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Full text of "The Triumph of Mechanics, Project Physics Text and Handbook Volume 3"

PROJECT PHYSICS 



Text and Handbook O 
The Triumph of Mechanics 



I I I I I I 





c^ 



PROJECT PH-YSICS 



Directors 

F. James Rutherford 
Gerald Holton 
Fletcher G. Watson 



Published by HOLT, RINEHART and WINSTON, Inc. New York. Toronto 




PROJECT PHYSICS 



Unit 



3 



Text and Handbook 

The Triumph of Mechanics 





Directors of Harvard Project Physics 

Gerald Holton, Department of Physics, Harvard 

University 
F. James Rutherford, Chairman of the Department 

of Science Education, New York University 
Fletcher G. Watson, Harvard Graduate School 

of Education 



Acknowledgments, Text Section 

The authors and publisher have made every effort 
to trace the ownership of all selections found in this 
book and to make full acknowledgment for their use. 
Many of the selections are in the pubbc domain. 

Grateful acknowledgment is hereby made to the 
following authors, publishers, agents, and individ- 
uals for use of their cop^xighted material. 



Special Consultant 
to Project Physics 

Andrew Ahlgren, University of Minnesota 



A partial list of staff and consultants to Hansard 
Project Physics appears in the Text and 
Handbook, Unit 1. 



This Text and Handbook, Unit 3 is one of the many 
instructional materials developed for the Project 
Physics Course. These materials include Text, Handbook, 
Resource Book, Readers, Programmed Instruction 
booklets. Film Loops, Transparencies, 16mm films, 
and laboratory equipment. 



Copyright © 1975, 1970, Project Physics 

All Rights Reserved 

ISBN 0-03-089638-X 

456789 039 987654321 

Project Physics is a registered trademark 



P. 1 Desagulier, J. T., The Newtonian System of the World, 

The Best Model of Government , an Allegorical Poem. 

P. 2 Descartes, Rene, The Philosophical Works of Descartes, 

trans. Haldane, Ehzabeth S. and Ross, G. R. T., Vol. 1, p. 

299, copyright 1931 by Cambridge University Press. 

P. 2 "A Free Inquiry into the received Notion of Nature," 

The Works of Robert Boyle, Vol. IV, p. 362. 

P. 3 Newton, Sir Isaac, The Principia, Motte's translation 

revised by Florian Cajori, preface to first edition (1687), p. 

xviii, copyright ©1961 by University of California Press. 

P. 5 Lucretius, On the Nature of Things, trans. H. Munro, 

George Bell & Sons, Book II, p. 51. 

P. 5 Bacon, Francis, Complete Essays Including the New 

Atlantis and Novum Organum, edit, by Finch, Henry L., 

Washington Square Press. 

P. 6 Lavoisier, Antoine, Elements of Chemistry, (1790), a 

facsimile of an original translation by Robert Kerr with new 

intro. by Professor Douglas McKie, Dover Publications, Inc. 

P. 22 Gerhardt, Die Philosophische Schriften von Gottfried 

Wilhelm Leibniz, 1875-90, Vol. IV, p. 286. Eng. Trans, in 

Dugas, R., Mechanics in the Seventeenth Century, Central 

Book Co., p. 407. 

P. 41 Darwin, Erasmus, Botanic Garden — The Economy of 

Vegetation, J. Moore, p. 49. 

P. 43 Savery, Thomas, in Hart, I. B., James Watt &■ the 

History of Steam Power, copvTight ©1961 by Crowell 

Collier-Macmillan & Co. 

P. 50 Joule, James Prescott, The Scientific Papers of James 

Prescott Joule, an illustrated facsimile of the 1884 edition, 

copyright ©1963 by Dawson of Pall Mall. 

P. 56 Ibid. 

P. 58 Helmholtz, H. L. F., Popular Scientific Lectures, 

edited by Morris Kline, copyright 1873, 1881, ©1962 by 

Dover Publications, Inc. 

P. 63 Poincare, Henri, La Science et I'Hypothese, E. 

Flammarion, excerpt trans, by McClaim, John. 

P. 78 Lord KeKin, "The Size of Atoms,'",\'flfj/re, Vol. I, p. 

551. 

P. 86 Frost, Robert, from "West-Running Brook," The 

Poetry of Robert Frost, ed. by Edwiu-d Connery Lathem, p. 

259, copyright ©1969 by Holt, Rinehaii and Winston. Inc., 

copyright ©1956 by Robert Frost. Reprinted by permission 

of Holt. Rinehart and Winston, Inc. 

P. 92 Lord Kelvin, "The Kinetic Theory of the Dissipation of 

Energy." Proceedings of the Royal Society of Edinburgh, 

Vol. 8, p. 325. 

P. 93 Nietzsche, Friedrich. "Der Wille zur Macht. " 

Nietzsche: An Anthology of His Works, ed. Manthey-Zorn. 

Otto, Washington Square Press, p. 90. 



IV 



p. 93 Shelley, Percy Bysshe, Hellas, ed. Wise, Thomas J., 
Reeves & Turner, pp. 51-52 not inclusive. 
Pp. 93-94 Poincare, Henri, Kinetic Theory, ed. Stephen 
Brush, Vol. 2. p. 206, copyright ©1966 by Pergamon Press. 
Pp. 120-121 Huygens, Christiaan, Treatise on Light, first 
published in Paris (1690), trans. Thompson, Silvanus P., 
copyright 1912 by Dover Publications, Inc. 
P. 134 Power, Henry, Experimental Philosophy (1664), 
reprinted 1966 by Johnson Reprint Corp., p. 192. 
P. 134 Donne, John, "The First Anniversary," Donne's 
Poetical Works, ed. Grierson, Herbert J., Oxford University 
Press (Clarendon Press imprint). Vol. 1, p. 237. 



Picture Credits, Text Section 

Cover photograph, p. 134 "Locomotive Wheels" by Farrell 

Grehan. 

P. 2 (watch assembly) Swiss Federation of Watch 

Manufacturers; (Blake drawing) Whitworth Art Gallery, 

University of Manchester. 

P. 3 Albert B. Gregory, Jr. 

P. 4, 135 (top margin) Pictorial Parade, N.Y.C. 

P. 5 (thunderhead) Peter M. Saunders. Woods Hole 

Oceanographic Institute, (bonfire) Colgate University. 

P. 7 (Lavoisier portrait) painted by Jacques Louis David. 

Courtesy of The Rockefeller University. 

P. 12 Boston Patriots Football Club. 

P. 14 (hockey players) Pictorial Parade, N.Y.C; (space 

vehicle) National Aeronautics & Space Administration; 

(galaxy) Mount Wilson and Palomar Observatories; 

(colliding balls) National Film Board of Canada. 

P. 17 The Boeing Company. 

P. 18 (galaxy) Mount Wilson and Palomar Observatories; 

(colliding cars) Henry Groskwsky, LIFE MAGAZINE, © 

Time Inc. 

P. 20 (Huygens) Royal Netherlands Academy of Sciences 

and Letters, Amsterdam. 

P. 21 (Descartes) the Franz Hals portrait in the National Art 

Museum, Copenhagen. 

P. 22 (Leibniz) Burndy Library, Norwalk, Conn. 

P. 27 "The Little Prince" ft-om THE LITTLE PRINCE by 

Antoine de Saint Exupery, copyright 1943 by Harcourt, 

Brace & World, Inc., reproduced with their permission. 

P. 28, 135 (margin, second from top) Russ Kinne — Photo 

Researchers. 

P. 35 Albert B. Gregory, Jr. 

P. 36 Dr. Harold E. Edgerton, M.I.T. 

P. 37 Dimitri Kessel, LIFE MAGAZINE, © Time Inc. 

P. 39 (camel & waterwheel) C. W, Kirkland, LIFE 

MAGAZINE, © Time Inc.; (reversible overshot 

waterwheel) Agricola, Georgius, De Re Metallica, 1556, 

Houghton Library, Harvard University; (windmill) M. G. 

Walker, LIFE MAGAZINE, © Time Inc. 



P. 40 (aeolipile) Science Museum, London. 

P. 43 Newcomen Society, London. 

P. 44 (Watt in his laboratory) Figuier, Louis, Les Merveilles 

de la Science, Furne, Jouvet et Cie, Paris. 

P. 46 (train) Boston & Maine Corporation. 

P. 48 (steamboat) Science Museum, London. 

P. 49 (top) Courtesy of General Electric Company; (bottom) 

American Institute of Physics. 

P. 52 Professor Keith R. Porter, Dept. of Biology, Harvard 

University. 

P. 55 "The Repast of the Lion," by Henri Rousseau, 

courtesy of The Metropolitan Museum of Art, bequest of 

Samuel A. Lewisohn, 1951. 

P. 57 (Goethe) painting by Angelica Kauffmann, 1787, 

Harvard College Observatory; (Schelling) pastel by 

Friedrich Tieck, 1801, collection of Dr. Hermann von 

Schelling. 

P. 58 Koenigsberger, Leo, Hermann von Helmholtz, 

Braunschweig Druck und Verlag, von Friedrich Vieweg und 

Sohn. 

P. 65 (wrecker) Harry Redl, LIFE MAGAZINE, © Time 

Inc.; (ocean liner) courtesy of Kenyon & Eckhardt, Inc. and 

Fugazy Travel Bureau, Inc. 

P. 70, 135 (margin, third firom top) Courtesy AMF-Voit. 

P. 71 (balloon) U.S. Air Force. 

P. 86 American Institute of Physics. 

P. 91 (Maxwell) Courtesy of Trinity College Library, 

Cambridge; (colliding balls) PSSC Physics, D. C. Heath & 

Co., 1965. 

P. 92 (light bulb) Dr. Harold E. Edgerton, M.I.T. ; (bonfire) 

Colgate University; (landscape) "Mount 

WHbamson — Clearing Storm," 1944, Ansel Adams. 

P. 93 Thompson, Silvanus P., Life of William Thomson, vol. 

1, MacmiUan and Co., Ltd., London. 

P. 94 P. E. Genereux, E. Lynn. 

P. 96 Greek National Tourist Office, N.Y.C. 

P. 100, 111, 135 (bottom margin) Magnum Photos Inc., 

N.Y.C. Werner Bischof 

P. 105 Union Pacific Railroad. 

P. 117 "Singer with Lyre" from red-figured amphora 

attributed to the Brygos painter. Courtesy of the Museum of 

Fine Arts, Boston. John Michael Rodocanachi Fund. 

P. 119 National Film Board of Canada. 

Pp. 120, 122-123, 125 (ripple tank photos) Courtesy, Film 

Studio Educational Development Center. 

P. 121 Courtesy, College of Engineering, University of 

California, Berkeley. 

P. 125 (radiotelescope) National Radio Astronomy 

Observatory, Green Bank, W.Va. 

P. 128 U.S.' Nav7. 

P. 130 Schaefer and Seawell 

P. 133 (anechoic chamber) Bell Telephone Laboratories; 

(concert haU) Hedrich-BIessing 

P. 143 U.S. Na\^. 



All photographs not credited above were made 
by the staff of Harvard Project Physics. 



Science is an adventure of the ivhole 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 extension of 
human possibilities .... 

I propose that science be taught at whatever level, from the loivest to the 

highest, in the humanistic way. It should be taught with a certain historical 

understanding, ivith a certain philosophical understanding, xvith 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 



Background The Project Physics Course is based on the ideas and 
research of a national curriculum development project that worked in 
three phases. First, the authors— a high school physics instructor, a 
university physicist, and a professor of science education — collaborated 
to lay out the main goals and topics of a new introductory physics 
course. They worked together from 1962 to 1964 with financial support 
from the Carnegie Corporation of New York, and the first version of 
the text was tried out in two schools with encouraging results. 

These preliminary results led to the second phase of the Project 
when a series of major grants were obtained from the U.S. Office of 
Education and the National Science Foundation, starting in 1964. 
Invaluable additional financial support was also provided by the 
Ford Foundation, the Alfred P. Sloan Foundation, the Carnegie 
Corporation, and Harvard University. A large number of collaborators 
were brought together from all parts of the nation, and the group 
worked together for over four years under the title Harvard Project 
Physics. At the Project's center, located at Harvard University, 
Cambridge, Massachusetts, the staff and consultants included college 
and high school physics instructors, astronomers, chemists, historians 
and philosophers of science, science educators, psychologists, 
evaluation specialists, engineers, film makers, artists and graphic 
designers. The instructors serving as field consultants and the students 
in the trial classes were also of vital importance to the success of 
Harvard 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 instructors and students in those schools 
reported their criticisms and suggestions to the staff in Cambridge, 
and these reports became the basis for the subsequent revisions of 
the course materials. In the Preface to the Text you will find a list of the 
major aims of the course. 

vi 



We wish it were possible to list in detail the contributions of each 
person who participated in some part of Harvard Project Physics. 
Unhappily it is not feasible, since most staff members worked on a 
variety of materials and had multiple responsibilities. Furthermore, 
every text chapter, experiment, piece of apparatus, film or other item 
in the experimental program benefitted from the contributions of a 
great many people. Beginning on page A21 of the Text Appendix is a 
partial list of contributors to Harvard Project Physics. There were, in 
fact, many other contributors too numerous to mention. These include 
school administrators in participating schools, directors and staff 
members of training institutes for teachers, instructors who tried the 
course after the evaluation year, and most of all the thousands of students 
who not only agreed to take the experimental version of the course, 
but who were also willing to appraise it critically and contribute their 
opinions and suggestions. 

The Project Physics Course Today. Using the last of the experimental 
versions of the course developed by Harvard Project Physics in 
1964-68 as a starting point, and taking into account the evaluation 
results from the tryouts, the three original collaborators set out to 
develop the version suitable for large-scale publication. We take 
particular pleasure in acknowledging the assistance of Dr. Andrew 
Ahlgren of the University of Minnesota. Dr. Ahlgren was invaluable 
because of his skill as a physics instructor, his editorial talent, his 
versatility and energy, and above all, his commitment to the goals of 
Harvard Project Physics. 

We would also especially like to thank Ms. Joan Laws whose 
administrative skills, dependability, and thoughtfulness contributed so 
much to our work. The publisher, Holt, Rinehart and Winston, Inc. 
of New York, provided the coordination, editorial support, and general 
backing necessary to the large undertaking of preparing the final 
version of all components of the Project Physics Course, including 
texts, laboratory apparatus, films, etc. Damon-Educational Division, a 
company located in Westwood, Massachusetts, worked closely with us 
to improve the engineering design of the laboratory apparatus and to 
see that it was properly integrated into the program. 

In the years ahead, the learning materials of the Project Physics 
Course will be revised as often as is necessary to remove remaining 
ambiguities, clarify instructions, and to continue to make the materials 
more interesting and relevant to students. We therefore urge all 
students and instructors who use this course to send to us (in care of 
Holt, Rinehart and Winston, Inc., 383 Madison Avenue, New York, 
New York 10017) any criticism or suggestions they may have. 



F. James Rutherford 
Gerald Holton 
Fletcher G. Watson 



VII 



Answers to End-of-Section Questions 



Chapter 9 

Q1 False 

02 No. Don't confuse mass with volume or mass 

with weight. 

Q3 Answer C 

Q4 No. Change speed to velocity and perform 

additions by vector techniques. 

Q5 (a), (c) and (d) (Their momenta before collision 

are equal in magnitude and opposite in direction.) 

Q6 Least momentum: a pitched baseball (small 

mass and fairly small speed) 

Greatest momentum: a jet plane in flight (very large 

mass and high speed) 

Q7 (a) about 4 cm/sec. Faster ball delivers more 

momentum to girl. 

(b) about 4 cm/sec. More massive ball delivers more 
momentum to girl. 

(c) about 1 cm/sec. With same gain in momentum 
more massive girl gains less speed. 

(d) about 4 cm/sec. Momentum change of ball is 
greater if its direction reverses. 

(These answers assume the mass of the ball is much 

less than the mass of the girl.) 

Q8 It can be applied to situations where only 

masses and speeds can be determined. 

Q9 Conservation of mass: No substances are added 

or allowed to escape. 

Conservation of momentum: No net force from 

outside the system acts upon any body considered 

to be part of the system. 

Q10 None of these is an isolated system. In cases 

(a) and (b) the earth exerts a net force on the system. 
In case (c) the sun exerts a net force on the system. 
Q11 Answer (c) (Perfectly elastic collisions can 
only occur between atoms or subatomic particles.) 
Q12 Answer (d) (This assumes mass is always 
positive.) 

Q13 Answer (c) 

Q14 (a) It becomes stored as the object rises. 

(b) It becomes "dissipated among the small parts" 
which form the earth and the object. 

Chapter 10 

Q1 Answer (b) 
Q2 Answer (b) 
Q3 Answer (c) 

Q4 Answer (c) The increase in potential energy 
equals the work done on the spring. 
Q5 Answer (e) You must do work on the objects 
to push them closer together. 

Q6 Answer (e) Kinetic energy increases as gravita- 
tional potential energy decreases. Their sum remains 
the same (if air resistance is negligible). 
Q7 Potential energy is greatest at extreme position 
where the speed of the string is zero. Kinetic energy 
is greatest at midpoint where the string is unstretched. 
Q8 The less massive treble string will gain more 
speed although both gain the same amount of kinetic 



VIII 



energy (equal to elastic potential energy given by 

guitarist). 

Q9 Multiply the weight of the boulder (estimated 

from density and volume) by the distance above 

ground level that it seems to be. (For further 

discussion see SG 10.15.) 

Q10 None. Centripetal force is directed inward 

along the radius which is always perpendicular to 

the direction of motion for a circular orbit. 

Q11 Same, if initial and final positions are identical. 

Q12 Same, if frictional forces are negligible. Less if 

frictional forces between skis and snow are taken 

into account. 

Q13 Answer (c) 

Q14 Answer (c) 

Q15 False. It was the other way around. 

Q16 Chemical, heat, kinetic or mechanical 

Q17 Answer (b) 

Q18 Answer (d) 

Q19 It is a unit of power, or rate of doing work, 

equal to 746 watts. 

Q20 Answer (d) 

Q21 Answer (b) 

Q22 Nearly all. A small amount was transformed 

into kinetic energy of the slowly descending weights 

and the water container would also have been 

warmed. 

Q23 Answer (a) 

Q24 Answer (e) 

Q25 The statement means that the energy which 

the lion obtains from eating comes ultimately from 

sunlight. He eats animals, which eat plants which 

grow by absorbed sunlight. 

Q26 Answer (c) 

Q27 Answer (a) 

Q28 Answer (c) 

Q29 Answer (c) 

Q30 AE is the change in the total energy of the 

system 

AW is the net work (the work done on the 
system — the work done by the system) 

AH is the net heat exchange (heat added to 
the system — heat lost by the system) 
031 1. heating (or cooling) it 

2. doing work on it (or allowing it to do work) 

Chapter 11 

01 Answer (c) 

02 True 

03 False 

04 Answer (b) 

05 In gases the molecules are far enough apart 
that the rather complicated intermolecular forces can 
safely be neglected. 

06 Answer (b) 

07 Answer (b) 

08 Answer (d) 

09 Answer (c) 

(continued on p. Al) 



Project Physics 



Text O 



The Triumph of Mechanics 




1 r I ri r 



...^ 



Things to Do and Use 



Experiments 

3-1 Collisions in One Dimension I 

3-2 Collisions in One Dimension II 

3-3 Collisions in Two Dimensions I 

3-4 Collisons in Two Dimensions II 

3-5 Conservation of Energy I 

3-6 Conservation of Energy II 

3-7 Measuring the Speed of a Bullet 

3-8 Energy Analysis of a Pendulum Swing 

3-9 Least Energy 

3-10 Temperature and Thermometers 

3-1 1 Calorimetry 

3-12 Ice Calorimetry 

3-13 Monte Carlo Experiment on Molecular 

Collisions 

3-14 Behavior of Gases 

3-15 Wave Properties 

3-16 Waves in a Ripple Tank 

3-17 Measuring Wavelength 

3-18 Sound 

3-19 Ultrasound 

Activities 

Is Mass Conserved? 

Exchange of Momentum Devices 

Student Horsepower 

Steam-powered Boat 

Problems of Scientific and Technological 

Growth 
Predicting the Range of an Arrow 
Drinking Duck 

Mechanical Equivalent of Heat 
A Diver in a Bottle 
Rockets 
How to Weigh a Car with a Tire Pressure 

Gauge 
Perpetual Motion Machines? 
Standing Waves on a Drum and Violin 
Moire Patterns 
Music and Speech Activities 
Measurement of the Speed of Sound 
Mechanical Wave Machines 
Resource Letter 

Fii! 
L18 
LI 9 
L20 
L21 
L22 
L23 
L24 
L25 
L26 
L27 
L28 
L29 
L30 
L31 



L32 
L33 
L34 
L35 

L36 



One-Dimensional Collisions I 

One-Dimensional Collisions II 

Inelastic One-Dimensional Collisions 

Two-Dimensional Collisions I 

Two-Dimensional Collisions II 

Inelastic Two-Dimensional Collisions 

Scattering of a Cluster of Objects 

Explosion of a Cluster of Objects 

Finding the Speed of a Rifle Bullet I 

Finding the Speed of a Rifle Bullet II 

Recoil 

Colliding Freight Cars 

Dynamics of a Billiard Ball 

A Method of Measuring Enegy — Nails 

Driven into Wood 
Gravitational Potential Energy 
Kinetic Energy 

Conservation of Energy — Pole Vault 
Conversation of Energy — Aircraft 

Take-off 
Reversibility of Time 



L37 Superposition 
L38 Standing Waves on a String 
L39 Standing Waves in a Gas 
L40 Vibrations of a Wire 
L41 Vibrations of a Rubber Hose 
L42 Vibrations of a Drum 
L43 Vibrations of a Metal Plate 
Programmed Instruction Booklets 
The Kinetic-Molecular Theory of Gases 
Waves 1 The Superposition Principle 
Waves 2 Periodic Waves 

1 Silence, Please 

by Arthur C. Clarke 

2 The Steam Engine Comes of Age 

by R. J. Forbes and E. J. Dijksterhuis 

3 The Great Conservation Principles 
by Richard P. Feynman 

4 The Barometer Story 
by Alexander Calandra 

5 The Great Molecular Theory of Gases 
by Eric M. Rogers 

6 Entropy and the Second Law of 

Thermodynamics 
by Kenneth W. Ford 

7 The Law of Disorder 
by George Gamow 

8 The Law 

by Robert M. Coates 

9 The Arrow of Time 
by Jacob Bronowski 

10 James Clerk !\Aaxwell 
by James R. Newman 

1 1 Frontiers of Physics Today: Acoustics 
by Leo L. Beranek 

1 2 Randomness and the Twentieth Century 
by Alfred M. Bork 

13 IVaves 

by Richard Stevenson and R. B. Moore 

14 What is a Wave? 

by Albert Einstein and Leopold Infeld 

1 5 l\/lusical Instruments and Scales 
by Harvey E. White 

1 6 Founding a Family of Fiddles 
by Carteen M. Hutchins 

1 7 The Seven Images of Science 
by Gerald Holton 

18 Scientific Cranks 
by Martin Gardner 

1 9 Physics and the Vertical Jump 
by Elmer L. Offenbacher 

Tr;: 

T19 One-Dimensional Collisions 

T20 Equal Mass Two-Dimensional Collisions 

T21 Unequal Mass Two-Dimensional 

Collisions 

T22 Inelastic Two-Dimensional Collisions 

T23 Slow Collisions 

T24 The Watt Engine 

T25 Superposition 

T26 Square Wave Analysis 

T27 Standing Waves 

T28 Two-Slit Interference 

T29 Interference Pattern Analysis 



Contents 



TEXT, UNIT 3 



Prologue 1 

Chapter 9 Conservation of Mass and Momentum 

Conservation of mass 5 

Collisions 9 

Conservation of momentum 1 

Momentum and Newton's laws of motion 1 5 

Isolated systems 18 

Elastic collisions 19 

Leibniz and the conservation law 21 

Chapter 10 Energy 

Work and kinetic energy 29 

Potential energy 31 

Conservation of mechanical energy 34 

Forces that do no work 37 

Heat energy and the steam engine 39 

James Watt and the Industrial Revolution 43 

The experiments of Joule 49 

Energy in biological systems 51 

Arriving at a general law 56 

A precise and general statement of energy conservation 60 

Faith in the conservation of energy 62 

Chaoter 11 The Kinetic Theory of Gases 

An overview of the chapter 69 

A model for the gaseous state 71 

The speeds of molecules 74 

The sizes of molecules 76 

Predicting the behavior of gases from the kinetic theory 79 

The second law of thermodynamics and the dissipation of energy 85 

Maxwell's demon and the statistical view of the second law of thermodynamics 88 

Time's arrow and the recurrence paradox 91 

Chapter 12 Waves 

Introduction 101 

Properties of waves 1 02 

Wave propagation 1 05 

Periodic waves 106 

When waves meet: the superposition principle 1 09 

A two-source interference pattern 110 

Standing waves 115 

Wave fronts and diffraction 1 20 

Reflection 1 22 

Refraction 1 26 

Sound waves 1 28 

Epilogue 134 

Acknowledgments iv 

Picture Credits Iv 

Answers to End-of-Section Questions A25 

Brief Answers to Study Guide Questions A36 

Index A44 



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UNIT 



3 



The Triumph of Mechanics 



CHAPTERS 

9 Conservation of Mass and Momentum 

10 Energy 

11 The Kinetic Theory of Gases 

12 Waves 



PROLOGUE The success of Isaac Newton in uniting the studies of 
astronomy and of terrestrial motion is one of the glories of the human mind. It 
was a turning point in the development of science and humanity. Never before 
had a scientific theory been so successful in finding simple order in observable 
events. Never before had the possibilities for using one's rational faculties for 
solving any kind of problem seemed so promising. So it is not surprising that 
after his death in 1 727 Newton was looked upon almost as a god, especially in 
England. Many poems like this one appeared: 

Newton the unparalled'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 all her secret Ways; 

'Gainst Mathematics she has no defence, 

And yields t' experimental Consequence; 

His tow'ring Genius, from its certain Cause 

Ev'ry Appearance a priori draws 

And shews th' Almighty Architect's unalter'd Laws. 

Newton's success in mechanics altered profoundly the way in which 
scientists viewed the universe. Physicists after Newton explained the motion of 
the planets around the sun by treating the solar system as a huge machine. Its 
"parts" were held together by gravitational forces rather than by nuts and bolts. 
But the motions of these parts relative to each other, according to Newton's 
theory, were determined once and for all after the system had first been put 
together. 

We call this model of the solar system the Newtonian world-machine. As is 
true for any model, certain things are left out. The mathematical equations 
which govern the motions of the model cover only the main properties of the 



(From J. T. Desagulier, The New- 
tonian System of the World, the 
Best Model of Government, an 
Allegorical Poem.) 



Beginning of Book 3, The Systems of the World,' 
in the 1713 edition of Newton's Principia. 



Unit 3 1 



2 Unit 3 



The Triumph of Mechanics 



i^ 0^ 




'*^^4'''*' 




real solar system. The masses, positions and velocities of the parts of the 
system, and the gravitational forces among them are well 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 observed motions. Moreover, it turned out that 
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 
h\s Principles of Philosophy (1644), Rene Descartes, the most influential 
French philosopher of the seventeenth century, had written: 

I do not recognize any difference between the machines that 
artisans 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 with 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. 




'The Ancient of Days" by William 
Blake, an English poet who had little 
sympathy with the Newtonian style 
of "natural philosophy." 



Robert Boyle (1627-1691), a British scientist, is known particularly for his 
studies of the properties of air. (See Chapter 1 1 .) Boyle, a pious man, 
expressed the "mechanistic" viewpoint 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 
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 perform all those many things, 
which he designed it should, by the meer contrivance of brute 
matter managed by certain laws of local motion, and upheld by his 
ordinary and general concourse, 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 best glorify the 
Creator by discovering and proclaiming these laws. 

Our main concern in this unit is with physics as it developed after Newton. 
In mechanics, Newton's theory was extended to cover a wide range of 



Prologue 



Unit 3 3 



phenomena, and new concepts were introduced. The conservation laws to be 
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 with 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 behavior 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 wrote: 



Ironically, Newton himself explicitly 
ejected the deterministic aspects of 
the "World-Machine" which his 
followers had popularized. 



I wish we could derive the rest of the phenomena of Nature by the 
same kind of reasoning from mechanical principles, for I 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 
from 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 Philosophy. 



Scientists after Newton strove to understand nature in many different 
areas, "by the same kind of reasoning from mechanical principles." We will see 
in this unit how wide was the success of Newtonian mechanics — but you will 
see also some evidence of limits to its applicability. 




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 
doesn't spoil our other reactions to 
the picture, but rather gives us an- 
other dimension of understanding it. 




9.1 Conservation of mass 

9.2 Collisions 

9.3 Conservation of nnomentum 

9.4 Momentum and Newton's laws of motion 

9.5 Isolated systems 

9.6 Elastic collisions 

9.7 Leibniz and the conservation law 



5 
9 

10 
15 
18 
19 
21 




CHAPTER NINE 



Conservation of 
Mass and Momentum 



9.1 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 twenty-four years before Newton's birth, the English philosopher 
Francis Bacon included the following among his basic principles of modem 
science in Novum Organum (1620): 

There is nothing more true in nature than the twin propositions 
that "nothing is produced from nothing" and "nothing is reduced 
to nothing" . . . the sum total of matter remains unchanged, 
without increase or diminution. 

This view 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 indestructible. For example, we may see a large 
boulder crushed 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 dissolved in acid? Does the amount of matter remain unchanged even in 
such chemical reactions? Or what of large-scale changes such as the forming 
of rain clouds or of seasonal variations? 




SG 9 1 




6 Unit 3 



Conservation of Mass and Momentum 




In some open-air chemical reactions, 
the mass of objects seems to de- 
crease, while in others it seems to 
increase. 



Note the closed flask shown in his 
portrait on p. 7. 



To test whether the total quantity of matter actually remains constant, we 
must know how to measure that quantity. Clearly it cannot simply be 
measured by its volume. For example, we might put water in a container, 
mark the water level, and then freeze the water. If so, we find that the volume 
of the ice is larger than the volume of the water we started with. This is true 
even if we carefully seal the container so that no water can possibly come in 
from the outside. Similarly, suppose we 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 our examples in Units 1 and 2, we 
assumed that the mass of a given object does not change. But a burnt match 
has a smaller mass than an unbumt one; and 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. But 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. He carefully 
weighed the flasks and their 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 Traite tlementaire de Chimie (1789): 



SG 9.2 




Conservation of mass was demon- 
strated in experiments on chemical 
reactions in closed flasks. 

The meaning of the phrase 'closed 
system" will be discussed in more 
detail in Sec. 9.5. 



We may lay it down as an incontestable axiom that in all the 
operations of art and nature, nothing is created; an equal quantity 
of matter exists both before and after the experiment, . . . and 
nothing takes place beyond changes and modifications in the 
combinations of these elements. Upon this principle, the whole art 
of performing chemical experiments depends. 



Lavoisier knew that if he put some material in a well-sealed botde 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. But at least one thing would remain 
unchanged — the total mass of all the different materials in the botde. 

In the years after Lavoisier's pioneering work, a vast number of similar 
experiments were performed with ever increasing accuracy. The result was 
always the same. As far as we now can measure with sensitive balances 
(having a precision of better than 0.000001%), mass isconserved — that is, it 
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 what we will call the /aw of conservation of mass. This law is 
basic to both physics and chemistry. 



T R A I T E 

£l£mentaire 
D E C HIMIE, 

PRtSENTt DANS UN ORDRE NOUVEAU 

ET d'apres les d^couvertes mooernes; 

Avec Figures : 

Tar M. Lavo 1 SI E R , it C Acaditnie iti 
Scienets, de la Societi RoyaU de Medcciae , dtl 
SocUus d' Agncullure de Parts O SOrUans , dt 
la Socitic Romaic de Lonjrei , de I'lnjliiut de 
Botogne t de la Societe Helveii^ue de BaJU , da 
celtes de PhUadelphie , Harlem , Mancheller , 
Padoue , Oc. 



TOME PREMIER. 




A PARIS. 

dm CucHET, Lihraire , rue & hotel Serpeme. 

M. DCC. LXXXIX. 

t*iu U PiivlUgi ie tAeadimii dii Scietuti dt U 
Soeleti RoyaU de Me'Jeeiite. 




Antoine Laurent Lavoisier (1743-1794) 
is known as the "father of modern 
chemistry" because he showed the 
decisive importance of quantitative 
measurements, confirmed the prin- 
ciple of conservation of mass in chem- 
ical reactions, and helped develop the 
present system of nomenclature for 
the chemical elements. He also 
showed that organic processes such 
as digestion and respiration are 
similar to burning. 

To earn money for his scientific re- 
search, Lavoisier invested in a private 
company which collected taxes for 
the French government. Because the 
tax collectors were allowed 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 company, and his association 
with the company was one of the rea- 
sons why Lavoisier was guillotined 
during the French Revolution. 

Also shown in the elegant portrait by 
David is Madame Lavoisier. She had 
been only fourteen at the time of her 
marriage. Intelligent as well as beauti- 
ful, she assisted her husband by 
taking data, translating scientific 
works from English into French, and 
making illustrations. About ten years 
after her husband's execution, she 
married another scientist. Count 
Rumford, who is remembered for his 
experiments which cast doubt on the 
caloric theory of heat. 



8 Unit 3 



Conservation of Mass and Momentum 



"The change in the total mass is 
zero" can be expressed symbolically 
as M, m, = where i! represents 
the sum of the masses of m, in all 
parts of the system. 



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 conserved. 
Rather, the earth, including its atmosphere gains and loses matter constantly. 
The most important addition occurs in the form of dust particles. These 
particles are detected by their impacts on satellites that are outside most of the 
atmosphere. Also, they create light and ionization when they pass through the 
atmosphere and are slowed down by it. The number of such particles is larger 
for those particles which are of smaller size. The great majority are ver\' thin 
particles on the order of 10 ""* cm diameter. Such small particles cannot be 
indixIduaUy detected from the ground when they enter the atmosphere. They 
are far too small to appear as meteorites, which result when particles at least 
several millimeters in diameter vaporize. The total estimated inflow of mass of 
all these particles, large and small, is about lO^g/sec over the whole surface of 
the earth. (Note: the mass of the earth is about 6 x lO^^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 this amount is comparatively small. 

The earth does lose mass by evaporation of molecules from the top of the 
atmosphere. The 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 temperature. At present the rate is 
probably less than 5 x lO^g/sec over the whole earth. This loss is ven' small 
compared with 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.) 



Try these end-of-section questions 
before qoing on. 



SG 9.3-9.7 



Q1 True or false: Mass is consented in a closed system only if there 
is no chemical reaction in the system. 

Q2 If 50 cm^ of alcohol is mixed with 50 cm^ 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 with the law of 
conservation of mass? 

Q3 Which one of the following statements is true? 

(a) Lavoisier was the first person to believe that the amount of 
material stuff in the universe did 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 experimentally. 



Section 9.2 



Unit 3 



9.2 Collisions 

Looking at moving things in the world around us easily leads to the 
conclusion that ever^'thing set in motion eventually stops. Every clock, 
ever)' machine eventually runs down. It appears that the amount of 
motion in the universe must be decreasing. The universe, bke 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 with their idea of 
the perfection of God, who surely would not construct such an imperfect 
mechanism. Some definition of "motion" was needed which would permit 
one to make the statement that "the quantity of motion in the universe is 
constant." 

Is there such a constant factor in motion that keeps the world 
machine going? To answer these questions most directly, we can do some 
simple laboratory experiments. We will use a pair of identical carts with 
nearly frictionless wheels, or better, two dry-ice discs or two air-track 
gliders. In the first experiment, a lump of putty is attached so that the 
carts will 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 see when you do the experiment, both carts stop in the 
collision: their motion ceases. But is there anything related to their 
motions which does not change? 

Yes, there is. If we add the velocity 1^.4 of one cart to the velocity Vg of 
the other cart, we 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. 

We might wonder whether this finding holds for all collisions. In 
other words, is there a "law of conservation of velocity"? The example 
above was a very special circumstance. Carts with equal masses approach 
each other with equal speeds. Suppose we make the mass of one of the 
carts twice the mass of the other cart. (We 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 move together in the direction of the initial velocity of the more 
massive object. Our guess that the vector sum of the velocities might be 
consened in all collisions is wrong. 

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. 

We have been trying to show that the "quantity of motion" is always 
the same before and after the collision. But our results indicate that the 
proper definition of "quantity of motion" may involve the mass of a body 



Note that in Units 1 and 2 we dealt 
mostly with phenomena in which 
this fact did not have to be faced. 




in syiDDOis, A-_ V Jai, v. ^ u 
in this particular case. 



5efcrt: v» + w 



•X^ V5=^' 




-- "a -, 



V'b 



Before: \7^ + Vg,=o 




A.t-cr: \t'%^o 








Afkr: \/t: + ^L-o 



'A + "& 



10 Unit 3 



Conservation of Mass and Momentunn 



In general symbols, 

Al my 0. 



as well as its speed. Descartes had suggested that the proper measure of a 
body's quantity of motion was the product of its mass and its speed. Speed 
does not involve direction and is considered always to have a positive 
value. The examples above, however, show that this product (a scalar and 
always positive) is not a conserved quantity. In the first and third 
collisions, for example, the products of mass and speed are zero for the 
stopped carts after the collision. But they obviously are not equal to zero 
before the collision. 

But if we make one very important change in Descartes' definition, 
we do obtain a conserved quantity. Instead of defining "quantity of 
motion" as the product of mass and speed, 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 well as the speed. On the 
next page the quantities mv are analyzed for the three collisions we have 
considered. In all three head-on collisions, the motion of both carts before 
and after collision is described by the equation: 



m^Vx +m^VB=mf,VjJ+msVB' 



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 9.8, 9.9 



before 
collision 



after 
collision 



Where m^ and m^ represents the masses of the carts, Vi and ig represent 
their velocities before the collision and v/ and Vb' represent their 
velocities after the collision. 

In 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 directiy to a powerful law. 



Q4 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? 

Q5 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 


(a) 


2 kg 




3 m/sec 


2 kg 


3 m/sec 


(b) 


2 




2 


3 


3 


(c) 


2 




3 


3 


2 


(d) 


2 




3 


1 


6 



9.3 Conservation of momentum 



Since the momentum of a system is 
the vector sum of the momentum 
of its parts, it is sometimes called 
the "total momentum" of the system. 
We will assume that "total" is 
understood. 



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. 



Analyses of Three Collisions 



^aCO) + mf,{0) = 



In Section 9.2 we discuss three examples of 
collisions between two carts. In each case the 
carts approached each other head-on, collided, 
and stuck together. We will show here that in 
each collision the motion of the carts before and 
after the collision is described by the general 
equation 

"7aV^a + mgi^B = rnj^V;,' + m^v^' 

where m^ and hIq represent the masses of the 
carts, i^A and v^ their velocities before collision, 
and v/a' and v^' their velocities after the collision. 

Example 1: Two carts with equal masses 
move with equal speeds — but in opposite 
directions — before the collision. The speed of the 
stuck-together carts after the collision is zero. 
Before collision, the product of mass and velocity 
has the same magnitude for each cart, but 
opposite direction. So their vector sum is 
obviously zero. After collision, each velocity is 
zero, so the product of mass and velocity is also 
zero. 

This simple case could be described in a 
few sentences. More complicated cases are 
much easier to handle by using an equation and 
substituting values in the equation. To show how 
this works, we will go back to the simple case 
above, even though it will seem like a lot of 
trouble for such an obvious result. We substitute 
specific values into the general equation given in 
the first paragraph above for two colliding bodies. 
In this specific case m^ = mg, v^ = -v^, and 
Va' = *^' = 0. Just before collision, the vector 
sum of the separate momenta is given by 
m^vl + m^Q, which in this case is equal to 
mA\r^ + rriAi-VA) or m^vl - m^v^ 
which equals zero. 

After the collision, the vector sum of the 
momenta is given by ivaVa' + msVB- Since 
both velocities after collision is zero, then 




Thus, before the collision, the vector sum of 
the products of mass and velocity is zero, and 
the same is true for the vector sum after the 
collision. The general equation is therefore 
"obeyed" in this case. 

Example 2: The carts move with equal 
speeds toward each other before the collision. 
The mass of one cart is twice that of the other. 
After the collision, the velocity of the stuck- 
together carts is found to be h the original 
velocity of the more massive cart. In symbols: 



^A = 2m B, v^B = -v.^ 
Before the collision: 



and Va' = V, 



Wa- 



m.v 



A" A 



= m^iVs = {2ms)VA + m^{-VA) 

= 2m £ A - m^A 

= m^VA 
After the collision: 

mAV^A + m^v^ = {2m^y^VA + m^v^A 
= %rn^A + \rn^^\ 
= m^VA 

Again, the sum of mv's is the same before and 
after the collision. Therefore, the general 
equation describes the collision correctly. 

Example 3: Two carts approach each other; 
the mass of one cart is twice that of the other. 
Before the collision, the speed of the less 
massive cart is twice that of the more massive 
cart. The speed of the stuck-together carts after 
the collision is found to be zero. In symbols: hIa 
= 2mB, Vs = -2irA and Va' = v^' = 0. 
Before the collision: 

mAVA + msVe = (2mB)VA + mB(-2 Va) 
= 2m ^v A - 2m ^v A 
= 

After the collision: 

Ar7A(0) + m^{0) = 

Again, the principle holds. Indeed, it holds for all 
collisions of this kind on which no external 
pushes or pulls are exerted, regardless of their 
masses and their initial velocities. 

In these examples all motion has been along 
a straight line. However, the principle is most 
useful for collisions that are not directly head-on 
and where the bodies go off at different angles. 
An example of such a collision is on page 23. 



12 Unit 3 



Conservation of Mass and Momentum 



SG 9.10,9.11 



F+able 




other cart 



Forces on cart 6 during collision. 



In general, for n objects the law can 
be written 



2(mA 



X ('"•''^)a 




Consider each of the collisions that we examined. The momentum of the 
system as a whole — the vector sum of the individual parts — is the same 
before and after collision. Moreover, the total momentum doesn't change 
during the collision, as the results of a typical experiment on page 10 
show. Thus, we can summarize the results of the experiments briefly: the 
momentum of the system is conserved. 

We arrived at this rule (or law, or principle) by observing the special 
case of collisions between two carts that stuck together after colliding. But 
in fact this laiv of conservation of momentum is a completely general, 
universal law. The momentum of any system is conserved if one condition 
is met: that no net force is acting on the system. 

To see just what this condition means, let us examine the forces 
acting on one of the carts. Each cart experiences three main forces. There 
is of course a downward puU F grav exerted by the earth and an upward 
push f^tabie cxcrted by the table. During the collision, there is also a push 
f^^from other cart excrtcd by the Other cart. The first two forces evidently 
cancel, since the cart is not accelerating up or down. Thus the net force 
on each cart is just the force exerted on it by the other cart as they 
collide. (To simplify, we assume that frictional forces exerted by the table 
and the air are small enough to neglect. That was the reason for using 
dry-ice disks, air-track gliders, or carts with "frictionless" wheels. This 
assumption makes it easier to discuss the law of conservation of 
momentum. But we wHl see that the law holds whether friction exists or 
not.) 

The two carts form a system of bodies, each cart being a part of the 
system. The force exerted by one cart on the other cart is a force exerted 
by one part of the system on another part. But it is not a force on the 
system as a whole. The outside forces acting on the carts (by the earth 
and by the table) exactly cancel. Thus, there is no net outside force. We 
can say that the system is "isolated." This condition must be met in order 
for the momentum of a system to stay constant, to be conserxed. 

If the net force on a system of bodies is zero, the momentum of the 
system will not change. This is the law of conservation of momentum for 
systems of bodies that are moving with linear velocity v. 

So far we have considered only cases in which two bodies collide 
direcdy and stick together. But the remarkable thing about the law of 
conservation of momentum is how universally it applies. For example: 

(a) It holds true no matter what kind of forces the bodies exert on 
each other. They may be gravitational forces, electric or magnetic forces, 
tension in strings, compression in springs, attraction or repulsion. The 
sum of the mi''s before is equal to the sum of mv's after any interaction. 

(b) It doesn't matter whether the bodies stick together or scrape 
against each other or bounce apart. They don't even have to touch. When 
two strong magnets repel or when an alpha paiticle is repelled by a 
nucleus, conservation of momentum still holds. 

(c) The law is not restricted to systems of only two objects; there can 
be any number of objects in the system. In those cases, the basic 
conservation equation is made more general simply by adding a term for 
each object to both sides of the equation. 



= 9 



Example of the Use of the Conservation of 
Momentum 

Here is an example that illustrates how one 
can use the law of conservation of momentum. 

(a) 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 
towards it with a speed of 1 ,000 m/sec. How fast 
does the capsule (with the meteorite stuck in it) 
move after being hit? 

aHa mass of the meteorite = 0.1 Kg 

/77b mass of the capsule = 1,000 Kg 

Va initial velocity of meteorite = 1 ,000 m/sec 

Vn initial velocity of capsule = 

v/ final velocity of meteorite 

Vq final velocity of capsule 

The law of conservation of momentum states that 

rriAVA + m^Ve = m^VA' + m^v^' 

Inserting the values given, 

(0.1 kg) (1000 m/sec) + (1000 kg) (0) = 

(0.1 kg)7,' + (1000 kg)7B 
100 kg m/sec = (0.1 kg)v^' + (1000 kg)\^ ' 

Since the meteorite sticks to the capsule, v^ = 
Va' so we can write 

100 kg m/sec = (0.1 kg)vA' + (1000 kg)^^' 
100 kg m/sec = (1000.1 kg)7^' 

Therefore 



Va = 



1 00 kg • m/sec 



1000.1 kg 
= 0.1 m/sec 
(in the original direction of the motion of the 
meteorite). 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 in the values 
only as a final step. Substituting v^' for v^' and 
letting 7b' = would leave the equation m,^^ = 
m.\ 7a' + /77b7a' = (^A + /77b)7^'. Solving for Va' 

71 , nn^Vf, 



[m^ +/77b) 

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. 

(b) An identical capsule at rest nearby is hit 
by a meteorite of the same mass as the other. 




CLAWfr.( 




But this meteorite, hitting another part of the 
capsule, does not penetrate. Instead it bounces 
straight back with almost no change of speed. 
(Some support for the reasonableness of this 
claim is given in SG 9.24.) How fast does the 
capsule move on 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 (a): 

rrif, =0.1 kg v^ =0 

(Vq = 1000 kg Vf,' = -1000 m/sec 

Va = 1000 m/sec Vb' = ? 

The law of conservation of momentum stated 
that m .^^ + m^Q = m^^^ ' + ^bTb'. Here 
(0.1 kg) (1000 m/sec) + (1000 kg) (0) = 
(0.1 kg) (-1000 m/sec) + (1000 kq)v^ 
100 kg m/sec = -100 kg m/sec + (1000 V.q)v^' 

200 kg ■ m/sec „ „ , 

^B = — TT^I^r; = 0-2 m/sec 

^ 1 000 kg 

Thus, the struck capsule moves on with about 
twice the speed of the capsule in (a). (A general 
symbolic approach can be taken to this solution, 
too. But 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 object is given less momentum if it 
absorbs the projectile than if it reflects it. (A 
goalie who catches the soccer ball is pushed 
back less than one who lets the ball bounce off 
his chest.) 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. 



14 Unit 3 



Conservation of Mass and Momentum 




SG 9.12-9.15 




One of the stroboscopic photographs 
that appears in the Handbook. 



(d) The size of the system is not important. The law apphes to a 
galaxy as well as to an atom. 

(e) The angle of the collision does not matter. All of our examples so 
far have involved collisions between two bodies moving along the same 
straight line. They were "one-dimensional collisions." But if two bodies 
make a glancing collision rather than a head-on collision, each will move 
off at an angle to the line of approach. The law of conservation of 
momentum applies to such two-dimensional collisions also. (Remember 
that momentum is a vector quantity.) The law of conservation of 
momentum also applies in three dimensions. The vector sum of the 
momenta is still the same before and after the collision. 

On page 13 is a worked-out example that will help you become 
familiar with the law of conservation of momentum. At the end of the 
chapter is a special page on the analysis of a two-dimensional collision. 
There are also stroboscopic photographs in the Project Physics Handbook 
and film loops of colliding bodies and exploding objects. These include 
collisions and explosions in two dimensions. The more of them you 
analyze, the more convinced you will be that the law of conservation of 
momentum applies to any isolated system. 

The worked-out example of page 13 displays a characteristic feature of 
physics. Again and again, physics problems are solved oy applying the 
expression of a general law to a specific situation. Both the beginning 
student and the veteran research physicist find it helpful, but also 
somewhat mysterious, that one can do this. It seems strange that a few 
general laws enable one to solve an almost infinite number of specific 
individual problems. Everyday life seems different. There you usually 
cannot calculate answers from general laws. Rather, you have to make 
quick decisions, some based on rational analysis, others based on 
"intuition." But the use of general laws to solve scientific problems 
becomes, with practice, quite natural also. 



Q6 Which of the following has the least momentum? Which has the 
greatest momentum? 

(a) a pitched baseball 

(b) a jet plane in flight 

(c) a jet plane taxiing toward the terminal 



Section 9.4 



Unit 



15 



07 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 

(b) 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. 



9.4 Momentum and Newton's laws of motion 



In Section 9.2 we developed the concept of momentum and the law of 
conservation of momentum by considering experiments with colliding 
carts. The law was an "empirical" law. That is, we arrived at it as a 
summary of experimental results, not from theory. The law was 
discovered — perhaps "invented" or "induced" are better terms — as a 
generalization from experiment. 

We can show, however, that the law of conservation of momentum 
follows directly from Newton's laws of motion. It takes only a bttle algebra. 
We will first put Newton's second law into a somewhat different form 
than we used before. 

Newton's second law expresses a relation between the net force Fnet 
acting on a body, the mass m of the body, and its acceleration a. We 
wrote this as Fnet = ^ci- But we can also write this law in terms of change 
of momentum of the body. Recalling that acceleration is the rate-of-change 
of velocity, a = Av/At, we can write: 



SG9.16 



FnPt — 



m 



or 



Av_ 
At 



FnetAt = mAv 



If the mass of the body is constant, the change in its momentum, 
A(mv), is the same as its mass times its change in velocity, m(Av). So 
then we can write: 

Fnet At = A(mv) 



If m is a constant, 

\(mv) mv mv 

m(v v) 

FAf is called the 'impulse." 
SG 9.17-9.20 



That is, the product of the net force on a body and the time interval 
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 derivation are given on page 16. Thus Newton's 
laws and the law of conservation of momentum are not separate, 
independent laws of nature. 



In Newton's second law. "change 
of motion' meant change of 
momentum-see Definition II at 
the beginning of the Principia. 



Deriving Conservation of IVIomentum from 
Newton's Laws 

Suppose two bodies with masses m,^ and 
a77b exert forces on each other (by gravitation or 
by mutual friction, etc.). ^ab is the force exerted 
on a body A by body B, and ^ba 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 ^B and ?ba are at every instant equal in 
magnitude and opposite in direction. Each body 
acts on the other for exactly the same time At. 
Newton's second law, applied to each of the 
bodies, says 

and p^^ At=A(m,Vs) 

By Newton's third law 



therefore 



P^sAt=-P„At 



Suppose that each of the masses iVj^ and 
^B are constant. Let Va and Vr stand for the 
velocities of the two bodies at some instant and 
let Va' and Vb' stand for their velocities at some 
later instant. Then we can write the last equation 
as 

mAVA'-rriAVA = - (m eVg' - m bVb) 
A little rearrangement of terms leads to 



and m,,v^' + m ^v u' ^m ,^v ^ + m^Va 

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. But this method works 
equally well for a system consisting of any 
number of bodies. For example, SG 9.21 shows 
you how to derive the law of conservation of 
momentum for a system of three bodies. 






Globular clusters of stars liKe this one 
contain tens of thousands of suns held 
together by gravitational attraction. 



Section 9.4 



Unit 3 17 



In all examples we considered each body to have constant mass. But 
a change of momentum can arise from a change of mass as well as from 
(or in addition to) a change of velocity. For example, as a rocket spews 
out exhaust gases, its mass decreases. The mass of a train of coal cars 
increases as it moves under a hopper which drops coal into the cars. In 
Unit 5 you wiD find that any body's mass increases as it moves faster and 
faster. (However, this effect is great enough to notice only at extremely 
high speeds.) The equation Fnet = ma is a form of Newton's second law 
that works in special cases where the mass is constant. But this form is 
not appropriate for situations where mass changes. Nor do the forms of 
the law of conservation of momentum that are based on Fnet = ^o. work 
in such cases. But other forms of the law can be derived for systems 
where mass is not constant. See. for example, the first pages of the article 
"Space Travel" in Reader 5. 

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 
conservation of momentum is often the preferred tool because it enables 
us to solve many problems which 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 we had a continuous record of the magnitude 
of the force. We 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) we could find the speed of the shell and 
the recoil speed of the cannon. But 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 laws to find the final speeds. 
However, we can use the law of conservation of momentum even if 
we know nothing about the force. The law of consei-vation of momentum 
is a law of the kind that says "before = after." Thus, it works in cases 
where we do not have enough information to apply Newton's laws during 
the whole inter\'al between "before" and "after." In the case of the cannon 
and shell the momentum of the system (cannon plus sheU) is zero 
initlallv. Therefore, by the law of consen-ation of momentum, the 
momentum will also be zero after the shell is fired. If we know the 
masses and the speed of one, after firing we can calculate the speed of 
the other. (The fibn loop tided "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 the 
Supplemental Unit entitled 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 9.21-9.24 




SG 9.25 



SG 9.26 
SG 9.27 



Q8 Since the law of conservation of momentum can be derived from 
Newton's laws, what good is it? 




;^ 



c_:'j^ 



%r^si 


•jOj- 1_^'' ,Xi 


fii«— — 


^^.^a* 


'^^?*' 




^ 


. --^/--..'SiSE 


"V-.V '-• ^HV 


?P 



18 Unit 3 

9.5 Isolated systems 



Conservation of Mass and Momentum 



SG 9.28-9.33 



There are important similarities between the conservation law of mass 
and that of momentum. We test both laws 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, we arrange an isolated system such 
as a sealed flask. Matter can neither enter or leave this system. When 
testing or using the law of conservation oi momentum, we arrange another 
kind of isolated system. Such a system is closed in the sense that each 
body in it experiences no net force from outside the system. 

Consider for example two dry-ice pucks colliding on a smooth 
horizontal table. The very low friction pucks form a very nearly closed or 
isolated system. We need not include in it the table and the earth, for 
their individual effects on each puck cancel. Each puck experiences a 
downward gravitational force exerted by the earth. But the table exerts an 
equally strong upward push. 

Even in this artificial example, the system is not entirely closed. 
There is always a little friction with the outside world. The layer of gas 
under the puck and air currents, for example, exert friction. AH outside 
forces are not completely balanced, and so the two pucks do not form a 
truly isolated system. Whenever 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 pounds. These forces are very small compared to the immense 
force (many tons) exerted by each car on the other when they collide. 
Thus, for many purposes, we can forget about the action of the road. For 
such purposes, the two skidding cars during the collision are nearly 
enough an isolated system. However, if friction with the road (or the table 
on which the pucks move) is too great to ignore, the law of conservation 
of momentum still holds, but we must apply it to a larger system, one 
which includes the road or table. In the case of the skidding cars or the 
pucks, the table or road is attached to the earth. So we would have to 
include the entire earth in a "closed system." 



Q9 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. 

Q10 Explain whether or not each of the following can be considered 
as an isolated system. 

(a) a baseball thrown horizontally 



Section 9.6 



Unit 



19 



(b) an artificial earth satellite 

(c) the earth and the moon 



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 form two pendula. One ball was 
released from rest at a certain height. It swung down and struck the 
other, which had been hanging at rest. 

After impact, the first ball stopped at the point of impact while the 
second ball swung from this point to the same height as that firom which 
the first ball had been released. When the second ball returned and struck 
the first, it now was the second ball which stopped at the point of impact 
as the first swung 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 die motion "transferred" from one ball to 
the other when they collided? Why didn't the first ball bounce back from 
the point of collision, or continue moving forward after the second ball 
moved away from the collision point? 

Our law of momentum conservation explains what is observed. But it 
would also allow quite different results. It 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 say how A and B share the momentum. 
The actual result is just one of infinitely many different outcomes that 
would all agree with conservation of momentum. For example, suppose 
(though it is never observed to happen) that ball A bounced back with ten 
times its initial speed. Momentum would stUl be conserved if ball B went 
ahead at eleven times A's initial speed. 

In 1668 three men reported to the Royal Society on the whole matter 
of impact. The three men were the mathematician John WaUls, 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 
also holds, in addition to the law of conservation of momentum. Not only 
was the vector sum of mi?s conserved, but so was the ordinary arithmetic 
sum of mi'2's! In modem algebraic form, the relationship he discovered 
can be expressed as 



hm^v/ + im^B^ = ^^.4^.4'^ + hmeVB^ 



■♦* 




\ 



ii 



^ 



• 



In general symbols. \1,\my^ 



Compare this equation with the 
conservation of momentum equation 
on page 10. 



20 Unit 3 



Conservation of Mass and Momentum 



SG 9.34-9.37 




Christiaan Huygens (1629-1695) was 
a Dutch physicist. He devised an im- 
proved telescope with which he dis- 
covered a satellite of Saturn and saw 
Saturn's rings clearly. He was the 
first to obtain the expression for 
centripetal acceleration (i/Vfl), he 
worked out a wave theory of light, and 
he invented a pendulum-controlled 
clock. His scientific contributions 
were major, and his reputation would 
undoubtedly have been greater were 
he not overshadowed by his con- 
temporary, Newton. 

Huygens, and others after him for 
about a century, did not use the 
factor ~. The quantity mv~ was called 
vis viva, Latin for "living force." 
Seventeenth- and eighteenth-century 
scientists were greatly interested 
in distinguishing and naming various 
'forces." They used the word 
loosely; it meant sometimes a push 
or a pull (as in the colloquial modern 
use of the word force), sometimes 
what we now call "momentum." and 
sometimes what we now call 
"energy." The term vis viva is no 
longer used. 



The scalar quantity hmv^ has come to be called kinetic energy. (The 
reason for the i which doesn't really affect the rule here, will 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 A's initial speed. A litde algebra will show that this is 
the only result that agrees with both conservation of momentum and 
conservation of kinetic energy. (See SG 9.33.) 

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 Section 
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 2m.{v/ + ^m^VB^. But both iTu^v/' and 
hmsVs^ are always positive numbers. Their sum cannot possibly equal zero 
(unless both v^ and Vb are zero, in which case there would be no 
collision — and not much of a problem). Kinetic energy is not conserved in 
this collision in which the bodies stick together. In fact, no collision in 
which the bodies stick together will show conservation of kinetic energy. 
It applies only to the collision of "perfectly hard" bodies that bounce back 
from 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, depending on the type of collision. 
It is conserved if the colliding bodies do not crumple or smash or dent or 
stick together or heat up or change physically in some other way. We call 
bodies that rebound without any such change "perfectly elastic." We 
describe collisions between them as "perfectly elastic collisions." In 
perfectly elastic collisions, both momentum and kinetic energy are 
conserved. 

Most collisions that we witness, are not perfectly elastic and kinetic 
energy is not conserved. Thus, the sum of the ^mv^'s after the collision is 
less than before the collision. Depending on how much kinetic energy is 
"lost," such collisions might be called "partially elastic," or "perfectly 
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 energy after the collision might be as much as, say, 96% of this 
value before the collision. Examples of true perfectly elastic collisions are 
found only in collisions between atoms or sub-atomic particles. 



SG 9.38-9.40 



Q11 Which phrases correctly complete the statement? Kinetic energy 
is conserved 

(a) in all collisions 



Section 9.7 



Unit 3 21 



(b) whenever momentum is consened 

(c) in some collisions 

(d) when the colliding objects are not too hard 
Q12 Kinetic energy is never negative because 

(a) scalar quantities are always positve 

(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 



9.7 Leibniz and the conservation law 

Rene Descartes believed that the total quantity of motion in the 
universe did not change. He wrote in his Principles of Philosophy: 

It is wholly rational to assume that God, since in the creation 
of matter He imparted different motions to its parts, and 
preserves all matter in the same way and conditions in which 
he created it, so He similarly preserves in it the same quantity 
of motion. 

Descartes proposed to define the quantity of motion of an object as 
the product of its mass and its speed. But as we saw in Section 1.1 this 
product is a conserved quantity only in very special cases. 

