of the 
moon to show that the "surface gravity" (acceleration 
due to gravity near the moon's surface;! is only about 
0.16 of what it is near the; earth's. (Mass of moon = 
7.34 X 10" kg; radius of moon = 1.74 x 10' m.) 

10. Use the equation for c(;ntrip(;tal force and the 
(;quation foi" giavitational force to derive an 



238 



CHAPTER 8 / STUUV GUIDE 



so 



expression for the period of a satellite orbiting a 
planet in terms of the radius of the orbit and mass 
of the planet. 

11. Using the formula for the gravitational force 
between two objects, calculate the force between a 
100-kg sphere and a 1,000-kg sphere placed 10 m 
apart. What are the accelerations of the two spheres? 
Could these accelerations ever be measured? 

12. Why were the discoveries of Neptune and Pluto 
triumphs for Newton's theory? 

13. By Newton's time, telescopic observations of 
Jupiter led to values for the orbited periods and radii 
of Jupiter's four large satellites. For example, the one 
named Callisto was found to have a period of 16.7 
days, and the radius of its orbit was calculated as 
1/80 AU. 

(a) From these data calculate the value of fcj,,,^,^,. 
(First convert days to years.) 

(b) Show that Jupiter's mass is about 1/1000 the 
mass of the sun. 

(c) How was it possible to have a value for the 
orbital radius of a satellite of Jupiter? 

14. What orbital radius must an earth satellite be 
given to keep it always above the same place on the 
earth, that is, in order to have a 24-hr period? (Hint: 
See Question 10.) 

15. Calculate the mass of the earth from the fact 
that a 1-kg object at the earth's surface is attracted to 
the earth with a force of 9.8 N. The distance from 
the earth s center to its surface is 6.4 x lo'' m. How 
many times greater is this than the greatest masses 
that you have had some experience in accelerating 
(for example, cars)? 

16. The mass of the earth can be calculated also 
from the distance and period of the moon. Show 
that the value obtained in this way agrees with the 
value calculated from measurements at the earth's 
surface. 

17. If tides are caused by the pull of the moon, why 
are there also tides on the side of the earth opposite 
the moon? 

18. The moon's orbit around the earth is a 



combination of two separate motions: a straight line 
and a faU toward the earth's center. Using diagrams, 
discuss each part of the moon's motion. 

19. Cavendish's value for G made it possible to 
calculate the mass of the earth and therefore its 
average density. The "density" of water is 1,000 
kg/m'; that is, for any sample of water, di\iding the 
mass of the sample by its volume gives 1,000 kg/m'. 

(a) What is the earth's average density? 

(b) The densest kind of rock known has a density of 
about 5,000 kg/m'. Most rock has a density of about 
3,000 kg/m'. What do you conclude from this about 
the internal structure of the earth? 

20. The manned Apollo 8 capsule was put into a 
nearly circular orbit 112 km above the moon's 
surface. The period of the orbit was 120.5 min. From 
these data, calculate the mass of the moon. (The 
radius of the moon is 1,740 km. Use a consistent set 
of units.) 

21. Mars has two satellites, Phobos and Deimos 
(Fear and Panic). A science-fiction story was once 
written in which the natives of Mars showed great 
respect for a groove in the ground. The groove 
turned out to be the orbit of Mars' closest moon 
"Bottomos." 

(a) If such an orbit were possible, what would the 
period be? 

(b) What speed would it need to have in order to go 
into such an orbit? 

(c) What would you expect to happen to an object in 
such an orbit? 

22. Using the values given in the table on p. 230, 
make a table of relative masses compared to the 
mass of the earth. 

23. The sun's mass is about 27,000,000 times greater 
than the moon's mass; the sun is about 400 times 
farther from the earth than the moon is. How does 
the gravitational force exerted on the earth by the 
sun compare with that exerted by the moon? 

24. The period of Halley's comet is about 75 yr. 
What is its average distance from the sun? The 
eccentricity of its orbit is 0.967. How far from the 
sun does it go? How close does it come to the sun? 



CHAPTER 8 / STUDY GUIDE 



239 



sg 



as. Accepting the validity of F_^ = Gm,m,//i' and 
recognizing that G is a universal constant, we are 
able to derive, and therefore to understand better, 
many particulars that previously seemed separate. 
For example, we can conclude: 

(a) that a, for a body of any mass m should be 
constant at a particular place on eiirth. 

(b) that a^ might be different at places on eai'th at 
different distances from the earth's center. 

(c) that at the earth's surface the weight of a body is 
related to its mass. 

(d) that the ratio R'/T^ is a constant for all the 
satellites of a body. 

(e) that tides occur about 6 hr apart. 

Describe briefly how each of these conclusions can 
be derived from the equation. 

26. The making of theories to account for 
observations is a major purpose of scientific study. 
Therefore, some reflection upon the theories 
encountered thus far in this course will be useful. 
Comment in a paragraph or more, with examples 
from Units 1 and 2 , on some of the statements 
below. Look at all the statements and select at least 
six, in any order you wash. 

(a) A good theory should summarize and not conflict 
with a body of tested obser\'ations. (Take, for 
example, Kepler's unwillingness to explain away the 
difference of 8 min of arc between his predictions 
and Tycho's observations.) 

(b) There is nothing more practical than a good 
theory. 

(c) A good theory should permit predictions of new 
observations which sooner or later can be made. 

(d) A good new theory should give almost the same 
predictions as older theories for the range of 
phenomena where the older theories worked well. 

(e) Every theory involves assumptions. Some involve 
also esthetic preferences of the scientist. 



(f) A new theory relates some previously unrelated 
observations. 

(g) Theories often involve abstract concepts derived 
from observation. 

(h) Empirical laws or "rules" organize many 
observations and reveal how changes in one quantity 
vary with changes in another, but such laws provide 
no explanation of the causes or mechanisms. 

(i) A theory never fits all data exacUy. 

(j) Predictions from theories may lead to the 
observation of new effects. 

(k) Theories that later had to be discarded may have 
been useful because they encouraged new 
observations. 

(1) Theories that permit quantitative predictions are 
preferred to qualitative theories. 

(m) An "unwritten text" lies behind the statement of 
every law of nature. 

(n) Communication between scientists is an essential 
part of the way science grows. 

(o) Some theories seem initially so strange that they 
are rejected completely or accepted only very slowly. 

(p) Models are often used in the making of a theor\' 
or in describing a theory. 

(q) The power of theories comes from their 
generality. 

27. What happened to Plato's problem? Was it 
solved? 

28. Why do we believe toda\' in a heliocentric 
system? Is it the same as either Copernicus' or 
Kepler's? VVliat is the experimental evidence? Has the 
geocentric system been disproved? 

29. Is Newton's work only of historical interest, or is 
it useful today? Elxplain. 

30. What were some of the major consequences of 
Nevv'ton's work on scientists' views of the world? 



240 



CHAPTER 8 / STUDY GUIDE 



EPILOGUE 



This unit started at the 
beginning of recorded history 



and followed human attempts to explain the cyclic motions 
observed in the heavens. We saw the long, gradual change from 
an earth-centered view to the modern one in which the earth is 
just another planet moving around the sun. We examined some 
of the difficulties encountered in making this change of 
viewpoint. We also tried to put into perspective Newton's 
synthesis of earthly and heavenly motions. From time to time, we 
suggested that there was an interaction of these new world views 
with the general culture. We stressed that all scientists are 
products of their times. They are limited in the degree to which 
they can abandon the teachings on which they were raised. 
Gradually, through the work of many scientists over the 
centuries, a new way of looking at heavenly motions arose. This 
in turn opened new possibilities for even more new ideas, and 
the end is not in sight. 

In addition, we looked at how theories are made and tested. 
We discussed the place of assumption and experiment, of 
mechanical models and mathematical description. In later parts 
of the course, you will come back to this discussion in more 
recent contexts. You will find that attitudes developed toward 
theory-making during the seventeenth-century scientific 
revolution remain immensely helpful today. 

In our study, we have referred to scientists in Greece, Egypt; 
Poland, Denmark, Austria, Italy, England, and other countries. 
Each; as Newton said of himself, stood on the shoulders of those 
who came earlier. For each major success there are many lesser 
advances or, indeed, failures. Science is a cumulative intellectual 
activity not restricted by national boundaries or by time. It is not 
constantly and unfailingly successful, but grows as a forest 
grows. New growth replaces and draws nourishment from the 
old, sometimes with unexpected changes in the different parts. 
Science is not a cold, calculated pursuit. It may involve 
passionate controversy, religious convictions, judgments of what 
beauty is, and sometimes wild private speculation. 

It is also clear that the Newtonian synthesis opened whole 
new lines of investigation, both theoretical and observational. In 
fact, much of our present science and also our technology had 
their effective beginnings with the work of Newton. New models, 
new mathematical tools, and a new self-confidence encouraged 
those who followed to attack new problems. A never-ending 
series of questions, answers, and more questions was well 
launched. The modern view of science is that it is a continuing 
exploration of ever more interesting fields. 

One problem remaining after Newton's work was the study of 
objects interacting not by gravitational forces, but by friction and 
collision. This study led, as the next unit will show, to the 





UNIT 2 I EPILOGUE 



241 



The engraving of the French Acad- 
eniv by Sebastian LeClerc 11698) 
reflects the activity of learned soci- 
eties at that time. The picture does 
not depict an actual scene, of 
course, but in allegory shows the 
excitement of communication that 
grew in an informal atmosphere. 
The dress is symbolic of the Greek 
heritage of the sciences. Although 
all the sciences are represented, 
the artist has put anatomy, botany, 
and zoology, symbolized by skele- 
tons and dried leaves, toward the 
edges, along with alchenw and the- 
ologv. .Mathematics and the physi- 
cal sciences, including astronomy, 
occupv the center stage. 



concepts of momentum and energy. It brought about a much 
broader view of the connection between different parts of 
science, such as physics, chemistiy, and biologv. Eventually, this 
line of study produced other statements as grand as Newton's 
law of universal gravitation. Among them were the conservation 
laws on which much of modern science and technology is based. 
An impoilant part of these laws describes how systems 
consisting of many interacting bodies work. That account will be 
the main subject of Unit 3. 

Nev\1on's influence was not limited to science alone. The 
centuiy following his death in 1727 was a period of further 
understanding and application of his discoveries and methods. 
His influence was felt especially in philosophy and literature, but 
also in many other fields outside science. Let us round out our 
\iew of Newton by considering some of these effects. 

The eighteenth centuiy is often called the Age of Reason or 
Centuiy of Enlightenment. "Reason " was the motto of the 
eighteenth-centuiy philosophers. However, their theories about 
impro\ang religion and society were not convincingly connected. 
Newtonian physics, religious toleration, and republican 
government were all advanced by the same movement. This does 
not mean there was really a logical link among these concepts. 
Nor were many eighteenth-centuiy thinkers in any field or nation 
much bothered by other gaps in logic and feeling. For example, 
they believed that all men are created equal." Yet they did little 




242 



UNIT 2 / EPILOGUE 



to remove the chains of black slaves, the ghetto walls 
imprisoning Jews, or the laws that denied rights to women. 

Still, compared with the previous century, the dominant theme 
of the eighteenth century was moderation, the "happy medium. " 
The emphasis was on toleration of different opinions, restraint 
of excess, and balance of opposing forces. Even reason was not 
allowed to question religious faith too strongly. Atheism, which 
some philosophers thought would logically result from unlimited 
rationality, was still regarded wath horror by most Europeans. 

The Constitution of the United States of America is one of the 
most enduring achievements of this period. Its system of "checks 
and balances" was designed specifically to prevent any one 
group from getting too much power. It attempted to establish in 
politics a state of equilibrium of opposing trends. This 
equilibrium, some thought, resembled the balance between the 
sun's gravitational pull and the tendency of a planet to fly off in 
a straight line. If the gravitational attraction upon the planet 
increased without a corresponding increase in planetary speed, 
the planet would fall into the sun. If the planet's speed increased 
without a corresponding increase in gravitational attraction, it 
would escape from the solar system. 

Political philosophers, some of whom used Newtonian physics 
as a model; hoped to create a similar balance in government. 
They tried to devise a system that would avoid the extremes of 
dictatorship and anarchy. According to James Wilson (1742-1798), 
who played a major role in writing the American Constitution, 

In government, the perfection of the whole depends on the 
bcdance of the parts, and the balance of the parts consists in 
the independent exercise of their separate power's, and, when 
their powers are separately exercised, then in their mutual 
influence and operation on one another. Each part acts and is 
acted upon, supports and is supported, regulates and is 
regulated by the rest. It might be supposed, that these powers, 
thus mutually checked and controlled, would remain in a state 
of inaction. But there is a necessity for movement in human 
ciffairs; and these powers are forced to move, though still to 
move in concert. They move, indeed, in a line of direction 
somewhat different from that, which each acting by itself would 
have taken; but, at the same time, in a line partaking of the 
natural directions of the whole — the ti-ue line of public liberty 
and happiness. 

Both Newton's life and his writings seemed to support the idea 
of political democracy. A former faiTn boy had penetrated to the 
outermost reaches of the human imagination. What he had 
found there meant, first of all, that only one set of laws governed 
heaven and earth. This smashed the old beliefs about "natural 
place " and extended a new democracy throughout the universe. 
Newton had shown that all matter, whether the sun or an 



UNIT 2 / EPILOGUE 343 



ordinal^' stone, was created equal; that is to say, all matter had 
the same standing before "the Laws of Nature and of Nature's 
God." (This phrase was used at the beginning of the Declaration 
of Independence to justify the desire of the people in the 
colonies to throw off their oppressive political system and to 
become an independent people.) All political thought at this time 
was heavily influenced by Nev\^onian ideas. The Principia 
seemed to offer a parallel to theories about democracy. It seemed 
logical that all people, like all natural objects, are created equal 
before nature's creator. 

In literature, too, many welcomed the new scientific \ieupoint. 
It supplied many new ideas, convenient figures of speech, 
parallels, and concepts which writers used in poems and essays. 
Newton's disco\'eiy that white light is composed of colors was 
referred to in many poems of the 1700's. (See Unit 4.) Samuel 
Johnson advocated that words drawn from the vocabulary of the 
natural sciences be used in literaiy works. He defined many such 
words in his Dictionary and illustrated their application in his 
"Rambler" essays. 

The first really powerful reaction against Newtonian cosmology 
was the Romantic movement. Romanticism was started in 
Germany about 1780 by young v\Titers inspired by Johann 
Wolfgang von Goethe. The most familiar examples of 
Romanticism in English literature are the poems and novels of 
Blake, Coleridge, Wordsworth, Shelley, Byron, and Scott. The 
Romantics scorned the mathematical \dew of nature. They 
believed that any whole thing, whether a single human being or 
the entire universe, is filled with a unique spirit. This spirit 
cannot be explained by reason; it can only he felt. The Romantics 
insisted that phenomena cannot be meaningfully analyzed and 
reduced to their separate parts by mechanical explanations. 

The Romantic philosophers in Germany regarded Goethe as 
their greatest scientist as well as their greatest poet. They 
pointed in particular to his theory of color, which flatly 
contradicted Newton's theory of light. Goethe held that white 
light does not consist of a mLxture of colors and that it is useless 
to "reduce" a beam of white light by passing it through a prism 
to study its separate spectral colors. Rather, he charged, the 
colors of the spectnjm are artificially produced by the prism, 
acting on and changing the light which is itself pure. 

In the judgment of all modern scientists, Neulon was right 
and Goethe wrong. This does not mean that Nature Philosophy, 
introduced by Friedrich Schelling in the early 1800's, was without 
any value. It encouraged speculation about ideas so general that 
they could not be easily tested by experiment. At the time, it was 
condemned by most scientists for just this reason. Today, most 
historians of science agree that Nature Philosophy exentually 
j:)layed an important role in making possible certain scientific 



244 UNIT 2 / EPILOGUE 



discoveries. Among these was the general principle of 
conservation of energy, which is described in Chapter 10. This 
principle asseri;ed that cdl the "forces of nature, " that is, the 
phenomena of heat, gravity, electricity, magnetism, and so forth, 
are forms of one underiying "force" (which we now call energy). 
This idea agreed well with the viewpoint of Nature Philosophy. It 
also could eventually be put in a scientifically acceptable form. 

Some modem artists, some intellectuals, and most members of 
the "counterculture" movements express a dislike for science. 
Their reasoning is similar to that of the Romantics. It is based on 
the mistaken notion that scientists claim to be able to find a 
mechanical explanation for everything. 

Even the Roman philosopher Lucretius (100-55 B.C.), who 
supported the atomic theory in his poem On the Nature of 
Things, did not go this far. To preserve some trace of "free will" 
in the universe, Lucretius suggested that atoms might swerve 
randomly in their paths. This was not enough for Romantics and 
also for some scientists. For example, Erasmus Darvvan, a 
scientist and grandfather of evolutionist Charles Darwin, asked 

Dull atheist, could a giddy dance 

Of atoms lawless hurl'd 
Construct so wonderful, so wise, 

So harmonised a world? 

The Nature Philosophers thought they could discredit the 
Newtonian scientists by forcing them to answer this question. To 
say "yes," they argued, would be absurd, and to say "no" would 
be disloyal to Newtonian beliefs. But the Newtonians succeeded 
quite well without committing themselves to any definite answer 
to Erasmus Darwin's question. They went on to discover 
immensely powerful and valuable laws of nature, which are 
discussed in the next units. 



UNIT 2 / EPILOGUE 245 



yMDT 




CHAPTER 9 Conservation of Mass and Momentum 

CHAPTER 10 Energ*' 

CHAPTER 11 The Kinetic Theory' of Gases 

CHAPTER 12 Waves 



& 



Isaac Nev\'tons development 
of mathematical principles of 
natural philosophy marks a turning point in the growth of 
human knowledge. The simple, elegant laws that njle not only 
terrestrial hut also astronomical phenomena were soon 
recognized as the premier example of the power of human 
reason. The fact that Newton had done so much to show that 
there is a rational order in natural events made it seem possible 
that any problem could be solved by reason. It is not surprising 
that after his death in 1727 Newton was looked upon almost as a 
god, especially in England. Many poems like this one appeared: 

Neuion the unparalleld'd, whose Name 
No Time will wear- out of the Book of Fame, 
Celestial Science has promoted more, 
Than all the Sages that have shone before. 
Nature compell'd his piercing Mind obeys, 
And gladly shows him iill her secit^t Ways; 
Gainst Mathematics she has no defence, 
And yields t' experimental Consequence; 



246 



UMIT 3 / PROLOGUE 



His tow 'ring Genius, from its certain Cause 

Ev'ry Appearance a priori draws 

And shews th' Almiglity Architect's unalter'd Laws. 

Newton's success in mechanics altered profoundly the way in 
w^hich scientists view^ed the universe. The motions of the sun 
and planets could now be considered as purely mechanical. As 
for any machine, w^hether a clock or the solar system, the 
motions of the parts were completely determined once the 
system had been put together. 

This model of the solar system is called the Newtonian world 
machine. As is true of any model, certain things are left out. The 
mathematical equations that govern the motions of the model 
cover only the main properties of the real solar system. The 
masses, positions, and velocities of the parts of the system, and 
the gravitational forces among them, are w^ell described. But the 
Newtonian model neglects the internal structure and chemical 
composition of the planets, heat, light, and electric and magnetic 
forces. Nevertheless, it serves splendidly to deal with obseived 
motions. Moreover, Newton's approach to science and many of 
his concepts became useful later in the study of those aspects he 
had to leave aside. 

The idea of a world machine does not trace back only to 
Newton's work. In Principles of Philosophy 11644), Rene Descartes, 
the most influential French philosopher of the seventeenth 
century, had written: 

I do not recognize any difference between the machines that 
cirtisans make and the different bodies that nature alone 
composes, unless it be that the effects of the machines depend 
only upon the adjustment of certain tubes or springs, or other 
instruments, that, having necessarily some proportion vvdth the 
hands of those who make them, are always so large that their 
shapes and motions can be seen, while the tubes and springs 
that cause the effects of natural bodies are ordinarily too small 
to be perceived by our senses. And it is certain that all the laws 
of Mechanics belong to Physics, so that all the things that are 
artificial, are at the same time natural. 

Robert Boyle (1627-1691), a British scientist, is knowm 
particularly for his studies of the properties of air. (See Chapter 
11.) Boyle, a pious man, expressed the 'mechanistic " vdewpoint 
even in his religious writings. He argued that a God who could 
design a universe that ran by itself like a machine was more 
wonderful than a God who simply created several different kinds 
of matter and gave each a natural tendency to behave as it does. 
Boyle also thought it was insulting to God to believe that the 
world machine would be so badly designed as to require any 
further divine adjustment once it had been created. He suggested 
that an engineer's skill in designing "an elaborate engine" is 



(From J. T. Desagulier, The New- 
tonian System of the World, the 
Best Model of Government, an Al- 
legorical Poem.) 



iS^ ^ 







UNIT 3 / PROLOGUE 



347 




"The Ancient of Days" by William 
Blake, an English poet who had lit- 
tle sympathy with the Nevx'ionian 
style of "Natural Philosophy. " 

Ironically I Neuron himself explic- 
itly rejected the deterministic as- 
pects of the "world machine" which 
his followers had popularized. 




A small area from the center of the 
picture has been enlarged to show 
what the picture is "really" like. 
Is the picture only a collection of 
dots? Knowing the underlying 
structure does not spoil your other 
reactions to the picture, but rather 
gives you another dimension of 
understanding i(. 



more deserving of praise if the engine never needs supervision or 
repair. "Just so, " he continued, 

... it more sets off the wisdom of God in the fabric of the 
universe, that He can make so vast a machine perfoim all those 
many things, which He designed it should, by the meer 
contrivance of biute matter managed by certain laws of local 
motion, and upheld by His ordinaiy and general concoui-se, 
than if He employed from time to time an intelligent overseer, 
such as nature is fancied to be, to regulate, assist, and controul 
the motions of the parts. . . . 

Boyle and many other scientists in the seventeenth and 
eighteenth centuries tended to think of God as a supreme 
engineer and physicist. God had set down the laws of matter and 
motion. Human scientists could hest glorify' the Creator by 
discovering and proclaiming these laws. 

This unit is mainly concerned with physics as it developed 
after Nevx'ton. In mechanics, Newton's theory was extended to 
cover a wide range of phenomena, and new concepts were 
introduced. The conser\^ation laws discussed in Chapters 9 and 
10 became increasingly important. These powerful principles 
offered a new way of thinking about mechanics. They opened up 
new areas to the study of physics, for example, heat and wave 
motion. 

Newtonian mechanics treated directly only a small range of 
experiences. It dealt udth the motion of simple bodies or those 
largely isolated from others, as are planets, projectiles, or sliding 
discs. Do the same laws work when applied to complex 
phenomena? Do real solids, liquids, and gases behave like 
machines or mechanical systems? Can their behaxior be 
explained by using the same ideas about matter and motion that 
Newton used to explain the solar system? 

At first, it might seem unlikely that everything can be reduced 
to matter and motion, the principles of mechanics. What about 
temperature, colors, sounds, odors, hardness, and so forth? 
Newton himself believed that the mechanical view would 
essentially show how to investigate these and all other 
properties. In the preface to the Principia he WTote: 

I wish we could derive the rest of the phenomena of Nature by 
the same kind of reasoning from mechanical principles, for 1 
am induced by many reasons to suspect that they may all 
depend upon certain forces by which the particles of bodies, by 
some causes hitherto unknown, are mutually impelled towards 
one another, and cohere according to regular figures, or are 
repelled and recede fiom one another. These forces being 
unknown, philosophers have hitherto attempted the search of 
Nature in vain; but I hope the principles here laid down will 
afford some light either to this or some truer method of 
Philosophv 



248 



UNIT 3 / PROLOGUE 




Gonseruation of Mass 
anil Momentum 



9.1 Conservation of mass 

9.2 Collisions 

9.3 Conservation of momentum 

9.4 Momentum and Neuton's laivs of motion 

9.5 Isolated systems 

9.6 Elastic collisions 

9.7 Leibniz and the consen^ation laiv 

9*1. I Conservation of mass 

The idea that despite ever-present, obvious change all around us 
the total amount of material in the universe does not change is 
really very old. The Roman poet Lucretius restated (in the first 
century B.C.) a belief held in Greece as early as the fifth century 
B.C.: 

. . . and no force can change the sum of things; for there is no 
thing outside, either into which any kind of matter can emerge 
out of the universe or out of which a new supply can arise and 
burst into the universe and change all the nature of things and 
alter their motions. [On the Nature of Things] 

Just 24 years before Newton's birth, the English philosopher 
Francis Bacon included the following among his basic principles 
of modern science in Novum Organum (1620): 

There is nothing more true in nature than the twin 
propositions that "nothing is produced fiom nothing" and 




SG 1 




CHAPTER 9 / COMSERVATION OF IVIASS AND IVIOMENTUM 



249 




In some open-air chemical reac- 
tions, the mass of objects seems to 
decrease, while in others it seems 
to increase. 

\ote the closed flask shown in his 
portrait on page 252. 

SG2 



"nothing is reduced to nothing" . . . the sum total ol matter 
i-emains unchanged, without increase or diminution. 

This \'ievv agrees with everyday observation to some extent. While 
the form in which matter exists may change, in much of our 
ordinary experience matter appears somehow indestnjctible. For 
example, you may see a large boulder ciushed to pebbles and 
not feel that the amount of matter in the universe has 
diminished or increased. But what if an object is burned to ashes 
or dissohed in acid? Does the amount of matter remain 
unchanged even in such chemical reactions? What of laige-scale 
changes such as the forming of rain clouds or seasonal 
xariations? 

I o test whether the total quantity of matter actually remains 
constant, you must know how to measure that quantity. Clearly, 
it cannot simply be measured by its volume. For example, you 
might put watei' in a container, mark the water level, and then 
freeze the water. If so, you will find that the volume of the ice is 
greater than the volume of the water you started with. This is 
Uxie even if you carefully seal the container so that no water can 
possibly come in from the outside. Similarly, suppose you 
compress some gas in a closed container. The volume of the gas 
decreases even though no gas escapes from the container. 

Following Newton, we regard the mass of an object as the 
proper measure of the amount of matter it contains. In all the 
examples in Units 1 and 2, we assumed that the mass of a given 
object does not change. However, a burnt match has a smaller 
mass than an unlournt one: an iron nail increases in mass as it 
rusts. Scientists had long assumed that something escapes from 
the match into the atmosphere and that something is added 
from the surroundings to the iron of the nail. Therefore, nothing 
is really "lost" or "created" in these changes. Not until the end 
of the eighteenth century was sound experimental evidence for 
this assumption provided. The French chemist Antoine Lavoisier 
produced this evidence. 

Lavoisier caused chemical reactions to occur in closed flasks (a 
"closed system"). He carefully weighed the flasks and theii- 
contents before and after the reaction. For example, he burned 
iron in a closed flask. The mass of the iron oxide produced 
equalled the sum of the masses of the iron and oxygen used in 
the reaction. With experimental evidence like this at hand, he 
could announce with confidence in I'rnite Elementaire de Chimic 
(1789): 

We may lay it down as an incontestable axiom that in all the 
operations of art and nature, nothing is crealed; an ec|ual 
quantity of matter exists both befoi"e and after the exptniment, 
. . . and nothing takes place beyond changes and modifications 
in the combinations of these elements. Upon this principle, the 
whole art of performing chemical o.xperimenls de|)eii(ls 



250 



UNIT 3 / THE TRIUMPH OF MECHANICS 



Lavoisier knew that if he put some material in a well-sealed 
bottle and measured its mass, he could return at any later time 
and find the same mass. It would not matter what had happened 
to the material inside the bottle. It might change from solid to 
liquid or liquid to gas, change color or consistency, or even 
undergo violent chemical reactions. At least one thing would 
remain unchanged: the total mass of all the different materials in 
the bottle. 

In the years after Lavoisier's pioneering work, a vast number of 
similar experiments were peiformed with ever-increasing 
accuracy. The result was always the same. As far as scientists 
now can measure udth sensitive balances (having a precision of 
better than 0.000001%), mass is conserved, that is, remains 
constant, in chemical reactions. 

To sum up, despite changes in location, shape, chemical 
composition, and so forth, the mass of any closed system remains 
constant. This is the statement of the law of conserx'ation of mass. 
This law is basic to both physics and chemistry. 

Obviously, one must know whether a given system is closed or 
not before applying this law to it. For example, it is perhaps 
surprising that the earth itself is not exactly a closed system 
within which all mass would be consei^^ed. Rather, the earth, 
including its atmosphere, gains and loses matter constantly. The 
most important addition occurs in the foiTn of dust particles. 
These particles are detected by their impacts on satellites that 
are outside most of the atmosphere, and by the light and 
ionization they create when they pass through the atmosphere 
and are slowed down by it. The great majority of these particles 
are very small (less than 10*' m in diameter) and cannot be 
detected individually when they enter the atmosphere. Particles 
larger than several millimeters in diameter appear as luminous 
meteorite trails when they vaporize in the upper atmosphere; 
these particles are only a small fraction of the total, both in 
terms of numbers and in terms of mass. The total estimated 
inflow of mass of all these particles, large and small, is about 10' 
g/sec over the whole surface of the earth. (The mass of the earth 
is about 6 X 10~"g.) This gain is not balanced by any loss of dust 
or larger particles, not counting an occasional spacecraft and its 
debris. The earth also collects some of the hot gas evaporating 
from the sun, but the amount is comparatively small. 

The earth does lose mass by evaporation of molecules from the 
top of the atmosphere. 7 he rate of this evaporation depends on 
how many molecules are near enough to the top of the 
atmosphere to escape without colliding with other molecules. 
Also, such molecules must have velocities high enough to escape 
the earth's gravitational pull. The velocities of the molecules are 
determined by the temperature of the upper atmosphere. 
Therefore, the rate of evaporation depends greatly on this 




Conservation of mass was demon- 
strated in e^qjeriments on chemical 
reactions in closed flasks. 

The meaning of the phrase "closed 
system" wiU be discussed in more 
detail in Sec. 9.5. 



"The change in the total mass is 
zero" can be expressed symboli- 
cally as AXni, = where S^ repre- 
sents the sum of the masses of m^ 
in all parts of the system. 







Meteorites have been found in all 
parts of the world. This meteorite 
fragment was one of several found 
in the Atacama desert of Chile in 
1822. 



CHAPTER 9 / CONSERVATION OF MASS AND MOMENTUM 



251 



Close Up I 

The Father of Modern Chemistry 



Antoine Laurent Lavoisier (1743- 
1794) is knowri as the "father of 
modern chemistry" because he 
showed the decisive importance of 
quantitative measurements, con- 
firmed the principle of conservation 
of mass in chemical reactions, and 
helped develop the present system 
of nomenclature for the chemical 



elements. He also showed that or- 
ganic processes such as digestion 
and respiration are similar to burning. 

To earn money for his scientific re- 
search, Lavoisier invested in a pri- 
vate company which collected taxes 
for the French government. Be- 
cause the tax collectors were al- 




lowed to keep any extra tax which 
they could collect from the public, 
they became one of the most hated 
groups in France. Lavoisier was not 
directly engaged in tax collecting, 
but he had married the daughter of 
an important executive of the com- 
pany, and his association with the 
company was one of the reasons 
why Lavoisier was guillotined dunng 
the French Revolution. 

Also shown in the elegant portrait by 
J. L. David IS Madame Lavoisier. 
She assisted her husband by taking 
data, translating scientific works from 
English into French, and making il- 
lustrations. About 10 years after her 
husband's execution, she married 
another scientist, Count Rumford, 
who IS remembered for his experi- 
ments which cast doubt on the ca- 
loric theory of heat. 



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ET d'aPRES LES D^COUVERTES MODERNESi 

Avec Figures ; 

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Sciences , Je la Soctetc RoyaU de MeJtcine , JtM 
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la Socuie RoyaU de LonJres , de t'hfiiiut de 
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celles de Ph:ladelphie , Harlem , Maachejltr , 
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Chei C u c H E T , Lihraire , rue & hotel Serpenie. 



M. D C C. L X X X I X. 



252 



LM I 3 I HL IRll \IPH OF MECHANICS 



temperature. At present, the rate is probably less than 5 x lO"* 
g/sec over the whole earth. This loss is very small compared to 
the addition of dust. (No water molecules are likely to be lost 
directly by atmospheric "evaporation"; they would first have to 
be dissociated into hydrogen and oxygen molecules.) 



1. TiTje or false: Mass is conserved in a closed system only if 
there is no chemical reaction in the system. 

2. If 50 cm^ of alcohol is mi\ed with 50 crn of water, the 
mixture amounts to only 98 cm^. An instrument pack on the 
moon weighs much less than on earth. Are these examples of 
contradictions to the law of conservation of mass? 

3. Which one of the following statements is true? 

(a) Lavoisier was the first person to believe that the amount of 
material in the universe does not change. 

(b) Mass is measurably increased when heat enters a system. 

(c) A closed system was used to establish the law of 
conservation of mass e^cperimentally. 

4. Five grams (5 g) of a red fluid at 12° C with a volume of 4 
mL are mi?ced with 10 g of a blue fluid at 5° C with a volume of 
8 ml. On the basis of this information only, and assuming that 
the fluids are minced in a closed system, what can you be sure 
of about the resulting mi}cture? 



Try these end-of-section questions 
before going on. 



SG 3-7 



9*2 Collisions 



Looking at moving things in the world around us easily leads to 
the conclusion that everything set in motion eventually stops. 
Every machine eventually runs down. It appears that the amount 
of motion in the universe must be decreasing. The universe, like 
any machine, must be running down. 

Many philosophers of the 1600 s could not accept the idea of a 
universe that was running down. The concept clashed vvdth their 
idea of the perfection of God, who surely would not construct 
such an imperfect mechanism. Some definition of "motion" was 
needed that would permit one to make the statement that "the 
quantity of motion in the universe is constant." 

Is there a constant "quantity of motion" that keeps the world 
machine going? To suggest an answer to this question, you can 
do some simple laboratory experiments. Use a pair of identical 
carts with nearly frictionless wheels; even better are two dry-ice 
discs or two air-track gliders. In the first experiment, a lump of 
putty is attached so that the carts wall stick together when they 
collide. The carts are each given a push so that they approach 
each other with equal speeds and collide head-on. As you will 




CHAPTER 9 / COIVSERVATIOM OF MASS AND MOMENTUM 



253 



Va 



3e(f re: \/f^ *- v^. 

Af+tr: V •. 



- "a - 



Vb 



l-x2V-J 



ijetcic : Va* v<>,=o 



A" v&" 



Va. 




*! * 7' \. 



- ^A- 



% 




Affer: Vi' + v4^o 



/n symbols, A Sv^ = A S ,V| = //7 
//jf.s particular case. 

In general s\'nibols, AS mi'. = 0. 



see when you do the experiment, both carts stop in the collision; 
their motion ceases. But is there anything related to their 
motions that does not change? 

Yes, there is. If you add the velocity v^ of one c;art to the 
velocitv' v„ of the other cart, you find that the vector sum does 
not change. The vector sum of the velocities of these oppositely 
moving carts is zero before the collision. It is also zero for the 
carts at rest after the collision. 

Does this finding hold for all collisions? In other vxoids, is 
there a "law of consenation of velocity"? The example aboxe was 
a veiy special circumstance. Cai'ts with equal masses approach 
each other with equal speeds. Suppose the mass of one of the 
carts is twice the mass of the other cart. (You can conveniently 
double the mass of one cart by putting another- cart on top of it.) 
Now let the carts approach each other with equal speeds and 
collide, as before. This time the carts do not come to rest. There 
is some motion remaining. Both objects moxe together in the 
direction of the initial velocity of the more massive object. The 
vector sum of the velocities is not conserved in all collisions. 

Another example of a collision will confirm this conclusion. 
This time let the first cart have twice the mass of the second, but 
only half the speed. When the carts collide head-on and stick 
together, they stop. The vector sum of the velocities is equal to 
zero after the collision. But it was not equal to zero before the 
collision. Again, there is no conservation of velocity'. 

These examples show that the "quantitv of riiotion" is always 
the same befor-e and after the collision. The results indicate that 
the proper definition of "quantity of motion" may invoke tin* 
mass of a body as well as its speed. Uescai'tes had suggested that 
the proper measui'e of a body's quant it\' of motion was the 
product of its mass and its speed. Speed does not involve 
direction and is considered always to have a positixe value. The 
examples above, however, show that this product (a scalar- and 
always positive) is not a conserved quantity. In the second and 
third collisions, for example, the products of mass and speed are 
zero for the stopped carets after the collision. Thex' obxiously ar-e 
not equal to zero before the collisiorn. 

If we make one x'eiy important change in Descar-tes' definition, 
we do obtain a conser-xed quantity. Instead of defining "quantity 
of motion" as the product of mass and speedy mv, we can define 
it (as Newton did) as the product of the mass and velocity', mv. 
In this way we include the idea of the direction of motion as xvell 
as the speed. In all thr^ee collisiorns, the motion of both carts 
before and after collision is described by the equation 



m,v, + m„v„ 



mA\ + m„v 



helor'c; 
collision 



after 
collision 



254 



UMT 3 / THE TRIl'MPH OF MECHAIVICS 



if m.^ and m,, represent the masses of the carts, v^ and v,, 
represent their velocities before the collision, and \\' and v^' 
represent their velocities aftei- the collision. 

In other words: The vector sum of the quantities mass X 
velocity is constant, or conserved, in all these collisions. This is a 
very important and useful equation, leading directly to a 
poweiliil law. 



In Unit 1, initial and final velocities 
were represented as v and v^. Here 
they are represented by v and v 
because we now need to add sub- 
scripts such as A and B. 



SG 8, 9 



5. Descartes defined the quantity of motion of an object as the 
product of its mass and its speed. Is his quantity^ of motion 
conserved as he believed it was? If not, how would you modify 
his definition so the quantity of motion would be conserved? 

6. Two carts collide head-on and stick together. In which of 
the following cases will the carts be at rest immediately after 
the collision? 







Cart A 




Cart B 




mass 


speed before 


mass 


speed before 




(kg) 


(m/secJ 


(kg) 


(m/secJ 


(a) 


2 


3 


2 


3 


(b) 


2 


2 


3 


3 


(c> 


2 


3 


3 


2 


(d) 


2 


3 


1 


6 



9*3 I Conservation of momentum 

The product of mass and velocity often plays an interesting role 
in mechanics. It therefore has been given a special name. Instead 
of being called "quantity of motion," as in Newton's time, it is 
now called momentum. The total momentum of a system of 
objects (for example, the two carts) is the vector sum of the 
momenta of all objects in the system. Consider each of the 
collisions examined. The momentum of the system as a whole, 
that is, the vector sum of the individual parts, is the same before 
and after collision. Thus, the results of the experiments can be 
summarized briefly: The momentum of the system is conserved. 

This rule (or law, or principle) follows from obseivations of the 
special case of collisions between two carts that stuck together 
after colliding. In fact, this law of conserx^ation of momentum is a 
completely general, universal law. The momentum of any system 
is consewed if one condition is met: that no net force is acting 
on the system. 

To see just what this condition means, examine the forces 
acting on one of the carts. Each cart experiences three main 
forces. There is, of course, a dovvoiward pull F exerted by the 
earth and an upward push F,^^,,,^ exerted by the table. During the 
collision, there is also a push F. ,. , exerted bv the other 

1 nom nthci" cart 



Since the momentum of a system 
is the vector sum of the momen- 
tum of its parts, it is sometimes 
called the "total momentum" of the 
system. We will assume that "total" 
is understood. 



SG 10, 11 



'table 



I Tothercart 



\J 



'gray 
Forces on cart B during collision. 



CHAPTER 9 / CONSERVATIOM OF MASS AND MOMENTUM 



255 



In general, for n objects the law can 
be written 



S (m,v),„f„„ = S (m,v,) 



7V afer 




SG 12-15 



cart. The first two forces evidently cancel, since the cart is not 
accelerating up or clown. Thus, the net force on each cai1 is just 
the force exerted on it b\' the other cai-t as they collide. (Assume 
that frictional forces e.xeited by the table and the air are small 
enough to neglect. That was the reason for using diy-ice disks, 
air-track gliders, or carts with ' frictionless" wheels. This 
assumption makes it easier" to discuss the law of conseiAation of 
momentum. Later, you v\ill see that the law holds whether- 
friction exists or rnot.i 

The two carets for-m a system of bodies, each cart being a part 
of the system. The for ce exerted by one cart on the other- cart is 
a force exerted by one part of the system on another- part. It is 
not a force on the system as a whole. The outside forces acting 
on the car-ts (by the ear-th and by the table) exactly cancel. Thus, 
there is no net outside force. The system is "isolated." This 
condition must be met in order for the momentum of a system 
to stay constant or- be conserved. 

If the net force on a system of bodies is zero, the momentum 
of the system will not change. This is the law of conserxation of 
momentum for systems of bodies that are moving with linear- 
velocity V. 

So far, you have consider^ed only cases in which two bodies 
collide directly and stick together. The remarkable thing about 
the law of conservation of momentum is how universally it 
applies. For example: 

1. It holds tr-ue no matter what kind of forces the bodies exert 
on each other-. They may be gr^avitational forces, electric or 
magnetic forceS; tension in strings, compression in springs, 
attr-action or r^epulsion. The sum of the mvs before is equal to 
the sum of ativ's after any interaction. 

2. It does not matter whether the bodies stick together- or 
scrape against each other or bounce apart. They do not even 
have to touch. When two strong magnets repel or when an alpha 
particle is r^epelled by a nucleus, conservation of momentum still 
holds. 

3. The law is not r^estricted to systems of onl\ two objects; 
there can be any number of objects in the system. In those cases, 
the basic conservation equation is made more general simpK' by 
adding a terin for each object to both sides of the etiuation. 

4. The size of the system is not important. Ihe law applies to 
a galaxy as well as to an atom. 

5. The angle of the collision does not matter-. .All of the 
examples so far- have involved collisions between two bodies 
moving along the same str^aight line. They wer-e "one-dimensional 
collisions." If two bodies make a glancing collision rather- than a 
head-on collision, each will move off at an angle to the line of 
appr-oach. The law of conser-vation of momentirm appli(\s to sirch 
"two-dimensional collisions ' also. iRemembei- that momentum 



256 



U,MT 3 / THE TRILMPH OF \II;(:HL\\IC:S 



(1) A space capsule at rest in space, far from the 
sun or planets, has a mass of 1 ,000 kg. A meteorite 
with a mass of 0.1 kg moves toward it with a speed 
of 1 ,000 m/sec. How fast does the capsule (with the 
meteorite stuck in it) move after being hit? 

m^ (mass of the meteorite) = 0.1 kg 

m^ (mass of the capsule) = 1 ,000 kg 

v^ (initial speed of meteorite) = 1 ,000 m/sec 

Vg (initial speed of capsule) = 

v^' (final speed of meteorite) 

i/g' (final speed of capsule) 



= 9 



The law of conservation of momentum states that 

Inserting the values given, 

(0.1 kg) (1,000 m/sec) + (1,000 kg) (0) = 

(0.1 kg)»/; + (1,000 kg)i/3' 
100 kg • m/sec = (0.1 kg)i/^' + (1,000 kg)i/g' 

Since the meteorite sticks to the capsule, v^' = i^^'; 
so we can write 

100 kg • m/sec = (0.1 kg)>// + (1,000 kg)\/^' 
100 kg • m/sec = (1,000.1 kg)i// 



Therefore, 



v/ = 



1 00 kg • m/sec 



1,000.1 kg 
= 0.1 m/sec 

(in the original direction of the motion of the meteo- 
rite). Thus, the capsule (with the stuck meteorite) 
moves on with a speed of 0.1 m/sec. 

Another approach to the solution is to handle the 
symbols first, and substitute the values as a final 
step. Substituting v^' for v^' and letting v^' = 
would leave the equation mj/^ = m^v^' + m^v^' 
= {m^ + m^)v;. Solving for (// 



v/ = 



m„\/„ 



(m^ + mj 

This equation holds true for any projectile hitting 
(and staying with) a body initially at rest that moves 
on in a straight line after collision. 