Gottfried Wilhelm Leibniz was aware of the error in Descartes' ideas 
on motion. In a letter in 1680 he wrote: 




Descartes (1596-1650) was the most 
important French scientist of the 
seventeenth century. In addition to 
his early contribution to the idea of 
momentum conservation, he is re- 
membered by scientists as the in- 
ventor of coordinate systems and the 
graphical representation of algebraic 
equations. His system of philosophy, 
which used the deductive structure 
of geometry as its model, is still in- 
fluential. 



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 same 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 he 
identified as "force" the quantity mv^ (which he called vis viva). We 
notice that this is just twice the quantity we now call kinetic energy. (Of 
course, whatever applies to mv^ applies equally to ^mv^.) 

As Huygens had pointed out, the quantity {k)'mv^ is conserved only 
in perfectly elastic collisions. In other collisions the total quantity of 
(i)wi;2 afjgj. collision is always less than before the collision. Still Leibniz 
was convinced that (^jmi^^ is always conserved. In order to save his 
conservation law, he 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. We will study 
some of these ideas in Chapter 11. 



22 I 'nit 3 



Conservation of Mass and Momentum 




Leibniz (1646-1716), a contemporary 
of Newton, was a German philosopher 
and diplomat, an advisor to Louis XIV 
of France and Peter the Great of Rus- 
sia. Independently of Newton he in- 
vented the method of mathematical 
analysis called calculus. A long public 
dispute resulted between the two 
great men concerning charges of 
plagiarism of ideas. 



Leibnitz extended conservation ideas to phenomena other than 
collisions. For example, when a stone is thrown straight upward, its 
quantity of (^)mv^ decreases as it rises, even without any collision. At 
the top of the trajectory, (^)mv^ 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 {^)mv^ is just one part of a more 
general, and really conserved quantity. In Chapter 10, this idea will lead 
us directly to the most powerful of all laws of science — the law of 
conservation of energy. 

013 According to Leibniz, Descartes' principle of conservation ofmv 
was 

(a) correct, but trivial. 

(b) another way of expressing the conservation of vis viva. 

(c) incorrect. 

(d) correct only in elastic collisions. 

Q14 How did Leibniz explain the apparent disappearance of the 
quantity (V)mv^ 

(a) during the upward motion of a thrown object? 

(b) when the object strikes the ground? 



A Collision in Two Dimensions 

The stroboscopic photograph shows a 
collision between two wooden discs on a 
"frictionless horizontal table" photographed from 
straight above the table. The discs are riding on 
tiny plastic spheres which make their motion 
nearly frictionless. 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^ = 2mA. 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. So 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: 7^ == 36.7 speed-units. Similarly 
we find that vT^' = 17.2 speed-units, and Vq' = 
11.0 speed-units. 

The total momentum before the collision is 
just m^v^x'. It Is represented by an arrow 36.7 
momentum-units long, drawn at right. 

The vector diagram shows the momenta 
^A^^^' and meVs' after the collision; /DaI^^' is 
represented by an arrow 17.2 momentum-units 
long. Since me = 2mA, the hIbV^' arrow is 22.0 
momentum-units long. 

The dotted line represents the vector sum of 
mA^^ ' and m^Vji'; that is, the total momentum 
after the collision. Measurement shows it to be 
34.0 momentum-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 inaccuracies? 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? 




m^V^'' ZZ.0 



'f^* '^B^b 




m,^- S6.7 



rnX- in 



*'■')'■■■<< — ■'»>■■■• I ■ ■ 
fi S- 16 V So 



f,UimUMHMVH 



9.1 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 
Two-dimensional Collisions I 
Two-dimensional Collisions II 
Inelastic Two-dimensional Collisions 
Scattering of a Cluster of Objects 
Explosion of a Cluster of Objects 

Transparencies 

One-dimensional Collisions 
Equal Mass Two-dimensional Collisions 
Unequal Mass Two-dimensional Collisions 
Inelastic Two-dimensional Collisions 

In addition, the Reader 3 articles "The Seven 
Images of Science" and "Scientific Cranks" are 
of general interest in the course. 

9.2 Certainly Lavoisier did not investigate every 
possible interaction. What justification did he 
have for claiming mass was conserved "in all the 
operations of art and nature"? 

9.3 It is estimated that every year at least 2000 
tons of meteoric dust fall on to the earth. The 
dust is mostly debris that was moving in orbits 
around the sun. 

(a) Is the earth (whose mass is about 6 x 10^' 
tons) reasonably considered to be a closed 
system with respect to the law of con- 
servation of mass? 

(b) How large would the system, including the 
earth, have to be in order to be completely 
closed? 

9.4 Would you expect that in your lifetime, when 
more accurate balances are built, you will see 
experiments which show that the law of con- 
servation of mass does not entirely hold for 
chemical reactions in closed systems? 

9.5 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 identical objects placed one on top 
of the other. (The reason is that the pull of 
gravity decreases with distance from the center 
of the earth.) Does this experiment contradict the 
law of conservation of mass? 

9.6 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 
almost from nothingness. Devise and describe an 
experiment by which you would test the law of 
conservation of mass for this demonstration. 

9.7 Consider the following chemical reaction, 
which was studied by Landolt in his tests of the 
law of conservation of mass. In a closed container. 



a solution of 19.4 g of potassium chromate in 
100.0 g of water is mixed with 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 liquid? 
(Assume the combined mass of all sub- 
stances in the system is conserved.) 

(b) If the remaining liquid (after removal of 
the yellow precipitate) is then heated to 
95°C, the water it contains will evaporate, 
leaving a white solid. What is the mass of 
this solid? (Assume that the water does not 
react with anything, either in (a) or in (b).) 

9.8 If a stationary cart is struck head-on by a 
cart with twice the mass, and the two carts stick 
together, they will move together with a speed 

■f as great as the moving cart had before collision. 
Show that this is consistent with the conservation 
of momentum equation. 

9.9 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: 

mAi'A + mnVB = rrij^Vj^' + m^Vs' 

(a) What quantities does the problem give for 
the equation? 

(b) Rearrange terms to get an expression for v^'. 

(c) Find the value of ly/. (Note v^' = v„'.) 

9.10 You have been given a precise technical 
definition for the word momentum. Look it up in 
a large dictionary and record i-ts 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 
here given? 

9.11 Benjamin Franklin, in correspondence with 
his friend James Bowdoin (founder and first 
president of the American Academy of Arts and 
Sciences), objected to the corpuscular theory of 
light by saying that a particle traveling with such 
immense speed (3 x lO** m/sec) would have the 
impact of a lO-kg ball fired from a cannon at 
100 m/sec. What mass did Franklin assign to the 
"light particle"? 

9.12 If powerful magnets are placed on top of 
each of two carts, and the magnets are so ar- 
ranged 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 the law of conservation of momentum 
apply? 

(c) Describe an arrangement for testing your 
answer to (b). 



24 Unit 3 



9.13 From the equation 

rUiV^ + rngVp — m,ti, ' + m^v,,' 

show that the change in momentum of object A is 
equal and opposite to the change of momentum of 
object B. Using the symbol Ap for change of 
momentum, rewrite the law of conservation of 
momentum for two bodies. What might it be for 3 
bodies? for n bodies? 

9.14 A person fires 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? 

9.15 If everyone in the world were to stand 
together in one field and jump up with an initial 
speed of 1 m/sec, 

(a) For how long would they be off the ground? 

(b) How high would they go? 

(c) What would be the earth's speed downward? 

(d) How far would it move? 

(e) How big would the field have to be? 

9.16 Did Newton arrive at the law of conserva- 
tion 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). 

9.17 If mass remains constant, then A(mv) = 
m(Av). Verify this relation by substituting some 
numerical values, for example for the case where 
m is 3 units and v changes from 4 units to 6 
units. 

9.18 (a) Why can ocean liners or planes not turn 

comers sharply? 
(b) In the light of your knowledge of the 
relationship between momentum and 
force, comment on reports about un- 
identified flying objects (UFO) turning 
sharp comers in full flight. 

9.19 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 seconds. What force does the snow 
exert on the girl? How far does she penetrate the 
drift? What happened to her momentum? 

9.20 During sports, the forces exerted on parts of 
the body and on the ball, etc., can be astonishingly 
large. To illustrate this, consider the forces in 
hitting a golf ball. Assume the ball's mass is .046 
kg. From the strobe photo on p. 27 of Unit 1, in 
which the time interval 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 ball during 
impact. 

9.21 The Text derives the law of conservation of 
momentum for two bodies from Newton's third 
and second laws. Is the principle of the conserva- 
tion of mass essential to this derivation? If so. 
where does it enter? 




9.22 Consider an isolated system of three bodies, 
A, B, and C. The forces acting among the bodies 
can be indicated by subscript: for example, the 
force exerted on body A by body B can be given 
the symbol P^g. By Newton's third law of motion, 
F*B^ = —F'^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^ab '^^^ac- 
Using these principles, show that the total 
momentum change of the system will be zero. 

9.23 In Chapter 4, SG 4.24 was about putting an 
Apollo capsule into an orbit around the moon. 

The question was: "Given the speed v„ neces- 
sary for orbit and the present speed i^, how long 
should the rocket engine with thrust F fire to 
give the capsule of mass m the right speed?" 
There you solved the problem by considering the 
acceleration. 

(a) Answer the question more directly by 
considering change in momentum. 

(b) What would be the total momentum of all 
the exhaust from the rocket? 

(c) If the "exhaust velocity" were v^, about 
what mass of fuel would be required? 

9.24 (a) Show that when two bodies collide their 

changes in velocity are inversely propor- 
tional to their masses. That is, if m^ and 
ttzb are the masses and Av^ and Av^ the 
velocity changes, show that numerically, 



AVu 



rriE 



(b)Show how it follows from conservation 
of momentum that if a light particle 
(like a B.B. pellet) bounces off a massive 
object (like a bowling ball), the velocity 
of the light particle is changed much 
more than the velocity of the massive 
object. 

(c) For a head-on elastic collision between 
a body of mass rUf, moving with velocity 
v_^ and a body of mass ma at rest, com- 
bining the equations for conservation of 
momentum and conservation of kinetic 
energy leads to the relationship v,^' = 
^Ai^A " wib) . (w^ + mg). Show that if 
body B has a much greater mass than 
body A, then v^' is almost exactly the 
same as z^^ — that is, body A bounces 
back with virtually no loss in speed. 



Utut 



25 



9.25 The equation rrif^Vp, + m^Va = rri/^Vj^' + m^v^' 
is a general equation applicable to countless 
separate situations. For example, consider a 10-kg 
shell fired from a 1000-kg cannon. If the shell is 
given a speed of 1000 m/sec, what would be the 
recoil speed of the cannon? (Assume the cannon 
is on an almost frictionless mount.) Hint: your 
answer could include the following steps: 

(a) If A refers^to the cannon and B to the shell, 
what are v^ and v^ (before firing)? 

(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. 

9.26 The engines of the first stage of the Apollo/ 
Saturn rocket develop an average thrust of 35 
million newtons for 150 seconds. (The entire 
rocket weighs 28 million newtons 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 6100 mUes/ 
hour. What would one have to know to 
compute its mass? 

9.27 Newton's second law can be written F'At = 
A(miO. 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 jumping 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 deformed under impact, slowly 
return to their original shape. Others are 
designed to have a somewhat pointed front- 
end bumper. 




a~~o 



9.28 A student in a physics class, having learned 
about elastic collisions and conservation laws, 
decides that he can make a self-propelled car. He 
proposes to fix a pendulum on a cart, using a 
"super-ball" 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 elastically. It is supposed to give the 
cart a series of bumps that propel it along. 

(a) Will his scheme work? (Assume the "super- 
ball" 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? 



^^ 



□ — 



\ t 
I 



9.29 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 the 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 automo- 
biles? 

(c) What simplifying assumptions have you 
made in answering (b)? 

9 '''^ Two pucks on a frictionless horizontal 
surface are joined by a spring. 

(a) Can they be considered an isolated system? 

(b) How do gravitational forces exerted by the 
earth affect your answer? 

(c) What about forces exerted by the pucks on 
the earth? 

(d) How big would the system have to be 
in order to be considered completely 
isolated? 

9 31 A hunter fires a gun horizontally at a target 
fixed to a hillside. Describe the changes of 
momentum to the hunter, 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? 

9.32 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? 

(Pool sharks will recognize that it depends upon 
the spin or "English" that the ball has. but to 
make the problem simpler, neglect this condition. 

(c) What is the change in momentum of the bal' 

(d) Is momentum conserved? 



26 



ml 6 



9.33 Discuss conservation of momentum for the 
system shown in this sketch from Le Petit 
Prince. What happens 

(a) if he leaps in the air? 

(b) if he runs around? 



rewrite the equations with m for m^ and rrif,' and 
Vb = 0; solve the simplified momentum equation 
for v/; substitute in the simplified kinetic energy 
equation; solve for v„'.) 
9.35 Fill in the blanks for the following motions: 



Object 



m V mv jmv'^ 

(kg) m/sec kgm/sec kgm^/sec^ 




.VJ^ 



Le petit prince sur I'asteroide B 612. 

9.34 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. 20 that this result is 
the only possible result that will be consistent 
with conservation of both momentum and kinetic 
energy. (That is, if m^ = mg and Vg = 0. then the 
result must be v,^' = 6 and Vg = ^^.^.) Combine the 
equations that express the two conservation laws 
and show that this is actually the case. (Hint: 



baseball 


0.14 


30.0 





_ 


hockey puck 


— 


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 











9 36 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. 

9.37 If both momentum and kinetic energy are 
conserved, say that a ball of mass m moving at 
speed V strikes, elastically, head-on, a second ball 
of mass 3m which is at rest. Using the principle 
of conservation of momentum and kinetic energy, 
find the speeds of the two balls after collision. 
9 38 Devise a way of giving a numerical estimate 
just how far from "perfectly elastic" a collision 
is — for example, the collision between a ball and 
the ground from which it bounces. 

9 39 Apply the law of conservation of momentum 
to discuss qualitatively a man swimming; a ship 
changing course; a man walking; a rocket taking 
off; a rifle being fired; a propeller plane in straight 
line motion, and while circling; a jet plane 
ascending; an apple dropping to earth; a comet 
being captured by the sun; a spaceship leaving 
earth; an atomic nucleus emitting a small 
particle. 

9.40 Describe the changes of kinetic energy involved 
in pole vaulting from the start of the vaulter's run to 
his landing. 



Unit 3 27 



10.1 Work and kinetic energy 

10.2 Potential energy 

10.3 Conservation of mechanical energy 

10.4 Forces that do no work 

10.5 Heat energy and the steam engine 

10.6 James Watt and the Industrial Revolution 

10.7 The experiments of Joule 

10.8 Energy in biological systems 

10.9 Arriving at a general law 

10.10 A precise and general statement of energy conservation 

10.11 Faith in the conservation of energy 



29 
31 
34 
37 
39 
43 
49 
51 
56 
60 
62 




CHAPTER TEN 



Energy 



10.1 Work and kinetic energy 



In everyday language we say that pitching, catching, and running on 
the baseball field is "playing," while sitting at a desk, reading, writing, and 
thinking is "working." But, in the language of physics, studying involves 
very little work, while playing baseball involves a great deal of work. The 
term "doing work" means something very definite in physics. It means 
"exerting a force on an object while the object moves in the direction of 
the force." When you throw a baseball, you exert a large force on it while 
it moves forward for about one meter. In doing so, you do a large amount 
of work. By contrast, in writing or in turning the pages of 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 floor 
straight upward to a conveyor belt at waist height. Here the language of 
common usage and the 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. And if the conveyor belt were twice as high above the floor, 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. 

With this example in mind, we can define work in a way that allows 
us to give a numerical value to the concept. The work W done on an 
object by a force P is defined as the product of the magnitude F of the 
force and the distance d that the object moves in the direction of F while 
the force is being exerted: 

W = Fd 

To lift a box weighing 100 newtons upward through 0.8 meters 
requires you to apply an upward force of 100 newtons. The work you do 
on the box is 100 newtons x 0.8 meters = 80 newton-meters. 




SG 10.1 



Note that work you do on a box 
does not depend on how fast you 
do your job. 



The way d is defined here, the W - F-i 
is correct. It does not, however, 
explicity 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 displacement along 
the direction of F: and this is entirelv 
correct 



Unit 3 29 



30 Unit 3 



Energy 



Note that work is a scalar quantity. 
A more general definition of work 
will be given in Sec. 10.4. 

The equation W - Fd implies that 
work is always a positive quantity 
However, by convention, when the 
force on a body and its 
displacement are in opposite 
directions, the work is negative. 
This implies that the body's energy 
would be decreased. The sign 
convention follows naturally from 
the more rigorous definition of 
mechanical work as W - Fl cos a 
where h is the angle between F and 



SG 10.2 



UI^U^ 


^r^^'-.,t,^^r''am^ 


^^ -J 


m- Jt^plJ?."*,- 


'^4 


w^~^ 


< -M 


«^3^^ ^JSk '^. 


mEL^^hS 





From our definition of work it follows that no work is done if there is 
no displacement. No matter how hard you 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 direction of the force. For example, suppose you are 
carrying 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 forward 
against the small resisting force of the air. 

Work is a useful concept in itself. But the concept is most useful in 
understanding the concept of energy. There are a great many forms 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 algebraically. We will also discuss how energy changes 
from one form to another. The general concept of energy is very- difficult 
to define. But to define some particular forms 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 energy transformed from one form to another. For example, 
when you throw a ball you do work on it. You also transform chemical 
energy, which your body obtains from food and oxygen, into energy of 
motion of the ball. When you lift a stone (doing work on it), you transform 
chemical energy into gravitational potential energy. If you release the 
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 transformed into heat. These are some of the forms energy 
takes; and 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 with Newton's laws of motion, to get an expression of 
this form of energy. Remember that a moving body has many attributes 
which are related by separate ideas. For example, we have studied speed v 
(Chapter 1), velocity v (Chapter 3), momentum mv (Chapter 9). We also 
saw how the seventeenth century thinkers groped for a clear idea of some 
conserved quantity in all motion. Now let us imagine that we exert a 
constant net force F on an object of mass m. This force accelerates the 
object over a distance d firom rest to a speed v. Using Newton's second 
law of motion, we can show in a few steps of algebra that 



Fd = imz'2 



The details of this derivation are given on the first half of page 32 "Doing 
Work on a Sled." 

We recognize Fd as 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 



Section 10.2 



Unit 3 31 



the object. So ^mv^ 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 Imi^ at that instant, and is called kinetic energy. We will use 
the symbol KE to represent kinetic energy. By definition then, 



KE 



imv 



Now it is clearer why we wrote \mv^ instead of just mv^ in Chapter 
9. If one is conserved, so must be the other — and conservation was all that 
we were concerned with there. But \mv^ also relates directly to the 
concept of work, and so provides a more useful expression for energy of 
motion. 

The equation Fd = ^mv^ was obtained by considering the case of an 
object initially at rest. In other words, the object had an initial kinetic 
energy of zero. But 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 = A(KE) 

The quantity A(KE) is by definition equal to (2"2^'^)finai ~ (^^^^)initiai. 
The proof of this general equation appears on the second half of page 32. 
Work is defined as the product of a force and a distance. Therefore, 
its units in the mks system are newtons x meters or new ton • meters. A 
newton- meter is also called z. joule (abbreviated J). The joule is the unit 
of work or of energy. 



Q1 If a force F is exerted 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 (b) Fd (c) Fid (d) ^Fd"" 

Q2 The kinetic energy of a body of mass m moving at a speed v is 
(a) \mv (b) kmy"^ (c) mv^ (d) ^mv"^ (e) mh)^ 



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 10.3. 



SG 10.3-10.8 



The name of the unit of energy and 
work commemorates J. P. Joule, a 
nineteenth-century English physicist, 
famous for his experiments showing 
that heat is a form of energy (see 
Sec. 10.7). There is no general 
agreement today whether the name 
should be pronounced like jool " 
or like "jowl.' The majority of 
physicists favor the former. 



10.2 Potential energy 

As we have seen in the previous section, doing work on an object can 
increase its kinetic energy. But work 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. And 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 gra\itational potential energy is transformed rapidly into 
kinetic energy of fall. In general terms, suppose a force P is used to 
displace an object upwards a distance d, without changing its KE. Then 



Doing Work on a 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 balancced by the upward push 
exerted by the ice, so F is 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 



V = at 



and 



d = hat^ 

where a is the acceleration of the body, t is 
the time interval during which it accelerates 




So 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 = v^,t 

If we consider a uniformly accelerated body 
whose speed changes from vo to v, the 
average speed during t is ^(v + vo). Thus 



d = —^xt 



By the definition of acceleration, a = AWf; 

r' , , 

D 




I II f9 



(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 = via. If 
we substitute this expression for t into the 
second equation, we obtain 

d = 2at = jS — r = 2 — 
a^ a 

The work done on the sled is W=Fd. 
From Newton's second law, F=ma, so 

W = Fd 

= ma X I — 
a 

The acceleration cancels out. giving 



therefore t = Avla = {v - vo)/a 
Substituting {v - vo)/a for t gives 



^^VJ;J^^V-V 



2 a 

{V +Vo) (V -Vo) 



2a 



2a 

The work W done is W^Fd, or, since F=^ma, 

W = ma X d 

V - V' 



= ma X 



2a 



Section 10.2 



Unit 3 33 



the increase in gravitational potential energy, A(P£)grav, is 

A(P£;^av - Fd 

Potential energy can be thought of as stored energy. As the book falls, 
its gravitational potential energy' decreases while its kinetic energy 
increases correspondingly. W^hen the book reaches its original height, all 
of the gravitational potential energy stored during the lift will have been 
transformed into kinetic energy. 

Many useful apphcations foUow from this idea of potential or stored 
energy. For example, the steam hammer used by construction crews is 
driven up by high-pressure steam ("pumping in" energy). When the 
hammer drops, the gravitational potential energ)' 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 reser\'oir. 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 rybber band or a spring, you increase its elastic 
potential energy. When you release the rubber 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 other kinds of 
forces. In an atom, the negatively charged electrons are attracted by the 
positively charged nucleus. If an externally applied force pulls an electron 
away from the nucleus, the electric potential energy increases. If the 
electron is pulled back and moves toward the nucleus, the potential 
energ\' decreases as the electron's kinetic energ\' increases. 

If two magnets are pushed together with north poles facing, the 
magnetic potential energy increases. When released, the magnets will 
move apart, gaining kinetic energ)' as they lose potential energy. 

Where is the potential energy located in all these cases? It might 
seem at first that it "belongs" to the body that has been moved. But this is 
not the most useful way of thinking about it. For without the other 
object — the earth, the nucleus, the other magnet — the work would not 
increase any potential form of energ>'. 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 aU of these interacting bodies. For example, you could give 
either magnet all the kinetic energy, just by releasing it 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 to the book alone. The work is done by an "outside" 



F 



3 



t — 

To lift the book at consiant speea. 
you must exert an upward force F 
equal in magnitude to the weight 
F, , of the book. The work you do 
in lifting the book through distance 
d is Fd. which is numerically equal 
to F d. See SG 10.9 and 10.10. 




A set mouse-trap contains elastic 
potential energy. 



SG 10.11 
SG 10.12 



SG 10.13 



34 Unit 3 



Energy 



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. 



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 just is 
converted from PE to KE. This is discussed in more detail in the next 
section. 

Q3 If a stone of mass m falls a vertical distance d, pulled by its 
weight Fgrav = w«g, the decrease in gravitational potential energy is (a) 
md (b) mOg (c) mUgd (d) ^md^ (e) d. 

04 When you compress a coU spring you do work on it. The elastic 
potential energy (a) disappears (b) breaks the spring (c) increases (d) 
decreases. 

Q5 Two electrically charged objects repel one another. To increase 
the electric potential energy, you must 

(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. 



The equations in this section are 
true only if friction is negligible. 
We shall extend the range later to 
include friction, which can cause the 
conversion of mechanical energy 
into heat energy. 



10.3 Conservation of mechanical energy 

In Section 10.1 we stated that the amount of work done on an object 
equals the amount of energy transformed from one form to another. For 
example, the chemical energy of a muscle is transformed into the kinetic 
energy of a thrown ball. Our statement implied that the amount of energy 
involved does not change — only its form changes. This is particularly 
ob\aous 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 transformed 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. Or consider a stone thrown upward. Between any two points in its 
path, the increase in gravitational potential energ\' equals the decrease in 
kinetic energy. For a stone falling or rising (without external forces such 
as friction), 

A(PE)g,av = -A(KE) 
This relationship can be rewritten as 

A(KE) + A(P£)grav - 
or still more concisely as 

A(K£ + P£g,av) = 
If {KE + PFgrav) represents the total mechanical energy of the system, 



Section 10.3 



Unit 3 35 



then the change in the system's total mechanical energy is zero. In other 
words, the total mechanical energy, A(K£ + P£grav) remains constant; it 
is conserved. 

A similar statement can be made for a vibrating guitar string. While 
the string is being pulled away from 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 
as the elastic potential energy increases. As it vibrates, there is a repeated 
transformation 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: 

ArP£)e.astic = -A(K£) 

In such an ideal case, the total mechanical energy (KE + PE elastic) 
remains constant; it is conserved. 

We 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. Potential energy also can be transformed into another form of 
potential energy without change in the total energy (KE + PE). We can 
write this rule in several equivalent ways: 



SG 10.14 




up lo nere we nave aiways con- 
sidered only changes in PE. There 
is some subtlety in defining an 
actual value of PE. See SG 10.15. 



or 



or 



or 



AK£ = -APE 



AKE > APE = 



A(K£ + P£) = 



KE + PE - constant 



These equations are different ways of expressing the law of conservation 
of mechanical energy when there is no "external" force. But suppose that 
an amount of work W 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 from 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 -i- suitcase + earth system does not change. 

The law of conservation of energy can be derived from Newton's laws 
of motion. Therefore, it tells us nothing that we could not, in principle, 
compute directly from Newton's laws of motion. However, there are 



36 Unit 3 



Energy 




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 spherical 
shape, and elastic potential energy 
is transformed into kinetic energy. 



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. Before 
long you win see how the law came to be very useful in understanding a 
huge variety of natural phenomena. 

A perfectly elastic collision is a good example of a situation where we 
often cannot apply Newton'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 coUlsion, 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 exactly the same after 
colliding as they were before. They have the same shape, same 
temperature, etc. In such a case, all of the elastic potential energy is 
converted back into kinetic energy. 

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 oi each object.) You may 
recall that the law of conservation of momentum also left us with useful 
but incomplete knowledge. We can use it to find the total momentum, but 
not the individual momentum vectors, of elastic objects in coUision. In 
Chapter 9 we saw how conservation of momentum and conservation of 
mechanical energy together limit the possible outcomes of perfectly elastic 
coUisions. For two coUiding 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. We 
usually are not interested in the detailed motion of each of every part of a 
complex system. We are not likely to care, for example, about the motion 
of every molecule in a rocket exhaust. Rather, we probably want to know 
only about the overall thrust and temperature. The principle of 
conservation of energy applies to total, defined systems, and such systems 
usually interest us most. 



Q6 As a stone falls frictionlessly 

(a) its kinetic energ>' is conser\'ed 

(b) its gravitational potential energ)' 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 

Q7 In what position is the elastic potential energy of the vibrating 
guitar string greatest? At which position is its kinetic energy greatest? 

Q8 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 a meter of bass string is greater than that of a 
meter of treble string.) 



Section 10.4 



Unit 3 37 



Q9 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 Forces that do no work 



In Section 10.1 we defined the work done on an object. It is the 
product of the magnitude of the force F applied to the object and the 
magnitude of the distance tfin the direction oif through which the 
object moves while the force is being applied. In all our examples so far, 
the object moved in the same direction as that of the force vector. 

But usually the direction of motion and the direction of the force are 
not the same. For example, suppose you carr\' a book at constant speed 
and 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 (by 
our definition of work). 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 — upwards 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 5*, and so the force does no work. This statement agrees entirely with 
the idea of work as energy being transformed from one form to another. 
Since the book's speed is constant, its kinetic energy is constant. And 
since its distance from the earth is constant, its gravitational potential 
energy is constant. So 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 transformation. For a 
circular orbit the centripetal force vector is perpendicular to the tangential 
direction of motion at any instant. So no work is being done. To put an 
artificial satellite into a circular orbit requires work. But once it is in orbit, 
the KE and PE stay constant and no further work is done on the satellite. 

WTien 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 forms. 
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 force and motion are in exactly the 
same direction. What about the more usual case, involving some angle 
between the force and the motion? 

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. Consider the example of a child sliding down a playground 




SG 10.16 



38 Unit 3 



Energy 




slide. The gravitational force Fgrav is directed down. So only the distance 
down determines the amount of work done by Fgrav It does not matter 
how long the sbde is, or what its shape is. 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 PE grav 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. 



sicota> noQfi 




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 10.17 



More generally, change in PEgrav depends only on change of position. 
The details of the path followed in making the change make no difference 
at all. The same is true 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 PEgrav 
depends only on change in height. For the example of the child on the 
slide, the gravitational potential energy decreases as his altitude decreases. 
If frictional forces are vanishingly small, aU the work goes into 
transforming PEgrav into KE. Therefore, the increases in KE depend only 
on the decrease in altitude. In other words, the child's speed when he 
reaches the ground will be the same whether he sbdes 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 simphcity to some physical laws, as we 
will see when we consider gravitational and electric fields in Chapter 14. 

Q10 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 H? 

Q1 1 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. 

Q1 2 A third skier went direcdy down a straight slope. How would 
his speed at the bottom compare with that of the skier who went off the 
jump? 



Section 10.5 



Unit 3 39 



Q13 No work is done when 

(a) a heavy box is pushed at constant speed along a rough horizontal 
floor 

(b) a nail is hammered into a board 

(c) there is no component of force parallel to the direction of motion 

(d) there is no component of force perpendicular to the direction of 
motion. 




10.5 Heat energy and the steam engine 

So far we have assumed that our equations for work and energy hold 
only if friction is absent or very small. Why? Suppose that factional forces 
do affect a suitcase or other object as it is being lifted. The object must do 
work against these forces as it moves. (This work in fact serves to warm 
up the stairs, the air, etc.). Consequendy, that much less work is available 
to increase PE or KE or both. How can we modify our expression of the 
law of conservation of mechanical energy to include these effects? 

Suppose that a book on a table has been given a shove and is sliding 
across the table top. If the surface is rough, it will exert a fairly large 
frictional force and the book will stop quickly. Its kinetic energy will 
rapidly disappear. But 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 conserved. 

However, close examination of the book and the tabletop show that 
they are warmer than before. The disappearance of kinetic energy of the 
book is accompanied 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, then 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 will see how theory was aided by practical 
knowledge of the relation of heat and work. This knowledge was gained in 
developing, for very practical reasons, the steam engine. 

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 greater coal supply. 
But water tended to seep in and flood these deeper mines. The need arose 
for an economical method for pumping water out of mines. The steam 
engine was developed initially to meet this very 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 






40 Unit 3 



Energy 




A model of Heron's aeolipile. Steam 
produced in the boiler escapes 
through the nozzles on the sphere, 
causing it to rotate. 



used directly to do work, as in a steam locomotive, or can be transformed 
into electrical energy. In typical twentieth-centur\' 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, 
too — internal combustion engines and turbines for example. But the steam 
engine remains a good model for the basic operation of this whole family 
of engines. 

The generation and transmission of electrical energy, and its 
conversion into mechanical energy, will be discussed in Chapter 15. Here 
we will focus on the central and historic link in the chain of energy 
conversion, the steam engine. 

Since ancient times it had been known that heat could be used to 
produce steam, which could then do mechanical work. The "aeolipile," 
invented by Heron of Alexandria about 100 a.d., 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 centur>', 
however, were commercially successful steam engines invented. 

Today we would say that a steam engine uses a supply of heat energy 
to do mechanical work. That is, it converts heat energy into mechanical 
energy. But many inventors in the eighteenth and nineteenth centuries 
did not think of heat in this way. They regarded heat as a thin, invisible 
substance that could be used over and over again to do work without 
being used up itself But they did not need to learn all the presently 
known laws of physics in order to become successful engineers. In fact, 
the sequence of events was just the opposite. Steam engines were 
developed first by inventors who knew relatively little about science. Their 
main interest lay in making money, or in improving the effectiveness and 
safety of mining. Later, scientists with both a practical knowledge of what 
would work and a curiosity about how it worked made new discoveries in 
physics. 

The first commercially successful steam engine was invented by 
Thomas Savery (1650-1715), an English militarv' engineer. Follow the 
explanation of it one sentence at a time, referring to the diagram on page 
41. In the Savery engine the water in the mine shaft is connected by a 
pipe and a valve D to a chamber called the cylinder. With valve D closed 
and valve B open, high-pressure steam from the boiler is admitted to the 
cylinder through valve A. This forces the water out of the cylinder and up 
the pipe. The water empties at the top and runs off at ground level. Valve 
A and valve B are closed. Valve D is opened, allowing an open connection 
between the cylinder and the water in the mine shaft. 

When valve C is opened, cold water pours over the cylinder. The 



Section 10.5 



Unit 3 41 




OJa-fe 
wine 



r in ^^\l^ \V\V.- 



J^'V. water pfjshQef op 
above QrouiAc/ (tvcl 



Schematic diagram of Savery engine. 



steam left in the cylinder cools and condenses to water. Since water 
occupies a much smaller volume than the same mass of steam, a partial 
vacuum forms in the cylinder. This vacuum allows the air pressure in the 
mine to force water from the mine shaft up the pipe into the cylinder. 

The same process, started by closing valve D and opening valves A 
and B, is repeated over and over. The engine is in effect a pump. It 
moves water from the mine shaft to the cylinder, then from the cylinder to 
the ground above. 

However, the Savery engine's use of high-pressure steam produced a 
serious risk of boiler or cylinder explosions. This defect was remedied by 
Thomas Newcomen (1663-1729), another Englishman. Newcomen 
invented an engine that used steam at lower pressure. His engine was 
superior in other ways also. For example, it could raise loads other than 
water. 

The Newcomen engine features a rocking beam. This beam connects 
to the load on one side and to a piston in a cylinder on the other side. 
When valve A is open, the cylinder is filled with steam at normal 
atmospheric pressure. The beam is balanced so that the weight of the load 
raises the piston to the upper end of the cylinder. While the piston is 
coming toward this position, valve A is still open and valve B is still 
closed. 

But when the piston reaches its highest position, valve A is closed 
and valve C is opened. Cooling water flows over the cylinder and the 
steam condenses, making a partial vacuum in the cylinder. This allows the 
pressure of the atmosphere to push the piston down. As the piston reaches 
the bottom of the cylinder, valve C is closed and valve B is opened briefly. 



In the words of Erasmus Darwin, 
the engine 
Bade with cold streams, the 

quick expansion stop. 

And sunk the immense of va- 
pour to a drop 
Press'd by the ponderous air 

the Piston falls 
Resistless, sliding through 

its iron walls; 
Quick moves the balanced 

beam, of giant-birth 
Wields his large limbs, and 

nodding shakes the earth. 



I 



42 Unit 3 



Energy 



Schematic diagram of Newcomen 
engine. In the original Newcomen 
engine the load was water being lifted 
from a mine shaft. 



pisroi^ 



VAL\/£ ^ 





Concjensec^ 

e>tean 




loa: 



l^lLLi 



The cooled and condensed steam runs off. The valve A Is opened, and the 
cycle begins all over again. 

Originally someone had to open and close the valves by hand at the 
proper times in the cycle. But later models did this automatically. The 
automatic method used the rhythm and some of the energy of the mo\ing 
SG 10.18 parts of the engine itself to control the sequence of operation. This idea, of 
using part of the output of the process to regulate the process itself, is 
caHed feedback . It is an essential part of the design of many modem 
mechanical and electronic systems. (See the article "Systems, Feedback, 
Cybernetics" in Unit 3 Reader. 

The Newcomen engine was widely used in Britain and other 
European countries throughout the eighteenth century. By modem 
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. 



Q14 When a book slides to a stop on the horizontal rough surface of 
a table 



Section 10.6 



Unit 3 43 



£/Af ENGJJV£%r ^^Ji/u^ Ha^erfm^A a/^m^r ?n^<i^J ^Fire^| 




At the left, a contemporary engraving 
of a working Newcomen steam engine. 
In July. 1698 Savery was granted a 
patent for A new invention for rais- 
ing of water and occasioning motion 
to all sorts of mill work by the impel- 
lent force of fire, which will be of great 
use and advantage 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 prevented New- 
comen from making much moneyfrom 
his superior engine during this period. 



(a) the kinetic energy of the book is transformed into potential 
energy. 

(b) heat is transformed into mechanical energy. 

(c) the kinetic energy of the book is transformed into heat energy. 

(d) the momentum of the book itself is conserved. 

Q15 True or false: The invention of the steam engine depended 
strongly on theoretical developments in the physics of heat. 

Q1 6 In Savery's steam engine, the energy of coal 

was changed (by burning) into energ\' which in turn 

was converted into the energy of the pump. 



10.6 James Watt and the Industrial Revolution 



A gready improved steam engine originated in the work of a 
Scotsman, James Watt. Watt's father was a carpenter who had a 
successful business selling equipment to ship owners. Watt was in poor 
health much of his life and gained most of his early education at home. In 



44 Unit 3 



Energy 




The actual model of the Newcomen 
engine that inspired Watt to conceive 
of the separation of condenser and 
piston. 



his father's attic workshop, he developed considerable skill in using tools. 
He wanted to become an instrument-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. As it turned out, this assignment had immense worldwide 
consequences. In acquainting himself with the model. Watt was impressed 
by how much steam was required to run 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 warming up the walls of its cylinders. The walls 
were then cooled again every 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. 
The steam in its cylinder, after pushing the piston up, 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 diagram opposite represents Watt's engine. With valve A open 
and valve B closed, steam under pressure enters 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 condenser is kept 
cool by water flowing over it, so the steam condenses. As steam leaves the 
cylinder, the pressure there decreases. Atmospheric 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. 

Watt's invention of the separate condenser might seem only a small 
step in the development of steam engines. But in fact it was a decisive 
one. Not having to reheat the cylinder again and again allowed huge fuel 



Watt in his workshop contemplating 
a model of a Newcomen engine. 
(A romanticized engraving from a 
nineteenth-century volume on tech- 
nology.) 



BSa'SFv 




Section 10.6 



Unit 3 45 




Schematic diagram of Watt engine. 



savings. Watt's engine could do more than twice as much work as 
Newcomen's with the same amount of fuel. This improvement enabled 
Watt to make a fortune by selling or renting his engines to mine owners. 
The fee that Watt charged for the use of his engines depended on 
their power. Power is defined as the rate of doing work (or the rate at 
which energy is transformed from one form to another). The mks unit of 
power is the joule-per-second, which is now fittingly called one watt: 

1 watt = 1 joule/sec 



Matthew Bouiton (Watf s business 
partner) proclaimed to Boswell (the 
biographer of Samuel Johnson): 
••| sell here, Sir, what all the world 
desires to have: POWER!' 



James Watt expressed the power of his engines in different units. 

One "foot-pound" is defined as the work done when a force of one 
pound is exerted on an object while the object moves a distance of one 
foot. (In mks units, this corresponds roughly to a force of 4 nevvtons while 
the object moves A meter. Thus. 1 foot-pound is approximately 5 newton- 
meters.) Watt found that a strong workhorse, working steadily, could lift a 
150-pound weight at the rate of almost four feet per second. In other 
words, it could do about 550 foot-pounds of work per second. Watt used 
this as a definition of a convenient unit for expressing the power of his 
engines: the horsepower. To this day the "horsepower" unit is used in 



46 Unit 3 Energy 

engineering — although it is now defined as precisely 746 watts. 



I 



Typical power ratings (in horsepower) 



SG 10.19-10.26 '^^" turning a crank 
Overshot waterwheel 
Turret windmill 
Savery steam engine (1702) 
Newcomen engine (1732) 
Smeaton's Long Benton engine (1772) 
Watt engine (of 1778) 

Cornish engine for London waterworks (1837) 
Electric power station engines (1900) 
Nuclear power station turbine (1970) 



0.06 h.p. 

3 

10 

1 

12 

40 

14 

135 

1000 

300,000 



[Adapted from R. J. Forbes, in C. Singer et al, History of Technology.] 




4 m 



A steam locomotive from the early 
part of the 20th century. 



Watt's invention, so superior to Newcomen's engine, 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. Nowadays 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 some 
greedy and cruel 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 — voluntarily or forced by poverty 
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 professional men. 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, machinery, 
and an emphasis on material goods. In some cases they confused research 
science itself with its technical apphcations (as is still done today). In 
some cases scientists were accused of explaining away all the awesome 
mysteries of nature. They denounced both science and technology, while 
often refusing to learn anything about them. In a poem by William Blake 
we find the questions: 



Section 10.6 



Unit 3 47 




And did the Countenance Divine 
Shine forth upon our clouded hills? 

And was Jerusalem builded here 
Among these dark Satanic mills? 



Elsewhere, Blake ad\ased his readers "To cast off Bacon, Locke, and 
Newton." John Keats was complaining about science when he included in 
a poem the line: "Do not all charms 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. We saw that Galilean 
and Newtonian physics also was attacked for distorting values. The same 
type of accusation can still be heard today. 

Steam engines are no longer widely used as direct sources of power in 
industry and transportation. But steam is indirectly 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- 



-.-r-^ 




Richard Trevithick's railroad at Euston 
Square, London, 1809. 



TREVITHICKS, 

rORTABIiB STEAM E-NGI>ilB . 



CatclLme who can . 



'.■i- 



% 





A nineteenth-century French steam cultivator. 



t 




lApchaaxleal rawer Subduing^ 
Anfaual 8pe<^ . 



■) ,. 






i*i 




Hfe:^ 



The "Charlotte Dundas," the first practical steamboat, built 
by William Symington, an engineer who had patented his own 
improved steam engine. It was tried out on the Forth and 
Clyde Canal in 1801. 



Section 10.7 



Unit S 49 



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. A description of 
the type of steam turbine used in modem power stations shows the 
change of scale since Heron's toy: 

The boiler at this station [in Brooklyn, New York] is as tall 
as a 14-story building. It weighs about 3,000 tons, more than a 
U.S. Navy destroyer. It heats steam to a temperature of 1,050° 
F and to a pressure of 1,500 pounds per square inch. It 
generates more than 1,300.000 pounds of steam an hour. This 
steam runs a turbine to make 150,000 kilowatts of electricity, 
enough to supply all the homes in a city the size of Houston, 
Texas. The boiler burns 60 tons (about one carload) of coal an 
hour. 

The 14-story boiler does not rest on the ground. It 
hangs — all 3,000 tons of it — from a steel framework. Some 
boilers are even bigger — as tall as the Statue of Liberty — and 
will make over 3,000.000 pounds of steam in one hour. This 
steam spins a turbine that will make 450,000 kilowatts of 
electricity — all of the residential needs for a city of over 
4,000.000 people! 



Below, a 200 thousand kilowatt tur- 
bine being assembled. Notice the 
thousands of blades on the rotor. 




Q17 The purpose of the separate condenser in Watt's steam engine 
is to 

(a) save the water so it can be used again 

(b) save fuel by not ha\1ng to cool and reheat the cylinder 

(c) keep the steam pressure as low as possible 

(d) make the engine more compact 

Q18 The history of the steam engine suggests that the social and 
economic effects of technology are 

(a) always beneficial to eveiyone 

(b) mostly undesirable 

(c) unimportant one way or another 

(d) verv' different for different levels of society 
Q19 What is horsepower? 



10.7 The experiments of Joule 




^•5<v 



In the steam engine a certain amount of heat does a certain amount 
of work. What happens to the heat in doing the work? 

Early in the nineteenth centun', most scientists and engineers thought 
that the amount of heat remained constant; and that heat could do work 
as it passed from a region at one temperature to a region at a lower 
temperature. For example, early steam engines condensed steam at high 



James Prescott Joule (1818-1889) 
Joule was the son of a wealthy Man- 
chester brewer. He is said to have 
become first interested in his arduous 
experiments by the desire to develop 
more efficient engines for the family 
brewery. 



50 Unit 3 



Energy 



The idea of heat as a conserved 
substance is consistent with many 
phenomena. An experiment showing 
this is •Calorimetry" in the 
Handbook. 



temperatures to water at low temperature. Heat was considered to be a 
substance called "caloric." The total amount of caloric in the universe was 
thought to be conserved. 

According to the caloric theory, heat could do work in much the same 
way that water can do work. Water falling from a high level to a low level 
can do work, with the total amount of water used remaining the same. 
The caloric explanation seemed reasonable. And most scientists accepted 
it, even though no one measured the amount of heat before and after it 
did work. 

A few scientists, however, disagreed. Some favored the view that heat 
is a form of energy. One who held this view was the English physicist 
James Prescott Joule. During the 1840's Joule conducted a long series of 
experiments designed to show that heat is a form of energy. He hoped to 
demonstrate in a variety of different experiments that the same decrease 
in mechanical energy always produced the same amount of heat. This, 
Joule reasoned, would mean that heat is a form of energy. 

For one of his early experiments he constructed a simple electric 
generator, which was driven by a falling weight. The electric current that 
was generated heated a wire. The wire was immersed in a container of 
water which it heated. From the distance that the weight descended he 
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 got hotter on compression. 

But his most famous experiments involved an apparatus in which 
slowly descending weights turned a paddle-wheel in a container of water. 
Owing to the friction between the wheel and the liquid, work was done on 
the hquid, raising its temperature. 

Joule repeated this experiment many times, constantly irnpro\ang 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 with a precision of a small fraction of a 
degree. And he allowed for the small amount of kinetic energy the 
descending weights had when they reached the floor. 

Joule pubhshed his results in 1849. He reported: 



1st. That the quantity of heat produced by the friction 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° Fahr. requires for its evolution the expenditure of a 
mechanical energy represented by the fall of 772 lb through 
the distance of one foot. 



Joule used the word "force" in- 
stead of "energy." The current 
scientific vocabulary was still being 

formed. 



The first statement is the evidence that heat is a form of energ\', 
contrary to the caloric theory. The second statement gives the numerical 
magnitude of the ratio he had found. This ratio related a unit of 
mechanical energy (the foot-pound) and a unit of heat (the heat required 



Section 10.8 



Unit 3 51 



to raise the temperature of one pound of water by one degree on the 
Fahrenheit scale). 

In the mks system, the unit of heat is the kilocalorie and the unit of 
mechanical energy is the joule. Joule's results are equivalent to the 
statement that 1 kilocalorie equals 4,150 joules. Joule's paddle-wheel 
experiment and other basically similar ones have since been performed 
with great accuracy. The currently accepted value for the "mechanical 
equivalent of heat" is 

1 kilocalorie = 4,184 joules 

We might, therefore, consider heat to be a form of energy. We will 
consider the nature of the "internal" energy associated with temperature 
further in Chapter 11. 

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 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 greatest 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 
designing engines. However, there are theoretical limits on efficiency. 
Thus, even a perfectly designed machine could never do work at 100% 
efficiency. We will hear more about this in Chapter 11. 

Q20 According to the caloric theory of heat, caloric 

(a) can do work when it passes between two objects at the same 
temperature 

(b) is another name for temperature 

(c) is produced by steam engines 

(d) is a substance that is conserved 
Q21 The kilocalorie is 

(a) a unit of temperature 

(b) a unit of energy 

(c) 1 kilogram of water at 1°C 

(d) one pound of water at 1°F 

Q22 In Joule's paddle-wheel experiment, was all the change of 
gravitational potential energy used to heat the water? 



This unit is called a British Thermal 
Unit (BJy)). 

SG 10.27, 10.28 

A kilocalorie is what some 
dictionaries call Marge calorie." It is 
the amount of heat required to raise 
the temperature of 1 kilogram of 
water by 1 Celsius ("centigrade"). 
This unit is identical to the "Calorie" 
(with a capital C) used to express the 
energy content of foods in dietetics. 



The efficiency of a steam engine is 
roughly 15-20%; for an automobile 
it is about 22%: and for a diesel 
engine it is as high as 40%. 

In Sec. 10.10 we mention some 
qualifications that must be placed 
on the simple idea of heat as a 
form of energy. 



10.8 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; bke 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 all animals, even 
carnivores, ultimately obtain their food energy from plant material. The 



52 Unit 3 



Energy 



Carbohydrates are molecules made 
of carbon, hydrogen, and oxygen 
A simple example is the sugar gib 
cose, the chemical formula for which 

is C-,H, .0,.. 



animal eaten by the lion has previously dined on plant material, or on 
another animal which had eaten plants. 

Green plants obtain energy from sunhght. 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 (H2O) and carbon dioxide (CO2). The 
energy used to synthesize carbohydrates is not lost; it is stored in the 
carbohydrate molecules as chemical energy. 

The process by which plants synthesize carbohydrates is called 
photosynthesis. It is still not completely understood and research in this 
field is lively. We know that the synthesis takes place in many small steps, 
and many of the steps are well understood. It is conceivable that we may 
learn how to photosynthesize 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 + energy 



glucose + oxygen 






Piif 




mm.. 



■ 'mm ■ .^^^i^'v^r^-iS v^ 

Electron micrograph of an energy- 
converting mitochondrion in a bat 
cell (200,000 times actual size). 



The energy stored in the glucose molecules is used by the animal that 
eats the plant. This energy maintains the body temperature, keeps its 
heart, lungs, and other organs operating, and enables various chemical 
reactions to occur in the body. The animal also uses it 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 ten 
steps, split glucose molecules into simpler molecules. In another sequence 
of reactions these molecules are oxidized (combined with oxygen), thereby 
releasing most of the stored energy and forming carbon dioxide and water. 



glucose + oxygen 



carbon dioxide + water + energy 



Proteins and fats are used to build and restore tissue and enzymes, 
and to pad delicate organs. They also can be used to pro\1de energy. Both 
proteins and fats can enter into chemical reactions which produce the 
same molecules as the split carbohydrates. From that point, the energy- 
releasing process is the same as in the case of cai'bohydrates. 

The released energy is used to change a molecule called adenosine 
diphosphate (ADP) into adenosin triphosphate (ATP). 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. 



Section 10.8 



Unit 



53 



The chemical and physical operations of the living body are in some 
ways like those in 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 kilocalories. (1 kilocalorie = 
10^ calories). However it could just as well be measured in joules or foot- 
pounds or British Thermal Units. The table in the margin gives the 
energy content of some foods. (The "calorie" or "large calorie" used by 
dieticians, is identical to what we have defined as the kilocalorie.) 

Much of the energy you obtain from food keeps your body's internal 
"machinery" running and keeps your body warm. Even when asleep your 
body uses about one kilocalorie 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. But efficiency probably never exceeds 25%, and usually is less. 
Studies of this sort are earned out in bioenergetics, one of the many 
fascinating and useful fields where physics and biology overlap. 

The table in the margin gives the results of experiments done in the 
United States of the rate at which a healthy young person of average build 
and metabohsm 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 the subject did nothing but sleep for eight 
hours a day and lie quietly the rest of the time, he would still need at 
least 1,700 kilocalories of energy each day. There are countries where 
large numbers of working people exist on less than 1,700 kilocalories a 
day. The U.N. Yearbook of National Accounts Statistics for 1964 shows 
that in India the average food intake was about 1,600 kilocalories per day. 
The United States average was 3,100 kilocalories per day. About half the 
population of Southeast Asia is at or below the stanation line. Vast 
numbers of people elsewhere in the world, including some parts of the 
United States, are also close to that line. It is estimated that if the 
available food were equally distributed among all the earths inhabitants, 
each would have about 2,400 kilocalories a day on the average. This is 
only a little more than the minimum required by a working person. 

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 
biUion or more. Furthermore, 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 



Approximate Energy Content of 
Various Foods (In Calories per 
kilogram) 



Butter 


7000 


Chocolate (sweetened) 


5000 


Beef (hamburger) 


4000 


Bread 


2600 


Milk (whole) 


700 


Apples (raw) 


500 


Lettuce 


150 



Adapted from U.S. Department of 
Agriculture, Agriculture Handbook No. 
8, June 1950. 



The chemical energy stored in food 
can be determined by burning the 
food in a closed container immersed 
in water and measuring the tem- 
perature rise of water. 



Approximate Rates of Using Energy 
During Various Activities (In Calories 
per hour) 



Sleeping 


70 


Lying down (awake) 


80 


Sitting still 


100 


Standing 


120 


Typewriting rapidly 


140 


Walking (3 mph) 


220 


Digging a ditch 


400 


Running fast 


600 


Rowing in a race 


1200 



Adapted from a handbook of the U.S. 
Department of Agriculture. 



SG 10.29-10.31 



SG 10.32 



54 Unit 3 



Energy 



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 connec- 
tion to physics is provided by the 
problem of inadequate world food 
supply: here, too. many physicists, 
with others, are presently trying to 
provide solutions through work 
using their special competence. 



the last ten years the increase in crop yield per acre in the poorer 
countries has averaged less than one percent per year, far less than the 
increase in population. The problem of supplying food energ>' for the 
world's hungry is one of the most difficult problems facing humanity 
today. 

In this problem of Ufe-and-death importance, what are the roles 
science and technology can play? Obviously, better agricultural practice 
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 fertihzers are not too difficult 
to build. But right here we meet a general law on the use of applications 
of scienc'e through technology: Before applying technology, study all the 
consequences that may be expected; otherwise you may create two new 
problems for every old one that you wish to "fix." 

In any particular country, the questions to ask include these: How 
will fertihzers interact with the plant being grown and with the soU? Will 
some of the fertilizer run off and spoil rivers and lakes and the fishing 
industry in that locality? How much water will be required? What variety 
of the desired plant is the best to use within the local ecological 
framework? How will the ordinary farmer be able to learn the new 
techniques? How will he be able to pay for using them? 

Upon study of this sort it may turn out that in addition to fertibzer, 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 adxisor for each 600 farms. Compare this 
with perhaps one advisor for 10,000 farms in Colombia, and one advisor 
per 100,000 farms in Indonesia. 

But for long-run 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 international 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 come in to 
help? It is usually true that one of the causes of some of the worse 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 sea water or brackish ground water are needed to allow irrigating 
fields with water from these plentiful sources. But before this wiD be 
economically possible, more basic knowledge will be needed on just how 
the molecules in hquids are structured, and how molecules move through 
membranes of the sort usable in de-salting equipment. Answers to such 
questions, and many like them, can only come through research in "pure" 
science, from trained research workers having access to adequate research 
facilities. 



Section 10.8 



Unit 3 55 



"The Repast of the Lion" 

by Henri Rousseau 

The Metropolitan Museum of Art 




Q23 Animals obtain the energy they need from food, but plants 

(a) obtain energy from sunlight 

(b) obtain energy from water and carbon dioxide 

(c) obtain energy from seeds 

(d) do not need a supply of energy 

Q24 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 quiedy without working 

(e) only one-fifth of the energy you obtain from food can be used to 
enable your body to do work on external objects 

Q25 Explain this statement: "The repast of the lion is sunlight." 



56 Unit 3 



Energy 



Joule began his long series of 
experiments 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 
competitive with steam engines. 



10.9 Arriving at a general law 

In Section 10.3 we introduced the law of conservation oi mechanical 
energy. This law apphes only in situations where no mechanical energy is 
transformed into heat energy or vice versa. But early in the nineteenth 
century, developments in science, engineering and philosophy suggested 
new ideas about energy. It appeared that all forms of energy (including 
heat) 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 battery 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. 
And in 1831, Michael Faraday, the great English scientist, discovered 
electromagnetic induction: the effect that 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 resulted from the same 
basic "force." This idea, though vague and imprecise, later bore fruit in 
the form of the law of conservation of energy. AH natural events involve a 
transformation 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 showing how to measure energy changes. 
Almost from 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 form to another. (The actual 
words "work" and "energy" were not used until later.) This 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 energv' transformed into heat energy. 

In 1843, Joule had 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 
[energy] is expended, an exact equivalent of heat is always 
obtained. 



Having said this. Joule got back to his work in the laboratorv-. He was 
basically a practical man who had litde time to speculate about a deeper 
philosophical meaning of his findings. But others, though using 
speculative arguments, were also concluding that the total amount of 



Section 10.9 



Unit 3 57 



energy in the universe is constant. 

A year before Joule's remark, for example, Julius Robert Mayer, a 
German physician, had proposed a general law of conservation of energy. 
Unlike Joule, Mayer had done no quantitative experiments. But he had 
observed body processes involving heat and respiration. And he had 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.) 

Mayer had been influenced strongly by the German philosophical 
school now known as Naturphilosophie or "nature-philosophy." This 
movement flourished in Germany during the late eighteenth and early 
nineteenth centuries. (See also the Epilogue to Unit 2.) Its most 
influential leaders were Johann Wolfgang von Goethe and Friedrich von 
Schelling. Neither of these men is known today as a scientist. Goethe is 
generally considered Germany's greatest poet and dramatist, while 
Schelling is remembered as a minor philosopher. But both men had great 
influence on the generation of German scientists educated at the 
beginning of the nineteenth century. The nature-philosophers were closely 
associated with the Romantic movement in literature, art, and music. The 
Romantics protested against the idea of the universe as a great machine. 
This idea, which had arisen after Newton's success in the seventeenth 
century, seemed morally empty and artistically worthless to them. The 
nature-philosophers also detested the mechanical world view. They refused 
to beheve that the richness of natural phenomena — including human 
intellect, emotions, and hopes — could be understood as the result of the 
motions of particles. 

At first glance, nature-philosophy would seem to have little to do with 
the law of conservation of energy. That law is practical and quantitative, 
whereas nature-philosophers tended to be speculative and qualitative. But 
nature-philosophy did insist on the value of searching for the underlying 
reality of nature. And this attitude did influence the discovery of the law 
of conservation of energy. Also, the nature-philosophers believed that the 
various phenomena of nature — gravity, electricity, magnetism, etc. — are 
not really separate from one another. Rather, they are simply different 
forms of one basic "force." This philosophy encouraged scientists to look 
for connections between different "forces" (or, in modern terms, between 
different forms of energy). It is perhaps ironic that in this way, it 
stimulated the experiments and theories that led to the law of conservation 
of energy. 

The nature-philosophers claimed that nature could be understood as it 
"really" is only by direct observation. But no comphcated "artificial" 
apparatus must be used — only feelings and intuitions. Goethe and 
Schelling were both very much interested in science and thought that 
their philosophy could uncover the hidden, inner meaning of nature. For 
Goethe the goal was "That I may detect the inmost force which binds the 
world, and guides its course." 

By the time conservation of energy was established and generally 
accepted, however, nature-philosophy was no longer popular. Scientists 
who had previously been influenced by it, including Mayer, now strongly 




Johann Wolfgang von Goethe (1749- 
1832) 

Goethe thought that his color theory 
(which most modern scientists con- 
sider useless) exceeded in importance 
all his literary works. 




J' 



Friedrich von Schelling (1775-1854) 



58 I' nit 3 



Energy 




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 scientifically 
respectable. 



SG 10.33 



opposed it. In fact, some hard-headed scientists at first doubted the law of 
conservation of energy simply because of their distrust of nature- 
phUosophy. 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 lesson we learned before: 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 
conserved. Some expressed the idea vaguely; others expressed it quite 
clearly. Some arrived at their belief mainly through philosophy: others 
from a practical concern with 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 discovery. 

The wide acceptance of the law of conservation of energy owes much 
to the influence of a paper published in 1847. This was two 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. 

Q26 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 



Energy Conservation on Earth 



^ 



Nuclear reactions inside 

the earth produce 

energy 

at a rate of 3 x lO^^w 



A. 




The nuclear reactions 
in the sun produce 
energy at a rate of 3.5 
X 1027W 

y 

The earth receives about 17 x lO^^W from the sun, of which 
about i 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 which is not reflected, . . . 




5 X lO^ew 
heats 
dry land 




3 X 1016W 
heats the 
air, producing 
winds, waves, etc. 



4 X 1016W 

evaporates 

water 



1.5 X 1013W 
is used by 
marine plants 
for 
photosynthesis 



Most of the energy given to watei 
Is given up again when the water 
condenses to clouds and rain; but 
every second about 10^^ Joules 
remains as gravitational potential 
energy of the fallen rain. 



'***^*««^^ 



\ 




Controlled nuclear 
reactions produce 2 
10i°W in electrical 
power 



electrochemistry 




light 



Some of this energy is 

used to produce 

lO^^W of hydroelectric 

power 

12 X 10" 
used in 
generating 
4 X 10"W of 
J electrical 

W P°''®'' 



mechanical 
power 



Ancient green plants 
have decayed an 
left a store of 
about 2.2 X lO^^ 
in the form of 
oil, gas, and 
coal. This store 
is being used at a 
rate of 5 x lO^^W 




Joules 






communication 




9 X 10"W is 

used in 

combustion 

engines. 

About \ 

of this is wasted 

as heat; less 

than 

3 X 10"W 

appears as y_ 

mechanical/ 

power . --O 





. . 3 X 1013W 
is used by 
land plants 
for 
photosynthesis 

\ Present-day green 
\ plants are being 
used as food for 
man and animals, 
at a rate 
of 2 X 1013W. 
Agriculture uses 
about To of this, 
and people 
ultimately consume 
3 X iQiiW as food. 
Direct use 
as raw 
materials 
for plastics 
and 

chemicals 
accounts 
for 
2 X IQiiW 



X 1012W 
is used 
for 

heating; 
this is 
equally 
divided 
between 
industrial 

and domestic ..J^»iji^>>%., 
uses. 






60 Unit 3 



Energy 



(c) stimulated speculation about the unity of natural phenomena 

(d) delayed progress in science by opposing Newtonian mechanics 
Q27 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 apparendy 
lost in steam engines 

(d) they made it possible to transmit energ)' over long distances 
Q28 The development of steam engines helped the discovery' of the 

law of conservation of energy because 

(a) steam engines produce 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 



10.10 A precise and general statement of energy conservation 



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. 



The word 'heat" is used rather 
loosely, even by physicists. This 
restriction on its meaning is not 
necessary in most contexts, but it 
is important for the discussion in 
this section. 



We can now try to pull many of the ideas in this chapter together into 
a precise statement of the law of conservation of energy. 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 works well for a great variety of phenomena, including the 
experiments of Joule. But difficulties arise with the idea of the heat 
"content" of a system. For example, when a solid is heated to its melting 
point, further heat input causes melting without increasing the 
temperature. (You may have seen this in the experiment on Calorimetry.) 
So simply adding the idea of heat as one form of a systems energy will 
not give us a complete general law. To get that, we must invent some 
additional terms with which to think. 

Instead of "heat," let us use the idea of an internal energy, an energy 
in the system that may 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.) 

Yet even these definitions do not permit a simple statement hke "heat 
input to a system increases its internal energy, and work done on a 
system increases its mechanical energy." For heat input to a system can 
have effects other than increasing inteniiil 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 



Section 10.10 



Unit 3 61 



mechanical energy. 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. 

As we mentioned before in discussing conservation laws, such laws 
can be expressed in two ways: (a) in terms of an isolated system, in 
which the total quantity of something does not change, or (b) in terms of 
how to measure the increases and decreases of the total quantity in an 
open (or non-isolated) system. The two ways of expressing the law are 
logicaDy related by the definition of "isolated." For example, conservation 
of momentum can be expressed either: (a) If no net outside force acts on 
a system, then the total mv of the system is constant; or (b) If a net 
outside force F acts on a system for a time At, the change in the total 
mv of the system is f^ x A^ In (a), the absence of the net force is a 
condition of isolation. In (b), one describes how the presence of a net 
force affects momentum. Form (b) is obviously more generally useful. 

A similar situation exists for the law of conservation of energy. We 
can say that the total energy of a system remains constant if the system is 
isolated. (By isolated we mean that no work is done on or by the system, 
and no heat passes between the system and its surroundings.) Or we can 
say that the change in energy of a now -isolated system is equal to the net 
work done on the system plus the net heat added to it. More precisely, we 
can let AW stand for the net work on the system, which is all the work 
done on the system minus all the work done by the system. We can let 
AH represent 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 total 
energy of the system, A£, is given by 



Special case of an isolated system: 




In general 

AW 



^ 



/ 



\ 



A£ 



AW + AH 



, AH 

\ 

AE = AW -^AH 1 

/ 
\ / 



This is a simple and useful form of the law of conservation of energy, and 
is sometimes called the first law of thermodynamics. 

This general expression includes as special cases the preliminary 
versions of the conservation law given earlier in the chapter. If there is no 
heat transfer at all, then AH = 0, and so A£ = AW. In this case, the 
change in energy 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 AE -- 
A H. Here the change in energy of a system is equal to the net heat 
transfer. 

We still lack a description of the part of the total energy of a 
system that we have called heat (or better, "internal" energy). So far we 
have seen only that an increase in internal energy is sometimes associated 
with an increase in temperature. We also mentioned the long-held 
suspicion that internal energy involves the motion of the "smaD parts" of 
bodies. We will take up this problem in detail in Chapter 11. 



Thermodynamics is the study of the 
relation between heat and mechani- 
cal energy. 



SG 10.34-10.38 



62 Unit 3 Energy 



Q29 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 express conservation of energy 
Q30 Define A£, AW, and AH for a system. 

Q31 What two ways are there for changing the total energy of a 



system 



10.11 Faith in the conservation of energy 

For over a century, the law of conservation of energy has stood as 
one of the most fundamental laws of science. We 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. 

The principle of conservation of energy has been immensely 
successsful. It is so firmly believed that it seems almost impossible that 
any new discovery 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 consider 
seriously the possibility that energy is not conserved. We have already 
pointed out 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. His faith in energy conservation was justified. Other 
evidence showed that "internal energy" changed by just the right amount 
SG 10.39, 10.40 ^^ explain observed changes in external energy. 

Another recent example is the "invention" of the neutrino by the 
physicist Wolfgang Pauli in 1933. Experiments had suggested that energy 
disappeared in certain nuclear reactions. But Paub 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 twenty 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 we 
believe it will always be justified. We believe that any apparent exceptions 
to the law will sooner or later be understood in a way which does not 



Section 10.11 Vnit i 63 

require us to give up the law. At most, they may lead us to discover new 
forms of energy making the law even more general and powerful. 

The French mathematician and philosopher Henri Poincare 
expressed this idea back in 1903 in his book Science and Hypothesis: 

. . . the principle of conservation of energy signifies simply 
that there is something which remains 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 discovery of conservation laws was one 
of the most important achievements of science. They 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. 

The list of known conservation laws has grown in recent years. The 
area of fundamental (or "elementar>'") particles 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 conservation laws as it now stands. One cannot 
say that the list is complete or eternal. But it does include the 
conservation laws that make up the working tool-kit of physicists today. 
Those which are starred are discussed in the basic text portions of this 
course. The others are treated in supplemental (optional) units, for 
example, the Supplemental Unit entided Elementary Particles. 

1. Linear momentum* 

2. Energy (including mass)* 

3. Angular momentum (including spin) 

4. Charge' 

5. Electron-family number 

6. Muon-family number 

7. Bary on -family number 

8. Strangeness number 

9. Isotopic spin 

Numbers 5 through 9 result from work in nuclear physics, high 
energy physics, or elementary or fundamental particle physics. If this 
aspect interests you, you will find the essay "Conservation Laws" (in 
the Reader entitled The Nucleus) worth reading at this stage. The first 
seven of the laws in the above listing are discussed in this selection. 



10.1 The Project Physics materials particularly 
appropriate for Chapter 10 include: 
Experiments 

Conservation of Energy 

Measuring the Speed of a Bullet 

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 Rifie Bullet- 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 
Reader Articles 

The Steam Engine Comes of Age 

The Great Conservation Principles 
Transparencies 

Slow Collisions 

The Watt Engine 

10.2 A man carries a heavy load across the level 
floor of a building. Draw an aiTow to represent 
the force he applies to the load, and one to rep- 
resent the direction of his motion. By the definition 
of work given, how much work does he do on the 
load? Do you feel uncomfortable about this 
result? Why? 

10.3 The speed of an object is always relative — 
that is. it will be different when measured from 
different reference frames. Since kinetic energy 
depends on speed, it too is only a relative quantity. 
If you are interested in the idea of the relativity 
of kinetic energy, consider this problem: An object 
of mass m is accelerated uniformly by a force F 
through a distance d. changing its speed from 

f, to V.,. The work done. Fd. is equal to the change 
in kinetic energy jmv.r-^mv,~. (For simplicity, 
assume the case of motion in only one direction 
along a straight line.) Now: describe this event 
from a reference frame which 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 uni- 
formly relative to one another)? 

i. the quantity -rmv- 
ii. the quantity Fd 
iii. the relationship Fd = M^rnv'-) 
(g) Explain why it is misleading to consider 
kinetic energy as something a body has. 
instead of only a quantity calculated from 
measurements. 

10.4 An electron of mass about 9.1 x lO^" 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 one joule? 

10.5 Estimate the kinetic energy of each of the 
following: (a) a pitched baseball (b) a jet plane 
(c) a sprinter in a 100-yard dash (d) the earth in 
its motion around the sun. 

10.6 A 200-kilogram iceboat is supported by a 
smooth surface of a frozen lake. The wind exerts 
on the boat a constant force of 400 newtons 
while the boat moves 900 meters. Assume that 
frictional forces are negligible, and that the boat 
starts from rest. Find the speed attained at the 
end of a 900 meter run by each of the following 
methods: 

(a) Use Newton's second law to find the ac- 
celeration of the boat. How long does it take 
to move 900 meters? How fast will 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). 

10.7 A 2-gram bullet is shot into a tree stump. 
It enters at a speed of 300 m sec and comes to 
rest after having penetrated 5 cm in a straight 
line. 

(a) What was the change in the bullets kinetic 
energy? 

(b) How much work did the tree do on the 
bullet? 

(c) What was the average force during impact? 

10.8 Refer back to SG 9.20. How much work 
does the golf club do on the golf ball? How much 
work does the golf ball do on the golf club? 

10.9 A penny has a mass of about 3.0 grams 
and is about 1.5 millimeters thick. You have 50 
pennies which you pile one abo\ e the other. 

(a) How much more gra\ itational potential 
energy has the top penny than the bottom 
one? 

(b) How much more ha\e all 50 pennies to- 
gether than the bottom one alone? 

10.10 (a) How high can you raise a book weigh- 

ing 5 newtons if you have available one 
joule of energy? 



64 Unit 3 



■!*^r:- ?5«F^- i^ggsiaasi^ 



(b) How many joules of energy are needed 
just to lift a 727 jet airliner weighing 
7 X 10^ newtons (fully loaded) to its 
cruising altitude of 10,000 meters? 

10.11 As a home experiment, hang weights on a 
rubber band and measure its elongation. Plot 
the force vs. stretch on graph paper. How could 
you measure the stored energy? 

10.12 For length, time and mass there are 
standards (for example, a standard meter). But 
energy is a "derived quantity," for which no 
standards need be kept. Nevertheless, assume 
someone asks you to supply him one joule of 
energy. Describe in as much detail as you can 
how you would do it. 

10.13 (a) Estimate how long it would take for 

the earth to fall up 1 meter 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 
meter from rest. 

10.14 The photograph below shows a massive 
lead wrecking ball being used to demolish a wall. 
Discuss the transformations of energy involved. 




10.15 This discussion will show that the PE of 
an object is relative to the frame of reference in 
which it is measured. The boulder in the photo- 
graph on page 37 was not lifted to its perch- 
rather the rest of the land has eroded away, 
leaving it where it may have been almost since 
the formation of the earth. Consider the question 
"What is the gravitational potential energy of the 
system boulder + earth?" You can easily calculate 
what the change in potential energy would be if 
the rock fell — it would be the product of the 
rock's weight and the distance it fell. But would 
that be the actual value of the gravitational energy 
that had been stored in the boulder-earth system? 
Imagine that there happened 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. Ap- 
parently 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 
system boulder + earth could have? 

(b) Is the system earth + boulder really isolated? 

(c) Is there a true absolute bottom of gravita- 
tional potential energy for any system that 
includes the boulder and the earth? 

These questions suggest that potential energy, 
like kinetic energy, is a relative quantity. 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 will 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? 

10.16 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. 

10.17 Discuss the conversion between kinetic and 
potential forms of energy in the system of a comet 
orbiting around the sun. 

10.18 Sketch an addition to one of the steam 
engine diagrams of a mechanical linkage that 
would open and close the valves automatically. 




Unit 



65 



10.19 Show that if a constant propelling force F 
keeps a vehicle moving at a constant speed v 
(against the friction of the surrounding) the 
power required is equal to Fv. 

10.20 The Queen Mary, one of Britain's largest 
steamships, has been retired to a marine museum 
on our west coast after completing 1,000 crossings 
of the Atlantic. Her mass is 81,000 tons (75 
million kilograms) and her maximum engine 
power of 234,000 horsepower (174 million watts) 
allows her to reach a maximum speed of 30.63 
knots (16 meters per second). 

(a) What is her kinetic energy at full speed? 

(b) Assume that at maximum speed all the 
power output of her engines goes into over- 
coming water drag. If the engines are 
suddenly stopped, how far will the ship 
coast before stopping? (Assume water drag 
is constant.) 

(c) What constant force would be required to 
bring her to a stop from full speed within 
1 nautical mile (2000 meters)? 

(d) The assumptions made in (b) are not valid 
for the following reasons: 

1. Only about 60% of the power deUvered to 
the propellor 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. 

3. If the propellers are not free-wheeling, 
they add an increased drag. 

Which of the above factors tend to increase, 
which to decrease the coasting distance? 

(e) Explain why tugboats are important for 
docking big ships. 

10.21 Devise an experiment to measure the 
power output of 

(a) a man riding a bicycle 

(b) a motorcycle 

(c) an electric motor. 

10.22 Refer to the table of "Typical Power 
Ratings" on p. 46. 

(a) What advantages would Newcomen's engine 
have over a "turret windmill"? 

(b) What advantage would you expect Watt's 
engine (1778) to have over Smeaton's 
engine (1772)? 

10.23 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 
can lift a load of one million pounds, using one 
bushel of coal as fuel. For example, Newcomen's 
engine had a duty of 4.3: it could perform 4.3 
million foot-pounds of work by burning a bushel 
of coal. Which do you think would have been 
more important to the engineers building steam 
engines -increasing the horsepower or increasing 
the duty? 

10.24 A modern term that is related to the "duty" 
of an engine is efficiency. The efficiency of an 



engine (or any device that transforms energy 
from one form to another) is defined as the 
percentage of the energy input that appears as 
useful output. 

(a) Why would it have been impossible to find 

a value for the efficiency of an engine before 
Joule? 

(b) The efficiency of "internal combustion" 
engines is seldom greater than 10%. For 
example, only about 10% of the chemical 
energy released in burning gasoline in an 
automobile engine goes into moving the 
automobile. What becomes of the other 90%? 

10.25 Engine A operates at a greater power than 
engine B does, but its efficiency is less. This 
means that engine A does (a) more work with 
the same amount of fuel, but more slowly (b) less 
work with the same amount of fuel, but more 
quickly (c) more work with the same amount of 
fuel and does it faster (d) less work with the same 
amount of fuel and does it more slowly. 

10.26 A table of rates for truck transportation is 
given below. How does the charge depend on the 
amount of work done? 

Truck Transportation (1965) 



Weight 


Moving rates (including pickup and 




del 


very) from Bosto 


n to: 




Chicago 


Denver 


Los Angeles 




(967 miles) 


(1969 miles) 


(2994 miles) 


100 lbs 


$ 18.40 


$ 24.00 


$ 27.25 


500 


92.00 


120.00 


136.25 


1000 


128.50 


185.50 


220.50 


2000 


225.00 


336.00 


406.00 


4000 


383.00 


606.00 


748.00 


6000 


576.00 


909.00 


1122.00 



10.27 Consider the following hypothetical values 
for a paddle-wheel experiment like Joule's: a 1- 
kilogram weight descends through a distance of 
1 meter, turning a paddle-wheel immersed in 5 
kilograms of water. 

(a) About how many times must the weight be 
allowed to fall in order that the temperature 
of the water will be increased by t Celsius 
degree? 

(b) List ways you could modify the experiment 
so that the same temperature rise would be 
produced with fewer falls of the weight? 
(There are at least three possible ways.) 

10.28 While traveling in Switzerland, Joule 
attempted to measure the difference in tempera- 
ture 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 meters high, such as Niagra Falls. Does 
it matter how much water goes down the fall? 

10.29 Find the power equivalent in watts or in 



66 Unit 3 



STUDY GUIDE 



horsepower of one of the activities Usted in the 
table on p. 53. 

10.30 About how many kilograms of hamburgers 
would you have to eat to supply the energy for a 
half-hour of digging? Assume that your body is 
20% efficient. 

10.31 When a person's food intake supplies less 
energy than he uses, he starts "burning" his own 
stored fat for energy. The oxidation of a pound 
of animal fat provides about 4,300 kilocalories of 
energy. Suppose that on your present diet of 4,000 
kilocalories a day you neither gain nor lose 
weight. If you cut your diet to 3,000 kilocalories 
and maintain your present physical activity, how 
long would it take to reduce your mass by 5 
pounds? 

10.32 jjj order to engage in normal light work, a 
person in India has been found to need on the 
average about 1,950 kilocalories of food energy a 
day, whereas an average West European needs 
about 3.000 kilocalories a day. Explain how each 
of the following statements makes the difference 
in energy need understandable. 

(a) The average adult Indian weighs about 110 
pounds; the average adult West European 
weighs about 150 pounds. 

(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. 

10.33 jsjq other concept in physics has the 
economic significance that "energy" does. Discuss 
the statement: "We could express energy in dollars 
just as weU as in joules or calories." 

'^^■^^ Show how the conservation laws for 
energy and for momentum can be applied to a 
rocket during the period of its lift off. 

10.35 Discuss the following statement: "During 
a typical trip, all the chemical energy of the 
gasoline used in an automobile is used to heat 
up the car, the road and the air." 

10.36 Show how all the equations we have given 
in Chapter 10 to express conservation of energy are 
special cases of the general statement AE = AW 

+ AH. Hint: let one or more of the terms equal 
zero.) 



10.37 (a) Describe the procedure by which a 

space capsule can be changed from a 
high circular 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 gravitational 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 energy. 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? 

10.38 Any of the terms in the equation A£ = AH 
+ AW can have negative values. 

(a) What would be true for a system for which 
i. A£ is negative? 

ii. AH is negative? 
iii. AW is negative? 

(b) Which terms would be negative for the 
following systems? 

i. a man digging a ditch 

ii. a car battery while starting a car 

iii. an electric light bulb just after it is 

turned on 
iv. an electric light bulb an hour after it is 

turned on 
V. a running refrigerator 
vi. an exploding firecracker 

10.39 In each of the following, trace the chain 
of energy transformations from the 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, climbs a hill at constant 
speed, and comes to stop at a traffic 
light. 

(c) A windmill pumps water out of a flooded 
field. 

10.40 Show how the law of conservation of 
energy applies to the motion of each of the 
situations listed in SG 9.39 and 9.40, p. 27. 



Unit .1 67 



11.1 An overview of the chapter 69 

11.2 A model for the gaseous state 71 

11.3 The speeds of molecules 74 

11.4 The sizes of molecules 76 

11.5 Predicting the behavior of gases from the kinetic theory 79 

11.6 The second law of thermodynamics and the 

dissipation of energy 85 

11.7 Maxwell's demon and the statistical view of the second 

law of thermodynamics 88 

11.8 Time's arrow and the recurrence paradox 91 



Bubbles of gas from high-pressure tanks expand as 
the pressure decreases on the way to the surface. 




CHAPTER ELEVEN 



The Kinetic Tlieory of Gases 



11.1 An overview of the chapter 



During the 1840's, many scientists recognized that heat is not a 
substance, but a form of energy which can be converted into other forms. 
Two of these scientists, James Prescott Joule and Rudolf Clausius, went a 
step further. They based this advance on the fact that 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 we will see that this 
idea is largely correct. It 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 century. If such small bits of matter really 
existed, they would be too small to observe even in the most powerful 
microscopes. Since scientists could not observe molecules, they could not 
check directly the hypothesis that heat is molecular kinetic energy. 
Instead, they had to derive from this hypothesis predictions about the 
behavior of measurably large samples of matter. Then they could test 
these predictions by experiment. For reasons which we will explain, it is 
easiest to test such hypotheses by observing the properties 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 the last major triumph of Newtonian mechanics. The 
method involved using a simple theoretical model of a gas. Newton's laws 
of motion were applied to the gas molecules assumed in this model as if 
they were tiny billiard balls. This method produced equations that related 
the easily observable properties 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: 

Unit 3 69 



SG11.1 



Molecules are the smallest pieces 
of a substance -they may be 
combinations of atoms cf simpler 
substances. 



70 Unit 3 The Kinetic Theory of Gases 

(1) explained rules that had been found previously by trlal-and-error 
methods. (An 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 
temperature, 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. 

But 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. But 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 theor>' 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 Newtonian theory, however, the reverse of 
any event is just as 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 discussing this 
problem from the viewpoint of kinetic theory, we will see how the concept 
of "randomness" entered physics. 

Modem physicists do not take too seriously the "billiard ball" idea of 
gas molecules — nor did most nineteenth centur\' physicists. All models 
oversimplify the actual facts. Therefore, the simple assumptions of a model 
often need adjustment in order to get a theory that agrees well with 
experimental data. Nevertheless the kinetic theor\' is still ven,' useful. 
Physicists are fond of it, and often present it as an example of how a 
physical theory should be developed. Section 11.5 gives one of the 
mathematical derivations from the model used in kinetic theory. This 
derivation is not given to be memorized in detail; it simply illustrates 
mathematical reasoning based on models. Physicists have found this 
method very useful in understanding many natural phenomena. 

Q1 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 



Section 11.2 



Unit 3 71 



Q2 True or false: In the kinetic theor\' of gases, as developed in the 
nineteenth centur>', it was assumed that Newton's laws of motion apply to 
the motion and collisions of molecules. 

Q3 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. 



11.2 A model for the gaseous state 

What are the differences between a gas and a hquid or solid? We 
know by observation that hquids 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 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 
between 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 a very simple model. The molecules are 
considered to behave like miniature billiard balls — that is, tiny spheres or 
clumps of spheres which 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 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. But now we are 
discussing a theoretical model of a gas. This model exists only in our 
imagination. Like the points, lines, triangles, and spheres studied in 
geometry, this theoretical model can be discussed mathematically. The 
results of such a discussion may help us to understand the real world of 
experience. 

In order to emphasize that our model is a theoretical one, we will use 
the word "particle" instead of "atom" or "molecule." There is now no 
doubt that atoms and molecules exist and have their own definite 
properties. The particles in the kinetic theory model, on the other hand, 
are idealized and imaginary. We imagine such objects as perfectly elastic 
spheres, whose supposed properties are hopefully similar to those of actual 
atoms and molecules. 

Our model represents the gas as consisting of a large number of very 




Balloon for carrying apparatus used 
for weather forecasting. 

Gases can be confined without a 
container. A star, for example, is a 
mass of gas confined by gravita- 
tional force. Another example is the 
earths atmosphere. 




Liquid 



^t 



SM 



\ ..^ 



^ ^ % 



A very simplified "model" of the three 
states of matter: 

(From General Chemistry, second edition, 
by Linus Pauling, W. H. Freeman and Com- 
pany, © 1953.) 



72 Unit 3 



The Kinetic Theory of Gases 



The word gas" was originally 
derived from the Greek word 
chaos: it was first used by the 
Belgian chemist Jan Baptista van 
Helmont (1580-1644). 



On the opposite page you will find 
a more detailed discussion of the 
idea of random fluctuations. 

SG 11.2 



The idea of disorder is elaborated in 
the Reader 3 articles "The Law of 
Disorder," "The Law," "The Arrow 
of Time," and "Randomness in the 
Twentieth Century." 



small particles in rapid, disordered motion. Let us define some of these 
terms. "A large number" means something hke a billion billion (lO^^) or 
more particles in a sample as small as a bubble in a soft drink. "Very 
small" means a diameter about a hundred-millionth of a centimeter (10"^° 
meter). "Rapid motion" means an average speed of a few hundred miles 
per hour. What is meant by "disordered" motion? Nineteenth-century 
kinetic theorists assumed that each individual molecule moved in a 
definite way, determined 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 collision with another particle. Therefore, we cannot 
make a definite prediction of the motion of any one individual particle. 
Instead, we must be content with describing the average behavior of large 
collections of particles. We still believe that from moment to moment each 
individual molecule behaves according to the laws of motion. But it turns 
out to be easier to describe the average behavior if we assume complete 
ignorance about any individual motions. To see why this is so, consider 
the results of flipping a large number of coins all at once. It would be very- 
hard to predict how a single coin would behave. But if you assume they 
behave randomly, you 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 bet 
that about as many are moving in one direction as in another. Further, 
the particles are equally likely to be found in any cubic centimeter of 
space inside the container. This is true no matter where such a region is 
located, and even though we do not know where a given particle is at any 
given time. "Disordered," then, means that velocities and positions are 
distributed randomly. Each molecule is just as hkely to be moving to the 
right as to the left (or in any other direction). And it is just as bkely to be 
near the center as near the edge (or any other position). 

In summary, we are going to discuss the properties of a model of a 
gas. The model is imagined to consist of a large number of very small 
particles in rapid, disordered motion. The particles move freely most of the 
tinie, exerting forces on one another only when they collide. The model is 
designed to represent the structure of real gases in many ways. However, 
it is simplified in order to make calculations manageable. By comparing 
the results of these calculations with the observed properties of gases, we 
can estimate the speeds and sizes of molecules. 



Q4 In the kinetic theory, particles are thought to exert 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 

Q5 Why was the kinetic theory first applied to gases rather than to 
liquids or solids? 



Averages and Fluctuations 

Molecules are too small, too numerous, and 
too fast for us to measure the speed of any one 
molecule, or 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 enables us to predict quite precisely the 
average speed of the molecules in a sample of 
gas, or the average kinetic energy, or the average 
distance the molecules move between collisions. 

Any measurement made on a sample of gas 
reflects 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 statistics to estimate just 
how different the averages are likely to be. We will 
call on two basic rules of statistics for random 
samples: 

1 . Large variations away from the average are 
less likely than small variations. (For example, 
if you toss 1 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 percent of the time and "tails" 50 
percent of the time. For small sets of tosses there 
will be many "fluctuations" (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 percent 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 180 tosses. 
Large fluctuations from 50% are less common still 
than for the smaller sets. 

Statistical theory shows that the average 
fluctuation 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 
quantities, 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. 



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74 Unit 3 The Kinetic Theory of Gases 

11.3 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 
SG 11.3 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) are immediately successful. 

In 1738, the Swiss 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. BemouUi assumed that 
Pressure is defined as the perpen- the pressure of a gas is simply a result of the impacts of individual 

dicular force on a surface divided molecules striking the wall of the container. If the density of the gas were 

by the area of the surface. twice as great there would be twice as many molecules per cubic 

centimeter. Thus, BemouUi 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 modern kinetic 
theory 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 (1687). Newton 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. (As it turned out, it was not.) 

The kinetic theory of gases was proposed again in 1820 by an English 
physicist, John Herapath. Herapath rediscovered Bernoulli's results on the 
relations between pressure and density of a gas and the speeds of the 
particles. But Herapath's work also was ignored by most other scientists. 
His earlier writings on the kinetic theory had been rejected for pubhcation 
by the Royal Society of London. Despite a long and bitter battle Herapath 
did not succeed in getting recognition for his theory. 

James Prescott Joule, however, did see the value of Herapath's work. 
In 1848 he read a paper to the Manchester Literary and Philosophical 
Society in which he tried to revive the kinetic theory. Joule showed how 
the speed of a hydrogen molecule could be computed (as Herapath had 
done). He reported a value of 2,000 meters per second 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 "hterary 
and philosophical society" for scientifically important papers. But evidence 
for the equivalence of heat and mechanical energy continued to mount. 
Several other physicists independently worked out the consequences of the 
hypothesis that heat energy in a gas is the kinetic energy' of molecules. 
Rudolf Clausius in Germany pubhshed a paper in 1856 on "The Nature of 
the Motion we call Heat." This paper established the basic principles of 
kinetic theory essentially in the form we accept today. Soon afterward, 
James Clerk Maxwell in Britain and Ludwig Bolzmann in Austria set forth 
the full mathematical details of the theory. 



Section 11.3 



Unit 3 75 



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. But some 
have much lower speeds and some much higher speeds. 

A simple example will help you to understand Maxwell's distribution 
of molecular speeds. Suppose a marksman shoots a gun at a practice 
target many times. Some bullets will probably hit the bullseye. Others will 
miss by smaller or larger amounts, as shown in (a) in the sketch below. 
We count the number of bullets scattered at various distances to the left 
and right of the bullseye in (b). Then we can make a graph showing the 
number of bullets at these distances as shown in (c). 



TARGET PRACTICE EXPERIMENT 
(a) Scatter of holes in target; (b) target 
marked off in distance intervals left 
and right of center; (c) graph of num- 
ber of holes per strip to left and right 
of center; (d) For a very large number 
of bullets and narrov^/ strips, the en- 
velope of the graph often closely 
approximates the mathematical curve 
called the "normal distribution" curve. 








This graph showing the distribution of hits illustrates a general 
principle of statistics, namely, if any quantity varies randomly about an 
average value, the graph showing the distribution of variations will 
resemble the one shown in (d) above in the margin. There will be a peak 
at the average value and a smooth decline on either side. A similar "bell- 
shaped curve," as it is called, describes the distribution of many kinds of 
physical measurements. The normal distribution law applies even to large 
groups of people. For example, consider the distribution of heights in a 
large crowd. Such a distribution results from the combined effect of a 
great many independent factors. A person's height, for example, depends 
upon many independent genes as well as environmental factors. Thus the 
distribution of heights will closely follow a normal distribution. The 
velocity of a gas molecule is determined by a very large number of 
independent collisions. So the distribution of velocities is also smoothly 
"bell-shaped." 

Maxwell's distribution law for molecular velocities in a gas is shown 
in the margin in graphical form for three different temperatures. The 
curve is not symmetrical since no molecule can have less than zero speed, 
but some have very large speeds. 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 




Maxwell's distribution of speeds in 
gases at different temperatures. 



76 Unit 3 The Kinetic Theory of Gases 

shifts to higher speeds. Then 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. But not until the 1920's was a direct 
experimental check possible. Otto Stern in Germany, 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 the next 
page.) Stern, Zartmann, and others found that molecular speeds are 
indeed distributed according to Maxwell's law. Virtually all of the 
individual molecules in a gas change speed as they collide again and 
again. Yet if a confined sample of gas is isolated, the distribution of 
speeds remains very much the same. For the tremendous number of 

SG 11.4 molecules in almost any sample of gas, the average speed has an 

SG 11.5 extremely stable value. 

Q6 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 exerted by molecules on one another 

Q7 The idea of speed distribution for gas molecules means that 

(a) each molecule always has the same speed 

(b) there is a wide range at speeds of gas molecules 

(c) molecules are moving fastest near the center of the gas 



11.4 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 with 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. But according to our 
kinetic-theory calculations, each of the gas molecules should have crossed 
the room hundreds of times by then. Something must be wrong with our 
kinetic-theory 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 hke 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 



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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 which 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 
revolution. Meanwhile the molecules in the drum continue moving to the right 
some moving fast and some moving slowly. 

Fastened to the inside of the drum is a sensitive film which 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 molecules 
which 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 indicates 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. 







ySfTeJ)\ 



78 



Unit 3 



The Kinetic Theory of Gases 



X'. 



The larger the molecules are, the more 
likely they are to collide with each 
other. 

SG 11.6 



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 one 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 could modify his model to 
explain why gases mix so slowly. Further, he derived a precise 
quantitative relationship between the molecule's 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, one doesn't 
know whether to blame a flaw in the model or 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 possible to work backwards and find fairly 
reliable values for molecular sizes. Here we can only report the result of 
these calculations. Typically, the diameter of gas molecules came out to be 
of the order of lO"^** meters to 10~^ meters. This is not far from the 
modem values — an amazingly good result. After all, no one previously had 
known whether a molecule was 1,000 times smaller or bigger than that. 
In fact, as Lord Kelvin remarked: 

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 modem times, losing 
all patience with it, have dismissed it to the realms of 
metaphysics, and made it smaller than 'anything we can 
conceive.' 



SG 11.7 
SG 11.8 



Kelvin showed that other methods could also be used to 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%). 



Q8 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. 



Section 11.5 



Unit 3 79 



11.5 Predicting the behavior of gases from the kinetic theory 



One of the most easily measured characteristics of a confined gas is 
pressure. Our experience with baUoons 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 34 feet. This fact was 
well known. Such pumps were widely used to obtain drinking water from 
wells and to remove water from mines. We already have seen one 
important consequence of this limited ability of pumps to lift water out of 
deep mines. This need provided the initial stimulus for the development of 
steam engines. Another consequence was that physicists became curious 
about why the lift pump worked at all. Also, why should there be a limit 
to its ability to raise water? 

Air Pressure. The puzzle was solved as a result of experiments by 
Torricelli (a student of GaUleo), Guericke, Pascal, and Boyle. By 1660, it 
was fairly clear that the operation of 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 water 
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 34 feet above the pool — but no 
higher. Atmospheric pressure at sea level is not great enough to support a 
column of water any higher. Mercury is almost 14 times as dense as 
water. Thus, ordinary pressure on a pool of mercury can support a column 
only T4 as high, about 2j feet (0.76 meter). This is a more convenient 
height for laboratory experiments. Therefore, much of the seventeenth- 
century research on air pressure was done with a column of mercury, or 
mercury "barometer." The first of these was designed by Torricelli. 

The height of the mercury column which 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 
strange at first. To understand it, we must understand the difference 
between pressure and force. Pressure is defined as the magnitude of the 
force acting perpendicularly on a surface divided by the area of that 
surface: P = FJA. Thus a large force may produce only a small 
pressure if it is spread 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 pressure if it is concentrated on a 
small area. Women's spike heel shoes have ruined many a wooden floor or 
carpet. The pressure at the place where the heel touched the floor was 
greater than that under an elephant's foot. 

In 1661 two English scientists, Richard Towneley and Henry Power, 
discovered an important basic relation. They found that the pressure 
exerted by a gas is directly proportional to the density of that gas. Using 
P for pressure and D for density, we can write this relationship as P ^ D 
or P -kD where k is some constant. For example, if the density of a 
given quantity of air is doubled (say by compressing it), its pressure also 
doubles. Robert Boyle confirmed this relation by extensive experiments. It 
is an empirical rule, now generally known as Boyle's Law. But the law 



SG 11.9 
SG 11.10 



.76 m 



c-r ^ c-j 0.0 m 



Torricelli's barometer is a glass tube 
standing in a pool of mercury. The 
top most part of the tube is empty of 
air. The air pressure on the pool sup- 
ports the weight of the column of 
mercury in the tube up to a height of 
about 27 feet (0.76 meter). 



Ul 



NOTICE 
XADIES 

Spike heels 



THE WUSEL'U fiNCS THAT 5riK£ 
HEELS CAU5E UN!;E'"Aii;aFl[ TAM- 
AJE TO OC'i;. PLOOW. HELP L'5 
P«V£NT iT e\ USl^^ A TAIt OP 

HEEL Ci^vEtS- perosir 50'^ANr 

WHEN NOU LEA\E THE \>L'S£L'M 

qetlt^n thew anp sour 50* 
PEfOSiT »MLL BE CETL'^NEC. 



7/..'"'^ 



l|^t< 



SG 11.11 
SG 11.12 



80 Unit 3 



The Kinetic Theory of Gases 



holds true only under special conditions. 

The effect of temperature on gas pressure. Boyle recognized that if the 
temperature ofa gas changes during an experiment, the relation P = kD 
no longer applies. For example, the pressure exerted by a gas in a 
closed container increases if the gas is heated, even though its density stays 
constant. 

Many scientists throughout the eighteenth century investigated the 
expansion of gases by heat. The experimental results were not consistent 
enough to estabhsh a quantitative relation between density (or volume) and 
temperature. But 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°) 



On the Celsius scale, water freezes 
at 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 
Hotness and Temperature in the 
Handbook. 



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 0- 

We call this equation 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 when pressure is so high, or 
temperature so low, that the gas is nearly changing to a liquid. 

Why does the number 273 appear in the ideal gas law? Simply because 
we are measuring temperature on the Celsius scale. If we had chosen to use 
the Fahrenheit scale, the equation for the ideal gas law would be 



P = k'D(t + 460°) 



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. 



where t is the temperature measured on the Fahrenheit scale. In other 
words, the fact that the number is 273 or 460 has no great importance. It just 
depends on our choice ofa particular scale for measuring temperature. 
However, it is important to note what would happen if ( were decreased to 
-273°C or -460°F. Then the entire factor invohing temperature would be 
zero. And, according to the ideal gas law, the pressure of any gas would also 
fall to zero at this temperature. The chemical properties of the gas no longer 
makes sense. 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 



Section 11.5 



Unit 



81 



this temperature. However, a series of cooling operations has produced 
temperatures less than 0.0001 degree above this limit. 

In view of the unique meaning of this lowest temperature, Lord Kelvin 
proposed a new temperature scale. He called it the absolute temperature 
scale, and put its zero at -273°C. Sometimes it is called the Kelvin 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 form: 



P = kDT 



T Is the temperature in degrees Kelvin and k is the proportionality constant. 

The equation P = kDT summsLnzes experimental facts about gases. 
Now we can see whether the kinetic-theory 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 conservation of momentum will describe the results of the impact. 
The detailed reasoning for this procedure is worked out on pages 82 and 83. 
This is a beautifully simple application of Newtonian mechanics. The result 
is clear: applying Newtonian mechanics to the kinetic molecular model of 
gases leads to the conclusion that P = hDiv^)^^ where (i^^)av is the average of 
the squared speed of the molecules. 

So we have two expressions for the pressure of a gas. One summarizes 
the experimental facts, P = kDT. The other is derived by Newton's laws 
from a theoretical model, P = W(v^)ay. The theoretical expression will agree 
with the experimental expression only if kT = K^^)av This would mean 
that the temperature of a gas is proportional to (v^)av The mass m of each 
molecule is a constant, so we can also say that the temperature is 
proportional to hm(v^)ay. In equation form, T ^ 2"z(i'^)av You should recall 
that hm(v^) is our expression for kinetic energy. 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 parts." We were aware that the higher the temperature of a gas, the . 
more rapidly its molecules are moving. But the conclusion T « 2^(x^^)av is a 
precise quantitative relationship derived from the kinetic model and 
empirical laws. 

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. 



This "absolute zero" point on the 
temperature scale has been found 
lo be 273.16 Celsius (459.69 F). 



For our purposes it is sufficiently 
accurate to say the absolute tem- 
perature of any sample (symbolized 
by the letter T and measured in 
degrees Kelvin, or K) is equal to 
the Celsius temperature ( plus 273 : 

Tt' 273 
The boiling point of water, for 
example, is 373 K on the absolute 
scale. 



SG 11.13, 11.14 



'Celsii 



327 



100-- 

0-- 
-78-- 



-183 -- 

-196 -- 



-273 -L 



lead melta 



^Absolute 

(Kelvin^ 



--600 



water boils 



--373 



water Freezes - -2 73 

carbon dioxide -|-l9 5 
freezes fdry ice) 

absolabe zero -*- 



Comparison of the Celsius and ab- 
solute temperature scales. 



Deriving an Expression For Pressure 
From the Kinetic Theory 

We begin with the model of a gas described in 
Section 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 ignored. 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 argument will require that one of 
the walls be movable. So let us arrange for that 
wall to be movable, but to fit snugly into the box. 

In SG 9.24 we 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 our wall, very little kinetic 
energy is transferred. If the collision is elastic, the 
particle 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 
opposite 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 {P ^ ma), by Newton's second law. 
As shown in Section 9.4, the force can also be 
written as 



P = 



At 



where limv) 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 7 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 \smv^. 
Since the particle collides elastically with the wall, it 
rebounds with the same speed. Therefore, the 
momentum in the x direction after the collision is 
m{-v^) = -TJ^x- The change in the 




momentum of the particle as a result of this 
collision is 



final 


initial 


change in 


momentum 


momentum 


momentum 


i-mv,) 


(mv,) 


= {-2mv,) 



Note that all the vector quantities considered 
in this derivation have only two possible 
directions: to the right or to the left. We can 
therefore indicate direction by using a + or a - 



-mv. 



mV^ 



-2mv^ 



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, 
2Uv^. If 2L/v^ = the time between collisions, 




2 



then vJ2L = the number of collisions per 
second. Thus, the change in momentum per 
second is given by 



(change in 

momentum per 

second) 



(change in (number of 

momentum in x collisions = 
one collision) per second) 

{-2mv^) X {vJ2L) = 



The net force equals the rate of change of 
momentum. Thus, the average force acting on 
the molecule (due to the wall) is equal to 
-mv^^L; and by Newton'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 



p _ F _ F _ mvl _ mv\ 



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 (i/x^)av 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 





Iv 




'^z 


v/ 


/ 


X 


/ 

/ 





is, a molecule moving with speed v will have 
three components: v^, Vy, and v^. If the 
motion is random, then there is no preferred 
direction of motion for a large collection of 
molecules, and (i^x^)av = (^.v^)av = (i^z^)av It can 
be shown from a theorem in geometry that v"^ = 
^x^ + ^y^ + ^z^- These last two expressions can 
be combined to give 



or 



(rx^)av = MV^)av 



By substituting this expression for (v^x^)av in the 
pressure formula, we get 



P = 



Nm X 1/3(1/2)3 



V 

3 V ^^ )av 

Notice now that Nm is the total mass of the gas 
and therefore Nm/V is just the density D. So 



where V (here L^) is the volume of the cubical 
container. 

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 



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 v. 



84 Unit 3 



The Kinetic Theory of Gases 



SG 11.15 

Brownian motion was named after 
the English botanist. Robert Brown 
who in 1827 observed the phenome 
non 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. 




This phenomenon can be 
demonstrated by means of the 
expansion cloud chamber, cooling of 
CO2 fire extinguisher, etc. The "wall" 
is here the air mass being pushed 
away. 



Diesel engines have no spark plugs; 
ignition is produced by temperature 
rise during the high compression of 
the air-fuel vapor mixture. 



called Brownian, Movement . 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. Hence the impact of the invisible 
molecules makes the visible particle "dance" in the viewfield of the 
microscope. The hotter the gas, the more lively the motion, as the equation 
Ta: h'm(v^)av prcdicts. 

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 hvely, 
random motion. In the words of the twentieth-century' physicist Max Bom, 
we live in a "restless universe." 

But we need a more extensive argument in order to make confident 
quantitative predictions from kinetic theor)'. We know by experience that 
when a gas is compressed or expanded very slowly, its temperature changes 
hardly at all. Thus Boyle's simple law (P = kD) applies. But when a gas is 
compressed or condensed rapidly, the temperature does change. Then, only 
the more general gas law (P = kDT) applies. Can our model explain this? 

In the model used on the special pages, 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 
we suddenly reduce the outside force that holds one wall in place. 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. But 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 work must come 
from somewhere. But the only available source of energy here is the kinetic 
energy (hmv^) of the particles. In fact, we can show that molecules 
colliding perfecdy elastically with a receding wall rebound with slighdy less 
speed (see SG 11.16). Therefore the kinetic energy of the particles must 
decrease. But the relationship T »: 2»i(^^)av imphes that the temperature of 
the gas will then drop. And this is exactly what happens! 

If we increase the outside force on the wall instead of decreasing it. just 
the opposite happens. The gas is suddenly compressed as the wall moves 
inward, doing work on the particles and increasing their kinetic energy. As 
kmv^ goes up, we expect the temperature of the gas to rise — which is just 
what happens when we compress a gas quickly. 

The model also predicts that, for slow motion of the wall, Boyle's law 
applies. However, the gas must not be insulated from its surroundings. 
Suppose we keep the suiToundings of the gas at a constant temperature — for 
example, by immersing the gas container in a large water bath. Small 
changes in the temperature of the gas will then be cancelled by exchange of 
heat with the surroundings. Whenever the kinetic energy of the molecules 



Section 11.6 Un,t 3 85 

momentarily decreases (as during expansion), the temperature of the gas wilJ 

drop below that of its surroundings. Unless the walls of the container are 

heat insulators, heat will then flow into the gas until its temperature rises to 

that of the surroundings. Whenever the kinetic energy momentarily 

increases (as during compression), the temperature of the gas will rise above 

that of its surroundings. Heat will then flow out of the gas until its 

temperature falls to the temperature of the surroundings. This natural 

tendency of heat to flow from hot bodies to cold bodies explains why the 

average kinetic energy of the particles remains nearly constant when gas is 

slowly compressed or expanded. SG1117-1122 



Q9 The relationship between the density and pressure of a gas 
expressed by Boyle's law, P = kD, holds true 

(a) for any gas under any conditions 

(b) for some gases under any conditions 

(c) only if the temperature is kept constant 

(d) only if the density is constant 

Q1 If a piston is pushed rapidly into a container of gas, what will 
happen to the kinetic energy of the molecules of gas? What will happen to 
the temperature of the gas? 

Q1 1 Which of the following conclusions result only when the ideal gas 
law and the kinetic theory model are both considered to apply? 

(a) P is proportional to T. 

(b) P is proportional to (i'^)av. 

(c) (i^2)av is proportional to T. 



11.6 The second law of thermodynamics and the dissipation of energy 



We have seen that the kinetic-theory model can explain the way a gas 
behaves when it is compressed or expanded, warmed 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 to us from the sun 
through the vacuum of space. This is not explainable in terms of the 
thermal motion of particles. But in most cases the model worked 
splendidly, explaining the phenomena of heat in terms of the ordinary 



86 Unit 3 



The Kinetic Theory of Gases 



SG 11.23 



"Our life runs down in sending up 

the clock. 
The brook runs down in sending 

up our life. 
The sun runs down 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] 




Sadi Carnot (1796-1832) 

Modern steam engines have a 
theoretical limit of about SS^'c 
efficiency-but in practice they 
seldom have better than 20°:. 



motions of particles. This was indeed a triumph of Newtonian mechanics. 
It fulfilled much of the hope Newton had expressed in the Principia: that 
all phenomena of nature could be explained in terms of the motion of the 
small parts of matter. In the rest of this chapter we will touch briefly on 
the further development of thermodynamic theory. (Additional discussion 
appears in several articles in Reader 3.) 

The first additional concept arose out of a basic philosophical theme of 
the Newtonian cosmology: the idea that the world is like a machine whose 
parts never wear out, and which never runs down. This idea inspired the 
search for conservation laws applying to matter and motion. So far in this 
text, it might seem 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 estabhshed in 
chemistry. In physics, the laws of conservation of momentum and of 
energy had been equally well established. 

Yet these successful conservation laws could not banish the suspicion 
that somehow the world is running down, the parts of the machine are 
wearing out. Energy may be conserved in burning fuel, but it loses its 
usefulness as the heat goes off into the atmosphere. Mass may be 
conserved in scrambling an egg, but its organized structure is lost. In 
these transformations, something is conserved, but something is also lost. 
Some processes are irreversible — they will not run backwards. There is no 
way to z/rjscramble an egg, although such a change would not \1olate 
mass conservation. There is no way to draw smoke and hot fumes back 
onto a blackened stick, forming a new, unbumed match. 

The first attempts to find quantitative laws for such irreversible 
processes were stimulated by the development of steam engines. During 
the eighteenth and nineteenth centuries, engineers steadily increased the 
efficiency of steam engines. Recall that efficiency refers to the amount of 
mechanical work obtainable from a given amount of fuel energy. (See 
Section 10.6.) In 1824 a young French engineer, Sadi Carnot, published a 
short book entitled Reflections on the Motive Power of Fire. Carnot raised 
the question: Is there a maximum possible efficiency of an engine? 
Conservation of energy, of course, requires a limit of 100%, since energy 
output can never be greater than energy input. But, by analyzing the flow 
of heat in engines, Carnot proved that the maximum efficiency actually is 
always less than 100%. That is, the useful energy output can never even 
be as much as the input energy. There is a fixed bmit on the amount of 
mechanical energy obtainable from a given amount of heat by using an 
engine. This limit can never be exceeded regardless of what 
substance — steam, air, or anything else — is used in the engine. 

In addition to this limit on efficiency even for ideal engines, real 
engines operate at still lower efficiency in practice. For example, heat 
leaks from the hot parts of the engine to the cooler parts. Usually, this 
heat bypasses the part of the engine where it could be used to generate 
mechanical energy. 

Carnot's analysis of steam engines shows that there is an unavoidable 
waste of mechanical energy, even under ideal circumstances. The total 



Section 11.6 



Unit 3 87 



amount of energy in the high-temperature steam is conserved as it passes 
through the engine. But while part of it is transformed into useful 
mechanical energy', the rest is discharged in the exhaust. It then joins the 
relatively low temperature pool of the surrounding world. Carnot reasoned 
that there always must be some such "rejection" of heat from any kind of 
engine. This rejected heat goes off into the surroundings and becomes 
unavailable for useful work. 

These conclusions about heat engines became the basis for the 
Second Law of Thermodynamics . This law has been stated in various 
ways, all of which are roughly equivalent. It expresses the idea that it is 
impossible to convert a given amount of heat fully into work. 

Carnot's analysis implies more than this purely negative statement, 
however. In 1852, Lord Kelvin asserted that the second law of 
thermodynamics applies even more generally. There is, he said, a 
universal tendency in nature toward the "degradation" or "dissipation" of 
energy. Another way of stating this principle was suggested by Rudolph 
Clausius, in 1865. Clausius introduced a new concept, entropy (from the 
Greek word for transformation). In thermodynamics, entropy is defined 
quantitatively in terms of temperature and heat transfer. But here we will 
find it more useful to associate entropy with disorder. Increases in entropy 
occur with increasing disorder of motion in the parts of a system. 

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 (and hence "ordered"). 
The ball strikes the floor and bounces several times. During each bounce, 
the mechanical energy of the ball decreases and the ball warms 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 all the motion left. According to the entropy concept, 
all motion of whole bodies will run down like this. In other words, 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 we will not go into the mathematics here). The general 
version of the second law of thermodynamics, as stated by Clausius, is 
therefore quite simple: the entropy of an isolated system always tends to 
increase. 

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 exhausts 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. (We 
say "forbidden" in the sense that Nature does not show that such things 
happen. Hence, the second law, formulated by the human mind, describes 



The first law of thermodynamics, 
or the general law of conservation 
of energy, does not forbid the full 
conversion of heat into mechanical 
energy. The second law is an 
additional constraint on what 
can hapoen in nature 

SG 11.24-11.26 




88 Unit 3 



The Kinetic Theory of Gases 



Two illustrations from Flammarion's 
novel, Le Fin du Monde. 




"La miserable race humaine perira 
par le froid." 




'Ce sera la fin." 



SG 11.27 



well what Nature does or does not do.) 

We haven't pointed it out yet, but all familiar processes are to some 
degree irreversible. Thus, Lord Kelvin predicted that all bodies in the 
universe would eventually reach the same temperature by exchanging 
heat with each other. When 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 hfe on earth would cease, 
and the universe would be dead. 

This general "heat-death" idea, based on predictions from 
thermodynamics, aroused some popular interest at the end of the 
nineteenth century. It appeared in several books of that time, such as H. 
G. Wells' The Time Machine. The French astronomer Camille Flammarion 
wrote a book describing ways in which the world would end. The 
American historian Henry Adams had learned about thermodynamics 
through the writings of one of America's greatest scientists, J. Willard 
Gibbs. Adams attempted to extend the application of the second law from 
physics to human history in a series of essays entided The Degradation of 
the Democratic Dogma. 



Q12 The presumed "heat death of the universe" refers to a state 

(a) in which all mechanical energy has been transformed into heat 
energy 

(b) in which all heat energy has been transformed into other forms of 
energy 

(c) in which the temperature of the universe decreases to absolute 
zero 

(d) in which the supply of coal and oil has been used up. 

Q13 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. 



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



Is there any way of avoiding the "heat death?" Is irreversibility a basic 
law of physics, or only an approximation based on our limited experience 
of natural processes? 

The Austrian physicist Ludwig Boltzmann investigated the theory of 



Section 11.7 



Unit 



89 



irreversibility. He concluded that the tendency toward dissipation of energy 
is not an absolute law of physics that holds rigidly always. Rather, it is 
only a statistical law. Think of a can of air containing about lO^^ 
molecules. Boltzmann argued that, of aU conceivable arrangements of the 
gas molecules at a given instant, nearly all would be almost completely 
"disordered." Only a relatively few aiTangements would have most of the 
molecules moxing in the same direction. And even if a momentarily 
ordered arrangement of molecules occurred by chance, it would soon 
become less ordered by collisions, etc. Fluctuations from complete disorder 
will of course occur. But the greater the fluctuations, the less likely it is to 
occur. For collections of particles as large as 10^2, the chance of a 
fluctuation large enough to be measurable is vanishingly small. It is 
conceivable that a cold ketde 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 stor\'. 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 
movement" of microscopic particles. Fluctuations are an important aspect 
of the world of ver\' small particles. But they are virtually undetectable for 
any large collection of molecules familiar to us in the everyday world. 

Still, the second law is different in character from all the other 
fundamental laws of physics we have studied so far. The difference is that 
it deals with probabilities, not uncertainties. 

Maxwell proposed an interesting "thought experiment" to show how 
the second law of themodynamics could be violated or disobeyed. It 
involved an imaginary being who could observe individual molecules and 
sort them out in such a way that heat would flow from cold to hot. 
Suppose a container of gas is divided by a diaphragm into tvvo parts, A 
and B. Initially the gas in A is hotter than the gas in B. This means that 
the molecules in A have greater average KE and therefore greater average 
speeds than those in B. However, the speeds are distributed according to 




Consider also a pool table — the 
ordered motion of a cue ball moving 
into a stack of resting ones gets soon 

"randomized." 



To illustrate Boltzmann s argument, 
consider a pack of cards when it is 
shuffled. Most possible arrange- 
ments of the cards after shuffling 
are fairly disordered. If we start 
with an ordered arrangement — for 
example, the cards sorted by suit 
and rank — then shuffling would 
almost certainly lead to a more 
disordered arrangement. (Never- 
theless it does occasionally happen 
that a player is dealt 13 spades- 
even if no one has stacked the 
deck.) 



Drawing by Steinberg; © 1953, The New Yorker Magazine, Inc. 



SrBiy££/:^ 



90 Unit 3 



The Kinetic Theory of Gases 



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. 



A 


B 


•■"-'} 




-1 


V 


*" ^" N 



Initially the average KE of molecules 
is greater in A. 



Maxwell's distribution (Section 11.3). Therefore many molecules in A have 
speeds less than the average in A. 

"Now conceive a finite being," Maxwell suggested, "who knows the 
paths and velocities of all the molecules by simple inspection but who can 
do no work except open and close a hole in the diaphragm by means of a 
slide without mass." (If the shde or door has no mass, no work will be 
needed to move it.) This "finite being" observes the molecules in A. When 
he sees one coming whose speed is less than the average speed of the 
molecules in B, he opens the hole and lets it go into B. Now the average 
speed of the molecules of B is even lower than it was before. Next, the 
"being" watches for a molecule of B with a speed greater than the average 
speed in A. When he sees one, he opens the hole to let the molecule go 
into A. Now the average speed in A is even higher than it was before. 
Maxwell concludes: 




Only fast molecules are allowed to go 
from B to A. 



J 



r 



Only slow molecules are allowed to 
go from A to S. 



■H 


* 


"1 


\ 


■■■ 


■■" 




\^' 


/I 




{ 


^ 


• — •- 


■-^ 


/ 1 






/ 




\ 


V L 








/ 


*^ 


J 


^ 




< 



As this continues, the average KE 
in A increases and the average KE 
in B decreases. 



Then the number of molecules in A and B are the same as at 
first, but the energy in A is increased and that in B 
diminished, that is, the hot system has got hotter and the cold 
colder and yet no work has been done, only the intelligence of 
a very observant and neat-fingered being has been employed. 



In the same way, a group of such beings could keep watch over a swarm 
of randomly moving molecules. By allowing passage to those molecules 
moving only in some given direction, they could establish a region of 
orderly molecular motion. 

The imaginary "being who knows the paths and velocities of all the 
molecules" has come to be known as "Maxwell's demon." Maxwell's 
thought experiment shows that if there were any way to sort out 
individual molecules, the tendency to increasing entropy could be 
reversed. Some biologists have suggested that certain large molecules, 
such as enzymes, may function in just this way. They may influence the 
motions of smaller molecules, building up ordered molecular systems in 
living beings. This would help to explain why growing plants or animals 
do not tend toward higher disorder, while lifeless objects do. 

As interesting as this suggestion is, it shows a misunderstanding of 
the second law of thermodynamics. The second law doesn't say that the 
order can never (or is extremely unlikely to) increase in anij system. It 
makes that claim only for an isolated, or closed system. The order of part 
of a closed system may increase, but only if the order of the other parts 
decreases by as much or more. This point is made nicely in the following 
passage from a UNESCO document on environmental pollution. 



SG 11.28 



Some scientists used to feel that the occuiTence, reproduction, 
and growth of order in living systems presented an exception 
to the second law. This is no longer believed to be so. True, 
the bving system may increase in order, but only by diffusing 
energy to the surroundings and by converting complicated 
molecules (carbohydrates, fats) called food into simple 



Section 11.8 



Unit 3 91 



molecules (CO2, H2O). For example, to maintain a healthy 
human being at constant weight for one year requires the 
degradation of about 500 kilograms (one half ton) of food, and 
the diffusion into the surroundings (from the human and the 
food) of about 500,000 kilocalories (two million kilojoules) of 
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 impUcations of human ecolog>' and environmental pollution 
without understanding that first. 



Q14 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. 

Q15 True or false? 

(a) Maxwell's demon was able to get around the second law of 
thermodynamics. 

(b) Scientists have made a Maxwell demon. 

(c) Maxwell believed that his demon actually existed. 




James Clerk Maxwell (1831-1879) 



11.8 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 obsened. 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 
thermodynamics is an absolutely valid law of physics because it agrees 
with all the experimental data. However, (2) kinetic theory 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 setded. 

The critics of kinetic theory pointed to two apparent contradictions 
between kinetic theory and the principle of dissipation of energy. These 
were the reversibility paradox and the recurrence paradox. 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 Newton's laws of motion are 
reversible in time. For example, if we watch a motion picture of a 




The reversibility paradox: Can a model 
based on reversible events explain a 
world in which so many events are irre- 
versible'' (Also see photographs on next 
page.) 



92 Unit 3 



The Kinetic Theory of Gases 




bouncing ball, it is easy to tell whether the film is being run forward or 
backward. We 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 we could not tell whether the film was being run 
forward or backward. In the kinetic theory, molecules are assumed to make 
perfectly elastic collisions. Imagine that we 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 backward. Either way would show valid sequences of 
collisions. But — and this is the paradox— consider motion pictures of 
interactions involving large objects, containing many molecules. One can 
immediately tell the difference between forward (true) and backward 
(impossible) time direction. For example, a smashed lightbulb does not 
reassemble itself in real life, though a movie run backward can make it 
appear to do so. 

The kinetic theory is based on laws of motion which 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 — ^fi"om 
past to future. This contradicts the possibility implied in Newton's laws of 
motion: that it does not matter whether we think of time as flowing 
forward or backward. As Lord Kehin expressed the paradox. 

If . . . the motion of every particle of matter in the 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 the fall in drops reforming 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 had previously yielded. Boulders would 
recover fi-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 true, 
living creatures would grow backwards, with conscious 
knowledge of the future, but no memory of the past, and 
would become again unborn. But the real phenomena of life 
infinitely transcend human science; and speculation regarding 
consequences of their imagined reversal is utterly unprofitable. 



Kelvin himself, and later Boltzmann, used statistical probability to explain 
why we do not observe such lai-ge-scale reversals. 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 



Section 11.8 



Unit 3 93 



kind are possible in principle, but for all practical purposes they are out of 
the question. 

The answer to Zermelo's argiynent is that his first claim is incorrect. 
The second law of thermodynamics is not an absolute law, but a statistical 
law. It assigns a very low probability to ever detecting any overall increase 
in order, but does not declare it impossible. 

However, another small possibility allowed in kinetic theory leads to a 
situation that seems unavoidably to contradict the dissipation of energy. 
The recurrence paradox revived an idea that appeared frequendy in 
ancient philosophies and 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 
who have died may be born again and go through the same life. The 
German philosopher Friedrich Nietzsche was convinced of the truth of 
this idea. He even tried to prove it bv appealing to the principle of 
conservation of energy. He 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 
foUows 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. 