(2) An identical capsule at rest nearby is hit by 
a meteorite of the same mass as the other. How- 
ever, this meteorite, hitting another part of the cap- 
sule, does not penetrate. Instead, it bounces straight 



6I0SC up C, 

Conservation of Momentum 





back with almost no change of speed. (Some sup- 
port for the reasonableness of this claim is given in 
SG 24.) How fast does the capsule move after being 
hit? Since all these motions are along a straight line, 
we can drop the vector notation from the symbols 
and indicate the reversal in direction of the meteorite 
with a minus sign. 
The same symbols are appropriate as in (1): 



m^ = 0.1 kg 
m^ = 1,000 kg 
i/„ = 1,000 m/sec 



V, =0 

I// = - 1 ,000 m/sec 

V ' = 7 



The law of conservation of momentum started 
\ha{mj/^ + m^v^ = mj/^' + m^v^'. Here, 

(0.1 kg) (1,000 m/sec) -i- (1,000 kg) (0) = 

(0.1 kg) (-1,000 m/sec) + (1,000 kg)v/g' 

100 kg • m/sec = - 100 kg • m/sec -i- (1,000 kg)\/3' 



V' = 



200 kg • m/sec 
1 ,000 kg 



= 0.2 m/sec 



Thus, the struck capsule moves on with about twice 
the speed of the capsule in (1 ). (A general symbolic 
approach to this solution can be taken, too. The 
result is valid only for the special case of a projectile 
rebounding perfectly elastically from a body of much 
greater mass.) 

There is a general lesson here. It follows from the 
law of conservation of momentum that a struck ob- 
ject is given less momentum if it absorbs the pro- 
jectile than if it reflects it. (A goalie who catches the 
soccer ball is pushed back less than one who lets 
the ball bounce off.) Some thought will help you to 
understand this idea: An interaction that merely 
stops the projectile is not as great as an interaction 
that first stops it and then propels it back again. 



CHAPTER 9 / CONSERVATION OF MASS AND MOMENTUM 



357 




One of the stroboscopic photo- 
graphs of two colliding objects ttuit 
appears in the Handbook. 



is a vector quantity.) The law of conseivation of iiionientiim also 
applies in three dimensions. The xector sum of the momenta is 
still the same hefore and after the collision. 

On page 257 is a vvorked-out exam|ile that \\ ill hi^lp you 
hecome familiar" with the law of consenation of momentum. On 
page 259 is an analysis of a two-dimensional collision. There are 
also strohoscopic photographs in th(^ ProjecA Physics lliuulhook 
and film loops of (U)lliding hodi(^s and (^\|)loding objects. Ilu^se 
include collisions and explosions In two dimensions. The more 
of them you analyze, the moi(> conxinced you will hv. that the 
law of conseiA'atlon of momentum applies to imv isolal(!d system. 

The worked-out example on page 257 displays a c:haracteristic 
featuie of physics. Again and again, physics problcMiis are solved 
by applying the expression of a geiwrnl law to a spcu'ific 
situation. Both the beginning student and th(? xeteian lesearch 
physicist find it helpful, but also somewhat mysterious, that one 
can do tiiis. It seems strange that a few general kiws enable one 
to solve an almost infinite number of specific individual 
problems. Everyday life seems diffei'ent. There you usually 
cannot calculate answers from general laws. Rather-, you have to 
make cjuick decisions, some based on r'ational analysis, other's 
based on "intuition." The use of general laws to solve scientific 
problems will become, with practice, quite natural also. 



7. State the law of conservation of momentum in terms of 

(a) change in the total momentum of a system; 

(b) the total initial momentum and final momentum; 

(c) the individual parts of a system. 

8. Which of the follow ing has the least momentum? Which has 
the greatest monmiUwn? 

(a) a pitched baseball 

(bl a Jet plane inflight 

(c) a jet plane taxiing toward the terminal 

9. A girl on ice skates is at rest on a horizontal sheet of 
smooth ice. As a result of catching a rubber ball moving 
horizontally toward her, she moves at 2 cm/sec. Give a rough 
estimate of what her speed would have been 

(a) if the rubber ball were thrown twice as fast. 
(bj if the rubber ball had twice the mass. 

(c) if the girl had twice the mass. 

(d) if the rubber ball were not caught by the girl, but bounced 
off and went straight back with no change of speed. 



258 



UMIT 3 / THE TIIIUMPH OF \IL;c:HA.\ICS 



Close Upl 



A Collision in Two Dimensions 



The stroboscopic photograph shows a collision 
between two wooden discs on a "frictionless hori- 
zontal table" photographed from straight above the 
table. The discs are riding on tiny plastic spheres 
which make their motion nearly fnctionless. Body 
B (marked x) is at rest before the collision. After the 
collision it moves to the left, and Body A (marked 
- ) moves to the right. The mass of Body B is known 
to be twice the mass of Body A: m^ = 2m^. We will 
analyze the photograph to see whether momentum 
was conserved. {Note: The size reduction factor of 
the photograph and the [constant] stroboscopic 
flash rate are not given here. But as long as all 
velocities for this test are measured in the same 
units, it does not matter what those units are.) 

In this analysis, we will measure in centimeters 
the distance the discs moved on the photograph. 
We will use the time between flashes as the unit of 
time. Before the collision, Body A (coming from the 
lower part of the photograph) traveled 36.7 mm in 
the time between flashes: v^ = 36.7 speed-units. 
Similarly, we find that v^' = 17.2 speed-units, and 
i?g' = 11.0 speed units. 

The total momentum before the collision is just 
mj/^'. It is represented by an arrow 36.7 momen- 
tum-units long, drawn at right. 

The vector diagram shows the momenta m^v^' 
and mj/g' after the collision; m^v^' is represented 
by an arrow 17.2 momentum-units long. Since m^ 
= 2m^, the m^v^' arrow is 22.0 momentum-units 
long. 

The dotted line represents the vector sum of m^v^' 
and rrigVg', that is, the total momentum after the 
collision. Measurement shows it to be 34.0 mo- 
mentum-units long. Thus, our measured values of 
the total momentum before and after the collision 
differ by 2.7 momentum-units. This is a difference 
of about 7%. We can also verify that the direction 
of the total is the same before and after the collision 
to within a small uncertainty. 

Have we now demonstrated that momentum was 
conserved in the collision? Is the 7% difference 
likely to be due entirely to measurement inaccura- 
cies? Or is there reason to expect that the total 
momentum of the two discs after the collision is 
really a bit less than before the collision? 




MbVb»22.0 



MaV^MbV^ 




MaV;= 17.2 



Momentum Scale (arbitrary units) 



M^V^-36.7 



10 



I? 



20 



CHAPTER 9 / COIVSERVATIOIV OF MASS AND MOMENTUM 



ZS9 



9«4r Momentum and Xeiiton's laivs of motion 



SG 16 



Section 9.2 developed the concept of momentum and tlic law of 
conservation of momentum by considering experiments with 
colliding carts. The law was an "empirical" law; that is, it was 
discovered (perhaps "invented" or "induced" are better temis) as 
a generalization from experiment. 

We can show, howevei", that the law of conseiAation of 
momentum also follows directly from Newton's laws of motion. It 
takes only a little algebra; that is, we can deduce the law from an 
established theoiy. It would also be possible to derive Newlon's 
laws fiom the conseivation law. Which is the fundamental law 
and which the conclusion is therefore a bit arbitrary. Newton's 
laws used to be considered the fundamental ones, but since 
about 1900 the conservation law has been assumed to be the 
fundamental one. 

Newton's second law expresses a relation between the net 
force F,,g, acting on a body, the mass m of the body, and its 
acceleration a. We wrote this as F,,^, = ma. We can also write this 
law in terms oi change of momentum of the body. Recalling that 
acceleration is the rate-of-change of velocity, a = AvA/, we can 
write 



See Chapter 20. 



Av 
Tt 



or 



If m is a constant, 

A(mv) = mv — mv 
= m(v' — v) 
= m A V 

FAf is called the "impulse." 



F„ At = mAv 

net 



If the mass of the body is constant, the change in its momentum, 
AIahv), is the same as its mass times its change in xelocitv', m(Av). 
Then we can write 



^n... ^t 



Al/riv) 



SG 17-20 



In Newton's second law, "change of 
motion" meant change of momen- 
tum. See Definition 11 at the begin- 
ning of the Principia. 



that is, the product of the net force on a hodv and the time 
inter\'al during which this force acts equals the change in 
momentum of the body. 

This statement of Newton's second law is more nearly what 
Newton used in the Principia. Together with Newton's third law, 
it enables us to derive the law of conservation of momentum for 
the cases we have studied. The details of the deiixation are given 
on page 262. Thus, Newton's laws and the law of conseivation of 
momentum are not separate, independent law^s of nature. 

In all the examples considered so far and in the deiixation 
above, we have consider ed each piece of the system to have a 
constant mass. But the definition perTnits a change of 



260 



UMT 3 / THE TIllUMPH OF MECHANICS 



momentum to arise from a change of mass as well as from a 
change of velocity. In many cases, it is convenient to consider 
objects whose mass is changing. For example, as a rocket spews 
out exhaust gases its mass is decreasing; the mass of a train of 
coal cars increases as it rolls past a hopper that drops coal into 
the cars. These cases can be analyzed using objects that do have 
constant masses, but this requires, for example, analyzing 
separately the case of each lump of coal that falls into the train. 
However, the law of conservation of momentum also is valid 
when the masses of the objects involved are not constant, as long 
as no net forces act on the system as a whole. Problems such as 
the one of the coal train or the rocket are much easier to analyze 
in this way than in the other. 

In one form or another, the law of conservation of momentum 
can be derived from Newton's second and third laws. 
Nevertheless, the law of conseI^^ation of momentum is often the 
preferred tool because it enables us to solve many problems that 
would be difficult to solve using Newton's laws directly. For 
example, suppose a cannon that is free to move fires a shell 
horizontally. Although it was initially at rest, the cannon is forced 
to move while firing the shell; it recoils. The expanding gases in 
the cannon barrel push the cannon backward just as hard as 
they push the shell forward. Suppose you had a continuous 
record of the magnitude of the force. You could then apply 
Newton's second law separately to the cannon and to the shell 
to find their respective accelerations. After a few more steps 
(involving calculus), you could find the speed of the shell and the 
recoil speed of the cannon. In practice, it is very difficult to get 
a continuous record of the magnitude of the force. For one thing, 
the force almost certainly decreases as the shell moves toward 
the end of the barrel. So it would be very difficult to use 
Newton's law^s to find the final speeds. 

However, you can use the law of conservation of momentum 
even if you know nothing about the force. The law of 
conservation of momentum is a law of the kind that says 'before 
= after." Thus, it works in cases where you do not have enough 
information to apply Newton's laws during the whole interval 
between "before" and "after. " In the case of the cannon and 
shell, the momentum of the system (cannon plus shell) is zero 
initially. Therefore, by the law of conservation of momentum, the 
momentum will also be zero after the shell is fired. If you know 
the masses and the speed of one, after firing you can calculate 
the speed of the other. (The film loop titled "Recoil" provides just 
such an event for you to analyze.) On the other hand, if both 
speeds can be measured afterwards, then the ratio of the masses 
can be calculated. In Unit 6, "The Nucleus, "you will see how 
just such an approach was used to find the mass of the neutron 
when it was originally discovered. 



SG 21-24 




SG25 



SG 26 
SG 27 



CHAPTER 9 / CONSERVATION OF MASS AND MOMENTUM 



261 



liiose Upl 



and 



F^^M = MmJ/J 



By Newton's third law, 

' AB ~ ~ '"b 

SO that 



Therefore, 



F.e^t= -F,,M 



^{m^v,) = -Mm^v^) 



Suppose that each of the masses m^ and m^ are 
constant. Let v^ and v^ stand for the velocities of 
the two bodies at some instant, and let v^' and v^' 
stand for their velocities at some later instant. Then 
we can write the last equation as 

A little rearrangement of terms leads to 



and 



nn,v^' - m,v^ = - m^v^' + m^v^ 



my' + mj/J = my, + my^ 



You will recognize this as our original expression 
of the law of conservation of momentum. 

Here we are dealing with a system consisting of 
two bodies. This method works equally well for a 
system consisting of any number of bodies. For 
example, SG 21 shows you how to derive the law 
of conservation of momentum for a system of three 
bodies. 




Deriving Conservation of Momentum 
from Newton's Lows 



Suppose two bodies with masses m^ and /n^ ex- 
ert forces on each other (by gravitation or by mutual 
friction, etc.). F^g is the force exerted on body A by 
body B, and Fg^ is the force exerted on body B by 
body A. No other unbalanced force acts on either 
body; they form an isolated system. By Newton's 
third law, the forces F^g and Fg^ are at every instant 
equal in magnitude and opposite in direction. Each 
body acts on the other for exactly the same time 
M. Newton's second law, applied to each of the 
bodies, says 

F^gA? = \{m^v^) 







Globular clusters of stars like this one contain 
tens of thousands of suns held together by gravi- 
tation attraction 




262 



LMT 3 IHL; TKIIMPH OF MECHANICS 



# 10. Since the law of conservation of momentum can be 
derived from Newton's laws, what good is it? 
11. What force is required to change the momenlum of an 
object by 50 kg ■ m/sec in 15 sec? 

9.5 I Isolated systems 

There are important similarities between the conservation law of 
mass and that of momentum. Both laws are tested by observing 
systems that are in some sense isolated from the rest of the 
universe. When testing or using the law of conservation of mass, 
an isolated system such as a sealed flask is used. Matter can 
neither enter nor leave this system. When testing or using the 
law of conservation of momentum, another kind of isolated 
system, one which experiences no net force from outside the 
system, is used. 

Consider, for example, two diy-ice pucks colliding on a 
smooth horizontal table. The very low-friction pucks form a veiy 
nearly closed or isolated system. The table and the earth do not 
have to be included since their individual effects on each puck 
cancel. Each puck experiences a downward graxitational force 
exerted by the earth. The table exeits an equally strong upwai'd 
push. 

Even in this artificial example, the system is not entirely 
isolated. There is always a little friction vvdth the outside world. 
The layer of gas under the puck and air currents, for example, 
exert friction. All outside forces are not completely balanced, and 
so the two pucks do not form a truly isolated system. Whenevei- 
this is unacceptable, one can expand or extend the system so 
that it includes the bodies that are responsible for the external 
forces. The result is a new system on which the unbalanced 
forces are small enough to ignore. 

For example, picture two cars skidding toward a collision on 
an icy road. The frictional forces exerted by the road on each car 
may be several hundred newtons. These forces are veiy small 
compared to the immense force (thousands of nevutons) exerted 
by each car on the other when they collide. Thus, for many 
purposes, the action of the road can be ignored. For such 
purposes, the two skidding cars during the collision are nearly 
enough an isolated system. However, if friction vvdth the road (or 
the table on which the pucks move) is too great to ignore, the 
law of conservation of momentum still holds, but in a lar^ger 
system — one which includes the i^oad or- table. In the case of the 
skidding cars or the pucks, the load or table is attached to the 
earth. So the entii^e ear^h would have to be included in a "closed 
system." 






SG 28-33 



CHAPTER 9 / CONSERVATION OF MASS AND MOMENTUM 



263 



/ 
t 



> 



/ 



4. 



12. Define what is meant by "closed" or "isolated" system for 
the purpose of the law of conservation of mass; for the 
purpose of the law of conservation of momentum. 

13. E}<.plain whether or not each of the following can be 
considered an isolated system. 

(a) a baseball thrown horizontally 

(b) an artificial earth satellite 

(c) the earth and the moon 

14. Three balls in a closed system have the following masses 
and velocities: 

ball A: 4 kg, 8 m/sec left 

ball B: 10 kg, 3 m/sec up 

ball C: 8 kg, 4 m/sec right 
Using the principles of mass and momentum conservation, 
what can you discover about the final condition of the system? 
What cannot be discovered? 



9.6 Elastic collisions 



In 1666, members of the recently formed Royal Society of London 
witnessed a demonstration. Two hardwood balls of equal size 
^ ^ were suspended at the ends of two strings to foiTn two pendula. 

One ball was released from rest at a certain height. It swung 
— ; - f • - down and stnjck the other, which had been hanging at rest. 

• After impact, the first ball stopped at the point of impact while 

\ the second ball swung ft^om this point to the same height as that 

from which the first ball had been released. When the second 
^ • ball returned and struck the first, it was now the second ball 

which stopped at the point of impact as the first sv\amg up to 
almost the same height from which it had started. This motion 
repeated itself for several swings. 
\ This demonstration aroused great interest among members of 

the Society. In the next few years, it also caused heated and 
often confusing arguments. Why did the balls rise each time to 
nearly the same height after each collision? Why was the motion 
"~P"^ ' "transferred" from one ball to the other when they collided? Why 

j i did the first ball not bounce back fixim the point of collision, or 

1 continue moving forward after the second ball moved away from 

^X^,. the collision point? 

The law of conservation of momentum explains what is 

~.«,— i„„ obseiA'ed, but it would also allow (juite different results. The law 

says only that the momentum of ball A just before it strikes ball 
B is equal to the total momentum of A and B just after collision. 
It does not sav how A and B share the momentum. The actual 



w result is just one of infinitelv manv different outt'omes that 



264 UNIT 3 / IHE TKIUVIPH OF MECHANICS 



would all agree with conservation of momentum. For example, 
suppose (though it has never been observed to happen) that ball 
A bounced back with 10 times its initial speed. Momentum 
would still be conseived /f ball B went ahead at 11 times As 
initial speed. 

In 1668, three men reported to the Royal Society on the whole 
matter of impact. The three men were the mathematician John 
Wallis, the architect and scientist Christopher Wren, and the 
physicist Christian Huygens. Wallis and Wren offered partial 
answers for some of the features of collisions; Huygens analyzed 
the problem in complete detail. 

Huygens explained that in such collisions another conservation 
law in addition to the law of conservation of momentum, also 
holds. Not only is the vector sum of mv's conseived, but so is the 
ordinary arithmetic sum of rnv^s! In modern algebraic form, the 
relationship he discovered can be expressed as 

t'^a^a' + r"^"^«' " t"^a^^" + T^b^'b" 

The scalar quantity VzmV^ has come to be called kinetic energy. 
(The reason for the Vz, which does not really affect the rule here, 
wdll become clear in the next chapter.) The equation stated 
above, then, is the mathematical expression of the conservation 
of kinetic energy. This relationship holds for the collision of two 
"perfectly hard" objects similar to those observed at the Royal 
Society meeting. There, ball A stopped and ball B went on at As 
initial speed. A little algebra will show that this is the only result 
that agrees with both conseivation of momentum and 
conservation of kinetic energy. (See SG 33.) 

But is the conservation of kinetic energy as general as the law 
of conservation of momentum? Is the total kinetic energy present 
conserved in any interaction occurring in any isolated system? 

It is easy to see that it is not. Consider the first example of Sec. 
9.2. Two carts of equal mass (and with putty between the 
bumping surfaces) approach each other with equal speeds. They 
meet, stick together, and stop. The kinetic energy of the system 
after the collision is 0, since the speeds of both carts are zero. 
Before the collision the kinetic energy of the system was Vzm^v,^' 
+ V2m,jVg\ Both Vzm^v^^ and Vzm^v^^ are always positive 
numbers. Their sum cannot possibly equal zero (unless both v^ 
and Vg are zero, in which case there would be no collision and 
not much of a problem). Kinetic energy is not conseived in this 
collision in which the bodies stick together. In fact, no collision 
in which the bodies stick together will show consen^ation of 
kinetic energy. It applies only to the collision of "perfectly hard" 
bodies that bounce back fi^om each other. 

The law of conservation of kinetic energy, then, is not as 
general as the law of conservation of momentum. If two bodies 
collide, the kinetic energy may or may not be conserved, 



In general symbols, AS, -my^ = 0. 

Compare this equation with the 
conservation of momentum equa- 
tion on page 254. 

SG 34-37 




Christian Huygens (1629-1695) was 
a Dutch physicist. He devised an 
improved telescope with which he 
discovered a satellite of Saturn and 
saw Saturn's rings clearly. Huygens 
was the first to obtain the e^cpres- 
sion for centripetal acceleration 
fv'/RA' he worked out a wave theory 
of light; and he invented a pendu- 
lum-controlled clock. His scientific 
contributions were major, and his 
reputation would undoubtedly have 
been greater had he not been over- 
shadowed by his contemporary, 
Newton. 



CHAPTER 9 / CONSERVATION OF MASS AND MOMENTUM 



265 



Huvgens, and others after him for 
about a century, did not use the 
factor '/-. The quantity niv^ was 
c^ed \is \i\'a, Latin for "living force." 
Seventeenth- and eighteenth-cen- 
tury scientists were greatly inter- 
ested in distinguishing and naming 
various "forces." They used the 
word loosely; it meant sometimes 
a push or a puU (as in the colloquial 
modern use of the word force I, 
sometimes what is now called "mo- 
mentum," and sometimes what is 
now called 'energ\'." The term vis 
viva is no longer used. 



depending on the type of collision. It is consened if the colliding 
bodies do not cnjmple or smash or dent or stick together or heat 
up or change physically in some other way. Bodies that inbound 
without any such change aie perfectly elastic. Collisions between 
them are perfectly elastic collisions. In perfectly elastic collisions, 
both momentum and kinetic energy are conserved. 

Most collisions are not peifectly elastic, and kinetic energ\' is 
not conseived. Thus, the sum of the V^mv^'s after the collision is 
less than that before the collision. Depending on how much 
kinetic energy is "lost," such collisions might bv. called "partially 
elastic" or "peifectly inelastic." The loss of kinetic energy' is 
greatest in perfectly inelastic collisions, when the colliding 
bodies remain together. 

Collisions between steel ball bearings, glass marbles, hardwood 
balls, billiard balls, or some rubber balls (silicone rubber) are 
almost perfectly elastic, if the colliding bodies are not damaged 
in the collision. The total kinetic energ\' after the collision might 
be as much as, say, 96% of this value before the collision. 
Examples of peifectly elastic collisions are found only in 
collisions between atoins or subatomic particles. 




Descartes (1596-16501 was the 
most important French scientist of 
the seventeenth centun'. In addi- 
tion to his early contribution to the 
idea of momentum conserx'ation, 
he is remembered by scientists 
as the inventor of coordinate sys- 
tems and the graphical representa- 
tion of algebraic equations. Des- 
cartes' system of philosophy, which 
used the deductive structure of 
geometry as its model, is still injhi- 
ential. 



# 15. Which phrases correctly complete the statement? Kinetic 
energy is conserxed 

(a) in all collisions. 

(b) whenever momentum is conserved. 

(c) in some collisions. 

(d) when the colliding objects are not too hard. 
16. Kinetic energy is never negative because 

(a) scalar quantities are always positive. 

(b) it is impossible to draw vectors with negative length. 

(c) speed is always greater than zero. 

(d) it is proportional to the square of the speed. 

9.T I Leibniz and the consen^ation laiv 

Rene Descartes believed that the total quantity of motion in the 
universe did not change. He v\Tote in Principles of Philosophy: 

It is wholly rational to assume tliat (iod, since in the ci-eation of 
matter He imparted different motions to its parts, and presenes 
all matter in the same way and conditions in which He created 
it, so He similarly preserves in it the same quantity of motion. 

Descailes proposed to define the quantity of motion of 
an object as the product of its mass and its speed. As you saw 
in Sec. 1.1, this product is a conserved quantity only in \'eiy 
special cases. 



266 



UNIT 3 / THE TRIUMPH OF MblCHAMCS 



Gottfried Wilhelm Leibniz was aware of the error in Descartes' 
ideas on motion. In a letter in 1680 he wrote: 

M. Descartes' physics has a great defect; it is that his rules of 
motion or laws of nature, which are to serve as the basis, are 
for the most part false. This is demonstrated. And his great 
principle, that the quantity of motion is conserved in the world, 
is an error. 

Leibniz, however, was as sure as Descartes had been that 
something involving motion was conserved. Leibniz called this 
something, which he identified as 'force, " the quantity mV^ 
(which he called vis viva). This is just twice the quantity now 
called kinetic energy. (Of course, whatever applies to mv^ applies 
equally to Vzmv^.) 

As Huygens had pointed out, the quantity Vzmx^ is consented 
only in perfectly elastic collisions. In other collisions, the total 
quantity of Vzmv^ after collision is always less than before the 
collision. Still, Leibniz was convinced that Vzmv^ is always 
conserved. In order to save his conservation law, Leibniz 
invented an explanation for the apparent loss of vis viva. He 
maintained that the vis viva is not lost or destroyed. Rather, it is 
merely "dissipated among the small parts " of which the colliding 
bodies are made. This was pure speculation, and Leibniz offered 
no supporting evidence. Nonetheless, his explanation anticipated 
modern ideas about the connection between energy and the 
motion of molecules. 

Leibniz extended conservation ideas to phenomena other than 
collisions. For example, when a stone is thrown straight upward, 
its quantity of Vamv" decreases as it rises, even without any 
collision. At the top of the trajectoiy, Vzmv^ is zero for an instant. 
Then it reappears as the stone falls. Leibniz wondered whether 
something applied or given to a stone at the start is somehow 
stored as the stone rises, instead of being lost. His idea would 
mean that Vzmv^ is just one part of a more general and really 
conseived quantity. 




Leibniz (1646-1716), a contempo- 
rary of Newton, was a German phi- 
losopher and diplomat, an advisor 
to Louis Xrv of France and Peter 
the Great of Russia. Independently 
of Newton, Leibniz invented the 
method of mathematical analysis 
called calculus. A long public dis- 
pute resulted between the two 
great men concerning charges of 
plagiarism of ideas. 



% 17. According to Leibniz, Descartes' principle of conservation 
of mv was 

(a) correct, but trivial. 

(b) another way of expressing the conservation of vis viva. 

(c) incorrect. 

(d) correct only in elastic collisions. 

18. How did Leibniz e}cplain the apparent disappearance of the 
quantity Vznv/ 

(a) during the upward motion of a thrown object? 

(b) when the object strikes the ground? 



CHAPTER 9 / CONSERVATION OF MASS AND MOMENTUM 



267 



study 
guide 



1. The Project Physics learning materials 
particularly appropriate for Chapter 9 include: 

Experiments 

Collisions in One Dimension 
Collisions in Two Dimensions 

Film Loops 

One-Dimensional Collisions. I 
One-Dimensional Collisions. II 
Inelastic One-Dimensional Collisions 
Tvvo-Dimensional Collisions. I 
Two-Dimensiontil Collisions. II 
Inelastic Tvvo-Dimensional Collisions 
Scattering of a (Cluster of Objects 
Elxplosion of a Cluster of Objects 

2. Certainly Lavoisier did not investigate everv' 
possible interaction. What justification did he have 
for claiming mass was conserved "in all the 
operations of art and nature"? 

3. It is estimated that ever\' year at least 1 ,800 
metric tons of meteoric dust fcdl to the earth. The 
dust is mostly debris that was ino\ang in orbits 
around the sun. 

(a) Is the earth (whose mass is about 5.4 x lO"' tons) 
reasonably considered to be a closed system with 
respect to the law of conservation of mass? 

(b) How large would the system, including the earth, 
have to be in order to be completely closed? 

4. Would you expect that in your lifetime , when 
more accurate balances are built , you will see 
experiments which show that the law of conservation 
of mass does not entirely hold for chemical reactions 
in closed systems? 

5. Dayton C. Miller, a renowned experimenter at 
Case Institute of Technology, was able to show that 
two objects placed side by side on an equal-arm pan 
balance did not exactly balance two otherwise 
id(!nti(al objects placed one on top of the other. I The 
reason is that the pull of gravity d(HT(!as(\s with 



distance from the center of the earth. i Does this 
experiment contradict the law of conservation of 
mass? 

6. A children's toy known as a Snake consists of a 
tiny pill of mercuric thiocyanate. When the pill is 
ignited, a large, serpent-like foam curls out idmost 
from nothingness. Devise and describe an 
experiment by which you would test the law of 
conservation of mass for this demonstration. 

7. Consider the following chemical reaction, which 
was studied by Landolt in his tcjsts of the law of 
conservation of mass. In a closed container, a 
solution of 19.4 g of potassiuin chromate in 100.0 g 
of water is mixed witli a solution of 33. 1 g of lead 
nitrate in 100.0 g of water. A bright yellow solid 
precipitate forms and settles to the bottom of the 
container. When removed from the liquid, this solid 
is found to have a mass of 32.3 g and is found to 
have properties different from either of the 
reactants. 

(a) What is the mass of the remaining licjuid? 
(Assume the combined inass of idl substances in the 
system is conserved.! 

(b) After removal of the yellow precipitate, the 
remaining liquid is heated to 95° C. The water 
evaporates, leaving a white solid with a mass of 20.2 
kg. Why does this result imply that the water did not 
react with anything in either la I or lb I? 

8. (a) Ten grams (10 g) of a solid are added to 50 g 
of a liquid on earth. What is their total mass? What 
would be their totid mass on the moon? 

(b) A mixture that weighs 50 N on earth weighs 
about 8 N on the moon. Why is this fact not a 
violation of conservation of mass? 

(c) Ten cubic centimeters 110 cm'i of a solid are 
added to 50 mL (cm^) of a liquid. The resulting 
mixture has a volume of 54 cm'. Why does this result 
not violate conservation of mass? 

9. (a) For an isolated system, state the principle of 
consenation of momentum in terms of the change 
in total momentum AP and of the totiU initiid and 
final momenta. 

(b) For the following disks, find the momentum of 
each disk and the totid monunitiim of th(! svstem: 



268 



C:H.\PTER 9 / STUDY GUIDE 



sg 



disk A: 5 kg, 8 m/sec west 
disk B: 6 kg, 25 m/sec north 
disk C: 10 kg, 2 m/sec east 
disk D: 4 kg, 5 m/sec east 

(c) If the disks in (b) all collide at the same instant 
and stick together, what is the final momentum of 
the system? What is the final speed of the group of 
disks? 

(d) Explain how a system could have zero total 
momentum and yet still have many massive objects 
in motion. 

10. A freight car of mass 10" kg travels at 2.0 m/sec 
and collides with a motionless freight car of mass 1.5 
X lO'' kg on a horizontal track. The two cars lock 
and roll together after impact. Find the velocity of 
the two cars after collision. Hints: The general 
equation for conservation of momentum for a two- 
body system is: 

(a) What quantities in the problem can be 
substituted in the equation? 

(b) Rearrange terms to get an expression for v^'. 

(c) Find the value of v^'. (Note: v^' = vj .} 

11. You have been given a precise technical 
definition for the word momentum. Look it up in a 
large dictionary and record its various uses. Can you 
find anything similar to our definition in these more 
general meanings? How many of the uses seem to 
be consistent with the technical definition given 
here? 

la. Benjamin Franklin, in correspondence with his 
friend James Bowdoin (founder and first president of 
the American Academy of /\rts and Sciences), 
objected to the corpuscular theory of light by saying 
that a particle traveling with such immense speed 
(3 X 10" m/sec) would have the impact of a 10-kg ball 
fired from a cannon at 100 m/sec. What mass did 
Franklin assign to the "light particle"? 

13. If powerful magnets are placed on top of each 
of two carts, and the magnets are so arranged that 
like poles face each other when one cart is pushed 
toward the other, the carts bounce away from each 
other without actually making contact. 



(a) In what sense can this be called a collision? 

(b) Will tlie law of conseiA/ation of momentum 
apply? 

(c) Describe an ai'rangement for testing your answer 
to (b). 

14. A person throws a fast ball vertically. Clearly, the 
momentum of the ball is not conserved; it first loses 
momentum as it rises, then gains it as it falls. How 
large is the "closed system" within which the ball's 
momentum, together with that of other bodies (tell 
which), is conserved. What happens to the rest of the 
system as the ball rises? as it falls? 

15. Did Newton arrive at the law of conservation of 
momentum in the Principia? If a copy of the 
Principia is available, read Corollary III and Definition 
II (just before and just after the three laws). 

16. (a) For how long would a force of 20 N have to 
be applied to cause a system to gain a momentum of 
80 kg • m/sec? 

(b) A 5-kg object initially travels udth a momentum 
of 50 kg • m/sec. If a 10-N force acts on the object for 
5 sec, what is the final speed? (Solve this problem 
first using Newton's second law directly and then 
using momentum formulas.) 

(c) What are the advantages of momentum formulas 
over Newton's laws? Is one more basic than the 
other? 

17. (a) Why can ocean liners or planes not turn 
corners sharply? 

(b) In the light of your knowledge of the relationship 
between momentum and force, comment on reports 
about unidentified flying objects (UFO's) turning 
sharp corners in full flight. 

18. A girl on skis (mass of 60 kg including skis) 
reaches the bottom of a hill going 20 m/sec. What is 
her momentum? She strikes a snowdrift and stops 
within 3 sec. What force does the snow exert on the 
girl? How far does she penetrate the drift? What 
happens to her momentum? 

19. During sports, the forces exerted on parts of the 
body and on the ball, etc., can be astonishingly lai-ge. 
To illustrate this, consider the forces in hitting a golf 



CHAPTER 9 / STUDY GUIDE 



269 



so 



ball. Assume the Ijall's mass is ().U4t) k^. From the 
strobe photo on p. 27 of Unit 1, in which the time 
intenal between strobe flashes was 0.01 sec, 
estimate: 

(a) the speed of the ball after impact; 

(b) the magnitude of the ball's momentum after 
impact; 

(c) how long the impact lasted; 

(d) the average force exerted on the b^ill during 
impact. 

20. We derived tlie law of conservation of 
momentum for two bodies from \'e\\1on's third and 
second laws. Is the principle of the conservation of 
mass essential to this derivation? If so, where does it 
it enter? 



31. Consider an isolated system of three bodies, A, 
B, and C. The forces acting among the bodies can be 
indicated by subscripts; for example, the force 
exerted on body A by body B can be given the symbol 
F,„. Bv i\evvlon's third law of motion, F = —F . 

•^o .. HA \B 

Since the system is isolated, the only force on each 
body is the sum of the forces exerted on it by the 
other two; for example, F^ = F^^ + F^, . Using these 
principles, show that the total momentum change of 
the system will be zero. 

aa. In Chapter 4, SG 33 was about putting an Apollo 
capsule into an orbit around the moon. TIk; question 
was: "Given the speed i'_ necessary' for orbit and the 
current speed v, how long should the engine with 
thrust F fire to give the capsule of mass m the right 
speed?" There you solved tiie problem by 
considering the acceleration. 

(a) .Answer the question more directly by considering 
change in niomoiitum. 



(b) What would be the total momentum of uU the 
exhaust from the rocket? 

(c) If the "exhaust velocity" were v , about what mass 
of fuel would be required? 

23. (a) Show that when two bodies collide their 
changes in velocity are inversely proportional to their 
masses; that is, if m^ and m„ are the masses and Av^ 
and AV|j the velocity changes, show that numerically, 



Ai'. 






(b) Show how it follows from consen'ation of 
momentum that if a light particle (like a B.B. pellet) 
bounces off a massive object (like a bowling balll, 
the velocity of the light particle is chiinged much 
more than the velocity of the massive object. 

(c) For a head-on elastic collision betiveen a body of 
mass m^ moving with velocity \ ^ and a body of mass 
m^ at rest, combining the equations for conservation 
of momentum and conservation of kinetic energy 
leads to the relationship v/ = v^ (m^ - mj I (m^ + 
ni^l. Show that if body B has a much greater mass 
than body A, then v/ is almost exactly the same as 
v^; that is, body A bounces back with virtually no loss 
in speed. 

a4. The equation m^' ^ + ni„v„ = "i^v/ + ni„v„' is a 
general equation applicable to countless separate 
situations. For example, consider a 10-kg shell fired 
from a 1 ,000-kg cannon. If the shell is given a speed 
of 1,000 m/sec, what would be the recoil speed of 
the cannon? lAssume the cannon is on £in £ilmost 
frictionless mount. I Hint: Your answer could include 
the following steps: 

(a) If A refers to the cannon and B to the shell, what 
are v^ and \^^ (before firingi? 

(b) What is the total momentum before firing? 

(c) What is the total momentum after firing? 

(d) Compare the magnitudes of the momenta of the 
cannon and of the shell after firing. 

(e) Compare the ratios of the speeds and of the 
masses of the shell and cannon after firing. 

25. The engines of the first stage of the Apollo 
Saturn rocket develop an average; thrust of 3.") million 



Z70 



CH.VPTER 9 / STUDY GUIDE 



sg 



N for 150 sec. (The entire rocket weighs 28 million 
N near the earth's surface.) 

(a) How much momentum will be given to the 
rocket during that interval? 

(b) The final speed of the vehicle is 9,760 km/hr. 
What would one have to know to compute its mass? 

26. Newlon's second law can be written FAt = 
A(mv). Use the second law to explain the following: 

(a) It is safer to jump into a fire net or a load of hay 
than onto the hard ground. 

(b) When jvimping down from some height, you 
should bend your knees as you come to rest, instead 
of keeping your legs stiff. 

(c) Hammer heads are generally made of steel rather 
than rubber. 

(d) Some cars have plastic bumpers which, 
temporarily defomed under impact, slowly return to 
their original shape. Others are designed to have a 
somewhat pointed front-end bumper. 



(S 






— ^^ /— \ 



37. A student in a physics class, having learned 
about elastic collisions and consei'vation laws, 
decides that he can make a self-propelled car. He 
proposes to fix a pendulum on a cart, using a 
"superball" as a pendulum bob. He fixes a block to 
the cart so that when the ball reaches the bottom of 
the arc, it strikes the block and rebounds elasticaUy. 
It is supposed to give the cart a series of bumps that 
propel it along. 

(a) Will his scheme work? (Assume the "superball " is 
perfectly elastic.) Give reasons for your answer. 

(b) What would happen if the cart had an initial 
velocity in the forward direction? 

(c) What would happen if the cart had an initial 
velocity in the backward direction? 



^I 



LD- 



\ ti 
i I' 



as. A police report of an accident describes two 
vehicles colliding (inelastically) at an icy intersection 
of country roads. The cars slid to a stop in a field 
as shown in tlie diagram. Suppose the masses of the 
cars are approximately the same. 

(a) How did the speeds of the two cars compare just 
before collision? 

(b) What information would you need in order to 
calculate the actual speeds of the automobiles? 

(c) What simplifying assumptions have you made in 
answering (b)? 

29. A person fires a gun horizontally at a target fixed 
to a hillside. Describe the changes of momentum to 
the person, the bullet, the target, and the earth. Is 
momentum conserved 

(a) when the gun is fired? 

(b) when the bullet hits? 

(c) during the bullet's flight? 

30. A billiard ball moving 0.8 m/sec collides with the 
cushion along the side of the table. The collision is 
head-on and can here be regarded as perfectly 
elastic. What is the momentum of the ball 

(a) before impact? 

(b) after impact? 

(c) What is the change in momentum of the ball? 

(d) Is moinentum conserved? 



CHAPTER 9 / STUDY GUIDE 



371 



sg 



31. Discuss conservation of momentum for the 
system shown in this sketch from Lc Petit Prince. 
What happens 

(a) if he leaps in the air? 

(b) if he runs around? 

32. (a) State the principle of consen'ation of kinetic 
energy. For what systems is this law applicable? 

(b) What does this principle tell you about the 
change in the total kinetic energy, and about the 
total initial and finid kinetic energies, of an isolated 
system of elastic interactions? 

(c) What is the total kinetic energy of a system of 
two carts with masses of 5 kg and 10 kg, traveling 
toward one another at 4 m/sec and 3 in/sec, 
respectively? 

(d) If the two carts in (c) rebound elastically, what is 
the total final kinetic energy? 

33. Fill in the blanks for the following motions: 







m 


V 


mv 


V2mv' 


Object 




(kg) 


m s 


kg ■ m/s 


kg ■ iW/^ 


baseball 




0.14 


30.0 


— 


— 


hockey pu 


ck 


— 


50.0 


8.55 


— 


superball 




0.050 


1.5 


— 


— 


light car 




1460 


— 


— 


1.79 X 10' 


mosquito 




— 


— 


2.0 X 10= 


4.0 X 10 '^ 


football 




— 


— 


— 


— 


player 













34. A system consists of three particles with masses 
of 4 g, 6 g, and 8 g traveling toward a single point 
with velocities of 20 cm/sec north, 3 cm/sec east, and 
10 cm/sec south. 

(a) Calculate the total initial mass, momentum, and 
kinetic energy of the system. 

(b) If the particles reach the point at the same 
instant, calculate where possible the final total mass, 
momentum, and kinetic energy of the system in 
each of the following situations: 111 the system is not 
closed and the collision is elastic; (2) tlie system is 
not closed and the collision is inelastic; (3) the 
system is closed and the collision is elastic; and (41 
the system is closed and the collision is inelastic. 

35. Two balls, one of which has three times the 
mass of the other, collide head-on, each moving with 




the same speed. The more massive ball stops; the 
other rebounds with twice its original speed. Show 
that both momentum and kinetic energy are 
conserved. 

36. A ball of mass m moving at speed v strikes, 
elastically, head-on, a second ball of mass 3ni which 
is at rest. Using the principle of conser\ation of 
momentum and kinetic energy, find the speeds of 
the two balls after collision. 

37. When one ball collides with a stationary ball of 
the same mass, the first ball stops and the second 
goes on with the speed the first ball had. The claim 
is made on p. 265 that this result is the only possible 
result that will be consistent with consen'ation of 
both momentum and kinetic energv'. (That is, if m^ 
— m,, and v^ = 0, then the result must be i/ = 
and I',/ = v^.l Combine the e(]uations that express 
the two conservation laws and show that this is 
actually the case. (Hint: RewTite the equations with m 
for m^ and m,/ and v„ = 0; solve the simplified 
momentum eciuation for i'/; siil)stitut(! in \.hv. 
sinipiilicd kiiuMic (;n(Mg\ e(|iiati()ii; soke for \,/.i 



272 



CHAPTER 9 / STUDY GUIDE 




1 



et^APTfej ] 




Energy 



10.1 Work and kinetic energv 

10.2 Potential energv 

10.3 Conservation of mechanical energy 

10.4 Forces that do no ivork 

10.5 Heat as energt' 

10.6 The steam engine and the Industrial Revolution 

10.7 The efficiency of engines 

10.8 Energy in biological systems 

10.9 Arriving at a general conservation law 

10.10 The laws of thermodji'namics 

10.11 Faith in the laivs of thermodynamics 



lO.l. I Work and kinetic energy 

In eveiyday language, pitching, catching, and running on the 
baseball field are "playing," while sitting at a desk, reading, 
writing, and thinking are "working." However, in the language of 
physics, "work" has been given a rather special definition, one 
that involves physical concepts of force and displacement 
instead of the subjective ones of reward or accomplishment. It is 
more closely related to the simple sense of effort or labor. The 
work done on an object is defined as the product of the force 
exerted on the object times the displacement of the object along 
the direction of the force. 

When you throw a baseball, you exert a large force on it while 
it moves forward for about 1 m. In doing so, you do a large 
amount of work. By contrast, in waiting or in turning the pages of 







CHAPTER 10 / ENERGY 



273 



SG 1 



Note that work you do on a box 
does not depend on how fa.st you 
do your job. 



The way d is defined here, the W 
= Fd is correct. It does not, how- 
ever, explicitly tell how to compute 
W if the motion is not in exactly the 
same direction as the force. The 
definition of d implies that it would 
be the component of the displace- 
ment along the direction of F; £ind 
this is entirely correct. 



Note that work is a scalar quantity'. 

The equation W = Fd implies that 
work is always a positive quantity. 
However, by convention, when the 
force on a body and its displace- 
ment are in opposite directions, 
the work is negative. This implies 
that the body's energy would be 
decreased. The sign convention fol- 
lows naturally from the more rig- 
orous definition of mechcUiiciil work 
as W = Fl cos 6, where 8 is the 
angle between F and 7. 



a book you exert only a small force over a short distance. This 
does not require much work, as the term "work" is understood 
in physics. 

Suppose you are employed in a factory to lift boxes from the 
flooi' straight upward to a conveyor belt at waist height. Here the 
language of common usage and that of physics both agree that 
you are doing work. If you lift two boxes at once, you do twice as 
much work as you do if you lift one box. If the conveyor belt 
were twice as high aboxe the (loor, you would do twice as much 
work to lift a box to it. The work you do depends on both the 
magnitude of the force you must exert on the box and the 
distance through which the box moves in the direction of the 
force. 

We can now define the work W done on an object by a force F 
as the product of the magnitude F of the force and the distance 
d that the object moves in the direction ofF while the force is 
being exerted; in symbols, 

W = Fd 

To lift a box weighing 100 N upward through 0.8 m requires 
you to apply a force of 100 N. The work you do on the box is 100 
N X 0.8 m = 80 N-m. 