SG 11.29 



The Worlds 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, "Hellas' 
(1822)] 




Lord Kelvin (1824-1907) 



If the number of molecules is finite, there is only a finite number of 
possible arrangements of molecules. Hence, 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 history of the universe is ever 
exacdy repeated, then the entire history of the universe will be repeated. 
And, 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. At about the same time, in 1 889, the French mathematician 
Henri Poincare pubfished 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: 



SG 11.30-11.32 



A bounded world, governed only by the laws of mechanics, will 
always pass through a state very close to its initial state. On the 
other hand, according to accepted experimental laws (if one 



94 Unit 3 



The Kinetic Theory of Gases 




attributes absolute validity to them, and if one is willing to press 
their consequences to the extreme), the universe tends toward a 
certain final state, from which it will never depart. In this final 
state, from which will be a kind of death, all bodies wiD be at rest 
at the same temperature. 

. . . the kinetic theories can extricate themselves from this 
contradiction. The world, according to them, tends at first toward 
a state where it remains for a long time without apparent change; 
and this is consistent with 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 state 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 body to 
a warm one, it will not be necessary to have the acute vision, the 
intelligence, and the dexterity of Maxwell's demon; it will suffice 
to have a httle patience. 



SG 11.33 




Record of a particle in Brownian 
motion. Successive positions, record- 
ed every 20 seconds, are connected by 
straight lines. The actual paths be- 
tween recorded positions would be 
as erratic as the overall path. 



Poincare was willing to accept the possibility of a violation of the second law 
after a very long time. But others refused to admit even this possibility. In 
1896, Zermelo published a paper attacking not only the kinetic theory but 
the mechanistic world view 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 right and partly wrong. Mach and Zermelo were 
correct in believing that Newton's laws of mechanics cannot fully describe 
molecular and atomic processes. (We 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 bilhard balls. But Boltzmann was right in 
defending the usefulness of the molecular model. The kinetic theory is very 
nearly correct except for those properties of matter which involve the 
structure of molecules themselves. 

In 1905, Albert Einstein pointed out that the fluctuations predicted by 
kinetic theory could be used to calculate the rate of displacement for particles 
in "Brownian" movement. Precise quantitative studies of Brownian 
movement confirmed Einstein's theoretical calculations. This new success of 
kinetic theory — along with discoveries in radioactivity and atomic 
physics — persuaded 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 survived. In new form, these issues still 
interest physicists today. 

This chapter concludes the application of Newtonian mechanics to 
individual particles. The story was mainly one of triumphant success. Toward 
the end, however, we have hinted that, like all theories, Newtonian 
mechanics has serious Umitations. These will be explored later. 

The last chapter in this unit covers the successful use of Newtonian 
mechanics in the case of mechanical wave motion. This will complete the list 
of possibilities of particle motion. In Unit 1 we treated the motion of single 
particles or isolated objects. The motion of a system of objects bound by a 



1700 



•Q 



O 



GEORGE WASHINGTON 




MARIE ANTOINETTE 

NAPOLEON BONAPARTE 



FREDERICK THE GREAT 

iifiaMfaiiiiiiinw 

. ■ i i ■ I'iflBUi^ffLl^ga^Mii 



ABRAHAM LINCOLN 



BENJAMIN FRANKLIN 
LEONHARD EULER 

JOSEPH BLACK 



SADI CARNOT 



JOSEPH PRIESTLEY 

I 

ANTOINE LAVOISIER 




PIERRE LAPLACE 

I 

BENJAMIN THOMPSON, COUNT RUMFORD 

I 

JOHN DALTON 

I 

JEAN BAPTISTE FOURIER 

I 

KARL FRIEDRICH GAUSS 





ALEXANDER POPE 

SAMUEL JOHNSON 




I MADAME DESTAEL 

JOHANN WOLFGANG VON GOETHE 

WILLIAM BLAKE 



■ EMILY BRONTE 

EUZABETH BARRETT BROWNING 



WILLIAM HOGARTH 



JEAN-AUGUSTE INGRES 




JOHANN SEBASTIAN BACH 



GEORGE FREDERICK HANDEL 



96 



Unit 3 



The Kinetic Theory of Gases 



force of interaction, such as the Earth and Sun, was treated in Unit 2 and in 
Chapters 9 and 10 of this unit. In this chapter we discussed the motions of a 
system of a very large number of separate objects. Finally, in Chapter 12 we 
will study the action of many particles going back and forth together as a 
wave passes. 



SG 11.34 
SG 11.35 



Ql 6 The kinetic energy of a falling stone is transformed into heat 
when the stone strikes the ground. Obviously this is an irreversible process; 
we never see the heat transform into kinetic energy of the stone, so that the 
stone rises off the ground. We belie\e that the process is irreversible because 

(a) Newton's laws of motion prohibit the reversed process. 

(b) the probability of such a sudden ordering of molecular motion is 
extremely small. 

(c) the reversed process would not conserve energy. 

(d) the reverse process would violate the second law of thermodynamics. 



The ruins of a Greek temple at Delphi 
are as elegant a testimony to the con- 
tinual encroachment of disorder, as is 
the tree to the persistent development 
of islands of order by living organisms. 




■'■'•'' The Project Physics materials particularly 
appropriate for Chapter 11 include: 
Experiments 

Monte Carlo Experiment on Molecular 

Collisions 
Behavior of Gases 
Activities 
Drinking Duck 

Mechanical Equivalent of Heat 
A Diver in a Bottle 
Rockets 

How to Weigh a Car with a Tire Pressure Gauge 
Perpetual-Motion Machines 
Film Loop 

Reversibility of Time 
Reader Articles 
The Barometer Story 
The Great Molecular Theory of Gases 
Entropy and the Second Law of Thermo- 
dynamics 
The Law of Disorder 
The Law 

The Arrow of Time 
James Clerk Maxwell 
Randomness and the Twentieth Century 

11.2 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. 

11.3 xhe 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 all 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. 

1 1 -4 Consider these aspects of the curves showing 
Maxwell's distribution of molecular speeds: 

(a) All show a peak. 

(b) The peaks move toward higher speed at 
higher temperatures. 

(c) They are not symmetrical, like normal 
distribution curves. 

Explain these characteristics on the basis of the 
kinetic model. 

11.5 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. 

Il-S How did Clausius modify the simple kinetic 
model for a gas? What was he able to explain 
with this new model? 

H-^ Benjamin Franklin observed in 1765 that 
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 mole- 
cule. Suppose that one cubic centimeter of oil 
forms a continuous layer one molecule thick that 
just covers an area on water of 1000 square 
meters. 

(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)? 

Il-S 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 apples in a bin, then the total volume of a 
material is approximately equal to the volume of 
one molecule times the number of molecules in 
the material. 

(a) Roughly how many molecules are there in 1 
cubic centimeter of water? (For this approx- 
imation, you can take the volume of a 
molecule to be d^ if its diameter is d.) 

(b) The density of a gas (at 1 atmosphere 
pressure and 0°C) is about 1/1000 the 
density of a liquid. Roughly how many mole- 
cules are there in 1 cc. of gas? Does this 
estimate support the kinetic model of a gas 
as described on p. 82? 

11-9 How high could water be raised with a lift 
pump on the moon? 

11.10 At sea level, the atmospheric pressure of 
air ordinarily can balance a barometer column of 
mercury of height 0.76 meters or 10.5 meters 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? 

^1-^^ How many atmospheres of pressure do you 
exert on the ground when you stand on flat-heeled 



Unit 3 97 



shoes? skis? skates? (1 atmosphere is about 
15 lbs/in.) 

11.12 From the definition of density, D = MIV 
(where M is the mass of a sample and V is its 
volume), write an expression relating pressure P 
and volume V of a gas. 

11.13 Show how all the proportionalities describ- 
ing gas behavior on p. 79 are included in the 
ideal gas law: P = kD (t + 273°) 

11.14 The following information appeared in a 
pamphlet published by an oil company: 

HOW'S YOUR TIRE PRESSURE? 
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 one 
pound for every 10-degree dip in outside air. If 
your tires register 24 pounds pressure on an 
80-degree day, for example, they'll have only 
19 pounds pressure when the outside air 
plunges to 30° Fahrenheit. 

If you keep your car in a heated garage at 
60°, and drive out into a 20 degrees-below-zero 
morning, your tire pressure drops from 24 
pounds to 18 pounds. 

Are these statements consistent with the ideal 
gas law? (Note: The pressure registered on a 
tire gauge is the pressure above normal atmos- 
pheric pressure of about 15 pounds/sq. in.) 

11-15 Distinguish between two uses of the word 
"model" in science. 

11.16 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 still rebound from a moving wall without 
changing speed as seen in the frame of reference 
of the moving piston. Show that the rebound 
speed as measured in the laboratory would be 
less from a retreating wall (as is claimed at the 
bottom of p. 84). 

(Hint: First write an expression relating the 
particle's speed relative-to-the-wall to its speed 
relative-to-the-laboratory.) 

11.1 / What would you expect to happen to the 
temperature of a gas that was released from a 



■ . 


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^m •D'Notri run. iNciM(i*ri 
^■^^ 0» «tO«l . ,vi IJOI> c.-.^.. 

^^^ "' t. h..^ 

^^^ "2 »'«• •«. »»ii<liiii m inn. .- 


1 



container in empty space (that is, with nothing 
to push back)? 

1118 List some of the directly observable 
properties of gases. 

11.19 What aspects of the behavior of gases can 
the kinetic molecular theory be used to explain 
successfully? 

11.20 Many products are now sold in spray cans. 
Explain in terms of the kinetic theory of gases 
why it is dangerous to expose the cans to high 
temperatures. 

11.21 When a gas in an enclosure is compressed 
by pushing in a piston, its temperature increases. 



I 




Explain this fact in two ways: 

(a) by using the first law of thermodynamics. 

(b) by using the kinetic theor>' of gases. 
The compressed air eventually cools down to 

the same temperature as the surroundings. 
Describe this heat transfer in terms of molecular 
collisions. 

11.22 From the point of view of the kinetic 
theory, how can one explain (a) that a hot gas 
would not cool itself down while in a perfectly 
insulating 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.) 



98 



lilt 3 



11.23 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 to- 
gether those whose sudden encounter normally 
forms the starting-point of substantial fabrics — 
earth and sea and sky and the races of living 
creatures. 

11.24 Clausius' statement of the second law of 
thermodynamics is: "Heat will not of its own 
accord pass from a cooler to a hotter body." Give 
examples of the operation of this law. Describe 
how a refrigerator can operate, and show that it 
does not contradict the Clausius statement. 

11.25 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 sea 
water and exhausted blocks of ice, using the heat 
from the water to run the ship.) 

11.26 Imagine a room that is perfectly insulated 
so that no heat can enter or leave. In the room is 
a refrigerator that is plugged into an electric 
outlet in the wall. If the door of the refrigerator 
is left open, what happens to the temperature of 
the room? 



11.27 Since there is a tendency for heat to flow 
from hot to cold, will the universe eventually 
reach absolute zero? 

11.28 Does Maxwell's demon get around the 
second law of thermodynamics? List the assump- 
tions in Maxwell's argument. Which of them do 
you believe are likely to be true? 

11.29 Since all the evidence is that molecular 
motions are random, one might expect that any 
given arrangement of molecules will recur if one 
just waited long enough. Explain how a paradox 
arises when this prediction is compared with the 
second law of thermodynamics. 

11.30 (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-reversibility of Newton's laws 
of motion is compared with the second 
law of thermodynamics. 

11.31 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? 

11.32 List the assumptions in the "recurrence" 
theory. Which of them do you believe to be true? 

1''.3? Some philosophical and religious systems 
of the Far East and the Middle East include the 
ideas of the eternal return. If you have read about 
some of these philosophies, discuss what analogies 
exist with some of the ideas in the last part of 
this chapter. Is it appropriate to take the existence 
of such analogies to mean there is some direct 
connection between these philosophical and 
physical ideas? 

11.34 Where did Newtonian mechanics run into 
difficulties in explaining the behavior of mole- 
cules? 

11.35 What are some advantages and dis- 
advantages of theoretical models? 



Unit 3 99 



12.1 Introduction 

12.2 Properties of waves 

12.3 Wave propagation 

12.4 Periodic waves 

12.5 When waves meet: the superposition principle 

12.6 A two-source interference pattern 

12.7 Standing waves 

12.8 Wave fronts and diffraction 

12.9 Reflection 

12.10 Refraction 

12.11 Sound Waves 



101 
102 
105 
106 
109 
110 
115 
120 
122 
126 
128 




CHAPTER TWELVE 



Waves 



12.1 Introduction 



The world is continually criss-crossed by waves of all sorts. Water 
waves, whether giant rollers in the middle of the ocean or gently-formed 
rain ripples on a still pond, are sources of wonder or pleasure. If the 
earth's crust shifts, violent waves in the solid earth cause tremors 
thousands of miles away. A musician plucks a guitar string and sound 
waves pulse against our ears. Wave disturbances may come in a 
concentrated bundle like the shock front from an airplane flying at 
supersonic speeds. Or the disturbances may come in succession like the 
train of waves sent out from a steadily vibrating source, such as a bell or a 
string. 

All of these are mechanical waves, in which bodies or particles 
physically move back and forth. But there are also wave disturbances in 
electric and magnetic fields. In Unit 4, you will learn that such waves are 
responsible for what our senses 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 this text we have considered motion in terms of individual 
particles. In this chapter we begin to study the cooperative motion of 
collections of particles in "continuous media" moving in the form of 
mechanical waves. We will see how closely related are the ideas of 
particles and waves which we use to describe events in nature. 

A comparison will help us here. Look at a black and white photograph 
in a newspaper or magazine with a magnifying glass. You will see that the 
picture is made up of many little black dots printed on a white page (up 
to 20,000 dots per square inch). Without the magnifier, you do not see the 
individual dots. Rather, you see a pattern with all possible shadings 
between completely black and completely white. 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 



SG 12.1 




A small section from the lower right 
of the photograph on the opposite 
page. 



Unit 3 101 



102 Unit 3 



The Kinetic Theory of Gases 



Waves should be studied in the 
laboratory. Most of this chapter is 
only a summary of some of what 
you will learn there. The articles 
'Waves" and "What is a Wave" in 
Reader 3 give additional discussion 
of wave behavior. Transparencies 
and film loops on waves are listed 
in SG 12.1. The Programmed 
Instruction booklets Waives 1 and 
Waves 2 may help you with the 
mathematics of periodic waves 
(see Sec. 12.4) and wave super- 
position (see Sec. 12.5). 



g 



Th« Project Physics Coui 





the first three units of the Text. In Unit 2 for example, we treated each 
planet as a particle undergoing the sun's gravitational attraction. We 
described the behavior of the solar system in terms of the positions, 
velocities, and accelerations of point-like objects. For someone interested 
only in planetary 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 we saw two different descriptions of a gas. One 
was in terms of the behavior of the individual particles making up the gas. 
We used Newton's laws of motion to describe what each individual 
particle does. Then we used average values of speed or energy to describe 
the behavior of the gas. But we also discussed concepts such as pressure, 
temperature, heat, and entropy. These 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 served a useful purpose and helped us to understand 
what we cannot directly see. 

Now we are about to study waves, and once again we find the 
possibility of using different points of view. Most of the waves discussed in 
this chapter can be described in terms of the behavior of particles. But we 
also want to understand waves as disturbances traveling in a continuous 
medium. We want, in other words, to see both the forest and the 
trees — the picture as a whole, not only individual dots. 



12.2 Properties of waves 



Suppose that two people are holding opposite ends of a rope. Suddenly 
one person snaps the rope up and down quickly once. That "disturbs" the 
rope and puts a hump in it which travels along the rope toward the other 
person. We can call the traveling hump one kind of a wave, a pulse. 

OriginaUy, the rope was motionless. The height of each point on the 
rope depended only upon its position along the rope, and did not change 
in time. But when one person snaps the rope, he creates a rapid change 
in the height of one end. This disturbance then moves away from its 
source. The height of each point on the rope depends upon time as well 
as position along the rope. 

The disturbance is a pattern of displacement along the rope. The 
motion of the displacement pattern from 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 wave moves. These 
four terms are common to all mechanical wave situations. 

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 moving 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 down as each wave 
passes. But they do not experience any net displacement on the average. 



Section 12.2 



Unit 3 103 



No material has moved from the wave source, either on the surface or 
among the particles of the hquid. The same holds for rope waves, sound 
waves in air, etc. 

As any one of these waves moves through a medium, the wave 
produces a changing displacement of the successive parts of the medium. 
Thus we can refer to these waves as waves of displacement. If we can see 
the medium and recognize the displacements, then we can see waves. But 
waves also may exist in media we cannot see (such as air). Or they may 
form as disturbances of a state we cannot detect with our eyes (such 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 wave passes. First, move the end 
of the spring from side to side, or up and down as in sketch (a) in the 
margin. A wave of side-to-side or up-and-down displacement will travel 
along the spring. Now push the end of the spring back and forth, along 
the direction of the spring itself, as in sketch (b). A wave of back-and- 
forth displacement will travel along the spring. Finally, twist the end of 
the spring clockwise and counterclockwise, as in sketch (c). A wave of 
angular displacement will travel along the spring. Waves like those in (a), 
in which the displacements are perpendicular to the direction the wave 
travels, are called transverse waves. Waves like those in (b), in which the 
displacements are in the direction the wave travels, are called longitudinal 
waves. And waves 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, 
however, transverse and torsional waves die out very quickly, and usually 
cannot be produced at all. So sound waves in air and water are 
longitudinal. The molecules of the medium are displaced back and forth 
along the direction that 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 down. 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. 

To describe completely transverse waves, such as those in ropes, you 
must specify the direction of displacement. Longitudinal and torsional 
waves do not require this specification. The displacement of a longitudinal 
wave can be in only one direction — along the direction of travel of the 
wave. Similarly, the angular displacements of a torsional wave can be 
around only one axis — the direction of travel of the wave. But the 
displacements of a transverse wave can be in any and all of an infinite 
number of directions. The only requirement is that they be at right angles 
to the direction of travel of the wave. You can see this by shaking one end 
of a rope randomly instead of straight up and down or straight left and 
right. For simplicity, our diagrams of rope and spring waves here show 
transverse displacements consistently in only one of all the possible planes. 

When the displacement pattern of a transverse wave does he in a 




|) ^^■v:> "1 RANSVfcRSE 

m" 'v?^, .^(Mlff (a) 



lOr^GITLIDINAL 

in mmmK<i\iK' (c) 

"Snapshots" of three types of waves. 
In (c), small markers have been put 
on the top of each coil in the spring. 



(a) 



r(b) 



(a) "Snapshot" representation of 
a sound wave progressing to the 
right. The dots represent the density 
of air molecules, (b) Graph of air 
pressure P vs. position x at the in- 
stant of the snapshot. 



104 Unit 3 



Waves 




Three of the infinitely many different 
polarization planes of a transverse 
v\/ave. 



single plane, we say the wave is polarized. For waves on ropes and 
springs, we can observe the polarization directly. Thus, in the photograph 
on the previous page, the waves the person makes are in the horizontal 
plane. However, whether we can see the wave directiy or not, there is a 
general test for polarization. The test involves finding some effect of the 
wave which depends on the angular position of a medium or obstacle 
through which it travels. An example of the principle is illustrated in the 
diagram below. Here, the transmission of a rope wave depends on the 
angle at which a slotted board is held. Each of the three sketches begins 
with the same wave approaching the obstacle (top line). Whether the 
wave passes through (bottom line) depends on the angle the slot makes 
with the plane of the rope's mechanical motion. 




The same short wave train on the rope 
approaches the slotted board in each 
of the three sketches (top). Depending 
on the orientation of the slot, the train 
of waves (a) goes entirely through the 
slot; (b) is partly reflected and partly 
transmitted with changed angles of 
rope vibration; or (c) is completely 
reflected. 



VVV-. 





• 


ir<Kr 


b 






In general, if some effect of a wave depends similarly on the angular 
position of an obstacle or medium, the wave must be polarized. Further, 
we can conclude that the wave is transverse rather than longitudinal or 
torsional. Some interesting and important examples of this principle will 
be presented in Chapter 13. 

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 clarifv" 
the difference between waves which propagate and those which do not. 
You probably have read some description of the great wheat plains of our 
Middle West, Canada, or Central Europe. Such descriptions usually 
mention the "beautiful, wind-formed waves that roll for miles across the 
fields." The medium for such a wave is the wheat, and 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 our rope and water waves. We will concentrate on waves that do 
originate at sources and propagate themselves. For the purposes of this 
chapter, ivaves are disturbances ivhich propagate in a medium. 



Section 12.3 



Unit 



105 



Q1 What kinds of mechanical waves can propagate in a solid? 

Q2 What kinds of mechanical waves can propagate in a fluid? 

Q3 What kinds of mechanical waves can be polarized? 

Q4 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? 



12.3 Wave propagation 



Waves and their behavior are perhaps best studied by beginning with 
large mechanical models and focusing our attention on pulses. Consider 
for example a freight train, with many cars attached to a powerful 
locomotive, but standing still. If the locomotive now 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 coupUngs are 
the medium. The "bump" traveling along the line of cars is the wave. The 
disturbance proceeds all the way 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 wave propagates. That, in turn, depends 
upon the type of wave and the characteristics of the medium. In any case, 
the effect of a disturbance is never transmitted instantly over 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. 



A very important point: energy 
transfer can occur without matter 
transfer. 



An engine starting abruptly can start 
a displacement wave along a line of 
cars. 




106 Unit 3 



Waves 



^U. 



.X-^' 



H 







A rough representation of the forces 
at the ends of a small section of rope 
as a transverse pulse moves past. 

SG 12.2 



The exact meaning of stiffness and 
density factors is different for 
different kinds of w/aves and 
different media. For tight strings, 
for example, the stiffness factor is 
the tension T in the string, and the 
density factor is the mass per unit 
length, m/l. The propagation speed 
V is given by 



The same comments apply also to transverse waves. The series of 
sketches in the margin represents a wave on a rope. Think of the sketches 
as frames of a motion picture film, taken at equal time intervals. The 
material of the rope does not travel along with 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 — 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 with 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 will 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. 

Q5 What is transferred along the direction of wave motion? 

Q6 On what two properites of a medium does wave speed depend? 



12.4 Periodic waves 

Many of the disturbances we have 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, we see a single wave running along the medium with a certain 
speed. We call this kind of wave a pulse. 

Now let us 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 
swing is virtually identical to every other swing, and the swing 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 caUed the amplitude A, as shown on page 107. 
The time taken to complete one vibration is called the period T. The 
number of vibrations per second is called the frequency f. 



Section 12.4 



Unit 3 107 



What happens when such a vibration is appUed 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, we observe a wave 
propagating along the rope. The wave takes the form 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. But 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 \ 
(lambda). 

If a single pulse or a wave crest moves fairly slowly through the 
medium, we can easily find its speed. In principle all we need is a clock 
and a meter stick. By timing the pulse or crest over a measured distance, 
we 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 indirectly 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 right, as it might look in snapshots taken 
every ^ period. Follow the progress of the crest that started out from the 
extreme left at t - 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 \ in one period of oscillation T. 
The speed v of the crest is therefore 

distance moved \ 



-^^ 



J>. 



corresponding time interval T 



All parts of the wave pattern propagate with the same speed. Thus the 
speed of any one crest is just the speed of the wave. We can say, 
therefore, that the speed v of the wave is 

wavelength k 



period of oscillation T 

But T =1//, where/ = frequency (see Text, Chapter 4, page 108). 
Therefore v = f\, or wave speed - frequency x wavelength. 

We can also write this relationship as X = v/f or/ == v/k. These 
expressions imply that, for waves of the same speed, the frequency and 
wavelength are inversely proportional. That is, a wave of twice 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 which are moving "in step" as the 
periodic wave passes. The crest points C and C have reached maximum 








The 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 frequently 
referred to as "sinusoidal.' 



108 Unit 3 



Waves 



displacement positions in the upward direction. The trough points D and 
D' have reached maximum displacement positions in the downward 
direction. The 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 which are vibrating identically when this 
wave passes. Note that C and C are a distance k apart, and so are D and 
D'. 



A "snapshot" of a periodic wave mov- 
ing to the right. Letters indicate sets 
of points with the same phase. 



'P,/'^"*^v 



./. 




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, 2A., 3X, . . . and nk (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 with one another. Each is separated from the next by a 
distance k. 

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 down, D begins to go up. Points such as these are caDed 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 A/2, 3X/2, 5X/2, . . . are one-half period out of phase. 

Q7 Of the wave variables — frequency, wavelength, period, amplitude 
and polarization — which ones describe 

(1) space properties of waves? 

(2) time properties of waves? 

08 A wave with the displacement as smoothly and simply varying 
firom point to point as that shown in the last illustration above is called a 
sine wave. How might the "wavelength" be defined for a periodic wave 
that isn't a sine wave? 

Q9 A vibration of 100 cycles per second produces a wave. 

(1) What is the wave frequency? 

(2) What is the period of the wave? 

(3) If the wave speed is 10 meters per second what is the 
wavelength? (If necessary, look back to find the relationship you need to 
answer this.) 

Q10 If points X and Y on a periodic wave are one-half period "out of 
phase" with each other, which of the following must he true? 



Section 12.5 



Unit 3 109 



(a) X oscillates at half the frequency at which Y osciUates. 

(b) X and Y always move in opposite directions. 

(c) X is a distance of one-half wavelength from Y. 



12.5 When waves meet: the superposition principle 

So far, we have considered single waves. What 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 waves 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 observed with aU types of waves. 
You can easily see thai it is true for surface ripples on water. (Look back, 
for example to the opening photograph for the chapter.) You could reason 
that it must be true fot sound waves also, since two conversations can 
take place across a table without distorting each other. (Note that when 
particles encounter each oiber, they collide. Waves can pass through each 
other.) 

But what happens during the time when the two waves overlap? 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 by 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 displacement of 0.4 cm upward and the other a 
maximum 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 behavior, 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. We can easily determine what the rope looks hke at any given 
instant. AU we need to do is add up the displacements caused by each 
wave 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 on page 110. 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. Check the 
diagrams with a ruler. You will find that the net displacement (solid line) 
is just the sum of the individual displacements (broken lines). 

The superposition 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 we would find by experiment that 
the superposition principle works equally well for three, ten, or any 
number of waves. Each wave makes its own contribution, and the net 
result is simply the sum of all the individual contributions. 

We can turn the superposition principle around. If waves add as we 








The superposition of two rope waves 
at a point. The dashed curves are the 
contributions of the individual waves. 



\ 



110 Unit 3 



Waves 



have described, then we can think of a complex wave as the sum of a set 
of simpler waves. In the diagram below (right), a complex pulse has been 
analyzed into a set of three simpler pulses. 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. 





SG 12.4-12.8 



Q11 Two periodic waves of amplitudes Aj and A^ pass through a 
point P. What is the greatest displacement of P? 

Q12 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? 



12.6 A two-source interference pattern 

The photograph at the right (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 the water level pattern 
at a given instant. 

The third photograph (far right) introduces a phenomenon which will 
play an important role in later parts of the course. It shows the pattern of 
ripples on a water surface which is disturbed by two vibrating sources. 
The two small sources go through their up and down motions together. 
That is, they are in phase. Each creates its own set of circular, spreading 
ripples. The photograph catches the pattern made by the overlapping sets 
of waves at one instant. It is called an interference pattern. 




The ripple tank shown in the photb- 
graph at the left is being used by 
students to observe a circular pulse 
spreading over a thin layer of water. 
When a vibrating point source is im- 
mersed at the edge of the tank, it 
produces periodic wave trains of 
crests and troughs, somewhat as 
shown in the "cut-away" drawing 
at the left below. The center figure 
below is an instantaneous photograph 
of the shadows of ripples produced 
by a vibrating point source. The crests 
and troughs on the water surface 
show up in the image as bright and 
dark circular bands. Below right, 
there were two point sources vibrat- 
ing in phase. The overlapping waves 
create an interference pattern. 




(«.) 



o o 







Pattern produced when two circular 

pulses, each of a crest and a trough, 

spread through each other. The small 

circles indicate the net displacement: 

• = double height peak 

® = average level 

O = double depth trough 



Diagram representing the separate 
pulses superposing as in the Figure 
at the left. The top sketch illustrates 
two crests about to arrive at the 
vertical line. The bottom sketch illus- 
trates a crest about to arrive together 
with a trough. 



Section 12.6 Unit 3 113 

We can interpret what we see in this photograph in terms of what we 
already know about waves. And we 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 
superposition principle. 

Suppose that two sources produce identical pulses at the same 
instant. Each pulse contains one crest and one trough. (See page 112 at 
the left.) 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 
we have placed a completely darkened circle wherever 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 on page 112 represents the spatial pattern of 
the water level at a given instant. 

At the points marked with darkened circles in (a), the two pulses 
arrive in phase, as indicated in (b). At 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, 
they 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 in (a) marked with half-darkened circles, the two pulses 
arrive completely out of phase, as shown in (c). Here the waves cancel 
and so are said to interfere destructively, leaving the water surface 
undisturbed. The lines N in (a) 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. Note that all 
points on these lines are one-crest-trough distance (^A) further from one 
source than from the other. 

When periodic waves of equal amplitude are sent out instead of single 
pulses, overlap occurs all over the surface. All along the central dotted hne 
there is a doubled disturbance amplitude. AU along the side lines the 
water height remains undisturbed. Depending on the wavelength and the 
distance between the sources, there can be many such lines of 
constructive and destructive interference. 

Now we can interpret the ripple tank interference pattern on page 
111. 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 are other bands 
where crest and trough follow one another, where the waves reinforce. 
These are called antinodal lines. 

Look closely at the diagram on page 114. It explains what is 
happening in the lower right hand photograph on page 111. Notice its 
symmetry. The central band labeled A o is an antinode where 



114 



Unit 3 



Waves 




Analysis of interference pattern similar 
to that of the lower right photograph 
on p. 1 1 1 set up by two in-phase peri- 
odic sources. (Here S, and S. are 
separated by four wavelengths.) The 
letters A and N designate antinodal 
and nodal lines. The dark circles in- 
dicate where crest is meeting crest, 
the blank circles where trough is 
meeting trough, and the half-dark 
circles where crest is meeting trough. 



SG 12.9 



reinforcement is complete. The other lines of maximum constructive 
interference are labeled A^, A2, A3, 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,, N2, etc. represent bands along which there is 
maximum destructive interference. Points on these lines move up and 
down much less than they would because of waves from either source 
alone. Compare the diagram with the photograph and identify antinodal 
lines and nodal lines. 

Whenever we find such an interference pattern, we know that it is set 
up by overlapping waves from two sources. For water waves, the 
interference pattern can be seen directly. But whether visible or not, all 
waves can set up interference patterns — including earthquake waves, 
sound waves, or x rays. For example, suppose two loudspeakers are 
working at the same frequency. By moving about and listening 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 k 
and the source separation S^S-z- From these we could calculate the angles 
at which the nodal and antinodal lines radiate out to either side of Aq. 
Conversely, we might know 5,82 and might have found these angles by 
probing around in the two-source interference pattern. If so, we can 
calculate the wavelength even if we can't see the crests and troughs of 
the waves directly. This is very useful, for most waves in nature can't be 
directly seen. So their wavelength has to be found in just this way: letting 
waves set up an interference pattern, probing for the nodal and antinodal 
lines, and calculating A. from the geometry. 



Section 12.7 



Unit o 115 



The figure at the right shows part of the pattern of the diagram on 
the opposite page. At any point P on an antinodal hne, the waves from 
the two sources arrive in phase. This can happen only if P is equally far 
from Si 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 - S2P) must equal n\, \ 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 (^\, |X, |X, etc.) farther from one source 
than from the other. This condition can be written S^Q -S-^Q =(n +5) A.. 

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 A, 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 the next page. 

This analysis allows us 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 \ 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. 



s,«. 



\ 



9 ^z 



^ 



^ i 



S,P-S.2P = n\ 
S.Q - S^Q = (n+ i)X 

Since the sound wave patterns in 
space are three-dimensional, the 
nodal or antinodal regions in this 
case are two-dimensional surfaces. 

For example thev are olanes. not 
lines. 



Q1 3 Are nodal lines in interference patterns regions of cancellation or 
reinforcement? 

Q14 What are antinodal lines? antinodal points? 

Q15 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. 

Q16 Under what circumstances do waves from two in-phase sources 
arrive at a point out of phase? 



12.7 Standing waves 



If both ends of a rope are shaken with the same frequency and same 
amplitude, an interesting thing happens. The 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. (With the aid of Transparency T-27, using the 
superposition principle, you can see that this effect is just what would be 
expected from the addition of the two oppositely traveling waves.) The 
important thing to remember is that the standing oscillation we observe is 



Calculating k from an Interference Pattern 

d = (S1S2) = separation between Si and S2. 

(Si and S2 may be actual sources that are 

in phase, or two slits through which a 

previously prepared wave front passes.) 
^ = 00 = distance fronn 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. 
L = OP = distance to point P on detection 

line measured from sources. 

Waves reaching P from Si have traveled 
farther than waves reaching P from S2. If the 
extra distance is k (or 2A, 3\, etc.), the waves 
will arrive at P in phase. Then P will be a point 
of strong wave disturbance. If the extra distance 
is hk (or |A., lA., etc.), the waves will arrive out of 
phase. Then P will be a point of weak or no 
wave disturbance. 

With P as center we draw an arc of a circle 
of radius PS2; it is indicated on the figure by the 
dotted line S2M. Then line segment PS2 = line 
segment PM. Therefore the extra distance that 
the wave from S travels to reach P is the length 
of the segment SM. 

Now if d is very small compared to f, as we 
can easily arrange in practice, the circular arc 
S2M will then be a very small piece of a large 
diameter circle — or nearly a straight line. Also, 
the angle S1MS2 is very nearly 90°. Thus, the 
triangle S1S2/W can be regarded as a right 
thangle. Furthermore, angle S1S2/W is equal to 
angle POO. Then right triangle S1S2/W is a 
similar triangle POO. 



S,M _ X 
SiS.,~OP 



or 



S,/W 



If the distance € is large compared to x, the 
distances /: and L are nearly equal, and we can 
write 



S,M 
~d~ 



But S^M is the extra distance traveled by the 
wave from source Si. For P to be a point of 
maximum wave disturbance, S:M must be equal 
to nk (where n = if P is at 0, and r? = I if P 
is at the first maximum of wave disturbance 
found to one side of 0, etc.). So the equation 
becomes 



and 



nk _x 
~d~l 

ne 



This important result says that if we measure 
the source separation d, the distance/, and the 
distance x from the central line to a wave 
disturbance maximum, we can calculate the 
wavelength k. 

d 

S, \ 



1 



\ 



\ 



M 






.\> 



i 



Q 



.\::^ 



1 



i 



Section 12.7 



i lUt 



117 



really the effect of two traveling waves. 

To make standing waves on a rope, there do not have 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 waves 
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 energy is then transmitted from the vibrating 
string to the body of the instrument. The sound waves sent forth from there 
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. The connection between length of string and 
musical tone was recognized over two thousand years ago. This relationship 
contributed greatly to the idea that nature is built on mathematical 
principles. Early in the development of musical instruments, people learned 
how to produce certain pleasing harmonies by plucking strings. These 
harmonies result if the strings are of equal tautness and diameter and if their 
lengths are in the ratios of small whole numbers. Thus the length ratio 2:1 
gives the octave, 3:2 the musical fifth, and 4:3 the musical fourth. This 
striking connection between music and numbers encouraged the 
Pythagoreans to search for other numerical ratios or harmonies in the 
universe. The Pythagorean ideal strongly affected Greek science and many 
centuries later inspired much of Kepler's work. In a general form, the ideal 
flourishes to this day in many beautiful appfications of mathematics to 
physical experience. 

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. We can imagine an unlimited variety of waves 
traveling back and forth. But, in fact, only certain wavelengths (or 
frequencies) can produce standing waves in a given medium. In the example 
of a stringed instrument, the two ends are fixed and so must be nodal points. 
This fact puts an upper limit on the length of standing waves possible on a 
fixed rope of length L. Such waves must be those for which one-half 
wavelength just fits on the rope (L = X/2). Shorter waves also can 
produce standing patterns having more nodes. But always, some whole 
number of one-half wavelengths must just fit on the rope (L = A./2). 

We can turn this relationship around to give an expression for all 
possible wavelengths of standing waves on a fixed rope: 




A vibrator at the left produces a wave 
train that runs along the rope and re- 
flects from the fixed end at the right. 
The sum of the oncoming and the re- 
flected waves is a standing wave 
pattern. 




Lyre player painted on a Greek vase 
in the 5th century B.C. 



SG 12.13 



n 



118 



Unit 3 



Waves 




Or simply. A.,, « 1/n. That is, if Aj is the longest wavelength possible, the other 
possible wavelengths will be ^X,, 3X1, . 1/^A.i. Shorter wavelengths 
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,/ 3c 1/A., we can rewrite the expression 
for all possible standing waves as 



/n=^n 



Film Loops 38—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 
paperbacks Waves and the Ear and 
Horns, Strings and Harmony. 

See the Reader 3 articles "Musical 
Instruments and Scales" and 
"Founding a Family of Fiddles." 



SG 12.16 



The lowest possible frequency of a standing wave is usually the one most 
strongly present when the string vibrates after being plucked or bowed. If/, 
represents this lowest possible frequency, then the other possible standing 
waves would have frequencies 2/,, Sf^, . . . nfy These higher frequencies are 
called "overtones" 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 
frequencies. Also, the overtones are not exactly simple multiples of the 
fundamental frequency. That is, the overtones are not strictly "harmonic." 
This effect is still greater in more comphcated systems 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 
superposition 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. 



Q17 When two identical waves of same frequency travel in opposite 
directions and interfere to produce a standing wave, what is the motion of the 
medium at 

(1) the nodes of the standing wave? 

(2) the places between nodes, called "antinodes" or loops, of the 
standing wave? 

Q18 If the two interfering waves have wavelength \, what is the 
distance between the nodal points of the standing wave? 

Q19 What is the wavelength of the longest travehng waves which can 
produce a standing wave on a string of length L? 

Q20 Can standing waves of any frequency, as long as it is higher than 
the fundamental, be set up in a bounded medium? 




In the Film Loop Vibration of a Drum, a 
marked rubber "drumhead" is seen vi- 
brating in several of its possible modes. 
Below are pairs of still photographs from 
three of the symmetrical modes and 
from an antisymmetrical mode. 








Diffraction of ripples around the edge 
of a barrier. 



P« 



Diffraction of ripples through a nar- 
row opening. 




Diffraction of ripples through two 
narrow openings. 



120 Vnit 3 

12.8 Wave fronts and diffraction 



Waves 



Waves can go around corners. For example, you can hear a voice coming 
from the other side of a hiU, even though there is nothing to reflect the sound 
to you. We are so used to the fact that sound waves do this that we scarcely 
notice it. This spreading of the energy of waves into what we would expect to 
be "shadow" regions is called diffraction. 

Once again, water waves will illustrate this behavior most clearly. From 
among all the arrangements that can result in diffraction, we will 
concentrate on two. The first is shown in the second photograph in the 
margin at the left. 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 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 were placed 
where the slit is. 

The bottom photograph shows the second barrier arrangement we want 
to investigate. Now there are two narrow slits in the barrier. The pattern 
resulting from superposition 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 sbt, with all sources in phase. 

We can describe these and all other effects of diffraction if we 
understand a basic characteristic of waves. It was first stated by Christian 
Huygens in 1678 and is now known as Huygens' principle. But in order to 
state the principle, we first need the definition of a wave front. 

For a water wave, a wave front is an imaginary line along the water's 
surface. 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 surface 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. 

The simplest wave fronts are straight lines parallel to each other, as in 
the top part of the center photograph at the left. Or they may be circular, 
as in the bottom part of the same photograph. Sound waves are somewhat 
different. Since a sound wave spreads not over a surface but in three 
dimensions, its wave fronts become very nearly spherical surfaces. At large 
distances from the source, however, the radius of a spherical wave front is 
also large. Thus, any small section of the wave front is nearly flat. AH 
circular and spherical wave fronts become virtually straight-line or flat- 
plane fronts at great distances from their sources. 

Now 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 the next 



Section 12.8 



Unit 3 121 



particle which is in the straight hne drawn from the (source), 
but that it also imparts some of it necessarily to all others 
which touch it and which oppose themselves to its movement. 
So it arises that around each particle there is made a wave of 
which that particle is the center. 

The diffraction patterns seen at sbts in a barrier are certainly 
consistent with Huygens' principle. The wave arriving at the barrier 
causes the water in the sht to oscillate. The oscillation of the water in the 
slit acts as a source for waves traveling out from it in all directions. When 
there are two sbts and the wave reaches both shts in phase, the oscillating 
water in each slit acts like a point source. The resulting interference 
pattern is similar to the pattern produced by waves from two point sources 
oscillating in phase. 

Or consider what happens behind a breakwater wall as in the aerial 
photograph of the harbor below. By Huygens' principle, water oscillation 
near the end of the breakwater sends circular waves propagating into the 
"shadow" region. 

We can understand all diffraction patterns if we keep both Huygens' 
principle and the superposition principle in mind. For example, consider a 
sht wider than one wavelength. In this case the pattern of diffracted 
waves contains nodal lines (see the series of four photographs in the 
margin). 

The figure on p. 122 helps to explain why nodal lines appear. There 
must be points Like P that are just A. farther from side A of the slit than 
from side B. That is, there must be points P for which AP differs from BP 




Each point on a wave front can be 
thought of as a point source of waves. 
The waves from all the point sources 
interfere constructively only along 
their envelope, which becomes the 
new wave front. 




WAVES^ 




BREAKWATER: 




Vf^'7rS' '»' />i/}.*il//' WAVES ! 

iiiiiii: ^ 



When part of the wave front is blocked, 
the constructive interference of waves 
from points on the wave front extends 
into the "shadow" region. 




When all but a very small portion of a 
wave front is blocked, the wave propa- 
gating away from that smail portion 
is nearly the same as from a point 
source. 



122 L'nit 3 



Waves 




by exactly \. For such a point, AP and OP differ by one-half wavelength, 
A./2. By Huygens' principle, we may think of points A and O as in-phase 
point sources of circular waves. But since AP and OP differ by XJ2, the 
two waves will arrive at P completely out of phase. So, according to the 
superposition principle, the waves from A and O will cancel at point P. 

But this argument also holds true for the pair of points consisting of 
the first point to the right of A and the first to the right of O. In fact, it 
holds true for each such matched pair of points, all the 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 A., then there can be no nodal point. This is obvious, 
since no point can be a distance k farther from one side of the sht than 
from the other. Slits of widths less than k behave nearly as point sources. 
The narrower they are, the more nearly their behavior resembles that of 
point sources. 

We can easily compute the wavelength of a wave from the 
interference pattern set up where diffracted waves overlap. For example, 
we can analyze the two-slit pattern on page 120 exactly as we analyzed 
the two-source pattern in Section 12.6. This is one of the main reasons for 
our interest in the interference of diffracted waves. By locating nodal lines 
formed beyond a set of slits, we can calculate k even for waves that we 
cannot see. 

For two-slit interference, the larger the wavelength compared to the 
distance between sbts, the more the interference pattern spreads out. That 
is, as k 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 around a 
corner, you hear the bass drums and tubas better than the piccolos and 
cornets — even though they actually are playing equally loud. 



Q21 What characteristic do all points on a wave front have in 
common? 

Q22 State Huygens' principle. 

Q23 Why can't there be nodal lines in a diffraction pattern from an 
opening less than one wavelength wide? 

Q24 What happens to the diffraction pattern from an opening as the 
wavelength of the wave increases? 

Q25 Can there be diffraction without interference? Interference 
without diffraction? 



12.9 Reflection 



We have seen that waves can pass through one another and spread 
around obstacles in their paths. Waves also are reflected, at least to some 



Section 12.9 



Unit 3 123 



degree, whenever they reach any boundary of the medium in which they 
travel. Echoes are familiar examples of the reflection of sound waves. All 
waves share the property of reflection. Again, the superposition principle 
will help us understand what happens when reflection occurs. 

Suppose that one end of a rope is tied tighdy to a hook securely 
fastened to a massive wall. From the other end, we send a pulse wave 
down the rope toward the hook. Since the hook cannot move, the force 
exerted by the rope wave can do no work on the hook. Therefore, the 
energy carried in the wave cannot leave the rope at this fixed end. 
Instead, the wave bounces back — is reflected — ^ideally with the same 
energy. 





What does the wave look like after it is reflected? The striking result 
is that the wave seems to flip upside down on reflection. As the wave 
comes in from left to right and encounters the fixed hook, it pulls up on 
it. By Newton's third law, the hook must exert a force on the rope in the 
opposite direction while reflection is taking place. The details of how this 
force varies in time are complicated. The net effect is that an inverted 
wave of the same form is sent back down the rope. 

Two-dimensional water-surface waves exhibit a fascinating variety of 
reflection phenomena. There may be variously shaped crest lines, variously 
shaped barriers, and various directions from which the waves approach the 
barrier. If you have never watched closely as water waves are reflected 
from a fixed barrier, you should do so. Any still pool or water-filled wash 
basin or tub will do. Watch the circular waves speed outward, reflect from 
rocks or walls, run through each other, and finally die out. Dip your 
fingertip into and out of the water quickly, or let a drop of water fall from 
your finger into the water. Now watch the circular wave approach and 
then bounce off a straight wall or a board. The long side of a tub is a 
good straight barrier. 

The sketches in the margin picture the results of reflection from a 
straight wall. Three crests are shown. You may see more or fewer than 
three clear crests in your observations, but that does not matter. 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 









--" / I 




SG 12.21 



124 Unit 3 



Waves 




two of them have been reflected. Notice the dashed curves in the last 
sketch. They attempt to 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 
imaginary source at point S' is called the image of the source S. 

We mention the reflection of circular waves first, because that is what 
one usually notices first when studying water waves. But it is easier to see 
a general principle for explaining reflection by obser\ang a straight wave 
front, reflected from a straight barrier. The ripple-tank photograph at the 
left shows one instant during such a reflection. (The wave came in from 
the upper left at an angle of about 45°.) The sketches below show in more 
detail what happens as the wave crests reflect from the straight barrier. 
The first sketch shows three crests approaching the barrier. The last 
sketch shows the same crests as they move away from the barrier after 
the encounter. The two sketches between show the reflection process at 
two different instants during reflection. 








SG 12.22 



The description of wave behavior is often made easier by drawing 
lines perpendicular to the wave fronts. Such lines, called rays, indicate the 
direction of propagation of the wave. Notice the drawing at the left, for 
example. Rays have been drawn for a set of wave crests just before 
reflection and just after reflection from a barrier. The straight-on direction, 
perpendicular to the reflecting surface, is shown by a dotted line. The ray 
for the incident crests makes an angle d^ with the straight-on direction. 
The ray for the reflected crests makes an angle ^r with it. The angle of 
reflection 6^ is equal to the angle of incidence 6^ : that is, dj. = d^. This is 
an experimental fact, which you can verify for yourself. 

Many kinds of wave reflectors are in use today, from radar antennae 
to infrared heaters. Figures (a) and (b) below show how straight-line 
waves reflect from two circular reflectors. A few incident and reflected 
rays are shown. (The dotted fines are perpendicular to the barrier surface.) 
Rays reflected from the half circle (a) head off in all directions. However, 
rays reflected from a small segment of the circle (b) come close to 



SG 12.23-12.25 




obok 



Section 12.9 



Unit 3 125 



meeting at a single point. And a barrier with the shape of a parabola (c) 
focuses straight-line waves precisely at a point. Similarly, a parabolic 
surface reflects plane waves to a sharp focus. An impressive example is a 
radio telescope. Its huge parabolic surface reflects faint radio waves from 
space to focus on a detector. 

The wave paths indicated in the sketches could just as well be 
reversed. For example, spherical waves produced at the focus become 
plane waves when reflected from a parabolic surface. The flashlight and 
automobile 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. 

Q26 What is a "ray"? 

Q27 What is the relationship between the angle at which a wave 
front strikes a barrier and the angle at which it leaves? 

Q28 What shape of reflector can reflect parallel wave fronts to a 
sharp focus? 

Q29 What happens to wave fronts originating at the focus of such a 
reflecting surface? 






Above: A ripple tank siiadow showing 
how circular waves produced at the 
focus of a parabolic wall are reflected 
from the wall into straight waves. 



Left: the parabolic surface of a radio 
telescope reflects radio waves from 
space to a detector supported at the 
focus. 



Below: the filament of a flashlight 
bulb is at the focus of a parabolic 
mirror, so the reflected light forms 
a nearly parallel beam. 




126 



Unit 3 



Waves 




\r 



fuv 



~\f 



V 



J\_A_ 



Pulses encountering a boundary be- 
tween two different media. The speed 
of propagation is less in medium 2. 






\, 



12.10 Refraction 

What happens when a wave propagates from one medium to another 
medium in which its speed of propagation is different? We begin with the 
simple situation pictured in the margin. Two one-dimensional pulses 
approach a boundary separating two media. The speed of the propagation 
in medium 1 is greater than it is in medium 2. We might imagine the 
pulses to be in a light rope (medium 1) tied to a relatively heavy rope 
(medium 2). Part of each pulse is reflected at the boundary. This reflected 
component is flipped upside down relative to the original pulse. You will 
recall the inverted reflection at a hook in a wall discussed earher. The 
heavier rope here tends to hold the boundary- point fixed in just the same 
way. But we are not particularly interested here in the reflected wave. We 
want to see what happens to that part of the wave which continues into 
the second medium. 

As shown in the figure, the transmitted pulses are closer together in 
medium 2 than they are in medium 1. Is it clear why this is so? The 
speed of the pulses is less in the heavier rope. 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 
pulse is itself squeezed 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 happens to a periodic wave at such a 
boundary. The figure at the left pictures this situation. For the sake of 
simplicity, we have assumed that all of the wave is transmitted, and none 
of it reflected. Just as the two pulses were brought closer and each pulse 
was squeezed narrower, the periodic wave pattern is squeezed together 
too. Thus, the wavelength X2 of the transmitted wave is shorter than the 
wavelength \i of the incoming, or incident, wave. 

Although the wavelength changes when the wave passes across the 
boundary, the fi"equency of the wave cannot change. If the rope is 
unbroken, the pieces immediately on either side of the boundary' must go 
up and down together. The frequencies of the incident and transmitted 
waves must, then, be equal. So we can simply label both of them /. 

We can write our wavelength, frequency, and speed relationship for 
both the incident and transmitted waves separately: 



Continuous wave train crossing the 
boundary between two different me- 
dia. The speed of propagation is less 
in medium 2. 



\if = I'l. and ^2/ = V2 

If we divide one of these equations by the other, eliminating the/'s, we 
get 



SG 12.26 



X, 



This equation tells that the ratio of the wavelengths in the two media 
equals the ratio of the speeds. 

The same sort of thing happens when water ripples cross a boundary. 



Section 12.10 



Unit 3 



127 



Experiments show that the ripples move more slowly in shallower water. A 
piece of plate glass is placed on the bottom of a ripple tank to make the 
water shallower there. This creates a boundary between the deeper and 
shallower part (medium 1 and medium 2). Figure (a) below shows the 
case where this boundary is parallel to the crest lines of the incident 
wave. As with rope waves, the wavelength of water waves in a medium is 
proportional to the speed in that medium. 




Water waves offer a possibility not present for rope waves. We can 
arrange to have the crest lines approach the boundary at any angle, not 
only head-on. The photograph below right shows such an event. A ripple 
tank wave approaches the boundary at the angle of incidence 0,. The 
wavelength and speed of course change as the wave passes across the 
boundary. But the direction of the wave propagation changes too. Figure 
(d) in the margin shows how this comes about. As each part of a crest 
line in medium 1 enters medium 2, its speed decreases and it starts to lag 
behind. In time, the directions of the whole set of crest lines in medium 2 
are changed from their directions in medium 1. 

This phenomenon is called refraction. It occurs whenever a wave 
passes into a medium in which the wave velocity is reduced. The wave 
fronts are turned (refracted) so that they are more nearly parallel to the 







V, 

— > 




1 


- 


sfty 


— 


rcty 



>^. 



(c) 



Xi 




SG 12.27-12.31 



Left: ripples on water (coming from 
the left) encounter the shallow region 
over the corner of a submerged glass 
plate. 

Right; ripples on water (coming from 
the left) encounter a shallow region 
over a glass plate placed at an angle 
to the wavefronts. 



128 



Unit 3 



Waves 



Aerial photograph of the refraction 
of ocean waves approaching shore. 



The slow^ing of star light by 
increasingly dense layers of the 
atmosphere produces refraction 
that changes the apparent position 
of the star. 




boundary. (See the photographs at the bottom of the previous page.) This 
accounts for something 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, when they come near the beach their crest-lines are nearly 
parallel to the shorehne. A wave's speed is steadily reduced as it moves 
into water that gets gradually more shallow. So the wave is refracted 
continuously as if it were always crossing a boundary between different 
media, as indeed it is. The refraction of sea waves is so great that wave 
crests can curl around a small island with an all-beach shoreline and 
provide surf on all sides. (See the photograph on page 139.) 

Q30 If a periodic wave slows down on entering a new medium, what 
happens to (1) its frequency? (2) its wavelength? (3) its direction? 

Q31 Complete the sketch in the margin to show roughly what 
happens to a wave train that enters a new medium where its speed is 
greater. 



Look again at the bottom figure in the 
margin of p. 103. 



12.11 Sound waves 

Sound waves are mechanical disturbances that propagate through a 
medium, such as the air. Typically, sound waves are longitudinal waves, 
producing changes of density and pressure in the medium through which 
they travel. The medium may 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. But here, of course, we will concentrate on sound 
as an example of wave motion. Sound has all the properties of wave 
motion that we have considered so far It exhibits refraction, diffraction. 



Section 12.11 unit 3 129 

and the same relations among frequency, wavelength, and propagation 
speed and interference. Only the property of polarization is missing, 
because sound waves are longitudinal, not transverse. 

Vibrating sources for sound waves may be as simple as a tuning fork 
or as complex as the human larynx with its vocal cords. Tuning forks and 
some special electronic devices produce a steady "pure tone." Most of the 
energy in such a tone is in simple harmonic motion at a single frequency. 
The "pitch" of a sound we hear goes up as the frequency of the wave 
increases. 

People can hear sound waves with frequencies between about 20 and 
20,000 cycles per second. Dogs can hear over a much wider range (15- 
50,000 cps). Bats, porpoises, and whales generate and respond to 
frequencies up to about 120,000 cps. 

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 power flow , such 
as 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 lO"'*^ watts per square 
centimeter (relative intensity = 1). Below this "threshold" level, the normal 
ear does not perceive sound. 

RELATIVE INTENSITY SOUND 

1 Threshold of hearing 

10' Normal breathing 

10^ Leaves in a breeze 

103 

10'* . Library 

10^ Quiet restaurant 

10® Two-person conversation 

10^ . Busy traffic 

10* Vacuum cleaner 

10* Roar of Niagara Falls 

10'" Subway train 

10" 

10'^ Propeller plane at takeoff 

10'^ Machine-gun fire 

10'^ Small jet plane at takeoff 

10'^ 

10'« Wind tunnel 

10'" Space rocket at lift-off 



Levels of noise intensity about 10^^ 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. 

It has always been fairly obvious that sound takes time to travel from 
source to receiver. Light and sound are often closely associated in the 
same event — lightning and thunder, for instance. In all such cases, we 
perceive the sound later. By timing echoes over a known distance, the 
French mathematician Marin Mersenne in 1640 first computed the speed 



SG 12.32 



Noise and the Sonic Boom 

The world seems to be increasingly loud with 
unpleasant, manmade noise. At worst it is a 
major nuisance and may be tiring, painful, and 
sometimes even physically harmful. Loud, 
prolonged noise can produce temporary 
deafness. Very loud noise, kept up for a long 
time, can produce some degree of permanent 
deafness, especially deafness with respect to 
high-frequency sounds. 

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 goes. 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 rug on 
the floor can absorb 90% of room noise. (A foot 
of fresh snow is an almost perfect absorber of 
noise outdoors. Cities and countrysides are 
remarkably hushed after a snowfall.) 

In the last few years a new kind of noise 
has appeared: 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 700 miles per hour. Many 
types of military airplanes can travel at two or 
three times this speed. Flying at such speeds, 
the planes unavoidably and continually produce 
sonic booms. SST (Supersonic Transport) planes 
are now in civilian use in some countries. The 
unavoidable boom raises important questions. 
What is the price of technological "progress"? 
Who gains, and what fraction of the population? 
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 his 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 it remains in the 
same position in a medium, the wave it produces 
spreads out symmetrically around it, as in 
diagram 1 . But 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.) 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 
below. 








.TW^JvVfic-iivv>;-%w., ..i.,:dSj^itoiJ«SNJi!»ixsaESsaaa 



All these concepts apply not only to water 
waves but also to sound waves, including 
those disturbances 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 3-dimensions) 
that spreads away from the source. 

Actually two cones of sharp pressure 
change are formed: one originating at the 
front of the airplane 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," or 
"bang-zone"), is as long as the supersonic flight 
path itself. In such an area, typically thousands 
of miles long and 50 miles 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 350-ton 
supersonic transport plane would break many 
thousands of dollars worth of windows, plaster 
walls, etc., and cause fright and annoyance to 
millions of people. Thus the supersonic flight 
of such planes may have to be confined to 
over-ocean use — though it may even turn out 
that the annoyance to people on shipboard, on 
islands, etc., is so great that over-ocean flights, 
too, will have to be restricted. 



AIR 
PKESSURt 



norma,! 



IC 30 30 40 so (,0 10 go 10 l«0 HO 

miNisecon<i5 

This curve represents the typical sonic boom 
from an airplane flying at supersonic speed 
(speed greater than about 700 mph). The pres- 
sure rises almost instantly, then falls relatively 
slowly to below-normal pressure, then rises 
again almost instantaneously. The second pres- 
sure rise occurs about 0.1 second after the 
first one, making the boom sound "double." 



Double-cone shock wave, or sonic boom, pro- 
duced by an airplane that is travelling (at 13- 
mile altitude) at three times the speed of sound. 
Building B is just being hit by shock wave, 
building A was struck a few seconds ago, and 
building C will be hit a few seconds later. 




132 



Unit 3 



Waves 



SG 12.33-12.35 



The article Silence, Please" in 
Reader 3 is an amusing fantasy 
about wave superposition. 



The acoustic properties of a hall 
filled with people are very different 
from those of the empty hall. 
Acoustical engineers sometimes fill 
the seats with felt-covered 
sandbags while making tests. 



of sound in air. But it took another seventy years before William Derham 
in England, comparing the flash and noise from cannons across 12 miles, 
came close to the modern measurements. 

Sound in air at 68°F moves at 1,125 feet per second (about 344 
meters per second or 770 mph). As for all waves, the speed of sound 
waves 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 sea water, their speed is about 4,890 ft/sec; in 
steel, about 16,000 ft/sec; in quartz, about 18,000 ft/sec. 

Interference of sound waves can be shown in a variety of ways. In a 
large hall with hard, sound-reflecting surfaces, there will 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 known as 
beats. When two notes of shghtly 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 we 
sometimes see hghtning without hearing thunder. Similar refraction of 
sound occurs in layers of water of different temperatures. .This sometimes 
causes problems in using sonar (sound navigation and ranging) devices at 
sea. Sonic refraction is used for a variety of purposes today. Geologists use 
them 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 waves travel through the earth and are 
received by detection devices at different locations. The path of the waves, 
as refracted by layers in the earth, can be calculated from the relative 
intensities of sound received. From knowledge of the paths, estimates can 
be made of the composition of the layers. 

We have already mentioned diffraction as a property of sound waves. 
Sound waves readily bend around corners and barriers to reach the 
listener within range. Sound waves reflect, as do rope or water waves, 
wherever they encounter a boundary between different media. Echo 
chamber effects (which can be artificially produced by electronics) have 
become famihar to listeners who enjoy popular music. The "live" sound of 
a bare room results from multiple reflections of waves which normally 
would be absorbed by furniture, rugs, and curtains. The architechtural 
accidents called "whispering galleries" show vividly how sound can be 
focused by reflection from curved surfaces. Laboratory rooms which 
greatly reduce reflections are called anechoic chambers. All these effects 
are of interest in the study of acoustics. Moreover, the proper acoustical 
design of public buildings is now recognized as an important function by 
most good architects. 

In this chapter we have explained the basic phenomena of mechanical 
waves, ending with the theory of sound propagation. These explanations 
were considered the final triumph of Newtonian mechanics as applied to 
the transfer of energy of particles in motion. Most of the general principles 



Section 12.11 



Unit 3 133 




An anechoic chamber being used for research in 
acoustics. Sound is almost completely absorbed 
during 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 accoustically de- 
signed for unamplified performances. 



of acoustics were discovered in the 1870's. Since then the study of 
acoustics has become involved with such fields as quantum physics. But 
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 
move back and forth or up and down in a medium. 

Q32 List five wave behaviors that can be demonstrated with sound 
waves. 

Q33 Why can't sound waves be polarized? 




134 Unit 3 The Triumph of Mechanics 

EPILOGUE Seventeenth-century scientists thought they could eventually 
explain all physical phenomena by reducing them to matter and motion. 
This mechanistic viewpoint became known as the Newtonian worldview or 
Newtonian cosmology, since its most impressive success was Newton's 
theory of planetary motion. Newton and other scientists of his time 
proposed to apply similar methods to other problems, as we mentioned in 
the Prologue to this unit. 

The early enthusiasm for this new approach to science is vividly 
expressed by Henry Power in his book Experimental Philosophy (1664). 
Addressing his fellow natural philosophers (or scientists, as we would now 
call them), he wrote: 

You are the enlarged and elastical Souls of the world, who, 
removing all former rubbish, and prejudicial resistances, do 
make way for the Springy Intellect to flye out into its desired 
Expansion . . . 

. . . This is the Age wherein (me-thinks) Philosophy comes 
in with a Spring-tide ... I see how all the old Rubbish must be 
thrown away, and carried away with so powerful an Inundation. 
These are the days that must lay a new Foundation of a more 
magnificent Philosophy, never to be overthrown: that will 
Empirically and Sensibly canvass the Phaenomena of Nature, 
deducing the causes of things from such Originals in Nature, as 
we observe are producible by Art, and the infallible 
demonstration of Mechanicks; and certainly, this is the way, and 
no other, to build a true and permanent Philolophy. 

In Power's day there were many people who did not regard the old 
Aristotelian cosmology as rubbish. For them, it provided a comforting sense 
of unity and interrelation among natural phenomena. They feared that this 
unity would be lost if everything was reduced simply to atoms moving 
randomly through space. The poet John Donne, in 1611, complained 
bitterly of the change already taking place in cosmology: 

And new Philosophy calls all in doubt. 

The Element of fire is quite put out; 

The Sun is lost, and th' earth, and no man's wit 

Can well direct him where to looke for it. 

And freely men confesse that this world's spent, 

When in the Planets, and the Firmament 

They seeke so many new; then see that this 

Is crumbled out againe to his Atomies. 

Tis all in peeces, all coherence gone; 

All just supply, and all Relation. . . 

Newtonian physics provided powerful methods for analyzing the world 
and uncovering the basic principles of motion for individual pieces of 
matter. But the richness and complexity of processes in the real world 
seemed infinite. Could Newtonian physics deal as successfully with these 
real events as with ideal processes in a hypothetical vacuum? Could the 
perceptions of colors, sounds, and smells really be reduced to 'nothing 



Epilogue 



Unit 3 135 



but" matter and motion? In the seventeenth century, and even in the 
eighteenth century, it was too soon to expect Newtonian physics to answer 
these questions. There was still too much work to do in establishing the 
basic principles of mechanics and applying them to astronomical problems. 
A full-scale attack on the properties of matter and energy had to wait until 
the nineteenth century. 

This unit covered several successful applications and extensions of 
Newtonian mechanics which were accomplished by the end of the 
nineteenth century. For example, we discussed the conservation laws, new 
explanations of the properties of heat and gases, and estimates of some 
properties of molecules. We introduced the concept of energy, linking 
mechanics to heat and to sound. In Unit 4 we will show similar links to 
light, electricity, and magnetism. We also noted that applying mechanics on 
a molecular level requires statistical ideas and presents questions about the 
direction of time. 

Throughout most of this unit we have emphasized the application of 
mechanics to separate pieces or molecules of matter. But scientists found 
that the molecular model was not the only way to understand the behavior 
of matter. Without departing from basic Newtonian cosmology, scientists 
could also interpret many phenomena (such as sound and light) in terms of 
wave motions in continuous matter. By the middle of the nineteenth century 
it was generally believed that all physical phenomena could be explained 
by a theory that was built on the use of either particles or waves. In the 
next unit, we will discover how much or how little validity there was in this 
belief. We will begin to see the rise of a new viewpoint in physics, based 
on the field concept. Then, in Unit 5, particles, waves, and fields will come 
together in the context of twentieth-century physics. 





STUDY GUIDE 



12.1 



The Project Physics materials particularly 



appropriate for Chapter 12 include: 

Experiments 

Sound 
Activities 

Standing Waves on a Drum and a Violin 

Moire' Patterns 

Music and Speech Activities 

Measurement of the Speed of Sound 

Mechanical Wave Machines 
Film Loops 

Superposition 

Standing Waves in a String 

Standing Waves in a Gas 

Four loops on vibrations 
Reader Articles 
Silence, Please 

Frontiers of Physics Today: Acoustics 
Waves 

What is a Wave 

Musical Instruments and Scales 
Founding a Family of Fiddles 

'^■^ Some waves propagate at such a high 
speed that we are usually not aware of any 
delay in energy transfer. For example, the delay 
between the flash and the "bang" in watching 
lightning or fireworks seems peculiar, because the 
propagation time for sounds produced near us is 
not noticeable. Give an example of a compression 
wave in a solid, started by an action at one end, 
that propagates so quickly that we are not 
aware of any delay before an effect at the 
other end. 

"'2.3 Describe the differences in phase of 
oscillation of various parts of your body as you 
walk. What points are exactly in phase? Which 
points are exactly 2" cycle out of phase? Are 
there any points i cycle out of phase? 

'2-* Pictured are two pulse waves (A and B) on 
a rope at the instants before and after they 




plot the shape of the rope at the end of each 
interval. 

12 5 

Repeat Exercise 12.3 for the two pulses 

(A and C) pictured at the top. 

12.6 Yhe wave below propagates to the right 
along a rope. What is the shape of the wave 
propagating to the left that could for an instant 
cancel this one completely? 





> 




overlap (t, and t,). Divide the elapsed time 
between t, and tj into four equal intervals and 



1 ? 7 

The velocity of a portion of rope at some 
instant as transverse waves are passing through 
it is the superposition of the velocities of waves 
passing through that portion. Is the kinetic energy 
of a portion of the rope the superposition of the 
kinetic energies of waves passing through that 
region? Justify your answer. 

'^" Graphically superpose the last three curves 
of the figure on p. 1 10 to find their sum (which 
should be the original curve). 

^■® What shape would the nodal regions have 
for sound waves from two loudspeakers? 

"'•^" Imagine a detection device for waves is 
moved slowly to either the right or left of the 
point labeled A„ in the figure on p. 114. Describe 
what the detection device would register. 

• 2. 1 I What kind of interference pattern would 
you expect to see if the separation between two 



136 Unit 3 



STUDY GUIDE 



in-phase sources were less than the wavelength 
X? Where would the nodal and an tinodal 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 separatior 
between the two in-phase sources is increased by 
one wavelength. 

12.12 Derive an equation, similar to nkl = dx„, 
for nodal points in a two-source interference 
pattern (where d is the separation of the sources, 
I the distance from the sources, and x„ the 
distance of the n"" node from the center line). 

12.13 If you suddenly disturbed a stretched 
rubber hose or slinky with a frequency that 
precisely matched a standing wave frequency, 
would standing waves appear immediately? If 
not, what factors would determine the time delay? 

12.14 Different notes are sounded with the same 
guitar string by changing its vibrating length 
(that is, pressing the string against a brass 
ridge). If the full 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"? 

12.15 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 com- 
pletely free to be displaced, so the standing waves 
must have an antinode near the open end. 

(a) What will be the wavelength of the funda- 
mental standing wave in a pipe of length L closed 
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 (b) for the case of a pipe open 
at both ends. 

12.16 Imagine a spherical blob of jello in which 
you can set up standing vibrations. What would 
be some of the possible modes of vibration? 
(Hint: what possible symmetrical nodal surfaces 
could there be?) 

12.17 Suppose that straight-line ripple waves 
approach a thin straight barrier which is a few 
wavelengths long and which is oriented with its 
length parallel to the wavefronts. What do you 
predict about the nature of the diffraction pattern 
along a straight line behind the barrier which is 
perpendicular to the barrier and passes through 
the center of the barrier? Why do people who 
design breakwaters need to concern themselves 
with diffraction effects? 

12.18 A megaphone directs sound along the 
megaphone axis if the wavelength of the sound is 



small compared to the diameter of the opening. 
Estimate the upper limit of frequencies which are 
diffracted at a cheerleader's megaphone opening. 
Can you hear what a cheerleader shouts even 
though you are far off the axis of the megaphone? 

12.19 Explain why it is that the narrower a slit 
in a barrier is, the more nearly it can act like a 
point source of waves. 

12.20 If light is also a wave, then why have you 
not seen light being diffracted by the slits, say 
those of a picket fence, or diffracted around the 
comer of houses? 

12.21 By actual construction with a ruler and 
compass on a tracing of the photograph on p. 127, 
show that rays for the reflected wave front appear 
to come from S'. Show also that this is consistent 
with 6, = 6,. 

12.22 A straight-line wave approaches a right- 
angle reflecting barrier as shown in the figure. 
Find the shape, size, and direction of propagation 
of the wave after it has been completely reflected 
by the barrier. 




12.23 With ruler and compass reproduce part (b) 
of the figure at the bottom of p. 124 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 larger than the one in 
the figure. (Hint: the dotted lines are along radii.) 

12.24 Convince yourself that a parabolic reflector 
will actually bring parallel wave-fronts to a sharp 
focus. Draw a parabola y = kx- (choosing any 
convenient value for k) and some parallel rays 
along the axis as in part (c) of the Figure at the 
bottom of p. 124. Construct line segments per- 
pendicular to the parabola where the rays hit it, 
and draw the reflected rays at equal angles on the 
other side of these lines. 

12.25 The /oca/ length of a curved reflector is the 
distance from the reflector to the point where 
parallel rays are focused. Use the drawing in 

SG 12.24 to find the focal length of a parabola 
in terms of k. 

12.26 Recalling that water surface waves travel 
slower in shallow water, what would you expect to 
happen to the shape of the following wave as it 



Unit 3 137 



continues to the right? Pay particular attention to 
the region of varying depth. Can you use the line 
of reasoning above to give at least a partial ex- 
planation of the cause of breakers near a beach? 

12.27 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. 



#;%:, 







12.28 On the opposite page is an aerial photograph 
of ocean waves entering from the upper right and 
encountering a small island. Describe the wave 
phenomena demonstrated by this encounter. 

12.29 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. 

6 



air 



3I 



ass 




(a) Indicate and label an angle equal to angle 
of incidence 6^. 

(b) Indicate and label an angle equal to angle 
of refraction 0^. 

(c) Label the wavelength in air X,. 

(d) Label the wavelength in glass K^^. 

(e) Show that vjv^ = Xa/^h- 

(f) If you are familiar with trigonometry, show 
that sin 6 J sin 6^ = A. JX,,. 



12.30 A periodic ripple-tank wave passes through 
a straight boundary between deep and shallow 
water. The angle of incidence at the boundary is 
45° and the angle of refraction is 30°. The 
propagation speed in the deep water is 0.35 m/sec. 
and the frequency of the wave is 10 cycles per 
sec. Find the wavelengths in the deep and 
shallow water. 

12.31 Look at Figure (d) on p. 127. Convince 
yourself that if a wave were to approach the 
boundary between medium 1 and medium 2 from 
below, along the same direction as the refracted 
ray in the figure, it would be refracted along the 
direction of the incident ray in the figure. This is 
another example of a general rule: if a wave 
follows a set of rays in one direction, then a wave 
can follow the same set of rays in the opposite 
direction. In other words, wave paths are 
reversible. 

12.32 Suppose that in an extremely quiet room 
you can barely hear a buzzing mosquito at a 
distance of one meter. 

(a) What is the sound power output of the 
mosquito? 

(b) How many mosquitoes would it take to supply 
the power for one 100- watt reading lamp? 

(c) If the swarm were at ten meters" distance, 
what would the sound be like? (Sound intensity 
diminishes in proportion to the square of the 
distance from a point source.) 

12.33 How can sound waves be used to map the 
floors of oceans? 

12.34 Estimate the wavelength of a 1000 cycles 
per second sound wave in air; in water; in steel 
(refer to data in text). Do the same if /= 10.000 
cps. Design the dimensions of an experiment to 
show two-source interference for 1000 cps sound 
waves. 

12.35 Waves reflect from an object in a definite 
direction only when the wavelength is small 
compared to the dimensions of the object. This is 
true for sound waves as well as for any other. 
What does this tell you about the sound fre- 
quencies a bat must generate if it is to catch a 
moth or a fly? Actually some bats can detect 

the presence of a wire about 0.12 mm in 
diameter. Approximately what frequency does 
that require? 



138 Unit 3 











Refraction, reflection, and diffraction of waves around Farallon Island, 
California. There are breakers all around the coast. The swell coming from 
top right rounds both sides of the island, producing a crossed pattern 
below. The small islet radiates' the waves away in all directions. (U.S. 
Navy photograph.) 



Vnit 3 



139 



Section 12.9 



Unit 3 125 



meeting at a single point. And a barrier with the shape of a parabola (c) 
focuses straight-line waves precisely at a point. Similarly, a parabolic 
surface reflects plane waves to a sharp focus. An impressive example is a 
radio telescope. Its huge parabolic surface reflects faint radio waves from 
space to focus on a detector. 

The wave paths indicated in the sketches could just as well be 
reversed. For example, spherical waves produced at the focus become 
plane waves when reflected from a parabolic surface. The flashhght and 
automobile 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. 

Q26 What is a "ray"? 

Q27 What is the relationship between the angle at which a wave 
front strikes a barrier and the angle at which it leaves? 

Q28 What shape of reflector can reflect parallel wave fronts to a 
sharp focus? 

Q29 What happens to wave fronts originating at the focus of such a 
reflecting surface? 







Above: A ripple tank shadow shovying 
how circular waves produced at the 
focus of a parabolic wall are reflected 
from the wall into straight waves. 



Left: the parabolic surface of a radio 
telescope reflects radio waves from 
space to a detector supported at the 
focus. 



Below: the filament of a flashlight 
bulb is at the focus of a parabolic 
mirror, so the reflected light forms 
a nearly parallel beam. 




126 



Unit 3 



Waves 



A 




"V 



fUV 



~U" 



V 



JUU 



Pulses encountering a boundary be- 
tween two different media. The speed 
of propagation is less in medium 2. 



v.— >- Vj-^ 



TA/W\Af 



HS^ 



12.10 Refraction 

What happens when a wave propagates from one medium to another 
medium in which its speed of propagation is different? We begin with the 
simple situation pictured in the margin. Two one-dimensional pulses 
approach a boundary separating two media. The speed of the propagation 
in medium 1 is greater than it is in medium 2. We might imagine the 
pulses to be in a light rope (medium 1) tied to a relatively heavy rope 
(medium 2). Part of each pulse is reflected at the boundary. This reflected 
component is flipped upside down relative to the original pulse. You will 
recall the inverted reflection at a hook in a wall discussed earlier. The 
heavier rope here tends to hold the boundary- point fixed in just the same 
way. But we are not particularly interested here in the reflected wave. We 
want to see what happens to that part of the wave which continues into 
the second medium. 

As shown in the figure, the transmitted pulses are closer together in 
medium 2 than they are in medium 1. Is it clear why this is so? The 
speed of the pulses is less in the heavier rope. 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 
pulse is itself squeezed 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 happens to a periodic wave at such a 
boundary. The figure at the left pictures this situation. For the sake of 
simplicity, we have assumed that all of the wave is transmitted, and none 
of it reflected. Just as the two pulses were brought closer and each pulse 
was squeezed narrower, the periodic wave pattern is squeezed together 
too. Thus, the wavelength A2 of the transmitted wave is shorter than the 
wavelength \i of the incoming, or incident, wave. 

Although the wavelength changes when the wave passes across the 
boundary, the frequency of the wave cannot change. If the rope is 
unbroken, the pieces immediately on either side of the boundary must go 
up and down together. The frequencies of the incident and transmitted 
waves must, then, be equal. So we can simply label both of them /. 

We can write our wavelength, frequency, and speed relationship for 
both the incident and transmitted waves separately: 



Continuous wave train crossing the 
boundary between two different me- 
dia. The speed of propagation is less 
in medium 2. 



^1/ - ^1- arid \2/ = ^2 

If we divide one of these equations by the other, eliminating the/'s, we 
get 



SG 12.26 



X2 ^2 



This equation tells that the ratio of the wavelengths in the two media 
equals the ratio of the speeds. 

The same sort of thing happens when water ripples cross a boundary. 



Section 12.10 



Unit 3 



127 



Experiments show that the ripples move more slowly in shallower water. A 
piece of plate glass is placed on the bottom of a ripple tank to make the 
water shaDower there. This creates a boundary between the deeper and 
shallower part (medium 1 and medium 2). Figure (a) below shows the 
case where this boundary is parallel to the crest lines of the incident 
wave. As with rope waves, the wavelength of water waves in a medium is 
proportional to the speed in that medium. 




Water waves offer a possibility not present for rope waves. We can 
arrange to have the crest lines approach the boundary at any angle, not 
only head-on. The photograph below right shows such an event. A ripple 
tank wave approaches the boundary at the angle of incidence 0j. The 
wavelength and speed of course change as the wave passes across the 
boundary. But the direction of the wave propagation changes too. Figure 
(d) in the margin shows how this comes about. As each part of a crest 
line in medium 1 enters medium 2, its speed decreases and it starts to lag 
behind. In time, the directions of the whole set of crest lines in medium 2 
are changed from their directions in medium 1. 

This phenomenon is called refraction. It occurs whenever a wave 
passes into a medium in which the wave velocity is reduced. The wave 
fronts are turned (refracted) so that they are more nearly parallel to the 





■— 1 


V, 

> 






' 


^ 




ray 



>. 



(c) 



Xt 











x_ 




y 


-^ 


^ 


Vf 


y 


/ 


'A 






SG 12.27-12.31 



Left: ripples on water (coming from 
the left) encounter the shallow region 
over the corner of a submerged glass 
plate. 

Right: ripples on water (coming from 
the left) encounter a shallow region 
over a glass plate placed at an angle 
to the wavefronts. 



128 



Unit 3 



Waves 



Aerial photograph of the refraction 
of ocean waves approaching shore. 



The slowing of star light by 
increasingly dense layers of the 
atmosphere produces refraction 
that changes the apparent position 
of the star. 




boundary. (See the photographs at the bottom of the previous page.) This 
accounts for something 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, when they come near the beach their crest-lines are nearly 
parallel to the shoreline. A wave's speed is steadily reduced as it moves 
into water that gets gradually more shallow. So the wave is refracted 
continuously as if it were always crossing a boundary between different 
media, as indeed it is. The refraction of sea waves is so great that wave 
crests can curl around a small island with an all-beach shoreline and 
provide surf on all sides. (See the photograph on page 139.) 

Q30 If a periodic wave slows down on entering a new medium, what 
happens to (1) its frequency? (2) its wavelength? (3) its direction? 

Q3-| Complete the sketch in the margin to show roughly what 
happens to a wave train that enters a new medium where its speed is 
greater. 



Look again at the bottom figure in the 
margin of p. 103. 



12.11 Sound waves 

Sound waves are mechanical disturbances that propagate through a 
medium, such as the air. Typically, sound waves are longitudinal waves, 
producing changes of density and pressure in the medium through which 
they travel. The medium may be a sohd, hquid, 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. But here, of course, we will concentrate on sound 
as an example of wave motion. Sound has all the properties of wave 
motion that we have considered so far. It exhibits refraction, diffraction. 



Section 12.11 i;,„^ 3 129 

and the same relations among frequency, wavelength, and propagation 
speed and interference. Only the property of polarization is missing, 
because sound waves are longitudinal, not transverse. 

Vibrating sources for sound waves may be as simple as a tuning fork 
or as complex as the human larynx with its vocal cords. Tuning forks and 
some special electronic devices produce a steady "pure tone." Most of the 
energy in such a tone is in simple harmonic motion at a single frequency. 
The "pitch" of a sound we hear goes up as the frequency of the wave 
increases. 

People can hear sound waves with frequencies between about 20 and 
20,000 cycles per second. Dogs can hear over a much wider range (15- 
50,000 cps). Bats, porpoises, and whales generate and respond to 
frequencies up to about 120,000 cps. 

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 power flow , such 
as 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 SOUND 

1 Threshold of hearing 

10' Normal breathing 

10^ Leaves in a breeze 

10=^ 

10^ Library 

10^ Quiet restaurant 

10* Two-person conversation 

10' Busy traffic 

10^ Vacuum cleaner 

10" Roar of Niagara Falls 

10'° Subway train 

10" 

10'- Propeller plane at takeoff 

10'^ Machine-gun fire 

10'^ Small jet plane at takeoff 

lO's 

10"^ Wind tunnel 

10'" Space rocket at lift-off 



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. 

It has always been fairly obvious that sound takes time to travel from 
source to receiver. Light and sound are often closely associated in the 
same event — lightning and thunder, for instance. In all such cases, we 
perceive the sound later. By timing echoes over a known distance, the 
French mathematician Marin Mersenne in 1640 first computed the speed 



SG 12.32 



Noise and the Sonic Boom 

The world seems to be increasingly loud with 
unpleasant, manmade noise. At worst it is a 
major nuisance and may be tiring, painful, and 
sometimes even physically harmful. Loud, 
prolonged noise can produce temporary 
deafness. Very loud noise, kept up for a long 
time, can produce some degree of permanent 
deafness, especially deafness with respect to 
high-frequency sounds. 

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 goes. 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 rug on 
the floor can absorb 90% of room noise. (A foot 
of fresh snow is an almost perfect absorber of 
noise outdoors. Cities and countrysides are 
remarkably hushed after a snowfall.) 

In the last few years a new kind of noise 
has appeared: 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 700 miles per hour. Many 
types of military airplanes can travel at two or 
three times this speed. Flying at such speeds, 
the planes unavoidably and continually produce 
sonic booms. SST (Supersonic Transport) planes 
are now in civilian use in some countries. The 
unavoidable boom raises important questions. 
What is the price of technological "progress"? 
Who gains, and what fraction of the population? 
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 his 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 it remains in the 
same position in a medium, the wave it produces 
spreads out symmetrically around it, as in 
diagram 1 . But 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.) In diagram 3, 
the source is moving through the medium faster 
than 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 
below. 



-9^ 








All these concepts apply not only to water 
waves but also to sound waves, including 
those disturbances 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 3-dimensions) 
that spreads away from the source. 

Actually two cones of sharp pressure 
change are formed: one originating at the 
front of the airplane 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," or 
"bang-zone"), is as long as the supersonic flight 
path itself. In such an area, typically thousands 
of miles long and 50 miles 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 350-ton 
supersonic transport plane would break many 
thousands of dollars worth of windows, plaster 
walls, etc., and cause fright and annoyance to 
millions of people. Thus the supersonic flight 
of such planes may have to be confined to 
over-ocean use — though it may even turn out 
that the annoyance to people on shipboard, on 
islands, etc., is so great that over-ocean flights, 
too, will have to be restricted. 



AIR 
PRESSURE 



normal 



'' 



10 30 30 40 so iO 10 g^ 10 l«0 do 

mi Mi seconds 

This curve represents the typical sonic boom 
from an airplane flying at supersonic speed 
(speed greater than about 700 mph). The pres- 
sure rises almost instantly, then falls relatively 
slowly to below-normal pressure, then rises 
again almost instantaneously. The second pres- 
sure rise occurs about 0.1 second after the 
first one, making the boom sound "double." 



Double-cone shock wave, or sonic boom, pro- 
duced by an airplane that is travelling (at 13- 
mile altitude) at three times the speed of sound. 
Building B is just being hit by shock wave, 
building A was struck a few seconds ago, and 
building C will be hit a few seconds later. 




132 



Unit 3 



Waves 



SG 12.33-12.35 



The article Silence, Please" in 
Reader 3 is an amusing fantasy 
about wave superposition. 



The acoustic properties of a hall 
filled with people are very different 
from those of the empty hall. 
Acoustical engineers sometimes fill 
the seats with felt-covered 
sandbags while making tests. 



of sound in air. But it took another seventy years before William Derham 
in England, comparing the flash and noise from cannons across 12 miles, 
came close to the modern measurements. 

Sound in air at 68°F moves at 1,125 feet per second (about 344 
meters per second or 770 mph). As for all waves, the speed of sound 
waves 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 sea water, their speed is about 4,890 ft/sec; in 
steel, about 16,000 ft/sec; in quartz, about 18,000 ft/sec. 

Interference of sound waves can be shown in a variety of ways. In a 
large hall with hard, sound-reflecting surfaces, there will 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 known 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 we 
sometimes see lightning without hearing thunder. Similar refraction of 
sound occurs in layers of water of different temperatures. .This sometimes 
causes problems in using sonar (sound navigation and ranging) devices at 
sea. Sonic refraction is used for a variety of purposes today. Geologists use 
them 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 waves travel through the earth and are 
received by detection devices at different locations. The path of the waves, 
as refracted by layers in the earth, can be calculated from the relative 
intensities of sound received. From knowledge of the paths, estimates can 
be made of the composition of the layers. 

We have already mentioned diffraction as a property of sound waves. 
Sound waves readily bend around corners and barriers to reach the 
listener within range. Sound waves reflect, as do rope or water waves, 
wherever they encounter a boundary between different media. Echo 
chamber effects (which can be artificially produced by electronics) have 
become famfliar to Hsteners who enjoy popular music. The "live" sound of 
a bare room results from multiple reflections of waves which normally 
would be absorbed by furniture, rugs, and curtains. The architechtural 
accidents called "whispering galleries" show vividly how sound can be 
focused by reflection from curved surfaces. Laboratory rooms which 
greatly reduce reflections are called anechoic chambers. All these effects 
are of interest in the study of acoustics. Moreover, the proper acoustical 
design of public buildings is now recognized as an important function by 
most good architects. 

In this chapter we have explained the basic phenomena of mechanical 
waves, ending with the theory of sound propagation. These explanations 
were considered the final triumph of Newtonian mechanics as applied to 
the transfer of energy of particles in motion. Most of the general principles 



Section 12.11 



Unit 3 133 




An anechoic chamber being used lor research in 
acoustics. Sound is almost completely absorbed 
during 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 accoustically de- 
signed for unamplified performances. 



of acoustics were discovered in the 1870's. Since then the study of 
acoustics has become involved with such fields as quantum physics. But 
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 
move back and forth or up and down in a medium. 

Q32 List five wave behaviors that can be demonstrated with sound 
waves. 

Q33 Why can't sound waves be polarized? 



134 l^nit 3 



The Triumph of Mechanics 




EPILOGUE Seventeenth-century scientists thought they could eventually 
explain all physical phenomena by reducing them to matter and motion. 
This mechanistic viewpoint became known as the Newtonian worldview or 
Newtonian cosmology, since its most impressive success was Newton's 
theory of planetary motion. Newton and other scientists of his time 
proposed to apply similar methods to other problems, as we mentioned in 
the Prologue to this unit. 

The early enthusiasm for this new approach to science is vividly 
expressed by Henry Power in his book Experimental Philosophy (1664). 
Addressing his fellow natural philosophers (or scientists, as we would now 
call them), he wrote: 

You are the enlarged and elastical Souls of the world, who, 
removing all former rubbish, and prejudicial resistances, do 
make way for the Springy Intellect to flye out into its desired 
Expansion . . . 

. . . This is the Age wherein (me-thinks) Philosophy comes 
in with a Spring-tide ... I see how all the old Rubbish must be 
thrown away, and carried away with so powerful an Inundation. 
These are the days that must lay a new Foundation of a more 
magnificent Philosophy, never to be overthrown: that will 
Empirically and Sensibly canvass the Phaenomena of Nature, 
deducing the causes of things from such Originals in Nature, as 
we observe are producible by Art, and the infallible 
demonstration of Mechanicks; and certainly, this is the way, and 
no other, to build a true and permanent Philolophy. 

In Power's day there were many people who did not regard the old 
Aristotelian cosmology as rubbish. For them, it provided a comforting sense 
of unity and interrelation among natural phenomena. They feared that this 
unity would be lost if everything was reduced simply to atoms moving 
randomly through space. The poet John Donne, in 1611, complained 
bitterly of the change already taking place in cosmology: 

And new Philosophy calls all in doubt. 

The Element of fire is quite put out; 

The Sun is lost, and th' earth, and no man's wit 

Can well direct him where to looke for it. 

And freely men confesse that this world's spent. 

When in the Planets, and the Firmament 

They seeke so many new; then see that this 

Is crumbled out againe to his Atomies. 

Tis all in peeces, all coherence gone; 

All just supply, and all Relation. . . 



Newtonian physics provided powerful methods for analyzing the world 
and uncovering the basic principles of motion for individual pieces of 
matter. But the richness and complexity of processes in the real world 
seemed infinite. Could Newtonian physics deal as successfully with these 
real events as with ideal processes in a hypothetical vacuum? Could the 
perceptions of colors, sounds, and smells really be reduced to "nothing 



Epilogue 



Unit 3 135 



but" matter and motion? In the seventeenth century, and even in the 
eighteenth century, it was too soon to expect Newtonian physics to answer 
these questions. There was still too much work to do in establishing the 
basic principles of mechanics and applying them to astronomical problems. 
A full-scale attack on the properties of matter and energy had to wait until 
the nineteenth century. 

This unit covered several successful applications and extensions of 
Newtonian mechanics which were accomplished by the end of the 
nineteenth century. For example, we discussed the conservation laws, new 
explanations of the properties of heat and gases, and estimates of some 
properties of molecules. We introduced the concept of energy, linking 
mechanics to heat and to sound. In Unit 4 we will show similar links to 
light, electricity, and magnetism. We also noted that applying mechanics on 
a molecular level requires statistical ideas and presents questions about the 
direction of time. 

Throughout most of this unit we have emphasized the application of 
mechanics to separate pieces or molecules of matter. But scientists found 
that the molecular model was not the only way to understand the behavior 
of matter. Without departing from basic Newtonian cosmology, scientists 
could also interpret many phenomena (such as sound and light) in terms of 
wave motions in continuous matter. By the middle of the nineteenth century 
it was generally believed that all physical phenomena could be explained 
by a theory that was built on the use of either particles or waves. In the 
next unit, we will discover how much or how little validity there was in this 
belief. We will begin to see the rise of a new viewpoint in physics, based 
on the field concept. Then, in Unit 5, particles, waves, and fields will come 
together in the context of twentieth-century physics. 





1 ? 1 

The Project Physics materials particularly 
appropriate for Chapter 12 include: 

Experiments 

Sound 
Activities 

Standing Waves on a Drum and a Violin 
Moire' Patterns 
Music and Speech Activities 
Measurement of the Speed of Sound 
Mechanical Wave Machines 
Film Loops 
Superposition 

Standing Waves in a String 
Standing Waves in a Gas 
Four loops on vibrations 

Reader Articles 

Silence, Please 

Frontiers of Physics Today: Acoustics 

Waves 

What is a Wave 

Musical Instruments and Scales 

Founding a Family of Fiddles 

^^•^ Some waves propagate at such a high 
speed that we are usually not aware of any 
delay in energy transfer. For example, the delay 
between the flash and the "bang" in watching 
lightning or fireworks seems peculiar, because the 
propagation time for sounds produced near us is 
not noticeable. Give an example of a compression 
wave in a solid, started by an action at one end, 
that propagates so quickly that we are not 
aware of any delay before an eff'ect at the 
other end. 

12.3 Describe the differences in phase of 
oscillation of various parts of your body as you 
walk. What points are exactly in phase? Which 
points are exactly 7 cycle out of phase? Are 
there any points 7 cycle out of phase? 

'2-^ Pictured are two pulse waves (A and B) on 
a rope at the instants before and after they 




plot the shape of the rope at the end of each 
interval. 

12 5 

Repeat Exercise 12.3 for the two pulses 

(A and C) pictured at the top. 

12.6 'Yhe wave below propagates to the right 
along a rope. What is the shape of the wave 
propagating to the left that could for an instant 
cancel this one completely? 






overlap (t, and t.^). Divide the elapsed time 
between t, and t^ into four equal intervals and 



12 7 

The velocity of a portion of rope at some 

instant as transverse waves are passing through 
it is the superposition of the velocities of waves 
passing through that portion. Is the kinetic energy 
of a portion of the rope the superposition of the 
kinetic energies of waves passing through that 
region? Justify your answer. 

12s 

Graphically superpose the last three curves 

of the figure on p. 1 10 to find their sum (which 
should be the original curve). 

^•^ What shape would the nodal regions have 
for sound waves from two loudspeakers? 

'^•'*' Imagine a detection device for waves is 
moved slowly to either the right or left of the 
point labeled A„ in the figure on p. 114. Describe 
what the detection device would register. 

12.11 What kind of interference pattern would 
you expect to see if the separation between two 



136 



Unit 3 



STUDY GUIDE 



in-phase sources were less than the wavelength 
X? Where would the nodal and an tinodal lines be 
if the two in-phase sources were separated by 
the distance X? By \/2? Convince yourself that 
one additional nodal line appears on each side of 
the central antinodal line whenever the separatior 
between the two in-phase sources is increased by 
one wavelength. 

12.12 Derive an equation, similar to nkl = dx„, 
for nodal points in a two-source interference 
pattern (where d is the separation of the sources, 
I the distance from the sources, and x„ the 
distance of the n"" node from the center line). 

12.13 If you suddenly disturbed a stretched 
rubber hose or slinky with a frequency that 
precisely matched a standing wave frequency, 
would standing waves appear immediately? If 
not, what factors would determine the time delay? 

12.14 Different notes are sounded with the same 
guitar string by changing its vibrating length 
(that is, pressing the string against a brass 
ridge). If the full 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"? 

12.15 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 com- 
pletely free to be displaced, so the standing waves 
must have an antinode near the open end. 

(a) What will be the wavelength of the funda- 
mental standing wave in a pipe of length L closed 
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 (b) for the case of a pipe open 
at both ends. 

12.16 Imagine a spherical blob of jello in which 
you can set up standing vibrations. What would 
be some of the possible modes of vibration? 
(Hint: what possible symmetrical nodal surfaces 
could there be?) 

12.17 Suppose that straight-line ripple waves 
approach a thin straight barrier which is a few 
wavelengths long and which is oriented with its 
length parallel to the wavefronts. What do you 
predict about the nature of the diffraction pattern 
along a straight line behind the barrier which is 
perpendicular to the barrier and passes through 
the center of the barrier? Why do people who 
design breakwaters need to concern themselves 
with diffraction effects? 

12.18 A megaphone directs sound along the 
megaphone axis if the wavelength of the sound is 



small compared to the diameter of the opening. 
Estimate the upper limit of frequencies which are 
diffracted at a cheerleader's megaphone opening. 
Can you hear what a cheerleader shouts even 
though you are far off the axis of the megaphone? 

12.19 Explain why it is that the narrower a slit 
in a barrier is, the more nearly it can act like a 
point source of waves. 

12.20 If light is also a wave, then why have you 
not seen light being diffracted by the slits, say 
those of a picket fence, or diffracted around the 
comer of houses? 

12.21 By actual construction with a ruler and 
compass on a tracing of the photograph on p. 127, 
show that rays for the reflected wave front appear 
to come from S'. Show also that this is consistent 
with 0. = 6»i. 

12.22 A straight-line wave approaches a right- 
angle reflecting barrier as shown in the figure. 
Find the shape, size, and direction of propagation 
of the wave after it has been completely reflected 
by the barrier. 




12.23 With ruler and compass reproduce part (b) 
of the figure at the bottom of p. 124 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 larger than the one in 
the figure. (Hint: the dotted lines are along radii.) 

12.24 Convince yourself that a parabolic reflector 
will actually bring parallel wave-fronts to a sharp 
focus. Draw a parabola y = kx- (choosing any 
convenient value for k) and some parallel rays 
along the axis as in part (c) of the Figure at the 
bottom of p. 124. Construct line segments per- 
pendicular to the parabola where the rays hit it, 
and draw the reflected rays at equal angles on the 
other side of these lines. 

12.25 The /oca/ length of a curved reflector is the 
distance from the reflector to the point where 
parallel rays are focused. Use the drawing in 

SG 12.24 to find the focal length of a parabola 
in terms of k. 

12.26 Recalling that water surface waves travel 
slower in shallow water, what would you expect to 
happen to the shape of the following wave as it 



"v. 



iM^^!*'?*??^!;^-' 



'". T^y ^ T '.'.' v itrr 



Unit 3 



137 



continues to the right? Pay particular attention to 
the region of varying depth. Can you use the line 
of reasoning above to give at least a partial ex- 
planation of the cause of breakers near a beach? 

12.27 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. 



Sho^liovj 







12.28 On the opposite page is an aerial photograph 
of ocean waves entering from the upper right and 
encountering a small island. Describe the wave 
phenomena demonstrated by this encounter. 

12.29 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. 

B 



our 



^lass 



/ 



(a) Indicate and label an angle equal to angle 
of incidence d^. 

(b) Indicate and label an angle equal to angle 
of refraction ^b- 

(c) Label the wavelength in air \i. 

(d) Label the wavelength in glass X,,. 

(e) Show that vJVf^ = >^^/K■ 

(f) If you are familiar with trigonometry, show 
that sin 0Jsin 0^ = X\; A.,,. 



12.30 A periodic ripple-tank wave passes through 
a straight boundary between deep and shallow 
water. The angle of incidence at the boundary is 
45° and the angle of refraction is 30°. The 
propagation speed in the deep water is 0.35 m, sec. 
and the frequency of the wave is 10 cycles per 
sec. Find the wavelengths in the deep and 
shallow water. 

12.31 Look at Figure (d) on p. 127. Convince 
yourself that if a wave were to approach the 
boundary between medium 1 and medium 2 from 
below, along the same direction as the refracted 
ray in the figure, it would be refracted along the 
direction of the incident ray in the figure. This is 
another example of a general rule: if a wave 
follows a set of rays in one direction, then a wave 
can follow the same set of rays in the opposite 
direction. In other words, wave paths are 
reversible. 

12.32 Suppose that in an extremely quiet room 
you can barely hear a buzzing mosquito at a 
distance of one meter. 

(a) What is the sound power output of the 
mosquito? 

(b) How many mosquitoes would it take to supply 
the power for one lOO-watt reading lamp? 

(c) If the swarm were at ten meters' distance, 
what would the sound be like? (Sound intensity 
diminishes in proportion to the square of the 
distance from a point source.) 

12.33 How can sound waves be used to map the 
floors of oceans? 

12.34 Estimate the wavelength of a 1000 cycles 
per second sound wave in air; in water; in steel 
(refer to data in text). Do the same if /= 10,000 
cps. Design the dimensions of an experiment to 
show two-source interference for 1000 cps sound 
waves. 

12.35 Waves reflect from an object in a definite 
direction only when the wavelength is small 
compared to the dimensions of the object. This is 
true for sound waves as well as for any other. 
What does this tell you about the sound fre- 
quencies a bat must generate if it is to catch a 
moth or a fly? Actually some bats can detect 

the presence of a wire about 0.12 mm in 
diameter. Approximately what frequency does 
that require? 



138 Unit 3 




Refraction, reflection, and diffraction of waves around Farallon Island, 
California. There are breakers all around the coast. The swell coming from 
top right rounds both sides of the island, producing a crossed pattern 
below. The small islet radiates' the waves away in all directions. (U.S. 
Navy photograph.) 



Unit 3 



139 



Project Physics 



Handbook 



3 



The Triumph of Mechanics 







1 I 

Hid j 







r 

1 






Picture Credits 

Cover (upper left) Cartoon by Charles Gary Solin 
and reproduced by his permission only. Rocket 
photograph courtesy of Douglas MissUe and Space 
Systems Division, (right) HRW photo by Lonny 
Kalfus. 

Page 5 HRW photo by Lonny Kalfus. 

Page 13 Photo by J. Ph. Charbonnier (Photo Re- 
searchers). 

Page 15 Wide World Photo. 

Page 16 Brunswick Corporation. 

Page 80 (cartoon) By permission of Johnny Hart 
and Field Enterprises, Inc. 

Pages 49, 62, 63 Cartoons by Charles Gary Solin 
and reproduced by his permission only. 

Page 58 "Black Country," The Mansell Collection, 
London. 

Page 59 Carl Rose, New Yorker Magazine, 3/28/70. 

Page 59 Cartoon Laffs from True, Fawcett Publi- 
cations, Inc., Greenwich, Conn., 1958. 

Page 65 "Physics and Music," Scientific Ameri- 
can, July 1948. 

Page 89 Linda Cronquist, Wheat Ridge High 
School, Wheat Ridge, Colorado. 

All photographs used with Film Loops courtesy of 
National Film Board of Canada. 

Photographs of laboratory equipment and of stu- 
dents using laboratory equipment were suppUed 
with the cooperation of the Project Physics staff and 
Damon Corporation. 



Contents 



Project Physics Handbook 



Units 



EXPERIMENTS 

3-1 Collisions in One Dimension I i 
3-2 Collisions in One Dimension II 
3-3 Collisions in Two Dimensions I 
3-4 Collisions in Two Dimensions II 
3-5 Conservation of Energy I 22 
3-6 Conservation of Energy II 25 
3-7 Measuring the Speed of a Bullet 



5 

13 
16 



26 



Music and Speech Activities 66 
Measurement of the Speed of Sound 
Mechanical Wave Machines 67 
Resource Letter 68 



66 



3-8 



of a Pendulum 



3-9 
3-10 
3-11 
3-12 
3-13 



31 



Energy Analysis 

Swing 28 

Least Energy 29 

Temperature and Thermometers 

Calorimetry 33 

Ice Calorimetry 37 

Monte Carlo Experiment on Molecular 

Collisions 38 

14 Behavior of Gases 43 

15 Wave Properties 47 

16 Waves in a Ripple Tank 48 

17 Measuring Wavelength 49 

18 Sound 51 



76 
77 



77 
78 



3-19 Ultrasound 53 

ACTIVITIES 

Is Mass Conserved? 56 

Exchange of Momentum Devices 56 

Student Horsepower 57 

Steam-powered Boat 57 

Problems of Scientific and Technological 

Growth 58 
Predicting the Range of an Arrow 59 
Drinking Duck 59 
Mechanical Equivalent of Heat 60 
A Diver in a Bottle 60 
Rockets 62 
How to Weigh a Car With a Tire Pressure 

Gauge 62 
Perpetual Motion Machines? 62 
Standing Waves on a Drum and a Violin 63 
Moire Patterns 64 



FILM LOOP NOTES 

LI 8 One-dimensional Collisions I 
L19 One-dimensional ColUsions II 
L20 Inelastic One-dimensional 

CoUisions 77 
L21 Two-dimensional ColUsions I 
L22 Two-dimensional Collisions II 
L23 Inelastic Two-dimensional 

Collisions 78 
L24 Scattering of a Cluster of Objects 79 
L25 Explosion of a Cluster of Objects 79 
L26 Finding the Speed of a Rifle 

Bullet I 81 
L27 Finding the Speed of a Rifle 

Bullet II 82 
L28 Recoil 82 

L29 Colhding Freight Cars 83 
L30 Dynamics of a Billiard Ball 84 
L31 A Method of Measuring Energy — Nails 

Driven into Wood 85 
L32 Gravitational Potential Energy 86 
L33 Kinetic Energy 87 
L34 Conservation of Energy — Pole 

Vault 87 
L35 Conservation of Energy — Aircraft 

Takeoff 89 
L36 Reversibihty of Time 90 
L37 Superposition 90 
L38 Standing Waves on a String 91 
L39 Standing Waves in a Gas 92 
L40 Vibrations of a Wire 93 
L41 Vibrations of a Rubber Hose 94 
L42 Vibrations of a Drum 95 
L43 Vibrations of a Metal Plate 95 



Unit 3/4 



EXPERIMENTS 



EXPERIMENT 3-1 COLLISIONS 
IN ONE DIMENSION— I 

In this experiment you will investigate the 
motion of two objects interacting in one dimen- 
sion. The interactions (explosions and colli- 
sions in the cases treated here) are called 
one-dimensional because the objects move 
along a single straight line. Your purpose is to 
look for quantities or combinations of quanti- 
ties that remain unchanged before and after 
the interaction — that is, quantities that are 
conserved. 

Your experimental explosions and colli- 
sions may seem not only tame but also artificial 
and unUke the ones you see around you in 
everyday life. But this is typical of many 
scientific experiments, which simplify the 
situation so as to make it easier to make mean- 
ingful measurements and to discover patterns 
in the observed behavior. The underlying laws 
are the same for all phenomena, whether or not 
they are in a laboratory. 

Two different ways of observing interac- 
tions are described here (and two others in Ex- 
periment 3-2). You will probably use only one of 
them. In each method, the friction between the 
interacting objects and their surroundings is 
kept as small as possible, so that the objects 
are a nearly isolated system. Whichever 
method you do follow, you should handle your 
results in the way described in the final sec- 
tion: Analysis of data. 

Method A— Dynamics Carts 

"Explosions" are easily studied using the low- 
friction dynamics carts. Squeeze the loop of 
spring steel flat and slip a loop of thread over it, 
to hold it compressed. Put the compressed loop 
between two carts on the floor or on a smooth 

IS 








Fig. 1 




table (Fig. 1). When you release the spring by 
burning the thread, the carts fly apart with ve- 
locities that you can measure from a strobe 
photograph or by any of the techniques you 
learned in earUer experiments. 

Load the carts with a variety of weights to 
create simple ratios of masses, say 2 to 1 or 3 to 
2. Take data for as great a variety of mass 
ratios as time permits. Because friction will 
gradually slow the carts down, you should 
make measurements on the speeds im- 
mediately after the explosion is over (that is, 
when the spring is through pushing). 

Since you are interested only in comparing 
the speeds of the two carts, you can express 
those speeds in any units you wish, without 
worrying about the exact scale of the photo- 
graph and the exact strobe rate. For example, 
you can use distance units measured directly 
from the photograph (in millimeters) and use 
time units equal to the time interval between 
strobe images. If you follow that procedure, the 
speeds recorded in your notes will be in 
mm/interval. 

Remember that you can get data from the 
negative of a Polaroid picture as well as from 
the positive print. 

Method B—Air Track 

The air track allows you to observe collisions 
between objects — "ghders" — that move with 
almost no friction. You can take stroboscopic 
photographs of the gUders either with the 
xenon strobe or by using a rotating slotted disk 
in front of the camera. 

The air track has three ghders: two small 
ones with the same mass, and a larger one 
which has just twice the mass of a small one. A 
small and a large ghder can be coupled to- 
gether to make one glider so that you can have 
colhsions between gliders whose masses are in 
the ratio of 1:1, 2:1, and 3:1. (If you add hght 
sources to the gliders, their masses will no 
longer be in the same simple ratios. You can 
find the masses from the measured weights of 
the ghder and light source.) 

You can arrange to have the gliders bounce 
apart after they colhde (elastic colhsion) or 
stick together (inelastic collision). Good tech- 



Experiment 3-2 



Unit 3/5 








Examine your table carefully. Search for 
quantities or combinations of quantities that 
remain unchanged before and after the in- 
teraction. 



nique is important if you are to get consistent 
results. Before taking any pictures, try both 
elastic and inelastic collisions with a variety of 
mass ratios. Then, when you have chosen one 
to analyze, rehearse each step of your proce- 
dure with your partners before you go ahead. 

You can use a good photograph to find the 
speeds of both carts, before and after they col- 
lide. Since you are interested only in compar- 
ing the speeds before and after each colhsion, 
you can express speeds in any unit you wish, 
without worrying about the exact scale of the 
photograph or the exact strobe rate. For exam- 
ple, you use distance units measured directly 
from the photograph (in milhmeters) and use 
time units equal to the time interval between 
strobe images. If you follow that procedure, the 
speeds recorded in your notes will be in 
mm/interval. 

Remember that you can get data from the 
negative of your Polaroid picture as well as 
from your positive print. 

Analysis of Data 

Assemble all your data in a table having col- 
umn headings for the mass of each object, Wa 
and mg, the speeds before the interaction, v,^ 
and Vb (for explosions, f a = ^b = 0). ^^^ the 
speeds after the collision, v/ and Vq'. 



1. Is speed a conserved quantity? That 
is, does the quantity (i^a + ^"8) equal the quan- 
tity (Vj,' + z^b')? 

2. Consider the direction as well as the 
speed. Define velocity to the right as positive 
and velocity to the left as negative. Is velocity 
a conserved quantity? 

3. If neither speed nor velocity is con- 
served, try a quantity that combines the mass 
and velocity of each cart. Compare (m^^v^ + 
WbZ'b) with (ruj^Vj^' + msVa') for each interac- 
tion. In the same way compare tu/v, mv, mhj, 
or any other likely combinations you can 
think of, before and after interaction. What 
conclusions do you reach? 



EXPERIMENT 3-2 COLLISIONS IN 
ONE DIMENSION— II 

Method A— Film Loops 

Film Loops 3-1, 3-2, and 3-3 show one- 
dimensional collisions that you cannot easily 
perform in your own laboratory, for they were 
filmed with a very high speed camera, produc- 
ing the effect of slow motion when projected at 
the standard rate. You can make measure- 
ments directly from the pictures projected onto 
graph paper. Since you are interested only in 
comparing speeds before and after a collision, 
you can express speeds in any unit you wish 
— that is, you can make measurements in any 
convenient distance and time units. 

Notes for these film loops are located on 
pages 76 to 77. If you use these loops, read 
the notes carefully before taking your data. 

Method B—Stroboscopic 
Photographs 

Stroboscopic photographs showing seven dif- 
ferent examples of one-dimensional collisions 
appear on the following pages.* They are use- 
ful here for studying momentum and again 
later for studying kinetic energy. 

''Reproduced by permission of National Film Board of 
Canada 



Unit 3/6 



Experiment 3-2 



For each event you should find the speeds 
of the balls before and after collision. From the 
values for mass and speed of each ball, you 
should calculate the total momentum before 
and after collision. You will use the same val- 
ues to calculate the total kinetic energy before 
and after collision. 

You should read Section I, before analyz- 
ing any of the events, in order to find out what 
measurements to make and how the colUsions 
were produced. After you have made your 
measurements, turn to Section II for questions 
to answer about each event. 

/. The Measurements You Will Make 

To make the necessary measurements you will 
need a metric ruler marked in millimeters, 
preferably of transparent plastic with sharp 
scale markings. Before starting your work, 
consult Fig. 1 for suggestions on improving 
your measuring technique. 

main tma.rk5 , fj. . 

end rclioLble.'^ too thick r , r 7 




-l.?%^^m\ 



\ 



Fig. 1 

Fig. 2 shows schematically that the collid- 
ing balls were hung from very long wires. The 
balls were released from rest, and their 
double-wire (bifilar) suspensions guided them 
to a squarely head-on colhsion. Stroboscopes il- 
luminated the 3 X 4 ft rectangle that was the 
field of view of the camera. The stroboscopes 
are not shown in Fig. 2. 

Notice the two rods whose tops reach into 
the field of view. These rods were 1 meter (± 2 
milUmeters) apart, measured from top center 
of one rod to top center of the other. The tops of 
these rods are visible in the photographs on 




Fig. 2 Set-up for photographing one-dimensional 
collisions. 



which you will make your measurements. This 
enables you to convert your measurements to 
actual distances if you wish. However, it is 
easier to use the lengths in milUmeters mea- 
sured directly off the photograph if you are 
merely going to compare momenta. 

The balls speed up as they move into the 
field of view. Likewise, as they leave the field of 
view, they slow down. Therefore successive 
displacements on the stroboscopic photograph, 
each of which took exactly the same time, will 
not necessarily be equal in length. Check this 
with your ruler. 

As you measure a photograph, number the 
position of each ball at successive flashes of the 
stroboscope. Note the interval during which 
the colhsion occurred. Identify the clearest 
time interval for finding the velocity of each 
ball (a) before the colhsion and (b) after the 
collision. Then mark this information close on 
each side of the interval. 



Experiment 3-2 Unit 3/7 



II. Questions to be Answered about 
Each Event 

After you have recorded the masses (or relative 
masses) given for each ball and have recorded 
the necessary measurements of velocities, 
answer the following questions. 

1. What is the total momentum of the system 
of two balls before the colUsion? Keep in mind 
here that velocity, and therefore momentum, 
are vector quantities. 

2. What is the total momentum of the system 
of two balls after the collision? 

3. Was momentum conserved within the 
Umits of precision of your measurements? 

Event 1 

The photographs of this Event 1 and all the 
following events appear below as Figs. 10 to 
16. This event is also shown as the first exam- 
ple in Film Loop LI 8, "One-Dimensional Colli- 
sions I." 

Figure 3 shows that ball B was initially at 
rest. After the colUsion both balls moved off to 
the left. The balls are made of steel. 



&VENT 1 



before 



fe 




3.-ft-e 



350 grams 552 ^rams 



Fig. 3 

Event 2 

This event, the reverse of Event 1, is shown as 
the second example in Film Loop LI 8, "One- 
Dimensional Colhsions I." 

Fig. 3 shows that ball B came in from the 
left and that ball A was initially at rest. The 
collision reversed the direction of motion of ball 
B and sent ball A off to the right. (The balls are 
of hardened steel.) 

As you can tell by inspection, ball B moved 
slowly after collision, and thus you may have 
trouble getting a precise value for its speed. 
This means that your value for this speed is the 



&V&NT 2. 

be-fore C3 — >■ 

350 ^ram^ 532 ^ra,ms> 




Fig. 4 



O 



least reliable of your four speed measurements. 
Nevertheless, this fact has only a small 
influence on the reliability of your value for the 
total momentum after colhsion. Can you ex- 
plain why this should be so? 

Why was the direction of motion of ball B 
reversed by the collision? 

If you have already studied Event 1, you 
will notice that the same balls were used in 
Events 1 and 2. Check your velocity data, and 
you will find that the initial speeds were nearly 
equal. Thus, Event 2 was truly the reverse of 
Event 1. Why, then, was the direction of mo- 
tion of ball A in Event 1 not reversed although 
the direction of ball B in Event 2 was reversed? 

Event 3 

This event is shown as the first example in 
Film Loop L19, "One-Dimensional Collisions 
n." Event 3 is not recommended unless you 
also study one of the other events. Event 3 is 
especially recommended as a companion to 
Event 4. 

Fig. 5 shows that a massive ball (A) en- 
tered from the left. A less massive ball B came 
in from the right. The directions of motion of 
both balls were reversed by the colhsion. (The 
balls were made of hardened steel.) 

When you compare the momenta before 
and after the colhsion you will probably find 
that they differed by more than any other event 
so far in this series. Explain why this is so. 

Event 4 

This event is also shown as the second ex- 



Unit 3/8 Experiment 3-2 



be-Pore 
Fig. 5 




l,80 kilogK-atn 532 g trains 



O o 



ample in Film Loop LI 9, "One-Dimensional 
Collisions II." 

Fig. 6 shows that two balls came in from 
the left, that ball A was far more massive than 
ball B, and that ball A was moving faster than 
ball B before collision. The colhsion occurred 
when A caught up with B, increasing B's speed 
at some expense to its own speed. (The balls 
were made of hardened steel.) 

Each ball moved across the camera's field 
from left to right on the same line. In order to 
be able to tell successive positions apart on a 
stroboscopic photograph, the picture was taken 

&VB-NT A- 



O o 



Fig. 6 



twice. The first photograph shows only the 
progress of the large ball A because ball B had 
been given a thin coat of black paint (of negh- 
gible mass). Ball A was painted black when the 
second picture was taken. It wUl help you to 
analyze the collision if you actually number 
white-ball positions at successive stroboscope 
flashes in each picture. 

Event 5 

This event is also shown as the first example in 



Film Loop L20, "Inelastic One-Dimensional 
Collisions." You should find it interesting to 
analyze this event or Event 6 or Event 7, but it 
is not necessary to do more than one. 



E-V^NT 5 



before 



& A 

o 




a-Fte^ 



-O 



Fig. 7 



Fig. 7 shows that ball A came in from the 
right, striking ball B which was initially at rest. 
The balls were made of a soft material (plas- 
ticene). They remained stuck together after the 
collision and moved off to the left as one. A 
colhsion of this type is called "perfectly 
inelastic ." 

Event 6 

This event is shown as the second example in 
Fibn Loop L20, "Inelastic One-Dimensional 
Collisions." 

Fig. 8 shows that balls A and B moved in 
from the right and left, respectively, before col- 
lision. The balls were made of a soft material 
(plasticene). They remained stuck together 
after the collision and moved off together to the 
left. This is another "perfectly inelastic" colh- 
sion, like that in Event 5. 

This event was photographed in two parts. 
The first print shows the conditions before col- 
lision, the second print, after collision. Had the 
picture been taken with the camera shutter 
open throughout the motion, it would be 
difficult to take measurements because the 
combined balls (A + B) — after collision — re- 
traced the path which ball B followed be- 
fore collision. You can number the positions of 
each ball before collision at successive flashes 
of the stroboscope (in the first photo); and you 
can do likewise for the combined balls (A + B) 
after the colhsion in the second photo. 



Experiment 3-2 Unit 3/9 



B A 

bc-for«i /^'^ — ^ „ 

445 grams GG2 grams 




A-+- B 



Fig.8 




A & 

4.79 kilogra^nr^ GfeO grams 
A+ B 



af-tcr 




Fig. 9 



Event 7 

Fig. 9 shows that balls A and B moved in from 
opposite directions before colhsion. The balls 
are made of a soft material (plasticene). They 
remain stuck together after colhsion and move 
off together to the right. This is another so- 
called "perfectly inelastic'' collision. 

This event was photographed in two parts. 
The first print shows the conditions before col- 
lision, the second print, after colhsion. Had the 
picture been made with the camera shutter 
open throughout the motion, it would be 



difficult to take measurements because the 
combined balls (A + B) trace out the same path 
as incoming ball B. You can number the posi- 
tions of each ball before colhsion at successive 
flashes of the stroboscope (in the first photo- 
graph), and you can do likewise for the com- 
bined balls (A -I- B) after collision in the second 
photograph. 

Photographs of the Events 

The photographs of the events are shown in 
Fig. 10 through 16. 




Fig. 10 Event 1,10 flashes/sec 



Unit 3I^0 Experiment 3-2 



before 



after 




Fig. 11 Event 2, 10 flashes/sec 



before 



after 




Fig. 12 Event 3. 10 flashes/sec 



Experiment 3-2 Unit SHA 



ball A 



ball B 




Fig. 13 Event 4, 10 flashes/sec 




Fig. 14 Event 5, 10 flashes/sec 



Unit 3^2 Experiment 3-2 



before 




after 




Fig. 15 Event 6, 10 flashes/sec 



before 



after 




Fig. 16 Event 7, 10 flashes/sec 



UnitSnZ 



EXPERIMENT 3-3 COLLISIONS 
IN TWO DIMENSIONS— I 

Collisions rarely occur in only one dimension, 
that is, along a straight line. In billiards, bas- 
ketball, and tennis, the ball usually rebounds 
at an angle to its original direction; and ordi- 
nary explosions (which can be thought of as 
collisions in which initial velocities are all 
zero) send pieces flying off in all directions. 




This experiment deals with colhsions that 
occur in two dimensions — that is, in a single 
plane — instead of along a single straight line. It 
assumes that you know what momentum is 
and understand what is meant by "conserva- 
tion of momentum" in one dimension. In this 
experiment you will discover a general form of 
the rule for one dimension that applies also to 
the conservation of momentum in cases where 
the parts of the system move in two (or three) 
dimensions. 



Two methods of getting data on two- 
dimensional colhsions are described below 
(and two others in Experiment 3-4), but you 
will probably want to follow only one method. 
Whichever method you use, handle your re- 
sults in the way described in the last section. 

Method A— Colliding Pucfcs 

On a carefully leveled glass tray covered with a 
sprinkhng of Dyhte spheres, you can make 
pucks coast with almost uniform speed in any 
direction. Set one puck motionless in the center 
of the table and push a second similar one to- 
ward it, a little off-center. You can make excel- 
lent pictures of the resulting two-dimensional 
glancing collision with a camera mounted di- 
rectly above the surface. 

To reduce reflection from the glass tray, 
the photograph should be taken using the 
xenon stroboscope with the hght on one side 
and almost level with the glass tray. To make 
each puck's location clearly visible in the 
photograph, attach a steel ball or a small white 
Styrofoam hemisphere to its center. 