From the definition of work, it follows that no w^ork is done if 
there is no displacement. No matter how hard \'ou push on a 
wall, no work is done if the wall does not move. Also, no work is 
done if the only motion is perpendicular to the dii^ection of the 
force. For example, suppose you are earning a book bag. You 
must pull up against the downward pull of gravity to keep the 
bag at a constant height. But as long as you are standing still you 
do no work on the bag. Even if you walk along with it steadily 
in a horizontal line, the only work you do is in moving it foiA\'ard 
against the small resisting force of the air. 

Work is a useftil concept in itself. The concept is most useful 
in understanding the concept of energy. There are a great many 
foiTHs of energy. A few of them will be discussed in this chapter. 
We will define them, in the sense of describing how they can be 
measured and how they can be expressed algebraicallv. We will 
also discuss how energy changes ft'om one form to another. The 
general concept of energy is veiy difficult to define. But to define 
some particular foiTiis of energy is easy enough. The concept of 
work helps greatly in making such definitions. 

The chief importance of the concept of work is that work 
represents an amount of energv' transformed fiom one form to 
another. For example, when you throw a ball you do work on it. 
You also transform chemical energv', which youi- bodv obtains 
from food and oxygen, into energy of motion of the ball. When 
you lift a stone (doing work on it), you transfomi chemical 
energ\' into graxitational potcMitial oncrgv. If vou icicase the 



274 



UNIT 3 / THE TRIUMPH OF MECHANICS 



stone, the earth pulls it downward (does work on it); gravitational 
potential energy is transformed into energy of motion. When the 
stone strikes the ground, it compresses the ground below it 
(does work on it); energy of motion is transfonned into heat. In 
each case, the work is a measure of how much energy is 
transferred. 

The form of energy we have been calling "energy of motion" is 
perhaps the simplest to deal with. We can use the definition of 
work, W = Fd, together vvdth Newton's laws of motion to get an 
expression of this form of energy. Imagine that you exert a 
constant net force F on an object of mass m. This force 
accelerates the object over a distance d from rest to a speed v. 
Using Newton's second law of motion, we can show in a few 
steps of algebra that 

Fd = YzmV" 



SG 2 



■> 



The details of this derivation are given on the first half of page 
276. 

Fd is the expression for the work done on the object by 
whatever exerted the force F. The work done on the object 
equals the amount of energy transformed from some form into 
energy of motion of the object. So Vzmv^ is the expression for the 
energy of motion of the object. The energy of motion of an object 
at any instant is given by the quantity VzmV^ at that instant and 
is called kinetic energy. The symbol KE is used to represent 
kinetic energy. By definition then, 

KE = Vznr/ 

Now it is clearer why Vznnv' instead of just m\r was used in 
Chapter 9: VzmV' relates directly to the concept of work and so 
provides a useful expression for energy of motion. 

The equation Fd — Vzmv^ was obtained by considering the 
case of an object initially at rest. In other words, the object had 
an initial kinetic energy of zero. The relation also holds for an 
object already in motion when the net force is applied. In that 
case, the work done on the object still equals the change in its 
kinetic energy: 

Fd = MKE) 

The quantity A(KE) is by definition equal to (Vzmv^)^,^,, 

- (V2mv");,^i(j3,. The proof of this general equation appears on the 

second half of page 276. 

Work is defined as the product of a force and a distance. 
Therefore, its units in the mks system are newtons x meters or 
neu4on-meters.A newton-meter is also called a joule (symbol J). 
The joule is the unit of work or of energ\/. 



The Greek word kinetos means 
"moving." 



The speed of an object must be 
measured relative to some refer- 
ence frame, so kinetic energy is a 
relative quantity also. See SG 3. 



SG 3-8 



The name of the unit of energy and 
work commemorates J. P. Joule, 
nineteentli-century English physi- 
cist famous for his experiments 
showing that heat is a form of en- 
ergy (see Sec. 10.7). There is no gen- 
eral agreement today whether the 
name should be pronounced like 
"jool" or like "jowl." The majority 
of physicists favor the former. 



CHAPTER 10 / ENERGY 



275 



Close Up I 

Doing Work on o Sled 

Suppose a loaded sled of mass m Is initially at 
rest on low-friction ice. You, wearing spiked shoes, 
exert a constant horizontal force F on the sled. The 
weight of the sled is balanced by the upward push 
exerted by the ice, so F is effectively the net force 
on the sled. You keep pushing, running faster and 
faster as the sled accelerates, until the sled has 
moved a total distance d. 

Since the net force F is constant, the acceleration 
of the sled is constant. Two equations that apply to 
motion starting from rest with constant acceleration 
are 



and 



V = at 



d = V2af 



Therefore, the work done in this case can be 
found from just the mass of the body and its final 
speed. With more advanced mathematics, it can be 
shown that the result is the same whether the force 
is constant or not. 

More generally, we can show that the change in 
kinetic energy of a body already moving is equal to 
the work done on the body. By the definition of 
average speed, 

d = vj 

If we consider a uniformly accelerated body whose 
speed changes from v^ to v, the average speed (i^^J 
during t is V2(v + vJ. Thus, 

V + V. 

d = — — ^ X t 




% 




„ 1-^ V^"T 9 



where a is the acceleration of the body, t is the time 
interval during which it accelerates (that is, the time 
interval during which a net force acts on the body), 
V is the final speed of the body, and d is the distance 
it moves in the time interval t. 

According to the first equation, t = vs. If we 
substitute this expression for t in the second equa- 
tion, we obtain 

d = Vzaf = Vza—, = V2— 
a a 

The work done on the sled is lA/ = Fd. From 
Newton's second law, F = ma. so 

W = Fd 

= ma X V2— 
a 

The acceleration cancels out, giving 
W = V2mv' 



Bythedefinitionof acceleration, a = Ai/ 1; therefore, 
t = Iv a = (v - vJ a. Substituting (v - v^) a for 
t gives 

V + v^ V - v^ 

d = : — - X 



2 a 

^ (V + V^) {V - v^) 

2a 

= ° 

2a 

The work (W) done is W = Fd, or, since F = ma, 

W = ma X d 

= ma X 

2a 

m , , 

= 2 (^ - ^0') 

= ^ 2m v^ - ^^2mv,^ 



276 



UMT 3 / THE TRIUMPH OF MECHANICS 



1. If a force F is everted on an object while the object moves a 
distance d in the direction of the force, the work done on the 
object is: 

(a) F (bj Fd (c) F/d (d) VzFd' 

2. The kinetic energy of a body of mass m moving at a speed v 
is: 

(a) Vzxnv (b) Vzm\r (c) mv" (d) 2mv^ (e) mV 



10.2 I Potential energ[^ 

As you saw in the previous section, doing work on an object can 
increase its kinetic energy. Work also can be done on an object 
without increasing its kinetic energy. For example, you might lift 
a book straight up at a small, constant speed, so that its kinetic 
energy stays the same. But you are still doing work on the book. 
By doing work you are using your body's store of chemical 
energy. Into what form of energy is it being transformed? 

The answer, as Leibniz suggested, is that there is "energy" 
associated with height above the earth. This energy is called 
gravitational potential energy. Lifting the book higher and higher 
increases the gravitational potential energy. You can see clear 
evidence of this effect when you drop the book. The gravitational 
potential energy is transformed rapidly into kinetic energy of fall. 
In general terms, suppose a force F is used to displace an object 
upwards a distance d, without changing its KE. Then, the 
increase in gravitationcd potential energy, AlPEIg,^^, is 

Potential energy can be thought of as stored energy. As the 
book falls, its gravitational potential energy decreases while its 
kinetic energy increases correspondingly. When the book reaches 
its original height, all of the gravitational potential energy stored 
during the lift vAW have been transfonned into kinetic energy. 

Many useful applications follow from this idea of potential or 
stored energy. For example, the steam hammer used by 
construction crews is driven by high-pressure steam ("pumping 
in " energy). When the hammer drops, the gravitational potential 
energy is converted to kinetic energy. Another example is the 
proposal to use extra available energy from electric power plants 
during low-demand periods to pump water into a high reservoir. 
When there is a large demand for electricity later, the water is 
allowed to run down and drive the electric generators. 

There are forms of potential energy other than gravitational. 
For example, if you stretch a rubber band or a spring, you 
increase its elastic potential energy. When you release the rubber 



lf= 



} — :'. 



■3 



h 



To lift the book at constant speed, 
you must exert an upward force F 
equal in magnitude to the weight 
F of the book. The work you do 

grav *^ 

in lifting the book through distance 
d is Fd, which is numerically equal 
to F d. See SG 9 and 10. 




A stretched bow contains elastic 
potential energy. When released, 
the resulting kinetic energy propels 
the arrow to the target. 



CHAPTER 10 / ENERGY 



277 



SG 11 
SG 12 
SG 13 




The work you have done on the 
earth-book system is equal to the 
energy you have given up from 
your store of chemical energy. 



band, it can deliver the stored energy to a projectile in the form 
of kinetic energy. Some of the work done in blowing up an 
elastic balloon is also stored as potential energy. 

Other forms of potential energy are associated with othei' kinds 
of forces. In an atom, the negatively charged electrons are 
attracted by the positixely charged nucleus. If an externally 
applied force pulls an electron away from the nucleus, the 
electric potential energy increases. If the election is pulled hack 
and moves toward the nucleus, the potential energy' decreases as 
the electron's kinetic energy increases. 

If two magnets are pushed together with north poles facing, 
the magnetic potential energy increases. When released, the 
magnets wdll move apart, gaining kinetic energy as they lose 
potential energy. 

Where is the potential energv' located in all these cases? It 
might seem at first that it "belongs" to the body that has been 
moved. But without the other object (the eailh, the nucleus, the 
other magnet) the work would not increase any potential form 
of energy. Rather, it would increase only the kinetic energy of the 
object on which work was done. The potential energy belongs 
not to one body, but to the whole system of interacting bodies! 
This is evident in the fact that the potential energy is available to 
any one or to all of these interacting bodies. For example, you 
could give either magnet all the kinetic energv' just by releasing 
one magnet and holding the other in place. Or suppose you 
could fix the book somehow to a hook that would hold it at one 
point in space. The earth would then "fall" up toward the book. 
Eventually the earth would gain just as much kinetic energy at 
the expense of stored potential energy as the book would if it 
were free to fall. 

The increase in gravitational potential energy "belongs" to the 
earth-book system, not the book alone. The work is done by an 
"outside" agent (you), increasing the total energy of the 
earth-book system. When the book falls, it is responding to 
forces exerted by one part of the system on another. The total 
energy of the system does not change; it is converted from PE to 
KE. This is discussed in more detail in the next section. 



3. If a stone of mass m falls a vertical distance d, pulled by its 
weight F^_ _^, = ma^ the decrease in gravitational potential 
energy is: 

(a) md (b) mag (c) ma^d (d) ¥21x10" (e) d 

4. When you compress a coil spring, you do work on it. The 
elastic potential energy: 

(a) disappears (b) breaks the spring (c) increases (d) decreases 

5. Tv\'o electrically charged objects repel one another. To 
increase the electric potential energv, you n^usl 



278 



UNIT 3 / THE TRIL iVlPH OF MECHANICS 



(a) make the objects move faster. 

(b) move one object in a circle around the other object. 

(c) attach a rubber band to the objects. 

(d) pull the objects farther apart. 

(e) push the objects closer together. 



10*3 I Conservation of mechanical energy 

In Sec. 10.1, you learned that the amount of work done on an 
object equals the amount of energy transfonned from one fonn 
to another. For example, the chemical energy of a muscle is 
transfonned into the kinetic energy of a thrown ball. This 
statement implies that the amount of energy involved does not 
change, only its form changes, rhis is particularly obvious in 
motions where no "outside" force is applied to a mechanical 
system. 

While a stone falls freely, for example, the gravitational 
potential energy of the stone-earth system is continually 
transfonned into kinetic energy. Neglecting air friction, the 
decrease in gravitational potential energy is, for any portion of 
the path, equal to the increase in kinetic energy. Consider a 
stone thrown upward. Between any two points in its path, the 
increase in gravitational potential energy equals the decrease in 
kinetic energy. For a stone falling or rising (without external 
forces such as friction), the only force applied is F The work 
done by this force is (wdth d positive for upward displacements) 



-^gravd 



= -AK£ 



This relationship can be rewritten as 

Ml 
or still more concisely as 



MKE) + A(f£)g,.^^. = 



MKE + PE„ 



= 



If (KE + f£grav' represents the total mechanical energy of the 
system, then the change in the system's total mechanical energy 
is zero. In other words, the total mechanical energy, A(KE + 
PE^^.J, remains constant; it is conserved. 

A similar statement can be made for a vdbrating guitar string. 
While the string is being pulled away ft^om its unstretched 
position, the string-guitar system gains elastic potential energy. 
When the string is released, the elastic potential energy 
decreases while the kinetic energy of the string increases. The 
string coasts through its unstretched position and becomes 
stretched in the other direction. Its kinetic energy then decreases 



The equations in this section are 
true only if friction is negligible. We 
shall extend the range later to in- 
clude friction, which can cause the 
conversion of mechanical energy 
into heat energy. 

SG 14 




CHAPTER 10 / ENERGY 



279 



Up to this point, we have alu'ays 
considered only changes in PE. 
There is some subtlety in defining 
an actual value of PE. See SG 15. 



SG 16 




During its contact with a golf club, 
a golf ball is distorted, as is shown 
in the high-speed photograph. As 
the ball moves away from the club, 
the ball recovers its normal spheri- 
cal shape, and elastic potential en- 
ergy is transformed into kinetic 
energy. 



as the elastic potential energy increases. As it vibrates, thei^ is a 
repeated transfoiTnation of elastic potential energy into kinetic 
energy and back again. The string loses some mechanical energy; 
for example, sound waves radiate away. Otherwise, the decrease 
in elastic potential energy over any part of the string's motion 
would be accompanied by an equal increase in kinetic energy, 
and vice versa: 



A(P£), 



= -MKE) 



In such an ideal case, the total mechanical energy (K£ + PE^.,.j^,J 
remains constant; it is conseived. 

Galileo's experiment with the pendulum (Sec. 3.5) can also be 
described in these terms. The gra\dtational potential energy is 
determined by the height to which the pendulum was originally 
pulled. That potential energy is converted to kinetic energy at 
the bottom of the swing and back to potential energy at the 
other side. Since the pendulum retains its initial energy, it will 
stop (KE — 0, PE — max) only when it returns to its initial 
height. 

You have seen that the potential energy of a system can be 
transformed into the kinetic energy of some part of the system, 
and vice versa. Suppose that an amount of work IV is done on 
part of the system by some external force. Then the energy of the 
system is increased by an amount equal to W. Consider, for 
example, a suitcase-earth system. You must do work on the 
suitcase to pull it away ft om the earth up to the second floor. 
This work increases the total mechanical energy of the 
earth-suitcase system. If you yourself are included in the system, 
then your internal chemical energy decreases in proportion to 
the work you do. Therefore, the total energy of the lifter + 
suitcase -f earth system does not change. 

The law of conservation of energy can be derived from 
Nev^on's laws of motion. Therefore, it tells nothing that could 
not, in principle, be computed directly ftom Newton's laws of 
motion. However, there are situations where there is simply not 
enough information about the forces involved to apply Newton's 
laws. It is in these cases that the law of conservation of 
mechanical energy demonstrates its usefulness. 

A perfectly elastic collision is a good example of a situation 
where we often cannot apply Newlon's laws of motion. In such 
collisions, we do not know and cannot easily measure the force 
that one object exerts on the other. We do know that during the 
actual collision, the objects distort one another. (See the 
photograph of the golf ball in the margin.) The distortions are 
produced against elastic forces. Thus, some of the combined 
kinetic energy of the objects is transformed into elastic potential 
energy as they distort one another. Then elastic potential energy 
is transformed back into kinetic energy as the objects separate. 
In an ideal case, both the objects and their surroundings are 



280 



UNIT 3 / THE TRIUMPH OF MECHANICS 



exactly the same after colliding as they were before. 

This is useful but incomplete knowledge. The law of 
conservation of mechanical energy gives only the total kinetic 
energy of the objects after the collision. It does not give the 
kinetic energy of each object separately. (If enough information 
were available, we could apply Newton's laws to get more 
detailed results, namely, the speed of each object.) You may 
recall that the law of conservation of momentum also supplies 
useful but incomplete knowledge. It can be used to find the total 
momentum, but not the individual momentum vectors, of elastic 
objects in collision. In Chapter 9, you saw how conservation of 
momentum and conseivation of mechanical energy together limit 
the possible outcomes of perfectly elastic collisions. For two 
colliding objects, these two restrictions are enough to give an 
exact solution for the two velocities after collision. For more 
complicated systems, conservation of energy remains important. 
Scientists usually are not interested in the detailed motion of 
every part of a complex system. They are not likely to care, for 
example, about the motion of every molecule in a rocket exhaust. 
Rather, they probably want to know only about the overall thiTist 
and temperature. These can be found from the overall 
conservation laws. 



• 6. As a stone falls frictionlessly, 

(a) its kinetic energy is conserved. 

(b) its gravitational potential energy is conserved. 

(c) its kinetic energy changes into gravitational potential 
energy. 

(d) no work is done on the stone. 

(e) there is no change in the total energy. 

7. In which position is the elastic potential energy of the 
vibrating guitar string greatest? In which position is its kinetic 
energy greatest? 

8. If a guitarist gives the same amount of elastic potential 
energy to a bass string and to a treble string, which one will 
gain more speed when released? (The mass of 1 m of bass 
string is greater than that of 1 m of treble string.) 

9. How would you compute the potential energy stored in the 
system shown in the margin made up of the top boulder and 
the earth? 

10,4 I Forces that do no ivork 

In Sec. 10.1, the work done on an object was defined as the 
product of the magnitude of the force F applied to the object 




CHAPTER 10 / ENERGY 



281 



The reason that your arm gets tired 
even though you do no work on the 
book is that muscles are not rigid, 
but are constantly relaxing and 
tightening up again. That requires 
chemical energy. When you carry 
a heavy load on your back or shoul- 
ders, the supporting force is mostly 
pro\ided by bones, not muscles. As 
a result, you can carry much bigger 
loads for greater distances. 



and the magnitude of the distance d in the direction of F 
through which the ohject moves while the force is being applied. 
In all the examples so far, the object moxed in the same 
direction as that of the force vector. 

Usually, the direction of motion and the direction of the force 
are not the same. For example, suppose you cany a book at 
constant speed horizontally, so that its kinetic energy does not 
change. Since there is no change in the book's energy, you are 
doing no work on the book Iby the definition of workl. You do 
apply a force on the book, and the book does move through a 
distance. But here the applied force and the distance are at right 
angles. You exert a vertical force on the book upwaid to balance 
its weight. But the book moves horizontally. Here, an applied 
force F is exerted on an object while the object moves at right 
angles to the direction of the force. Therefore, F has no 
component in the direction of d, and so the force does no work. 
This statement agrees entirely with the idea of woi'k as energy 
being transformed from one form to another. Since the book's 
speed is constant, its kinetic energy is constant. Since its 
distance from the earth is constant, its gravitational potential 
energy is constant. Therefore, there is no transfer of mechanical 
energy. 

A similar reasoning, but not so obvious, applies to a satellite in 
a circular orbit. The speed and the distance from the earth are 
both constant. Therefore, the kinetic energy and the gravitational 
potential energy are both constant, and there is no energy 
transfoiTiiation. For a circular orbit, the centripetal force vector is 
perpendicular to the tangential direction of motion at any 
instant. No work is being done. To put an artificial satellite into a 
circular orbit requires work. Once it is in oibit, however, the KE 
and PE stay constant, and no further work is done on the 
satellite. 

When the orbit is eccentric, the force vector is usually not 
perpendicular to the direction of motion. In such cases, energy is 
continually transformed between kinetic and gravitational 
potential fomis. The total energy of the system remains constant, 
of course. 

Situations where the net force is exactly perpendicular- to the 
motion are as rare as situations where the for^ce and motion are 
in exactly the same direction. What about the more usual case, 
involving some angle between the force and the motion? 



J£Cof/D ncxiR 










rif^T fuy-in 



282 



UMT 3 / THE TRIUMPH OF MECHAiVICS 



In general, the work done on an object depends on how far 
the body moves in the direction of the force. As stated before, the 
equation W — Fd properly defines work only if d is the distance 
moved in the direction of the force. The gravitationcd force F 
is directed down. So only the distance down determines the 
amount of work done by F^_^^, Change in gravitational potential 
energy depends only on change in height, near the earth's 
surface, at least. For example, consider raising a suitcase from 
the first floor to the second floor. The same increase in Pfig,^, of 
the suitcase-earth system occurs regardless of the path by which 
the suitcase is raised. Also, each path requires the same amount 
of work. 

More generally, change in P£g,^, depends only on change of 
position. The details of the path foflowed in making the change 
make no difference at all. The same is tnae for changes in elastic 
potential energy and electric potential energy. The changes 
depend only on the initial and final positions, and not on the 
path taken between these positions. 

An interesting conclusion follows from the fact that change in 
PE^^.^^. depends only on change in height. For example, consider a 
child on a slide. The gravitational potential energy' decreases as 
her altitude decreases. If frictional forces are vanishingly small, 
all the work goes into transfomiing P£ into KE. Therefore, the 
increases in KE depend only on the decreases in altitude. In 
other words, the child's speed when she reaches the ground will 
be the same whether she slides down or jumps off the top. A 
similar principle holds for satellites in orbit and for electrons in 
TV tubes. In the absence of friction, the change in kinetic energy 
depends only on the initial and final positions, and not on the 
path taken between them. This principle gives great simplicity to 
some physical laws, as you wdll see when you study gravitational 
and electric fields in Chapter 14. 



10. How much work is done on a satellite during each 
revolution if its mass is m, its period is T, its speed is v, and 
its orbit is a circle of radius R? 

11. Two skiers were together at the top of a hill. While one 
skier skied down the slope and went off the jump, the other 
rode the ski-lift down. Compare their changes in gravitational 
potential energy. 

12. A third skier went directly down a straight slope. How 
would this skier's speed at the bottom compare with that of 
the skier who went off the jump? 

13. No work is done when 

(a) a heavy bo^ is pushed at constant speed along a rough 
horizontal floor. 



If frictional forces also have to be 
overcome, additional work will be 
needed, and that additional work 
may depend on the path chosen, 
for example, whether it is long or 
short. 



SG 17 





CHAPTER 10 / ENERGY 



283 



See, for example, Elxperiment 3.11 
on mixing hot and cold liquids. 




Benjamin Thompson was born in 
Wohurn, Massachusetts, in 1753. 
After several years as a shopkeep- 
er's apprentice, he married a rich 
widow and moved to Concord (then 
called Rumfordj. During the Revo- 
lution, Thompson was a Tory; he 
left with the British army for Eng- 
land when Boston was taken bv the 
rebels. In 17H3, Thompson left Eng- 
land and ultimatelv settled in Ba- 
varia. There he designed fortifica- 
tions and built munitions, and 
served as an administrator. The 
King of Bavaria was sufficiently im- 
pressed to make him a Count in 
1790, and Thompson took the 
name Rumford. In 1 793 he re- 
turned to England and continued 
to work on scientific experiments. 
Rumford was one of the founders 
of the Royal Institution. In 1804 
he married Lavoisier's widow; the 
marriage was an unhappy one, and 
they soon separated. Rumford died 
in Trance in 1H14, leaving his f;state 
to institutions in the United States. 



(b) a iiiiil is Ivuumcrcd into n board. 

(c) there is no conipoiuutt of force parallel to the direction of 
motion. 

(d) there is no component of force perpendicular to the 
direction ofn}otion. 



10.^ I Heat as energy 



Suppose that a book on a table has been given a push and is 
sliding across the tabletop. If the surface is rough, it will exert a 
fairly large frictional force, and the book will stop quickly. Ihe 
book's kinetic energy wall rapidly disappear. No corresponding 
increase in potential energy will occur, since there is no change 
in height. It appears that, in this example, mechanical energy is 
not consented. 

However, close examination of the bocjk and the tabletop 
shows that they are wairmer than before. The disappearance of 
kinetic energy of the book is a(xx)mpanied by the appearance of 
heat. This suggests, but by no means proves, that the kinetic 
energy of the book was transformed into heat. If so, heat must be 
one form of energy^ This section deals with how the idea of heat 
as a form of energy gained acceptance during the nineteenth 
century. You wdll see how theory was aided by practical 
knowledge of the relation of heat and work. 

Until the middle of the nineteenth centuiy, heat was generally 
thought to be some kind of fluid, called caloric fluid. No heat is 
lost or gained overall when hot and cold bodies are mixed. 
(Mixing equal parts of boiling and nearly freezing water produces 
water at just about 50°C.) One could therefore conclude that the 
caloric fluid is conserAed in that kind of experiment. Some 
substances, like wood or coal, seem to lock up that "fluid" and 
can release it during combustion. 

Although the idea that the heat content of a substarnce is 
r'epresented by a quantity of conserAed fluid was an appai'ently 
useful one, it does not adequately explain some phenomena 
involving heat. Friction, for example, was known to produce heat. 
But it was difficult to understand how the conseiAation of ciiloric 
fluid applied to friction. 

In the late eighteenth century, while boring cannon for the 
Elector of Bavaria, Benjamin Thompson, Count Rumford, 
observed that a great deal of heat was generated. Some of the 
cannon sha\ings wer^e hot eriough to glow. Rumfbrti made some 
careful measurements by immersing the cannon in w ater- and 
measuring the rate at which the temperature rose. His results 
showed that so mirch heat evohed that [he cannon would haxe 
melted had it not been cooled. From rnaru' such experiments, 
Rumford concluded that heat is not a conserved fluid but is 
generated when work is done and continues to appear \\ ilhout 



284 



UNIT 3 / THE TKIl'MPH OF MECHANICS 



limit as long as work is done. His estimate of the ratio of heat to 
work was within an order of magnitude of the presently 
accepted value. 

Rumford's experiments and similar work by Davy at the Royal 
Institution did not convince many scientists at the time. The 
reason may have been that Rumford could give no clear 
suggestion of just what heat is, at least not in terms that were 
compatible with the accepted models for matter at that time. 

Nearly a half-centuiy later, James Prescott Joule repeated on a 
smaller scale some of Rumford's experiments. Starting in the 
1840's and continuing for many years. Joule refined and 
elaborated his apparatus and his techniques. In all cases, the 
more careful he was, the more exact was the proportionality of 
the quantity of heat (as measured by a change in temperature 
and the amount of work done). 

For one of his early experiments, Joule constructed a simple 
electric generator, which was driven by a falling weight. The 
electric current that was generated heated a wire. The ware was 
immersed in a container of water, which it heated. From the 
distance that the weight descended Joule calculated the work 
done (the decrease in gravitational potential energy). The product 
of the mass of the water and its temperature rise gave him a 
measure of the amount of heat produced. In another experiment, 
he compressed gas in a bottle immersed in water, measuring the 
amount of work done to compress the gas. He then measured 
the amount of heat given to the water as the gas grew hotter on 
compression. 

Joule's most famous experiments involved an apparatus in 
which slowly descending weights turned a paddle wheel in a 
container of water. Owang to the friction between the wheel and 
the liquid, work was done on the liquid, raising its temperature. 

Joule repeated this experiment many times, constantly 
improving the apparatus and refining his analysis of the data. He 
learned to take very great care to insulate the container so that 
heat was not lost to the room. He measured the temperature rise 
vvdth a precision of a small fraction of a degree. He even allowed 
for the small amount of kinetic energy the descending weights 
had when they reached the floor. 

Joule published his results in 1849. He reported: 

1st. That the quantity of heat produced by the fiiction of 
bodies, whether solid or liquid, is always proportional to the 
quantity of [energy] expended. And 2nd. That the quantity of 
heat capable of increasing the temperature of a pound of water 
... by 1° Fcihr. requires for its evolution the expenditure of a 
mechanical energy represented by the fall of 772 lb through the 
distance of one foot. 




James Prescott Joule (1818-1889). 
Joule was the son of a wealthy 
Manchester brewer. He is said to 
have become first interested in his 
arduous e^cperiments by the desire 
to develop more efficient engines 
for the family brewery. 



Joule used the word 'force " instead 
of "energy." The currently used sci- 
entific vocabulary was still being 
formed. 

This unit is called a British Thermal 
Unit (BTU). 



The first statement is the evidence that heat is a fomi of 
energy, contraiy to the caloric theoiy. The second statement 



CHAPTER 10 / EIVERGV 



285 



The metric system uses prefixes to 
specify some multiplication factors 
for the fundamental units. The fol- 
lowing table shows the common 
prefixes. 



Factor 



Prefix 



Symbol 



10 

10' 

10" 

10' 

10' 

10^^ 

10^' 

10" 

10"' 

10'- 

10"' 

10 ■ 

10 

10" 



deca- 
hecto- 
kilo- 
mega- 

giga- 

tera- 

peta- 

deci- 

centi- 

milli- 

micro- 

nano- 

pico- 

femto- 



da 

h 

k 

M 

G 

T 

P 

d 

c 

m 

M- 
n 

P 
f 



gives the numerical magnitude of the ratio he had found. Ihis 
ratio related a unit of mechanical enei'g\' I the foot-pound i and a 
unit of heat (the heat required to raise the tempeiatuie of 1 lb 
of water by 1° on the Fahrenheit scale). 

By the time Joule did his experiments, the idea of a caloric 
fluid seemed to have outlived its usefulness. The idea that heat is 
a form of energy was slowly being accepted. Joule's experiments 
were a strong argument in favor of lliat idea, ^'ou will look at its 
development more closely in Sec. 10. iJ. 

Until recently, it was traditional to measure heat in units based 
on temperature changes, and mechanical energy in units based 
on work. This made comparison of results in one kind of 
experiment easy, but it obscured the fundamental equivalence of 
heat and other forms of energy. All types of energy are expressed 
in joules IJ): ^^ 

1 J = 1 kg sec" = 1 N-m Inewton-meterl 
A 
However, you will often see energies expressed in terms of other 

units. A few of them are listed here. 



Unit Name 


Symbol 


Definition 


Conversion 


kilowatt hour 


kWh 


A watt (W) is 1 J per 


1 kWh = 3.60 






second, so 1 J = 1 W ■ 


MJ 






sec. A kWh is the amount 








of energy delivered in 








an hour if 1 kJ is 








delivered per second. 




Calorie 


Cal 


The energy required to heat 


4.18 kJ 


(or kilocalorie) 


(or kcal) 


1 kg of water by 1 C. 




British Thermal 


BTU 


The energy required to heat 


1.06 kJ 


Unit 




0.454 kg by 0.556=C. 





14. When a book slides to a stop on the horizontal rough 
surface of a table 

(a) the kinetic energy of the book is transformed into potential 
energy. 

(b) heat is transformed into mechanical energ\^. 

(c) the kinetic energy of the book is transformed into heat 
energ}^. 

(d) the momentum of the book itself is conscned. 

15. The kilocalorie is 

(a) a unit of temperature. 

(b) a unit of energy. 

(c) 1 kg of water at 1°C. 

16. In Joule's paddle-wheel esperimenl, was all the change of 
gr'avitatioi}al potential energy used to heat the water.' 



286 



UMT 3 / THE TRIL'VIPH OF MECHAMCS 



10.6 I The steam engine and the Industrial 
Revolution 

Until about 200 years ago, most work was done by people or 
animals. Work was obtained from wind and water also, but these 
were generally unreliable as sources of energy. For one thing, 
they were not always available when and where they were 
needed. In the eighteenth century, miners began to dig deeper 
and deeper in search of a greater coal supply. Water tended to 
seep in and flood these deeper mines. The need arose for an 
economical method of pumping water out of mines. The steam 
engine was developed initially to meet thisveiy practical need. 

The steam engine is a device for converting the energy of some 
kind of fuel into heat energy. For example, the chemical energy 
of coal or oil, or the nuclear energy of uranium, is converted to 
heat. The heat energy in turn is converted into mechanical 
energy. This mechanical energy can be used directly to do work, 
as in a steam locomotive, or transformed into electrical energy. 
In typical twentieth-century industrial societies, most of the 
energy used in factories and homes comes from electrical energy. 
Falling water is used to generate electricity in some parts of the 
country. But steam engines still generate most of the electrical 
energy used in the United States today. There are other heat 
engines, such as internal combustion engines, for example. The 
steam engine remains a good model for the basic operation of 
this whole family of engines. 

Since ancient times, it has been known that heat can be used 
to produce steam, which can then do mechanical work. The 
aeolipile, invented by Heron of Alexandria about 100 ad , worked 
on the principle of Newton's third law. (See margin.) The rotating 
lawn sprinkler works the same way except that the driving force 
is water pressure instead of steam pressure. 

Heron's aeolipile was a toy, meant to entertain rather than to 
do any useful work. Perhaps the most "useful" application of 
steam to do work in the ancient world was another of Heron's 
inventions. This steam-driven device astonished worshippers in a 
temple by causing a door to open when a fire was built on the 
altar. Not until late in the eighteenth century, however, were 
commercially successful steam engines invented. 

The first commercially successful steam engine was invented 
by Thomas Savery (1650-1715), an English military engineer. In 
Savery's engine, water is lifted out of a mine by alternately filling 
a tank with high-pressure steam, driving the water up and out 
of the tank, and then condensing the steam, drawing more water 
into the tank. 

Unfortunately, inherent in the Saveiy engine's use of high- 
pressure steam was a serious risk of boiler or cylinder 
explosions. This defect was remedied by Thomas Newcomen 
(1663-1729), another Englishman. Newcomen invented an engine 







CHAPTER 10 / ENERGY 



287 



/A 




A model of Heron's aeolipile. 
Steam produced in the boiler es- 
capes through the nozzles on the 
sphere, causing the sphere to ro- 
tate. 



Stec 




that used steam at lower pressure. His engine was superior in 
other ways also. For example, it could raise loads other than 
water. Instead of using the steam to force water into and out of a 
cylinder, Newcomen used the steam to force a piston hack and 
forth. The motion of the piston could then be used to drive a 
pump or other engine. 

The Newcomen engine was widely used in Britain and other 
European countries throughout the eighteenth century. By 
modern standards, it was not a very good engine. It burned a 
large amount of coal but did only a small amount of work at a 
slow, jerky rate. But the great demand for machines to pump 
water from mines produced a good market, even for Newcomen's 
uneconomical engine. 

The work of Scotsman James Watt led to a greatly improved 
steam engine. Watt's father was a carpenter who had a 
successful business selling equipment to shipowners. Watt was 
in poor health much of his life and gained most of his early 
education at home. In his father's attic workshop, Watt 
developed considerable skill in using tools. He wanted to become 
an insti-ument maker and went to London to learn the trade. 
Upon his return to Scotland in 1757, he obtained a position as 
instrument maker at the University of Glasgow. 

In the winter of 1763-1764, Watt was asked to repair a model 
of Newcomen's engine that was used for demonstration lectures 
at the university. This assignment had immense worldwide 
consequences. In acquainting himself with the model, Watt was 
impressed by how^ much steam was required to iTjn the engine. 
He undertook a series of experiments on the behavior of steam 
and found that a major problem was the temperature of the 
cylinder walls. Newcomen's engine wasted most of its heat in 
w^arming up the walls of its cylinders. The walls were then 
cooled again eveiy time cold water was injected to condense the 
steam. 

Early in 1765, Watt remedied this wasteful defect by devising a 
modified type of steam engine. In retrospect, it sounds like a 
simple idea. After pushing the piston up, the steam was admitted 
to a separate container to be condensed. With this system, the 
cylinder could be kept hot all the time, and the condenser could 
be kept cool all the time. 

The separate condenser allowed huge fuel savings. Watt's 
engine could do twice as much work as Newcomen's with the 
same amount oi fuel. Watt also added many other refinements, 
such as automatically controlled valves that were opened and 
closed by the reciprocating action of the piston itself, as well as a 
governor that controlled the amount of steam reaching the 
engine, to maintain a constant speed for the engine. 

This idea, of using pait of the output of the process to regulate 
the process itself, is call jd /eed/jac/c. It is an essential pail of the 



288 



UNIT 3 / THE TRIUMPH OF MECHAMCS 



;:K-V 



^//i( HI^GJJVB %r ^/{a/'/^/fa /fa/f/'/^/f^y/ ^7/^i77ivr ?/u?</,'^ ' //Fiic^^J 




design of many modern mechanical and electronic systems. 

Watt and his associates were good businessmen as well as 
good engineers. They made a fortune manufacturing and selling 
the improved steam engines. 

Watt's inventions stimulated the development of engines that 
could do many other jobs. Steam drove machines in factories, 
railway locomotives, steamboats, and so forth. Watt's engine gave 
an enormous stimulus to the growth of industry in Europe and 
America. It thereby helped transform the economic and social 
structure of Western civilization. 

The widespread development of engines and machines 
revolutionized mass production of consumer goods, 
construction, and transportation. The average standard of living 
in Western Europe and the United States rose sharply. It is 
difficult for most people in the industrially "developed" countries 
to imagine what life was like before the Industrial Revolution. But 
not all the effects of industrialization have been beneficial. The 
nineteenth-century factory system provided an opportunity for 



Above, a contemporary engraving 
of a working Newcomen steam en- 
gine. In July 1698, Savery was 
granted a patent for "A new inven- 
tion for raising of water and occa- 
sioning motion to all sorts of mill 
work by the impellant force of fire, 
which wilt be of great use and ad- 
vantage for drayning mines, serving 
townes with water, and for the 
working of all sorts of mills where 
they have not the benefitt of water 
nor constant windes." The patent 
was good for 35 years and pre- 
vented Newcomen from making 
much money from his superior en- 
gine during this period. 



CHAPTER 10 / ENERGY 



289 




The actual model of the Newcomen 
engine that inspired Watt to con- 
ceive of the separation of con- 
denser and piston. 



With valve A open and valve B 
closed, steam under pressure en- 
ters the cylinder and pushes the 
piston upward. When the piston 
nears the top of the cylinder, valve 
A is closed to shut off the steam 
supply. Then valve B is opened, so 
that steam leaves the cylinder and 
enters the condenser. The con- 
denser is kept cool by water flow- 
ing over it, so the steam conden- 
ses. As steam leaves the cylinder, 
the pressure there decreases. At- 
mospheric pressure (helped by the 
inertia of the flywheel) pushes the 
piston down. When the piston 
reaches the bottom of the cylinder, 
valve B is closed, and valve A is 
opened, starting the cycle again. 
Valves A and B are connected to 
the piston directly, so that the mo- 
tion of the piston itself operates 
them. 



some greedy and ci\iel employers to treat workers almost like 
slaves. These employers made huge profits, while keeping 
employees and their families on the edge of starvation. This 
situation, which was especially serious in England early in the 
nineteenth century, led to demands for reform. New laws 
eventually eliminated the worst excesses. 

More and more people left the farms, eithei- xoluntaiily or 
forced by poveit}' and new land laws, to work in factories. 
Conflict grew intense between the working class, made up of 
employees, and the middle class, made up of employers and 
professionals. At the same time, some artists and intellectuals 
began to attack the new tendencies of their society. They saw 
this society becoming increasingly dominated by commerce, 
machineiy, and an emphasis on material goods. In some cases, 
they confused research science itself with its technical 
applications (as is still done today). Scientists were sometimes 
accused of explaining away all the awesome mysteries of nature. 
These artists denounced both science and technology, while 
often refusing to learn anything about them. A poem by William 
Blake contains the questions: 



VALVE 
THROTTLE ^ 





ir^^=®1__i=^ 



VALVE 

B 



CY UNDER 



BOILER 



CONDENSCR 



FLYWHEEL 



cohdensed 



Steam 



290 



UMT 3 / THE TRIUMPH OF MECHANICS 




Ccr,nectcd 
to flywheel 

And did the Countenance Divine 

Shine forth upon our clouded hUls? 
And was Jerusalem builded here 

Among these dark Satanic mills? 

Elsewhere, Blake advised his readers "To cast off Bacon, Locke, 
and Nevvi^on." John Keats was complaining ahout science when 
he included in a poem the line: "Do not all channs fly / At the 
mere touch of cold philosophy?" These attitudes are part of an 
old tradition, going back to the ancient Greek opponents of 
Democritus' atomism. Galilean and Newtonian physics also were 
attacked for distorting values. The same type of accusation can 
still be heard today. 

Steam engines are no longer wddely used as direct sources of 
power in industry and transportation. Indirectly, however, steam 
is still the major source of power. The steam turbine, invented 
by the English engineer Charles Parsons in 1884, has now largely 
replaced older kinds of steam engines. At present, steam turbines 
drive the electric generators in most electric-power stations. 
These steam-run generators supply most of the power for the 
machinery of modern civilization. Even in nuclear power 
stations, the nuclear energy is generally used to produce steam, 
which then drives turbines and electric generators. 

The basic principle of the Parsons turbine is simpler than that 
of the Newcomen and Watt engines. A jet of high-pressure steam 
strikes the blades of a rotor, driving the rotor around at high 
speed. The steam expands after passing through the rotor, so the 
next rotor must be larger. This accounts for the characteristic 
shape of turbines. Large electric-power station turbines, such as 
"Big Allis" in New York City, use more than 500,000 kg of steam 
an hour and generate electrical energy at a rate of 150 million 
joules per second. 

The usefulness of an engine for many tasks is given by the rate 
at which it can deliver energy, that is, by its power. The unit of 
power is the watt, symbol W. It is defined as 1 W = 1 J/sec 



Waff's "governor." If the engine 
speeds up for some reason, the 
hea\y balls swing out to rotate in 
larger circles. They are pivoted 
at the top, so the sleeve below is 
pulled up. The cam that fits against 
the sleeve is therefore also pulled 
up; this forces the throttle to move 
donn and close a bit. The reduc- 
tion in steam reaching the engine 
thus slows it down again. The op- 
posite happens when the engine 
starts to slow down. The net result 
is that the engine tends to operate 
at nearly a stable level. 




A steam locomotive from the early 
part of the twentieth century. 




CHAPTER 10 / ENERGY 



291 




Watt, of course, used pounds and 
feet to express these results. 

In some contexts, the horsepower 
is defined as 73o.36 \V. More often, 
it is defined as 745.56 VV. This am- 
biguity of traditional units is one of 
the reasons for replacing them with 
metric ones. 



Matthew Boulton (Watt's business 
partner) proclaimed to Boswell (the 
biographer of Samuel Johnson): "I 
sell here, Sir, what all the world 
desires to have: POWER! " 



As with energv', there are many common units of power v\ith 
traditional definitions. Before the steam engine, the standard 
source of power was the workliorse. Watt, in order to rate his 
engines in a unit people could understand, measured the power 
output of a horse. He found that a strong horse, working steadily, 
could lift an object of 75 kg mass, which weighed about 10 N, at 
a rate of about 1 m/sec. The horse in this case thus did work at a 
rate of about 750 W. The "horsepower" unit is still used, but its 
value is now given by definition (see margin), not by experiment. 

SOME POWER RATINGS 

Person turning a crank 0.06 h.p. 50 W 

Overshot waterwheel 3 2 kW 

Turret windmill 10 7 kW 

Savery steam engine (1702) 1 0.7 kW 

Newcomen engine (1732) 12 9 kW 

Smeaton's Long Benton engine (1772) 40 30 kW 

Watt engine (of 1778) 14 10 kW 

Cornish engine for London water works (1837) 135 100 kW 

Electric power station engines (1900) 1,000 0.7 MW 

Nuclear power station turbine (1970) 300,000 200 MW 

(Adapted from R. J. Forbes, in C. Singer ef a/., History of Technology.) 



292 



UNIT 3 / THE TRIUMPH OF MECHANICS 




Richard Trevithick's railroad at 
Euston Square, London, 1809. 



TRE>^1TH1CKS, 

rORTABLB STEAM ENGIKE. 



Catcb. jne who can , 





Mechanic al I'mver Subdaing 
Ankoal Speed . 



/ I) y 




A nineteenth-century French steam 
cuhivator. 



The "Charlotte Dundas," the first 
practical steamboat, built by Wil- 
liam Symington, an engineer who 
had patented his own improved 
steam engine. It was tried out on 
the Forth and Clyde Canal in 1801. S 




CHAPTER 10 / ENERGY 



293 




Above, a 200, 000 -kilowatt turbine 
being assembled. Notice the thou- 
sands of blades on the rotor. 