The large puck has twice the mass of the 
small puck. You can get a greater variety of 
masses by stacking pucks one on top of the 
other and fastening them together with tape 
(but avoid having the colhsions cushioned by 
the tape). 

Two people are needed to do the experi- 
ment. One experimenter, after some prelimi- 
nary practice shots, launches the projectile 
puck while the other experimenter operates the 
camera. The resulting picture should consist of 
a series of white dots in a rough "Y" pattern. 

Using your picture, measure and record all 
the speeds before and after colhsion. Record 
the masses in each case too. Since you are in- 
terested only in comparing speeds, you can use 
any convenient speed units. You can simplify 
your work if you record speeds in mm/dot in- 
stead of trying to work them out in cm/sec. Be- 
cause friction does slow the pucks down, find 
speeds as close to the impact as you can. You 
can also use the "puck" instead of the kUogram 
as your unit of mass. 



UnitanA 



Experiment 3-3 



Method B-— Colliding 
Disk Magnets 

Disk magnets will also slide freely on Dylite 
spheres as described in Method A. 

The difference here is that the magnets 
need never touch during the "colUsion." Since 
the interaction forces are not really instan- 
taneous as they are for the pucks, the magnets 
follow curving paths during the interaction. 
Consequently the "before" velocity should be 
determined as early as possible and the "after" 
velocities should be measured as late as possi- 
ble. 

Following the procedure described above 
for pucks, photograph one of these "collisions." 
Again, small Styrofoam hemispheres or steel 
balls attached to the magnets should show up 
in the strobe picture as a series of white dots. 
Be sure the paths you photograph are long 
enough so that the dots near the ends form 
straight lines rather than curves. 




Using your photograph, measure and re- 
cord the speeds and record the masses. You can 
simplify your work if you record speeds in 
mm/dot instead of working them out in cm/sec. 
You can use the disk instead of the kilogram as 
your unit of mass. 

Analysis of Data 

Whichever procedure you used, you should 
analyze your results in the following way. Mul- 
tiply the mass of each object by its before-the- 
collision speed, and add the products. 



1 . Do the same thing for each of the ob- 
jects in the system after the collision, and add 
the after-the-collision products together. Does 
the sum before the colUsion equal the sum 
after the colHsion? 



Imagine the collision you observed was an 
explosion of a cluster of objects at rest; the 
total quantity mass-times-speed before the 
explosion will be zero. But surely, the mass- 
times-speed of each of the flying fragments 
after the explosion is more than zero! "Mass- 
times-speed" is obviously not conserved in an 
explosion. You probably found it wasn't con- 
served in the experiments with pucks and 
magnets, either. You may already have sus- 
pected that you ought to be taking into account 
the directions of motion. 

To see what is conserved, proceed as fol- 
lows. 

Use your measurements to construct a 
drawing like Fig. 1 , in which you show the di- 
rections of motion of all the objects both before 
and after the collision. 



^ 



\ 



-^ 



Fig. 1 



Have all the direction lines meet at a 
single point in your diagram. The actual paths 
in your photographs will not do so, because the 
pucks and magnets are large objects instead of 
points, but you can still draw the directions of 
motion as lines through the single point P. 

On this diagram draw a vector arrow 
whose magnitude (length) is proportional to 
the mass times the speed of the projectile 
before the collision. (You can use any conve- 



Experiment 3-3 



Unit 3/1 5 



nient scale.) In Fig. 2, this vector is marked 
rrij^Vf,. Before an explosion there is no motion at 
all, and hence, no diagram to draw. 




Fig. 2 



Below your first diagram draw a second 
one in which you once more draw the direc- 
tions of motion of all the objects exactly as 
before. On this second diagram construct the 
vectors for mass-times-speed for each of the 
objects leaving P after the collision. For the 
colhsions of pucks and magnets your diagram 
will resemble Fig. 3. Now construct the "after 
the colhsion" vector sum. 



^aV, 




/^/T^/r p 



Fig. 3 

The length of each of your arrows is given 
by the product of mass and speed. Since each 
arrow is drawn in the direction of the speed, 
the arrows represent the product of mass and 
velocity mv which is called moTnentum. The 
vector sums "before" and "after" collision 
therefore represent the total momentum of the 
system of objects before and after the colhsion. 
If the "before" and "after" arrows are equal, 
then the total momentum of the system of in- 
teracting objects is conserved. 



2. How does this vector sum compare 
with the vector sum on your before-the- 
colhsion figure? Are they equal within the 
uncertainty? 



3. Is the principle of conservation of 
momentum for one dimension different from 
that for two, or merely a special case of it? 
How can the principle of conservation of 
momentum be extended to three dimensions? 
Sketch at least one example. 

4. Write an equation that would express 
the principle of conservation of momentum 
for collisions of (a) 3 objects in two dimen- 
sions, (b) 2 objects in three dimensions, (c) 3 
objects in three dimensions. 



jmm 




A 3,000-pound steel ball swung by a crane against the 
walls of a condemned building. What happens to the 
momentum of the ball? 



Unit3n6 



EXPERIMENT 3-4 COLLISIONS IN 
TWO DIMENSIONS— II 

Method A^Film Loops 

Several Film Loops (L21, L22, L23, L24, and 
L25) show two-dimensional collisions that you 
cannot conveniently reproduce in the labora- 
tory. Notes on these films appear on pages 77- 
79. Project one of the loops on the chalkboard 
or on a sheet of graph paper. Trace the paths of 
the moving objects and record their masses 
and measure their speeds. Then go on to the 
analysis described in the notes for Film Loop 
L21 on p. 77. 

Method BStroboscopic 
Photographs 

Stroboscopic photographs* of seven different 
two-dimensional colhsions in a plane are used 
in this experiment. The photographs (Figs. 5 to 
12) are shown on the pages immediately fol- 
lowing the description of these events. They 
were photographed during the making of Film 
Loops L21 through L25. 

/. Material Needed 

1. A transparent plastic ruler, marked in mil- 
limeters. 

2. A large sheet of paper for making vector 
diagrams. Graph paper is especially conve- 
nient. 

3. A protractor and two large drawing trian- 
gles are useful for transferring direction vec- 
tors from the photographs to the vector dia- 
grams. 

//. How the Collisions were 
Produced 

Balls were hung on 10-meter wires, as shown 
schematically in Fig. 1 . They were released so 
as to collide directly above the camera, which 
was facing upward. Electronic strobe lights 
(shown in Fig. 4) illuminated the rectangle 
shown in each picture. 

Two white bars are visible at the bottom of 
each photograph. These are rods that had their 
tips 1 meter (± 2 millimeters) apart in the ac- 
tual situation. The rods make it possible for you 

"Reproduced by permission of National Film Board of 
Canada 




Fig. 1 Set-up for photographing two-dimensional 
collisions. 

to convert your measurements to the actual 
distance. It is not necessary to do so, if you 
choose instead to use actual on-the-photograph 
distances in milhmeters as you may have done 
in your study of one-dimensional collisions. 

Since the balls are pendulum bobs, they 
move faster near the center of the photographs 
than near the edge. Your measurements, 
therefore, should be made near the center. 

///. A Sample Procedure 

The purpose of your study is to see to what ex- 
tent momentum seems to be conserved in 
two-dimensional colhsions. For this purpose 
you need to construct vector diagrams. 

Consider an example: in Fig. 2, a 450 g and 
a 500 g ball are moving toward each other. 
Ball A has a momentum of 1.8 kg-m/sec, in 
the direction of the ball's motion. Using the 
scale shown, you draw a vector 1.8 units long, 
parallel to the direction of motion of A. Simi- 
larly, for ball B you draw a momentum vector 
of 2.4 units long, parallel to the direction of 
motion of B. 



Experiment 3-4 Unit 3IM 



A50 



\ 



6<sa(e 




Fig. 2 Two balls moving in a plane. Their individual 
momenta, which are vectors, are added together vector- 
ially in the diagram on the lower right. The vector sum is 
the. total momentum of the system of two balls. (Your 
own vector drawings should be at least twice this size.) 



\ 
\ 

A -/ 



A. 9 m/sec ' 
/ 



> ^ 



^ 



Scale 
! icq. m/%ec 
|_r 1 




3-0 rr\/6<SC 



The system of two balls has a total momen- 
tum before the collision equal to the vector sum 
of the two momentum vectors for A and B. 

The total momentum after the colhsion is 
also found the same way, by adding the 
momentum vector for A after the colhsion to 
that for B after the collision (see Fig. 3). 

This same procedure is used for any event 
you analyze. Determine the momentum (mag- 
nitude and direction) for each object in the 
system before the collision, graphically add 
them, and then do the same thing for each 
object after the collision. 

For each event that you analyze, consider 
whether momentum is conserved. 

Events 8, 9, 10, and 11 

Event 8 is also shown as the first example in 



be-Fore 




af-ter 



Fig. 3 The two balls collide and move away. Their indi- 
vidual momenta after collision are added vectorially. 
The resultant vector is the total momentum of the sys- 
tem after collision. 



Unit 3m 



Experiment 3-4 



Fibn Loop L22, "Two-Dimensional Collisions: 
Part II," as well as on Project Physics Trans- 
parency T-20. 

Event 10 is also shown as the second ex- 
ample in Film Loop L22. 

Event 11 is also shown in Film Loop L21, 
"Two-Dimensional Colhsions: Part I," and on 
Project Physics Transparency T-21 . 

These are all elastic collisions. Events 8 
and 10, are simplest to analyze because each 
shows a colUsion of equal masses. In Events 8 
and 9, one ball is initially at rest. 

A small sketch next to each photograph in- 
dicates the direction of motion of each ball. The 
mass of each ball and the strobe rate are also 
given. 

Events 12 and 13 

Event 12 is also shown as the first example in 
Film Loop L23, "Inelastic Two-Dimensional 
Collisions." 

Event 13 is also shown as the second ex- 
ample in Film Loop L23. A similar event is 
shown and analyzed in Project Physics Trans- 
parency T-22. 

Since Events 12 and 13 are similar, there 
is no need to do both. 

Events 12 and 13 show inelastic collisions 
between two plasticene balls that stick to- 
gether and move off as one compound object 
after the collision. In 13 the masses are equal; 
in 12 they are unequal. 

Caution: You may find that the two objects 
rotate sUghtly about a common center after the 
colhsion. For each image after the collision, 
you should make marks halfway between the 
centers of the two objects. Then determine the 
velocity of this "center of mass," and multiply 
it by the combined mass to get the total 
momentum after the colUsion. 

Event 14 

Do not try to analyze Event 14 unless you have 
done at least one of the simpler events 8 
through 13. 

Event 14 is also shown on Film Loop L24 
"Scattering of a Cluster of Objects." 

Figure 4 shows the setup used in photo- 
graphing the scattering of a cluster of balls. 



The photographer and camera are on the floor, 
and four electronic stroboscope Ughts are on 
tripods in the lower center of the picture. 

You are to use the same graphical methods 
as you used for Events 8 through 13 to see if the 
conservation of momentum holds for more 
than two objects. Event 14 is much more com- 
plex because you must add seven vectors, 
rather than two, to get the total momentum 
after the colhsion. 

In Event 14, one ball comes in and strikes a 
cluster of six balls of various masses. The balls 
were initially at rest. Two photographs are in- 
cluded: Print 1 shows only the motion of ball A 
before the event. Print 2 shows the positions of 
all seven balls just before the colhsion and the 
motion of each of the seven balls after the colli- 
sion. 

You can analyze this event in two different 
ways. One way is to determine the initial 
momentum of ball A from measurements 
taken on Print 1 and then compare it to the 
total final momentum of the system of seven 
balls from measurements taken on Print 2. The 
second method is to determine the total final 
momentum of the system of seven balls 




Fig. 4 Catching the seven scattered balls to avoid 
tangling in the wires from which they hang. The photog- 
rapher and the camera are on the floor. The four strobo- 
scopes are seen on tripods in the lower center of the 
picture. 



Experiment 3-4 



Unit3n9 



on Print 2, predict the momentum of ball A, 
and then take measurements of Print 1 to see 
whether ball A had the predicted momentum. 
Choose one method. 

The tops of prints 1 and 2 lie in identical 
positions. To relate measurements on one print 
to measurements on the other, measure a ball's 
distance relative to the top of one picture with a 
rule; the ball would lie in precisely the same 
position in the other picture if the two pictures 
could be superimposed. 

There are two other matters you must con- 
sider. First, the time scales are different on the 



two prints. Print 1 was taken at a rate of 5 
flashes/second, and Print 2 was taken at a rate 
of 20 flashes/second. Second, the distance 
scale may not be exactly the same for both 
prints. Remember that the distance from the 
center of the tip of one of the white bars to 
center of the tip of the other is 1 meter (± 2 
mm) in real space. Check this scale carefully 
on both prints to determine the conversion 
factor. 

The stroboscopic photographs for Events 
8 to 14 appear in Figs. 5 to 12. 




AQ3G73 



H' 



B O ^'^^ 9 



Fig. 5 Event 8, 20 flashes/sec 



B Q 540c\ 



Fig. 6 Event 9, 20 flashes/sec 



Unit 3120 Experiment 3-4 



2>G7g 5G7g 



Fig. 7 Event 10, 20 flashes/sec 



5593 2)GI_3 



Fig. 8 Event 1 1 , 20 flashes/sec 



559 c» 



A 




B 



Fig. 9 Event 12, 20 flashes/sec 




Experiment 3-4 



Unit 3/21 




5COg 



500g 



Fig. 10 Event 13, 10 flashes/sec 




AjO. 6OO3 



Fig. 11 Event 14, print 1, 5 
flashes/ sec 



1 


• 




r 

B BAIL A 


546 g 


BALL B 


366 S 


BALL C 


539 g 


BALL D 


357 g 


BALL E 


220 g 


BALL F 


1.0 2 kg 


BALL G 


1.78 kg 

1 




X3 



Fig. 12 Event 14, print 2, 20 
flashes/sec 



Unit 3/22 



EXPERIMENT 3-5 
CONSERVATION OF ENERGY— I 

In the previous experiments on conservation of 
momentum, you recorded the results of a 
number of coUisions involving carts and glid- 
ers having different initial velocities. You 
found that within the limits of experimental 
uncertainty, momentum was conserved in 
each case. You can now use the results of these 
colhsions to learn about another extremely 
useful conservation law, the conservation of 
energy. 

Do you have any reason to believe that the 
product of w and v is the only conserved quan- 
tity? In the data obtained from your photo- 
graphs, look for other combinations of quan- 
tities that might be conserved. Find values for 
m/v, m'h), and mv'^ for each cart before and after 
collision, to see if the sum of these quantities 
for both carts is conserved. Compare the re- 
sults of the elastic collisions with the inelastic 
ones. Consider the "explosion" too. 

Is there a quantity that is conserved for one 
type of collision but not for the other? 

There are several alternative methods to 
explore further the answer to this question; you 
will probably wish to do just one. Check your 



results against those of classmates who use 
other methods. 

Method A— Dynamics Carts 

To take a closer look at the details of an elastic 
colhsion, photograph two dynamics carts as 
you may have done in the previous experiment. 
Set the carts up as shown in Fig. 1. 

The mass of each cart is 1 kg. Extra mass 
is added to make the total masses 2 kg and 4 
kg. Tape a light source on each cart. So that 
you can distinguish between the images 
formed by the two lights, make sure that one of 
the bulbs is slightly higher than the other. 

Place the 2-kg cart at the center of the 
table and push the other cart toward it from the 
left. If you use the 12-slot disk on the strobo- 
scope, you should get several images during 
the time that the spring bumpers are touching. 
You will need to know which image of the 
right-hand cart was made at the same instant 
as a given image of the left-hand cart. Match- 
ing images will be easier if one of the twelve 
slots on the stroboscope disk should be sUghtly 
more than half-covered with tape. (Fig. 2.) Im- 
ages formed when that slot is in front of the 
lens will be fainter than the others. 



^^. 







'O 




/V 



-"'X- 



^un o-f{ in cthfer^ of h.hk \ \ 



r-iv r'd 



y 






Fig. 1 



Experiment 3-5 



Unit 3/23 




Fig. 2 

Compute the values for the momentuni, 
(mv), for each cart for each time interval while 
the springs were touching, plus at least three 
intervals before and after the springs touched. 
List the values in a table, making sure that you 
pair off the values for the two carts correcdy. 
Remember that the lighter cart was initially at 
rest while the heavier one moved toward it. 
This means that the first few values ofmv for 
the lighter cart will be zero. 

On a sheet of graph paper, plot the momen- 
tum of each cart as a function of time, using 
the same coordinate axes for both. Connect 
each set of values with a smooth curve. 

Now draw a third curve which shows the 
sum of the two values ofmv, the total momen- 
tum of the system for each time interval. 



1. Compare the final value ofmv for the 
system with the initial value. Was momen- 
tum conserved in the colUsion? 

2. What happened to the momentum of 
the system while the springs were touching 
— was momentum conserved during the 
collision? 



Now compute values for the scalar quan- 
tity mv"^ for each cart for each time interval, 
and add them to your table. On another sheet of 
graph paper, plot the values of mv^ for each 
cart for each time interval. Connect each set of 
values with a smooth curve. 

Now draw a third curve which shows the 
sum of the two values ofmv'^ for each time in- 
terval. 



3. Compare the final value ofmv"^ for the 
system with the initial value. Is mv^ a con- 
served quantity? 



4. How would the appearance of your 
graph change if you multiphed each quantity 
by i? (The quantity ^v"^ is called the kinetic 
energy of the object of mass m and speed v.) 



Compute values for the scalar quantity 
imi;^ for each cart for each time interval. On a 
sheet of graph paper, plot the kinetic energy of 
each cart as a function of time, using the same 
coordinate axes for both. 

Now draw a third curve which shows the 
sum of the two values of ^v^, for each time 
interval. 



5. Does the total amount of kinetic 
energy vary during the collision? If you found 
a change in the total kinetic energy, how do 
you explain it? 



Method B — Magnets 

Spread some Dyhte spheres (tiny plastic beads) 
on a glass tray or other hard, flat surface. A 
disc magnet will shde freely on this low- 
friction surface. Level the surface carefully. 

Put one magnet puck at the center and 
push a second one toward it, shghtly off center. 
You want the magnets to repel each other with- 
out actually touching. Try varying the speed 
and direction of the pushed magnet until you 
find conditions that make both magnets move 
off after the collision with about equal speeds. 

To record the interaction, set up a camera 
directly above the glass tray (using the motor- 
strobe mount if your camera does not attach 
directly to the tripod) and a xenon stroboscope 
to one side as in Fig. 3. Mount a steel ball or 
Styrofoam hemisphere on the center of each 
disk with a small piece of clay. The ball will 
give a sharp reflection of the strobe Ught. 

Take strobe photographs of several inter- 
actions. There must be several images before 
and after the interaction, but you can vary 
the initial speed and direction of the moving 
magnet, to get a variety of interactions. Using 
your photograph, calculate the "before" and 
"after" speeds of each disk. Since you are 
interested only in comparing speeds, you can 
use any convenient units for speed. 



Experiment 3-5 




^EfjDfj sr/^oee 



1. Is m^i^ a conserved quantity? Is iwz;2 a 
conserved quantity? 



If you find there has been a decrease in the 
total kinetic energy of the system of interacting 
magnets, consider the following: the surface is 
not perfectly frictionless and a single magnet 
disk pushed across it wiU slow down a bit. 
Make a plot of iwi;^ against time for a moving 
puck to estimate the rate at which kinetic 
energy is lost in this way. 



2. How much of the loss in \vnv'^ that you 
observed in the interaction can be due to 
friction? 

3. What happens to your results if you 
consider kinetic energy to be a vector quan- 
tity? 



When the two disks are close together (but 
not touching) there is quite a strong force be- 
tween them pushing them apart. If you put the 
two pucks down on the surface close together 
and release them, they will fly apart: the kinet- 
ic energy of the system has increased. 

If you have time to go on, you should try to 
find out what happens to the total quantity 
imf^ of the disks while they are close together 
during the interaction. To do this you will need 
to work at a fairly high strobe rate, and push 
the projectile magnet at fairly high speed 
— without letting the two magnets actually 



touch, of course. Close the camera shutter be- 
fore the disks are out of the field of view, so that 
you can match images by counting backward 
from the last images. 

Now, working backward from the last in- 
terval, measure v and calculate ^v^ for each 
puck. Make a graph in which you plot ^v^ for 
each puck against time. Draw smooth curves 
through the two plots. 

Now draw a third curve which shows the 
sum of the two ^mv^ values for each time inter- 
val. 



4. Is the quantity imf ^ conserved during 
the interaction, that is, while the repelling 
magnets approach very closely? 

Try to explain your observations. 



Method C— Inclined Air TraclfS 

Suppose you give the glider a push at the bot- 
tom of an inclined air track. As it moves up the 
slope it slows down, stops momentarily, and 
then begins to come back down the track. 

Clearly the bigger the push you give the 
glider (the greater its initial velocity vO, the 
higher up the track it will chmb before stop- 
ping. From experience you know that there is 
some connection between i^j and d, the dis- 
tance the ghder moves along the track. 

According to physics texts, when a stone is 
thrown upward, the kinetic energy that it has 
initially (imf i^) is transformed into gravita- 
tional potential energy (mUgh) as the stone 
moves up. In this experiment, you will test to 
see whether the same relationship apphes to 
the behavior of the ghder on the inchned air 
track. In particular, your task is to find the ini- 
tial kinetic energy and the increase in potential 
energy of the air track ghder and to compare 
them. 

The purpose of the first set of measure- 
ments is to find the initial kinetic energy ^i^,^ 
You cannot measure v^ directly, but you can 
find it from your calculation of the average 
velocity v^^ as follows. In the case of uniform 
acceleration v^v = K^i + vd, and since t;, = at 
the top of the track, fgv = ^Vi or u, = 2rav Re- 
member that f av = Ad/At, sof, = 2Ad/At; Ad and 
At are easy to measure with your apparatus. 



Experiment 3-6 



Unit 3/25 



To measure Ad and M three people are 
needed: one gives the ghder the initial push, 
another marks the highest point on the track 
that the glider reaches, and the third uses a 
stopwatch to time the motion from push to rest. 

Raise one end of the track a few centime- 
ters above the tabletop. The launcher should 
practice pushing until he can reproduce a push 
that will send the ghder nearly to the raised 
end of the track. 

Record the distance traveled and time 
taken for several trials, and weigh the ghder. 
Determine and record the initial kinetic 
energy. 

To calculate the increase in gravitational 
potential energy, you must measure the verti- 
cal height h through which the ghder moves 
for each push. You will probably find that you 
need to measure from the tabletop to the track 




Fig. 4 

at the initial and final points of the gUder's mo- 
tion (see Fig. 4), since h =hf-h,. Calculate the 
potential energy increase, the quantity maji 
for each of your trials. 

For each trial, compare the kinetic energy 
loss with the potential energy increase. Be sure 
that you use consistent units: m in kilograms, v 
in meters/second, a^ in meters/second^, h in 
meters. 



1 . Are the kinetic energy loss and the po- 
tential energy increase equal within your ex- 
perimental uncertainty? 

2. Explain the significance of your result. 



Here are more things to do if you have time 
to go on: 

(a) See if your answer to 1 continues to be true 
as you make the track steeper and steeper. 

(b) When the ghder rebounds from the rubber 
band at the bottom of the track it is momentar- 
ily stationary — its kinetic energy is zero. The 
same is true of its gravitational potential 
energy, if you use the bottom of the track as the 
zero level. And yet the ghder will rebound from 
the rubber band (regain its kinetic energy) and 



go quite a way up the track (gaining gravita- 
tional potential energy) before it stops. See if 
you can explain what happens at the rebound 
in terms of the conservation of mechanical 
energy. 

(c) The ghder does not get quite so far up the 
track on the second rebound as it did on the 
first. There is evidently a loss of energy. See if 
you can measure how much energy is lost each 
time. 

EXPERIMENT 3-6 
CONSERVATION OF ENERGY— II 

Method A— Film Loops 

You may have used one or more of Film Loops 
L18 through L25 in your study of momentum. 
You will find it helpful to view these slow- 
motion films of one and two-dimensional colh- 
sions again, but this time in the context of the 
study of energy. The data you collected previ- 
ously will be sufficient for you to calculate the 
kinetic energy of each ball before and after the 
colhsion. Remember that kinetic energy imv^ is 
not a vector quantity, and hence, you need only 
use the magnitude of the velocities in your cal- 
culations. 

On the basis of your analysis you may wish 
to try to answer such questions as these: Is 
kinetic energy consumed in such interactions? 
If not, what happened to it? Is the loss in ki- 
netic energy related to such factors as relative 
speed, angle of impact, or relative masses of 
the colhding balls? Is there a difference in the 
kinetic energy lost in elastic and inelastic colh- 
sions? 

The film loops were made in a highly con- 
trolled laboratory situation. After you have de- 
veloped the technique of measurement and 
analysis from film loops, you may want to turn 
to one or more loops deahng with things out- 
side the laboratory setting. Film Loops L26 
through L33 involve freight cars, bilhard balls, 
pole vaulters, and the like. Suggestions for 
using these loops can be found on pages 
81-87. 

Method B — Stroboscopic 
Photographs of Collisions 

When studying momentum, you may have 



Unit 3/26 



Experiment 3-7 



taken measurements on the one-dimensional 
and two-dimensional collisions shown in 
stroboscopic photographs on pages 9-12 
and 19-21. If so, you can now easily re- 
examine your data and compute the kinetic 
energy ^v^ for each ball before and after the 
interaction. Remember that kinetic energy is a 
scalar quantity, and so you will use the mag- 
nitude of the velocity but not the direction in 
making your computations. You would do well 
to study one or more of the simpler events (for 
example. Events 1, 2, 3, 8, 9, or 10) before at- 
tempting the more complex ones involving in- 
elastic colhsions or several balls. Also you may 
wish to review the discussions given earUer for 
each event. 

If you find there is a loss of kinetic energy 
beyond what you would expect from measure- 
ment error, try to explain your results. Some 
questions you might try to answer are these: 
How does kinetic energy change as a function 
of the distance from impact? Is it the same be- 
fore and after impact? How is energy conserva- 
tion influenced by the relative speed at the time 
of colUsion? How is energy conservation 
influenced by the angle of impact? Is there a 
difference between elastic and inelastic in- 
teractions in the fraction of energy conserved? 

EXPERIMENT 3-7 MEASURING 
THE SPEED OF A BULLET 

In this experiment you will use the principle of 

the conservation of momentum to find the 

speed of a bullet. Sections 9.2 and 9.3 in the 

Text discuss colhsions and define momentum. 

You will use the general equation of the 

principle of conservation of momentum for 

— ^ -> — * 

two-body collisions: WaI^a + ^b^b = 'm.fjJ'h + 

The experiment consists of firing a projec- 
tile into a can packed with cotton or a heavy 
block that is free to move horizontally. Since all 
velocities before and after the collision are in 
the same direction, you may neglect the vector 
nature of the equation above and work only 
with speeds. To avoid subscripts, call the mass 
of the target M and the much smaller mass of 
the projectile m. Before impact the target is at 
rest, so you have only the speed v of the projec- 



tile to consider. After impact both target and 
embedded projectile move with a common 
speed v' . Thus the general equation becomes 



or 



mv =(M -\- m)v' 

y _ (M + m)v' 
m 



Both masses are easy to measure. Therefore, if 
the comparatively slow speed v' can be found 
after impact, you can compute the high speed v 
of the projectile before impact. There are at 
least two ways to find v' . 

Method A^Air Track 

The most direct way to find f ' is to mount the 
target on the air track and to time its motion 
after the impact. (See Fig. 1.) Mount a small 
can, lightly packed with cotton, on an air-track 
ghder. Make sure that the glider will still ride 
freely with this extra load. Fire a "bullet" (a 
pellet fi-om a toy gun that has been checked for 
safety by your instructor) horizontally, parallel 
to the length of the air track. If M is large 




Fig. 1 

enough, compared to m, the ghder's speed will 
be low enough so that you can use a stopwatch 
to time it over a meter distance. Repeat the 
measurement a few times until you get consis- 
tent results. 



1. What is your value for the bullet's 
speed? 

2. Suppose the collision between bullet 
and can was not completely inelastic, so that 
the bullet bounced back a little after impact. 
Would this increase or decrease your value for 
the speed of the bullet? 

3. Can you think of an independent way 



Experiment 3-7 



Unit 3/27 



to measure the speed of the bullet? If you can, 
go on and make the independent measure- 
ment. Then see if you can account for any 
differences between the two results. 



Method B— Ballistic Pendulum 

This was the original method of determining 
the speed of bullets, invented in 1742 and is 
still used in some ordnance laboratories. A 
movable block is suspended as a freely swing- 
ing pendulum whose motion reveals the bul- 
let's speed. 

Obtaining the Speed Equation 

The collision is inelastic, so kinetic energy is 
not conserved in the impact. But during the 
nearly frictionless swing of the pendulum after 
the impact, mechanical energy is conserved 
— that is, the increase in gravitational potential 
energy of the pendulum at the end of its 
upward swing is equal to its kinetic energy 
immediately after impact. Written as an equa- 
tion, this becomes 

(M + m)a^ JM^jn)v:! 

where h is the increase in height of the pen- 
dulum bob. 

Solving this equation for v' gives: 

V' = V2aji 
Substituting this expression for v' in the 
momentum equation above leads to 



')V2a^ 



Now you have an equation for the speed v of 
the bullet in terms of quantities that are known 
or can be measured. 

A Useful Approximation 

The change h in vertical height is hard to 
measure accurately, but the horizontal dis- 
placement d may be 10 centimeters or more 
and can be found easily. Therefore, let's see if 
h can be replaced by an equivalent expression 
involving d. The relation between h and d can 
be found by using a little plane geometry. 

In Fig. 2, the center of the circle, O, repre- 
sents the point from which the pendulum is 
hung. The length of the cords Is l. 




Fig. 2 



D 

In the triangle OBC, 

P =d' + (I - hy 
so l^ =d^ + P - 2lh + h' 

and 2lh =d' + h^ 

For small swings, h is small compared with / 
and d, so you may neglect h'^ in comparison 
with d^, and write the close approximation 

2lh = d^ 
or h = dy2l 

Putting this value of h into your last equation 
for V above and simpUfylng gives: 



V = 



(M + m)d 



If the mass of the projectile is small com- 
pared with that of the pendulum, this equation 
can be simplified to another good approxima- 
tion. How? 

Finding the Projectile's Speed 

Now you are ready to turn to the experiment. 
The kind of pendulum you use will depend on 
the nature and speed of the projectUe. If you 
use pellets from a toy gun, a cylindrical card- 
board carton stuffed lightly with cotton and 
suspended by threads from a laboratory stand 
will do. If you use a good bow and arrow, stuff 



Unit 3/28 



Experiment 3-8 



6lio(ihft 





and explain any difference between the two v 
values. 



r 1^ 



-J> 



Fig. 3 

straw into a fairly stiff corrugated box and 
hang it from the ceiling. To prevent the target 
pendulum from twisting, hang it by parallel 
cords connecting four points on the pendulum 
to four points directly above them, as in Fig. 3. 

To measure d, a hght rod (a pencil or a soda 
straw) is placed in a tube clamped to a stand. 
The rod extends out of the tube on the side to- 
ward the pendulum. As the pendulum swings, 
it shoves the rod back into the tube so that the 
rod's final position marks the end of the swing 
of the pendulum. Of course the pendulum must 
not hit the tube and there must be sufficient 
friction between rod and tube so that the rod 
stops when the pendulum stops. The original 
rest position of the pendulum is readily found 
so that the displacement d can be measured. 

Repeat the experiment a few times to get 
an idea of how precise your value for d is. Then 
substitute your data in the equation for v, the 
bullet's speed. 



1. What is your value for the bullet's 
speed? 

2. From your results, compare the kinet- 
ic energy of the bullet before impact with 
that of the pendulum after impact. Why is 
there such a large difference in kinetic 
energy? 

3. Can you describe an independent 
method for finding v? If you have time, try it, 



EXPERIMENT 3-8 ENERGY 
ANALYSIS OF A PENDULUM 
SWING 

According to the law of conservation of energy, 
the loss in gravitational potential energy of a 
simple pendulum as it swings from the top of 
its swing to the bottom is completely trans- 
ferred into kinetic energy at the bottom of the 
swing. You can check this with the following 
photographic method. A one-meter simple 
pendulum (measured from the support to the 
center of the bob) with a 0.5 kg bob works well. 
Release the pendulum from a position where it 
is 10 cm higher than at the bottom of its swing. 
To simplify the calculations, set up the 
camera for 10: 1 scale reduction. Two different 
strobe arrangements have proved successful: 
(1) tape an AC blinky to the bob, or (2) attach 
an AA cell and bulb to the bob and use a 
motor-strobe disk in front of the camera lens. 
In either case you may need to use a two-string 
suspension to prevent the pendulum bob from 




Fig. 1 

spinning while swinging. Make a time expo- 
sure for one swing of the pendulum. 

You can either measure directly from your 
print (which should look something like the 
one in Fig. 1), or make pinholes at the center of 
each image on the photograph and project the 
hole images onto a larger sheet of paper. Cal- 
culate the instantaneous speed i^ at the bottom 



Experiment 3-9 



Unit 3/29 



of the swing by dividing the distance traveled 
between the images nearest the bottom of the 
swing by the time interval between the images. 
The kinetic energy at the bottom of the swing 
^mv'-, should equal the change in potential 
energy from the top of the swing to the bottom. 
If Ah is the difference in vertical height be- 
tween the bottom of the swing and the top, 
then 

V = V2a,Ah 

If you plot both the kinetic and potential 
energy on the same graph (using the bottom- 
most point as a zero level for gravitational po- 
tential energy), and then plot the sum of KE + 
PE, you can check whether total energy is con- 
served during the entire swing. 



EXPERIMENT 
ENERGY 



3-9 LEAST 



Concepts such as momentum, kinetic energy, 
potential energy, and the conservation laws 
often turn out to be unexpectedly helpful in 
helping us to understand what at first glance 
seem to be unrelated phenomena. This exper- 
iment offers just one such case in point: How 




Fig. 1 



can we explain the observation that if a chain 
is allowed to hang freely from its two ends, 
it always assumes the same shape? Hang a 
three-foot length of beaded chain, the type used 
on hght sockets, from two points as shown in 
Fig. 1. What shape does the chain assume? At 
first glance it seems to be a parabola. 

Check whether it is a parabola by finding 
the equation for the parabola which would go 
through the vertex and the two fixed points. 
Determine other points on the parabola by 
using the equation. Plot them and see whether 
they match the shape of the chain. 

The following example may help you plot 
the parabola. The vertex in Fig. 1 is at (0,0) and 
the two fixed points are at (-8, 14.5) and (8, 
14.5). All parabolas symmetric to the y axis 
have the formula y = kx^, where fe is a con- 
stant. For this example you must have 14.5 = 
k(8y, or 14.5 = 64/?. Therefore, k = 0.227, and 
the equation for the parabola going through the 
given vertex and two points is y = 0.227x^ By 
substituting values for x, we calculated a table 
of X and y values for our parabola and plotted 
it. 

A more interesting question is why the 
chain assumes this particular shape, which is 
called a catenary curve. You will recall that the 
gravitational potential energy of a body mass m 
is defined as mUgh, where Ug is the acceleration 
due to gravity, and h the height of the body 
above the reference level chosen. Remember 
that only a difference in energy level is mean- 
ingful; a different reference level only adds a 
constant to each value associated with the 
original reference level. In theory, you could 
measure the mass of one bead on the chain, 
measure the height of each bead above the ref- 
erence level, and total the potential energies 
for all the beads to get the total potential 
energy for the whole chain. 

In practice that would be quite tedious, so 
you will use an approximation that will still 
allow you to get a reasonably good result. (This 
would be an excellent computer problem.) 
Draw vertical parallel lines about 1-inch apart 
on the paper behind the chain (or use graph 
paper). In each vertical section, make a mark 
beside the chain in that section (see Fig. 2 on 
the following page.) 



Unit 3/30 



Experiment 3-9 




Fig. 2 

The total potential energy for that section 
of the chain will be approximately Magh„^,, 
where h,„, is the average height which you 
marked, and M is the total mass in that section 
of chain. Notice that near the ends of the chain 
there are more beads in one horizontal interval 
than there are near the center of the chain. To 
simplify the solution further, assume that M is 
always an integral number of beads which you 
can count. 

In summary, for each interval multiply the 
number of beads by the average height for that 
interval. Total all these products. This total is a 
good approximation to the gravitational poten- 
tial energy of the chain. 

After doing this for the freely hanging 
chain, pull the chain with thumbtacks into 
such different shapes as those shown in Fig. 3. 
Calculate the total potential energy for each 



Fig. 3 




shape. Does the catenary curve (the freely- 
formed shape) or one of these others have the 
minimum total potential energy? 

If you would like to explore other instances 
of the minimization principle, we suggest the 
following: 

1 . When various shapes of wire are dipped into 
a soap solution, the resulting film always forms 
so that the total surface area of the film is a 
minimum. For this minimum surface, the total 
potential energy due to surface tension is a 
minimum. In many cases the resulting surface 
is not at all what you would expect. An excel- 
lent source of suggested experiments with soap 
bubbles, and recipes for good solutions, is the 
paperback Soap Bubbles and the Forces that 
Mould Them, by C. V. Boys, Doubleday Anchor 
Books. Also see "The Strange World of Surface 
Film," The Physics Teacher, Sept., 1966. 

2. Rivers meander in such a way that the work 
done by the river is a minimum. For an expla- 
nation of this, see "A Meandering River," in 
the June 1966 issue of Scientific American. 

3. Suppose that points A and B are placed in a 
vertical plane as shown in Fig. 4, and you want 
to build a track between the two points so that 
a ball will roll from A to B in the least possible 




Fig. 4 

time. Should the track be straight or in the 
shape of a circle, parabola, cycloid, catenary, 
or some other shape? An interesting property of 
a cycloid is that no matter where on a cycloidal 
track you release a ball, it will take the same 
amount of time to reach the bottom of the 
track. You may want to build a cycloidal track 
in order to check this. Don't make the track so 
steep that the ball sUps instead of roUing. 

A more complete treatment of "The Princi- 
ple of Least Action" is given in the Feynman 
Lectures on Physics, Vol. II, p. 19-1. 



Unit 3/31 



EXPERIMENT 3-10 
TEMPERATURE AND 
THERMOMETERS 

You can usually tell just by touch which of two 
similar bodies is the hotter. But if you want to 
tell exactly how hot something is or to com- 
municate such information to somebody else, 
you have to find some way of assigning a 
number to "hotness." This number is called 
temperature, and the instrument used to get 
this number is called a thermometer. 

It's not difficult to think of standard units 
for measuring intervals of time and distance 
— the day and the foot are both familiar to us. 
But try to imagine yourself living in an era be- 
fore the invention of thermometers and tem- 
perature scales, that is, before the time of 
Galileo. How would you describe, and if possi- 
ble give a number to, the "degree of hotness" of 
an object? 

Any property (such as length, volume, 
density, pressure, or electrical resistance) that 
changes with hotness and that can be mea- 
sured could be used as an indication of tem- 
perature; and any device that measures this 
property could be used as a thermometer. 

In this experiment you wUl be using ther- 



mometers based on properties of liquid- 
expansion, gas-expansion, and electrical resis- 
tance. (Other common kinds of thermometers 
are based on electrical voltages, color, or gas- 
pressure.) Each of these devices has its own 
particular merits which make it suitable, from 
a practical point of view, for some applications, 
and difficult or impossible to use in others. 

Of course it's most important that readings 
given by two different types of thermometers 
agree. In this experiment you will make your 
own thermometers, put temperature scales on 
them, and then compare them to see how well 
they agree with each other. 

Defining a Temperature Scale 

How do you make a thermometer? First, you 
decide what property (length, volume, etc.) 
of what substance (mercury, air, etc.) to use 
in your thermometer. Then you must decide 
on two fixed points in order to arrive at the 
size of a degree. A fixed point is based on a 
physical phenomenon that always occurs at 
the same degree of hotness. Two convenient 
fixed points to use are the melting point of 
ice and the boiling point of water. On the 
Celsius (centigrade) system they are assigned 




Any quantity that varies with hotness can be used to establish a temperature scale 
(even the time it takes for an alka seltzer tablet to dissolve in water!). Two "fixed 
points" (such as the freezing and boiling points of water) are needed to define the 
size of a degree. 



Unit 3/32 



Experiment 3-10 



the values 0°C and 100°C at ordinary atmo- 
spheric pressure. 

When you are making a thermometer of 
any sort, you have to put a scale on it against 
which you can read the hotness-sensitive 
quantity. Often a piece of centimeter-marked 
tape or a short piece of ruler will do. Submit 
your thermometer to two fixed points of hot- 
ness (for example, a bath of boiling water and a 
bath of ice water) and mark the positions on 
the indicator. 

The length of the column can now be used 
to define a temperature scale by saying that 
equal temperature changes cause equal 
changes along the scale between the two 
fixed-point positions. Suppose you marked the 
length of a column of liquid at the freezing 
point and again at the boiling point of water. 
You can now divide the total increase in length 
into equal parts and call each of these parts 
"one degree" change in temperature. 

On the Celsius scale the degree is 1/100 of 
the temperature range between the boiling and 
freezing points of water. 

To identify temperatures between the fixed 
points on a thermometer scale, mark off the 
actual distance between the two fixed points on 
the vertical axis of a graph and equal intervals 
for degrees of temperature on the horizontal 
axis, as in Fig. 1. Then plot the fixed points (x) 
on the graph and draw a straight line between 
them. 

Now, the temperature on this scale, cor- 
responding to any intermediate position 1, can 
be read off the graph. 

Other properties and other substances can 




T 100 



Fig. 1 



be used (the volume of different gases, the 
electrical resistance of different metals, and so 
on), and the temperature scale defined in the 
same way. All such thermometers will have to 
agree at the two fixed points — but do they agree 
at intermediate temperatures? 

If different physical properties do not 
change in the same way with hotness, then the 
temperature values you read from thermome- 
ters using these properties will not agree. Do 
similar temperature scales defined by different 
physical properties agree anywhere besides at 
the fixed points? That is a question that you 
can answer from this lab experience. 



Comparing Thermometers 

You will make or be given two "thermometers" 
to compare. Take readings of the appropriate 
quantity — length of liquid column, volume 
of gas, electrical resistance, thermocouple 
voltage, or whatever — when the devices are 
placed in an ice bath, and again when they are 
placed in a boiling water bath. Record these 
values. Define these two temperatures as 0° 
and 100° and draw the straight-Line graphs that 
define intermediate temperatures as described 
above. 

Now put your two thermometers in a series 
of baths of water at intermediate tempera- 
tures, and again measure and record the 
length, volume, resistance, etc. for each bath. 
Put both devices in the bath at the same time 
in case the bath is cooHng down. Use your 
graphs to read off the temperatures of the 
water baths as indicated by the two devices. 

Do the temperatures measured by the two 
devices agree? 

If the two devices do give the same read- 
ings at intermediate temperatures, then you 
could apparently use either as a thermometer. 
But if they do not agree, you must choose only 
one of them as a standard thermometer. Give 
whatever reasons you can for choosing one 
rather than the other before reading the follow- 
ing discussion. If possible, compare your re- 
sults with those of classmates using the same 
or different kinds of thermometers. 

There will of course be some uncertainty 



Experiment 3-11 



Unit 3/33 



in your measurements, and you must decide 
whether the differences you observe between 
two thermometers might be due only to this 
uncertainty. 

The relationship between the readings 
from two different thermometers can be dis- 
played on another graph, where one axis is the 
reading on one thermometer and the other axis 
is the reading on the other thermometer. Each 
bath will give a plot point on this graph. If the 
points fall along a straight Une, then the two 
thermometer properties must change with 
both in the same way. If, however, a fairly reg- 
ularly smooth curve can be drawn through the 
points, then the two thermometer properties 
probably depend on hotness in different ways. 
(Figure 2 shows possible results for two ther- 
mometers.) 






let , 
water 



Fig. 2 



'C ' 
loo 






r 



60 



©ia^3 



® ba-iU 4- 



> 



on dt^ict A 



100 



Discussion 

If we compare many gas thermometers— at 
constant volume as well as pressure, and use 
different gases, and different initial volumes 
and pressures— we find that they all behave 
quantitatively in very much the same way with 
respect to changes in hotness. If a given hot- 
ness change causes a 10% increase in the pres- 
sure of gas A, then the same change will also 
cause a 10% increase in gas B's pressure. Or, if 
the volume of one gas sample decreases by 
20% when transferred to a particular cold 
bath, then a 20% decrease in volume will also 
be observed in a sample of any other gas. This 



means that the temperatures read from differ- 
ent gas thermometers all agree. 

This sort of close similarity of behavior be- 
tween different substances is not found as con- 
sistently in the expansion of hquids or sohds, or 
in their other properties — electrical resistance, 
etc. — and so these thermometers do not agree, 
as you may have just discovered. 

This suggests two things. First, that there 
is quite a strong case for using the change in 
pressure (or volume) of a gas to define the 
temperature change. Second, the fact that in 
such experiments all gases do behave quan- 
titatively in the same way suggests that there 
may be some underlying simphcity in the be- 
havior of gases not found in liquids and sohds, 
and that if one wants to learn more about the 
way matter changes with temperature, one 
would do well to start with gases. 

EXPERIMENT 3-11 CALORIIVIETRY 

Speedometers measure speed, voltmeters mea- 
sure voltage, and accelerometers measure 
acceleration. In this experiment you wOl use a 
device called a calorimeter. As the name sug- 
gests, it measures a quantity connected with 
heat. 

Unfortunately heat energy cannot be 
measured as directly as some of the other 
quantities mentioned above. In fact, to mea- 
sure the heat energy absorbed or given off by a 
substance you must measure the change in 
temperature of a second substance chosen as a 
standard. The heat exchange takes place in- 
side a calorimeter, a container in which mea- 
sured quantities of materials can be mixed to- 
gether without an appreciable amount of heat 
being gained from or lost to the outside. 

A Preliminary Experiment 

The first experiment will give you an idea of 
how good a calorimeter's insulating ability 

really is. 

Fill a calorimeter cup about half full of ice 
water. Put the same amount of ice water with 
one or two ice cubes floating in it in a second 
cup. In a third cup pour the same amount of 
water that has been heated to nearly boiling. 
Measure the temperature of the water in each 



Unit 3/34 



Experiment 3-11 




cup, and record the temperature and the time 
of observation. 

Repeat the observations at about five- 
minute intervals throughout the period. Be- 
tween observations, prepare a sheet of graph 
paper with coordinate axes so that you can 
plot temperature as a function of time. 

Mixing Hot and Cold Liquids 

(You can do this experiment while continuing 
to take readings of the temperature of the 
water in your three cups.) You are to make sev- 
eral assumptions about the nature of heat. 
Then you will use these assumptions to predict 
what will happen when you mix two samples 
that are initially at different temperatures. If 
your prediction is correct, then you can feel 
some confidence in your assumptions — at 
least, you can continue to use the assumptions 
untU they lead to a prediction that turns out to 
be wrong. 

First, assume that, in your calorimeter, 
heat behaves like a fluid that is conserved 
— that is, it can flow from one substance to 
another but the total quantity of heat H present 
in the calorimeter in any given experiment is 
constant. This implies that heat lost by warm 
object just equals heat gained by cold object. 



Or, in symbols 



^H, = ^H, 



Next, assume that, if two objects at differ- 
ent temperatures are brought together, heat 
will flow from the warmer to the cooler object 
until they reach the same temperature. 

Finally, assume that the amount of heat 
fluid AH which enters or leaves an object is 
proportional to the change in temperature AT 
and to the mass of the object, m. In symbols, 

AH = cmAT 

where c is a constant of proportionality that 
depends on the units — and is different for dif- 
ferent substances. 

The units in which heat is measured have 
been defined so that they are convenient for 
calorimeter experiments. The calorie is defined 
as the quantity of heat necessary to change the 
temperature of one gram of water by one Cel- 
sius degree. (This definition has to be refined 
somewhat for very precise work, but it is ade- 
quate for your purpose.) In the expression 

AH = cmAT 

when m is measured in grams of water and T 
in Celsius degrees, H will be the number of 



Experiment 3-11 



Unit 3/35 



calories. Because the calorie was defined this 
way, the proportionality constant c has the 
value 1 cal/gC° when water is the only sub- 
stance in the calorimeter. (The calorie is 
1/1000 of the kilocalorie — or Calorie.) 

Checking the Assumptions 

Measure and record the mass of two empty 
plastic cups. Then put about ^ cup of cold water 
in one and about the same amount of hot water 
in the other, and record the mass and tempera- 
ture of each. (Don't forget to subtract the mass 
of the empty cup.) Now mix the two together in 
one of the cups, stir gently with a thermome- 
ter, and record the final temperature of the 
mixture. 

Multiply the change in temperature of the 
cold water by its mass. Do the same for the hot 
water. 



1. What is the product (mass x tempera- 
ture change) for the cold water? 

2. What is this product for the hot water? 

3. Are your assumptions confirmed, or 
is the difference between the two products 
greater than can be accounted for by un- 
certainties in your measurement? 



Predicting from the Assumptions 

Try another mixture using different quantities 
of water, for example } cup of hot water and ^ 
cup of cold. Before you mix the two, try to pre- 
dict the final temperature. 



4. What do you predict the temperature 
of the mixture will be? 

5. What final temperature do you ob- 
serve? 

6. Estimate the uncertainty of your 
thermometer readings and your mass mea- 
surements. Is this uncertainty enough to ac- 
count for the difference between your pre- 
dicted and observed values? 

7. Do your results support the assump- 
tions? 



measurable change in temperature by this 
time. If you are to hold to your assumption of 
conservation of heat fluid, then it must be that 
some heat has gone from the hot water into the 
room and from the room to the cold water. 



8. How much has the temperature of the 
cold water changed? 

9. How much has the temperature of the 
water that had ice in it changed? 



IVIelting 

The cups you filled with hot and cold water at 
the beginning of the period should show a 



The heat that must have gone from the 
room to the water-ice mixture evidently did not 
change the temperature of the water as long as 
the ice was present. But some of the ice melted, 
so apparently the heat that leaked in was used 
to melt the ice. Evidently, heat was needed to 
cause a "change of state" (in this case, to melt 
ice to water) even if there was no change in 
temperature. The additional heat required to 
melt one gram of ice is called "latent heat of 
melting." Latent means hidden or dormant. 
The units are cal/g — there is no temperature 
unit here because no temperature change is 
involved in latent heat. 

Next, you will do an experiment mixing 
materials other than liquid water in the 
calorimeter to see if your assumptions about 
heat as a fluid can still be used. Two such ex- 
periments are described below, "Measuring 
Heat Capacity" and "Measuring Latent Heat." 
If you have time for only one of them, choose 
either one. Finally, do "Rate of Cooling" to 
complete your preliminary experiment. 

Measuring Heat Capacity 

(While you are doing this experiment, continue 
to take readings of the temperature of the 
water in your three test cups.) Measure the 
mass of a small metal sample. Put just enough 
cold water in a calorimeter to cover the sample. 
Tie a thread to the sample and suspend it in a 
beaker of boiling water. Measure the tempera- 
ture of the boiling water. 

Record the mass and temperature of the 
water in the calorimeter. 

When the sample has been immersed in 
the boiling water long enough to be heated 
uniformly (2 or 3 minutes), hft it out and hold 



Unit 3/36 



Experiment 3-11 



it just above the surface for a few seconds to let 
the water drip off, then transfer it quickly to 
the calorimeter cup. Stir gently with a ther- 
mometer and record the temperature when it 
reaches a steady value. 



10. Is the product of mass and tempera- 
ture change the same for the metal sample 
and for the water? 

11. If not, must you modify the assump- 
tions about heat that you made earUer in the 
experiment? 



In the expression Mi - cm^T, the constant 
of proportionality c (called the "specific heat 
capacity") may be different for different mate- 
rials. For water the constant has the value 
1 cal/gC°. You can find a value of c for the 
metal by using the assumption that heat 
gained by water equals the heat lost by sample. 
Or, writing subscripts w and s for water and 
metal sample, AH^, = -AH,. 



Then 
and 



c^,m„,,At^, = -Csm.Ms 
-c„m„At„. 



c. = 



-^'"''if'-"'a' 

Tn,,At., 



12. What is your calculated value for the 
specific heat capacity c^ for the metal sample 
you used? 



If your assumptions about heat being a 
fluid are valid, you now ought to be able to pre- 
dict the final temperature of any mixture of 
water and your material. 

Try to verify the usefulness of your value. 
Predict the final temperature of a mixture of 
water and a heated piece of your material, 
using different masses and different initial 
temperatures. 



13. Does your result support the fluid 
model of heat? 



Measuring Latent Heat 

Use your calorimeter to find the "latent heat of 
melting" of ice. Start with about i cup of water 



that is a little above room temperature, and re- 
cord its mass and temperature. Take a small 
piece of ice from a mixture of ice and water 
that has been standing for some time; this will 
assure that the ice is at 0°C and will not have to 
be warmed up before it can melt. Place the 
small piece of ice on paper toweling for a mo- 
ment to dry off water on its surface, and then 
transfer it quickly to the calorimeter. 

Stir gently with a thermometer until the 
ice is melted and the mixture reaches an 
equilibrium temperature. Record this tempera- 
ture and the mass of the water plus melted ice. 



14. What was the mass of the ice that you 
added? 



The heat given up by the warm water is: 
AH„. = c,,Tn„,At„, 



I „. V^y.IIl.y.<.*«,^. 



The heat gained by the water formed by the 
melted ice is: 

H, = Cu-WjAt, 

The specific heat capacity c„. is the same in 
both cases — the specific heat of water. 

The heat given up by the warm water first 
melts the ice, and then heats the water formed 
by the melted ice. If we use the symbol AH^ for 
the heat energy required to melt the ice, we can 
write: 

-AH,, = AH;. + AH, 
So the heat energy needed to melt the ice is 

AHt = -AH„.-AH, 

The latent heat of melting is the heat en- 
ergy needed per gram of ice, so 



latent heat of melting = 



AH, 
m 



15. What is your value for the latent heat 
of melting of ice? 



When this experiment is done with ice 
made from distilled water with no inclusions of 
liquid water, the latent heat is found to be 80 
calories per gram of ice. How does your result 
compare with the accepted value? 



Experiment 3-12 



Unit 3/37 



Rate of Cooling 

If you have been measuring the temperature of 
the water in your three test cups, you should 
have enough data by now to plot three curves 
of temperature against time. Mark the temper- 
ature of the air in the room on your graph too. 



16. How does the rate at which the hot 
water cools depend on its temperature? 

17. How does the rate at which the cold 
water heats up depend on its temperature? 



Weigh the amount of water in the cups. 
From the rates of temperature change (de- 
grees/minute) and the masses of water, cal- 
culate the rates at which heat leaves or enters 
the cups at various temperatures. Use this in- 
formation to estimate the error in your earUer 
results for latent or specific heat. 



EXPERIMENT 3-12 ICE 
CALORIMETRY 

A simple apparatus made up of thermally in- 
sulating styrofoam cups can be used for doing 
some ice calorimetry experiments. Although 
the apparatus is simple, careful use will give 
you excellent results. To determine the heat 
transferred in processes in which heat energy 
is given off, you will be measuring either the 
volume of water or the mass of water from a 
melted sample of ice. 

You will need either three cups the same 
size (8 oz or 6 oz), or two 8 oz and one 6 oz cup. 
Also have some extra cups ready. One large 
cup serves as the collector, A, (Fig. 1), the sec- 
ond cup as the ice container, I, and the smaller 
cup (or one of the same size cut back to fit in- 



C^P .1^ 





Fig. 1 



Fig. 2 



side the ice container as shown) as the cover, 
K. 

Cut a hole about i-inch in diameter in the 
bottom of cup I so that melted water can drain 
out into cup A. To keep the hole from becoming 
clogged by ice, place a bit of window screening 
in the bottom of I. 

In each experiment, ice is placed in cup I. 
This ice should be carefully prepared, free of 
bubbles, and dry, if you want to use the known 
value of the heat of fusion of ice. However, you 
can use ordinary crushed ice, and, before doing 
any of the experiment, determine experimen- 
tally the effective heat of melting of this non- 
ideal ice. (Why should these two values differ?) 

In some experiments which require some 
time to complete (such as Experiment b), you 
should set up two identical sets of apparatus 
(same quantity of ice, etc.), except that one 
does not contain a source of heat. One will 
serve as a fair measure of the background ef- 
fect. Measure the amount of water collected in 
it during the same time, and subtract it from 
the total amount of water collected in the ex- 
perimental apparatus, thereby correcting for 
the amount of ice melted just by the heat leak- 
ing in from the room. An efficient method for 
measuring the amount of water is to place the 
arrangement on the pan of a balance and lift 
up cups I and K at regular intervals (about 10 
min.) whUe you weigh A with its contents of 
melted ice water. 

(a) Heat of melting of ice. Fill a cup about Ho 4 
full with crushed ice. (Crushed ice has a larger 
amount of surface area, and so will melt more 
quickly, thereby minimizing errors due to heat 
from the room.) Bring a small measured 
amount of water (say about 20 cc) to a boil in a 
beaker or large test tube and pour it over the ice 
in the cup. Stir briefly with a poor heat conduc- 
tor, such as a glass rod, until equilibrium has 
been reached. Pour the ice-water mixture 
through cup I. Collect and measure the final 
amount of water (wif) in A. If tuo is the original 
mass of hot water at 100°C with which you 
started, then Wf - rwo is the mass of ice that was 
melted. The heat energy absorbed by the melt- 
ing ice is the latent heat of melting for ice, Li, 



Unit 3/38 



times the mass of melted ice: Li(mf-mo). This 
will be equal to the heat energy lost by the boil- 
ing water cooling from 100°C to 0°C, so we can 
write 



LiirUf - rrio) = nioAT 



and 



Li = 



nic 



rrif - TUo 



100C° 



Note: This derivation is correct only if 
there is still some ice in the cup afterwards. If 
you start with too little ice, the water will come 
out at a higher temperature. 

For crushed ice which has been standing 
for some time, the value of Lj will vary between 
70 and 75 calories per gram. 

(b) Heat exchange and transfer by conduction 
and radiation. For several possible experi- 
ments you will need the following additional 
apparatus. Make a small hole in the bottom of 
cup K and thread two wires, soldered to a light- 
bulb, through the hole. A flashlight bulb which 
operates with an electric current between 300 
and 600 milhamperes is preferable; but even a 
GE #1130 6-volt automobile headUght bulb 
(which draws 2.4 amps) has been used with 
success. (See Fig. 2.) In each experiment, you 
are to observe how different apparatus affects 
heat transfer into or out of the system. 

1 . Place the bulb in the ice and turn it on for 5 
minutes. Measure the ice melted. 

2. Repeat 1, but place the bulb above the ice 
for 5 minutes. 

3. and 4: Repeat 1 and 2, but cover the inside 
of cup K with aluminum foil. 

5. and 6: Repeat 3 and 4, but in addition cover 
the inside of cup I with aluminum foil. 
7. Prepare "heat absorbing" ice by freezing 
water to which you have added a small amount 
of dye, such as India ink. Repeat any or all of 
experiments 1 through 6 using this "specially 
prepared" ice. 

Some questions to guide your observations: 
Does any heat escape when the bulb is im- 
mersed in the ice? What arrangement keeps in 
as much heat as possible? 



EXPERIMENT 3-13 

MONTE CARLO EXPERIMENT 

ON MOLECULAR COLLISIONS 

A model for a gas consisting of a large number 
of very small particles in rapid random motion 
has many advantages. One of these is that it 
makes it possible to estimate the properties of a 
gas as a whole from the behavior of a compara- 
tively small random sample of its molecules. In 
this experiment you will not use actual gas par- 
ticles, but instead employ analogs of molecular 
collisions. The technique is named the Monte 
Carlo method after that famous (or infamous) 
gambling casino in Monaco. The experiment 
consists of two games, both of which involve 
the concept of randomness. You will probably 
have time to play only one. 

Game I Collision Probability for a 
Gas of Marbles 

In this part of the experiment, you will try to 
find the diameter of marbles by rolling a "bom- 
barding marble" into an array of "target mar- 
bles" placed at random positions on a level 
sheet of graph paper. The computation of 
the marble diameter will be based on the 
proportion of hits and misses. In order to as- 
sure randomness in the motion of the bom- 
barding marble, start at the top of an inclined 
board studded with nails spaced about an 
inch apart — a sort of pinball machine (Fig. 1). 




Fig. 1 



Experiment 3-13 



Unit 3/39 



To get a fairly even, yet random, distribution 
of the bombarding marble's motion, move its 
release position over one space for each release 
in the series. 

First you need to place the target marbles 
at random. Then draw a network of crossed 
grid Unes spaced at least two marble diameters 
apart on your graph paper. (If you are using 
marbles whose diameters are half an inch, 
these grid hnes should be spaced 1.5 to 2 
inches apart.) Number the grid hnes as shown 
in Fig. 2. 



9 
8 
7 

t ' 

^ 4 

3 f 
S 
I 







■A -' 



_4„^.-_-. 



T I j 







>- f— 



/ 



8 9 



3 ^ S 6 

X — 

Fig. 2 Eigiit consecutive two-digit numbers in a 
table of random numbers were used to place the 
marbles. 



One way of placing the marbles at random 
is to turn to the table of random numbers at the 
end of this experiment. Each student should 
start at a different place in the table and then 
select the next eight numbers. Use the first two 
digits of these numbers to locate positions on 
the grid. The first digit of each number gives 
the X coordinate, the second gives the y 
coordinate — or vice versa. Place the target 
marbles in these positions. Books may be 
placed along the sides of the graph paper and 
across the bottom to serve as containing walls. 



With your array of marbles in place, make 
about fifty trials with the bombarding marble. 
From your record of hits and misses compute 
R, the ratio between the number of runs in 
which there are one or more hits to the total 
number of runs. Remember that you are count- 
ing "runs with hits," not hits, and hence, sev- 
eral hits in a single run are still counted as 
"one." 

inferring the size of the marbies. How does the 
ratio R lead to the diameter of the target ob- 
ject? The theory applies just as well to deter- 
mining the size of molecules as it does to mar- 
bles, although there would be lO^** or so 
molecules instead of 8 "marble molecules." 

If there were no target marbles, the bom- 
barding marble would get a clear view of the 
full width, sayD, of the back wall enclosing the 
array. There could be no hit. If, however, there 
were target marbles, the 100% clear view 
would be cut down. If there were N target mar- 
bles, each with diameter d, then the clear path 
over the width D would be reduced by N x d. 

It is assumed that no target marble is hid- 
ing behind another. (This corresponds to the 
assumption that the sizes of molecules are ex- 
tremely small compared with the distances be- 
tween them.) 

The blocking effect on the bombarding 
marble is greater than just Nd, however. The 
bombarding marble will miss a target marble 
only if its center passes more than a distance of 
one radius on either side of it. (See Fig. 3 on 
next page.) This means that a target marble 
has a blocking effect over twice its diameter 
(its own diameter plus two radii), so the total 
blocking effect of N marbles is 2Nd. There- 
fore the expected ratio R of hits to total trials 
is 2NdlD (total blocked width to total width). 
Thus: 



R _2Nd 



which we can rearrange to give an expression 
for d: 



d = 



RD 

2N 



Unit 3140 Experiment 3-13 





d 

1 




Fig. 3 « > 

A projectile will clear a target only if it passes outside 
a center-to-center distance d on either side of it. 
Therefore, thinking of the projectiles as points, the 
effective blocking width of the target is 2d. 

To check the accuracy of the Monte Carlo 
method compare the value for d obtained from 
the formula above with that obtained by direct 
measurement of the marbles. (For example, 
hne up your eight marbles against a book. 
Measure the total length of all of them together 
and divide by eight to find the diameter d of one 
marble.) 



1. What value do you calculate for the 
marble diameter? 

2. How well does your experimental pre- 
diction agree with direct measurement? 



Game II Mean Free Path Between 
Collision Squares 

In this part of the experiment you play with 
blacked-in squares as target molecules in place 
of marble molecules in a pinball game. On a 
sheet of graph paper, say 50 units on a side 
(2,500 squares), you will locate by the Monte 
Carlo method between 40 and 100 molecules. 
Each student should choose a different number 
of molecules. 



You will find a table of random numbers 
(from to 50) at the end of this experiment. 
Begin anywhere you wish in the table, but then 
proceed in a regular sequence. Let each pair of 
numbers be the x and y coordinates of a point 
on your graph. (If one of the pair is greater than 
49, you cannot use it. Ignore it and take the 
next pair.) Then shade in the squares for which 
these points are the lower left-hand corners. 
You now have a random array of square target 
"molecules." 




Fig. 4 



(03j 02) 



Rules of the game. The way a bombarding par- 
ticle passes through this array, it is bound to 
colhde with some of the target particles. There 
are five rules for this game of collision. All of 
them are illustrated in Fig. 5. 
(a) The particle can travel only along lines of 
the graph paper, up or down, left or right. They 
start at some point (chosen at random) on 




r — r 



Fig. 5 



Experiment 3-13 



Unit 3/41 



the left-hand edge of the graph paper. The par- 
ticle initially moves horizontally from the start- 
ing point until it colhdes with a blackened 
square or another edge of the graph paper. 

(b) If the particle strikes the upper left-hand 
corner of a target square, it is diverted upward 
through a right angle. If it should strike a low- 
er left-hand comer it is diverted downward, 
again through ninety degrees. 

(c) When the path of the particle meets an 
edge or boundary of the graph paper, the parti- 
cle is not reflected directly back. (Such a rever- 
sal of path would make the particle retrace its 
previous paths.) Rather it moves two spaces to 
its right along the boundary edge before re- 
versing its direction. 

(d) There is an exception to rule (c). Whenever 
the particle strikes the edge so near a corner 
that there isn't room for it to move two spaces 
to the right without meeting another edge of 
the graph paper, it moves two spaces to the left 
along the boundary. 

(e) Occasionally two target molecules may oc- 
cupy adjacent squares and the particle may hit 
touching corners of the two target molecules at 
the same time. The rule is that this counts as 
two hits and the particle goes straight through 
without changing its direction. 

Finding the "mean free path." With these colli- 
sion rules in mind, trace the path of the particle 
as it bounces about among the random array of 
target squares. Count the number of collisions 
with targets. Follow the path of the particle 
until you get 51 hits with target squares (colli- 
sions with the edge do not count). Next, record 
the 50 lengths of the paths of the particle be- 
tween collisions. Distances to and from a 
boundary should be included, but not distances 
along a boundary (the two spaces introduced to 
avoid backtracking). These 50 lengths are the 



free paths of the particle. Total them and di- 
vide by 50 to obtain the mean free path, L, for 
your random two-dimensional array of square 
molecules. 

In this game your molecule analogs were 
pure points, i.e., dimensionless. In his inves- 
tigations Clausius modified this model by 
giving the particles a definite size. Clausius 
showed that the average distance L a molecule 
travels between collisions, the so-called "mean 
free path," is given by 



L = 



V 

Na 



where V is the volume of the gas, N is the 
number of molecules in that volume, and a is 
the cross-sectional area of an individual 
molecule. In this two-dimensional game, the 
particle was moving over an area A, instead 
of through a volume V, and was obstructed 
by targets of width d, instead of cross-section- 
al area a. A two-dimensional version of 
Clausius's equation might therefore be: 



L = 



2Nd 



where N is the number of blackened square 
"molecules." 



3. What value of L do you get from the 
data for your runs? 

4. Using the two-dimensional version of 
Clausius's equation, what value do you esti- 
mate for d (the width of a square)? 

5. How does your calculated value of d 
compare with the actual value? How do you 
explain the difference? 



Unit 3/42 



Experiment 3-13 



TABLE OF 1000 RANDOM TWO-DIGIT NUMBERS 
(FROM to 50) 



03 47 


44 22 


30 30 


22 00 


00 49 


22 17 


38 30 


23 21 


20 11 


24 33 


16 22 


36 10 


44 39 


46 40 


24 02 


19 36 


38 21 


45 33 


14 23 


01 31 


33 21 


03 29 


08 02 


20 31 


37 07 


03 28 


47 24 


11 29 


49 08 


10 39 


34 29 


34 02 


43 28 


03 43 


43 40 


26 08 


28 06 


50 14 


21 44 


47 21 


32 44 


11 05 


05 05 


05 50 


23 29 


26 00 


09 05 


27 31 


08 43 


04 14 


18 18 


04 02 


48 39 


48 22 


38 18 


15 39 


48 34 


50 28 


37 21 


15 09 


23 42 


31 08 


19 30 


06 00 


20 18 


30 24 


15 33 


10 07 


14 29 


05 24 


35 12 


11 12 


11 04 


01 10 


25 39 


48 50 


24 44 


03 47 


34 04 


44 07 


12 13 


42 10 


40 48 


45 44 


42 35 


41 26 


41 10 


23 05 


06 36 


08 43 


37 35 


12 41 


02 02 


19 11 


06 07 


42 31 


23 47 


47 25 


10 43 


12 38 


16 08 


18 39 


03 31 


49 26 


07 12 


17 31 


17 31 


35 07 


44 38 


40 35 


31 16 


10 47 


38 45 


28 40 


33 34 


24 16 


42 38 


19 09 


41 47 


50 41 


32 43 


45 37 


30 38 


22 01 


30 14 


02 17 


45 18 


29 06 


13 27 


46 24 


27 42 


03 09 


08 32 


24 02 


05 49 


18 05 


22 00 


23 02 


44 43 


43 20 


00 39 


05 03 


49 37 


23 22 


33 42 


26 29 


00 20 


12 03 


10 05 


02 39 


11 27 


39 32 


13 30 


36 45 


09 03 


46 40 


22 07 


03 03 


05 39 


03 46 


35 24 


22 49 


17 33 


35 01 


01 32 


18 09 


47 03 


39 41 


36 23 


19 41 


16 20 


38 36 


29 48 


07 27 


48 14 


34 13 


07 48 


39 12 


20 18 


19 42 


38 23 


33 26 


15 29 


20 02 


21 45 


04 31 


48 13 


23 32 


37 30 


09 24 


45 11 


27 07 


39 43 


13 05 


47 45 


47 45 


00 06 


41 18 


05 02 


03 09 


18 00 


14 21 


49 17 


30 37 


25 15 


04 49 


24 19 


40 23 


24 17 


17 16 


20 46 


06 18 


45 07 


06 28 


49 44 


10 08 


43 00 


38 26 


34 41 


11 16 


05 26 


50 25 


38 47 


39 38 


42 45 


10 08 


16 06 


43 18 


34 48 


27 03 


21 19 


13 42 


16 04 


00 18 


16 46 


13 13 


16 29 


44 10 


29 18 


22 45 


41 23 


03 10 


35 30 


24 36 


38 09 


25 21 


08 40 


20 46 


39 14 


37 31 


34 50 


20 14 


21 46 


38 46 


12 27 


20 44 


46 06 


01 41 


30 49 


18 48 


39 43 


13 04 


24 15 


08 22 


13 29 


04 05 


42 29 


50 47 


01 50 


01 48 


18 14 


04 43 


27 46 


23 07 


19 28 


07 10 


23 19 


41 45 


25 27 


19 10 


09 47 


34 45 


08 45 


25 21 


49 21 


18 46 


16 40 


35 14 


41 28 


41 15 


44 17 


04 33 


15 22 


12 45 


39 07 


34 27 


14 47 


35 33 


42 29 


47 47 


40 33 


42 45 


07 08 


38 15 


08 25 


22 06 


07 26 


32 44 


03 42 


42 34 


33 27 


10 45 


18 40 


11 48 


48 03 


07 16 


32 25 


20 25 


44 22 


39 28 


06 09 


04 26 


14 35 


36 03 


15 22 


02 07 


46 48 


45 12 


47 11 


30 19 


33 32 


34 25 


45 17 


13 26 


03 37 


33 35 


08 13 


15 26 


09 18 


34 25 


42 38 


40 01 


43 31 


30 33 


39 11 


49 41 


27 44 


11 39 


06 19 


47 23 


15 06 


22 08 


50 44 


50 11 


18 16 


00 41 


07 47 


34 25 


28 10 


50 03 


22 35 


49 36 


44 21 


25 12 


19 44 


31 51 


49 18 


40 36 


00 27 


22 12 


31 04 


32 17 


08 23 


38 32 


01 47 


43 53 


44 04 


10 27 


16 00 


16 33 


39 00 


01 50 


07 28 


35 02 


38 00 


46 47 


33 29 


28 41 


09 23 


47 48 


37 32 


07 02 


07 48 


07 41 


22 13 


37 27 


27 12 


34 21 


07 04 


49 34 


05 03 


36 07 


10 15 


21 48 


14 44 


39 39 


15 09 


23 23 


37 31 


00 25 


17 37 


13 41 


13 39 


40 14 


19 48 


34 18 


08 18 


08 06 


44 26 


12 45 


32 24 


24 30 


29 13 


34 39 


27 44 


11 20 


37 40 


36 46 


35 22 


09 09 


07 45 


29 12 


48 35 


05 38 


43 11 


45 18 


28 14 


04 37 


48 38 


43 12 


14 08 


04 04 


18 17 


10 33 


04 32 


27 37 


33 42 


34 41 


07 41 


49 14 


31 38 


08 31 


38 30 


42 10 


08 09 


17 32 


46 15 


15 43 


15 31 


46 45 


42 34 


46 31 


29 03 


08 32 


11 06 


20 21 


24 16 


13 17 


29 34 


42 31 


16 00 


02 48 


10 34 


32 14 


25 39 


29 31 


18 37 


28 50 


07 28 


08 24 


20 15 


60 11 


21 31 


20 49 


07 35 


41 16 


16 17 


43 36 


20 26 


39 38 


00 49 


14 10 


29 01 


49 28 


21 30 


40 15 


01 07 


16 04 


19 09 


36 12 



Unit 3/43 



EXPERIMENT 3-14 BEHAVIOR 
OF GASES 

Air is elastic or springy. You can feel this when 
you put your finger over the outlet of a bicycle 
pump and push down on the pump plunger. 
You can tell that there is some connection be- 
tween the volume of the air in the pump and 
the force you exert in pumping, but the exact 
relationship is not obvious. About 1660, Robert 
Boyle performed an experiment that disclosed 
a very simple relationship between gas pres- 
sure and volume, but not until two centuries 
later was the kinetic theory of gases developed, 
which accounted for Boyle's law satisfactorily. 
The purpose of these experiments is not 
simply to show that Boyle's Law and Gay 
Lussac's Law (which relates temperature and 
volume) are "true." The purpose is also to show 
some techniques for analyzing data that can 
lead to such laws. 

/. Volume and Pressure 

Boyle used a long glass tube in the form of a J 
to investigate the "spring of the air." The short 
arm of the J was sealed, and air was trapped in 
it by pouring mercury into the top of the long 
arm. (Apparatus for using this method may be 
available in your school.) 

A simpler method requires only a small 
plastic syringe, calibrated in cc, and mounted 
so that you can push down the piston by piling 
weights on it. The volume of the air in the 
syringe can be read directly from the calibra- 
tions on the side. The pressure on the air due to 




Fig. 1 



the weights on the piston is equal to the force 
exerted by the weights divided by the area of 
the face of the piston: 

P — w 

" A 

Because "weights" are usually marked 
with the value of their mass, you will have to 
compute the force from the relation Fgrav = 
magrav (It will help you to answer this question 
before going on: What is the weight, in new- 
tons, of a 0.1 kg mass?) 

To find the area of the piston, remove it 
from the syringe. Measure the diameter (2R) of 
the piston face, and compute its area from the 
familiar formula A = ttR^. 

You wUl want to both decrease and in- 
crease the volume of the air, so insert the pis- 
ton about halfway down the barrel of the 
syringe. The piston may tend to stick shghtly. 
Give it a twist to free it and help it come to its 
equilibrium position. Then record this position. 

Add weights to the top of the piston and 
each time record the equilibrium position, after 
you have given the piston a twist to help over- 
come friction. 

Record your data in a table with columns 
for volume, weight, and pressure. Then re- 
move the weights one by one to see if the vol- 
umes are the same with the piston coming up 
as they were going down. 

If your apparatus can be turned over so 
that the weights pull out on the plunger, obtain 
more readings this way, adding weights to in- 
crease the volume. Record these as negative 
forces. (Stop adding weights before the piston 
is puUed all the way out of the barrel!) Again 
remove the weights and record the values on 
returning. 

Interpreting your results. You now have a set of 
numbers somewhat like the ones Boyle re- 
ported for his experiment. One way to look for a 
relationship between the pressure P„. and the 
volume V is to plot the data on graph paper, 
draw a smooth simple curve through the 
points, and try to find a mathematical expres- 
sion that would give the same curve when plot- 
ted. 



Unit 3/44 



Experiment 3-14 



Plot volume V (vertical axis) as a function 
of pressure P^, (horizontal axis). If you are will- 
ing to believe that the relationship between ?„. 
and V is fairly simple, then you should try to 
draw a simple curve. It need not actually go 
through all the plot points, but should give an 
overall "best fit." 

Since V decreases as P^ increases, you can 
tell before you plot it that your curve represents 
an "inverse" relationship. As a first guess at 
the mathematical description of this curve, try 
the simplest possibility, that 1/V is proportional 
to Pu,. That is, IfV oc P,,,. A graph of proportional 
quantities is a straight hne. If lA^ is propor- 
tional to P„„ then a plot of W value against 
P^, will he on a straight hne. 

Add another column to your data table for 
values of IfV and plot this against P^.. 



1. Does the curve pass through the 
origin? 

2. If not, at what point does your curve 
cross the horizontal axis? (In other words, 
what is the value of P„, for which 1/V would be 
zero?) What is the physical significance of the 
value of P,„? 



In Boyle's time, it was not understood that 
air is really a mixture of several gases. Do you 
believe you would find the same relationship 
between volume and pressure if you tried a va- 
riety of pure gases instead of air? If there are 
other gases available in your laboratory, flush 
out and refill your apparatus with one of them 
and try the experiment again. 



3. Does the curve you plot have the same 
shape as the previous one? 



//. Volume and Temperature 

Boyle suspected that the temperature of his air 
sample had some influence on its volume, but 
he did not do a quantitative experiment to find 
the relationship between volume and tempera- 
ture. It was not until about 1880, when there 
were better ways of measuring temperature, 
that this relationship was established. 

You could use several kinds of equipment 
to investigate the way in which volume 



changes with temperature. Such a piece of 
equipment is a glass bulb with a J tube of mer- 
cury or the syringe described above. Make sure 
the gas inside is dry and at atmospheric pres- 
sure. Immerse the bulb or syringe in a beaker 
of cold water and record the volume of gas and 
temperature of the water (as measured on a 
suitable thermometer) periodically as you 
slowly heat the water. 

A simpler piece of equipment that will give 
just as good results can be made from a piece of 
glass capillary tubing. 

Equipment note: assembling a 
constant-pressure gas 
thermometer 

About 6" of capillary tubing makes a ther- 
mometer of convenient size. The dimensions of 
the tube are not critical, but it is very important 
that the bore be dry. It can be dried by heating, 
or by rinsing with alcohol and waving it 
frantically — or better still, by connecting it to a 
vacuum pump for a few moments. 
Filling with air. The dry capillary tube is dipped 
into a container of mercury, and the end sealed 
with fingertip as the tube is withdrawn (Fig. 2), 
so that a pellet of mercury remains in the lower 
end of the tube. 




Fig. 2 



Experiment 3-14 



Unit 3/45 




Fig. 3 

The tube is held at an angle and the end 
tapped gently on a hard surface until the mer- 
cury pellet sUdes to about the center of the tube 
(Fig. 3). 

One end of the tube is sealed with a dab of 
silicone sealant; some of the sealant will go up 
the bore, but this is perfectly all right. The 
sealant is easily set by immersing it in boil- 
ing water for a few moments. 

Taking measurements. A scale now must be 
positioned along the completed tube. The scale 
will be directly over the bore if a stick is placed 
as a spacer next to the tube and bound together 
with rubber bands (Fig. 4). (A long stick makes 
a convenient handle.) The zero of the scale 
should be ahgned carefully with the end of the 
gas column, that is, the end of the silicone seal. 





Fig. 5 

In use, the thermometer should be com- 
pletely immersed in whatever one wishes to 
measure the temperature of, and the end 
tapped against the side of the container gently 
to allow the mercury to slide to its final resting 
place (Fig. 5). 

Filling with some other gas. To use some gas 
other than air, begin by connecting a short 
length of rubber tubing to a fairly low-pressure 
supply of gas. As before, trap a pellet of mer- 
cury in the end of a capillary tube, but this time 
do not tap it to the center. Leave it flat so that 
it will be pushed to the center by the gas pres- 
sure (Fig. 6). Open the gas valve shghtly for 
a moment to flush out the rubber tube. With 
your finger tip closing off the far end of the 
capillary tube to prevent the mercury being 




Fig. 4 



Fig. 6 



Unit 3/46 



Experiment 3-14 




Fig. 7 

blown out, work the rubber connecting tube 
over the capillary tube. Open the gas valve 
slightly, and very cautiously release your finger 
very slightly for a brief instant until the pellet 
has been pushed to about the middle of the 
tube. 

Remove from the gas supply, seal off as be- 
fore (the end that was connected to gas sup- 
ply), and attach scale. Plot a graph of volume 
against temperature. 

Interpreting your results. 



5. With any of the methods mentioned 
here, the pressure of the gas remains con- 
stant. If the curve is a straight line, does this 
"prove" that the volume of a gas at constant 
pressure is proportional to its temperature? 

6. Remember that the thermometer you 
used probably depended on the expansion of a 
liquid such as mercury or alcohol. Would your 
graph have been a straight line if a different 
type of thermometer had been used? 

7. If you could continue to cool the air, 
would there be a lower limit to the volume it 
would occupy? 



Draw a straight line as nearly as possible 
through the points on your V-T graph and ex- 
tend it to the left until it shows the approximate 
temperature at which the volume would be 
zero. Of course, you have no reason to assume 
that gases have this simple linear relationship 
all the way down to zero volume. (In fact, air 



would change to a liquid long before it reached 
the temperature indicated on your graph for 
zero volume.) However, some gases do show 
this linear behavior over a wide temperature 
range, and for these gases the straight Hne al- 
ways crosses the T-axis at the same point. 
Since the volume of a sample of gas cannot be 
less than 0, this point represents the lowest 
possible temperature of the gases — the "abso- 
lute zero" of temperature. 



8. What value does your graph give for 
this temperature? 



///. Questions for Discussion 

Both the pressure and the temperature of a gas 
sample affect its volume. In these experiments 
you were asked to consider each of these fac- 
tors separately. 



9. Were you justified in assuming that 
the temperature remained constant in the 
first experiment as you varied the pressure? 
How could you check this? How would your 
results be affected if, in fact, the temperature 
went up each time you added weight to the 
plunger? 

10. In the second experiment the gas 
was at atmospheric pressure. Would you ex- 
pect to find the same relationship between 
volume and temperature if you repeated the 
experiment with a different pressure acting 
on the sample? 



Gases such as hydrogen, oxygen, nitrogen, 
and carbon dioxide are very different in their 
chemical behavior. Yet they all show the same 
simple relationships between volume, pres- 
sure, and temperature that you found in these 
experiments, over a fairly wide range of pres- 
sures and temperatures. This suggests that 
perhaps there is a simple physical model that 
will explain the behavior of all gases within 
these limits of temperature and pressure. 
Chapter 11 of the Text describes just such a 
simple model and its importance in the de- 
velopment of physics. 



Unit 3/47 



EXPERIMENT 3-15 WAVE 
PROPERTIES 

In this laboratory exercise you will become 
familiar with a variety of wave properties in 
one- and two-dimensional situations.' Using 
ropes, springs, Slinkies, or a ripple tank, you 
can find out what determines the speed of 
waves, what happens when they collide, and 
how waves reflect and go around corners. 

Waves in a Spring 

Many waves move too fast or are too small to 
watch easily. But in a long "soft" spring you 
can make big waves that move slowly. With a 
partner to help you, pull the spring out on a 
smooth floor to a length of about 20 to 30 feet. 
Now, with your free hand, grasp the stretched 
spring two or three feet from the end. Pull the 
two or three feet of spring together toward the 
end and then release it, being careful not to let 
go of the fixed end with your other hand! 
Notice the single wave, called a pulse, that 
travels along the spring. In such a longitudi- 
nal pulse the spring coils move back and forth 
along the same direction as the wave travels. 
The wave carries energy, and hence, could be 
used to carry a message from one end of the 
spring to the other. 

You can see a longitudinal wave more eas- 
ily if you tie pieces of string to several of the 
loops of the spring and watch their motion 
when the spring is pulsed. 

A transverse wave is easier to see. To 
make one, practice moving your hand very 
quickly back and forth at right angles to the 
stretched spring, until you can produce a pulse 
that travels down only one side of the spring. 
This pulse is called "transverse" because the 
individual coils of wire move at right angles to 
(transverse to) the length of the spring. 

Perform experiments to answer the follow- 
ing questions about transverse pulses. 



1. Does the size of the pulse change as it 
travels along the spring? If so, in what way? 

2. Does the pulse reflected from the far 
end return to you on the same side of the 
spring as the original pulse, or on the opposite 
side? 

3. Does a change in the tension of the 
spring have any effect on the speed of the 
pulses? When you stretch the spring farther, 
in effect you are changing the nature of the 
medium through which the pulses move. 



Next observe what happens when waves 
go from one material into another — an effect 
called refraction. To one end of your spring at- 
tach a length of rope or rubber tubing (or a dif- 
ferent kind of spring) and have your partner 
hold the end of this. 



4. What happens to a pulse (size, shape, 
speed, direction) when it reaches the bound- 
ary between the two media? The far end of 
your spring is now free to move back and forth 
at the joint which it was unable to do before 
because your partner was holding it. 



Have your partner detach the extra spring 
and once more grasp the far end of your origi- 
nal spring. Have him send a pulse on the same 
side, at the same instant you do, so that the two 
pulses meet. The interaction of the two pulses 
is called interference. 



*Adapted from R. F. Brinckerhoff and D. S. Taft, Modern 
Laboratory Experiments in Physics, by permission of Sci- 
ence Electronics, Nashua, N.H. 



5. What happens (size, shape, speed, di- 
rection) when two pulses reach the center of 
the spring? (It will be easier to see what hap- 
pens in the interaction if one pulse is larger 
than the other.) 

6. What happens when two pulses on op- 
posite sides of the spring meet? 

As the two pulses pass on opposite sides of 
the spring, can you observe a point on the 
spring that does not move at all? 

7. From these two observations, what 
can you say about the displacement caused by 
the addition of two pulses at the same point? 



Unit 3/48 



Experiment 3-16 



By vibrating your hand steadily back and 
forth, you can produce a train of pulses, a 
periodic wave. The distance between any two 
neighboring crests on such a periodic wave is 
the wavelength. The rate at which you vibrate 
your hand will determine the frequency of the 
periodic wave. Use a long spring and produce 
short bursts of periodic waves so you can ob- 
serve them without interference by reflections 
from the far end. 



8. How does the wavelength seem to de- 
pend on the frequency? 



You have now observed the reflection, re- 
fraction, and interference of single waves, or 
pulses, traveling through different materials. 
These waves, however, moved only along one 
dimension. So that you can make a more realis- 
tic comparison with other forms of traveling 
energy, in the next experiment you will turn to 
these same wave properties spread out over a 
two-dimensional surface. 



EXPERIMENT 3-16 
RIPPLE TANK 



WAVES IN A 



In the laboratory one or more ripple tanks will 
have been set up. To the one you and your 
partner are going to use, add water (if neces- 
sary) to a depth of 6 to 8 mm. Check to see that 
the tank is level so that the water has equal 
depth at all four corners. Place a large sheet of 
white paper on the table below the ripple tank, 
and then switch on the hght source. Distur- 
bances on the water surface are projected onto 
the paper as hght and dark patterns, thus al- 
lowing you to "see" the shape of the distur- 
bances in the horizontal plane. 

To see what a single pulse looks Hke in a 
ripple tank, gently touch the water with your 
fingertip — or, better, let a drop of water fall into 
it from a medicine dropper held only a few mil- 
limeters above the surface. 

For certain purposes it is easier to study 
pulses in water if their crests are straight. To 
generate single straight pulses, place a three- 
quarter-inch dowel, or a section of a broom 
handle, along one edge of the tank and roll it 
backward a fraction of an inch. A periodic 



wave, a continuous train of pulses, can be 
formed by rolling the dowel backward and for- 
ward with a uniform frequency. 

Use straight pulses in the ripple tank to ob- 
serve reflection, refraction, and diffraction, 
and circular pulses from point sources to ob- 
serve interference. 

Reflection 

Generate a straight pulse and notice the direc- 
tion of its motion. Now place a barrier in the 
water so that it intersects that path. Generate 
new pulses and observe what happens to the 
pulses when they strike the barrier. Try differ- 
ent angles between the barrier and the incom- 
ing pulse. 



1. What is the relationship between the 
direction of the incoming pulse and the 
reflected one? 

2. Replace the straight barrier with a 
curved one. What is the shape of the reflected 
pulse? 

3. Find the point where the reflected 
pulses run together. What happens to the 
pulse after it converges at this point? At this 
point — called the focus — start a pulse with 
your finger, or a drop of water. What is the 
shape of the pulse after reflection from the 
curved barrier? 



Refraction 

Lay a sheet of glass in the center of the tank, 
supported by coins if necessary, to make an 
area of very shallow water. Try varying the 
angle at which the pulse strikes the boundary 
between the deep and shallow water. 



4. What happens to the wave speed at the 
boundary? 

5. What happens to the wave direction at 
the boundary? 

6. How is change in direction related to 
change in speed? 



Interference 

Arrange two point sources side by side a few 
centimeters apart. When tapped gently, they 
should produce two pulses on the spring. You 



Experiment 3-17 



Unit 3/49 



will see the action of interference better if you 
vibrate the two point sources continuously 
with a motor and study the resulting pattern of 
waves. 



diffraction pattern depend on the length of 
the waves? 



7. How does changing the wave fre- 
quency affect the original waves? 

Find regions in the interference pattern 
where the waves from the two sources cancel 
and leave the water undisturbed. Find the re- 
gions where the two waves add up to create a 
doubly great disturbance. 

8. Make a sketch of the interference pat- 
tern indicating these regions. 

9. How does the pattern change as you 
change the wavelength? 



Diffraction 

With two-dimensional waves you can observe a 
new phenomenon — the behavior of a wave 
when it passes around an obstacle or through 
an opening. The spreading of the wave into the 
"shadow" area is called diffraction. Generate 
a steady train of waves by using the motor driv- 
en straight-pulse source. Place a small barrier 
in the path of the waves so that it intercepts 
part but not all of the wave front. Observe what 
happens as the waves pass the edge of the bar- 
rier. Now vary the wavelength of the incoming 
wave train by changing the speed of the motor 
on the source. 



10. How does the interaction with the 
obstacle vary with the wavelength? 

Place two long barriers in the tank, leav- 
ing a small opening between them. 

11. How does the angle by which the 
wave spreads out beyond the opening depend 
on the size of the opening? 

12. In what way does the spread of the 



EXPERIMENT 3-17 MEASURING 
WAVELENGTH 

There are three ways you can conveniently 
measure the wavelength of the waves gener- 
ated in your ripple tank. You should try them 
all, if possible, and cross-check the results. If 
there are differences, indicate which method 
you believe is most accurate and explain why. 

A. Direct 

Set up a steady train of pulses using either 
a single point source or a straightline source. 
Observe the moving waves with a stroboscope, 
and then adjust the vibrator motor to the low- 
est frequency that will "freeze" the wave pat- 
tern. Place a meter stick across the ripple tank 
and measure the distance between the crests of 
a counted number of waves. 

B. Standing Waves 

Place a straight barrier across the center of 
the tank parallel to the advancing waves. 
When the distance of the barrier from the 
generator is properly adjusted, the superposi- 
tion of the advancing waves and the waves 
reflected from the barrier will produce stand- 
ing waves. In other words, the reflected waves 
are at some points reinforcing the original 
waves, while at other points there is always 
cancellation. The points of continual cancella- 
tion are called nodes. The distance between 
nodes is one-half wavelength. 

C. Interference Pattern 

Set up the ripple tank with two point 
sources. The two sources should strike the 
water at the same instant so that the two 
waves will be exactly in phase and of the same 
frequency as they leave the sources. Adjust 
the distance between the two sources 




Unit 3/50 



Experiment 3-17 




Fig. 1 An interference pattern in water. Two point 
sources vibrating in phase generate waves in a ripple 
tank. A and C are points of maximum disturbance (in 
opposite directions) and B is a point of minimum distur- 
bance. 



and the frequency of vibration until a distinct 
pattern is obtained, such as in Fig. 1. 

As you study the pattern of ripples you will 
notice hnes along which the waves cancel al- 
most completely so that the ampUtude of the 
disturbance is almost zero. These lines are 
called nodal lines, or nodes. You have already 
seen nodes in your earUer experiment with 
standing waves in the ripple tank. 

At every point along a node the waves ar- 
riving from the two sources are half a 
wavelength out of step, or "out of phase." This , 
means that for a point (such as B in Fig. 1) 
to be on a line of nodes it must be i or li or 
2i . . . wavelengths farther from one source 
than from the other. 

Between the hnes of nodes are regions of 
maximum disturbance. Points A and C in Fig. 
1 are on lines down the center of such re- 
gions, called antinodal lines. Reinforcement of 
waves from the two sources is a maximum 
along these lines. 

For reinforcement to occur at a point, the 
two waves must arrive in step or "in phase." 
This means that any point on a hne of an- 
tinodes is a whole number of wavelengths 0, 1, 
2, . . . farther from one source than from the 
other. The relationship between crests, 
troughs, nodes and antinodes in this situation 
is summarized schematically in Fig. 2. 




Fig. 2 

Analysis of interference pattern similar to that of Fig. 1 
at the top of the left column set up by two in-phase 
periodic sources. (Here S, and Sj are separated by four 
wavelengths.) The letters A and N designate antinodal 
and nodal lines. 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. 



Experiment 3-18 



Unit 3/51 



Most physics textbooks develop the mathe- 
matical argument of the relationship of wave- 
length to the geometry of the interference 
pattern. (See, for example, p. 119 in Unit 3 of 
the Project Physics Text.) If the distance be- 
tween the sources is d and the detector is at 
a comparatively greater distance L from the 
sources, then d, L, and k are related by the 
equations 



or 



^ _xd 



where x is the distance between neighboring 
antinodes (or neighboring nodes). 

You now have a method for computing the 
wavelength \ from the distances that you can 
measure precisely. Measure x, d, and L in your 
ripple tank and compute K. 

EXPERIMENT 3-18 SOUND 

In previous experiments you observed how 
waves of relatively low frequency behave in 
different media. In this experiment you wdll try 
to determine to what extent audible sound ex- 
hibits similar properties. 

At the laboratory station where you work 
there should be the following: an oscillator, a 
power supply, two small loudspeakers, and a 
group of materials to be tested. A loudspeaker 
is the source of audible sound waves, and your 
ear is the detector. First connect one of the 
loudspeakers to the output of the oscillator and 
adjust the oscillator to a frequency of about 
4000 cycles per second. Adjust the loudness so 
that the signal is just audible one meter away 
from the speaker. The gain-control setting 
should be low enough to produce a clear, pure 
tone. Reflections from the floor, tabletop, and 
hard-surfaced walls may interfere with your 
observations so set the sources at the edge of a 
table, and put soft material over any unavoid- 
ably close hard surface that could cause 
reflective interference. 

You may find that you can locahze sounds 
better if you make an "ear trumpet" or stetho- 
scope from a small funnel or thistle tube and a 
short length of rubber tubing (Fig. 1). Cover 




Fig. 1. Sound from the speaker can be detected by 
using a funnel and rubber hose, the end of which is 
placed to the ear. The Oscillator's banana plug jacks 
must be inserted into the -8V, +8V and ground holes of 
the Power Supply. Insert the speaker's plugs into the 
sine wave — ground receptacles of the Oscillator. Select 
the audio range by means of the top knob of the Oscil- 
lator and then turn on the Power Supply. 

the ear not in use to minimize confusion when 
you are hunting for nodes and maxima. 

Transmission and Refiection 

Place samples of various materials at your sta- 
tion between the speaker and the receiver to 
see how they transmit the sound wave. In a 
table, record your quaUtative judgments as 
best, good, poor, etc. 

Test the same materials for thefr ability to 
reflect sound and record your results. Be sure 
that the sound is really being reflected and is 
not coming to your detector by some other path. 
You can check how the intensity varies at the 
detector when you move the reflector and ro- 
tate it about a vertical axis (see Fig. 2). 




Fig. 2 





Unit 3/52 



Experiment 3-18 



If suitable materials are available to you, also 
test the reflection from curved surfaces. 



1. On the basis of your findings, what 
generalizations can you make relating 
transmission and reflection to the properties 
of the test materials? 



Refraction 

You have probably observed the refraction or 
"bending" of a wave front in a ripple tank as 
the wave slowed down in passing from water of 
one depth to shallower water. 

You may observe the refraction of sound 
waves using a "lens" made of gas. Inflate a 
spherical balloon with carbon dioxide gas to a 
diameter of about 4 to 6 inches. Explore the 
area near the balloon on the side away from the 
source. Locate a point where the sound seems 
loudest, and then remove the balloon. 



2. Do you notice any difference in loud- 
ness when the balloon is in place? Explain. 



Diffraction 

In front of a speaker set up as before place a 
thick piece of hard material about 25 cm long, 
mounted vertically about 25 cm directly in 
front of the speaker. Slowly probe the area 
about 75 cm beyond the obstacle. 



3. Do you hear changes in loudness? Is 
there sound in the "shadow" area? Are there 
regions of silence where you would expect to 
hear sound? Does there seem to be any pat- 
tern to the areas of minimum sound? 



For another way to test for diffraction, use 
a large piece of board placed about 25 cm in 
front of the speaker with one edge aligned with 
the center of the source. Now explore the area 
inside the shadow zone and just outside it. 

Describe the pattern of sound interference 
that you detect. 



4. Is the pattern analogous to the pattern 
you observed in the ripple tank? 



Wavelength 

(a) Standing wave method Set your 
loudspeaker about i meter above and facing 



toward a hard tabletop or floor or about that 
distance from a hard, smooth plaster wall or 
other good sound reflector (see Fig. 3). Your 
ear is most sensitive to the changes in intensity 
of faint sounds, so be sure to keep the volume 
low. 

Explore the space between the source and 
reflector, listening for changes in loudness. 
Record the positions of minimum loudness, or 
at least find the approximate distance between 
two consecutive minima. These minima are 
located i wavelength apart. 



5. Does the spacing of the minima de- 
pend on the intensity of the wave? 

Measure the wavelength of sound at sev- 
eral different frequencies. 

6. How does the wavelength change 
when the frequency is changed? 



(h) Interference Method Connect the two loud- 
speakers to the output of the oscillator and 
mount them at the edge of the table about 25 
cm apart. Set the frequency at about 4,000 
cycles/sec to produce a high-pitched tone. Keep 
the gain setting low during the entire experi- 
ment to make sure the oscillator is producing 
a pure tone, and to reduce reflections that 
would interfere with the experiment. 

Move your ear or "stethoscope" along a hne 
parallel to, and about 50 cm from, the line join- 
ing the sources. Can you detect distinct max- 
ima and minima? Move farther away from the 
sources; do you find any change in the pattern 
spacing? 









r 


fl| 


! . 1 


• v.v • 

I-' ■ 


f 






^ 


1 


1 



Fig. 3 



Experiment 3-19 



Unit 3/53 



7. What effect does a change in the 
source separation have on the spacing of the 
nodes? 

8. What happens to the spacing of the 
nodes if you change the frequency of the 
sound? To make this experiment quantita- 
tive, work out for yourself a procedure similar 
to that used with the ripple tank. (Fig. 2.) 



Measure the separation d of the source 
centers and the distance x between nodes and 
use this data to calculate the wavelength k. 



9. Does the wavelength change with fre- 
quency? If so, does it change directly or in- 
versely? 



Calculating the Speed of Sound 

The relationship between speed v, wave- 
length X, and frequency / is i; = \f. The os- 
cillator dial gives a rough indication of the 
frequency (and your teacher can advise you on 
how to use an oscilloscope to make precise fre- 
quency settings). Using your best estimate of X, 
calculate the speed of sound. If you have time, 
extend your data to answering the following 
questions: 



10. Does the speed of the sound waves 
depend on the intensity of the wave? 

11. Does the speed depend on the fre- 
quency? 



EXPERIMENT 3-19 ULTRASOUND 

The equipment needed for this experiment is 
an oscillator, power supply, and three ul- 
trasonic transducers — crystals that transform 
electrical impulses into sound waves (or vice 
versa), and several materials to be tested. The 
signal from the detecting transducer can be 
displayed with either an oscilloscope (as in Fig. 
1) or an amplifier and meter (Fig. 2). One or 
two of the transducers, driven by the oscillator 
are sources of the ultrasound, while the third 
transducer is a detector. Before you proceed, 
have the teacher check your setup and help you 




Fig. 1 Complete ultrasound equipment. Plug the +8v, 
-8v, ground jacks from the Amplifier and Oscillator 
into the Power Supply. Plug the coaxial cable attached 
to the transducer to the sine wave output of the Oscil- 
lator. Plug the coaxial cable attached to a second 
transducer into the input terminals of the amplifier. Be 
sure that the shield of the coaxial cable is attached 
to ground. Turn the oscillator range switch to the 5K- 
50K position. Turn the horizontal frequency range 
switch of the oscilloscope to at least 10kHz. Turn on 
the Oscillator and Power Supply. Tune the Oscillator 
for maximum reception, about 40 kilocycles. 

get a pattern on the oscilloscope screen or a 
reading on the meter. 

The energy output of the transducer is 
highest at about 40,000 cycles per second, and 




Fig. 2 Above, ultrasound transmitter and receiver. The 
signal strength is displayed on a microammeter con- 
nected to the receiver amplifier. Below, a diode con- 
nected between the amplifier and the meter, to rectify 
the output current. The amplifier selector switch should 
be turned to ac. The gain control on the amplifier should 
be adjusted so that the meter will deflect about full-scale 
for the loudest signal expected during the experiment. 
The offset control should be adjusted until the meter 
reads zero when there is no signal. 



Unit 3/54 



Experiment 3-19 



the oscillator must be carefuUy "tuned" to that 
frequency. Place the detector a few centime- 
ters directly in front of the source and set the 
oscillator range to the 5-50 kilocycle position. 
Tune the oscillator carefully around 40,000 
cycles/second for maximum deflection of the 
meter or the scope track. If the signal output is 
too weak to detect beyond 25 cm, plug the de- 
tector transducer into an amplifier and connect 
the output of the amplifier to the oscilloscope or 
meter input. 

Transmission and Reflection 

Test the various samples at your station to see 
how they transmit the ultrasound. Record your 
judgments as best, good, poor, etc. Hold the 
sample of the material being tested close to the 
detector. 

Test the same materials for their ability to 
reflect ultrasound. Be sure that the ultrasound 
is really being reflected and is not coming to 
your detector by some other path. You can 
check this by seeing how the intensity varies at 
the detector when you move the reflector. 

Make a table of your observations. 



1. What happens to the energy of ul- 
trasonic waves in a material that neither 
reflects nor transmits well? 



Diffraction 

To observe diffraction around an obstacle, put 
a piece of hard material about 3 cm wide 8 or 
10 cm in front of the source (see Fig. 3.) Ex- 
plore the region 5-10 cm behind the obstacle. 




2. Do you find any signal in the "shadow" 
area? Do you find minima in the regions 
where you would expect a signal to be? Does 
there seem to be any pattern relating the 
areas of minimum and maximum signals? 



Put a larger sheet of absorbing material 10 
cm in front of the source so that the edge ob- 
structs about one-half of the source. 

Again probe the "shadow" area and the 
area near the edge to see if a pattern of maxima 
and minima seems to appear. 

Measuring Wavelength 

(a) Standing Wave Method 

Investigate the standing waves set up be- 
tween a source and a reflector, such as a hard 
tabletop or metal plate. Place the source about 
10 to 15 cm from the reflector with the detec- 
tor. 



3. Does the spacing of nodes depend on 
the intensity of the waves? 



Find the approximate distance between 
two consecutive maxima or two consecutive 
minima. This distance is one half the wave- 
length. 

(b) Interference Method For sources, connect 
two transducers to the output of the oscillator 
and set them about 5 cm apart. Set the oscil- 
lator switch to the 5-50 kilocycle position. For a 
detector, connect a third transducer to an oscil- 




Flg. 3 Detecting diffraction of ultrasound around a 
barrier. 



Fig. 4 Set-up for determination of wavelength by the 
interference method. 



Experiment 3-19 



Unit 3/55 



loscope or amplifier and meter as described in 
Part A of the experiment. Then tune the oscil- 
lator for maximum signal from the detector 
when it is held near one of the sources (about 
40,000 cycles/sec). Move the detector along a 
line parallel to and about 25 cm in front of a 
line connecting the sources. Do you find dis- 
tinct maxima and minima? Move closer to the 
sources. Do you find any change in the pattern 
spacing? 



4. What effect does a change in the sep- 
aration of the sources have on the spacing of 
the nulls? 



To make this experiment quantitative, 
work out a procedure for yourself similar to 
that used with the ripple tank. Measure the ap- 
propriate distances and then calculate the 
wavelength using the relationship 



derived earher for interference patterns in a 
ripple tank. 



5. In using that equation, what assump- 
tions are you making? 



The Speed of Ultrasound 

The relationship between speed v, wavelength 
A., and frequency / is i; = \f. Using your best 
estimate of \, calculate the speed of sound. 



6. Does the speed of the ultrasound 
waves depend on the intensity of the wave? 

7. How does the speed of sound in the in- 
audible range compare with the speed of au- 
dible sound? 



Unit 3/56 



ACTIVITIES 



IS MASS CONSERVED? 

You have read about some of the difficulties in 
establishing the law of conservation of mass. 
You can do several different experiments to 
check this law. 

Alka-Seltzer. 

You will need the following equipment: Alka- 
Seltzer tablets; 2-liter flask, or plastic one- 
gallon jug (such as is used for bleach, distilled 
water, or duplicating fluid); stopper for flask or 
jug; warm water; balance (sensitivity better 
than 0.1 g) spring scale (sensitivity better than 
0.5 g). 

Balance a tablet and 2-liter flask contain- 
ing 200-300 cc of water on a sensitive balance. 
Drop the tablet in the flask. When the tablet 
disappears and no more bubbles appear, read- 
just the balance. Record any change in mass. 
If there is a change, what caused it? 

Repeat the procedure above, but include 
the rubber stopper in the initial balancing. 
Immediately after dropping in the tablet, place 
the stopper tightly in the mouth of the flask. 
(The pressure in a 2-liter flask goes up by no 
more than 20 per cent, so it is not necessary to 
tape or wire the stopper to the flask. Do not use 
smaller flasks in which proportionately higher 
pressure would be built up.) Is there a change 
in mass? Remove the stopper after all reaction 
has ceased; what happens? Discuss the differ- 
ence between the two procedures. 

Brightly Colored Precipitate. 

You will need: 20 g lead nitrate; 1 1 g potassium 
iodide; Erlenmeyer flask, 1000 cc with stopper; 
test tube, 25 x 150 mm; balance. 

Place 400 cc of water in the Erlenmeyer 
flask, add the lead nitrate, and stir until dis- 
solved. Place the potassium iodide in the test 
tube, add 30 cc of water, and shake until dis- 
solved. Place the test tube, open and upward, 
carefully inside the flask and seal the flask 
with the stopper. Place the flask on the balance 
and bring the balance to equilibrium. Tip the 
flask to mix the solutions. Replace the flask on 



the balance. Does the total mass remain con- 
served? What does change in this experiment? 

Magnesium Flash Bulb. 

On the most sensitive balance you have avail- 
able, measure the mass of an unflashed mag- 
nesium flash bulb. Repeat the measurement 
several times to make an estimate of the preci- 
sion of the measurement. 

Flash the bulb by connecting it to a battery. 
Be careful to touch the bulb as little as possi- 
ble, so as not to wear away any material or 
leave any fingerprints. Measure the mass of 
the bulb several times, as before. You can get a 
feeling for how small a mass change your bal- 
ance could have detected by seeing how large a 
piece of tissue paper you have to put on the 
balance to produce a detectable difference. 

EXCHANGE OF MOMENTUM 
DEVICES 

The four situations described below are more 
complex tests for conservation of momentum, 
to give you a deeper understanding of the gen- 
erality of the conservation law and of the im- 
portance of your frame of reference, (a) Fasten 
a section of HO gauge model railroad track to 
two ring stands as shown in Fig. 1. Set one 
truck of wheels, removed from a car, on the 
track and from it suspend an object with mass 
roughly equal to that of the truck. Hold the 
truck, pull the object to one side, parallel to the 
track, and release both at the same instant. 
What happens?" 



Fig. 1 




Activities 



Unit 3/57 



Predict what you expect to see happen if 
you released the truck an instant after releas- 
ing the object. Try it. 

Try increasing the suspended mass, 
(b) Fig. 2 shows a similar situation, using an 
air track supported on ring stands. An object 
of 20 g mass was suspended by a 50 cm string 
from one of the small air-track gliders. (One 
student trial continued for 166 swings.) 

■w.mf —111 




(c) Fasten two dynamics carts together with 
four hacksaw blades as shown in Fig. 3. Push 
the top one to the right, the bottom to the left, 
and release them. Try giving the bottom cart a 
push across the room at the same instant you 
release them. 

What would happen when you released the 
two if there were 10 or 20 bearing balls or small 
wooden balls hung as pendula from the top 
cart? 




Fig. 3 

(d) Push two large rubber stoppers onto a short 
piece of glass tubing or wood (Fig. 4). Let the 
"dumbbell" roll down a wooden wedge so that 
the stoppers do not touch the table until the 
dumbbell is almost to the bottom. When the 
dumbbell touches the table, it suddenly in- 
creases its hnear momentum as it moves off 
along the table. Principles of rotational mo- 
mentum and energy are involved here that 
are not covered in the Text, but even without 




Fig. 4 

extending the Text, you can deal with the 
"mysterious" increase in hnear momentum 
when the stoppers touch the table. 

Using what you have learned about con- 
servation of momentum, what do you think 
could account for this increase? (Hint: set the 
wedge on a piece of cardboard supported on 
plastic beads and try it.) 

STUDENT HORSEPOWER 

When you walk up a flight of stairs, the work 
you do goes into frictional heating and in- 
creasing gravitational potential energy. The 
A(PE)grav, in joules, is the product of your 
weight in newtons and the height of the stairs 
in meters. (In foot-pounds, it is your weight in 
pounds times the height of the stairs in feet.) 

Your useful power output is the average 
rate at which you did the hfting work — that is, 
the total change in (PEjgrav, divided by the time 
it took to do the work. 

Walk or run up a flight of stairs and have 
someone time how long it takes. Determine the 
total vertical height that you lifted yourself by 
measuring one step and multiplying by the 
number of steps. 

Calculate your useful work output and 
your power, in both units of watts and in horse- 
power. (One horsepower is equal to 550 
foot-pounds/sec which is equal to 746 watts.) 

STEAM-POWERED BOAT 

You can make a steam-propelled boat that will 
demonstrate the principle of Heron's steam 
engine (Text Sec. 10.5) from a small tooth- 
powder or talcum-powder can, a piece of can- 
dle, a soap dish, and some wire. 

Place the candle in the soap dish. Punch 
a hole near the edge of the bottom of the can 
with a needle. Construct wire legs which are 
long enough to support the can horizontally 



Unit 3/58 



Activities 



Vs?n 



H— ■ 



ii 



.-^- 



over the candle and the soap dish. Rotate the 
can so that the needle-hole is at the top. Half 
fill the can with water, replace the cover, and 
place this "boiler" over the candle and Ught the 
candle. If this boat is now placed in a large pan 
of water, it will be propelled across the pan. 

Can you explain the operation of this boat 
in terms of the conservation of momentum? of 
the conservation of energy? 

PROBLEMS OF SCIENTIFIC AND 
TECHNOLOGICAL GROWTH 

The Industrial Revolution of the eighteenth 
and nineteenth centuries is rich in examples of 
man's disquiet and ambivalence in the face of 
technological change. Instead of hving among 



pastoral waterwheel scenes, men began to live 
in areas with pollution problems as bad or 
worse than those we face today, as is shown in 
the scene at Wolverhampton, England in 1866. 
As quoted in the Text, WiUiam Blake lamented 
in "Stanzas from Milton," 

And did the Countenance Divine 
Shine forth upon our clouded hills? 

And was Jerusalem builded here 
Among these dark Satanic mills? 

Ever since the revolution began, we have 
profited from advances in technology. But we 
also still face problems like those of pollution 
and of displacement of men by machines. 

One of the major problems is a growing 
lack of communication between people work- 
ing in science and those working in other 
fields. When C. P. Snow published his book 
The Two Cultures and the Scientific Revolu- 
tion in 1959, he initiated a wave of debate 
which is still going on. 

In your own community there are probably 
some pollution problems of which you are 
aware. Find out how science and technology 




A portrayal of the scene near Wolverhampton, England in 1866, called "Black Country." 



Activities 



Unit 3/59 



"^^=3^ 



r^^^- 




^r- 



"Honk, honk, honk, honk, honk, honk, honk. 









cough, cough, cough, cough, cough, cough, cough. 



honk, honk, honk, honk, honk, honk, honk." 



may have contributed to these problems — and 
how they can contribute to solutions! 

PREDICTING THE RANGE OF AN 
ARROW 

If you are interested in predicting the range of 
a projectile from the work you do on a sling- 
shot while drawing it back, ask your teacher 
about it. Perhaps he will do this with you or tell 
you how to do it yourself. 

Another challenging problem is to estimate 
the range of an arrow by calculating the work 
done in drawing the bow. To calculate work, 
you need to know how the force used in draw- 
ing the string changed with the string dis- 
placement. A bow behaves even less according 
to Hooke's law than a slingshot; the force vs. 
displacement graph is definitely not a straight 
line. 

To find how the force depends on string 
displacement, fasten the bow securely in a vise 
or some soUd mounting. Attach a spring bal- 
ance to the bow and record values of force (in 
newtons) as the bowstring is drawn back one 
centimeter at a time from its rest position 
(without having an arrow notched). Or, have 
someone stand on a bathroom scale, holding 
the bow, then pull upwards on the string; the 
force on the string at each position will equal 
this apparent loss of weight. 

Now to calculate the amount of work done, 
plot a force vs. displacement graph. Count 
squares to find the "area" (force units times 



displacement units) under the graph; this is 
the work done on the bow — equal to the elastic 
potential energy of the drawn bow. 

Assume that all the elastic potential 
energy of the bow is converted into the kinetic 
energy of the arrow and predict the range of the 
arrow by the same method used in predicting 
the range of a shngshot projectile. 

A recent magazine article stated that an 
alert deer can leap out of the path of an ap- 
proaching arrow when he hears the twang of 
the bowstring. Under what conditions do you 
think this is possible? 

DRINKING DUCK 

A toy called a Drinking Duck (No. 60,264 
in Catalogue 671, about $1.00, Edmund 
Scientific Co., Barrington, New Jersey 08007) 
demonstrates very well the conversion of heat 




' "What's this scheme of yours for on economical method 
of launching a satellite?" 



Unit 3/60 



Activities 



energy into energy of gross motion by the proc- 
esses of evaporation and condensation. The 
duck will continue to bob up and down as long 
as there is enough water in the cup to wet his 
beak. 




Rather than dampen your spirit of adven- 
ture, we won't tell you how it works. First, see 
if you can figure out a possible mechanism for 
yourself. If you can't, George Gamow's book. 
The Biography of Physics, has a very good ex- 
planation. Gamow also calculates how far the 
duck could raise water in order to feed himself. 
An interesting extension is to replace the water 
with rubbing alcohol. What do you think will 
happen? 

Lest you think this device useful only as a 
toy, an article in the June 3, 1967, Saturday 
Review described a practical apphcation being 
considered by the Rand Corporation. A group of 
engineers built a 7-foot "bird" using Freon 11 
as the working fluid. Their intention was to in- 
vestigate the possible use of large-size ducks 
for irrigation purposes in the Nile River Valley. 

MECHANICAL EQUIVALENT 
OF HEAT 

By dropping a quantity of lead shot from a 
measured height and measuring the resulting 
change in temperature of the lead, you can get 
a value for the ratio of work units to heat 
units — the "mechanical equivalent of heat." 
You will need the following equipment: 
Cardboard tube Lead shot (1 to 2 kg) 

Stoppers Thermometer 



Close one end of the tube with a stopper, 
and put in 1 to 2 kg lead shot that has been 
cooled about 5°C below room temperature. 
Close the other end of the tube with a stopper in 
which a hole has been drilled and a thermome- 
ter inserted. Carefully roll the shot to this end 
of the tube and record its temperature. Quickly 
invert the tube, remove the thermometer, and 
plug the hole in the stopper. Now invert the 
tube so the lead falls the fuU length of the tube 
and repeat this quickly one hundred times. 
Reinsert the thermometer and measure the 
temperature. Measure the average distance 
the shot falls, which is the length of the tube 
minus the thickness of the layer of shot in the 
tube. 

If the average distance the shot falls is h 
and the tube is inverted N times, the work you 
did raising the shot of mass m is: 

AW ^ N X mag x h 

The heat AH needed to raise the temperature of 
the shot by an amount AT is: 

AH = cmAT 

where c is the specific heat capacity of lead, 
0.031 cal/gC°. 

The mechanical equivalent of heat is 
AW/ AH. The accepted experimental value is 
4.184 newton-meters per kilocalorie. 

A DIVER IN A BOTTLE 

Descartes is a name well known in physics. 
When we graphed motion in Text Sec. 1.5, we 
used Cartesian coordinates, which Descartes 
introduced. Using Snell's law of refraction, 
Descartes traced a thousand rays through a 
sphere and came up with an explanation of the 
rainbow. He and his astronomer friend Gas- 
sendi were a bulwark against Aristotehan 
physics. Descartes belonged to the generation 
between Galileo and Newton. 

On the lighter side, Descartes is known for 
a toy called the Cartesian diver which was very 
popular in the eighteenth century when very 
elaborate ones were made. To make one, first 
you will need a column of water. You may 



Activities 



Unit 3/61 



find a large cylindrical graduate about the 
laboratory, the taller the better. If not, you can 
improvise one out of a gallon jug or any other 
tall glass container. FUl the container almost to 
the top with water. Attach a piece of glass tub- 
ing that has been fire-pohshed on each end. 
Lubricate the glass tubing and the hole in the 
stopper with water and carefully insert the 
glass tubing. Fit the rubber stopper into the top 
of the container as shown in Fig. 1. 

Next construct the diver. You may limit 
yourself to pure essentials, namely a small pill 
bottle or vial, which may be weighted with 
wire and partially filled with water so it just 
barely floats upside down at the top of the 
water column. If you are so inclined, you can 
decorate the bottle so it looks like a real under- 
water swimmer (or creature, if you prefer). The 
essential things are that you have a diver that 
just floats and that the volume of water can be 
changed. 

Now to make the diver perform, just blow 
momentarily on the rubber tube. According to 
Boyle's law, the increased pressure (trans- 
mitted by the water) decreases the volume of 
trapped air and water enters the diver. The 
buoyant force decreases, according to Archi- 
medes' principle, and the diver begins to sink. 

If the original pressure is restored, the 
diver rises again. However, if you are lucky, 
you will find that as you cautiously make him 
sink deeper and deeper down into the column 
of water, he is more and more reluctant to re- 
turn to the surface as the additional surface 
pressure is released. Indeed, you may find a 
depth at which he remains almost stationary. 
However, this apparent equilibrium, at which 
his weight just equals the buoyant force, is un- 
stable. A bit above this depth, the diver will 
fireely rise to the surface, and a bit below this 
depth he will sink to the bottom of the water 
column from which he can be brought to the 
surface only by vigorous sucking on the tube. 

If you are mathematically inclined, you 
can compute what this depth would be in terms 
of the atmospheric pressure at the surface, the 
volume of the trapped air, and the weight of the 
diver. If not, you can juggle with the volume of 
the trapped air so that the point of unstable 




X 



rubber tufce. 
'1^ rubber stopper 




If 



p'li! bottle. 



■Qir 



ujire weio'i"^ to k?ep 
6i\it<- naiit- sidfe up 




Fig. 1 

equilibrium comes about halfway down the 
water column. 

The diver raises interesting questions. 
Suppose you have a well-behaved diver who 
"floats" at room temperature just halfway 
down the water column. Where will he "float" 
if the atmospheric drops? Where will he "float" 
if the water is cooled or is heated? Perhaps the 
ideal gas law is not enough to answer this 
question, and you may have to do a bit of read- 
ing about the "vapor pressure" of water. 