A schematic model of an inte- 
grated energy supply. 



9 17. The purpose of the separate condenser in Watts steam 
engine is to 

(a) save the water so it can be used again. 

(b) save fuel by not having to cool and reheat the cylinder. 

(c) keep the steam pressure as low as possible. 

(d) make the ei^gine more compact. 

18. The history of the steam engine suggests that the social 
and economic effects of technology are 

(a) always beneficial to everyone. 

(b) mostly undesirable. 

(c) unimportant one way or another. 

lO.T I The efficiency of engines 

Joule's finding a value for the "mechanical equivalent of heat" 
made it possible to describe engines in a new way. The concept 
of efficiency applies to an engine or to any device that transforms 
energy from one form to another. Efficiency is defined as the 
percentage of the input energy that appears as useful output. 
Since energy is conserved, the gr^eatest possible efficiency is 100% 
when all of the input energy appears as useful output. Obviously, 
efficiency must be considered as seriously as power output in 



C^RAVITATION 




EARTM CORE 



EA6TM CRUSTS 



OCEAN /C 




MECHANICAL 



aEOI.06.iCAL' 
iroRASE 



INTEGRATED ENCRSV SuPfLY AV>OEl. 




Electrical^ 



-^ 



J. MATES lAU 

V use. 



294 



UNIT 3 / THE TRIUMPH OF XHiCHAMCS 



designing engines. Fuel is, after all, a part of the cost of running 
an engine, and the more efficient an engine is, the cheaper it is 
to run. 

Watt's engine was more efficient than Newcomen's, which in 
turn was more efficient than Saveiy's. Is there any limit to 
improvements in efficiency? 

The law of energy conseivation clearly imposes a limit of 100%. 
No engine can put out more work than is put into it. Even before 
that law had been formulated, a young French engineer, Sadi 
Carnot, established that there is in practice an even lower limit. 
The reasons for this limit are just as fundamental as the law of 
energy conservation. 

Carnot started vvath the postulate that heat does not by itself 
flow from a cold body to a hot one. It then follows that if heat 
does flow from cold to hot, some other change must take place 
elsewhere. Some work must be done. Using an elegant argument, 
which is summarized on page 300, Carnot showed that no 
engine can be more efficient than an ideal, reversible engine and 
that all such engines have the same efficiency. 

Since all reversible engines have the same efficiency, one has 
only to choose a simple engine and calculate its efficiency to find 
an upper limit to the efficiency of any engine. Carnot did the 
calculation and found that the ratios of heat and work in a 
reversible engine depend only on the temperature of the hot 
substance from which the engine obtains its heat and on the 
temperature of the cold substance that extracts the waste heat 
from the engine. The temperatures used in this case are called 
absolute, or Kelvin ; temperatures: 

T (absolute, in °K) = T (Celsius, in °C) + 273 

On the Kelvin scale, water freezes at 273°K. Absolute zero (0°K) is 
-273°C. 

The expression found by Carnot for the efficiency of reversible 
engines is 




Sadi Carnot (1796-1832). Son of 
one of Napoleon's most trusted 
generals, Sadi Carnot was one of 
the new generation of e}cpert ad- 
ministrators who hoped to produce 
a new enlightened order in Europe. 
He died of cholera in Paris at the 
age of 36. 



See also Sec. 11.5. 



hot body, temperature Tj 
heat in, H^ 

I engine | >work, W 

heat out, H, = H, 



W 



cold body, temperature T, | 

work out W 

heat in H, 



efficiency 



Although Carnot did not wiite the 
formula this way, we are making 
use of the fact that heat and energy 
are equivalent. 






CHAPTER 10 / ENERGY 



295 



SG 24-29 The relation involved can also be written as follows: 



You can feel some of thi waste heat 
by feeling the exhaust from a car. 



In the MKS system of units, the 
coefficient of performance yields a 
"pure" number (without units!. 
Many American engineers measure 
heat in BTL' and electric energ\' in 
kilowatt-hours (kVVhl, so that coef- 
ficients of performance for air con- 
ditioners or refrigerators are often 
rated in BTU/kWh. 



See Sec. 10.10. 



SG 25 



H, 
T. 



W 



T. 



This result is called Carnot's theorem. Notice that unless T^ is 
zero, an unattainably low temperature, no engine can have an 
efficiency of 1 (or 100%); that is, every engine must return some 
"waste" heat to the outside before returning to get more energy 
from the hot body. 

In steam engines, the "hot body" is the steam fresh from the 
boiler, and the waste heat is extracted from the condenser. In an 
internal combustion engine (a car engine, for example), the hot 
body is the gas inside the cylinder just as it explodes, and the 
cold substance is the exhaust. Any engine that derives its 
mechanical energy from heat must also be cooled to remove the 
"waste" heat. If there is any friction, or other inefficiency, in the 
engine, it will add further heat to the waste and reduce the 
efficiency to below the theoretical limit. 

A refrigerator or air conditioner is also called a "heat engine." 
It uses work (in the form of electrical or mechanical power) to 
move heat fiom a cold body (from inside the freezing 
compartment! to a hot one (the outside room). Carnot's relations 
also provide an upper limit to how much heat can be extracted 
by such an engine for a given amount of work, or in practical 
terms, for how big your electric bill will be. 



hot bodv, T, 



heat out, H^ = W -\- H^ 

I engine ] < work, W 

heat in, H, 



T 



cold body, T, 



coefficient of performance = 



heat in H, 



work in W 



T, - T., 



The generalization of Carnot's theorem is now known as the 
second law of thermodynamics. This law is recognized as one of 
the most poweiful laws of physics. Even in simple situations it 
can help explain natural phenomena and the fundamental limits 
of technolog\'. 

1 he coefficient of performance of an air conditioner depends 
on the reciprocal of the temperature difference between the 
inside of a house and the outside. The bigger that difference, the 
moi-e work it will take to move the same amount of heat tmm 
inside to outside. An air conditioner mounted in the sun 
therefore needs more electricity than one mounted in the shade 
on the same house. 



296 



UNIT 3 / THE TRIUMPH OF MECHANICS 



If you bum oil at home, the furnace requires some inefficiency 
to burn cleanly, so some heat is lost out the chimney. (Also, a 
very efficient furnace would be unaffordably expensive.) Usually 
about half the energy content of the oil is converted to useful 
heat in a house. 

If you install "flameless electric heat," which works just like an 
electric blanket or toaster put in under the floor, the power 
company has to burn oil in a boiler, use the steam to generate 
electricity, and deliver the electricity to your home. Because 
metals melt above a certain temperature and because the cooling 
water never gets below freezing, Carnot's theorem makes it 
impossible to make the efficiency greater than about 0.6. Since 
the power company's boiler also loses some of its energy out the 
chimney, and since the electricity loses some of its energy on the 
way from the power plant, only about one-quarter to one-third 
of the energy originally in the oil actually makes it to your home. 
Obviously, electric heating wastes a lot of oil or coal. 

Because of the limits placed by Carnot's theorem on heat 
engines, it is sometimes important not only to give the actual 
efficiency of a heat engine but also to specify how close it comes 
to the maximum possible. One freezer might have a much larger 
coefficient of perfomiance than another, but only because it does 
not operate at as low a temperature as the other. 

The more carefully you look at a process, the more information 
is seen to be important. At first, you could probably be satisfied 
with any sort of engine. A closer examination will lead you to 
understand the equivalence of heat and energy, so that energy 
use and efficiency become important criteria in choosing an 
engine. Carnot's investigation showed the importance not only of 
energy but also of temperature. 



The coldest temperature feasible 
for T^ is about 280°K. (Why?) The 
hottest possible temperature for T^ 
is about 780°K. So the maximum 
efficiency is 0.64. 



• 19. The efficiency of a heat engine is the ratio of 

(a) the work output to the heat input. 

(b) the work output to the heat output. 

(c) the heat output to the heat input. 

20. A heat engine is most efficient when it works between 
objects that have 

(a) a large temperature difference. 

(b) a small temperature difference. 

(c) a large size. 

10.8 I Energy in biological systems 

All living things need a supply of energy to maintain life and to 
carry on their normal activities. Human beings are no exception; 



CHAPTER 10 / ENERGY 



297 



Carbohydrates are molecules made 
of carbon, hydrogen, and oxygen. 
A simple example is the sugar glu- 
cose, the chemical formula for 
which is C H. 0„. 



?y • t /. -• - VI ..( » ' -' .' ;'/ ;*>",- 









f 



:::^im- 



■^:^^. 



f .S^^^-^'C- 



■4 



Electron micrograph of an energy- 
converting mitochondrion in a bat 
cell (200,000 times actual size). A 
chloroplast, which is the part of a 
plant cell that converts CO^ and 
HJD to carbohydrates, looks very 
much the same, e^ccept it is green, 
not colorless. 



like all animals, we depend on food to supply us with energy. 

Most human beings are omnivores; that is, they eat both 
animal and plant materials. Some animals are herbivores, eating 
only plants, while others are carnivores, eating only animal flesh. 
But iill animals, even carnivoi'es, ultimately obtain their food 
energy fiom plant material. The animal eaten by the lion has 
previously dined on plant material or on another animal that has 
eaten plants. 

Green plants obtain energy from sunlight. Some of that energy 
is used by the plant to perform the functions of life. Much of the 
energy is used to make carbohydrates out of water (H,0) and 
carbon dioxide (CO J. The energy used to synthesize 
carbohydrates is not lost; it is stored in the carbohydrate 
molecules as chemical energv'. 

The process by which plants synthesize carbohydrates is 
called photosynthesis. This process is still not completely 
understood, and research in this field is lively. The synthesis 
takes place in many small steps, and many of the steps are well 
understood. It is conceivable that scientists may learn how to 
photosvnthesize carbohydrates without plants, thus producing 
food economically for the rapidly increasing world population. 
The overall process of producing carbohydrates (the sugar 
glucose, for example) by photosynthesis can be represented as 
follows: 

carbon dioxide + water + energ\' —> glucose -t- oxygen 

The energy stored in the glucose molecules is used by the 
animal that eats the plant. This energy maintains the body 
temperature, keeps the heart, lungs, and other organs operating, 
and enables various chemical reactions to occur in the body. The 
animal also uses the energy to do work on external objects. The 
process by which the energy stored in sugar molecules is made 
available to the cell is very complex. It takes place mostly in tiny 
bodies called mitochondria, which are found in all cells. Each 
mitochondrion contains enzymes which, in a series of about 10 
steps, split glucose molecules into simpler molecules. In another 
sequence of reactions, these molecules are oxidized (combined 
with oxygen), thereby releasing most of the stoi'ed energ)' and 
forming carbon dioxide and water: 

glucose + oxygen -^ carbon dioxide + water + energv' 

Proteins and fats are used to build and restore tissue and 
enzymes, and to pad delicate organs. They also can be used to 
provide energy. Both proteins and fats can enter into chemical 
reactions that produce the same molecules as the split 
carbohydrates. From that point, the energy-releasing process is 
the same as in the case of carbohydrates. 

The released energy is used to change a molecule called 
adenosine diphosphate i ADPi into adenosine triphosphate (ATPi. 



29H 



UNIT 3 / THE TRIUMPH OF MECHANICS 



In short, chemical energy originally stored in glucose molecules 
in plants is eventually stored as chemical energy in ATP 
molecules in animals. The ATP molecules pass out of the 
mitochondrion into the body of the cell. Wherever energy is 
needed in the cell, it can be supplied by an ATP molecule. As it 
releases its stored energy, the ATP changes back to ADP. Later, 
back in a mitochondrion, the ADP is reconverted to energy-rich 
ATP. 

The overall process in the mitochondrion involves breaking 
glucose, in the presence of oxygen, into carbon dioxide and 
water. The energy released is transferred to ATP and stored there 
until needed by the animal's body. 

The chemical and physical operations of the living body are in 
some ways like those of an engine. Just as a steam engine uses 
chemical energy stored in coal or oil, the body uses chemical 
energy stored in food. In both cases, the fuel is oxidized to 
release its stored energy. The oxidation is vigorous in the steam 
engine, and gentle, in small steps, in the body. In both the steam 
engine and the body, some of the input energy is used to do 
work; the rest is used up internally and eventually "lost " as heat 
to the surroundings. 

Some foods supply more energy per unit mass than others. 
The energy stored in food is usually measured in Calories. 
However, it could just as well be measured in joules. The table 
on page 302 gives the energy content of some foods. (The 
"Calorie" or "large calorie" used by dieticians is identical to the 
kilocalorie of chemists.) 

Much of the energy you obtain from food keeps your body's 
internal "machinery " running and keeps your body vv^arm. Even 
when asleep, your body uses about 1 Cal every minute. This 
amount of energy is needed just to keep alive. 

To do work, you need more energy. Yet only a fraction of this 
energy can be used to do work; the rest is wasted as heat. Like 
any engine, the body of humans or other animals is not 100% 
efficient. Its efficiency when it does work varies with the job and 
the physical condition and skill of the worker. Efficiency 
probably never exceeds 25% and usually is less. Studies of this 
sort are carried out in bioenergetics, one of many fascinating and 
useful fields where physics and biology overlap. 

The table on page 302 gives the results of experiments done in 
the United States to determine the rate at which a healthy young 
person of average build and metabolism uses energy in various 
activities. The estimates were made by measuring the amount of 
carbon dioxide exhaled. Thus, they show the total amount of 
food energy used, including the amount necessary just to keep 
the body functioning. 

According to this table, if you did nothing but sleep for eight 
hours a day and lie quietly the rest of the time, you would still 



The chemical energy stored in food 
can be determined by burning the 
food in a closed container im- 
mersed in water and measuring the 
temperature rise of water. 



SG 29-31 



CHAPTER 10 / ENERGY 



299 



mose up I 



Cornot's Proof 



Carnot's proof of maximum efficiency of ideal, 
reversible engines starts with the premise that when 
a cold object is in touch with a warmer one, the cold 
object does not spontaneously cool itself further and 
so give more heat to the warm object. However, an 
engine placed between the two bodies can move 
heat from a cold object to a hot one. Thus, a re- 
frigerator can cool a cold bottle further, ejecting heat 
into the hot room. You will see that this is not simple. 
Carnot proposed that during any such experiment, 
the net result cannot be only the transfer of a given 
quantity of heat from a cold body to a hot one. 

The engines considered in this case all work in 
cycles. At the end of each cycle, the engine itself 
is back to where it started. During each cycle, it has 
taken up and given off heat, and it has exerted 
forces and done work. 

Consider an engine, labeled R in the figure, 
which suffers no internal friction, loses no heat be- 
cause of poor insulation, and runs so perfectly that 
it can work backwards in exactly the same way as 
forwards (Fig. A). 



Hoi" Object 



Hot Object 



i 
® 

i 



H, 



H2 



>W 







Hi 

H2 



Cold Object 



Cold Object 



Hot 


H'l 




{"7\ 




i 


H2 




cold 





Now suppose someone claims to have invented 
an engine, labeled Z in the next figure, which is 
even more efficient than the ideal engine R. That 
is, in one cycle it makes available the same amount 
of work, W, as the R engine does, but takes less 
heat energy, /-/', from the hot object to do it (/-/', 
< /-/,). Since heat and energy are equivalent and 
since Hg = H, - Wand H\ = H\ - W, it will also 
be true that H\ < H^ (Fig. B). 

Suppose the two engines are connected so that 
the work from one can be used to drive the other. 
For example, the Z engine can be used to make 
the R engine work like a refrigerator (Fig. C). 



300 



LMIT 3 / THE TRIUMPH OF MECHANICS 



At the end of one cycle, both Z and R are back 
where they started. No work has been done; the Z 
engine has transferred some heat to the cold object; 
and the R engine has transferred some heat to the 
hot object. The net heat transferred is H, - H\, 
and the net heat taken from the cold object is H^ 
- H\. These are, in fact, the same: 

H^- H[ = {H, - W) - {H\ - W) 
= H, - H\ 

Because Z is supposed to be more efficient than R, 
this quantity should be positive; that is, heat has 
been transferred from the cold object to the hot 
object. Nothing else has happened. But, according 
to the fundamental premise, this is impossible. 

The only conclusion is that the Z engine was im- 
properly "advertised" and that it is either impossible 
to build or in fact it is less efficient than R. 

As for two different reversible engines, they must 
have the same efficiency. Suppose the efficiencies 
were different; then one would have to be more 
efficient than the other. What happens when the 
more efficient engine is used to drive the other re- 
versible engine as a refrigerator? The same argu- 
ment just used shows that heat would be transferred 
from a cold body to a hot one. This is impossible. 
Therefore, the two reversible engines must have 
the same efficiency. 

To actually compute that efficiency, you must 
know the properties of one reversible engine; all 
reversible engines working between the same tem- 
peratures must have that same efficiency. (Carnot 
computed the efficiency of an engine that used an 
ideal gas instead of steam.) 



Mot 



i 



y 



Cold 



4 



HI H, 



© ^ .0 

Hi Hz 



r 



CHAPTER 10 / ENERGY 301 



7,000 


29 


5,000 


21 


4,000 


17 


2,600 


11 


700 


3 


500 


2 


150 


0.6 



need at least 1,700 Calories of energy each day. There are 
countries where large numbers of working people exist on less 
than 1,700 Calories a day. The U.N. Yearbook of \ational 
Accounts Statistics shows that in India the average food intake 
was about 1,600 Calories per day. The United States average was 
3,100 Calories per day. About hdf the population of Southeast 
Asia is at or below the starvation line. Vast numbers of people 

elsewhere in the world, including some parts of the United 

SG 32 States, are also close to that line. It is estimated that if the 

available food were equally distributed among all the earth's 
inhabitants, each would have about 2,400 Calories a day on the 
axerage. This is only a little more than the minimum required by 
a working person. 

APPROXIMATE ENERGY CONTENT OF VARIOUS FOODS 

Cal kg MJ kg 

Butter 

Chocolate (sweetened) 

Beef (hamburger) 

Bread 

Milk (whole) 

Apples (raw) 

Lettuce 

Adapted from a U.S. Department of Agriculture handbook. 

APPROXIMATE RATES OF USING ENERGY DURING VARIOUS ACTIVITIES 

Calhr W 

Sleeping 

Lying down (awake) 
Sitting still 
Standing 

Typewriting rapidly 
Walking (5 km hr) 
Digging a ditch 
Running fast 
Rowing in a race 

Adapted from a U.S. Department of Agriculture handbook. 

It is now estimated that at the current rate of increase, the 
population of the world may double in 30 years. Thus, by the 
year 2000 it would be 7 billion or more. Furthennore, the rate at 
which the population is increasing is itself increasing. 
Meanwhile, the production of food supply per person has not 
increased markedly on a global scale. For example, in the last 10 
years the increase in crop yield per acre in the poorer countries 
has averaged less than 1% per year — far less than the increase 
in population. The problem of supplying food energy' for the 
world's hungiy is one of the most difficult problems facing 
humanity toda\'. 

302 UNIT 3 / THE TKIL'MPH OF MECHANICS 



70 


80 


80 


90 


100 


120 


120 


140 


140 


160 


220 


250 


400 


460 


600 


700 


1,200 


1,400 



In this problem of life-and-death importance, what are the 
roles science and technology can play? Obxaously, better 
agricultural practices should help, both by opening up new land 
for farming and by increasing production per acre on existing 
land. The application of fertilizers can increase crop yields, and 
factories that make fertilizers are not too difficult to build. 
However, it is important to study all the consequences before 
applying science through technology; othei^vdse you may create 
two new problems for every old one that you wash to "fix." 

In any particular country, the questions to ask include these: 
How will fertilizers interact with the plant being grown and with 
the soil? Will some of the fertilizer run off and spoil rivers and 
lakes and the fishing industiy in that locality? How much water 
will be required? What variety of the desired plant is the best to 
use within the local ecological fi^amework? How^ will the ordinary 
farmer be able to learn the new techniques? How wdll the farmer 
be able to pay for using them? 

Upon study of this sort it may turn out that in addition to 
fertilizer, a country may need just as urgently a better system of 
bank loans to small farmers and better agricultural education to 
help the farmer. Such training has played key roles in the rapid 
rise of productivity in the richer countries. Japan, for example, 
produces 7,000 college graduate agriculturalists each year. All of 
Latin America produces only 1,100 per year. In Japan there is 
one farm advisor for each 600 farms. Compare this with perhaps 
one advisor for 10,000 famis in Colombia, and one advisor per 
100,000 farms in Indonesia. 

For long-term solutions, the problem of increasing food 
production in the poorer countries goes far beyond changing 
agricultural practices. Virtually all facets of the economies and 
cultures of the affected countries are involved. Important factors 
range from internal economic aid and internal food pricing 
policies to urbanization, industrial growth, public health, and 
family-planning practice. 

Where, in all this, can the research scientist's contribution 
help? It is usually tnae that one of the causes of some of the 
worst social problems is ignorance, including the absence of 
specific scientific knowledge. For example, knowledge of how 
food plants can grow efficiently in the tropics is lamentably 
sparse. Better ways of removing salt from seawater or brackish 
groundwater are needed to allow irrigating fields with water from 
these plentiful sources. Before this will be economically possible, 
more basic knowledge wall be needed on just how molecules 
move through membranes of the sort usable in desalting 
equipment. Answers to such questions, and many like them, can 
only come through research in "pure" science by trained 
research workers haxdng access to adequate research facilities. 



The physics of energy transforma- 
tions in biological processes is one 
example of a lively interdisciplinary 
field, namely biophysics (where 
physics, biology, chemistry, and 
nutrition all enter). Another con- 
nection to physics is provided by 
the problem of inadequate world 
food supply; here, too, many physi- 
cists, with others, are presently 
trying to provide solutions through 
work using their special compe- 
tence. 



CHAPTER 10 / ENERGY 



303 



"The Repast of the Lion" by Henri 
Rousseau. The Metropolitan Mu- 
seum of Art. 



• Zl. Animals obtain the energy they need from food, but plants 

(a) obtain energy from sunlight. 

(b) obtain energy from water and carbon dioside. 

(c) obtain energy from seeds. 

(d) do not need a supply of energy. 

22. The human body has an efficiency of about 20%. This 
means that 

(a) only one-fifth of the food you eat is digested. 

(b) four-fifths of the energy you obtain from food is destroyed. 

(c) one-fifth of the energy you obtain from food is used to run 
the "machinery" of the body. 

(d) you should spend 80% of each day lying quietly without 
working. 

(e) only one-fifth of the energy you obtain from food can be 
used to enable your body to do work on e^cternal objects. 

23. E?cplain this statement: "The repast of the lion is sunlight." 




304 



UNIT 3 / THE TRIUMPH OF MECHANICS 



1.0 .9 I Arriving at a general conservation laiv 

In Sec. 10.3, the law of conseiA^ation of mechanical energy was 
introduced. This law applies only in situations where no 
mechanical energy is transfomied into heat energy or vice versa. 
Early in the nineteenth centuiy, developments in science, 
engineering, and philosophy suggested new ideas about energy. 
It appeared that all fomis of energy (including heati could be 
transformed into one another with no loss. Therefore, the total 
amount of energy in the universe must be constant. 

Volta's invention of the electric batteiy in 1800 showed that 
chemical reactions could produce electricity. It was soon found 
that electric currents could produce heat and light. In 1820, Hans 
Christian Oersted, a Danish physicist, discovered that an electric 
current produces magnetic effects. In 1831, Michael Faraday, the 
great English scientist, discovered electromagnetic induction: 
When a magnet moves near a coil or a wire, an electric current is 
produced in the coil or wire. To some thinkers, these discoveries 
suggested that all the phenomena of nature were somehow 
united. Perhaps all natural events result from the same basic 
"force." This idea, though vague and imprecise, later bore fruit in 
the form of the law of conseiA'ation of energy. All natural events 
involve a transfonnation of energy from one form to another. But 
the total quantity of energy does not change during the 
transformation. 

The invention and use of steam engines helped to establish 
the law of conservation of energy by showdng how to measure 
energy changes. Almost fiom the beginning, steam engines were 
rated according to a quantity termed their "duty." This term 
referred to how heavy a load an engine could lift using a given 
supply of fuel. In other words, the test was how much work an 
engine could do for the price of a ton of coal. This very practical 
approach is typical of the engineering tradition in which the 
steam engine was developed. 

The concept of work began to develop about this time as a 
measure of the amount of energy transformed from one fomi to 
another. (The actual words "work" and "energy" were not used 
until later.) This concept made possible quantitative statements 
about the transformation of energy. For example. Joule used the 
work done by descending weights as a measure of the amount 
of gravitational potential energy transformed into heat energy. 

In 1843, Joule stated that whenever a certain amount of 
mechanical energy seemed to disappear, a definite amount of 
heat always appeared. To him, this was an indication of the 
conservation of what we now call energy. Joule said that he was 

. . . satisfied that the grand agents of nature are by the Creator's 
fiat indestructible; and that, wherever mechanical [energ\'] is 
expended, an exact equivalent of heat is always obtained. 



Joule began his long series of ex- 
periments by investigating the "duty" 
of electric motors. In this case, 
duty was measured by the work the 
motor could do when a certain 
amount of zinc was used up in the 
battery that ran the motor. Joule's 
interest was to see whether motors 
could be made economically com- 
petitive with steam engines. 



CHAPTER 10 / ENERGY 



305 




Julius Robert von Mayer (1814- 
18781 was one of the first to realize 
that heat is a form of energy. He 
worked out the mechanical equiva- 
lent of heat. 




Friedrich von Schelling (1775-1854) 



One of the great successes of the 
Naturphilosophic was Oersted's dis- 
covery of the connection between 
elect^icit^' and magnetism (see Unit 
4, Sec. 14.11). See also Unit 2 Epi- 
logue. 



Having said this, Joule got back to his work in the laboratoiy. He 
was basically a practical man who had little tinier to s|)ecnlate 
about a deeper philosophical meaning ot his tindings. But others, 
though using speculative arguments, were also concluding that 
the total amount of energy in the universe is constant. 

A year before Joule's remark, for example, Julius Robert Mayer, 
a GeiTnan physician, had proposed a general law of conservation 
of energy. Mayer had done no quantitative experiments, but he 
had obseived body processes involving heat and respiration. He 
had also used other scientists' published data on the thermal 
properties of air to calculate the mechanical equivalent of heat. 
(Mayer obtained about the same value that Joule did.i 

Mayer had been influenced by the German philosophical 
school now known as Naturphilosophie or "Nature Philosophy. ' 
This school flourished during the late eighteenth and ear-ly 
nineteenth centuries. According to Nature Philosophy, the 
various phenomena and forces of nature — such as graxdty, 
electricity, and magnetism — are not r^eally separate from one 
another but are all manifestations of some unifying "basic ' 
natural force. This philosophy therefore encouraged experiments 
searching for that underiving force and for connections between 
different kinds of forces observed in nature. 

The most influential thinkers of the school of Nature 
Philosophers were Johann Wolfgang von Goethe and Friedrich 
von Schelling. Neither of these men is knowii today as a scientist. 
Goethe is generally considered Germany's greatest poet and 
dramatist, while Schelling is remembered as a minor 
philosopher. Both men had great influence on the generation of 
German scientists educated at the beginning of the nineteenth 
century. The Nature Philosopher's were closely associated with 
the Romantic movement in liter^ature, art, and music. The 
Romantics protested against the idea of the universe as a great 
machine. This idea seemed morally emptv and artistically 
worthless to them. The Nature Philosophers also detested the 
mechanical world view. They refused to believe that the richness 
of natural phenomena, including human intellect, emotions, and 
hopes, could be understood as the result of the motions of 
particles. 

The Nature Philosophers claimed that nature could be 
understood as it "really " is only by direct observation. But no 
complicated "artificial ' apparatus must be used, only feelings 
and intuitions. For Goethe the goal of his philosophy was: "That 
I may detect the inmost force which binds the world, and guides 
its course." 

Although its emphasis on the irnity of nature led the follower's 
of Naturphilosophie to some \'ery useful insights, such as the 
general concept of the conservation of energy, its romantic and 
aritiscientific bias made it less and less influential. Scientists who 



306 



UNIT 3 / THE TRIUMPH OF MECHANICS 



had previously been influenced by it, including Mayer, now 
strongly opposed it. In fact, some hard-headed scientists at first 
doubted the law of conservation of energy simply because of 
their distrust of Nature Philosophy. For example, William Barton 
Rogers, founder of the Massachusetts Institute of Technology, 
wrote in 1858: 

To me it seems as if many of those who are discussing this 
question of the conservation of force are plunging into the fog 
of mysticism. 

However, the law was so quickly and successfully put to use in 
physics that its philosophical origins were soon forgotten. 

This episode is a reminder of a familiar lesson: In the ordinary 
day-to-day work of scientists, experiment and mathematical 
theory are the usual guides. But in making a truly major advance 
in science, philosophical speculation often also plays an 
important role. 

Mayer and Joule were only two of at least a dozen people who, 
between 1832 and 1854, proposed in some form the idea that 
energy is conseiA^ed. Some expressed the idea vaguely; others 
expressed it quite clearly. Some arrived at the belief mainly 
through philosophy; others from a practical concern vvdth 
engines and machines or from laboratory investigations; still 
others from a combination of factors. Many, including Mayer and 
Joule, worked quite independently of one another. The idea of 
energy conservation was somehow "in the air," leading to 
essentially simultaneous, separate discoveries. 

The initial udde acceptance of the law of conservation of 
energy owed much to the influence of a paper published in 1847. 
This was 2 years before Joule published the results of his most 
precise experiments. The author, a young German physician and 
physicist named Hermann von Helmholtz, entitled his work "On 
the Conservation of Force." Helmholtz boldly asserted the idea 
that others were only vaguely expressing, namely, "that it is 
impossible to create a lasting motive force out of nothing." He 
restated this theme even more clearly many years later in one of 
his popular lectures: 

We arrive at the conclusion that Nature as a whole possesses a 
store of force which cannot in any way be either increased or 
diminished, and that, therefore, the quantity of force in Nature 
is just as eternal and unalterable as the quantity of matter. 
Expressed in this form, I have named the general law The 
Principle of the Conservation of Force.' 

Any machine or engine that does work (provides energy) can 
do so only by drawing from some source of energy. The machine 
cannot supply more energy than it obtains from the source. 
When the source runs out, the machine will stop working. 
Machines and engines can only transform energy; they cannot 
create it or destroy it. 




Hermann von Helmholtz 
(1821-1894) 



Helmholtz's paper, "Zur Erhaltung 
der Kraft," was tightly reasoned 
and mathematically sophisticated. 
It related the law of conservation of 
energy to the established principles 
of Newtonian mechanics and 
thereby helped make the law sci- 
entifically respectable. 



CHAPTER 10 / ENERGY 



307 



Energy Conservation on Earth 



Nuclear reactions inside 

the earth produce 

energy 

at a rate of 3 x lO'^W 






. / y The nuclear reactions 

•*- :^ ' . In the sun produce 

energy at a rate of 3.5 
X 10"W 

The earth receives about 17 x 10'^W from the sun, of which 
about 33% is immediately reflected, mostly by clouds 
and the oceans; the rest is absorbed, converted to heat, and 
ultimately radiated into outer space as infrared radiation. Of that 
part of the solar energy that is not reflected,... 



5 X 10'^W 
heats 
dry land 



•^ . ^ 



..3 X 10'^W 
heats the 
air, producing 
winds, waves, etc. 



.4 X 10'^W 
evaporates 
water 



f^ost of the energy given to water 
is given up again when the water 
condenses to clouds and rain; but 
every second about 10'* J 
remains as gravitational potential 
energy of the fallen rain. 



Controlled nuclear 
reactions produce 2 x 
lO'^W in electrical 
power 

electrochemistry 



Some of this energy is 

used to produce 

ICW of hydroelectric 

power 

12 X 10"W is 
used in 
generating' 
4 X 10"Wof 
electrical 
power 



...1.5 X 10'^W 
is used by 
marine plants 
for 
photosynthesis 

Ancient green plants 
have decayed and 
left a store of 
about 2.2 X 10" J 
in the form of 
oil, gas, and 
coal. This store 
is being used at a 
rate of 5 x 10'^W. 







mechanical 
power 



9 X 10"W is 

used in 

combustion 

engines. 

About 75% 

of this is wasted 

as heat; less 

than 

3 X 10"W 

appears as 

mechanical 

power 



.3 X 10'^W 
is used by 
land plants 
for 
photosynthesis 

Present-day green 
plants are being 
used as food for 
people and animals, 
at a rate 
of 2 X 10"W. 
Agriculture uses 
about 10% of this, 
and people 
ultimately consume 
3 X lO^Was food. 

Direct use 

as raw 

materials 

for plastics 

and 

chemicals 

accounts 

for 

2 X 10"W 



3 X lO'^W 

is used 

for 

heating; 

this is 

equally 

divided 

between 

industrial 

and domestic 

uses. 



308 



UMT 3 / THE TRIUMPH OF MECHAMCS 



24. The significance of German Nature Philosophy in the 
history of science is that it 

(a) was the most extreme form of the mechanistic viewpoint. 

(b) was a reaction against excessive speculation. 

(c) stimulated speculation about the unity of natural 
phenomena. 

(d) delayed progress in science by opposing Newtonian 
mechanics. 

25. Discoveries in electricity and magnetism early in the 
nineteenth century contributed to the discovery of the law of 
conservation of energy because 

(a) they attracted attention to the transformation of energy 
from one form to another. 

(b) they made it possible to produce more energy at less cost. 

(c) they revealed what happened to the energy that was 
apparently lost in steam engines. 

(d) they made it possible to transmit energy over long 
distances. 

26. The development of steam engines helped the discovery of 
the law of conservation of energy because 

(a) steam engines produced a large amount of energy. 

(b) the caloric theory could not explain how steam engines 
worked. 

(c) the precise idea of work was developed to rate steam 
engines. 

(d) the internal energy of a steam engine was always found to 
be conserved. 



SG 33 



10.1.0 The lairs of thermoclvnainics 



Two laws, one precise and one general, summarize many of the 
ideas in this chapter. Both of these laws are called laws of 
thermodynamics . 

The first law of thermodynamics is a general statement of the 
conservation of energy and is based on Joule's finding that heat 
and energy are equivalent. It would be pleasingly simple to call 
heat "internal" energy associated with temperature. We could 
then add heat to the potential and kinetic energy of a system, 
and call this sum the total energy that is conserved. In fact, this 
solution works well for a great variety of phenomena, including 
the experiments of Joule. Difficulties arise with the idea of the 
heat "content" of a system. For example, when a solid is heated 
to its melting point, ftzither heat input causes melting without 



If you do not want to know what 
the detailed difficulties are , you can 
skip to the conclusion in the last 
paragraph on the next page. 



CHAPTER 10 / ENERGY 



309 



Special case of an isolated systeni: 




\ 




In general: 

AW ^ 
/ 



AH 



\ 



AF - AW -^AH 



\ 



/ 



\ 



\ 



/ 



The word "heat" is used rather 
loosely, even by physicists. This re- 
striction on its meaning is not nec- 
essary in most contexts, but it is 
important for the discussion in this 
section. 



increasing the temperature. lYou may have seen this in the 
experiment "Caloiimetiy. "I Simply adding the idea of heat as one 
form of a system's energy/ will not give a complete general law. 

Instead of "heat, " we can use the idea of an internal energy — 
energy in the system that ma\' take forms not directly related to 
temperature. We can then use the word "heat " to refer only to 
a transfer of energy between a system and its surroundings. (In a 
similar way, the term work is not used to describe something 
contained in the system. Rather, it describes the transfer of 
energy from one system to another.) 

Even these definitions do not pennit a simple statement such 
as "Heat input to a system increases its internal energy, and 
work done on a system increases its mechanical energy." Heat 
input to a system can have effects other than increasing internal 
energy. In a steam engine, for example, heat input increases the 
mechanical energy of the piston. Similarly, work done on a 
system can have effects other than increasing mechanical energ\'. 
In rubbing your hands together, for example, the work you do 
increases the internal energy of the skin of your hands. 

Therefore, a general conservation law of energy must include 
both work and heat transfer. Further, it must deal with change in 
the total energy of a system, not with a "mechanical" part and 
an "internal" part. 

In an isolated system, that is, a system that does not exchange 
energy with its surroundings, the total energy must remain 
constant. If the system exchanges energy with its surroundings, 
it can do so in only one of two ways: Work can be done on or by 
the system, or heat can be passed to or from the system. In the 
latter case, the change in energ\' of the system must equal the 
net energy gained or lost by the surroundings. More precisely, 
Aw stands for the net work on the system, which is all the work 
done on the system minus all the work done by the system. 
Similarly, AH represents the net heat transfer to the system, or 
the heat added to the system minus the heat lost by the system. 
Then the change in the total energy of the system, AE, is given 

by 

A£ = AW + AH 



This is the mathematical formula 
tion of the first law of thermody 
namics. 



This general expression includes as special cases the 
preliminary versions of the conserv ation law gi\'en earlier in the 
chapter. If there is no heat transfer at all, then AH = 0, and so 
Mi = Aw. In this case, the change in energ\' of a system equals 
the net work done on it. On the other hand, if work is done 
neither on nor by a system, then AW = 0, and AjE = AH. Here 
the change in energy of a system is equal to the net heat 
transfer. 

We still need a description of that part of the total enei'gv' of a 
.system called "heat" lor better', 'internal" energ\'). So far, we ha\e 



310 



UNIT 3 / THE TRIUMPH OF MECHyXMICS 



seen only that an increase in internal energy is sometimes 
associated wath an increase in temperature. We also mentioned 
the long-held suspicion that internal energy involves the motion 
of the "small parts" of bodies. We will take up this problem in 
detail in Chapter 11. 

The second law of thermodynamics is a general statement of 
the limits of the heat engine and is based on Carnot's theorem. 
You saw that a reversible engine is the most efficient engine and 
the most effective refrigerator. Any other engine is not as efficient 
or effective. In order to fonnulate that idea generally and 
precisely, a new function, the entropy, must be introduced. 

The change in entropy of a system, AS, is defined as the heat 
gained by the system, AH, divided by the temperature of the 
system, T: 

AS = AH/T 

Although this equation defines only changes of entropy, once a 
standard state for the system for which S = is chosen, the total 
entropy for any state of the system can be determined. 

Since the entropy is defined for any state of the system, an 
engine that works in a cycle (as any heat engine does) must have 
the same entropy at the end of a cycle as it does at the start. 
This is not necessarily true of the boiler (hot object) and the 
condenser (cold object), since these are not returned to their 
initial states. 

If the engine is a reversible one, the change in entropy of the 
cold object is 

AS3 = H,/T, 

Since the hot object loses heat, its change in entropy is 

ASj = -H^/T^ 

The total entropy change of the whole universe in this operation 
is 

AS . = AS, + AS, + /is . 

univei'se 1 2 engine 

H H 

= — ^ + 

Carnot's theorem (page 296) says that the difference must be 
zero: 



AS. 



universe 



= 



What about a less ideal engine? You know it must be less 
efficient than the reversible one; so for the engine X, 

H\ > H, and H\ > H, 

The total entropy change of the universe is now 



AS 



univei'se 



H H 

— + 



SG 34-38 



Ml 



© 



-►w 



»2 



TZ 



M 



© 



-►w 



Hi 



Tz 



CHAPTER 10 / ENERGY 



311 




It is a fairly straightforward matter to show that this time 



nudolf Clausius (1822-1888) 



AS > 

universe 

the entropy increases. 

Although proven only for these simple heat engines, the results 
that 

As = (reversible processes) 
As > (any other process) 

are general ones. They are, in fact, mathematical formulations of 
the second law of thermodvnamics. 

Rudolf Clausius, who first formulated the second law in the 
form given here, paraphrased the two laws in 1850, as follows: 
' rhe energy of the universe remains constant, but its entropy 
seeks to reach a maximum. " 



• 27. The first law of thermodynamics is 

(a) true only for steam engines. 

(b) true only when there is no friction. 

(c) a completely general statement of conservation of energy. 

(d) the only way to e?cpress conservation of energy. 

28. Define AE, AW, AH, and AS for a system. 

29. What two ways are there for changing the total energy of a 
system^ 

30. The second law of thermodynamics says that the entropy 
of the universe 

(a) cannot increase. 

(b) cannot decrease. 

(c) must increase. 

(d) must decrease. 

I.O.H. I Faith in the laivs of thermodynaiiiics 

For over a century, the law of conservation of energy has stood 
as one of the most fundamental laws of science. You will 
encounter it again and again in this course, in studying 
electricity and magnetism, the structure of the atom, and nuclear 
physics. Throughout the other sciences, from chemistry to 
biology, and throughout engineering studies, the same law 
applies. Indeed, no other law so clearly brings together the 
various scientific fields, giving all scientists a common set of 
concepts. 



312 



UIVIT 3 / THE TRIUMPH OF MECHAiMCS 



The principle of conseivation of energy has been immensely 
successful. It is so fimily believed that it seems almost impossible 
that any new discoveiy could disprove it. Sometimes energy 
seems to appear or disappear in a system, without being 
accounted for by changes in known forms of energy. In such 
cases, physicists prefer to assume that some hitherto unknown 
kind of energy is involved; rather than to consider seriously the 
possibility that energy is not conserved. You have already read 
Leibniz's proposal that energy could be dissipated among "the 
small parts" of bodies. He advanced this idea specifically in order 
to maintain the principle of conservation of energy in inelastic 
collisions and frictional processes. Leibniz's faith in energy 
consei-vation was justified. Other evidence showed that "internal 

energy ' changed by just the right amount to explain obseived 

changes in external energy. SG 39, 40 

Another similar example is the "invention " of the neutrino by 
the physicist Wolfgang Pauli in 1933. Experiments had suggested 
that energy disappeared in ceitain nuclear reactions. Pauli 
proposed that a tiny particle, named the "neutrino" by Enrico 
Fermi, was produced in these reactions. Unnoticed, the neutrino 
carried off some of the energy. Physicists accepted the neutrino 
theory for more than 20 years even though neutrinos could not 
be detected by any method. Finally, in 1956, neutrinos were 
detected in experiments using the radiation from a nuclear 
reactor. (The experiment could not have been done in 1933, since 
no nuclear reactor existed until nearly a decade later.) Again, 
faith in the law of conservation of energy turned out to be 
justified. 

The theme of 'conservation " is so powerful in science that 
scientists believe it will always be justified. Any apparent 
exceptions to the law will sooner or later be understood in a way 
which does not require us to give up the law. At most, these 
exceptions may lead us to discover new fomis of energy, making 
the law even more general and poweiful. 

The French mathematician and philosopher Henri Poincare 
expressed this idea in 1903 in his book Science and Hypothesis: 

■ . . the principle of conseivation of energy signifies simply that 
there is something which reniiuns constant. Indeed, no matter 
what new notions future experiences will give us of the world, 
we are sure in advance that there will be something which will 
remain constant, and which we shall be able to call energy. 

Today, it is agreed that the discoveiy of conservation laws was 
one of the most important achievements of science. These laws 
are powerful and valuable tools of analysis. All of them basically 
affirm that, whatever happens within a system of interacting 
bodies, certain measurable quantities will remain constant as 
long as the system remains isolated. 

CHAPTER 10 / ENERGY 313 



The list of knovvn conservation laws has grown in recent years. 
The area ot tnndainental lor "elementary") pailicles has yielded 
much of this new knowledge. Some of the newer laws are 
imperfectly and incompletely understood. Others are on 
uncertain ground and are still being argued. 

Below is a list of conseivation laws to date. This list is not 
complete or eternal, but it does include the conservation laws 
that make up the working tool-kit of physicists today. (Those 
laws that are starred are discussed in the basic text portions of 
this couree. The others are treated in optional supplemental 
units, for example, the supplemental unit entitled "Elementary 
Particles.") 

Conservation Laws 



1. Linear momentum* 

2. Energy (including mass)* 

3. Angular momentum (including spin) 

4. Charge* 

5. Electron-family number 

6. Muon-family number 

7. Baryon-family number 

8. Strangeness number 

9. Isotopic spin 



The first law of thermodynamics, 
or the general law of conservation 
of energ\', does not forbid the full 
conversion of heat into mechanical 
energy. The second law is an addi- 
tional constraint on what can hap- 
pen in nature. 



See the film loop on the irreversi- 
bility of time (Film Loop 36). 



Numbers 5-9 result from work in nuclear physics, high-energy 
physics, or elementary or fundamental particle physics. 

The second law of themiodynamics has a status rather 
different from the conseivation laws. It, too, is an extremely 
successful and powerful law. It, too, has continued to stand as 
one of the fundamental laws of science. Unlike the conseivation 
laws or the laws of motion, the second law of thermodynamics 
gives no precise results; it only says certain things are 
impossible. For example, it is impossible to make the entropy of 
the universe (or of an isolated system) decrease, it is impossible 
to make heat flow ft om a cold body to a hot one without doing 
work on something. 

In other words, the processes in\ oh ing heat happen in one 
direction only: The entropy increases; heat flows from hot 
objects to cold ones. Thus, the second law is connected in some 
fundamental way with the notion that time proceeds in one 
direction only. To word it differently, when a movie taken of real 
events is run backward, what you see cannot, in detail, be found 
to occur in the real world. Ihese ideas will be examined in more 
detail in the next chapter. 



314 



UNIT 3 / THE TFUIJIVIPH OF MECHANICS 



study 
snide 



1. The Project Physics materials particularly 
appropriate for Chapter 10 include: 

Experiments 

Conservation of Energy 
Measuring the Speed of a BuUet 
Temperature and Thermometers 
Calorimetry 
Ice Calorimetry 

Activities 

Student Horsepower 
Steam Powered Boat 
Predicting the Range of an Arrow 

Film Loops 

Finding the Speed of a Rifle Bullet. I 

Finding the Speed of a Rifle BuUet. II 

Recoil 

Colliding Freight Cars 

Dynamics of a Billiard Ball 

A Method of Measuring Energy — Nail Driven 

into Wood 
Gravitational Potential Energy 
Kinetic Energy 

Conservation of Energy: Pole Vault 
Conservation of Energy: Aircraft Takeoff 

2. A person carries a heavy load across the level 
floor of a buOding. Draw an arrow to represent the 
force applied to the load, and one to represent the 
direction of motion. By the definition of work given, 
how much work is done on the load? Do you feel 
uncomfortable about this result? Why? 

3. Kinetic energy, like speed, is a relative quantity; 
that is, kinetic energy is different when measured 

in different frames of reference. An object of mass m 
is accelerated uniformly in a straight line by a force 
F through a distance d. Its speed changes from v, 
to v^. The work done is equal to the change in kinetic 
energy: Fd = Vz mv \ — Vz mv \. Describe this event 
from a frame of reference that is itself moving with 
speed u along the same direction. 

(a) What are the speeds as observed in the new 
reference frame? 



(b) Are the kinetic energies observed to have the 
same value in both reference frames? 

(c) Does the change in kinetic energy have the same 
value? 

(d) Is the calculated amount of work the same? 
Hint: By the Galilean relativity principle, the 
magnitude of the acceleration, and therefore force, 
will be the same when viewed from frames of 
reference moving uniformly relative to each other. 

(e) Is the change in kinetic energy still equal to the 
work done? 

(f) Which of the following are "invariant" for changes 
in reference frame (moving uniformly relative to one 
another)? 

(1) the quantity Vzuwr 

(2) the quantity Fd 

(3) the relationship Fd = A(»/zmv^) 

(g) Eljcplain why it is misleading to consider kinetic 
energy as something a body has, instead of only a 
quantity calculated from measurements. 

4. An electron of mass about 9.1 x 10"^' kg is 
traveling at a speed of about 2 x lO" m/sec toward 
the screen of a television set. What is its kinetic 
energy? How many electrons like this one would be 
needed for a total kinetic energy of 1 J? 

5. A 5-kg object fravels uniformly at 4 m/sec. Over 
what distance must a 4-N force be applied to give 
the object a total kinetic energy of 80 J? 

6. Estimate the kinetic energy of each of the 
following: 

(a) a pitched baseball 

(b) a jet plane 

(c) a sprinter in a 90-m dash 

(d) the earth in its motion around the sun. 

7. A 200-kg iceboat is supported by the smooth 
surface of a frozen lake. The wind exerts on the boat 
a constant force of 400 N while the boat moves 900 
m. Assume that frictional forces are negligible and 
that the boat starts from rest. Find the speed 
attained at the end of a 900-m run by each of the 
following methods: 



CHAPTER 10 / STUDY GUIDE 



315 



sg 



(a) Use Newton's second law to find the acceleration 
of the boat. How long does it tiike to nio\e 900 in? 
How fast vvill it be mo\ing then? 

(b) Find the final speed of the boat by equating the 
work done on it by the wind and the increase in its 
kinetic energy. Compare your result with your 
answer in (a). 

8. A 2-g bullet is shot into a tree stump. It enters at 
a speed of 300 nVsec and comes to rest after having 
penetrated 5 cm in a straight line. 

(a) What was the change in the bullet's kinetic 
energ\'? 

(b) How much work did the tree do on the bullet? 

(c) What was the average force during impact? 

9. Refer to S(i 20 in Chapter 9. How much work 
does the golf club do on the golf ball? How much 
work does the golf ball do on the golf club? 

10. A penny has a mass of about 3.0 g and is about 
1.5 mm thick. You have 50 pennies which you pile 
one on top of the other. 

(a) How much more gravitational potentiid energv' 
has the top penny than the bottom one? 

(b) How much more gravitational potential energy 
have all 50 pennies together than the bottom one 
alone? 

11. (a) How high can you raise a book weighing 5 N 
if you have available 1 J of energy? 

(b) How many joules of energy are needed just to lift 
a jet airliner weighing 7 x lo' N (fully loaded I to its 
cruising idtitude of 10,000 m? 

la. I'here are standards for length, time, and mass 
(for example, a standard meter i. But energv' is a 
"derived qucintit\'" for which no standards need be 
kept. Nevertheless, assume someone asks you to 
supply 1 J of energy. Describe in as much detail as 
you can how you would do it. 

13. As a home experiment, hang weights on a 
rubber band and measure its elongation. Plot the 
force versus stretch on graph paper. 

(a) How can you measure the stored energy? 

(b» Show that over a straight section of the graph the 
stored energv' is equal to Va k(A,v)", where A,v is the 



change in length of the rubber band over that section 
of the graph and k is the slope of the line (the 
change in force divided Ijy the change in length). 

(c) If the weights on the rubber band bob up and 
down, discuss the "flow of energV' ' from kinetic, 
gravitational potenticd energy, and the rubber band's 
potential energy. 

14. (a) Estimate how long it would take for the 
earth to fall up 1 m to a 1-kg stone if this stone were 
somehow rigidly fixed in space. 

(b) Estimate how far the earth will actually move up 
while a 1-kg stone falls 1 m from rest. 

(c) Why is the gravitational potential energy assigned 
to the system rather than to the rock alone? 

15. The photograph below shows a massive lead 
wrecking ball being used to demolish a wall. Discuss 
the transformations of energV' involv ed. 




16. This di.scussion will show that the I^L of an 
object is relative to the frame of reference in which 
it is measured. The boulder in the photograph on 
page 281 was not lifted to its perch. Hather. the rest 
of the land has eroded away, leaving the boulder 
where it may have been almost sinct; tin* formation 
of the earth. Consider the question "What is the 
gravitational potential energV' of the boulder-earth 
system? ' You can easily calculate what the change in 
potential energ\' would be, if the rock fell. It would 
be the product of the rocks weight and the distiince 
it fell. But would that be the actual value of the 
gravitational (mergv' that had been stored in the 
boulder-earth svsteni? Imagine that there hap|)ened 



316 



CHAPTER 10 / STUDY GUIDE 



sg 



to be a deep mine shaft nearby and the boulder fell 
into the shaft. It would then fall much farther, 
reducing the gravitational potential energy much 
more. Apparently, the amount of energy stored 
depends on how far you imagine the boulder can 
fall. 

(a) What is the greatest possible decrease in 
gravitational potential energy the isolated 
boulder-earth system could have? 

(b) Is the boulder-earth system really isolated? 

(c) Is there a true absolute bottom of gravitational 
potential energy for any system that includes the 
boulder and the earth? 

The value of PE depends on the location of the 
(resting) frame of reference from which it is 
measured. This is not a serious problem, because we 
are concerned only with changes in energy. In any 
given problem, physicists wiU choose some 
convenient reference for the "zero-level" of potential 
energy, usually one that simplifies calculations. What 
would be a convenient zero-level for the gravitational 
potential energy of 

(a) a pendulum? 

(b) a roller coaster? 

(c) a weight oscillating up and down a spring? 

(d) a planet in orbit around the sun? 

17. The figure below (not drawn to scale) shows a 
model of a carnival "loop-the-loop." A car starting 
from a platform above the top of the loop coasts 
down and around the loop without falling off the 
track. Show that to traverse the loop successfully, 
the car must start from a height at least one-half a 
radius above the top of the loop. Hint: The car's 
weight must not be greater than the centripetal force 
required to keep it on the circular path at the top 
of the loop. 




19. Sketch an addition to one of the steam-engine 
diagrams of a mechanical linkage that would open 
and close the valves automatically (page 290). 

20. Show that if a constant propelling force F keeps 
a vehicle moving at a constant speed v (against the 
friction of the surroundings) the power required is 
equal to Fv. 





18. Discuss the conversion between kinetic and 
potential forms of energy in the system of a coinet 
orbiting the sun. 



21. The Queen Mary, one of Britain's largest 
steamships, has been retired after completing 1,000 
crossings of the Atlantic. Its mass is 75 million 
kilograms. A maximum engine power of 174 million 
watts is allowed the Queen Mary to reach a 
maximum speed of 30.63 knots (16 m/sec). 

(a) What is the kinetic energy at full speed? 

(b) Assume that at maximum speed all the power 
output of the engines goes into overcoming water 
drag. If the engines are suddenly stopped, how far 
will the ship coast before stopping? (Assume water 
drag is constant.) 

(c) Wliat constant force would be required to bring 
the ship to a stop from full speed within 1 nautical 
mUe (2,000 m)? 

(d) The assumptions made in (b) are not valid for the 
following reasons: 

(1) Only about 60% of the power delivered to the 
propeller shafts results in a forward thrust to the 
ship; the rest results in friction and turbulence, 
eventually warming the water. 

(2) Water drag is less for lower speed than for 
high speed. 



CHAPTER 10 / STUDY GUIDE 



317 



ss 



(3) If the propellers are not free-wheeling, they 

add an increased drag. 
Which of the above factors tends to increase, which 
to decrease the coasting distance? 

(e) Explain why tugboats are important for docking 
big ships. 

22. Devise an experiment to measure the power 
output of 

(a) a person riding a bicycle. 

(b) a motorcycle. 

(c) an electric motor. 

23. (a) A skier of 70 kg mass experiences a pull on 
a ski lift from an engine transmitting 140 W to the 
cable. Neglecting friction, how high can the engine 
pull the skier in 500 sec? 

(b) What size light bulb (in watts) produces as much 
heat as the human body at rest? 

24. One hundred joules (100 J) of heat is put into 
two engines. Engine A can lift 5 N a distance of 10 m 
in 10 sec. Engine B pulls with a force of 2 N for 5 
sec a distance of 20 m. Calculate the efficiency and 
power of each engine. 

25. Refer to the table of "Typical Power Ratings ' on 
page 292. 

(a) What advantages would Mewcomen's engine ha\'e 
over a "turret windmill"? 

(b) What advantage would you expect Watt's engine 
(17781 to have over Smeaton's engine (1772)? 

26. Besides "horsepower," another term used in 
Watt's day to describe the performance of steam 
engines was "duty. ' The duty of a steam engine was 
defined as the distance in feet that an engine could 
lift a load of 1 miUion pounds, using 1 bushel of coal 
as fuel. For example, N'ewcomen's engine had a duty 
of 4.3; it could perform 4.3 million foot-pounds of 
work by burning a bushel of coal. 

A bushel of coal contains about 900 MJ of energy. 
A bushel is 36 liters (L). What was the efficiency of 
Newcomen's engine? 

27. The introduction of the steam engine had both 
positive and negative effects, although all of these 
effects were not predicted at the time. 



(a) List several actual effects, both beneficial and 
undesirable, of the steam engine and of the gasoline 
internal combustion engine. 

(b) List several predicted effects, both beneficial and 
undesirable, of nuclear power and of solar power. 

28. (a) Find the maximum efficiency of an engine 
tliat makes use of the temperature differences in the 
ocean. In the tropics, the surface waters are about 
15°C, and the bottom waters are about 5°C. 

(b) Cooling 1 metric ton of water by 1°C produces 
about 4 MJ of energy. At what rate must warm 
surface water be pumped through an engine that is 
cooled by bottom water in order that the engine 
produce 1 MW of mechanical power? 

29. (a) Find the miiximum coefficient of 
performance of an air conditioner operating on a 
day when it is 40°C outside and 21°C inside. 

(b) The coils on the outside of an air conditioner 
have to be considerably wiirmer than 40°C and those 
inside must be cooler than 21°C. Othenvise, heat 
would be exchanged too slowly. Suppose the coils 
are, respectively, 10°C warmer and cooler than the 
ideal. How much is the coefficient of performance 
decreased? 

(c) What happens to the coefficient of performance 
of this air conditioner when the outside coils are put 
into the sun and heat up another 10°C? 

30. A table of rates for truck transportation is given 
below. How does the charge depend on the amount 
of work done? 

TRUCK TRANSPORTATION 

Weight Moving rates (including pickup and 

(kg I delivery) from Boston to: 





Chicago 


Denver 


Los Angeles 




(1,550 km) 


(3,150 km) 


(4,800 km) 


50 


$ 18.40 


$ 24.00 


$ 27.25 


225 


92.00 


120.00 


136.25 


450 


128.50 


185.50 


220.50 


900 


225.00 


336.00 


406.00 


1,800 


383.00 


606.00 


748.00 


2,700 


576.00 


909.00 


1,122.00 



31. Beginning with the postulate that heat does not 
flow h\' its(!lf from a cold bodx' to a hot l)()dy, arrange 



318 



CHAPTER 10 / STUDY GUIDE 



ss 



the following steps in the order used by Clausius in 
his formula for the maximum efficiency of an 
engine: 

(a) No engine can be more efficient than an ideal 
reversible engine. 

(b) Choose one ideal reversible engine and calculate 
its efficiency. 

(c) Work must be done to cause heat to flow from a 
cold to a hot body. 

(d) The ratio of heat to work depends only on the 
temperature differences of the reservoirs for ideal 
reversible engines. 

(e) All reversible engines have the same efficiency. 

(f) The efficiency of an ideal reversible engine is the 
maximum possible for any real engine; in reality, 
the efficiency is much less. 

32. Elxplain why all ideal reversible engines have the 
same efficiency and why this efficiency is the 
maximum possible for an engine. What is an ideal 
engine? What is a reversible engine? 

33. Assuming that no real engine can be perfectly 
reversible, why does the formula for the nitudmum 
efficiency of an engine imply that absolute zero can 
never be reached? 

34. Consider the following hypothetical values for a 
paddle-wheel experiment like Joule's: A 1-kg weight 
descends through a distance of 1 m, turning a paddle 
wheel immersed in 5 kg of water. 

(a) About how many times must the weight be 
allowed to fall in order that the temperature of the 
water increase by 0.5°C? 

(b) How could you modify the exjDeriment so that 
the same temperature rise would be produced with 
fewer falls of the weight? (Hint: There are at least 
three possible ways.) 

35. While traveling in Switzerland, Joule attempted 
to measure the difference in temperature of the 
water at the top and at the bottom of a waterfall. 
Assuming that the amount of heat produced at the 
bottom is equal to the decrease in gravitational 
potential energy, calculate roughly the temperature 
difference you would expect to observe between the 
top and bottom of a waterfall about 50 m high, such 



as Niagara Falls. Does it matter how much water 
goes down the fall? 

36. Because a nuclear power plant's interior must 
be kept to much closer tolerances than fossil-fueled 
plants, its operating temperature is kept lower. 
Compare the efficiencies of a nuclear power plant 
that produces steam at 600°K with a fossil-fueled 
plant that produces steam at 750°K. Both are cooled 
by water at 300°K. 

(a) If both plants produce 1 MW of electrical power, 
at what rate does each plant dump heat into the 
environment? 

(b) It takes 4.2 MJ to raise the temperature of 1 
metric ton of water by 1°C (or 1°K). How many tons 
of water must flow through each of the two plants if 
the emerging water is to be no warmer than 303°K? 

37. About how many kilograms of hamburgers 
would you have to eat to supply the energy for 30 
min of digging? Assume that your body is 20% 
efficient. 

38. If your food intake supplies less energy than you 
use, you start "burning" your own stored fat for 
energy. The oxidation of 0.45 kg of animal fat 
provides about 4,300 Calories of energy. Suppose that 
on your present diet of 4,000 Calories a day, you 
neither gain nor lose weight. If you cut your diet to 
3,000 Calories and maintain your present physical 
activity, how long would it take to reduce your mass 
by 22.5 kg? 

39. In order to engage in normal light work, a 
person in India has been found to need on the 
average about 1 ,950 Calories of food energy a day, 
whereas an average West European needs about 
3,000 Calories a day. Explain how each of the 
following statements makes the difference in energy 
need understandable. 

(a) The average adult Indian weighs about 49.5 kg; 
the average adult West European weighs about 67.5 
kg- 

(b) India has a warm climate. 

(c) The age distribution of the population for which 
these averages have been obtained is different in the 
two areas. 



CHAPTER 10 / STUDY GUIDE 



319 



so 



40. No other concept in physics has the economic 
significance that "energy" does. Discuss the 
statement: "We could express energy in dollars just 
as well as in joules or calories." 

41. Show how the conservation laws for energy and 
for momentum can be applied to a rocket during 
the period of its lift-off. 

42. Discuss the following statement: "During a 
typical trip , all the chemicid energy of the gasoline 
used in an automobile is used to heat up the car, the 
road, and the air." 

43. If you place a hot body and a cold one in 
thermal contact, heat will flow spontaneously. 
Suppose an amount of heat // flows from a body at 
temperature T, to a body at T,. What is the entropy 
change of the universe? 

44. (a) Describe the procedure by which a space 
capsule can be changed from a high circulai' orbit to 
a lower circular orbit. 

(b) How does the kinetic energy in the lower orbit 
compare with that in the higher orbit? 

(c) How does the gra\itational potential energy for 
the lower orbit compare with that of the higher 
orbit? 

(d) It can be shown (by using calculus) that the 
change in gravitational potential energy' in going 
from one circular orbit to another will be twice the 
change in kinetic energj'. How, then, will the total 
energy for the lower circular orbit compare with 
that for the higher orbit? 

(e) How do you account for the change in total 
energy? 

45. Any of the terms in the equation A£ = AH + 
Aw can have negative values. 

(a) What would be true of a system for wliich 

(1) Aii is negative? 

(2) AH is negative? 

(3) Aw is negative? 

(b) Which terms would be negative for the following 
systems? 

(1) a person digging a ditch 



(2) a car battery while starting a car 

(3) iin electric light bulb just after it is turned on 

(4) an electric light bulb an hour after it is turned 
on 

(5) a running reft'igerator 

(6) an exploding firecracker 

46. In each of the following, trace the chain of 
energy transformations from tlie sun to the energy 
in its final form. 

(a) A pot of water is boiled on an electric stove. 

(b) An automobile accelerates from rest on a level 
road I climbs a hQl at constant speed, and comes to a 
stop at a trafflc light. 

(c) A windmill pumps water out of a flooded field. 

47. Review what you read about the second law of 
thermodvnamics in Sec. 10.10. For the engine X 
considered in the discussion of that law, 

H[ > H, and H', - H, 

Refer to the difference between H\ and H^ as h: 

H[ - H^ = h 

thus, h is a positive number. 

(a) Use conser\'ation of energy to show that H' , — H, 
= h also. 

(b) Use the following equation (Carnot's theorem) 

H, H, 

T. ~ T, 



to show that for one cycle of engine X, 

AS..„..„ 



h_ _ jl 
T. T, 



(c) Prove that AS^^.^^^^ must be positive. 

48. An ice cube (10 g) melts in a glass of water (100 
g). Both are nearly at 0°C. Neglect temperature 
changes. Melting the ire requires 3.4 MJ of energy 
(which comes from cooling the water). What is the 
entropy change of the ice? of the water? of the 
universe? 



320 



CHAPTER 10 / STUDY GUIDE 




The Kinelic Theory of Gooes 



11.1 An overview of the chapter 

11.2 A model for the gaseous state 

11.3 The speeds of molecules 

11.4 The sizes of molecules 

11.5 Preilicting the behaiior of gases from the kinetic theorj' 

11.6 The second laiv of thermodj'namics and the dissipation of 
energv 

11.7 Maxwell's demon and the statistical vieiv of the second laiv 
of thermodji'namics 

11.8 Time's arrow and the recurrence paradox 



iLl I An overview of the chapter 

During the 1840 's, many scientists recognized that heat is not a 
substance, but a fonn of energy tliat can be converted into other 
forms. Two of these scientists, James Prescott Joule and Rudolf 
Clausius, went a step further. Heat can produce mechanical 
energy, and mechanical energy can produce heat; therefore, they 
reasoned, the "heat energy" of a substance is simply the kinetic 
energy of its atoms and molecules. In this chapter, you will see 
that this idea is largely correct. This idea forms the basis of the 
kinetic-molecular theory of heat. 

However, even the idea of atoms and molecules was not 
completely accepted in the nineteenth centuiy. If such small bits 
of matter really existed, they would be too small to obseive even 
under the most powerful microscopes. Since scientists could not 



SG 1 



Molecules are the smallest pieces 
of a substance; they may be com- 
binations of atoms of simpler sub- 
stances. 



CHAPTER 11 / THE KIIVETIC THEORY OF GASES 



321 



observe molecules, they could not check directly the h\pothesis 
that heat is molecular kinetic energ\'. Instead, they had to derive 
from this hypothesis predictions about the behaxior of 
measurably large samples of matter. Then they could test these 
predictions by experiment. For reasons that will be explained, 
it is easiest to test such hypotheses by obsemng the profDeities 
of gases. Therefore, this chapter deals mainly with the kinetic 
theory as applied to gases. 

The development of the kinetic theory of gases in the 
nineteenth century led to one of the last major- triumphs of 
Newtonian mechanics. The method involved using a simple 
theoretical model of a gas. In this model Neuron's laws of 
motion were applied to the gas molecules which were pictured 
as tiny balls. This method produced equations that related the 
easily observable pr operties of gases, such as pressure, density, 
and temperature, to properties not directly observable, such as 
the sizes and speeds of molecules. For* example, the kinetic 
theory: 

1. explained rules that had been found previously by trial-and- 
error methods. lAn example is "Boyle's law," which relates the 
pressure and the volume of a gas.) 

2. predicted new relations. (One surprising result was that the 
friction between layers of gas moving at different speeds 
increases with temperatur e, but is independent of the density of 
the gas.) 

3. led to values for the sizes and speeds of gas molecules. 

Thus, the successes of kinetic theory showed that Newtonian 
mechanics provided a way for understanding the effects and 
behavior of invisible molecules. 

Applying Newtonian mechanics to a mechanical model of 
gases resulted in some predictions that did not agree with the 
facts; that is, the model is not valid for all phenomena. According 
to kinetic theory, for example, the energy of a group of molecules 
should be shared equally among all the different motions of the 
molecules and their atoms. The properties of gases predicted 
from this "equal sharing" principle clearly disagreed with 
experimental evidence. Newtonian mechanics could be applied 
successfully to a wide range of motions and collisions of 
molecules in a gas. But it did not work for the motions of atoms 
inside molecules. It was not until the twentieth century that an 
adequate theory of the behavior of atoms, "quantum mechanics," 
was developed. (Some ideas from quantum mechanics are 
discussed in Unit 5.) 

Kinetic theory based on Newtonian mechanics also had 
trouble dealing with the fact that most phenomena are not 
reversible. An inelastic collision is an irreversible process. Other 
examples are the mixing of two gases or scrambling an egg. In 
NevYlonian theory, however, the reverse of anv event is just as 



322 UNIT 3 / THE TRILMPH OF MECHANICS 



reasonable as the event itself. Can irreversible processes be 
described by a theory based on Newtonian theory? Or do they 
involve some new fundamental law^ of nature? In studying this 
problem from the viewpoint of kinetic theory, you will see how 
the concept of "randomness" entered physics. 



1. Early forms of the kinetic molecular theory were based on 
the assumption that heat energy is 

(a) a liquid. 

(b) a gas. 

(c) the kinetic energy of molecules. 

(d) made of molecules. 

2. True or false: In the kinetic theory of gases, as developed in 
the nineteenth century, it was assumed that Newton's laws of 
motion apply to the motion and collisions of molecules. 

3. True or false: In the twentieth century, Newtonian 
mechanics was found to be applicable not only to molecules 
but also to the atoms inside molecules. 




Balloon for carrying weatherfore- 
casting apparatus. 



1. 1. .2 I A model for the gaseous state 

What are the differences between a gas and a liquid or solid? 
You know by observation that liquids and solids have definite 
volume. Even if their shapes change, they still take up the same 
amount of space. A gas, on the other hand, will expand to fill any 
container (such as a room). If not confined, it will leak out and 
spread in all directions. Gases have low densities compared to 
those of liquids and solids, typically about 1,000 times smaller. 
Gas molecules are usually relatively far apart from one another, 
and they only occasionally collide. In the kinetic theory model, 
forces betw^een molecules act only over very short distances. 
Therefore, gas molecules are considered to be moving freely most 
of the time. In liquids, the molecules are closer together; forces 
act among them continually and keep them from flying apart. In 
solids, the molecules are usually even closer together, and the 
forces between them keep them in a definite orderly 
arrangement. 

The initial model of a gas is very simple. The molecules are 
considered to behave like miniature balls, that is, tiny spheres or 
clumps of spheres that exert no force at all on each other except 
when they make contact. Moreover, all the collisions of these 
spheres are assumed to be perfectly elastic. Thus, the total 
kinetic energy of two spheres is the same before and after they 
collide. 

Note that the word "model" is used in two different senses in 
science. In Chapter 10, we mentioned the model of Newcomen's 




-P 



> 



Q 



J '-^tj 



/ 



a 



^ 



■^ 






:> 



J 



A very simplified "model" of the 
three states of matter: solid, liquid, 
gas. (From General Chemistry, 
second edition, by Linus Pauling, 
W. H. Freeman and Company, © 
1953.) 



CHAPTER 11 / THE KINETIC THEORY OF GASES 



323 



Gases can be confined without a 
container. A star, for example, is a 
mass of gas confined by grjivita- 
tional force. Another example is the 
earth's atmosphere. 



The word "gas" was originally de- 
med from the Greek word chaos; 
it was first used by the Belgian 
chemist Jan Baptista van Helmont 
(1580-1644). 



engine which James Watt was given to repair. That was a working 
model. It actually did function, although it was much smaller 
than the original engine, and contained some parts made of 
different materials. Now we are discussing a theoretical model of 
a gas. This model exists only in the imagination. Like the points, 
lines, triangles, and spheres studied in geometiy, this theoretical 
model can be discussed mathematically. The results of such a 
discussion may help you to understand the real world. 

This theoretical model represents the gas as consisting of a 
large number of very small particles in rapid, disordered motion. 
"A large number" means something like a billion billion iio"*) or 
more particles in a sample as small as a buiible in a soft diink. 
"Very small" means a diameter about a hundred-millionth of a 
centimeter (10 ~*" m). "Rapid motion" means an average speed of 
a few hundred kilometers pei' hour. What is meant by 
"disordered" motion? Nineteenth-century kinetic theorists 
assumed that each individual molecule mo\'ed in a definite way, 
detemiined by Newton's laws of motion. Of course, in practice 
it is impossible to follow a billion billion particles at the same 
time. They move in all directions, and each particle changes its 
direction and speed during collisions with other particles. 
Therefore, we cannot make a definite prediction of the motion of 
any one individual par'ticle. Instead, we must be content with 
describing the average behavior of large collections of particles. 
From moment to moment, each individual molecule behaves 
according to the laws of motion. But it is easier to describe the 
average behavior- if we assume complete ignorance about any 
individual motions. 

To see why this is so, consider the r'esults of flipping a lar-ge 
number of coins all at once. It would be very har-d to predict 
how a single coin would behave. But if you assume the coins 
behave randomly, voir can confidently predict that flipping a 
million coins will give approximately 50% heads and 50% tails. 
The same principle applies to molecules bouncing around in a 
container. You can safely assume that about as many are moving 
in one direction as in another-. Further-more, the molecules ar-e 
equally likely to be found in any cubic centimeter of space inside 
the container. This is true no matter- where such a r-egion is 
located, and even though we do not know wher-e a given 
molecule is at any given time. "Disordered," then, means that 
velocities and positions are distributed randomlv. Each molecule 
is just as likely to be moving to the right as to the left lor- in any 
other direction). It is just as likely to be near the center as near 
the edge (or any other position). 



• 4. What kind of a model is a test model of a bridge made of 
balsa wood? a computer program that simulates the forces 



324 



UMT 3 / THE TRIUMPH OF MECHA.MCS 



Averages ond Fluctuations 

Molecules are too small, too numerous, and too 
fast for us to measure the speed of any one mol- 
ecule, its kinetic energy, or how far it moves before 
colliding with another molecule. For this reason, the 
kinetic theory of gases concerns itself with making 
predictions about average values. The theory en- 
ables us to predict quite precisely the average 
speed of the molecules in a sample of gas, the 
average kinetic energy, or the average distance the 
molecules move between collisions. 

Any measurement made on a sample of gas re- 
flects the combined effect of billions of molecules, 
averaged over some interval of time. Such average 
values measured at different times, or in different 
parts of the sample, will be slightly different. We 
assume that the molecules are moving randomly. 
Thus, we can use the mathematical rules of statis- 
tics to estimate just how different the averages are 
likely to be. We will call on two basic rules of sta- 
tistics for random samples: 

1. Large variations away from the average are 

less likely to occur than are small variations. (For 



30-Toss Se-ta 






20 



5C 



90 IOC %Heads 



90 -Toss Sets 



30 



■■■■■ 
4.0 ro fco 



80 



90 ]0D "/o Head; 



I80-To;s Sets 



MmmsuL 



10 20 3C 40 5C fee 70 PC 90 100 % HeadS 



Giose Upl 



example, if you toss 10 coins, you are less likely 

to get 9 heads and 1 tail than to get 6 heads and 

4 tails.) 

2. Percentage variations are likely to be smaller 

for large samples. (For example, you are likely 

to get nearer to 50% heads by flipping 1 ,000 coins 

than by flipping just 10 coins.) 

A simple statistical prediction is the statement that 
if a coin is tossed many times, it will land "heads" 
50% of the time and "tails" 50% of the time. For 
small sets of tosses there will be many "fluctua- 
tions" (variations) to either side of the predicted 
average of 50% heads. Both statistical rules are 
evident in the charts at the right. The top chart 
shows the percentage of heads in sets of 30 tosses 
each. Each of the 10 black squares represents a 
set of 30 tosses. Its position along the horizontal 
scale indicates the percentage of heads. As we 
would expect from rule 1 , there are more values 
near the theoretical 50% than far from it. The second 
chart is similar to the first, but here each square 
represents a set of 90 tosses. As before, there are 
more values near 50% than far from it. And, as we 
would expect from rule 2, there are fewer values far 
from 50% than in the first chart. 

The third chart is similar to the first two, but now 
each square represents a set of 1 80 tosses. Large 
fluctuations from 50% are less common still than 
for the smaller sets. 

Statistical theory shows that the average fluctua- 
tion from 50% shrinks in proportion to the square 
root of the number of tosses. We can use this rule 
to compare the average fluctuation for sets of, say, 
30,000,000 tosses with the average fluctuation for 
sets of 30 tosses. The 30,000,000-toss sets have 
1,000,000 times as many tosses as the 30-toss 
sets. Thus, their average fluctuation in percent of 
"heads" should be 1,000 times smaller! 

These same principles hold for fluctuations from 
average values of any randomly distributed quan- 
tities, such as molecular speed or distance between 
collisions. Since even a small bubble of air contains 
about a quintillion (10'®) molecules, fluctuations in 
the average value for any isolated sample of gas 
are not likely to be large enough to be measurable. 
A measurably large fluctuation is not impossible, 
but extremely unlikely. 



CHAPTER 11 / THE KINETIC THEORY OF GASES 



325 



,;:'^^ 




Daniel Bernoulli (1700-1782) 



SG3 



Pressure is defined as the perpen- 
dicular force on a surface divided 
by the area of the surface. The unit 
of pressure, N/mS has been given 
the name pascal (svmbol Pa) after 
an eighteenth-century physicist. At- 
mospheric pressure is about 100 
kPa. 



acting on a bridge? What are the differences between 
"theoretical" and "working" models? 

5. Head the following description of a model of a gas and give 
a suitable numerical estimation for each underlined phrase: 
"a lare,e number of small particles in rapid, disordered 
motion. " 

6. In the kinetic theory, particles are thought to e}cert 
significant forces on one another 

(a) only when they are far apart. 

(b) only when they are close together. 

(c) all the time. 

(d) never. 

7. Why was the kinetic theory first applied to gases rather than 
to liquids or solids? 

1. 1. .3 I The speeds of molecules 

The basic idea of the kinetic theory is that heat is related to the 
kinetic energy of molecular motion. This idea had been 
frequently suggested in the past. However, many difficulties 
stood in the way of its general acceptance. Some of these 
difficulties are well worth mentioning. They show that not all 
good ideas in science (any more than outside of science i are 
immediately successful. 

In 1738, the Swdss mathematician Daniel Bernoulli showed 
how a kinetic model could explain a well-known property of 
gases. This property is described by Boyle's law: As long as the 
temperature does not change, the pressure of a gas is 
proportional to its density. Bernoulli assumed that the pressure 
of a gas is simply a result of the impacts of individual molecules 
striking the wall of the container. If the density of the gas were 
twice as great, there would be twice as many molecules per 
cubic centimeter. Thus, Bernoulli said, there would be twice as 
many molecules striking the wall per second and hence twice 
the pressure. Bernoulli's proposal seems to have been the first 
step toward the modem kinetic theoiy of gases. Yet it was 
generally ignored by other scientists in the eighteenth century. 
One reason for this was that Newton had proposed a different 
theory in his Principia (16871. Newlon showed that Boyle's law 
could be explained by a model in which particles at rest exert 
forces that repel neighboring particles. Newton did not claim 
that he had proved that gases really are composed of such 
repelling particles. But most scientists, impressed by Newton's 
discoveries, simply assumed that his treatment of gas pressure 
was also right, lit was not.) 

The kinetic theory of gases was proposed again in 1820 by 
English physicist John Heiapath. Herapath ledisctneied 



326 



UMT 3 / THE TRILMPH OF MECH^XIVICS 



Bernoulli's findings on the relations between pressure and 
density of a gas and the speeds of the particles. Herapath's work 
also was ignored by most other scientists. 

James Prescott Joule, however, did see the value of Herapath's 
work. In 1848, he read a paper to the Manchester Literaiy and 
Philosophical Society in which he tried to revive the kinetic 
theory. Joule shov^ed how the speed of a hydrogen molecule 
could be computed (as Herapath had done). He reported a value 
of 2,000 mb^sec at 0°C, the freezing temperature of water. This 
paper, too, was ignored by other scientists. For one thing, 
physicists do not generally look in the publications of a "literary 
and philosophical society " for scientifically important papers. 
However, evidence for the equivalence of heat and mechanical 
energy continued to mount. Several other physicists 
independently u^orked out the consequences of the hypothesis 
that heat energy in a gas is the kinetic energy of molecules. 
Rudolf Clausius in Germany published a paper in 1856 on "The 
Nature of the Motion we call Heat." This paper established the 
basic principles of kinetic theoiy essentially in the form accepted 
today. Soon afterward, James Clerk Maxwell in Britain and 
Ludwig Boltzmann in Austria set forth the full mathematical 
details of the theory. 

The Maxwell velocity distribution. It did not seem likely that all 
molecules in a gas would have the same speed. In 1859, Maxwell 
applied the mathematics of probability to this problem. He 
suggested that the speeds of molecules in a gas are distributed 
over all possible values. Most molecules have speeds not very far 
from the average speed. Some have much lower speeds and 
some much higher speeds. 



(b) 



• t * 



• • • 

• • • 

• • • 



Cc) 



• • • 

• • • • 

• • • • 



• • • • 
• « • • 



distonc* from cercHer 



dis+oi'nce fnom ceo+^r- 




Ludwig Boltzmann (1844-1906) 



A simple example will help you to understand Maxwell's 
distribution of molecular speeds. Suppose a person shoots a gun 
at a practice target many times. Some bullets will probably hit 
the bull's-eye. Others will miss by smaller or larger amounts, as 
shown in (a) in the sketch above. The number of bullets scattered 
at various distances from the center are counted. A graph of the 
results is shown in (b). This graph shows the distribution of hits 
for one set of shots. Another target may give a different 
distribution, but it is likely to have the same general shape. If you 
plot the distribution of hits for a veiy large number of shots, you 
will get a distribution like the one in (c). 



Target practice e}cperiment: (a) 
scatter of holes in target; (b) graph 
showing nuniber of holes in each 
half-ring of the bull's eye; (c) graph 
showing that the distribution be- 
comes smooth for a very large 
number of shots and for very nar- 
row rings. 



CHAPTER 11 / THE KINETIC THEORY OF GASES 



327 




Ma^cwell's distribution of speeds in 
gases at different temperatures. 



SG4 
SG5 



For still larger numbers of shots, the distribution spread you 
see in (c) will be too small to be noticed. Since the number of 
molecules in a gas is very large indeed, the graph shov\ing the 
distribution of molecular speeds is smooth at any scale that can 
be drawn. 

The actual shape of the curve in (c) is determined by many 
things about the gun, the person, and so on. Other processes 
give rise to other shapes of curves. The speeds of the molecules 
in a gas are determined by the collisions they have with each 
other. Maxwell used a very clever mathematical argument to 
deduce a distribution of molecular speeds. He was then able to 
argue, not rigorovisly, that the average number of molecules at 
each speed is not changed by the molecular collisions. 
Somewhat later, Ludwig Boltzmann published a rigorous proof 
that Maxwell's distribution is the only one that remains 
unchanged by collisions. 

Maxwell's distribution law for molecular speeds in a gas is 
shown in the margin in graphical form for three different 
temperatures. For a gas at any given temperature, the "tail " of 
each curve is much longer on the right (high speeds) than on the 
left (low speeds). As the temperature increases, the peak of the 
curve shifts to higher speeds, and the speed distribution 
becomes more broadly spread out. 

What evidence do we have that Maxwell's distribution law 
really applies to molecular speeds? Several successful predictions 
based on this law gave indirect support to it. Not until the 1920 s 
was a direct experimental check possible. Otto Stern in GeiTnany, 
and later Zartmann in the United States, devised a method for 
measuring the speeds in a beam of molecules. (See the 
illustration of Zartmann's method on page 330.) Stern, Zartmann, 
and others found that molecular speeds are indeed distributed 
according to Maxwell's law. 



8. In the kinetic theory of gases, it is assumed that the 
pressure of a gas on the walls of the container is due to 

(a) gas molecules colliding with one another. 

(b) gas molecules colliding against the walls of the container. 

(c) repelling forces everted by molecules on one another. 

9. The idea of speed distribution for gas molecules means that 

(a) each molecule always has the same speed. 

(b) there is a wide range of speeds of gas molecules. 

(c) molecules are moving fastest near the center of the gas. 



328 



UMT 3 / THE TRIUMPH OF MECHANICS 



1. 1. .-^ The sizes of molecules 



Is it reasonable to suppose that gases consist of molecules 
moving at speeds up to several hundred meters per second? If 
this model were correct, gases should mix vvdth each other very 
rapidly. But anyone who has studied chemistry knows that they 
do not. Suppose hydrogen sulfide or chlorine is generated at the 
end of a large room. Several minutes may pass before the odor 
is noticed at the other end. According to kinetic-theory 
calculations, each of the gas molecules should have crossed the 
room hundreds of times by then. Therefore, something must be 
wrong with the kinetic-theoiy model. 

Rudolf Clausius recognized this as a valid objection to his own 
version of the kinetic theory. His 1856 paper had assumed that 
the particles are so small that they can be treated like 
mathematical points. If this were true, particles would almost 
never collide with one another. However, the observed slowness 
of diffusion and mixing convinced Clausius to change his model. 
He thought it likely that the molecules of a gas are not 
vanishingly small, but of a finite size. Particles of finite size 
moving very rapidly would often collide with one another. An 
individual molecule might have an instantaneous speed of 
several hundred meters per second, but it changes its direction 
of motion every time it collides with another molecule. The more 
often it collides with other molecules, the less likely it is to move 
very far in any one direction. How often collisions occur depends 
on how crowded the molecules are and on their size. For most 
purposes, you can think of molecules as being relatively far apart 
and of very small size. But they are just large enough and 
crowded enough to get in one another's way. Realizing this, 
Clausius was able to modify his model to explain why gases mix 
so slowly. In addition, he derived a precise quantitative 
relationship between the molecules' size and the average 
distance they moved between collisions. 

Clausius now was faced with a problem that plagues every 
theoretical physicist. If a simple model is modified to explain 
better the observed properties, it becomes more complicated. 
Some plausible adjustment or approximation may be necessary 
in order to make any predictions from the model. If the 
predictions disagree with experimental data, is this because of a 
flaw in the model or a calculation error introduced by the 
approximations? The development of a theory often involves a 
compromise between adequate explanation of the data and 
mathematical convenience. 

Nonetheless, it soon became clear that the new model was a 
great improvement over the old one. It turned out that certain 
other properties of gases also depend on the size of the 
molecules. By combining data on several such properties, it was 



%/ 



^ 



c 



The larger the molecules are, the 
more likely they are to collide with 
each other. 



CHAPTER 11 / THE KINETIC THEORY OF GASES 



329 



:/Ai^;/ 



OV'tM 



1 



"--J 



geeesAj^ to ' 

P£rWe BEfiM 

To VAC.lM}Ki ■PuH'PS ' 




Pmuhi 






.1^ 



Direct Measurement of Molecular 
Speeds 

A narrow beam of molecules is formed by letting molecules of a hot gas 
pass through a series of slits. In order to keep the beam from spreading 
out, collisions with randomly moving molecules must be avoided. Therefore, 
the source of gas and the slits are housed in a highly evacuated chamber. 
The molecules are then allowed to pass through a slit in the side of a 
cylindrical drum that can be spun very rapidly. The general scheme is shown 
in the drawing above. 

As the drum rotates, the slit moves out of the beam of molecules. No 
more molecules can enter until the drum has rotated through a whole rev- 
olution. Meanwhile, the molecules in the drum continue moving to the right, 
some moving quickly and some moving slowly. 

Fastened to the inside of the drum is a sensitive film that acts as a 
detector. Any molecule striking the film leaves a mark. The faster molecules 
strike the film first, before the drum has rotated very far. 

The slower molecules hit the film later, after the drum has rotated farther. 
In general, molecules of different speeds strike different parts of the film. 
The darkness of the film at any point is proportional to the number of mol- 
ecules that hit it there. Measurement of the darkening of the film shows the 
relative distribution of molecular speeds. The speckled strip at the right 
represents the unrolled film, showing the impact position of molecules over 
many revolutions of the drum. The heavy band in- 
dicates where the beam struck the film before the 
drum started rotating. (It also marks the place to 
which infinitely fast molecules would get once the 
drum was rotating.) 

A comparison of some experimental results with 
those predicted from theory is shown in the graph. 
The dots show the experimental results, and the 
solid line represents the predictions from the kinetic 
theory. 






mHSPE£2> 


LOH ^fteu 




MOL£CULES 


MOLfCUL£S 




i '■■'.'.•'</!,•''■'•"'•■•'.• 












1 •■.■.•: -•':c'!-i^?^':'^^< 




^ 






•a 






\^t 


^^rrx. 






/ >^ 






/' ^^ 




r^ C 


Kr. 




/ 


V 




y^ 


^^^^ 






^/spun \ 



possible to vv^ork backwards and find fairly reliable values for 
molecular sizes. Here, only the result of these calculations is 
reported. Typically, the diameter of gas molecules came out to be 
of the order of lO'" to 10 ' m. This is not far from the modern 
values, an amazingly good result. After all, no one previously had 

knouTi whether a molecule was 1,000 times smaller or bigger 

than that. In fact, as Lord Kelvin remarked: SG 6 

The idea of an atom has been so constantly associated with 

incredible assumptions of infinite strength, absolute rigidity, 

mystical actions at a distance and indivisibility, that chemists 

and many other reasonable naturalists of modern times, losing 

all patience with it, have dismissed it to the realms of 

metaphysics, and made it smaller than anvtbing we can 

conceive.' 

SG 7 
Kelvin showed that other methods could also be used to SG 8 

estimate the size of atoms. None of these methods gave results as 

reliable as did the kinetic theory. But it was encouraging that 

they all led to the same order of magnitude (within about 50%j. 



# 10. In his revised kinetic-theory model Clausius assumed that 
the particles have a finite size, instead of being mathematical 
points, because 

(a) obviously everything must have some size. 

(b) it was necessary to assume a finite size in order to 
calculate the speed of molecules. 

(c) the size of a molecule was already well known before 
Clausius' time. 

(d) a finite size of molecules could account for the slowness of 
diffusion. 

11. Why were many people skeptical of the existence of atoms 
at the time of Clausius? How did Clausius estimate of the size 
of atoms reinforce atomic theory? 

1. 1. .5 I Predicting the behavior of gases from the 
kinetic theory 

One of the most easily measured characteristics of a confined gas 
is pressure. Experience with balloons and tires makes the idea 
of air pressure seem obvious, but it was not always so. 

Galileo, in his book on mechanics, Two New Sciences (1638), 
noted that a lift-type pump cannot raise water more than 10 m. 
This fact was well known. Such pumps were widely used to 
obtain drinking water from wells and to remove water from 
mines. You already have seen one important consequence of this 

CHAPTER 11 / THE KINETIC THEORY OF GASES 331 



.76 m 



0.0 m 



SG 9 
SG 10 



Torricelli's barometer is a glass 
tube standing in a pool of mercury. 
The topmost part of the tube is 
empty of air. The air pressure on 
the pool supports the weight of the 
column of mercury in the tube up 
to a height of about 0.76 m. The 
MKS unit of pressure is the \'/m', 
which has been given the name 
pascal (symbol Pa). 




Because the force acts on a very 
small surface, the pressure under a 
thin, high heel is greater than that 
under an elephant's foot. 



SG 11 
SG 36, 37 



SG 12 



limited ability of pumps to lift water out of deep mines. Ihis 
need provided the initial stimulus foi- the de\(?lopment of steam 
engines. Another' consecjuenee was that physicists became 
curious about \\h\' the lift pumj) worked at all. Also, why should 
there be a limit to its abilit\' to raise water? 

Air pressure. The |3uzzle was sohed as a result of experiments 
by Torricelli (a student of Galileo), Guericke, Pascal, and Boyle. 
By 1660, it was fairly clear that the o[3eiation o( a "lift" pump 
depends on the pressure of the air. The pump merely reduces 
the pressure at the top of the pipe. It is the pressure exerted by 
the atmosphere on the pool of v\ater below which forces water 
up the pipe. A good pump can reduce the pressure at the top of 
the pipe to nearly zero. Then the atmospheric pressure can force 
water up to about 10 m iibo\e the pool, but no higher. 
Atmospheric pressure at sea lexel is not great enough to support 
a column of water any higher. Mercurv is almost 14 times as 
dense as water. Thus, ordinary pressure on a pool of mercury 
can support a column only '/i4 as high, about 0.76 m. This is a 
more convenient height for laboratory experiments. Therefore, 
much of the se\'enteenth-century resear'ch on air pr^essur'e was 
done with a column of mercury, or mercury' "barometer." The 
first of these barometers w^as designed by Torricelli. 

The height of the mer'cury column that can be supported by 
air pressure does not depend on the diameter of the tube; that 
is, it depends not on the total amount of mercury, but only on its 
height. This may seem str'ange at first. To understand it, you 
must understand the difference between pressure and force. 
Pressure is defined as the magnitude of the force acting 
perpendicularly on a surface dixided by the area of that surface: 
P — FJ_/A. Thus, a large for^ce may produce only a small 
pressure if it is spiead over a large area. For example, you can 
walk on snow without sinking in it if you wear snowshoes. On 
the other' hand, a small force can produce a very large pr'essure if 
it is concentrated on a small area. Women's spike heel shoes 
ha\'e rnjined many a wooden floor' or carpet. The pressure at the 
place wher'e the heel touched the floor was greater' than that 
under an elephant's foot. 

In 1661, two English scientists, Richard Towneley and Heniy 
Power, discovered an important basic r^elation. They found that 
the pressure e;<erted by a i^as is directly proportional to the 
density of that gas. Using P for pressui'e and D for density, this 
relationship is F ^ D or P = kD where k is some constant. For 
example, if the density of a given quantity of air is doubled Isay, 
by compressing it I, its pressure also doubles. Robert Boyle 
confirmed this relation by extensixe experiments. It is an 
empirical i\i\e, now generally known as Boyle s law. The law 
holds true only under special conditions. 

The effect of temperature on gas pressure. Boyle r^ecognized 



332 



UiMT 3 / THE TRIl'MPH OF MECHANICS 



that if the temperature of a gas changes during an experiment, 
the relation P = kD no longer applies. For example, the pressure 
exerted by . gas in a closed container increases if the gas is 
heated, even though its density stays constant. 

Many scientists throughout the eighteenth centuiy investigated 
the expansion of gases by heat. The experimental results were 
not consistent enough to establish a quantitative relation 
between density (or volume) and temperature. Eventually; 
evidence for a surprisingly simple general law appeared. The 
French chemist Joseph-Louis Gay-Lussac (1778-1850) found that 
all the gases he studied (air, oxygen, hydrogen, nitrogen, nitrous 
oxide, ammonia, hydrogen chloride, sulfur dioxide, and carbon 
dioxide) changed their volume in the same way. If the pressure 
remained constant, then the change in volume was proportional 
to the change in temperature. On the other hand, if the volume 
remained constant, the change in pressure was proportional to 
the change in temperature. 

A single equation summarizes all the experimental data 
obtained by Boyle, Gay-Lussac, and many other scientists. It is 
known as the ideal gas law: 

P = kD(t + 273°) 
Here, t is the temperature on the Celsius scale. The 
proportionality constant k depends only on the kind of gas (and 
on the units used for P, D, and t). 

This equation is called the ideal gas law because it is not 
completely accurate for real gases except at very low pressures. 
Thus, it is not a law of physics in the same sense as the law of 
conservation of momentum. Rather, it simply gives an 
experimental and approximate summary of the observed 
properties of real gases. It does not apply w^hen pressure is so 
high, or temperature so low, that the gas nearly changes to a 
liquid. 

The number 273 appears in the ideal gas law simply because 
temperature is measured on the Celsius scale. The fact that the 
number is 273 has no great importance. It just depends on the 
choice of a particular scale for measuring temperature. However, 
it is important to note what would happen if t were decreased 
to — 273°C. Then the entire factor involving temperature would 
be zero. And, according to the ideal gas law, the pressure of any 
gas would also fall to zero at this temperature. Real gases 
become liquid long before a temperature of — 273°C is reached. 
Both experiment and thermodynamic theory indicate that it is 
impossible actually to cool anything, gas, liquid, or solid, down 
to precisely this temperature. However, a series of cooling 
operations has produced temperatures less than 0.0001° above 
this limit. 

In view of the unique meaning of this lowest temperature, 
Lord Kelvin proposed a new temperature scale. He called it the 



On the Celsius scale, water freezes 
at 0° and boils at 100°, when the 
pressure is equal to normal atmos- 
pheric pressure. On the Fahrenheit 
scale, water freezes at 32° and boils 
at 212°. Some of the details involved 
in defining temperature scales are 
part of the experiment "Tempera- 
ture and Thermometers" in the 
Handbook. 



If the pressure were kept constant, 
then according to the ideal gas law, 
the volume of a sample of gas 
would shrink to zero at — 273°C. 



This "absolute zero" point on the 
temperature scale has been found 
to be -273.16° Celsius. 



CHAPTER 11 / THE KINETIC THEORY OF GASES 



333 



For our purposes, it is sufiiciently 
accurate to say the absolute tem- 
perature of any sample (symbol- 
ized by the letter T and measured 
in degrees Kelvin, or °K) is equal to 
the Celsius temperature t plus 273°: 

T = t + 273° 

The boiling point of water, for ex- 
ample, is 373°K on the absolute 
scale. 

SG 13, 14 



'Celsius 



327- 



IOO-- 

27-- 
0-- 

-76- 



leodmcll^ 



(IC«lvi0) 



wofcK boils 

v^Aarm room 
water ft-tfczcs 

COrbondfOKide 
•freezes (dry ce) 



-I8S 

-196 



-rioo 

^273 

195 

490 



-273 -'- absolute zero 

Comparison of the Celsius and ab- 
solute temperature scales. 



absolute temperature scale and put its zero at -273°C. The 
absolute scale is sometimes called the KeKin scale. The 
temperature of - 273°C is now referred to as 0"K on the absolute 
scale and is called the absolute zero of temperature. 
The ideal gas law may now be written in simpler fomi: 

P = kDT 

T is the temperature in degrees Kelvin, and k is the 
proportionality' constant. 

The equation P — kDT summarizes experimental facts about 
gases. Now you can see whether the kinetic-theoiy model offers 
a theoretical explanation for these facts. 

Kinetic explanation of gas pressure. According to the kinetic 
theory, the pressure of a gas results from the continual impacts 
of gas particles against the container wall. This explains why 
pressure is proportional to density: The greater the density, the 
greater the number of particles colliding with the wall. But 
pressure also depends on the speed of the individual particles. 
This speed determines the force exerted on the wall during each 
impact and the frequency of the impacts. If the collisions with 
the wall are perfectly elastic, the law of consenation of 
momentum will describe the r esults of the impact. The detailed 
reasoning for this procedure is worked out on pages 336 and 337. 
This is a beautifully simple application of Newtonian mechanics. 
The r^esult is clear: Applying Newtonian mechanics to the kinetic 
molecular model of gases leads to the conclusion that P = 
M)D(v^l^^ whei^e (v^).,, is the average of the squared speed of the 
molecules. 

So there are two expressions for the pressure of a gas. One 
summarizes the experimental facts, P = kDT. The other is 
derived by Neuton's laws from a theoretical model, P — VaDlv^)^^. 
The theoretical expression will agree with the experimental 
expression only if kT = V:i{v^)_^^. This would mean that the 
temperature of a gas is proportional to lv^i_^^. The mass m of each 
molecule is a constant, so the temperature is also proportional 
to Vzmlv^),^. Thus, the kinetic theory leads to the conclusion that 
the temperature of a gas is proportional to the average kinetic 
energy of its molecules! We already had some idea that raising 
the temperature of a material somehow affected the motion of its 
"small par'ts." We wer'e awar^e that the higher- the temper-atur-e 
of a gas, the more rapidly its molecules are moving. But the 
conclusion T ^ V2mi\^}^^^ is a pr-ecise quantitative r'elationshij:) 
derived from the kinetic model and empir ical hivvs. 

This relationship makes possible other quantitative predictions 
from kinetic theory. We know by experience that when a gas is 
compressed or- condensed rapidly, the temperature changes, and 
the general gas law (P = kDT) applies. Can the model explain 
this result? 



334 



UNIT 3 / THE TRIUMPH OF MECHA.MCS 



In the model used on pages 336-37; particles were bouncing 
back and forth between the walls of a box. Every collision with 
the wall was perfectly elastic, so the particles rebounded with no 
loss in speed. Suppose the outside force that holds one wall in 
place is suddenly reduced. What will happen to the wall? The 
force exerted on the wall by the collisions of the particles will 
now be greater than the outside force. Therefore, the wall will 
move outward. 

As long as the wall was stationary, the particles did no work 
on it, and the wall did no work on the particles. Now the wall 
moves in the same direction as the force exerted on it by the 
particles. Thus, the particles must be doing work on the wall. 
The energy needed to do this w^ork must come from somewhere. 
The only av^ailable source of energy here is the kinetic energy 
Wzmxr) of the particles. In fact, it can be shown that molecules 
colliding perfectly elastically with a receding wall rebound with 
slightly less speed. Therefore, the kinetic energy of the particles 
must decrease. The relationship T ^ Vimlv^)^^ implies that the 
temperature of the gas will drop. This is exactly what happens! 

If the outside force on the wall is increased instead of 
decreased, just the opposite happens. The gas is suddenly 
compressed as the w^all moves inward, doing work on the 
particles and increasing their kinetic energy. As Vzni\r goes up, 
the temperature of the gas should rise, which is just what 
happens when a gas is compressed quickly. 

Many different kinds of experimental evidence support this 
conclusion and therefore also support the kinetic-theory model. 
Perhaps the best evidence is the motion of microscopic particles 
suspended in a gas or liquid, called Brownian motion. The gas 
or liquid molecules themselves are too small to be seen directly, 
but their effects on a larger particle (for example, a particle of 
smoke) can be observed through the microscope. At any instant, 
molecules moving at very different speeds are striking the larger 
particle from all sides. Nevertheless, so many molecules are 
taking part that their total effect nearly cancels. Any remaining 
effect changes in magnitude and direction from moment to 
moment. Therefore, the impact of the invisible molecules makes 
the visible particle "dance" in the viewiield of the microscope. 
The hotter the gas, the more lively the motion, as the equation 
T <x Vamlv^)^^ predicts. 

This experiment is simple to set up and fascinating to watch. 
You should do it as soon as you can in the laboratory. It gives 
visible evidence that the smallest parts of all matter in the 
universe are in a perpetual state of lively, random motion. 




SG 16 



This phenomenon can be demon- 
strated by means of the expansion 
cloud chamber, cooling of CO, fire 
extinguisher, etc. Here the "waH" is 
the air mass being pushed away. 



Diesel engines have no spark plugs; 
ignition is produced by tempera- 
ture rise during the high compres- 
sion of the air-fuel vapor mixture. 



SG 15 



Brownian motion was named after 
the English botanist, Robert Brown, 
who in 1827 observed the phenom- 
enon while looking at a suspension 
of the microscopic grains of plant 
pollen. The same kind of motion of 
particles ("thermal motion") exists 
also in liquids and solids, but there 
the particles are far more con- 
strained than in gases. 



• 12. The relationship between the density and pressure of a gas 
e;sipressed by Boyle 's law, P = kD, holds true 

CHAPTER 11 / THE KINETIC THEORY OF GASES 



335 



Close Up I 



Deriving on Expression for Pressure 
from the Kinetic Theory 



We begin with the model of a gas described in 
Sec. 11.2: "a large number of very small particles 
in rapid, disordered motion." We can assume here 
that the particles are points with vanishingly small 
size, so that collisions between them can be ig- 
nored. If the particles did have finite size, the results 
of the calculation would be slightly different. But the 
approximation used here is accurate enough for 
most purposes. 

The motions of particles moving in all directions 
with many different velocities are too complex as 
a starting point for a model. So we fix our attention 
first on one particle that is simply bouncing back 
and forth between two opposite walls of a box. 
Hardly any molecules in a real gas would actually 
move like this. But we will begin here in this simple 
way and later in this chapter extend the argument 
to include other motions. This later part of the ar- 
gument will require that one of the walls be movable. 
Therefore, we will arrange for that wall to be mov- 
able, but to fit snugly into the box. 

In SG 24 of Chapter 9, you saw how the laws of 
conservation of momentum and energy apply to 
cases like this. When a very light particle hits a more 
massive object, like the wall, very little kinetic en- 
ergy is transferred. If the collision is elastic, the par- 
ticle will reverse its direction with very little change 
in speed. In fact, if a force on the outside of the wall 
keeps it stationary against the impact from inside, 
the wall will not move during the collisions. Thus no 
work is done on it, and the particles rebound without 
any change in speed. 




How large a force will these particles exert on the 
wall when they hit it? By Newton's third law the 
average force acting on the wall is equal and op- 
posite to the average force with which the wall acts 
on the particles. The force on each particle is equal 
to the product of its mass times its acceleration {F 
= mi), by Newton's second law. As shown in Sec. 
9.4, the force can also be written as 



F = 



limv) 



where A{mv) is the change in momentum. Thus, to 
find the average force acting on the wall we need 
to find the change in momentum per second due 
to molecule-wall collisions. 

Imagine that a particle, moving with speed v^ (the 
component of v in the x direction) is about to collide 
with the wall at the right. The component of the 
particle's momentum in the x direction is mv^. Since 
the particle collides elastically with the wall, it re- 
bounds with the same speed. Therefore, the mo- 
mentum in the x direction after the collision is 







m{-vj = -mv^. The change in the momentum of 
the particle as a result of this collision is 



final 


initial _ change in 


momentum 


momentum momentum 


i-mvj - 


{mvj = {-2mvJ 



Note that all the vector quantities considered in 
this derivation have only two possible directions: to 



-mv^ 



rtWx 



-^wv^ 



336 



UMT 3 / THE TRIUMPH OF MECHANICS 



the right or to the left. We can therefore indicate 
direction by using a + or a - sign, respectively. 

Now think of a single particle of mass m moving 
In a cubical container of volume L^ as shown In the 
figure. 







The time between collisions of one particle with 
the right-hand wall Is the time required to cover a 
distance 2L at a speed of v;, that is, 2Llv^. If 2Llv^ 
equals the time between collisions, then vJ2L 
equals the number of collisions per second. Thus, 
the change in momentum per second Is given by 



(change in (number of 

momentum in x collisions 
one collision) per second) 

[-2n)v) X {vJ2L) 



(change In 
momentum 
per second) 



- mv 



The net force equals the rate of change of momen- 
tum. Thus, the average force acting on the molecule 
(due to the wall) is equal to -mvf/L, and by New- 
ton's third law, the average force acting on the wall 
(due to the molecule) is equal to +mv^^/L. So the 
average pressure on the wall due to the collisions 
made by one molecule moving with speed v^ is 

r, _ F _ F _ mv] _ mv^ 

where v (here L^) Is the volume of the cubical con- 
tainer. 

Actually, there are not one but N molecules in the 
container. They do not all have the same speed, 
but we need only the average speed in order to find 
the pressure they exert. More precisely, we need 
the average of the square of their speeds In the x 



direction. We call this quantity {v^)^^. The pressure 
on the wall due to N molecules will be N times the 
pressure due to one molecule, or 



P = 



V 



In a real gas, the molecules will be moving in all 
directions, not just In the x direction; that is, a mol- 




ecule moving with speed v will have three compo- 
nents: v^, v^, and v^. If the motion is random, then 
there is no preferred direction of motion for a large 
collection of molecules, and (v ^) = (v^) = (v ^) . 

' \ X 'av V y 'av V z 'av 

It can be shown from Pythagoras' theorem that v^ 
= I// -I- v^^ + y^. These last two expressions can 
be combined to give 

or 



(y ') = hs/) 
V X 'av o^ 'av 

By substituting this expression for (i/^ 
sure formula, we get 



in the pres- 



P = 



Hm X y)^ 



_ 1 



/V/77, 



3 y ^ '^-^ 

Notice now that A/m is the total mass of the gas, 
and therefore himlV is just the density D. So 

P = ^(^')av 

This is our theoretical expression for the pressure 
P exerted on a wall by a gas in terms of its density 
D and the molecular speed y. 



CHAPTER 11 / THE KIIVETIC THEORY OF GASES 



337 



(a) for any gas under any conditions. 

(b) for some gases under any conditions. 

(c) only if the temperature is kept constant. 
(dj only if the density is constant. 

13. Using the concept of work and the kinetic theory of gases, 
explain why the temperature of a gas and the kinetic energy 
of its molecules both increase if a piston is suddenly pushed 
into the container. 

14. What are the limits under which the ideal gas law 
describes the behavior of real gases? 

11. .6 I The second law of thermodi'nainics and 
the dissipation of ener^' 



SG 23 



"Our life runs down in sending up 

the clock. 
The brook runs down in sending up 

our life. 
The sun runs dorni in sending up 

the brook. 
And there is something sending up 

the sun. 
It is this backward motion toward 

the source, 
Against the stream, that most we 

see ourselves in. 
It is from this in nature we are 

from. 
It is most us." 
[Robert Frost, West-Running Brook] 



You have seen that the kinetic-theory model can explain the way 
a gas hehaves when it is compressed or expanded, wanned or 
cooled. In the late nineteenth century, the model was refined to 
take into account many effects we have not discussed. There 
proved to be limits beyond which the model breaks down. For 
example, radiated heat comes from the sun through the vacuum 
of space, rhis is not explainable in terms of the thennal motion 
of particles. But in most cases the model worked splendidly, 
explaining the phenomenon of heat in terms of the ordinary 
motions of particles. This was indeed a triumph of Newtonian 
mechanics. It fulfilled much of the hope Newton had expi'essed 
in the Principia that all phenomena of nature could be explained 
in terms of the motion of the small parts of matter. 

A basic philosophical theme of the Newtonian cosmology is 
the idea that the world is like a machine whose parts never wear 
out and which never ixins doun. This idea inspired the search 
for conservation laws applying to matter and motion. So far in 
this text, you have seen that this search has been successful. We 
can measure "matter" by mass, and "motion" by momentum or 
by kinetic energy. By 1850, the law of conservation of mass had 
been firmly established in chemistry. In physics, the laws of 
conseivation of momentum and of energy had been equally well 
established. 

Yet these successful conseivation laws could not banish the 
suspicion that somehow the world is running dov\Ti, the paits of 
the machine are wearing out. Energy may be conserved in 
burning fuel, but it loses its usefiilness as the heat goes off into 
the atmosphei-e. Mass may be conserved in scrambling an egg, 
but the organized structure is lost. In these transformations, 
something is consened, but something is also lost. Some 
processes are irreversible; that is, they will not run backward. 
There is no way to unscramble an egg, although such a change 
would not \iolate mass conser-vation. There is no wa\' to draw 



338 



UNIT 3 / THE TRIUMPH OF MECHANICS 



smoke and hot fumes back into a blackened stick, forming a new, 
unburned match. 

Section 10.10 discussed one type of irreversible motion, that 
involving heat engines, which was governed by the second law of 
themiodynamics. That law can be stated in several equivalent 
ways: Heat will not by itself flow from a cold body to a hot one. It 
is impossible to fully convert a given amount of heat into work. 
The entropy of an isolated system, and therefore of the universe, 
tends to increase. 

The processes of scrambling an egg, of mixing smoke and air, 
or of wearing down a piece of machinery do not, at first sight, 
seem to obey the same laws as do heat engines. However, these 
processes also are governed by the second law. Heat, as you have 
seen, is represented by the disordered motions of atoms and 
molecules. Converting (ordered) mechanical work into heat thus 
leads to an increase in disordered motion. 

The second law says that it is never possible to reverse this 
increase entirely. Heat can never be turned entirely into work. 
When the entropy of a system increases, the disordered motion 
in the system increases. 

For example, think of a falling ball. If its temperature is very 
low, the random motion of its parts is very low, too. Thus, the 
motion of all particles during the falling is mainly downward 
("ordered"). The ball strikes the floor and bounces several times. 
During each bounce, the mechanical energy of the ball 
decreases, and the ball waiTns up. Now the random thermal 
motion of the parts of the heated ball is far more vigorous. 
Finally, the ball as a whole lies still (no "ordered" motion). The 
disordered motion of its molecules (and of the molecules of the 
floor where it bounced) is the only motion left. As with the 
bouncing ball, all motions tend from ordered to disordered. In 
fact, entropy can be defined mathematically as a measure of the 
disorder of a system (though it is not necessaiy to go into the 
mathematics here). 

Irreversible processes are processes for which entropy 
increases. For example, heat will not flow by itself from cold 
bodies to hot bodies. A ball lying on the floor will not somehow 
gather the kinetic energy of its randomly moving parts and 
suddenly leap up. An egg will not unscramble itself. An ocean 
liner cannot be powered by an engine that takes heat from the 
ocean water and ejects ice cubes. All these and many other 
events could occur without violating any principles of Newtonian 
mechanics, including the law of conservation of energy. But they 
do not happen; they are "forbidden" by the second law of 
thermodynamics. (They are "forbidden" in the sense that such 
things do not happen in nature.) 

All familiar processes are to some degree irreversible. Thus, 
Lord Kelvin predicted that all bodies in the universe would 



SG 24-26 




CHAPTER 11 / THE KINETIC THEORY OF GASES 



339 




TVvo illustrations fron^ Flanima- 
rion's novel, La Fin du Monde: 
ItopI "La miserable race hurnaine 
perira par le froid." ihottorn) "Ce 
sera la fin. " 

SG 27 



eventually reach the same temperature by exchanging heat vvltli 
one another. Wlien this happened, it would be impossible to 
produce any useful work from heat. After all, work can only be 
done by means of heat engines when heat flows from a hot body 
to a cold body. Finally, the sun and other stars would cool, all 
life on earth would cease, and the universe would be dead. 
This general "heat-death" idea, based on predictions from 
thermodvnamics, aroused some popular interest at the end of 
the nineteenth century. I'he idea appeai-ed in several books of 
that time, such as H. G. Wells' The Time Machine. The French 
astronomer Camille Flammarion vviote a book describing ways in 
which the world uould end. 



15. The presumed "heat death of the universe" refers to a 
state in which 

(a) all mechanical energy has been transformed into heat 
energy. 

(b) all heat energy has been transformed into other forms of 
energy. 

(c) the temperature of the universe decreases to absolute 
zero. 

(d) the supply of coal and oil has been used up. 

16. What is a reversible process? 

17. Which of the following statements agrees with the second 
law of thermodynamics? 

(a) Heat does not naturally flow from cold bodies to hot 
bodies. 

(b) Energy tends to transform itself into less useful forms. 

(c) No engine can transform all its heat input into mechanical 
energy. 

(d) Most processes in nature are reversible. 

18. If a pot of water placed on a hot stove froze, Newton's laws 
would not have been violated. Why would this event violate the 
second law of thermodynamics? If an extremely small group 
of water molecules in the pot cooled for a moment, would this 
violate the second law? 



1.1 .7 I MaxivelFs demon and the statistical liew 
of the second law of thermodynamics 

Is there any way of avoiding the "heat death .^ Is irre\ ersibility a 
basic law of physics, or is it only an approximation based on 
limited experience of natural processes? 



340 



UMT 3 / THE TRIUMPH OF MKCHAIVICS 



The Austrian physicist Ludwig Boltzmann investigated the 
theory of irreversibility. He concluded that the tendency toward 
dissipation of energy is not an absolute law of physics that 
always holds. Rather, it is only a statistical law. Think of a 
container of air containing about 10"^ molecules. Boltzmann 
argued that, of all conceivable arrangements of the gas molecules 
at a given instant, nearly all would be almost completely 
"disordered." Only a relatively few arrangements would have 
most of the molecules moving in the same direction. Even if a 
momentarily ordered arrangement of molecules occurred by 
chance, it would soon become less ordered by collisions. 




Drawing by Steinberg; © 1963, The New 
Yorker Magazine, Inc. 



Fluctuations from complete disorder vvdll, of course, occur. But 
the greater the fluctuations, the less likely they are to occur. For 
collections of particles as large as 10^^, the chance of a 
fluctuation large enough to be measurable is vanishingly small. It 
is conceivable that a cold kettle of water will heat up on its own 
after being struck by only the most energetic molecules in the 
surrounding air. It is also conceivable that air molecules will 
"gang up" and strike only one side of a rock, pushing it uphill. 
But such events, while conceivable, are utterly improbable. 

For small collections of particles, however, it is a different 
story. For example, it is quite probable that the average height of 
people on a bus will be considerably greater or less than the 
national average. In the same way, it is probable that more 
molecules will hit one side of a microscopic particle than the 
other side. Thus, we can observe the "Brownian" motion of 
microscopic particles. Fluctuations are an important aspect of 
the world of very small particles. Hou^ever, they are virtually 
undetectable for any large collection of molecules familiar in the 
everyday world. 

The second law is different in character from all the other 
fundamental laws of physics you have studied so far. The 
difference is that it deals with probabilities, not certainties. 

Maxwell proposed an interesting "thought experiment" to 
show a possible violation of the second law. Suppose a container 



Consider also a pool table. The or- 
dered motion of a cue ball moving 
into a stack of resting ones soon 
gets "randomized." 



To illustrate Boltzmann's argu- 
ment, consider a pack of cards 
when it is shuffled. Most possible 
arrangements of the cards after 
shuffling are fairly disordered. If 
you start with an ordered arrange- 
ment, for example, the cards sorted 
by suit and rank, then shuffling 
would almost certainly lead to a 
more disordered arrangement. 
(Nevertheless, it does occasionally 
happen that a player is dealt 13 
spades, even if no one has stacked 
the deck.) 




A living systen^ Itkti Ihis tree ap- 
pears to contradict the second law 
of thermodynamics by bringing 
order out of disorder. (See page 
342). 



CHAPTER 11 / THE KIIVETIC THEORY OF GASES 



341 




Initially the average KE of molecules 
is greater in A. 




Only fast molecules are allowed to go 
from B to A. 




Only slow molecules are allowed to 
go from A to B 




As this continues, the average KE 
in A increases and the average KE 
in fi decreases. 

How Maxwell's "demon could use 
a small, massless door to increase 
the order of a system and make 
heat flow from a cold gas to a hot 
gas. 



SG 28 



of gas is divided by a diaphragm into two parts, the gas in one 
part being hotter than in the other. "Now conceive of a finite 
being, " Maxwell suggested, "who knows the patiis and velocities 
of all the molecules but who can do no work except open and 
close a hole in the diaphragm." This "finite being, " now known 
as 'Maxwell's demon, ' can make the hot gas hotter and the cold 
gas cooler just by letting fast molecules move in only one 
direction through the hole (and slow molecules in the other), as 
is shown in the diagram. 

There is, of course, no such fanciful demon (even in machine 
form) that can obseive and keep track of all the molecules in a 
gas. (If somehow it could be made to exist, one might find that 
the demon's entropy is affected by its actions. For example, its 
entropy might increase enough to compensate for the decrease 
in entropy of the gas. This is what happens in other systems 
where local order is created; the entropy elsewhere must 
increase.) 

Some biologists have suggested that certain large molecules, 
such as enzymes, may function as "Maxwell's demons." Large 
molecules may influence the motions of smaller molecules to 
build up the ordered stnjctures of living systems. I'his result is 
different from that of lifeless objects and is in apparent violation 
of the second law of thennodvTiamics. This suggestion, however, 
shows a misunderstanding of the law. The second law does not 
say that the order can never increase in any system. It makes 
that claim only for closed or isolated systems. Any system that 
can exchange energy with its surroundings can increase its 
order. 

There is some evidence, in fact, that the flow of energy through 
a system that is not closed tends to produce order in the system. 
The law governing these processes seems to be, again, the 
second law of thermodynamics. The existence of life therefore 
may be a result of the energy flow from the sun to the earth. Far 
from being a violation of the second law, life would be a 
manifestation of it. Life does have its cost in terms of the effect 
on the rest of the total system. This point is expressed vividly in 
the following passage from a UNESCO document on 
environmental pollution. 

Some scientists used to feel that tine occurrence, reproduction, 
and grovvtli of older in living systems presented an exception to 
the second law. This is no longer believed to be so. True, the 
living system may increase in order, but only by diffusing 
energy to the surroundings and by converting complicated 
molecules (carbohydrates, fats) called food into simple 
molecules (CO,, H,OI. For example, to maintain a healthy 
human being at constant weight for one yeai- i-equires the 
degradation of about 500 kilograms (one half ton) of food, and 
the diffusion into the surroundings (ft-om the human and the 
foodi of about 500,000 kiloc'alories (tvvo million kilojoulesl of 



342 



UNIT 3 / THE TRIUMPH OF MECH/WICS 



energy. The "order" in the human may stay constant or even 
increase, but the order in the surroundings decreases much, 
much more. Maintenance of life is an expensive process in 
terms of generation of disorder, and no one can understand the 
full implications of human ecology and environmental pollution 
without understanding that first. 



19. In each of the following pairs, which situation is more 
ordered? 

(a) an unbroken egg; a scrambled egg. 

(b) a glass of ice and warm water; a glass of water at uniform 
temperature. 

20. True or false? 

(a) Ma^iwell's demon was able to get around the second law of 
thermodynamics. 

(b) Scientists have made a Ma^<well's demon. 

(c) Ma^cwell believed that his demon actually existed. 



H .8 I Time's arrow and the recurrence 
paradox 

Late in the nineteenth century, a small but influential group of 
scientists began to question the basic philosophical assumptions 
of Newtonian mechanics. They even questioned the very idea of 
atoms. The Austrian physicist Ernst Mach argued that scientific 
theories should not depend on assuming the existence of things 
(such as atoms) which could not be directly observed. Typical 
of the attacks on atomic theory was the argument used by the 
mathematician Ernst Zermelo and others against kinetic theory. 
Zermelo believed that (1) the second law of themiodynamics is 
an absolutely valid law of physics because it agrees with all the 
experimental data. However, 12) kinetic theoiy allows the 
possibility of exceptions to this law (due to large fluctuations). 
Therefore, (3) kinetic theory must be wrong. It is an interesting 
historical episode on a point that is still not quite settled. 

The critics of kinetic theory pointed to two apparent 
contradictions between kinetic theory and the principle of 
dissipation of energy. These contradictions were the reversibility 
paradox and the recurrence parado^. Both paradoxes are based 
on possible exceptions to the second law; both could be thought 
to cast doubt on the kinetic theory. 

The reversibility paradox was discovered in the 1870 s by Lord 
Kelvin and Josef Loschmidt, both of whom supported atomic 
theory. It was not regarded as a serious objection to the kinetic 
theory until the 1890's. The paradox is based on the simple fact 
that Neu1:on's laws of motion are reversible in time. For example, 




The reversibility paradox: Can a 
model based on reversible events 
e^cplain a world in which so many 
events are irreversible? (Also see 
photographs on ne^ct page.) 



CHAPTER 11 / THE KINETIC THEORY OF GASES 



343 





M^^ 







if you watch a motion picture of a bouncing ball, it is easy to tell 
whether the film is being nm forward or backward. You know 
that the collisions of the ball with the floor are inelastic and that 
the ball rises less high after each bounce. If, however, the ball 
made perfectly elastic bounces, it would rise to the same height 
after each bounce. Then you could not tell v\'hether the film was 
being run forward or backward. In the kinetic theory, molecules 
are assumed to make perfectly elastic collisions. Imagine that 
you could take a motion picture of gas molecules colliding 
elastically according to this assumption. When showing this 
motion picture, there would be no way to tell whether it was 
being run forward or backwaid. Either way would show valid 
sequences of collisions. But here is the paradox: Consider motion 
pictures of interactions invoking large objects, containing many 
molecules. You can immediately tell the difference between 
forward (true) and backward (impossible) time direction. For 
example, a smashed light bulb does not reassemble itself in real 
life, though a mo\ie run backward can make it appear to do so. 

The kinetic theory is based on laws of motion that are 
reversible for each individual molecular interaction. How, then, 
can it explain the existence of irreversible processes on a large 
scale? The existence of such processes seems to indicate that 
time flows in a definite direction, that is, from past to future. 
This contradicts the possibility', implied in Nev\ton's laws of 
motion, that it does not matter whether we think of time as 
flowing fonvard or backward. As Lord Kelvin expressed the 
paradox, 

If . . . the motion of every particle of matter in tfie universe were 
precisely reversed at any instant, the course of nature would 
be simply reversed for ever after. The bursting bubble of foam at 
the foot of a waterfall would reunite and descend into the 
water; the thermal motions would reconcentrate their energy, 
and throw the mass up tlie fall in drops refonning into a close 
column of ascending water. Heat which had been generated 
by the friction of solids and dissipated by conduction, and 
radiation with absorption, would come again to the place of 
contact, and throw the moving body back against the force to 
which it bad previously yielded. Boulders would recover li-om 
the mud the materials required to rebuild them into their 
previous jagged forms, and would become reunited to the 
mountain peak from which they had formerly broken away. 
And if also the materialistic hypothesis of life were Uwe, living 
creatures would grow backwai'ds, with conscious knowledge of 
the future, but no memory of the past, and would become 
again unboiri. But the real phenomena of life infinitely 
transcend human science; and speculation regarding 
consequences of their imagined revei-sal is utterly un[)r{)fitable. 

Kelvin himself, and later Boltzmann, used statistical probability 
to explain whv we do not obseive such large-scale revei-sais. 



344 



UNIT 3 / THE THILMPH OF MECHANICS 



There are almost infinitely many possible disordered 
arrangements of water molecules at the bottom of a waterfall. 
Only an extremely small number of these arrangements would 
lead to the process described above. Reversals of this kind are 
possible in principle^ but for all practical purposes they are out of 
the question. 

The answer to Zemielo's argument is that his first claim is 
incorrect. The second law of thermodynamics is not an absolute 
law, but a statistical law. It assigns a veiy low probability to ever 
detecting any overall increase in order, but does not declare it 
impossible. 

However, another small possibility allowed in kinetic theoiy 
leads to a situation that seems unavoidably to contradict the 
dissipation of energy. The recurrence paradox revived an idea 
that appeared ft equently in ancient philosophies and is present 
also in Hindu philosophy to this day: the myth of the "eternal 
return." According to this myth, the long-range history of the 
world is cyclic. All historical events eventually repeat themselves, 
perhaps many times. Given enough time, even the matter that 
people were made of will eventually reassemble by chance. Then 
people w^ho have died may be born again and go through the 
same life. The Gemian philosopher Friedrich Nietzsche was 
convinced of the truth of this idea. He even tried to prove it by 
appealing to the principle of conservation of energy. Nietzsche 
wrote: 

If the universe may be conceived as a definite quantity of 
energy, as a definite number of centres of energy — and every 
other concept remains indefinite and therefore useless — it 
follows therefrom that the universe must go through a 
calculable number of combinations in the great game of chance 
which constitutes its existence. In infinity [of time], at some 
moment or other, every possible combination must once have 
been realized; not only this, but it must have been realized an 
infinite number of times. 

If the number of molecules is finite, there is only a finite number 
of possible arrangements of molecules. Therefore, somewhere in 
infinite time the same combination of molecules is bound to 
come up again. At the same point, all the molecules in the 
universe would reach exactly the same arrangement they had at 
some previous time. All events following this point would have 
to be exactly the same as the events that followed it before. That 
is, if any single instant in the histoiy of the universe is ever 
e^cactly repeated, then the entire histoiy of the universe wdll be 
repeated. As a little thought shows, it would then be repeated 
over and over again to infinity. Thus, energy would not endlessly 
become dissipated. Nietzsche claimed that this view of the 
eternal return disproved the "heat-death" theory. 



SG 29 



The World's great age begins anew, 

The golden years return. 
The earth doth like a snake renew 

His winter weeds outworn . . . 
Another Athens shall arise 

And to remoter time 
Bequeath, like sunset to the skies, 

The splendour of its prime . . . 
[Percy Bysshe Shelley, "HeUas" 
(1822)] 




Lord Kelvin (1824-1907) 



CHAPTER 11 / THE KINETIC THEORY OF GASES 



345 



SG 30-32 



SG 33 




Record of a particle in Brownian 
motion. Successive positions, re- 
corded every 20 sec, are connected 
by straight lines. The actual paths 
between recorded positions would 
be as erratic as the overall path. 



At about the same time, in 1889, the French mathematician 
Henri Poincare published a theorem on the possibility of 
recurrence in mechanical systems. According to Poincare, even 
though the universe might undergo a heat death, it would 
ultimately come alive again: 

A bounded vvoild, governed only by the laws of nieclianics, will 
always pass through a state veiy close to its initial slate. On the 
other hand, according to accepted experimental laws (if one 
attributes absolute validity to them, and if one is willing to 
pi-ess their consequences to the extreme), the universe tends 
toward a certain final state, from wliich it will nevei- depait. In 
this final state, from which v\ill be a kind of death, all bodies 
will be at rest at the same temperature. 

. . . the kinetic theories can extricate themselves fi'om this 
contradiction. The world, according to them, tends at fii-st 
toward a slate where it remains for a long time without 
apparent change; and this is consistent v\ath experience; but it 
does not remain that way forever; ... it merely stays there for 
an enormously long time, a time which is longer the more 
numerous are the molecules. This stale will not be the final 
death of the universe, but a sort of slumber, from which it will 
awake after millions of centuries. 

According to this theory, to see heat pass from a cold bod\' to 
a war-m one, it will rnot be necessary to have the acute vision, 
the intelligence, and the dexterity of Maxwell's demon; it will 
suffice to have a little patience. 

Poincare was willing to accept the possibility of a violation of the 
second law after a very long time. Others refused to admit exen 
this possibility. In 1896, Zermelo published a paper attacking not 
only the kinetic theory but the mechanistic world \dew in 
general. This view, he asserted, contradicted the second law. 
Boltzmann replied, repeating his earlier explanations of the 
statistical nature of irreversibility. 

The final outcome of the dispute between Boltzmann and his 
critics was that both sides were partly i ight and partly wrong. 
Mach and Zermelo were correct in believing that Newton's laws 
of mechanics cannot fully descr ibe molecular and atomic 
processes. IWe will come back to this subject in Unit 5.) For 
example, it is only approximately valid to describe gases in terms 
of collections of frantic little balls. Hut Boltzmann was right in 
defending the usefulness of the molecular- model. I'he kinetic 
theory is veiy nearly correct except for those properties of matter 
that involve the strTicture of molecules themselxes. 

In 1905, Alber-t Einstein pointed out that the fluctuations 
predicted by kinetic theory could be used to calculate the rate of 
displacement for particles in "BrovviTian" motion. Precise 
quantitative studies of Brownian motion confir-med Kinstein's 
theoretical calculations. This new success of kinetic theory, along 
with discoveries in r'adioac'tixity and atomic phvsi(\s, persuaded 



346 



UNIT 3 / THE TRIUMPH OF MECHANICS 



1700 



1750 



1800 



1736 



WATT 



GEORGE WASHINGTON 

MABJii^NTOINETTE 

NAPOLEON BONAPARTE 



1819 



Q 1 



FREDERICK THE GREAT 



lvictoria 



ABRAHAM LINCOLf 



1850 



Ml^ 



T, _ 3 



BENJAMIN FRANKLIN] 
LEONHARDEULER 

JOSEPH BLAC» 
^JOSEPH PRIESTLEY 



'^^ADI CARN( 




ANTOINE LAVOISIER 



PIERRE LAPLACE 

BENJAMIN THOMPSON, COUNT RUMFORD 
JOHN DALTON 



JEAN BAPTISTE FOURIER 

KARL FRIEORICH GAUSS 



^OLTAIRE ^^^Hl 

LiJiUIUHUMk 



JOHN STUART MILL' 




FRIEDRICH VON SCHELLING 



4 ALEXANDER POPEi 

SAMUEL JOHNSOr 



ROBERT BURNS 



I^adamedestaI 



CHARLES DICKENS 

JOHN KEATS 
I 
^^^■L EMILY BRONTE' 



LiOHANN WOLFGANG VON GOETHE 



ELIZABETH BARRETT BROWNIN( 



WILLIAM BLAKE 




^WILLIAM HOGART 



JEANAUGUSTE INGRES 



^JOSHUA REYNOLDS 

^THOMAS GAINSBOROUGH 
iJOHN SINGLETON COPLEY' 



(.FERDINAND DELACROIX 




FRANCISCO GOYA 1 



JACQUES LOUIS DAVID 



-^^TbNIO VIVALDI 

^OHANN SEBASTIAN BACH 

I 

-4GE0RGE FREDERICK HANDEL 



WOLFGANG MOZART ~ FRANZ SCHUBERT ^ 
LUDWIG VAN BEETHOVEN 

HECTOR BERLIOZ 



almost all the critics that atoms and molecules do exist. But the 
problems of irreversibility and of whether- the laws of physics 
must distinguish between past and future survixed. In a new 
form, these issues still interest physicists today. 

This chapter concludes the application of \e\\1onian 
mechanics to individuid particles. The stor^' was mainly one of 
triumphant success. However, like all theories, Newtonian 
mechanics has serious limitations. These will be explored later. 

The last chapter in this unit coxer^s the successful use of 
Newtonian mechanics in the case of mechanical wave motion. 
Wave motion completes the list of possibilities of particle motion. 
In Unit 1, you studied the motion of single particles or- isolated 
objects. The motion of a system of objects bound by a force of 
interaction, such as the earth and sun, was tr^eated in Unit 2 and 
in Chapter's 9 and 10 of this unit. In this chapter', you observed 
the motions of a system of a very large number of separate 
objects. Finally, in Chapter' 12 you will study the action of many 
particles going back and foi'th together- as a wave passes. 



SG34 
SG35 



Zl. The kinetic energy of a falling stone is transformed into 
heat when the stone strikes the ground. Obviously, this is an 
irreversible process; you never see the heat transform into 
kinetic energy of the stone, so that the stone rises off the 
ground. Scientists believe that the process is irreversible 
because 

(a) New'ton's laws of motion prohibit the reversed process. 

(b) the probability of such a sudden ordering of molecular 
motion is e^ctreinely small. 

(c) the reversed process would not conserx'c energy. 

(d) the reversed process would violate the second law of 
thermodvnamics. 



The ruins of a Greek temple at Del- 
phi are an elegant testimony to 
the continual encroachment of dis- 
order. 




*v 







348 



UNIT 3 / THE TRIUMPH OF MECHANICS 



study 
guide 



1. The Project Physics materials particularly 
appropriate for Chapter 11 include: 

Experiments 

Monte Carlo Elxperiment on Molecular 

Collisions 
Behavior of Gases 

Activities 

Drinking Duck 

Mechanical Equivalent of Heat 

A Diver in a Bottle 

Bockets 

How to Weigh a Car with a Tire Pressure Gauge 

Perpetual-Motion Machines 

Film Loop 

Beversibility of Time 

2. The idea of randomness can be used in 
predicting the results of flipping a large number of 
coins. Give some other examples where randomness 
is useful. 

3. The examples of early kinetic theories given in 
Sec. 11.3 include only quantitative models. Some of 
the underlying ideas are thousands of years old. 
Compare the kinetic molecular theory of gases to 
these Greek ideas expressed by the Roman poet 
Lucretius in about 60 B c : 

If you think that the atoms can stop and by their 
stopping generate new motions in things, you are 
wandering far from the path of truth. Since the 
atoms are moving freely through the void, they 
must aU be kept in motion either by their own 
weight or on occasion by the impact of another 
atom. For it must often happen that two of them 
in their course knock together and immediately 
bounce apart in opposite directions, a natural 
consequence of their hardness and solidity and the 
absence of anything behind to stop them. . . . 

It clearly follows that no rest is given to the 
atoms in their course through the depths of space. 
Driven along in an incessant but variable 
movement, some of them bounce far apart after a 



collision while others recoil only a short distance 
from the impact. From those that do not recoil 
far, being driven into a closer union and held there 
by the entanglement of their own interlocking 
shapes, are composed firmly rooted rock, the 
stubborn strength of steel and the like. Those 
others that move freely through larger tracts of 
space, springing far apart and carried far by the 
rebound — these provide for us thin air and blazing 
sunlight. Besides these, there are many other 
atoms at large in empty space which have been 
thrown out of compound bodies and have 
nowhere even been granted admittance so as to 
bring their motions into harmony. 

4. What is a distribution? Under what conditions 
would you expect a measured distribution to be 
similar to an ideal or predicted distribution? 

5. Consider these aspects of the curves showing 
Maxwell's distribution of molecular speeds: 

(a) AU show a peak. 

(b) The peaks move toward higher speed at higher 
temperatures. 

Explain these characteristics on the basis of the 
kinetic model. 

6. The measured speed of sound in a gas turns out 
to be nearly the same as the average speed of the gas 
molecules. Is this a coincidence? Discuss. 

7. How did Clausius modify the simple kinetic 
model for a gas? What was he able to explain with 
this new model? 

8. Benjamin Franklin observed in 1765 tliat a 
teaspoonful of oil would spread out to cover half an 
acre of a pond. This helps to give an estimate of the 
upper limit of the size of a molecule. Suppose that 
1 cm^ of oil forms a continuous layer one molecule 
thick that just covers an area on water of 1,000 m'. 

(a) How thick is the layer? 

(b) What is the size of a single molecule of the oil 
(considered to be a cube for simplicity)? 

9. Knowing the size of molecules allows us to 
compute the number of molecules in a sample of 
material. If we assume that molecules in a solid or 
liquid are packed close together, something like 



CHAPTER 11 / STUDY GUIDE 



349 



sg 



apples in a bin, then the total volume of a material is 
approximately equal to the volume of one molecule 
times th(! number of molecules in the material. 

(a) Houghly how many molecules are there in 1 cm^ 
of water? I For this approximation, you can take the 
volume of a molecule to be d' if its diameter is d.l 

(b) The density of a gas lat atmospheric pressure 
and 0°CI is about 1 1000 the densit\' of a liquid. 
Roughly how many molecules are there in 1 cm' of 
gas? Does this estimate support the kinetic model 
of a gas as described on p 336? 

10. How high could water be raised with a lift pump 
on the moon? 

11. At sea level, the atmospheric pressure of air 
ordinarily can balance a barometer column of 
mercur\' of height 0.76 m or 10.5 m of water. Air is 
approximately a thousand times less dense than 
liquid water. What can you say about the minimum 
height to which the atmosphere goes above the 
earth? 

12. (a) The pressure of a gas is 100 N/m'. If the 
temperature is doubled while the density is cut to 
one-third, what is the new pressure? 

(b) The temperature of a gas is 100°C. If the pressure 
is doubled and the density is also doubled by cutting 
the volume in half, what is the new temperature? 

13. What pressure do you exert on the ground when 
you stand on flat-heeled shoes? skis? skates? 

14. From the definition of density' , D = A//V' (where 
M is the mass of a sample and \' is its volume), write 
an expression relating pressure P and volume V of 

a gas. 

15. State the ideal gas law. What three 
proportionalities are contained in this law? What are 
the limitations of this law? 

16. Show how all the proportionalities describing 
gas behavior on p. 332 are included in the ideal gas 
law: P = kD (t + 273°). 

17. The following information appeared in a 
pamphlet published by an oil company: 

HOWS VOIH I'IHE PHESSURF? 



If you last checked the pressure in your tires on a 
warm day, one cold morning you may find your 
tires seriously underinflated. 

The Rubber Manufacturers Association warns 
that tire pressures drop approximately 12 kPa for 
every 10-deg dip in outside air. If your tires 
register 165 kPa pressure on a 30°C day, for 
example, they'll have only 130 kPa pressure when 
the outside air plunges to 0°C. 

If you keep your car in a heated garage at 15°C, 
and drhe out into a — 30°C morning, your tire 
pressure drops from 165 kJ'a to 125 kPa. 

Are these statements consistent with the ideal gas 
law? {i\'otc: The pressure registered on a tire gauge is 
the pressure above normal atmospheric pressure 
of about 100 kPa.i 



■ 




1 


1 


w 




1 


^B ' "■ SOT t: TUIf. INCINMOT! 
^^^^ 0. STO«f . ,vl IJO f Co— - 


1 



18. Distinguish between two uses of the word 
"model" in science. 

19. If a light particle rebounds from a massive, 
stationary wall with almost no loss of speed, then, 
according to the principle of Galilean relativity, it 
would stUl rebound from a moxing wall without 
changing speed as seen in the fnimc of reference of 
the mo\ing wall. Show that th(! reljound speed as 
measured in the laboratory would be less from a 
refreating wall (as is claimed at the bottom of p. 335). 
(Hint: First write an expression relating the particle's 
speed relative-to-the-wiill to its speed relati\c-to-the- 
laboratory.) 

20. What would you expect to happen to the 
temperature of a gas that v\'as released from a 
container in empty space (that is, with nothing to 
push back)? 



350 



CHAPTER 11 / STUDY GUIDE 



sg 



21. List some of the directly observable properties 
of gases. 

22. What aspects of the behavior of gases can the 
kinetic molecular theory be used to explain 
successfully? 

23. Many products are now sold in spray cans. 
Elxplain in terms of the kinetic theory of gases why it 
is dangerous to expose the cans to high 
temperatures. 

24. When a gas in an enclosure is compressed by 
pushing in a piston, its temperature increases. 
Elxplain this fact in two ways: 

(a) by using the first law of thermodynamics. 

(b) by using the kinetic theory of gases. 

The compressed air eventually cools down to the 
same temperature as the surroundings. Describe this 
heat transfer in terms of molecular collisions. 




25. From the point of view of the kinetic theory, 
how can you explain: (a) that a hot gas would not 
cool itself down while in a perfectly insulated 
container? (b) how a kettle of cold water, when put 
on the stove, reaches a boiling temperature. (Hint: At 
a given temperature the molecules in and on the 
walls of the solid container are also in motion, 
although, being part of a solid, they do not often get 
far away.) 

26. In the Principia, Newton expressed the hope that 
all phenomena could be explained in terms of the 
motion of atoms. How does Newton's view compare 
with this Greek view expressed by Lucretius in about 
60 B c ? 

I will now set out in order the stages by which 
the initial concentration of matter laid the 
foundations of earth and sky, of the ocean depths 
and the orbits of sun and moon. Certainly the 
atoms did not post themselves purposefully in due 
order by an act of intelligence, nor did they 
stipulate what movements each should perform. 
But multitudinous atoms, swept along in 
multitudinous courses through infinite time by 
mutual clashes and their own weight, have come 
together in every possible way and realized 
everything that could be formed by their 
combinations. So it comes about that a voyage of 
immense duration, in which they have experienced 
every variety of movement and conjunction, has 
at length brought together those whose sudden 
encounter normally forms the starting-point of 
substantial fabrics — earth and sea and sky and the 
races of living creatures. 

27. In Sec. 11.6, three statements of the second law 
of thermodynamics are given. In Chapter 10, we 
showed that the first and third laws are equivalent. 
Show that the first and second laws are also 
equivalent, by using an argument analogous to 
Carnot's; that is, show that if an engine violates 
either statement, then this engine (perhaps together 
with a reversible engine) also violates the other. 

28. There is a tremendous amount of internal 
energy in the oceans and in the atmosphere. What 
would you think of an invention that purported to 
draw on this source of energy to do mechanical 
work? (For example, a ship that sucked in seawater 



CHAPTER 11 / STUDY GUIDE 



351 



sg 



and exhausted blocks of ice, using the heat from the 
water to run the ship.) 

29. Compute the entropy change associated with the 
bouncing ball of page 339. Assume that the biill has 

a mass of 0.1 kg and fidls from a height of 1 m. How 
much energy is converted to heat? if the process 
happens nearly at room temperature (a bit less than 
300°K), what is the entropy change of the universe? 
Does it matter how much of the heat goes to the ball 
and how much goes elsewhere? 

30. Since there is a tendency for heat to flow from 
hot to cold, wiil the universe eventually reach 
absolute zero? 

31. Does Maxwell's demon get around the second 
law of thermodynamics? List the assumptions in 
Maxwell's argument. Which of them do you believe 
are likely to be true? 

32. Since all the e\idence is that molecular motions 
are random, you might expect that any given 
arrangement of molecules vviU recur if you just wait 
long enough. Elxplain how a paradox arises when this 
prediction is compared with the second law of 
thermodynamics. 

33. (a) Explain what is meant by the statement that 
Newton's laws of motion are time-reversible. 

(b) Describe how a paradox arises when the time- 
reversibUity of Neuion's laws of motion is compared 
with the second law of thermodynamics. 

34. Why is the melting of an ice cube an irreversible 
process even though it could easUy be refrozen? 

35. if there is a finite probability of an exact 
repetition of a state of the universe, there is also a 
finite probability of its exact opposite, that is, a state 
where molecules are in the same position but with 
reversed velocities. What would this imply about the 
subsequent history of the universe? 

36. List the assumptions in the "recurrence" theory. 
Which of them do you believe to be true? 

37. Some philosophical and religious systems of the 
Far East and the Middle East include the idea of the 
eternal return. If you have read about some of these 
philosophies, discuss what analogies exist to some 
of the ideas in the last part of this chapter. Is it 



appropriate to take the existence of such analogies to 
mean that there is some direct connection between 
these philosophical and physical ideas? 

38. Where did Newtonian mechanics run into 
difficulties in explaining the behavior of molecules? 

39. What are some advantages and disadvantages of 
theoretical models? 

40. At any point in a fluid, the upward force on a 
column of fluid at rest must be sufficient to support 
the weight of the fluid above it. The pressure of a 
fluid must therefore increase with the depth of the 
fluid. Consider a fluid whose density is D. By how 
much does the pressure increase if the depth below 
the surface is increased by ;t? (Consider the force on 
a column 1 m"^ in area.) 




41. Archimedes' principle: The fact that pressure 
increases with depth implies that an object (say, a 
balloon) forcibly immersed in a fluid uill experience 
a strong upward force from the fluid. The reason 
is that the fluid touching the bottom of the balloon 
has a greater pressure than has the fluid touching 
the top, so the upward force on the object exerted by 
the fluid below is greater than the downward force 
exerted by the fluid above it. This difference is called 
the "buoyant force." 

You can find the buoyant force as follows. 
C()nsid(!r a cube, length /. on a side, immersed in a 



352 



CHAPTER 11 / STUDY GUIDE 



so 



fluid of density D. In SG 36 you showed that the (c) What is the force exerted on the bottom of the 

pressure increases by Dg^ as tlie depth increases cube (magnitude and direction)? 

^ (d) What is the net force exerted by the fluid on tlie 

(a) Suppose the pressure at the top surface of the cube? 

cube is p. What is the pressure at the bottom? 

(e) Show that this force is equal to the weight of the 

(b) What is the force exerted on the top of the cube fluid displaced by the object, 
(magnitude and direction)? 



CHAPTER 11 / STUDY GUIDE 353 




Wwes 



12.1 What is a nave? 
12J3 Properties ofivaves 

12.3 Wave propagation 

12.4 Periodic naves 

12.5 When u'aves meet: the superposition principle 

12.6 A tivo-source interference pattern 

12.7 Standing naves 

12.8 Wave fronts and diiiraction 

12.9 Reflection 

12.10 Refraction 

12.11 Sound n'ai'es 



SG 1 




A small 
right of 
364. 



section from the lower 
the photograph on page 



'L2*'L I Uliat is a wave? 

The world is continually criss-crossed by waves of all sorts. 
Water waves, whether giant rollers in the middle of the ocean or 
gently fomied rain I'ipples on a still pond, are sources of wonder 
or pleasure. If the earth's crust shifts, violent waxes in the solid 
earth cause tremors thousands of kilometers away. A musician 
plucks a guitar string, and sound waxes pulse against the ears. 
Wave disturbances may come in a concentrated bundle, like the 
shock front from an aiiplane flying at supersonic speeds. Or the 
disturbances niiiy come in succession like the train of waxes sent 
out from a steadily vibrating source, such as a bell or a string. 

All of these examples are mcchnnicnl waxes, in which l)odies or 
particles physically moxe back and forth. Ihere are also xxaxe 
disturbances in electric and magnetic fields. In Unit 4, you will 
learn that such waxes are responsible for xvhat x'our sens(\s 



354 



UNIT 3 / THE TRIUMPH OF MECHAMCS 



experience as light. In all cases involving waves, however, the 
effects produced depend on the flow of energy as the wave 
moves forward. 

So far in ths text, you have considered motion in terms of 
Individual particles. In this chapter, you will study the 
cooperative motion of collections of particles in "continuous 
media" moving in the fonn of mechanical waves. You udll see 
how^ closely related are the ideas of particles and waves used to 
describe events in nature. 

A comparison vvdll help here. Look at a black and vv^hite 
photograph in a newspaper or magazine with a magnifying glass. 
You wdll see that the picture is made up of many little black dots 
printed on a white page (up to 3,000 dots per square centimeter). 
Without the magnifier, you do not see the indixddual dots. 
Rather, you see a pattern with all possible shadings between 
completely black and completely vuhite. These two views 
emphasize different aspects of the same thing. In much the same 
way, the physicist can sometimes choose between two (or more) 
ways of viewing events. For the most part, a particle view has 
been emphasized in the first three units of Project Physics. In 
Unit 2, for example, each planet was treated as a particle 
undergoing the sun's gravitational attraction. The behavior of the 
solar system was described in temis of the positions, velocities, 
and accelerations of point-like objects. For someone interested 
only in planetaiy motions, this is fine. But for someone 
interested in, say, the chemistry of materials on Mars, it is not 
very helpful. 

In the last chapter, you saw two different descriptions of a gas. 
One was in terms of the behaxdor of the individual particles 
making up the gas. Newton's laws of motion described what 
each individual particle does. Then average values of speed or 
energy described the behavior of the gas. You also studied 
concepts such as pressure, temperature, heat, and entropy. 
These concepts refer directly to a sample of gas as a whole. This 
is the viewpoint of thermodynamics, which does not depend on 
assuming Newton's laws or even the existence of particles. Each 
of these viewpoints sensed a useful purpose and helped you to 
understand what you cannot directly see. 

To study waves, you once again see the possibility of using 
different points of view. Most of the waves discussed in this 
chapter can be described in temis of the behavior of particles. 
But you also can understand waves as disturbances traveling in a 
continuous medium. You can, in other words, see both the forest 
and the trees; the picture as a whole, not only individual dots. 



13.2 I Properties of ivaves 



Waves should be studied in the lab- 
oratoiy. Most of this chapter is only 
a summary of some of what you 
will learn there. Film loops on 
waves are listed in SG 1. 



Suppose that tw^o people are holding opposite ends of a rope. 
Suddenly one person snaps the rope up and dovvTi quickly once. 



CHAPTER 12 / WAVES 



355 




1> 



(a) 



(b) 



"Snapshots" of three types of waves 
on a spring. In (c), small markers 
have been put on the top of each 
coil in the spring. 



That "disturbs" the rope and puts a hump in it which travels 
along the rope toward the other person. The traveling humjD is 
one kind of a wave, called a pulse. 

Originally, the rope was motionless. Ihe height of each point 
on the rope depended only upon its position along the rope and 
did not change in time. When one person snaps the rope, a 
rapid change is created in the height of one end. This 
disturbance then moves away from its source. I'he height of each 
point on the rope depends upon time as well as position along 
the rope. 

The disturbance is a pattern of disphicemcnt along the rope. 
The motion of the displacement pattern ft om one end of the 
rope toward the other is an example of a wave. The hand 
snapping one end is the source of the wave. The rope is the 
medium in which the vv^ave moves. 

Consider another example. When a pebble falls into a pool of 
still liquid, a series of circular crests and troughs spreads over 
the surface. This mo\ang displacement pattern of the liquid 
surface is a wave. The pebble is the source; the moving pattern of 
crests and troughs is the wave; and the liquid surface is the 
medium. Leaves, sticks, or other objects floating on the surface of 
the liquid bob up and dowoi as each wave passes. But they do 
not experience any net displacement on the average. No material 
has moved from the wave source, either on the surface or among 
the parficles of the liquid. The same holds for rope waves, sound 
waves in air, etc. 

As any one of these weaves mox'es through a medium, the wave 
produces a changing displacement of the successive parts of the 
medium. Thus, we can refer to these weaves as waves of 
displacement. If you can see the medium and recognize the 
displacements, then you can see waves. But waves also may exist 
in media you cannot see (such as air) or may form as 
disturbances of a state you cannot detect with your eyes Isuch as 
pressure or an electric field). 

You can use a loose spring coil to demonstrate three different 
kinds of motion in the medium through which a vva\e passes. 
First, move the end of the spiing from side to side, oi- up and 
dov\Ti as in sketch la) in the margin. A wave of side-to-side or up- 
and-down displacement uill travel along the spring. \ovv push 
the end of the spring back and forth, idong the direction of the 
spring itself, as in sketch (b). A wiive of back-and-forth 
displacement will travel along the spring. Finally, twist the end of 
the spring clockwise and counterclockwise, as in sketch Ic). A 
wave of angular displacement will travel along the spring. Waves 
like those in la), in which the displacements are perpendicular 
to the dir'ection the wave travels, are called transverse waves. 
Waves like those in lb), in which the displacements ar-e in the 
direction the wave tr'avels, ar'e called l()ns,itudinal waxes. Waves 



356 



I'NIT 3 / THE TRIUMPH OF MECHAMCS 



like those in (c), in which the displacements are twisting in a 
plane perpendicular to the direction the wave travels, are called 
torsional waves. 

All three types of wave motion can be set up in solids. In 
fluids, hovuever, transverse and torsional waves die out very 
quickly and usually cannot be produced at all. Therefore, sound 
waves in air and water are longitudinal. The molecules of the 
medium are displaced back and forth along the direction in 
which the sound travels. 

It is often useful to make a graph of wave patterns in a 
medium. However, a graph on paper always has a transverse 
appearance, even if it represents a longitudinal or torsional wave. 
For example, the graph at the right represents the pattern of 
compressions in a sound wave in air. The sound waves are 
longitudinal, but the graph line goes up and douoi. This is 
because the graph represents the increasing and decreasing 
density of the air. It does not represent an up-and-down motion 
of the air. 




A. unpolarized wave on a rope 



B. polarized wave on a rope 

To describe completely transverse waves, such as those in 
ropes, you must specify the direction of displacement. When the 
displacement pattern of a transverse wave lies in a single plane, 
the wave is polarized. For waves on ropes and springs, you can 
observe the polarization directly. Thus, in the photograph on the 
previous page, the waves the person makes are in the horizontal 
plane. Although there are few special properties associated with 
polarized waves on ropes, you will see (in Sec. 13.7) that for light 
waves, for example, polarization can have important effects. 

All three kinds of wave (longitudinal, transverse, and torsional) 
have an important characteristic in common. The disturbances 
move away from their sources through the media and continue 
on their own. We stress this particular characteristic by saying 
that these waves propagate. This means more than just that they 
"travel" or "move." An example will clarify the difference between 
waves that propagate and those that do not. You probably have 
read some description of the great wheat plains of the Middle 
West, Canada, or Central Europe. Such descriptions usually 
mention the "beautiful, vwnd-formed waves that roll for miles 
across the fields." The medium for such a wave is the wheat, and 



/ 



\ 



(a) 



(b) 



(a) "Snapshot" representation of a 
sound wave progressing to the 
right. The dots represent the den- 
sity of air molecules, (b) Graph of 
air pressure P versus position x 
at the instant of the snapshot. 



CHAPTER 12 / WAVES 



357 




An engine starting abruptly can 
start a displacement wave along a 
line of cars. 

A very important point: Energy 
transfer can occur without matter 
transfer. 



the disturbance is the swaying motion of the wheat. This 
disturbance does indeed travel, but it does not propagate: that is, 
the disturbance does not originate at a source and then go on 
by itself. Rather, it must be continually fanned by the wind. 
When the wind stops, the disturbance does not roll on, but 
stops, too. The traveling "waves" of swaying wheat are not at all 
the same as rope and water waves. This chapter will concentrate 
on waves that do originate at sources and propagate themselves. 
For the purposes of this chapter, waves are disturbances which 
propagate in a medium. 



1. What kinds of mechanical waves can propagate in a solid? 

2. What kinds of mechanical waves can propagate in a fluid? 

3. What kinds of mechanical waves can be polarized? 

4. Suppose that a mouse runs along under a rug, causing a 
bump in the rug that travels with the mouse across the room. 
Is this moving disturbance a propagating wave? 








A rough representation of the 
forces at the ends of a small sec- 
tion of rope as a transverse pulse 
moves past. 



1.2«3 I Wave propagation 

Waves and their behavior are perhaps best studied by beginning 
with large mechanical models and focusing your attention on 
pulses. Consider, for example, a freight train, with many cars 
attached to a powerful locomiotive, but standing still. If the 
locomotive starts abruptly, it sends a displacement wave running 
down the line of cars. The shock of the starting displacement 
proceeds from locomotive to caboose, clacking through the 
couplings one by one. In this example, the locomotive is the 
source of the disturbance, while the freight cars and their 
couplings are the medium. The "bump" traveling along the line 
of cars is the wave. The disturbance proceeds all the v\'a\' from 
end to end, and with it goes energy of displacement and motion. 
Yet no particles of matter are transferred that far; each car only 
jerks ahead a bit. 

How long does it take for the effect of a disturbance created at 
one point to reach a distant point? The time interval depends 
upon the speed with which the disturbance or waxe propagates. 
This speed, in turn, depends upon the type of waxe and the 
characteristics of the medium. In any case, the effect of a 
disturbance is ne\'er transmitted instantly oxer any distance. 
Each part of the medium has inertia, and each portion of the 
medium is compressible. So time is needed to transfer energy 
from one part to the next. 

The same comments apply also to transverse waxes. The series 
of sketches in the margin represents a wave on a rope. Think of 
the skettiies as frames of a motion picture film, taken at (njual 



358 



U\IT 3 / THE TRIUMPH OF MECHANICS 



time intervals. The material of the rope does not travel along wdth 
the wave. But each bit of the rope goes through an up-and-down 
motion as the wave passes. Each bit goes through exactly the 
same motion as the bit to its left, except a little later. 

Consider the small section of rope labeled X in the diagrams. 
When the pulse traveling on the rope first reaches X, the section 
of rope just to the left of X exerts an upward force on X. As X is 
moved upward, a restoring downward force is exerted by the 
next section. The further upward X moves, the greater the 
restoring forces become. Eventually, X stops moving upward and 
starts down again. The section of rope to the left of X now exerts 
a restoring (downward) force, while the section to the right 
exerts an upward force. Thus, the trip down is similar, but 
opposite, to the trip upward. Finally, X returns to the equilibrium 
position, and both forces vanish. 

The time required for X to go up and down, that is, the time 
required for the pulse to pass by that portion of the rope, 
depends on two factors. These factors are the magnitude of the 
forces on X, and the mass of X. To put it more generally: The 
speed wdth which a wave propagates depends on the stiffness 
and on the density of the medium. The stiffer the medium, the 
greater will be the force each section exerts on neighboring 
sections. Thus, the greater vvdll be the propagation speed. On the 
other hand, the greater the density of the medium, the less it will 
respond to forces. Thus, the slower will be the propagation. In 
fact, the speed of propagation depends on the ratio of the 
stiffness factor and the density factor. 



SG 2 



# 5. What is transferred along the direction of wave motion? 

6. On what two properties of a medium does wave speed 
depend? 

7. If a spring is heated to make it less stiff, does it carry waves 
faster or slower? // the boxcars in a train are unloaded, does 

the longitudinal start-up wave travel faster or slower? 

JiA»4t \ Periodic ivaves 

Many of the disturbances considered up to now have been 
sudden and short-lived. They were set up by a single disturbance 
like snapping one end of a rope or suddenly bumping one end 
of a train. In each case, you see a single wave njnning along the 
medium with a certain speed. This kind of wave is a pulse. 

Now consider periodic waves, continuous regular rhythmic 
disturbances in a medium, resulting from periodic vibrations of a 
source. A good example of a periodic vibration is a swinging 
pendulum. Each svvdng is virtually identical to every other swing, 



The exact meaning of stiffness and 
density factors is different for dif- 
ferent kinds of waves and different 
media. For tight strings, for exam- 
ple, the stiffness factor is the ten- 
sion T in the string, and the density 
factor is the mass per unit length, 
m/l. The propagation speed v is 
given by 

\ m/l 



CHAPTER 12 / WAVES 



359 









A 




./VA 







t=#- 



t =T 



T/je wave generated by a simple 
harmonic vibration is a sine wave. 
A "snapshot" of the displacement 
of the medium would show it has 
the same shape as a graph of the 
sine function familiar in trigonom- 
etry. This shape is frequentlv re- 
ferred to as "sinusoidal. 



and the suing repeats over and over again in time. Another 
example is the up-and-down motion of a weight at the end of a 
spring. The maximum displacement from the position of 
equilibrium is called the amplitude; A, as shown in the margin. 
Ihe time taken to complete one vibration is called the period, T. 
The number of vibrations per second is called the frequency, f 

What happens when such a vibration is applied to the end of a 
rope? Suppose that one end of a rope is fastened to the 
oscillating (vibrating) weight. As the weight vibrates up and 
down, you observe a wave propagating along the rope. The wave 
takes the fonn of a series of moving crests and troughs along the 
length of the rope. The source executes "simple harmonic 
motion" up and down. Ideally, every point along the length of 
the rope executes simple harmonic motion in turn. The wave 
travels to the right as crests and troughs follow one another. 
Each point along the rope simply oscillates up and down at the 
same frequency as the source. The amplitude of the wave is 
represented by A. The distance between any two consecutive 
crests or any two consecutive troughs is the same all along the 
length of the rope. This distance, called the wavelength of the 
periodic wave, is represented by the Greek letter X (lambda). 

If a single pulse or a wave crest moves fairly slowly through 
the medium, you can easily find its speed. In principle all you 
need is a clock and a meter stick. By timing the pulse or crest 
over a measured distance, you can get the speed. But it is not 
always simple to observe the motion of a pulse or a wave crest. 
As is shown below, however, the speed of a periodic wave can be 
found indir^ectly from its frequency and wavelength. 

As a wave progresses, each point in the medium oscillates with 
the frequency and period of the source. The diagram in the 
margin illustrates a periodic wave moving to the r ight, as it might 
look in snapshots taken every V4 period. Follow the progress of 
the crest that started out from the extreme left at f = 0. The time 
it takes this crest to move a distance of one wavelength is equal 
to the time required for one complete oscillation; that is, the 
crest moves one wavelength X in one period of oscillation 7'. The 
speed V of the crest is ther efore 

distance moved X 

corresponding time interval T 

All par1s of the wave patterrr propagate with the same speed. 
Thus, the speed of any one crest is just the speed of the wave. 
Therefore, the speed v of the wave is 

wavelength _ X 

period of oscillation 7' 

But 7" - 1/f, where/ = frequency (see Project Physics, Chapter 
4, page 112). Therefore, v = fK, or wave speed = frequency X 
v\'av(>length. 



360 



UNIT 3 / THE TRIUMPH OF MECHANICS 



We can also write this relationship as X = v/forf = v/X. These 
expressions imply that, for waves of the same speed, the 
frequency and wavelength are inversely proportional; that is, a 
wave of twdce the frequency would have only half the 
wavelength, and so on. This inverse relation of frequency and 
wavelength will be useful in other parts of this course. 

The diagram below represents a periodic wave passing 
through a medium. Sets of points are marked that are moving "in 
step" as the periodic wave passes. The crest points C and C 
have reached maximum displacement positions in the upward 
direction. The trough points D and D' have reached maximum 
displacement positions in the downward direction. 1 he points C 
and C have identical displacements and velocities at any instant 
of time. Their vibrations are identical and in unison. The same 
is true for the points D and D'. Indeed there are infinitely many 
such points along the medium that are vibrating identically 
when this w^ave passes. Note that C and C are a distance X 
apart, and so are D and D'. 



A "snapshot" of a periodic wave 
moving to the right. Letters indi- 
cate sets of points with the same 
phase. 






Points that move "in step," such as C and C, are said to be in 
phase with one another. Points D and D' also move in phase. 
Points separated from one another by distances of X, 2X, 3X, . . ., 
and nX (where n is any whole number) are all in phase with one 
another. These points can be anywhere along the length of the 
wave. They need not correspond with only the highest or lowest 
points. For example, points such as P, P', P", are all in phase w^ith 
one another. Each point is separated from the next one by a 
distance X. 

Some of the points are exactly out of step. For example, point 
C reaches its maximum upward displacement at the same time 
that D reaches its maximum downward displacement. At the 
instant that C begins to go dowoi, D begins to go up. Points such 
as these are one-half period out of phase with respect to one 
another; C and D' also are one-half period out of phase. Any two 
points separated from one another by distances of X/2, 3X/2, 
5X/2, . . . are one-half period out of phase. 



SG3 



• 8. Of the wave variables frequency^ wavelength, period, 
amplitude, and polarization, which ones describe 



CHAPTER 12 / WAVES 



361 



(a) space properties of waves? 

(b) time properties of waves? 

9. A wave with the displacement as smoothly and simply 
varying from point to point as that shown in the illustration on 
page 361 is called a "sine" wave. How might the "wavelength" 
be defined for a periodic wave that is not a sine wave? 

10. A vibration of 100 Hz (cycles per second) produces a wave. 
(a) What is the wave frequency? 

(h) What is the period of the wave? 

(c) If the wave speed is 10 m/sec, what is the wavelength? (If 
necessary, look back to find the relationship you need to 
answer this.) 

11. If points X and Y on a periodic wave are one-half period 
"out of phase" with each other, which of the following must be 
true? 

(a) X oscillates at half the frequency at which Y oscillates. 

(b) X and Y always move in opposite directions. 

(c) X is a distance of one-half wavelength from Y. 







=XV 




The superposition of ti\'o rope 
waves at a point. The dashed 
curves are the contributions of the 
individual waves. 



1.2«o I ¥llien Yvaves meet: the superposition 
principle 

So far, you have considered single waves. VV^hat happens when 
two waves encounter each other in the same medium? Suppose 
two waves approach each other on a rope, one traveling to the 
right and one traveling to the left. The series of sketches in the 
margin shows what would happen if you made this experiment. 
The weaves pass through each other without being modified. After 
the encounter, each wave looks just as it did before and is 
traveling just as it was before. This phenomenon of passing 
through each other unchanged can be obseived with all tvpes of 
waves. You can easily see that it is true foi" surface ripples on 
w^ater. (Look, for example, at the photograph on page 364.) You 
could reason that it must be true for sound waves also, since two 
conversations can take place across a tiible without distorting 
each other. (Note that when particles encounter each other, they 
collide. Waves can pass through each other.) 

What happens during the time? vxhen the two waxes o\'erlap? 
The displacements they provide add up. At each instant, the 
displacement of each point in the overlap region is just the sum 
of the displacements that would be caused b\' each of the two 
waves separately. An example is shown in the margin. Two waves 
travel toward each other on a rope. One has a maximum 
displacn'ment of 0.4 (^m ujiward and \he other a maxiiiuini 



362 



UMT 3 / THE TRIUMPH OF MECHAIVICS 



displacement of 0.8 cm upward. The total maximum upward 
displacement of the rope at a point where these two waves pass 
each other is 1.2 cm. 

What a wonderfully simple hehavior, and how easy it makes 
everything! Each wave proceeds along the rope making its own 
contribution to the rope's displacement no matter what any 
other wave is doing. You can easily determine what the rope 
looks like at any given instant. All you need to do is add up the 
displacements caused by each vv^ave at each point along the rope 
at that instant. This property of waves is called the superposition 
principle. Another illustration of wave superposition is shown in 
the margin. Notice that when the displacements are in opposite 
directions, they tend to cancel each other. One of the two 
directions of displacement may always be considered negative. 

The supeiposition principle applies no matter how many 
separate waves or disturbances are present in the medium. In 
the examples just given, only two waves are present. But you 
would find by experiment that the superposition principle works 
equally well for three, ten, or any number of waves. Each wave 
makes its owoi contribution, and the net result is simply the sum 
of all the individual contributions. 

If waves add as just described, then you can think of a 
complex wave as the sum of a set of simpler waves. In 1807, the 
French mathematician Jean-Baptiste Fourier advanced a very 
useful theorem. Fourier stated that any continuing periodic 
oscillation, however complex, could be analyzed as the sum of 
simpler, regular wave motions. This, too, can be demonstrated by 
experiment. The sounds of musical instruments have been 
analyzed in this way also. Such analysis makes it possible to 
"imitate " instruments electronically by combining just the right 
proportions of simple vibrations. 






^W 



a*b 



^/w a-^b^c 



IZ. Two periodic waves of amplitudes A, and A^ pass through a 

point P. What is the greatest possible displacement ofP? 

13. What is the displacement of a point produced by two waves 

together if the displacements produced by the waves 

separately at that instant are +5 cm and —6 cm? What is the 

special property of waves that makes this simple result 

possible? 



SG 4-8 



12.6 I A tivo-source interference pattern 

The photograph on page 364 (center) shows ripples spreading 
from a vibrating source touching the water surface in a "ripple 
tank." The drawing to the left of it shows a "cut-away" view of 



CHAPTER 12 / WAVES 



363 



Close Up I 



Woves in o Ripple Tank 







When something drops in the water 
produces periodic wave trains of ere; 
and troughs, somewhat as shown in h 
"cut-away" drawing at the left below 
The center figure below is an inste 
taneous photograph of the shadows 
ripples produced by a vibrating po 
source. The crests and troughs on I 
water surface show up in the image 
bright and dark circular bands. In !• 
photo below nght. there were two po 
sources vibrating in phase. The overia 
ping waves create an interference p< 
tern. 



'S^ 



c 




364 



UNIT 3 / THE THIl'VIPH OF MECHAMCS 



the water level pattern at a given instant. The third photograph 
(far right) introduces a phenomenon that will play an important 
role in later parts of the course. It shows the pattern of ripples 
on a water surface disturbed by two vibrating sources. The two 
small sources go through their up and down motions together; 
that is, they are in phase. Each source creates its ov^^ set of 
circular, spreading ripples. The photograph catches the pattern 
made by the overlapping sets of waves at one instant. This 
pattern is called an interference pattern. 

You can interpret what you see in this photograph in terms of 
what you already know about waves. You can predict how the 
pattern will change with time. First, tilt the page so that you are 
viewing the interference pattern from a glancing direction. You 
will see more clearly some nearly straight gray bands. This 
feature can be explained by the supeiposition principle. 

Suppose that two sources produce identical pulses at the 
same instant. Each pulse contains one crest and one trough. (See 
margin.) In each pulse the height of the crest above the 
undisturbed or average level is equal to the depth of the trough 
below. The sketches show the patterns of the water surface after 
equal time intervals. As the pulses spread out, the points at 
which they overlap move too. In the figure, a completely 
darkened circle indicates where a crest overlaps another crest. A 
half-darkened circle marks each point where a crest overlaps a 
trough. A blank circle indicates the meeting of two troughs. 
According to the superposition principle, the water level should 
be highest at the completely darkened circles (where the crests 
overlap). It should be lowest at the blank circles, and at average 
height at the half-darkened circles. Each of the sketches in the 
margin represents the spatial pattern of the water level at a given 
instant. 

At the points marked uath darkened circles in the figure, the 
two pulses arrive in phase. At the points indicated by blank 
circles, the pulses also arrive in phase. In either case, the waves 
reinforce each other, causing a greater amplitude of either the 
crest or the trough. Thus, the waves are said to interfere 
constructively. In this case, all such points are at the same 
distance from each source. As the ripples spread, the region of 
maximum disturbance moves along the central dotted line in (a). 

At the points marked with half-darkened circles, the two 
pulses arrive completely out of phase. Here the waves cancel and 
so are said to interfere destructively, leaving the water surface 
undisturbed. The lines N show the path along which the 
overlapping pulses meet when they are just out of phase. All 
along these lines there is no change or displacement of the water 
level. 

When periodic waves of equal amplitude are sent out instead 
of single pulses, overlap occurs all over the surface, as is shown 



o o 




N N 

Pattern produced when two circu- 
lar pulses, each of a crest and a 
trough, spread through each other. 
The small circles indicate the net 
displacement: 

# = double height peak 

O = average level 

O — double depth trough 



CHAPTER 12 / WAVES 



365 




Analysis of interference pattern similar to that of the lower right photo- 
graph on page 364 set up by two in-phase periodic sources. (Here S, and 
S^ are separated bv four wavelengths.) The dark circles indicate where 
crest is meeting crest, the blank circles where trough is meeting trough, 
and the half dark circles where crest is meeting trough. The other lines 
of maxinujm constructive interference are labeled A^^ A^, A^, etc. Points 
on these lines move up and down much more than they would because 
of waves from either source alone. The lines labeled N^, N.,, etc., repre- 
sent bands along which there is maximum destructive interference. 
Points on these lines move up and down much less than they would be- 
cause of waves from either source alone. Compare the diagram with the 
photograph and identify antinodal lines and nodal lines. 



on this page. All along the central dotted line, there is a doubled 
disturbance amplitude. All along the lines labeled N, the water 
height remains undisturbed. Depending on the wa\'elength and 
the distance between the sources, there can be many such lines 
of constructive and destructive interference. 

Now you can interpret the ripple tank interference pattern 
shown at the lower nght on page 364. The gray bands are areas 
where waves cancel each other, called nodal lines. These bands 
correspond to lines N in the simple case of pulses instead of 
periodic waves. Between these bands aie other bands where 
crest and trough follow one another, wiiere the waves reinforce. 
These are called antinodal lines. 

Such an interference pattern is set up by overlapping weaves 
from two sources. For water waves, the interference pattern can 
be seen directly. But whether \asible or not, all waves, including 
eai-thfjuake waves, sound waves, or X rays, can set up 
int(M lorcnce |iatteriis. I'or e.\am[ile, suppose two l()U(ls[)eak(MS 



366 



LMIT 3 / THE TRIIMPH OF MECHANICS 



are working at the same frequency. By changing your position in 
front of the loudspeakers, you can find the nodal regions where 
destructive interference causes only little sound to be heard. You 
also can find the antinodal regions where a strong signal comes 
through. 

The beautiful symmetry of these interference patterns is not 
accidental. Rather, the whole pattern is determined by the 
wavelength \ and the source separation SjS,. From these, you 
could calculate the angles at which the nodal and antinodal 
lines radiate out to either side of A^. Conversely, you might know 
SjS2 and might have found these angles by probing around in the 
two-source interference pattern. If so, you can calculate the 
wavelength even if you cannot see the crests and troughs of the 
waves directly. This is very useful, for most waves in nature 
cannot be directly seen. Their wavelength has to be found by 
letting weaves set up an interference pattern, probing for the 
nodal and antinodal lines, and calculating \ from the geometry. 

The figure at the right shows part of the pattern of the diagram 
on the opposite page. At any point P on an antinodal line, the 
waves from the two sources arrive in phase. This can happen 
only if P is equally far from S^ and S,, or if P is some whole 
number of wavelengths farther from one source than from the 
other. In other words, the difference in distances (S^P — S,P) 
must equal nX, A. being the wavelength and n being zero or any 
whole number. At any point Q on a nodal line, the waves from 
the two sources arrive exactly out of phase. This occurs because 
Q is an odd number of half-wavelengths (VzK, Vzk, %X, etc.) 
farther from one source than from the other. This condition can 
be written S^Q - S,Q -- (n + VzW. 

The distance from the sources to a detection point may be 
much larger than the source separation d. In that case, there is a 
simple relationship between the node position, the wavelength 
\, and the separation d. The wavelength can be calculated from 
measurements of the positions of nodal lines. The details of the 
relationship and the calculation of wavelength are described on 
page 369. 

This analysis allows you to calculate from simple 
measurements made on an interference pattern the wavelength 
of any wave. It applies to water ripples, sound, light, etc. You will 
find this method very useful in later units. One important thing 
you can do now is find X. for a real case of interference of waves 
in the laboratory. This practice will help you later in finding the 
wavelengths of other kinds of waves. 



SG 9 



r9 



Since the sound wave patterns in 
space are three-dimensional, the 
nodal or antinodal regions in this 
case are two-dimensional surfaces; 
that is, they are planes, not lines. 



• 14. Are nodal lines in interference patterns regions of 
cancellation or reinforcement? 
15. What are antinodal lines? antinodal points? 



CHAPTER 12 / WAVES 



367 




A vibrator ut the left produces a 
wave train that runs along the rope 
and reflects from the Ji.xed end at 
the right. The sum of the oncoming 
and the reflected waves is a stand- 
ing wave pattern. 



16. Nodal points in an interference pattern are places where 

(a) the waves arrive "out of phase." 

(b) the waves arrive "in phase." 

(c) the point is equidistant from the wave sources. 

(d) the point is one -half wavelength from both sources. 

17. Under what circumstances do waves from two in- phase 
sources arrive at a point out of phase? 




Lyre player painted on a Greek 
vase in the 5th century B.C. 



12 .T I standing^ waves 

If both ends of a rope are shaken with the same frequency and 
same amplitude, an interesting thing happens. Ihe interference 
of the identical waves coming from opposite ends results in 
certain points on the rope not moving at all! In between these 
nodal points, the rope oscillates up and down. But there is no 
apparent propagation of wave patterns in either direction along 
the rope. This phenomenon is called a standing wave or 
stationary wave. The important thing to remember is that the 
standing oscillation you observe is really the effect of two 
traveling waves. 

To make standing waves on a rope, there do not ha\'e to be 
two people shaking the opposite ends. One end can be tied to a 
hook on a wall. The train of waves sent down the rope by 
shaking one end will reflect back from the fixed hook. These 
reflected waves interfere with the new, oncoming \va\es and can 
produce a standing pattern of nodes and oscillation. In fact, you 
can go further and tie both ends of a string to hooks and pluck 
(or bow) the string. From the plucked point a pair of waves go 
out in opposite directions and then reflect back from the ends. 
The interference of these reflected waves traveling in opposite 
directions can produce a standing pattern just as before. The 
strings of guitars, violins, pianos, and all other stringed 
instruments act in just this fashion. The energy given to the 
strings sets up standing waves. Some of the energ\' is then 
transmitted from the vibrating string to the body of the 
instiTjment. The sound waves sent forth from the body are at 
essentially the same frequency as the standing waves on the 
string. 

The vibration frequencies at which standing waves can exist 
depend on two factors. One is the speed of wave propagation 
along the string. The other is the length of the string. I'he 
connection between length of string and musical tone was 
recognized over 2,000 years ago. This relationship contributed 
greatly to the idea that nature is built on mathematical 
principles. Eariy in the development of musical instruments, 
people learned how to produce certain pleasing harmonies by 
|)lu(king strings. These harriionies icsult if the strings ar-e of 



368 



UNIT 3 / THE TRIUMPH OF MECH/\1\ICS 



Giose ttn L 



Colculoting X from on Interference Pattern 



d = (8,82) = separation between 8, and S^. (8, 
and 82 may be actual sources that are in 
phase, or two slits through which a previously 
prepared wave front passes.) 

€ = OQ = distance from sources to a far-off line 
or screen placed parallel to the two sources. 

X = distance from center axis to point P along the 
detection line. 

/. = OP = distance to point P on detection line 
measured from sources. 

Waves reaching P from 8, have traveled farther 
than waves reaching P from S^. If the extra distance 
is \ (or 2k, 3\, etc.), the waves will arrive at P in 
phase. Then P will be a point of strong wave dis- 
turbance. If the extra distance is V2K (or ^/zK, %\, 
etc.), the waves will arhve out of phase. Then P will 
be a point of weak or no wave disturbance. 

With P as center, draw an arc of a circle of radius 
P82; it is indicated on the figure by the dotted line 
S^M. Then line segment P82 equals line segment 
PM. Therefore, the extra distance that the wave 
from 8 travels to reach P is the length of the seg- 
ment 8M. 

Now if d is very small compared to ^, as you can 
easily arrange in practice, the circular arc S^M will 
be a very small piece of a large-diameter circle, or 
nearly a straight line. Also, the angle 8,M82 is very 
nearly 90°. Thus, the triangle 8,82M can be re- 
garded as a right triangle. Furthermore, angle 
S,82M is equal to angle POQ. Then the right triangle 
S^S^M is similar to triangle POQ. 



8,M 
8,8. 



X 

OP 



or 



8,M 



If the distance ( is large compared to x, the dis- 
tances € and L are nearly equal. Therefore, 

S,M _ X 

~d~ ~ e 

But 8,M is the extra distance traveled by the wave 
from source 8,. For P to be a point of maximum 
wave disturbance, 8,M must be equal to nk (where 
n = if P is at Q, and n = 1 if P is at the first 
maximum of wave disturbance found to one side of 
Q, etc.). 80 the equation becomes 



nk 
d 


_ X 


k = 


dx 

n£ 



and 



This important result says that if you measure the 
source separation d, the distance (, and the dis- 
tance X from the central line to a wave disturbance 
maximum, you can calculate the wavelength k. 




I 



CHAPTER 12 / WAVES 



369 



370 



equal tautness and diameter and if their lengths are in the ratios 
of small whole numhers. Thus, the length ratio 2:1 gi\es the 
octave, 3:2 the musical fifth, and 4:3 the musical fouilh. This 
striking connection between music and numbei"s encouraged the 
Pythagoreans to search for other numeric^al ratios or harmonies 
in the univei'se. I'he P\'thagorean ideal strongly affected Greek 
science and many centuries later inspired much of Kepler's 
work, hi a general fomi, the ideal flourishes to this day in many 

beautiful applications of mathematics to jjhvsical experience. 

SG 13 Using the superposition principle, we can now define the 

harmonic relationship much more precisely. First, we must 
stress an important fact about standing wave patterns produced 
by reflecting waves from the boundaries of a medium. You can 
imagine an unlimited variet\' of waves traveling back and foith. 
But, in fact, only certain wavelengths (or frequencies) can produce 
standing waves in a given medium. In the example of a stringed 
instiTiment, the two ends are fixed and so must be nodal points. 
This fact puts an upper limit on the length of standing waxes 
possible on a fixed rope of length L. Such waves must be those 
for which one-half wavelength just fits on the rope IL — \/2). 
Shorter weaves also can produce standing patterns ha\ing more 
nodes. But always, some whole number of one-half wavelengths 
must just fit on the rope (L = X/2, 2\/2, 3X/2, etc. I so that L = 
nk/2. 

This relationship can be used to give an expression for all 
possible wavelengths of standing waves on a fixed rope: 

2L 

K = — 

n 

or simply X^ ^ l/n. That is, if X, is the longest wavelength 
possible, the other possible wavelengths will be VzX,, VbX,, . . ., 
1/nX,. Shorter waxelengths correspond to higher frequencies. 
Thus, on any bounded medium, only certain frequencies of 
standing waves can be set up. Since frequency /is inversely 
proportional to wavelength, f^f 1/X, we can rewiite the 
expression for all possible standing waves on a plucked string as 

f„^n 

In other circumstances, / may depend on n in some other way. 
The lowest possible frequency of a standing wa\ e is usually the 
one most stitjngly present when the string vibrates after being 
plucked or bowed. If/ represents this lowest possible frequency, 
then the other possible standing waves would have fi'equencies 
2/,, 3/,, . . ., nj\. These higher frequencies are called "o\ertones" 
of the "fundamental" frequency/. On an "ideal" string, there are 
in principle an unlimited number of such frequencies, all simple 
multiples of the lowest frequency. 

In real media, there are practical upper limits to the possible 
fre(|uencies. -Also, th(> o\ertones aie not cxactK simple nuiltiples 

UMT 3 / THE TRIUMPH OF MECHANICS 



I 



of the fundamental frequency; that is, the ov^ertones are not 
strictly "harmonic. " This effect is still greater in systems more 
complicated than stretched strings. In a saxophone or other 
wind instrument, an air column is put into standing wave 
motion. The overtones produced may not be even approximately 
harmonic. 

As you might guess from the superposition principle, standing 
waves of different frequencies can exist in the same medium at 
the same time. A plucked guitar string, for example, oscillates in 
a pattern which is the supeiposition of the standing waves of 
many overtones. The relative oscillation energies of the different 
instruments determine the "quality" of the sound they produce. 
Each type of instrument has its own balance of overtones. This is 
why a violin sounds different from a trumpet, and both sound 
different from a soprano voice, even if all are sounding at the 
same fundamental frequency. 



Film Loops 3&-43 show a variety of 
standing waves, including waves on 
a string, a drum, and in a tube of 
air. 

Mathematically inclined students 
are encouraged to pursue the topic 
of waves and standing waves, for 
example, Science Study Series pa- 
perbacks Waves and the Ear and 
Horns, Strings, and Harmony. 



SG 16 



# 18. When two identical waves of the same frequency travel in 
opposite directions and interfrre to produce a standing wave, 
what is the motion of the medium at 

(a) the nodes of the standing wave? 

(b) the places between nodes, called antinodes or loops, of the 
standing wave? 

19. If the two interfering waves have wavelength X, what is the 
distance between the nodal points of the standing wave? 

20. What is the wavelength of the longest traveling waves that 
can produce a standing wave on a string of length L? 

21. Can standing waves of any frequency, as long as it is higher 
than the fundamental, be set up in a bounded medium? 

12.8 I Wave fronts and diffraction 

Waves can go around corners. For example, you can hear a voice 
coming from the other side of a hill, even though there is 
nothing to reflect the sound to you. You are so used to the fact 
that sound waves do this that you scarcely notice it. This 
spreading of the energy of waves into what you might expect to 
be "shadow" regions is called diffraction. 

Once again, water waves wall illustrate this behavior most 
clearly. From among all the arrangements that can result in 
diffraction, we udll concentrate on two. The first is showoi in the 
second photograph in the margin on page 373. Straight water 
waves (coming from the top of the picture) are diffracted as they 
pass through a narrow slit in a straight barrier. Notice that the 
slit is less than one wavelength wide. The wave emerges and 



CHAPTER 12 / WAVES 



371 



Close Up I 



Vibration of o Drum 



I 



In the Film Loop "Vibration of a Drum," a 
marked rubber "drumhead" Is seen vibrat- 
ing In several of Its possible modes. Below 
are pairs of still photographs from three of 
the symmetrical modes and from an antl- 
symmetrlcal mode. 





I 



372 



UNIT 3 / THE TRIUIVIPH OF MECHANICS 



spreads in all directions. Also notice the pattern of the diffracted 
wave. It is basically the same pattern a vibrating point source 
would set up if it w^ere placed vv^here the slit is. 

The bottom photograph shows a second barrier arrangement. 
Now there are two narrow slits in the barrier. The pattern 
resulting from supeiposition of the diffracted waves from both 
slits is the same as that produced by two point sources vibrating 
in phase. The same kind of result is obtained when many narrow 
slits are put in the barrier; that is, the final pattern just matches 
that which would appear if a point source were put at the center 
of each slit, vdth all sources in phase. 

You can describe these and all other effects of diffraction if you 
understand a basic characteristic of weaves. This characteristic 
was first stated by Christian Huygens in 1678 and is now known 
as Huygens' principle. In order to understand the principle, you 
first need the definition of a wave front . 

For a water wave, a wave ft ont is an imaginary line along the 
water's surface and every point along this line is in exactly the 
same stage of vibration; that is, all points on the line are in phase. 
Crest lines are wave fronts, since all points on the water's suiface 
along a crest line are in phase. Each has just reached its 
maximum displacement upward, is momentarily at rest, and will 
start downward an instant later. 

Since a sound vv^ave spreads not over a surface but in three 
dimensions, its wave fronts are not lines but surfaces. The w^ave 
ft'onts for sound waves from a very small source are very nearly 
spherical surfaces, just as the wave ftonts for a veiy small source 
of water waves are circles. 

Huygens' principle, as it is generally stated today, is that every 
point on a wave front may be considered to behave as a point 
source for waves generated in the direction of the wave s 
propagation. As Huygens said: 

There is the further consideration in the emanation of these 
waves, that each particle of matter in which a wave spreads, 
ought not to communicate its motion only to tlie next particle 
which is in the straight line drawn from the (source), but that it 
also imparts some of it necessanly to ail otliers whicli touch 
it and which oppose themselves to its movement. So it arises 
that around each particle there is made a wave of which tliat 
particle is the center. 

The difft^action patterns seen at slits in a barrier are certainly 
consistent with Huygens' principle. The wave arriving at the 
barrier causes the water in the slit to oscillate. The oscillation of 
the water in the slit acts as a source for waves traveling out ftom 
it in all directions. When there are two slits and the wave 
reaches both slits in phase, the oscillating water in each slit acts 
like a point source. The resulting interference pattern is similar 




Diffraction of ripples around the 
edge of a barrier. 




Diffraction of ripples through a 
narrow opening. 




Diffraction of ripples through two 
narrow openings. 



CHAPTER 12 / WAVES 



373 




Each point on a wave front can be 
thought of as a point source of 
naves. The waves from all the point 
sources interfere constructively 
onlv alor}g their envelope, which 
becomes the new wave front. 




When part of the wave front is 
blocked, the constructive interfer- 
ence of waves from points on the 
wave front e,x/ends into the 
"shadow" region. 



\ 



When all but a ver\' small portion 
of a wave front is blocked, the wave 
propagating away from that small 
portion is nearly the same as that 
from a point source. 



to the pattern produced by \va\es IV(jni two point soiiices 
oscillating in phase. 

Consider what happens behind a breakwater wall as in ihe 
aerial photograph of the harbor above. By Huygens' principle, 
water- oscillation near the end of the breakwater sends cnr-cular 
wiues propagating into the "shadow" region. 

You can understand all diffraction patterns if you keep both 
Huygens' principle and the superposition principle? in mind. For 
example, consithM- a slit v\ider- than one waxeknigth. in tliis case 
the pattern of diffracted waves contains nodal lines Isee the 
series of four- photogr'aphs in the margin on page 375 1. 

The figirre on page 375 helps to explain why nodal lines 
appear. There must be points like P that ar-e jirst X farther- from 
side A of the slit than fr-om side 15; that is, there nurst be points P 
for which AF differs from BP b\' exactly \. For- sirch a point, AP 
and OP difler- by one-half wavelength, K/2. By Huygens' principle, 
you ma\' think of points A and O as in-phase point sour'ces of 
circular- waves. Birt since AP arid ()l^ differ- by \/2, the two waxes 
will arrive at P completely out of phase. So, according to the 
sirperpositiorn jorinciple, the vxaxes fr-om ,\ and () will (-ancel at 
[loint I'. 

This argument also holds tr-ue for the pair of points consisting 
of the fir\st point to the right of A and the fir-st to the r-ight of (). 
in fact, it holds true iov each such matched pair- of points, all tiie 
way across the slit. The waves originating at each such pair of 
points all cancel at point P. Thus, P is a nodal point, located on a 
nodal line. On the other hand, if the slit width is less than K. 
then there can be no nodal point. This is olnioirs, since no point 
can be a distance \ farilier- tVom one sid(^ of the slit tlian fi-om 



374 



UiMT 3 / 1 HE TRIIMPH OF .VIECH^V.MCS 



the other. Slits of widths less than \ hehave nearly as point 
sources. The narrower they are, the more nearly their behavior 
resembles that of point sources. 

You can easily compute the wavelength of a wave from the 
interference pattern set up where diffiacted wtwes overlap. For 
example, you can analyze the two-slit pattern on page 373 
exactly as you analyzed the two-souice pattern in Sec. 12.6. This 
is one of the main reasons for interest in the interference of 
diffracted waves. By locating nodal lines formed beyond a set of 
slits, you can calculate \ even for wiives that you cannot see. 

For two-slit inteiference, the larger the wavelength compared 
to the distance between slits, the more the interference pattern 
spreads out. That is, as X increases or d decreases, the nodal and 
antinodal lines make increasingly large angles with the straight- 
ahead direction. Similarly, for single-slit diffraction, the pattern 
spreads when the ratio of wavelength to the slit width increases. 
In general, diffraction of longer wavelengths is more easily 
detected. Thus, when you hear a band playing aiound a corner, 
you hear the bass drums and tubas better than the piccolos and 
cornets, even though they actually are playing equally loudly. 



A o B 




O 22. What characteristic do all [joints on a wave front have in 
common? 

23. State Huygens' principle. 

24. Can there be nodal lines in a diffraction pattern from an 
opening less than one wavelength wide? Explain. 

25. What happens to the diffraction pattern from an opening 
as the wavelength of the wave increases? 

26. Can there be diffraction without interference? inteiference 
without diffraction? 




SG 21 



12.9 Reflection 



You have seen that waves can pass through one another and 
spread around obstacles in their paths. Waves also are leflected, 
at least to some degree, whenever they reach any boundaiy of 
the medium in which they travel. Echoes are familial' examples 
of the reflection of sound waves. All waves share the property of 
reflection. Again, the supeiposition piinciple will help you 
understand what happens when reflection occurs. 

Suppose that one end of a rope is tied tightly to a hook 
securely fastened to a massive wall. From the other end, a pulse 
wave is sent down the rope towaid the hook. Since the hook 
cannot move, the force exerted by the rope wave can do no work 
on the hook. Theiefore, the energv carried in the w^jve cannot 




CHAPTER 12 / WAVES 



375 






leave the rope at this fixed end. histead, the wave bounces back, 
is reflected, ideally v\ith the same energy. 

What does the wave look like after it is reflected.^ 1 li(> .striking 
result is that the wave seems to fiip upside down on rellection. 
As the wave comes in from left to light and encounters the fixed 
hook, it pulls up on it. By Newton's third law, the hook must 
exert a force on the I'ope in the opposite direction while 
reflection is taking place. The details of how this force \aries in 
time are complicated. The net effect is that an inverted wave of 
the same form is sent back down the rope. 

SG 22 The sketches in the margin on the left show the results of 

reflection of water waves from a straight wall. You can see if the 
sketches are accurate by trying to reproduce the effect in your 
sink or bathtub. Wait until the water is still, then dip your 
fingertip briefly into the water or let a drop fall into the water. In 
the upper sketch, the outer crest is approaching the barrier at 
the right. The next two sketches show the positions of the crests 
after first one and then two of them have been reflected. Notice 
the dashed curves in the last sketch. They show that the 
reflected wave appears to originate from a point S' that is as far 
behind the barrier as S is in front of it. The imaginaiy source at 
point S' is called the image of the source S. 

Reflection of circular waves is studied first, because that is 
what you usually notice first when studying water waves. But it 
is easier to see a general principle for explaining reflection by 

-. ^^ observing a straight wave front, reflected ft'om a straight barrier. 

V \ \ The ripple-tank photograph on page 377 shows one instant 

; 'j ', during such a reflection. (The wave came in from the upper left 

/ /' ,' at an angle of about 45°.) The sketches below show in more 

III "-' 

/ / detaU what happens as the wave crests reflect from the straight 
r' barrier. 







The description of wave behavior is often made easier by 
drawing lines peipendicular to the vvaxe fronts. Such lines, called 
rays, indicate the direction of propagation of the wave. Notice the 
last drawing in the margin, for example. Rays have been drawn 
for a set of wave crests just befoi'e reflection and just after 
reflection horn a barriei'. The straight-on direction, peipendicular 
to the reflecting surface, is shouTi by a dotted line. The ray for 
the incident crests makes an angle 6, with the straight-on 
direction. The ray for the reflected crests makes an angle H, with 
it. The angle of reflection 6, is equal to the angle of incidence B,: 
that is, 6, = 0.. This is an experimental fact, which \ ou can \'erifv 
for vourself. 



376 



UIVIT 3 / THE TRIUVIPH OF MECHANICS 



Manv kinds of wave reflectors are in use today, from radar 
antennae to infrared heateis. Figures la) and (b) below show how 
straight-line waves reflect from two circular reflectois. A few 
incident and reflected rays are shown. (The dotted lines are 
peipendicular to the barrier suiface.) Rays reflected from the 
half-circle (a) head off in all directions. However, rays reflected 
from a smafl segment of the circle (b) come close to meeting at a 
single point. A barrier with the shape of a parabola (c) focuses 
straight-line waves piecisely at a point. Similarly, a parabolic 
surface reflects plane waves to a sharjD focus. An impressixe 
example is a radio telescope. Its huge parabolic suiface reflects 
faint radio waves fi^om space to focus on a detector. 




The wave piiths indicated in the sketches could just as well be 
reversed. For example, spherical waves pioduced at the focus 
become plane waves when reflected from a parabolic suiface. 
The flashlight and automoliile headlamp are familiar applications 
of this principle. In them, white-hot wires placed at the focus of 
parabolic reflectors produce almost parallel beams of light. 



Z7. What is a "ray"'::' 

28. What is the relationship between the angle at which a wave 

front strikes a barrier and the angle at which it leaves'::* 

Z9. What shape of reflector can reflect parallel wave fronts to 

a sharp focus':' 

30. What happens to wave fronts originating at the focus of 

such a reflecting surface':' 




m 





Above: A ripple tank shadow show- 
ing how circular waves produced 
at the focus of a parabolic wall are 
reflected from the wall into 
straight waves. 



12.10 I Refraction 

What happens when a wave propagates from one medium to 
another medium in which its speed of propagation is different? 
Look at the simple situation pictured on page 378. Two one- 



CHAPTER 12 / WAVES 



377 




Above: The fiUirnent of a flashlight 
bulb is at the focus of a parabolic- 
mirror, so the reflected light forms 
a nearly parallel beam. 



1 



A_A 



V 



~V 



v 



A 



Pulses encountering a boundary 
between two different media. The 
speed of propagation is less in me- 
dium Z. 




Continuous wave train crossing the 
boundary between two different 
media. The speed of propagation is 
less in medium Z. 



SG 2fi 



dimensional pulses approach a lioiindaiy sepaiating two media, 
rhe speed ot the propiigation in medium 1 is greatei- tlian it is 
in medium 2. Imagine the pulses to be in a light rope (medium 
II tied to a relatively hea\y rope Imedium 2i. Part of each pulse is 
reflected at the boundaiy. Tiiis rellected component is Hipped 
upside down relative to the original pulse. Recall the inverted 
reflection at a hook in a wall discussed earlier. The heavier rope 
here tends to hold the boundaiy point fixed in just the same 
way. But what happens to that part of the wave that continues 
into the second medium.-' 

As shown in the figure, the transmitted pulses aie closei' 
together in medium 2 than they are in medium 1. The speed of 
the pulses is less in \hv. heavier lope. So the second pulse is 
catching up with the first while the second pulse is still in the 
light rope and the first is already in the heavy rope. In the same 
way, each separate [lulse is itself s(|ueezed into a narrower form: 
that is, while the front of the pulse is entering the region of less 
speed, the back part is still moving with greater speed. 

Something of the same sort hafipens to a periodic wave at 
such a boundary. This situation is pictured in the figur-e below. 
For the sake of simplicity, assume that all of the wave is 
transmitted and none of it is reflected. Just as the two pulses 
wer e brought closer- and each pulse was squeezed narrower-, the 
periodic wave patter^n is squeezed together, too. Thus, the 
wiivelength \., of the transmitted waxe is shor-ter- than the 
wavelength X, of the incoming, or incident, wave. 

Although the wavelength changes when the wave passes 
across the boundary, the frequency of the wave cumnot change. If 
the rope is unbroken, the pieces immediately on either- side of 
the boundaiy must go up and down together. The frequencies of 
the incident and transmitted weaves must, then, be e(|ual. V\'e (-an 
simply label both of them/! 

The wavelength, frequency, and speed relationship for- both the 
incident and transmitted vviives can be written separately: 

\j/ = V, and \, f — v^ 

Dividing one of these equations bv the other and eliminating the 



1 his etiualicjn tells that the ratio of th(> wavelengths in the two 
media equals the ratio of the speeds. 

I'he same sort of thing hap|iens when water- ripples cr-oss a 
boundary. Experiments show that the ripples move more slowly 
in shallower water. A piece of plate glass is placed on the bottom 
of a r-i|)ple tank to make the water- shallower- there. This cr-(?at(\s 
a boundary' betwecMi the dee|)er- and sbailowei- part imiulium 1 



378 



UNIT 3 / I hi; IHllMPH OI MIICHAMCS 



(a) 




water 



and medium 21. Figure (a) iibove shows the case where this 
boundaiy is parallel to the crest lines of the incident wave. As 
with rope waves, the wavelength ot water waves in a medium is 
proportional to the speed in that medium. 

Water waves offer a possibility not present for rope waves. The 
crest lines can approach the boundaiy at any angle, not only 
head-on. Photograph (c) shows such an event. A ripple-tank wave 
approaches the boundary at an angle. The wavelength and 
speed, of course, change as the wave passes acioss the 
boundaiy. The direction of the wave propagation also changes. 
As each part of a ciest line in medium 1 enters medium 2, its 
speed decreases, and it starts to lag behind. In time, the 




(b) 




(c) 





Top: Ripples on water (coming 
from the left) encounter the shal- 
low region over the corner of a 
submerged glass plate. Bottom: 
Ripples on water (coming from the 
left) encounter a shallow region 
over a glass plate placed at an an- 
gle to the wave fronts. 



Aerial photograph of the refr^action 
of ocean waves approaching shore. 



CHAPTER 12 / WAVES 



379 



SG 27-31 



The slowing of star light by increas- 
ingly dense layers of the atmos- 
phere produces refraction that 
changes the apparent position of 
the star. 



directions of the whole set of ci^st lines in metliuiii 2 are 
changed from their directions in medium 1. 

This phenomenon is called refraction. Refraction occurs 
whenever a wave passes into a medium in which the wave 
velocitv' is reduced. The wave fronts are turned (refracted) so that 
they are more nearly parallel to the houndaiy. See photographs 
(b) and (c). This accounts for st^mething that you may have 
noticed if you have been at an ocean beach. No matter in what 
direction the waves are moving far from the shore, v\'hen they 
come near the beach their crest lines are nearly parallel to the 
shoreline. A wave's speed is steadily reduced as it mo\'es into 
water that gets gradually more shallow. So the wa\ e is refracted 
continuously as if it vv^ere always crossing a boundary between 
different media, as indeed it is. The refraction of sea wa\'es is so 
great that waxe crests can curl around a small island with an all- 
beach shoreline and proxide suif on all sides. 




31. If a periodic wave slows down on entering a new medium, 
what happens to (a) its frequency? (b) its wavelength? (c) its 
direction? 

32. Complete the sketch in the margin to show roughly what 
happens to a wave train that enters a new inedium in which its 
speed is greater. 



12.1.1 Sound ivaves 



Look again at the bottom figure in 
the margin of page 357. 



1 Hz = 1/sec, or one cycle (or os- 
cillation) per second. 



Sound waves are mechanical disturbances that propagate 
through a medium, such as the air. Typically, sound waves are 
longitudinal waves, producing changes of densitx' and pressure 
in the medium through which they traxel. The medium can be a 
solid, liquid, or gas. If the waves strike the ear, they can produce 
the sensation of hearing. The biology and psychology of hearing, 
as well as the physics of sound, are important to the science of 
acoustics. Here, of course, we will concentrate on sound as an 
example of wave motion. Sound has all the properties of wave 
motion considered so far. It exhibits refraction, diffraction, and 
the same relations among frequency, wavelength, and 
propagation speed and interference. Only the propeity of 
polarization is missing, because sound waxes arc longitudinal, 
not transxerse. 

Vibrating sources for sound wax^es may be as simple as a 
tuning fork or as complex as the human lannx xxith its xocal 
cords. Tuning forks and some special electronic dexices produce 
a steady "pure tone. " Most of the energy in such a tone is in 
simple hannonic motion at a single frequenrx The "pitch" of a 



380 



UNIT 3 THE TRH VIPH OF VIECR^VMCS 



sound goes up as the frequency of the wave increases. 

People can hear sound waves udth frequencies between about 
20 and 20,000 Hz. Dogs can hear over a much wider range 
(15-50,000 Hz). Bats, porpoises, and whales generate and respond 
to frequencies up to about 120,000 Hz. 

Loudness (or "volume") of sound is, like pitch, a psychological 
variable. Loudness is strongly related to the intensity of the 
sound. Sound intensity is a physical quantity. It is defined in 
terms of the energy carried by the wave and is measured in the 
number of watts per square centimeter transmitted through a 
surface perpendicular to the direction of motion of a wave front. 
The human ear can perceive a vast range of intensities of sound. 
The table below illustrates this range. It begins at a level of 
10"^'' watts per square centimeter (relative intensity = 1). Below 
this "threshold" level, the normal ear does not perceive sound. 



Relative Intensity 

1 

10' 

10' 

10' 

10' 

10' 

10' 

10' 

10« 

10' 

10'° 

10" 

10'^ 

10'^ 

10'^ 

10'' 

10" 

10" 



Sound 
Threshold of hearing 
Normal breathing 
Leaves in a breeze 

Library 

Quiet restaurant 

Two-person conversation 

Busy traffic 

Vacuum cleaner 

Roar of Niagara Falls 

Subway train 

Propeller plane at takeoff 

Machine-gun fire 

Small jet plane at takeoff 

Wind tunnel 

Space rocket at lift-off 



It is customary to measure loud- 
ness in decibels (db). The number 
of decibels is 10 times the exponent 
in the relative intensity of the sound. 
Thus, a jet plane at takeoff makes 
noise at the 140-db level. 



Levels of noise intensity about lO'" times threshold intensity can 
be felt as a tickling sensation in the ear. Beyond 10^^ times 
threshold intensity, the sensation changes to pain and may 
damage the unprotected ear. 

Often the simplest way of reducing noise is by absorbing it 
after it is produced but before it reaches your ears. Like all 
sound, noise is the energy of back and forth motion of the 
medium through which the noise travels. Noisy machinery can 
be muffled by padded enclosures in which the energy of noise is 
changed to heat energy, which then dissipates. In a house, a 
thick iTjg on the floor can absorb 90% of room noise. (Thirty 
centimeters of fresh snow is an almost perfect absorber of noise 
outdoors. Cities and countrysides are remarkably hushed after 
a snowfall.) 

It has always been fairly obvious that sound takes time to 
travel from source to receiver. By timing echoes over a known 
distance, the French mathematician Marin Mersenne in 1640 first 



Since many popular music con- 
certs produce steady sound levels 
near this intensity (and above it for 
the performers), there are many 
cases of impaired hearing among 
young people. 



CHAPTER 12 / WAVES 



381 



Close Upl 



The Sonic Doom 



In the last decades a new kind of noise has ap- 
peared: the sonic boom. An explosion-like sonic 
boom is produced whenever an object travels 
through air at a speed greater than the speed of 
sound (supersonic speed). Sound travels in air at 
about 340 m sec. Many types of military airplanes 
can travel at two or three times this speed. Flying 
at such speeds, the planes unavoidably and con- 
tinually produce sonic booms. SST (Supersonic 
Transport) planes such as the Concorde are now 
in civilian use in some countries. The unavoidable 
boom raises important questions. What are the 
consequences of technological "progress"? Who 
gains, and what fraction of the population do they 
represent? Who and how many pay the price? Must 
we pay it: must SST's be used? How much say has 
the citizen in decisions that affect the environment 
so violently? 

The formation of a sonic boom is similar to the 
formation of a wake by a boat. Consider a simple 
point source of waves. If the source remains in the 
same position in a medium, the wave it produces 
spreads out symmetrically around it. as in Diagram 
1. If the source of the disturbance is moving 
through the medium, each new crest starts from a 
different point, as in Diagram 2. 

Notice that the wavelength has become shorter 
in front of the object and longer behind it. This is 
called the Doppler effect. The Doppler effect is the 
reason that the sound an object makes seems to 
have a higher pitch when it is moving toward you 
and a lower pitch when it is moving away from you. 
In Diagram 3, the source is moving through the 
medium faster tfian the wave speed. Thus, the 
crests and the corresponding troughs overlap and 
interfere with one another. The interference is 
mostly destructive everywhere except on the line 
tangent to the wave fronts, indicated in Diagram 4. 
The result is a wake that spreads like a wedge away 
from the moving source, as in the photograph. 

All these concepts apply not only to water waves 
but also to sound waves, including those disturb- 
ances set up in air by a moving plane as the wind 
and body push the air out of the way. If the source 



of sound is moving faster than the speed of sound 
wave, then there is a cone-shaped wake (in three 
dimensions) that spreads away from the source. 

Actually, two cones of sharp pressure change are 
formed. One cone originates at the front of the air- 
plane and one at the rear, as indicated in the graph 
at the right. 

Because the double shock wave follows along 
behind the airplane, the region on the ground 
where people and houses may be struck by the 
boom (the "sonic-boom carpet") is as long as the 
supersonic flight path itself. In such an area, typi- 
cally thousands of kilometers long and 80 km wide, 
there may be millions of people. Tests made with 
airplanes flying at supersonic speed have shown 
that a single such cross-country flight by a 315-ton 
supersonic transport plane would break many thou- 
sands of dollars worth of windows, plaster walls, 
etc., and cause fright and annoyance to millions of 
people. Thus, the supersonic flight of such planes 
has been confined to over-ocean use. It may even 
turn out that the annoyance to people on shipboard, 
on islands, and on coastal areas near the flight 
paths is so great that over-ocean flights, too, will 
have to be restricted. 



3H2 



UMT 3 / THE TRIUMPH OF MECHANICS 




Double-cone shock wave, or sonic boom, pro- 
duced by an airplane that is traveling (at 21 -km 
altitude) at three times the speed of sound. 
Building B is just being hit by the shock wave, 
building A was struck a few seconds ago. and 
building C will be hit a few seconds later. (Draw- 
ing is not to scale.) 



AiR 



normal 




l€> 30 30 40 SO iO 10 g^ to IflO «» 

This curve represents the typical sonic boom 
from an airplane flying at supersonic speed 
(speed greater than about 340 misec). The pres- 
sure rises almost instantly, then falls relatively 
slowly to below-normal pressure, then nses again 
almost instantaneously. The second pressure 
rise occurs about 0. 1 sec after the first one, mak- 
ing the boom sound "double." 




^^.-<*=^$i?^/- 



>ite^..^^-' 



r^.- 



iimii^W^iM-ff. ' 



CHAPTER 12 / WAVES 



383 



SG 32 



SG 33-35 



Since the two frequencies are dif- 
ferent, the waves cannot stay in 
phase or out of phase. They there- 
fore alternate between destructi\e 
and constructive interference. See 
SG8. 



The acoustic properties of a hall 
filled with people are \'er\' different 
from those of the empty hall. 
Acoustical engineers sometimes fill 
the seats with felt-covered sand- 
bags while making tests. 



computed the speed of sound in air. It took another 70 years 
before William Derham in England, comparing the flash and 
noise from cannons across 20 km, came close to the modern 
measurements. 

Sound in air at 20°C moves at about 344 m/sec. As for all waves, 
the speed of sound waxes depends on the properties of the 
medium: the temperature, density, and elasticity. Sound waves 
generally travel faster in liquids than in gases, and faster still in 
solids. In seawater, their speed is about 1,500 m/sec; in steel, 
about 5,000 m/sec; in quartz, about 5,500 m/sec. 

Interference of sound waves can be shown in a xariety of ways. 
In a large hall with hard, sound-reflecting surfaces, there udll be 
dead" spots. At these spots, sound waves coming together after 
reflection cancel each other. Acoustic engineers must consider 
this in designing the shape, position, and materials of an 
auditorium. Another interesting and rather different example of 
sound interference is the phenomenon knovvTj as beats. When 
two notes of slightly different frequency are heard together, they 
interfere. This interference produces beats, a rhythmic pulsing 
of the sound. Piano tuners and string players use this fact to 
tune two strings to the same pitch. They simply adjust one string 
or the other until the beats disappear. 

Refraction of sound by different layers of air explains why you 
sometimes see lightning without hearing thunder. Similar 
refraction of sound occurs in layers of water of different 
temperatures. Geologists use the refraction of sound waves to 
study the earth's deep structure and to locate fossil fuels and 
minerals. Very intense sound waves are produced in the ground 
(as by dynamite blasts!. The sound waxes traxel through the 
earth and are received by detection dexices at different locations. 
The path of the waves, as refracted by layers in the earth, can be 
calculated from the relatixe intensities and times of sound 
received. From knowledge of the paths, estimates can be made of 
the composition of the layers. 

As mentioned, diffraction is a property of sound xvaxes. Sound 
xvaves readily bend around comeis and barriers to reach the 
listener xxithin range. Sound xvaves reflect, as do rope or water 
waves, xvherexer they encounter a boundaiy betxxeen different 
media. The architectural accidents ciilled ' xxhispeiing galleries " 
show vividly how sound can be focused by reflection from 
curved surfaces. All these effects are of interest in the study of 
acoustics. Moreox'er, the proper acoustical design of public 
buildings is now recognized as an important tunction hx' most 
good architects. 

In this chapter, you hax e studied the basic phenomena of 
mechanical waves, ending wdth the theory of sound propagation. 
The explanations of these phenomena xxere considered tiie final 
triumph of \e\\1onian mechanics as applied to the tiansfcr of 



384 



UNIT 3 / THE TRIUMPH OF MECHA.MCS 




An "anechoic chamber" being used for research in 
acoustics. Sound is almost completely absorbed dur- 
ing multiple reflections among the wedges of soft 
material that cover the walls. 



The concert hall of the University of Illinois Krannert 
Center for the Performing Arts was acoustically de- 
signed for unamplified performances. 



energy of particles in motion. Most of the general principles of 
acoustics were discovered in the 1870's. Since then, the study of 
acoustics has become involved with such fields as quantum 
physics. Perhaps its most important influence on modern 
physics has been its effect on the imagination of scientists. The 
successes of acoustics encouraged them to take seriously the 
power of the wave viewpoint, even in fields far from the original 
one — the mechanical motion of particles that mo\'e back and 
forth or up and dovvai in a medium. 



# 33. List five wave behaviors that can be demonstrated with 
sound waves. 



34. Can sound waves be polarized':' E^cplain. 



CHAPTER 12 / WAVES 



385 



study 

guide 

1. The Project Physics materials particularly 
appropriate for (]hapter 12 int-kide: 

Experiment^i 

Sound 

Activities 

Standing Waves on a Uruni and a Violin 
Moire Patterns 
Music and Speech Acti\ities 
Measurement of the Speed of Sound 
MechaniciU \\'a\'e Machines 

Film Loops 

Superposition 
Standing Waves in a String 
Standing Waves in a Gas 
Four Loops on X'ibrations 

2. Some waxes propagate at such a high speed that 
you are usually not a\vai-e of any delay '\n energy 
transfer. Give an example of a compression wave in a 
solid, started by an action at one end, that 
propagates so quickly that you are not aware of any 
delay before an effect is exliibited at the other end. 

3. Describe the diff(!rences in phase of oscillation 
of various parts of your l)ody as you walk. What 
points are exactly in phase? Which points are exactly 
Vz cycle out of phase? Ai-e there any points 'A cycle 
out of phase? 




Q+t| 




4. Pictured are two pulse waxes lA and H) on a 
rope at tiu; instants l)(!f()r(! and aft(!r tiu^y ()\(;rlap I/, 
and tj. Divide the elapsed time between f, and t, into 
four equal intervals and plot the shape of the rope 
at the end of each intei'v^U. 




5. Repeat SG 4 for the two pulses (A and CI 
pictured above. 



att2 



6. The wave above propagates to the right along a 
rope. What is the shape of the wa\'e propagating to 
the left that could for an instant cancel this one 
completely? 

7. The velocity of a portion of ropt: at some instant 
as transverse waves are passing through it is the 
superposition of the xclocities of waxes passing 
through that j)ortion. is th(! kin(;tic energ\' of a 
portion of the rope the superposition of the kinetic 
energies of xvax'es passing through that rc^gion? 
Justify your ansxxer. 

». Shoxxn in the; figure arc two xxaxes of slightly 
diff(M('nt frc(|U('n(x . find liu'lr sum gra|)hi(allx . Wni 



386 



C;FL\I»THK 12 / STUDY GUIDE 



will probably find that the sum has "beats." Can you 
find a way to determine the frequency of the beats 
if you know the fi-equencies of the waves? 

9. What sliape would the nodal regions have for 
sound waves from two loudspeakers? 

10. Imagine a detection device for waves is moved 
slowly to either the right or left of the point labeled 
A,, in the figure on page 366. Describe what the 
detection device would register. 



ss 



14. (a) What is the speed of sound in air if middle C 
(256 Hz) has a wavelength of 1.34 m? 

(b) What is the wavelength in water of middle C if 
sound waves travel at 1 ,500 m/sec in water? 

(c) What is the period of middle C in air and in 
water? 

15. Different notes are sounded with the same 
guitar string by changing its vibrating length (that is, 




11. What kind of interference pattern would you 
expect to see if the separation between two in-pliase 
sources were less than the wavelength \? Where 
would the nodal and antinodal lines be if the two in- 
phase sources were separated by the distance X? by 
X/2? Convince yourself that one additional nodal line 
appears on each side of the central antinodal line 
whenever the separation between the two in-phase 
sources is increased by one wavelength. 

13. Derive an equation, similar to nkd = d^^, for 
nodal points in a two-source interference pattern 
(where d is the separation of the sources, € the 
distance from the sources, and s„ the distance of the 
nth node from the center line). 

13. If you suddenly disturbed a stretched rubber 
hose or Slinky with a frequency that precisely 
matched a standing wave frequency, would standing 
waves appeal' immediately? If not, what factors 
would determine the time delav? 



pressing the string against a brass ridge). If the fuU 
length of the string is L, what lengths must it be 
shortened to in order to sound (a) a "musical 
fourth," (b) a "musical fifth," (c) an "octave"? 

16. An oscilloscope displays a picture of an electric 
wave so that its amplitude and frequency can be 
easily measured. What is the frequency of the 
incoming waves if eight complete cycles cover 10 cm 
of the oscilloscope screen and the electron beam is 
moving across the screen at 100 cm/sec? 

17. Standing sound waves can be set up in the air in 
an enclosure (like a bottle or an organ pipe). In a 
pipe that is closed at one end, the air molecules at 
the closed end are not free to be displaced, so the 
standing wave must have a displacement node at the 
closed end. At the open end, however, the molecules 
are almost completely free to be displaced, so the 
standing waves must have an antinode near the open 
end. 



CHAPTER 12 / STUDY GUIDE 



387 



sg 



(a) What will be the \va\'(;longth of the fundamental 
standing wave in a pipe of length L eloscnl at one 
end? (Hint: What is the longest wave that has a node 
and an antinode a distance L apart?) 

(b) What is a general expression for possible 
wavelengths of standing waves in a pipe closed at 
one end? 

(c) Answer (a) and lb) for the case of a pipe open at 
both ends. 

18. Imagine a spherical blob of Jello in whicii you 
can set up standing \'ibrations. What would be some 
of the possible modes of \ibration? (Hint: What 
possible symmetrical nodal surfaces could there be?) 

19. Suppose that straight-line ripple waves 
approach a thin straight barrier that is a few 
wavelengths long and that is oriented with its length 
pariillel to the wave fronts. What do you predict 
about the nature of the diffraction pattern along a 
straight line behind the barrier tiiat is j^erpendicular 
to the barri(M' and passes through the centCM* of the 
barrier? Why do people who design breakwaters 
need to concern themselves with diffraction effects? 

20. A megaphone directs sound along the 
megaphone axis if the waxelength of the sound is 
small compared to the diameter of the opening. 
Estimate the upper limit of frequencies that ai^e 
diffracted at a cheerleader's megaphone opening. 
Can you hear what a cheerleader shouts e\en though 
you are far off the axis of the megaphone? 

21. Explain why it is that the narrower a slit in a 
barrier is, the more nearly it can act like a point 
source of vvax'es. 

22. If light is also a wave, why luut; you not seen 
light Ijcing diffracted by the; slits of a picket fence, or 
diffracted around the corner of a house? 

23. Assuming that light is a waxe phenomenon, 
what is the wa\elength of green light if the first node; 
in a diffraction pattern is found 10 cm from th(; 
center line at a distanc-c; of 4 m from the slits which 
have a separation distance of 2.5 x lu ' cm? 

24. By actual construction with a ruler and compass 
on a tracing of the photograjjli on page 'M7, show 



that rays for the reflecttul wa\e front appc^ar to come 
from S'. Show also that this is consistent with 

e, = e,. 

25. A sti-aight-line wave approaches a right-angle 
reflecting barrier as shown in the figure. Find the 
shape, size, and direction of jjropagation of the wave 
after it has been completely reflected by th(! barrier. 




26. With ruler and compass reproduce part (b) of 
the figure at the bottom of page 377 and find the 
distance from the circle's center to the point P in 
terms of the radius of the circle r. Make the radius 
of your circle much lai'ger than the one in the 
figure. (Hint: The dotted lines are along radii.) 

27. Convince yourself that a parabolic reflector will 
actually bring parallel wave fronts to a sharp focus. 
Draw a parabola y = k^^ (choosing any convenient 
value for k) and some parallel rays along the axis as 
in part Ir) of the figure at the bottom of page 377. 
Construct line segments perpendicular to the 
parabola where the rays hit it, and di'aw the reflected 
rays at equal angles on the other side of these lines. 

28. The focal length of a curved reflector is the 
distance from the reflector to the ]3oint where 
pariUlel rays are focused. L'se \hv. drawing in S(i 24 to 
fine! the fociil length of a parabola in terms of k. 

29. K(!calling that water surface; waxes tra\(!l slower 
in shallow water, what would you expect to happen 
to the shape of the following wave as it continues 

to the right? Pay particular attention to the region of 
\'aiying depth. Can you use the; line of reasoning 
above to gi\e at least a partial explanation of the 
cause of breakers near a beach? 



3HH 



CHAPTEK 12 / STUDY GLIDE 



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30. If the frequency of a wave traveling in a medium 
is increased, what will happen to its speed? What 
determines the speed of waves in a medium? 

31. A straight-line wave in a ripple tank approaches 
a boundary between deep and shallow water as 
shown. Describe the shape of the wave as it passes 
through the boundary and then as it continues in the 
shallow water. 




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32. The diagram below shows two successive 
positions. AB and CD, of a wave train of sound or 
light, before and after crossing an air-glass 
boundary. The time taken to go from AB to DC is 
one period of the wave. 

(a) Indicate