After demonstrating the performance of 
your large-scale model by blowing or sucking 
in the rubber tube, you can mystify your audi- 
ence by making a small scale model in a bottle. 
A plastic bottle with flat sides can act Uke a 
diaphragm which increases the pressure with- 
in as the sides are pushed together. The bottle 



Unit 3/62 



Activities 



and diver are tightly sealed. In this case, add a 
rubber tube leading to a holeless stopper. Your 
classmates blowing as hard as they will cannot 
make the diver sink; but you, by secredy 
squeezing the bottle, can make him perform at 
your command. 




ROCKETS 

If it is legal to set off rockets in your area, and 
their use is supervised, they can provide excel- 
lent projects for studying conversion from 
kinetic to potential energy, thrust, etc. 

Ask your teacher for instructions on how to 
build small test stands for taking thrust data to 
use in predicting the maximum height, range, 
etc. of the rockets. (Estes Industries, Box 227, 
Penrose, Colorado 81240, will send a very 
complete free catalogue and safety rules on re- 
quest.) 

HOW TO WEIGH A CAR WITH A 
TIRE PRESSURE GAUGE 

Reduce the pressure in all four of your auto 
tires so that the pressure is the same in each 
and somewhat below recommended tire pres- 
sure. 

Drive the car onto four sheets of graph 
paper placed so that you can outline the area of 
the tire in contact with each piece of paper. 
The car should be on a reasonably flat surface 
(garage floor or smooth driveway). The 
flattened part of the tire is in equilibrium be- 
tween the vertical force of the ground upward 
and the downward force of air pressure wdthin. 

Measure the air pressure in the tires, and 
the area of the flattened areas. If you use inch 
graph paper, you can determine the area in 
square inches by counting squares. 



Pressure P (in pounds per square inch) is 
defined as F/A, where F is the downward force 
(in pounds) acting perpendicularly on the 
flattened area A (in square inches). Since the 
tire pressure gauge indicates the pressure 
above the normal atmospheric pressure of 15 
Ib/in^ you must add this value to the gauge 
reading. Compute the four forces as pressure 
times area. Their sum gives the weight of the 
car. 

PERPETUAL-MOTION MACHINES? 

You must have heard of "perpetual-motion" 
machines which, once started, will continue 
running and doing useful work forever. These 
proposed devices are inconsistent with laws of 
thermodynamics. (It is tempting to say that 
they violate laws of thermodynamics — but this 
implies that laws are rules by which Nature 
must run, instead of descriptions men have 
thought up.) We now believe that it is in princi- 
ple impossible to buUd such a machine. 

But the dream dies hard! Daily there are 
new proposals. Thus S. Raymond Smedile, in 
Perpetual Motion and Modern Research for 
Cheap Power (Science Publications of Boston, 
1962), maintains that this attitude of "it can't 
be done" negatively influences our search for 
new sources of cheap power. His book gives 
sixteen examples of proposed machines, of 
which two are shown here. 

Number 5 represents a wheel composed of 
twelve chambers marked A. Each chamber 
contains a lead ball B, which is free to roll. As 
the wheel turns, each ball rolls to the lowest 
level possible in its chamber. As the balls roll 
out to the right edge of the wheel, they create a 
preponderance of turning effects on the right 
side as against those balls that roll toward the 
hub on the left side. Thus, it is claimed the 
wheel is driven clockwise perpetually. If you 
think this will not work, explain why not. 

Number 7 represents a water-driven wheel 
marked A. D represents the buckets on the 
perimeter of the waterwheel for receiving 
water draining from the tank marked F. The 
waterwheel is connected to pump B by a belt 
and wheel. As the overshot wheel is operated 
by water dropping on it, it operates the pump 



Activities Unit 3/63 




rwL 




Number 5 



Number 7 



which sucks water into C from which it enters 
into tank F. This operation is supposed to go on 
perpetually. If you think otherwise, explain 
why. 

If such machines would operate, would the 
conservation laws necessarily be wrong? 

Is the reason that true perpetual motion 
machines are not found due to "theoretical" or 
"practical" deficiencies? 



STANDING WAVES ON A DRUM 
AND A VIOLIN 

You can demonstrate many different patterns 
of standing waves on a rubber membrane 
using a method very similar to that used in 
Film Loop 42, "Vibrations of A Drum." If you 
have not yet seen this loop, view it if possible 
before setting up the demonstration in your 
lab. 





The cartoons above (and others of the same style which are scattered through the 
Handbook) were drawn in response to some ideas in the Project Physics Course by a 
cartoonist who was unfamiliar with physics. On being informed that the drawing on the 
left did not represent conservation because the candle wasn't a closed system, he 
offered the solution at the right. (Whether a system is "closed " depends, of course, 
upon what you are trying to conserve.) 



Unit 3/64 



Activities 



Fig. 1 shows the apparatus in action, pro- 
ducing one pattern of standing waves. The 





Fig. 1 

drumhead in the figure is an ordinary 7-inch 
embroidery hoop with the end of a large balloon 
stretched over it. If you make your drumhead 
in this way, use as large and as strong a balloon 
as possible, and cut its neck off with scissors. A 
flat piece of sheet rubber (dental dam) gives 
better results, since even tension over the en- 
tire drumhead is much easier to maintain if the 



rubber is not curved to begin with. Try other 
sizes and shapes of hoops, as well as other 
drumhead materials. 

A 4-inch, 45-ohm speaker, lying under the 
drum and facing upward toward it, drives the 
vibrations. Connect the speaker to the output 
of an oscillator. If necessary, ampUfy the oscil- 
lator output. 

Turn on the oscillator and sprinkle salt or 
sand on the drumhead. If the frequency is near 
one of the resonant frequencies of the surface, 
standing waves will be produced. The salt will 
collect along the nodes and be thrown off from 
the antinodes, thus oudining the pattern of the 
vibration. Vary the frequency until you get a 
clear pattern, then photograph or sketch the 
pattern and move on to the next frequency 
where you get a pattern. 

When the speaker is centered, the vibra- 
tion pattern is symmetrical around the center 
of the surface. In order to get antisymmetric 
modes of vibration, move the speaker toward 
the edge of the drumhead. Experiment with 
the spacing between the speaker and the 
drumhead until you find the position that gives 
the clearest pattern; this position may be dif- 
ferent for different frequencies. 

If your patterns are distorted, the tension of 
the drumhead is probably not uniform. If you 
have used a balloon, you may not be able to 
remedy the distortion, since the curvature of 
the balloon makes the edges tighter than the 
center. By pulling gently on the rubber, how- 
ever, you may at least be able to make the ten- 
sion even all around the edge. 

A similar procedure, used 150 years ago 
and still used in analyzing the performance of 
violins, is shown in these photos reprinted from 
Scientific American, "Physics and Music." 

MOIRE PATTERNS 

You are probably noticing a disturbing visual 
effect from the patterns in Figs. 1 and 2 on the 
opposite page. Much of "op art" depends on 
similar effects, many of which are caused by 
moire patterns. 

If you make a photographic negative of 
the pattern in Fig. 1 or Fig. 2 and place it 
on top of the same figure, you can use it to 



Activities 



Unit 3/65 




Chladni Plates indicate the vibration of the body of a violin. These patterns were 
produced by covering a violin-shaped brass plate with sand and drawing a violin 
bow across its edge. When the bow caused the plate to vibrate, the sand concentrated 
along quiet nodes between the vibrating areas. Bowing the plate at various points, 
indicated by the round white marker, produces different frequencies of vibration 
and different patterns. Low tones produce a pattern of a few large areas; high tones 
a pattern of many small areas. Violin bodies have a few such natural modes of vibra- 
tion which tend to strengthen certain tones sounded by the strings. Poor violin bodies 
accentuate squeaky top notes. This sand-and-plate method of analysis was devised 
150 years ago by the German acoustical physicist Earnst Chladni. 



study the interference pattern produced by two 
point sources. The same thing is done on 
Transparency 28, Two Slit Interference. 

Long before op art, there was an increasing 
number of scientific apphcations of moire 
patterns. Because of the great visual changes 




caused by very small shifts in two regular over- 
lapping patterns, they can be used to make 
measurements to an accuracy of + 0. 000000 1 % . 
Some specific examples of the use of moire 
patterns are visuahzation of two- or multiple- 
source interference patterns, measurement of 




Fig. 1 



Fig. 2 



Unit 3/66 



Activities 



small angular shifts, measurements of diffu- 
sion rates of solids into liquids, and representa- 
tions of electric, magnetic, and gravitation 
fields. Some of the patterns created still cannot 
be expressed mathematically. 

Scientific American, May 1963, has an ex- 
cellent article; "Moire Patterns" by Gerald 
Oster and Yasunori Nishijima. The Science of 
Moire Patterns, a book by G. Oster, is avail- 
able from Edmund Scientific Co., Barrington, 
N.J. Edmund also has various inexpensive sets 
of different patterns, which save much draw- 
ing time, and that are much more precise than 
hand-drawn patterns. 

MUSIC AND SPEECH ACTIVITIES 

(a) Frequency ranges: Set up a microphone 
and oscilloscope so you can display the pres- 
sure variations in sound waves. Play different 
instruments and see how "high C" differs on 
them. 

(b) Some beautiful oscilloscope patterns result 
when you display the sound of the new com- 
putermusic records which use sound-synthe- 
sizers instead of conventional instruments. 

(c) For interesting background, see the follow- 
ing articles in Scientific American: "Physics 
and Music," July 1948; "The Physics of Vio- 
lins," November 1962; "The Physics of Wood 
Winds," October 1960; and "Computer Music," 
December 1959. 

(d) The Bell Telephone Company has an in- 
teresting educational item, which may be 
available through your local Bell Telephone 
office. A 33i LP record, "The Science of 
Sounds," has ten bands demonstrating differ- 
ent ideas about sound. For instance, racing 
cars demonstrate the Doppler shift, and a so- 
prano, a piano, and a factory whistle all sound 
alike when overtones are filtered out electroni- 
cally. The record is also available on the Folk- 
ways label FX 6136. 

MEASUREMENT OF THE SPEED 
OF SOUND 

For this experiment you need to work outside in 
the vicinity of a large flat wall that produces a 
good echo. You also need some source of loud 
pulses of sound at regular intervals, about one 



a second or less. A friend beating on a drum or 
something with a higher pitch will do. The im- 
portant thing is that the time between one 
pulse and the next doesn't vary, so a met- 
ronome would help. The sound source should 
be fairly far away from the wall, say a couple of 
hundred yards in front of it. 

Stand somewhere between the reflecting 
wall and the source of pulses. You will hear 
both the direct sound and the sound reflected 
from the wall. The direct sound will reach you 
first because the reflected sound must travel 
the additional distance from you to the wall 
and back again. As you approach the wall, this 
additional distance decreases, as does the time 
interval between the direct sound and the 
echo. Movement away from the wall increases 
the interval. 




If the distance from the source to the wall 
is great enough, the added time taken by the 
echo to reach you can amount to more than the 
time between drum beats. You will be able to 
find a position at which you hear the echo of 
one pulse at the same time you hear the direct 
sound of the next pulse. Then you know that 
the sound took a time equal to the interval be- 
tween pulses to travel from you to the wall and 
back to you. 

Measure your distance from the source. 
Find the time interval between pulses by 
measuring the time for a large number of 
pulses. Use these two values to calculate the 
speed of sound. 

(If you cannot get far enough away from 
the wall to get this synchronization, increase 
the speed of the sound source. If this is impos- 
sible, you may be able to find a place where you 
hear the echoes exactly halfway between the 
pulses as shown at the top of the opposite page. 
You will hear a pulse, then an echo, then the 
next pulse. Adjust your position so that these 
three sounds seem equally spaced in time. At 



Activities 



Unit 3/67 



<5 






icho 



J U 



this point you know that the time taken for the 
return trip from you to the wall and back is 
equal to ha// the time interval between pulses.) 

MECHANICAL WAVE MACHINES 

Several types of mechanical wave machines 
are described below. They help a great deal in 
understanding the various properties of waves. 

(a) Slinky 

The spring called a SHnky behaves much bet- 
ter when it is freed of friction with the floor or 
table. Hang a Slinky horizontally from strings 
at least three feet long tied to rings on a wire 
stretched from two soUd supports. Tie strings 
to the Slinky at every fifth spiral for proper 
support. 

Fasten one end of the Slinky securely and 
then stretch it out to about 20 or 30 feet. By 
holding onto a ten-foot piece of string tied to the 
end of the Slinky, you can illustrate "open- 
ended" reflection of waves. 

See Experiment 3-15 for more details on 
demonstrating the various properties of waves. 



Stre.tc.hed Ulire. 



rr 



II' 



2 



^ 



Oufiain nin^S 



V V y^^v V 9 



r{ 



! i j i I 






)u/5< 




1/ 






\ 



tciiJ 



(b) Rubber Tubing and Welding Rod 

Clamp both ends of a four- foot piece of rubber 
tubing to a table so it is under slight tension. 
Punch holes through the tubing every inch 
with a hammer and nail. (Put a block of wood 
under the tubing to protect the table.) 

Put enough one-foot lengths of welding rod 
for all the holes you punched in the tubing. Un- 
clamp the tubing, and insert one rod in each of 
the holes. Hang the rubber tubing vertically, as 
shown below, and give its lower end a twist to 
demonstrate transverse waves. Performance 
and visibility are improved by adding weights 
to the ends of the rods or to the lower end of the 
tubing. 

(c) A Better Wave Machine 

An inexpensive paperback. Similarities in 
Wave Behavior, by John N. Shive of Bell Tele- 
phone Laboratories, has instructions for build- 
ing a better torsional wave machine than that 
described in (b) above. The book is available 
from Garden State-Novo, Inc., 630 9th Avenue, 
New York, N.Y. 10036. 



s/;^jk 



V 




Unit 3/68 



Activities 



Resource Letter TLA-1 on Technology, Literature, and Art * 
since World War II 

William H. Davenport 

Department of Humanities, Harvey Mudd College, Claremont, California 91711 

(Received 10 December 1969) 



I. INTRODUCTION 

This resource letter lists materials for collateral 
reading in classes in physics and other sciences as 
well as in new cross-discipline courses; it also offers 
professors and students alike an opportunity to 
see how modern science and technology appear to 
artists and writers — in other words, to see them- 
selves as others see them. A sampling of books and 
articles in the increasingly publicized area of cross 
relationships between technology and society in 
general would require a separate resource letter, 
which may one day materialize. The reader will 
note that the proper distinctions between science 
and technology are often blurred, as indeed they 
are daily by the public. The basic hope in this 
letter is to promote the mutual communication 
and understanding between disciplines so neces- 
sary for personal growth and so vital in such areas 
as top-level decision making. 

As an earlier letter has said, the following 
listings are suggestions, not prescriptions. They are 
samplings guided by personal taste and experi- 
ence, offered with the notion of tempting readers to 
go farther and deeper on their own. 

By agreement with the editors, the listings 
contain a minimum of overlap with earlier letters, 
two of which are cited below; indeed, they are 
intended to pick up where the former left off. 
Anyone wishing bibliography on the "Two 
Cultures" argument, missing the presence of 
earlier "must" items by Mumford, Giedion, 
Holton, Bronowski, Barzun et al., or seeking 
historical material can probably find what he 
wants in the earlier letters. This bibliography is 
arranged alphabetically by author or editor, 
except where no such identification exists; in the 

* Prepared at the request of the Committee on Resource 
Letters of the American Association of Physics Teachers, 
supported by a grant from the National Science Founda- 
tion and published in the American Journal of Physics, 
Vol. 38, No. 4, April 1970, pp. 407-414. 



latter instances, titles are listed in the appropriate 
alphabetical sequence. 

II. RELEVANT RESOURCE LETTERS 

1. "Science and Literature" (SL-1). Marjorib Nicol- 

SON. Amer. J. Phys. 33, 175 (1965). 

2. "Resource Letter CoIR-1 on Collateral Reading for 

Physics Courses," Alfred M. Bork and Arnold B. 
Arons. Amer. J. Phys. 35, 71 (1967). 

m. TECHNOLOGY, LITERATURE, AND ART 

SINCE WORLD WAR II: INTERPLAY 

AND CROSS RELATIONS 

1. "Two Cultixres in Engineering Design," Anonymous. 

Engineering 197, 373 (13 Mar. 1964). A model essay 
demonstrating that dams and highway-lighting 
standards can be and should be both useful and beauti- 
ful. 

2. Poetics of Space. Gaston Bachelard. (Orion Press, 

New York, 1964). A physicist-philosopher justifies 
poetry as an answer to technology and formulas. In a 
provocative discussion of the "spaceness" of cellars, 
attics, and closets and of their relative effects on us, of 
which we are generally unaware, the author makes us 
see the familiar in a new light. He offers stimulating 
contrasts with common notions of space in physics and 
in the public mind, as influenced by Apollo missions. 

3. "Science: Tool of Culture," Cyril Bibby. Saturday 

Rev. 48, No. 23, 51 (6 June 1964). Pits the scientists 
and creative artists against the purely verbal scholars, 
asks for more science (better taught) in schools, and 
appeals to adminbtrators to change their methods of 
training teachers, so that science will appear not as 
an ogre but as a fairy godmother. 

4. Voices from the Crowd (Against the H-Bomb). David 

Boulton, Ed. (Peter Owen, London, 1964.) This 
anthology of poetry and prose, stemming from the 
Campaign for Nuclear Disarmament, is a good 
example of direct testimony of the effect of the Bomb 
on thinking and writing. Among the literary people 
included: Priestley, Comfort, Russell, Read, Osborne, 
Braine (Room at the Top). 

5. Poetry and Politics: 1900-1960. C. M. Bowra. (Uni- 

versity Press, Cambridge, England, 1966.) Discusses 
in part the effect of Hiroshima on poets, notably Edith 
Stilwell, whose form and vision were radically affected 
by the event, and the Russian, Andrei Voznesensky, 
whose poem on the death of Marilyn Monroe foresaw a 
universal disaster. 



Activities 



unit 3/69 



6. "Science as a Humanistic Discipline," J. Bronowski. 
Bull. Atomic Scientists 24, No. 8, 33 (1968). The 
author of Science and Human Values here covers the 
history of humanism, values, choice, and man as a 
unique creature. It is the duty of science to transmit 
this sense of uniqueness, to teach the world that man 
is guided by self-created values and thus comfort it 
for loss of absolute purpose. 

7. "Artist in a World of Science," Pearl Buck. Saturday 

Rev. 41, No. 38, 15-16, 42-44 (1958). Asks for artists 
to be strong, challenges writers to use the findings of 
science and illuminate them so that "human beings 
will no longer be afraid." 

8. The Novel Now. Anthony Burgess. (W. W. Norton 

and Co., -New York, 1967.) Prominent British novelist 
discusses the aftermath of nuclear war as a gloomy 
aspect of fictional future time and advances the thesis 
that comparatively few good novels came out of the 
war that ended with Hiroshima, although a good deal 
of ordinary fiction has the shadow of the Bomb in it. 

9. Beyond Modem Sculpture: Effects of Science & 

Technology on the Sculpture of this Centxiry. Jack 
BuRNHAM. (George BrazUler Inc., New York, 1969.) 
"Today's sculpture is preparing man for his replace- 
ment by information-processing energy " Bumham 
sees an argument for a mechanistic teleological 
interpretation of life in which culture, including art, 
becomes a vehicle for qualitative changes in man's 
biological status. [See review by Charlotte Willard in 
Saturday Rev. 52, No. 2, 19 (1969).] 

10. Cultures in Conflict. David K. Cornelius and 
Edwin St. Vincent, Eds. (Scott, Foresman and 
Company, Glenview, 111., 1964). A useful anthology 
of primary and secondary materials on the con- 
tinuing C. P. Snow debates. 

11. "The Computer and the Poet," Norman Cousins. 
Saturday Rev. 49, No. 30, 42 (23 July 1966). Suggests 
editorially (and movingly) that poets and program- 
mers should get together to "see a larger panorama of 
possibilities than technology alone may inspire" and 
warns against the "tendency to mistake data for 
wisdom." 

12. Engineers and Ivory Towers. Hardy Cross. Robert 
C. Goodpasture, Ed. (McGraw-Hill Book Co., 
New York, 1952). A sort of common-sense bible 
covering the education of an engineer, the full life, 
and concepts of technological art. 

13. Engineering: Its Role and Function in Himian 
Society. William H. Davenport and Daniel 
Rosenthal, Eds. (Pergamon Press, Inc., New York, 
1967). An anthology with four sections on the view- 
point of the humanist, the attitudes of the engineer, 
man and machine, and technology and the future. 
Many of the writers in this bibliography are repre- 
sented in an effort to present historical and con- 
temporary perspectives on technology and society. 

14. "Art and Technology— The New Combine," Douglas 
M. Davis. Art in Amer. 56, 28 (Jan.-Feb. 1968). 



Notes a new enthusiasm among many modern artists 
because of the forms, effects, and materials made 
possible by the new technology. Envisions full 
partnership between artist and machine in the 
creative process. 

15. So Human an Animal. Ren^ Dubcs. (Charles 
Scribner's & Sons, New York, 1968). Dubos, a promi- 
nent microbiologist, won a Pulitzer Prize for this work, 
and it deserves wide reading. Motivated by humanistic 
impulses, writing now like a philosopher and again 
like a poet, he disciLsses man'.s threatened dehuman- 
ization under technological advance. Man can adjust, 
Dubos says — at a price But first he mu.st understand 
himself as a creature of heredity and environment 
and then learn the .science of life, not merely science. 

16. The Theatre of the Absurd. Martin E.sslin. (Anchor 
Books-Doubleday and Co., Inc., Garden City, N. J., 
1961). The drama director for the British Broad- 
casting Company explains the work of Beckett, 
lonesco, Albee, and others a.s a reaction to loss of 
values, reason, and control in an age of totalitarianism 
and of that technological development, the Bomb. 

17. Engineering and the Liberal Arts. Samuel C. Flor- 
MAN. (McGraw-Hill Book Co., New York, 1968). 
The subtitle tells the story: A Technologist's Guide to 
History, Literature, Philosophy, Art, and Music. 
Explores the relationships between teclinology and the 
liberal arts — historical, aesthetic, functional. Useful 
reading lists are included. 

18. The Creative Process. Brewster Ghiseun, Ed. 

(University of California Press, Berkeley, 1952; 
Mentor Books, The New American Library, Inc., 
New York, paperback, 1961). Mathematicians, 
musicians, painters, and poets, in a symposium on the 
personal experience of creativity. Of use to those 
interested in the interplay between science and art. 

19. Postwar British Fiction: New Accents and Attitudes. 
James Gindin. (University of California Press, 
Berkeley, 1963). Traces the comic or existentialist 
view of the world in recent British novels as resulting 
in part from the threat of the hydrogen bomb. 

20. The Poet and the Machine. Paul Ginestier. Martin 
B. Friedman, Transl. (University of North Carolina 
Press, Chapel Hill, 1961; College and University 
Press, New Haven, Conn., paperback, 1964). Con- 
siders through analysis of generous examples from 
modern and contemporary poetry the effect of the 
machine on subject matter, form, and attitude. An 
original approach to the value, meaning, and influence, 
as the author puts it, of the poetry of our technology- 
oriented era. 

21. "Nihilism in Contemporary Literature," Charles I. 
Glicksberg. Nineteenth Century 144, 214 (Oct., 
1948). An example of the extreme view that man is 
lost in a whirlpool of electronic energy, that cosmic 
doubts, aloneness, and fear of cataclysmic doom have 
led to a prevailing mood of nihilism in writing. 

22. "Impact of Technological Change on the Humanities," 



Unit 3/70 



Activities 



Maxwell H. Goldberg. Educational Record 146, 
No. 4, 388-399 (1965). It is up to tlie humanities to 
soften the impact of advancing technology upon the 
pressured individual. One thing they can do is to 
help us pass the almost unlimited leisure time proph- 
esied for the near future under automation and, thus, 
to avoid Shaw's definition of hell. 

23. "A Poet's Investigation of Science," Robert Graves. 
Saturday Rev. 46, 82 (7 Dec. 1963). The dean of 
English poets in a lecture at the Massachusetts 
Institute of Technology takes technologists to task 
good-humoredly but with a sting, too. He is concerned 
about the upset of nature's balance, the weakening of 
man's powers through labor-saving devices, synthetic 
foods, artificial urban life, dulling of imagination by 
commercialized art, loss of privacy — ^all products or 
results of technology. Graves finds no secret mystique 
among advanced technologists, only a sense of fate 
that makes them go on, limited to objective views and 
factual accuracy, forgetting the life of emotions and 
becoming diminished people. 

24. Social History of Ait. Arnold Hauser. (Vintage 
Books, Random House, Inc., 4 vols.. New York, 
1951). In the fourth volume of this paperback edition 
of a standard work, considerable space is given to the 
cultural problem of technics and the subject of film 
and technics. 

25. "Automation and Imagination," Jacquetta Hawkes. 
Harper's 231, 92 (Oct. 1965). Prominent archaeologist 
fears loss of man's imaginative roots under years of 
technical training. While the technological revolution 
sweeps on toward a total efficiency of means, she says, 
we must control the ends and not forget the sig- 
nificance of the individual. 

26. The Future as Nightmare. Mark R. Hillegas. 

(Oxford University Press, New York, 1967). A study 
that begins with Wells and ends with recent science 
fiction by Ray Bradbury, Kurt Vonnegut, and Walter 
Miller, Jr. The latter three are worried about the 
mindless life of modern man with his radio, TV, and 
high-speed travel; the need to learn nothing more than 
how to press buttons; the machine's robbing man of 
the pleasure of working with his hands, leaving him 
nothing useful to do, and lately making decisions 
for him; and, of course, the coming nuclear holocaust. 

27. Science and Culture. Gerald Holton, Ed. (Beacon 
Press, Boston, 1967). Almost all of the 15 essays in 
this outstanding collection appeared, several in 
different form, in the Winter 1965 issue of Daedalus. 
Of particular relevance to the area of this bibliography 
are Herbert Marcuse's view of science as ultimately 
just technology; Gyorgy Kepes' criticism of modern 
artists for missing vital connections with technological 
reality; Ren6 Dubos' contention that technological 
applications are becoming increasingly alienated 
from human needs; and Oscar Handlin's documenta- 
tion of the ambivalent attitude of modern society 
toward technology. 



28. "The Fiction of Anti-Utopia," Irving Howe. New 
Republic 146, 13 (23 Apr. 1962). An analysis of the 
effect on modern fiction of the splitting apart of 
technique and values and the appearance of technical 
means to alter human nature, both events leading to 
the American dream's becoming a nightmare. 

29. The Idea of the Modern. Irving Howe, Ed. (Horizon 
Press, New York, 1967). A perspective on post- 
Hiroshima literature and its relation to technology 
calls for a frame of reference on modernism in art and 
literature in general. A useful set of ideas is contained 
in this volume, the summing-up of which is that 
"nihilism lies at the center of all that we mean by 
modernist literature." 

30. The Machine. K. G. P. Hult6n, Ed. (Museum d 
Modern Art, New York, 1968) . A metal-covered book 
of pictorial reproductions with introduction and 
running text, actually an exhibition catalogue, offering 
clear visual evidence of the interplay of modern art and 
modern technology in forms and materials. 

31. Literature and Science. Aldous Huxley. (Harper & 
Row, Publishers, New York, 1963). A literary and 
highly literate attempt to show bridges between the 
two cultures. Technological know-how tempered by 
human understanding and respect for nature will 
dominate the scene for some time to come, but only if 
men of letters and men of science advance together. 

32. The Inland Island. Josephine Johnson. (Simon & 
Schuster, Inc., New York, 1969). One way to avoid 
the evils of a technological society is to spend a year on 
an abandoned farm, study the good and the cruel 
aspects of nature, and write a series of sketches about 
the experience. Escapist, perhaps, but food for 
thought. 

33. The Sciences and the Himianities. W. T. Jones. 

(University of California Press, Berkeley, 1965). A 
professor of philosophy discusses conflict and recon- 
ciliation between the two cultures, largely in terms of 
the nature of reality and the need to understand each 
other's language. 

34. "The Literary Mind," Alfred Kazin. Nation 201, 
203 (20 Sept. 1965). Advances the thesis that it is 
more than fear of the Bomb that produces absurdist 
and existentialist writing, it is dissatisfaction that 
comes from easy self -gratifications: "Art has become 
too easy." 

35. "Imagination and the Age," Alfred Kazin. Reporter 
34, No. 9, 32 (5 May 1966). Analyzes the crisis 
mentality behind modern fiction, the guilt feelings 
going back to Auschwitz and Hiroshima. Salvation 
from the materialism of modem living lies in language 
and in art. 

36. New Landscape in Science and Art. Gyorgy Kepes. 

(Paul Theobald, Chicago, 1967). Like the earlier 
Vision in Motion by L. Moholy-Nagy (Paul Theobald, 
Chicago, 1947), this work will make the reader see 
more, better, and differently. Essays and comments by 
Gabo, Giedion, Gropius, Rossi, Wiener, and others 



Activities 



Unit 3/71 



37. 



39 



plus lavish illustration assist Kepes, author of the 
influential Language of Vision and head of the 
program on advanced visual design at the Massachu- 
setts Institute of Technology, to discuss morphology 
in art and science, form in engineering, esthetic 
motivation in science — in short, to demonstrate that 
science and its applications belong to the humanities, 
that, in Frank Lloyd Wright's words, "we must look 
to the artist brain ... to grasp the significance to 
society of this thing we call the machine." 
"If You Don't Mind My Saying So . . .," Joseph 
Wood Krutch. Amer. Scholar 37, 572 (Autumn, 
1968). Expresses fear over extending of experimenta- 
tion with ecology and belief that salvation does not lie 
with manipulation and conditioning but may come 
from philosophy and art. 
38. The Scientist vs the Humanist. George Levine and 
Owen Thomas, Eds. (W. W. Norton, New York, 
1963). Among the most relevant items are I. I. 
Rabi's "Scientist and Humanist"; Oppenheimer's 
"The Tree of Knowledge"; Howard Mumford Jones's 
"The Humanities and the Common Reader" (which 
treats technological jargon) ; and P. W. Bridgman's 
"Quo Vadis." 

Death in Life: Survivors of Hiroshima. Robert J. 
LiFTON. (Random House, Inc., New York, 1967). 
Chapter 10, "Creative Response: A-Bomb Litera- 
ture," offers samples of diaries, memoirs., and poems 
by survivors, running the gamut from protest to re- 
construction. See, also, Lifton's "On Death and Death 
Symbolism: The Hiroshima Disaster" in Amer. 
Scholar 257 (Spring, 1965). 

"The Poet and the Press," Archibald MacLeish. 
Atlantic 203, No. 3, 40 (March, 1959). Discusses the 
"divorce between knowing and feeling" about 
Hiroshima as part of a social crisis involving the decay 
of the life of the imagination and the loss of individual 
freedom. The danger here is growing acquiescence to a 
managed order and satisfaction with the car, TV, and 
the material products of our era. 
"The Great American Frustration," Archibald 
MacLeish. Saturday Rev. 51, No. 28, 13 (13 July 
1968). Prior to Hiroshima, it seemed that technology 
would serve human needs; after the event, it appeared 
tliat technology is bound to do what it can do. We 
are no longer men, but consumers filled with frustra- 
tions that produce the satirical novels of the period. 
We must try to recover the management of tech- 
nology and once more produce truly educated men. 
"The New Poetry," Frank MacShane. Amer. Scholar 
37, 642 (Autumn, 1968). Frequently, the modern poet 
writes of confrontation of man and machine. He is 
both attracted and repelled by technological change, 
which both benefits and blights. 
The Machine in the Garden: Technology and the 
Pastoral Ideal in America. Leo Marx. (Oxford 
University Press, New York, 1964; Galaxy, Oxford 
Univ. Press, New York, paperback, 1967). One of the 
three most significant contemporary works on the 



40 



41 



42 



43 



interplay of literature and technology [along with 
Sussman (69) and Sypher (72)], this study concen- 
trates on 19th-century American authors and their 
ambivalent reactions to the sudden appearance of the 
machine on the landscape. Whitman, Emerson, 
Thoreau, Hawthorne, Melville, and others reveal, 
under Marx's scrutiny, the meaning inherent in 
productivity and power. Whitman assimilated the 
machine, Emerson welcomed it but disliked ugly 
mills, Thoreau respected tools but hated the noise and 
smoke, Hawthorne and Melville noted man's growing 
alienation with the green fields gone, Henry Adams set 
the theme for the "ancient war between the kingdom of 
love and the kingdom of power . . . waged endlessly 
in American writing ever since." The domination of 
the machine has divested of meaning the older notions 
of beauty and order, says Marx, leaving the American 
hero dead, alienated, or no hero at all. Aptly used 
quotations, chronological order, and clarity of per- 
spective and statement (with which all may not agree) 
make this a "must" for basic reading in this special 
category. Furthermore, there are links to Frost, 
Hemingway, Faulkner, and other modern writers. 

44. Technology and Culture in Perspective. Ilene 
Montana, Ed. (The Church Society for College Work, 
Cambridge, 1967). Includes "Technology and Democ- 
racy" by Ilarvey Cox, "The Spiritual Meaning of 
Technology and Culture," by Walter Ong, and 
"The Artist's Response to the Scientific World," by 
Gyorgy Kepes. 

45. "Science, Art and Technology," Charles Morris. 
Kenyon Rev. 1, No. 4, 409 (Autumn, 1939). A tight 
study of three forms of discourse — scientific, aesthetic, 
and technological and a plea that the respective 
users acquire vision enough to see that each comple- 
ments the other and needs the other's support. 

46. "Scientist and Man of Letters," Herbert J. Muller. 
Yale Rev. 31, No. 2, 279 (Dec, 1941). SimUarities and 
differences again. Science has had some bad effects on 
literature, should be a co-worker; literature can give 
science perspective on its social function. 

47. The Myth of the Machine. Lewis Mumford. (Har- 
court, Brace & World, Inc., New York, 1967). 
Important historical study of human cultural develop- 
ment, that shows a major shift of emphasis from 
human being to machine, questions our commitment 
to technical progress, and warns against the down- 
playing of literature and fine arts so vital to complete 
life experience. See also his earlier Art and Technics 
(Columbia University Press, New York, 1952). 

48. "Utopia, the City, and the Machine," Lewis Mum- 
ford. Daedalus 94, No. 2, of the Proceedings of the 
American Academy of Arts and Sciences, 271 (Spring, 
1965). The machine has become a god beyond chal- 
lenge. The only group to understand the dehumanizing 
effects and eventual price of technology are the 
avant-garde artists, who have resorted to caricature. 

49. "Utopias for Reformers," Francois Bloch-Laine. 



Unit 3/72 



Activities 



Daedalus 94, No. 2, 419 (1965). DLsciLsses the aim of 
two Utopias — technological and democratic — to in- 
crease man's fulfillment by different approaches, 
which mu.st be combined. 

50. "Utopia and the Good Life," George Kateb, 
Daedalus 94, No. 2, 454 (1965). Describes the thrust 
of teclmology in freeing men from routine drudgery 
and setting up leisure and abundance, with attendant 
problems, however. 

51. Utopia and Utopian Thought. Frank Manuel, Ed. 

(Houghton Mifflin Co., Boston, 1966). A gathering of 
the preceding three, and other materials in substan- 
tially the s;ime form. 

52. Aesthetics and Technology in Building. Pier Luigi 
Nkrvi. (Harvard University Pres.s, Cambridge, 
Mass., 1965). "Nervi's thesis is that good architecture 
is a synthesis of technology and art," according to an 
expert review by Carl W. Condit, in Technol. and 
Culture 7, No. 3, 432 (Summer, 1966), which we also 
recommend. 

53. Liberal Learning for the Engineer. Sterling P. 
Olmsted. (Amer. Soc. Eng. Educ, Washington, D. C, 
1968). The most recent and comprehensive report on 
the state of liberal studies in the engineering and 
technical colleges and institutes of the U. S. Theory, 
specific recommendations, bibliography. 

54. Road to Wigan Pier. George Orwell. (Berkley 
Publishing Corporation, New York, 1967). A paper- 
back rei.ssue of the 1937 work by the author of 1984. 
Contains a 20-page digression, outspoken and con- 
troversial, on the evils of the machine, which has made 
a fully human life impossible, led to decay of taste, 
and acquired the status of a god beyond criticism. 

55. "Art and Technology: 'Cybernetic Serendipity'," 
S. K. OvERBECK. The Alicia Patterson Fund, 535 
Fifth Ave., New York, N. Y., 10017, SKO-1 (10 June 
1968). The first of a dozen illustrated newsletter- 
articles by Overbeck that are published by the Fund. 
The series (space limitations forbid separate listings) 
describes various foreign exhibitions of computer 
music, electronic sculpture, and sound and light, 
which, in turn, recall "Nine Evenings: Theater and 
Engineering" staged in New York, fall 1966, by 
Experiments in Art and Technology, with outside 
help, "to familiarize the artist with the realities of 
technology while indulging the technician's penchant 
to transcend the mere potentialities of his discipline." 

56. "Myths, Emotions, and the Great Audience," James 
Parsons. Poetry 77, 89 (Nov., 19.50). Poetry is 
important to man's survival because it is a myth 
maker at a time when "it is the developing r.ationale of 
as.sembly-linc production that all society be hitched 
to the machine." 

57. "Public and Private Problems in Modem Drama," 
Ronald Peacock. Tulane Drama Rev. 3, No. 3, 58 
(March, 1959). The dehumanizing effects of techno- 
cratic society as seen in modern plays going back as 
far as Georg Kaiser's Gas (1918). 



58. "The American Poet in Relation to Science," Norman 
Holmes Pearson. Amer. Quart. 1, No. 2, 116 
(Summer, 1949). Science and technology have done a 
service to poets by forciiig them into new modes of ex- 
pression; however, the poet remains the strongest force 
in the preservation of the freedom of the individual. 

59. Science, Faith and Society. Michael Polanyi. (Uni- 
versity of Chicago Press, Chicago, 1964). Originally 
published by Oxford University Press, London, 
in 1946, this work appears in a new format with a 
new introduction by the author, which fits the present 
theme, inasmuch as it considers the idea that all great 
discoveries are beautiful and that scientific discovery 
is like tlie creative act in the fine arts. 

60. Avant-Garde: The Experimental Theater in France. 
Leonard C. Pronko. (University of California Press, 
Berkeley, 1966). A keen analysis of the work of 
Beckett, lonesco. Genet, and others, which no longer 
reflects a rational world but the irrational world of the 
atom bomb. 

61. "Scientist and Humanist: Can the Minds Meet?" 
I. I. Rabi. Atlantic 197, 64 (Jan., 1956). Discusses 
modern antiintellectualism and the urge to keep up 
with the Russians in technology. Calls for wisdom, 
which is unobtainable as long as sciences and human- 
ities remain separate disciplines. 

62. "Integral Science and Atomized Art," Eugene 
Rabinowitch. Bull. Atomic Scientists 15, No. 2, 65 
(February, 1959). Through its own form and expres- 
sion, art could help man find the harmony now 
threatened by the forces of atomism and fear of 
nuclear catastrophe. 

63. "Art and Life," Sir Herbert Read. Saturday 
Evening Post 232, 34 (26 Sept. 1959). Modern 
violence and restlessness stem in great part from a 
neurosis in men who have stopped making things by 
hand. Production, not grace or beauty, is the guiding 
force of technological civilization. Recommends the 
activity of art to release creative, rather than de- 
structive, forces. 

64. The New Poets: American and British Poetry Since 
World War II. M. L. Rosenthal. (Oxford University 
Press, New York, 1967). Detects a dominant concern 
among contemporary poets with violence and war and 
links it to a general alienation of sensibility, due in 
great part to the fact that human values are being 
displaced by technology. 

65. "The Vocation of the Poet in the Modem Worid," 
Delmore Schwartz. Poetry 78, 223 (July, 1951). 
The vocation of the poet today is to maintain faith in 
and love of poetry, until he is destroyed as a human 
being by the doom of a civilization from which he has 
become alienated. 

66. "Science and Literature," Elizabeth Sewell. 
Commonweal 73, No. 2, 218 (13 May 1966). Myth 
and the simple affirmation of the human mind 
and body are the only two forms of imagination 
capable of facing modern enormities. The two terminal 
points of our technological age were Auschwitz and 



Activities 



Unit 3/73 



Hiroshima; in literature about them, we may yet see 
"the affirmation of simple humanity." 

67. "Is Technology Taking Over?" Charles E. Silber- 
MAN. Fortune 73, No. 2, 112 (Feb., 1966). A brisk 
discussion of familiar topics: art as defense; technology 
as an end; dehumanization and destruction; mass 
idleness; meaninglessness. Technology may not deter- 
mine our destiny, but it surely affects it and, in 
enlarging choice, creates new dangers. As the author 
points out, however, borrowing from Whitehead, 
the great ages have been the dangerous and distxirbed 
ones. 

68. "One Way to Spell Man," Wallace Stegner. 
Saturday Rev. 41, No. 21, 8; 43 (24 May 1958). 

Finds a real quarrel between the arts and technology 
but not between the arts and science, the latter two 
being open to exploitation by the technology of mass 
production. Reminds us that nonscientific experience 
is valid, and nonverifiable truth important. 

69. Victorians and the Machine : The Literary Response to 
Technology. Herbert L. Sussman. (Harvard Uni- 
versity Press, Cambridge, Mass., 1968). Does for 
English writers of the 19th century what Leo Marx 
[43] did for the Americans, with substantially similar 
conclusions. Writers stressed are Carlyle, Butler, 
Dickens, Wells, Ruskin, Kipling, and Morris, whose 
thought and art centered on the effects of mechaniza- 
tion on the intellectual and aesthetic life of their day. 
A major study of the machine as image, symbol, 
servant, and god — something feared and respected, 
ugly and beautiful, functional and destructive — as 
seen by the significant Victorian literary figures, this 
work also helps explain the thrust of much con- 
temporary writing. 

70. "The Poet as Anti-Specialist," May Swenson. 
Saturday Rev. 48, No. 5, 16 (30 Jan. 1965). A poet 
tells how her art can show man how to stay human in a 
technologized age, compares and contrasts the 
languages of science and poetry, wonders about the 
denerving and desensualizing of astronauts "trained to 
become a piece of equipment." 

71. "The Poem as Defense," Wylie Sypher. Amer. 
Scholar 37, 85 (Winter, 1967). The author is not 
worried about opposition between science and art, 
but about opposition between both of them together 
and technology. Technique can even absorb criticism 
of itself. Technological mentality kills the magic of 
surprise, grace, and chance. If Pop art, computer 
poetry, and obscene novels are insolent, society is even 
more so in trying to engineer people. 

72. Literature and Technology. Wylie Sypher. (Random 
House, Inc., New York, 1968). The best, almost the 
only, general study of its kind, to be required reading 
along with Leo Marx [43] and Herbert Sussman [69]. 
Develops the thesis that technology dreads waste and, 
being concerned with economy and precaution, lives 
by an ethic of thrift. The humanities, including art, 
exist on the notion that every full life includes waste— 
of virtue, intention, thinking, and work. The thesis b 



illustrated by examples from literature and art. 
Although, historicallv, technology minimizes indi- 
vidual participation and resultant pleasure, Sypher 
concedes that lately "technology has been touched by 
the joy of finding in its solutions the play of intellect 
that satisfies man's need to invent." 

73. Dialogue on Technology. Robert Theobald, Ed. 

(The Bobbs-Merrill Co., Inc., Indianapolis, 1967). 
Contains essays on the admiration of technique, 
human imagination in the space age, educational 
technology and value systems, technology and 
theology, technology and art. 

74. Science, Man and Morals. W. H. Thorpe. (Cornell 
University Press, Ithaca, N. Y., 1965). Brings out 
interplay among science, religion, and art, accenting an 
over-all tendency toward wholeness and unity. Traces 
modem plight in some degree to the Bomb. 

75. "Modem Literature and Science," I. Traschen. 
College English 25, 248 (Jan., 1964). Explores the 
common interests of scientist and poet in their search 
for truth as well as their differences, which produce 
alienation and literary reaction. 

76. "The New English Realism," Ossia Trilling. Tulane 
Drama Rev. 7, No. 2, 184 (Winter, 1962). Ever since 
Osborne's "Look Back in Anger," the modern British 
theatre has shown a realism based on revolt against 
class structure and the dilemma of threatening nuclear 
destruction, although scarcely touching on the new 
technology itself. 

77. "The Poet in the Machine Age," Peter Viereck. 
J. History Ideas 10, No. 1, 88 (Jan., 1949). A classifi- 
cation of antimachine poets, who for esthetic, pious, 
instinctual, or timid reasons have backed away, and 
promachine poets, who, as materialists, cultists, or 
adapters, have used the new gadgets to advantage. 
We must try to unite the world of machinery and the 
world of the spirit, or "our road to hell will be paved 
with good inventions." 

78. The Industrial Muse. With introduction by Jeremy 
Warburg, Ed. (Oxford University Press, New York, 
1958). An amusing and informative anthology of verse 
from 1754 to the 1950's dealing in all moods with 
engines, factories, steamboats, railways, machines, 
and airplanes. 

79. "Poetry and Industrialism," Jeremy Warburg. 
Modem Language Rev. 53, No. 2, 163 (1958). Treats 
the problem of imaginative comprehension as the 
modem poet strives to assimilate the new technology, 
make statements, and find terms for a new form of 
expression. 

80. Reflections on Big Science. Alvin Weinberg. (The 
MIT Press, Cambridge, Mass., 1967). The director of 
Oak Ridge National Laboratory devotes his first 
chapter, "The Promise of Scientific Technology; 
The New Revolutions," to nuclear energy, cheap 
electricity, technology of information, the Bomb, and 
dealing with nuclear garbage. He calls upon the 
humanists to restore meaning and purpose to our lives. 



Unit 3/74 



Activities 



81. The Theater of Protest and Paradox. George Well- 
WARTH, (New York University Press, 1964). A dis- 
cussion of contemporary playwrights, e.g., lonesco, 
who finds a machine-made preplanned city essentially 
drab; and Dtirrenmatt, whose "The Physicists" 
teaches the lesson that mankind can be saved only 
through suppression of technical knowledge. 

82. Flesh of Steel: Literature and the Machine in Ameri- 
can Culture. Thomas Reed West. (Vanderbilt 
University Press, Nashville, Tenn., 1967). A con- 
sideration of the writings of Sherwood Anderson, 
Dos Passes, Sandburg, Sinclair Lewis, Mumford, and 
Veblen which, while conceding that most of them are 
antimachine most of the time, preaches the positive 
virtues of the Machine: law, order, energy, discipline, 
which, at a price, produce a city like New York, 
where artists and writers may live and work on their 
own terms who could not exist if the machine stopped. 

83. "The Discipline of the History of Technology," 
Lynn White, Jr. Eng. Educ. 54, No. 10, 349 (June, 
1964). Technologists have begun to see that they 
have an intellectual need for the knowledge of the 
tradition of what they are doing. Engineers, too, 
must meet the mark of a profession, namely, the 
knowledge of its history. Even the humanists are 
realizing what this explosive new discipline can con- 
tribute to their personal awareness. 

84. Drama in a World of Science. Glynne Wickham. 

(University of Toronto Press, Toronto, Canada, 
1962). Treats the renascence of English theatre in the 
50's, the Bomb as topic, the individual confused by 
technology and its tyranny as protagonist, and mass 
conformity, violence, or apathy as themes. 

85. "The Scientist and Society." J. Tuzo Wilson. 
Imperial OU Rev., 20-22 (Dec, 1963). The humanist 
who pretends to have no interest in science and the 
technocrat who relies completely on science are 
equally deluded. Calls for tolerance and under- 
standing among all intellectual disciplines. The 
scientist must reconsider his position vis-d-^s the 
humanities and the arts. 

86. "Science is Everybody's Business," J. Tuzo Wilson. 
Amer. Scientist 52, 266A (1964). Includes new direc- 
tions in technology. 

87. "On the History of Science," J. Tuzo Wilson, 
Saturday Rev. 47, No. 18, 50 (2 May 1964). Suggests 
new university departments to train scientifically 
literate humanists. 

88. "The Long Battle between Art and the Machine." 
Edgar Wind, Harper's 228, 65 (Feb., 1964). Con- 
templation of fake-modem buildings, dehumanized 
music, and mass-produced furniture raises once more 
the old question of whether the artist uses the machine 
or becomes its slave. 

Postscript to Sec. m 

Since most of the foregoing material is critical or 
expository, except for quoted illustration, readers 



may wish to make a start with firsthand creative 
literary pieces. Here are some suggestions (unless 
otherwise indicated, items are available in various 
paperback editions; see the current issue of 
Paperbound Books in Print, R. R. Bowker Co., 
New York) . 

Plays 

On the theme of machine replacing man, there 
are two early modern classics for background : 

89. R.U.R. Karel Capek. 

90. The Adding Machine. Elmer Rice. 

Three British plays deal directly with the Bomb, 
and the fourth, the only one available in paper, 
alludes to it : 

91. The Tiger and the Horse. Robert Bolt. In Three 
Plays (Mercury Books, London, 1963). 

92. The Offshore Island. Marghanita Laski. (Cresset 
Press, London, 1959). 

93. Each His Own Wilderness. Doris Lessing. In 
New English Dramatists, E. Martin Browne, Ed. 
(Penguin Plays, London). 

94. Look Back in Anger. John Osborne. 

Two recent plays dealing with physicists : 

95. The Physicists. Friedrich Durrenmatt. 

96. In the Matter of J. Robert Oppenheimer. Heinar 

KiPPHARDT. 

Fiction 
A quartet of Utopian or anti-Utopian novels : 

97. Brave New World. Aldous Huxley. 

98. Nineteen Eighty-Four. George Orwell. 

99. Walden II. B. F. Skinner. 

100. We. E. Zamiatan. 

A quartet of science fiction : 

101. Fahrenheit 4^1. Ray Bradbury. 

102. Canticle for Leibowitz. Walter Miller, Jr. 

103. Player Piano. Kurt Vonnegut, Jr. 

104. Cat's Cradle. Kurt Vonnegut, Jr. 

A trio of short stories : 

105. "By the Waters of Babylon," Stephen V, Benet. 

106. "The Portable Phonograph," Walter Van Tilburq 
Clark, In The Art of Modem Fiction, R. West and R. 
Stallman, Eds., alternate ed. (Holt, Rinehart, & 
Winston, Inc., New York, 1949), 

107. "The Machine Stops," E. M. Forster. In Modem 
Short Stories, L. Brown, Ed. (Harcourt, Brace 
A World, Inc., New York, 1937). 



Activities 



Unit 3/75 



Poetry 

See Ginestier [20], Warburg [78], and Boulton 
[4] above. Also: 



108. The Modem Poets. John M. Brinnin and Bill 
Read, Eds. (McGraw-Hill Book Co., New York, 
1963). Contains poems by Hoffman, Lowell, Moss, and 
Nemerov pertaining to the Bomb. 

109. Weep Before God. John Wain. (The Macmillan 
Company, London, 1961). Sections VI-VII consider 
the Machine. 

110. Wildtrack. John Wain. (The Macmillan Company, 
London, 1965). Pages 10-12 satirize Henry Ford and 
the assembly line. 

HI. Today's Poets. Chad Walsh, Ed. (Charles Scrib- 
ner's Sons, New York, 1964). The Introduction 
mentions the Bomb, and a poem by Gil Orlovitz 
spoofs the computer. 



ACKNOWLEDGMENTS 

I wish to thank Professor Gerald Holton of 
Harvard for suggesting that I prepare this letter 
and for helping it on its way. 

For help in research or bibliography, I owe 
much to Dr. Emmanuel Mesthene (Director), 
Charles Hampden-Turner, and Tom Parmenter 
of the Harvard Program on Technology and 
Society, where I spent the sabbatical year 
1968-69; to Professor Leo Marx of Amherst 
College; and to Professor Wylie Sypher of 
Simmons College. 

In the later stages, I received helpful advice 
from Professor Arnold Arons of the University of 
Washington and from Professor Joel Gordon of 
Amherst. 

None of the above is responsible for errors, 
omissions, or final choices. 



Unit 3/76 



FILM LOOP NOTES 



FILM LOOP 18 
ONE-DIMENSIONAL COLLISIONS I 

Two different head-on collisions of a pair of 
steel balls are shown. The balls hang from 
long, thin wires that confine each ball's motion 
to the same circular arc. The radius is large 
compared with the part of the arc, so the curva- 
ture is hardly noticeable. Since the collisions 
take place along a straight line, they can be 
caUed one-dimensional. 




In the first example, ball B, weighing 350 
grams, is initially at rest. In the second exam- 
ple, ball A, with a mass of 532 grams, is the one 
at rest. 

With this film, you can make detailed 
measurements on the total momentum and 
energy of the balls before and after colhsion. 
Momentum is a vector, but in this one- 
dimensional case you need only worry about its 
sign. Since momentum is the product of mass 
and velocity, its sign is determined by the sign 
of the velocity. 

You know the masses of the baUs. Veloc- 
ities can be measured by finding the distance 
traveled in a known time. 

After viewing the film, you can decide on 
what strategy to use for distance and time 
measurements. One possibility would be to 
time the motion through a given distance with 
a stopwatch, perhaps making two Unes on the 
paper. You need the velocity just before and 
after the collision. Since the balls are hanging 



from wires, their velocity is not constant. On 
the other hand, using a small arc increases the 
chances of distance-time uncertainties. As 
with most measuring situations, a number of 
conflicting factors must be considered. 

You will find it useful to mark the crosses 
on the paper on which you are projecting, 
since this will allow you to correct for projector 
movement and film jitter. You might want to 
give some thought to measuring distances. You 
may use a ruler with marks in milHmeters, so 
you can estimate to a tenth of a milhmeter. Is it 
wise to try to use the zero end of the ruler, or 
should you use positions in the middle? Should 
you use the thicker or the thinner marks on the 
ruler? Should you rely on one measurement, or 
should you make a number of measurements 
and average them? 

Estimate the uncertainty in distance and 
time measurements, and the uncertainty in 
velocity. What can you learn from this about 
the uncertainty in momentum? 

When you compute the total momentum 




Film Loop Notes 



Unit 3/77 



before and after collision (the sum of the 
momentum of each ball), remember that you 
must consider the direction of the momentum. 

Are the differences between the momen- 
tum before and after coUision significant, or 
are they within the experimental error already 
estimated? 

Save the data you collect so that later you 
can make similar calculations on total kinetic 
energy for both balls just before and just after 
collision. 

FILM LOOP 19 ONE-DIMENSION 
COLLISIONS II 

Two different head-on collisions of a pair of 
steel balls are shown, with the same setup as 
that used in Film Loop 18, "One-Dimensional 
Collisions I." 

In the first example, ball A with a mass of 
1.8 kilograms collides head on with ball B, with 
a mass of 532 grams. In the second example, 
ball A catches up with ball B. The instructions 
for Film Loop 18, "One-Dimension Collisions 
I" may be followed for completing this inves- 
tigation also. 

FILM LOOP 20 INELASTIC 
ONE-DIMENSIONAL COLLISIONS 

In this film, two steel balls covered with plas- 
ticene hang from long supports. Two collisions 
are shown. The two balls stick together after 
colhding, so the collision is "inelastic." In the 
first example, ball A, weighing 443 grams, is at 
rest when ball B, with a mass of 662 grams, 
hits it. In the second example, the same two 
balls move toward each other. Two other films, 
"One-Dimensional Collisions I" and "One- 
Dimensional Colhsions 11" show collisions 
where the two balls bounce off each other. 
What different results might you expect firom 
measurements of an inelastic one-dimensional 
coUision? 

The instructions for Film Loop 18, 
"One-Dimensional Collisions I" may be fol- 
lowed for completing this investigation. 

Are the differences between momentum 
before and after colhsion significant, or are 
they within the experimental error already es- 
timated? 



Save your data so that later you can make 
similar calculations on total kinetic energy for 
both balls just before and just after the colh- 
sion. Is whatever difference you may have ob- 
tained explainable by experimental error? Is 
there a noticeable difference between elastic 
and inelastic collisions as far as the conserva- 
tion of kinetic energy is concerned? 

FILM LOOP 21 TWO- 
DIMENSIONAL COLLISIONS I 

Two hard steel balls, hanging from long, thin 
wires, collide. Unlike the collisions in Film 
Loops 18 and 20, the balls do not move along 
the same straight line before or after the colh- 
sions. Although strictly the balls do not all 
move in a plane, as each motion is an arc of a 
circle, to a good approximation everything oc- 
curs in one plane. Hence, the colhsions are 
two-dimensional. Two colhsions are filmed in 
slow motion, with ball A having a mass of 539 




Unit 3/78 



Film Loop Notes 



grams, and ball B having a mass of 361 grams. 
Two more cases are shown in Film Loop 22. 

Using this film, you can find both the 
momentum and the kinetic energy of each ball 
before and after the colhsion, and thus study 
total momentum and total kinetic energy con- 
servation in this situation. Thus, you should 
save your momentum data for later use when 
studying energy. 

Both direction and magnitude of momen- 
tum should be taken into account, since the 
balls do not move on the same line. To find 
momentum you need velocities. Distance 
measurements accurate to a fraction of a mil- 
limeter and time measurements to about a 
tenth of a second are suggested, so choose 
measuring instruments accordingly. 

You can project directly onto a large piece 
of paper. An initial problem is to determine 
lines on which the balls move. If you make 
many marks at the centers of the balls, run- 
ning the film several times, you may find that 
these do not form a perfect hne. This is due 
both to the inaccuracies in your measurements 
and to the inherent difficulties of high speed 
photography. Cameras photographing at a rate 
of 2,000 to 3,000 frames a second "jitter," be- 
cause the film moves so rapidly through the 
camera that accurate frame registration is not 
possible. Decide which line is the "best" ap- 
proximation to determine direction for veloc- 
ities for the balls before and after collision. 

You will also need the magnitude of the 
velocity, the speed. One possibihty is to mea- 
sure the time it takes the ball to move across 
two lines marked on the paper. Accuracy sug- 
gests a number of different measurements to 
determine which values to use for the speeds 
and how much error is present. 

Compare the sum of the momentum before 
collision for both balls with the total momen- 
tum after collision. If you do not know how to 
add vector diagrams, you should consult your 
teacher or the Programmed Instruction Book- 
let Vectors II. The momentum of each object is 
represented by an arrow whose direction is that 
of the motion and whose length is proportional 
to the magnitude of the momentum. Then, if 



the head of one arrow is placed on the tail of the 
other, moving the hne parallel to itself, the vec- 
tor sum is represented by the arrow which joins 
the "free" tail to the "free" head. 

What can you say about momentum con- 
servation? Remember to consider measure- 
ment errors. 

FILM LOOP 22 
TWO-DIMENSIONAL COLLISION 11 

Two hard steel balls, hanging from long thin 
wires, collide. Unhke the colUsions in Film 
Loops 18 and 20, the balls do not move along 
the sam.e straight line before or after the colli- 
sions. Although the balls do not strictly all 
move in a plane, as each motion is an arc of a 
circle, everything occurs in one plane. Hence, 
the colhsions are two-dimensional. Two colh- 
sions are filmed in slow motion, with both balls 
having a mass of 367 grams. Two other cases 
are shown in Film Loop 21 . 

Using this film you can find both the kinet- 
ic energy and the momentum of each ball be- 
fore and after the collision, and thus study total 
momentum and total energy conservation in 
this situation. Follow the instructions given for 
Film Loop 21 , "Two-dimensional Colhsions I," 
in completing this investigation. 

FILM LOOP 23 INELASTIC 
TWO-DIMENSIONAL COLLISIONS 

Two hard steel balls, hanging from long, thin 
wires, colhde. Unhke the colhsions in Film 
Loops 18 and 20, the balls do not move along 
the sam^e straight hne before or after the colh- 
sion. Although the balls do not strictly all move 
in a plane, as each motion is an arc of a circle, 
to a good approximation the motion occurs in 
one plane. Hence, the collisions are two- 
dimensional. Two collisions are filmed in slow 
motion. Each ball has a mass of 500 grams. 
The plasticene balls stick together after colli- 
sion, moving as a single mass. 

Using this fOm, you can find both the kinet- 
ic energy and the momentum of each ball be- 
fore and after the collision, and thus study total 
momentum and total energy conservation in 
this situation. Follow the instructions given 



Film Loop Notes 



Unit 3/79 



{oTFilm Loop 21 , "Two-dimensional Collisions 
I," in completing this investigation. 

FILM LOOP 24 SCATTERING 
OF A CLUSTER OF OBJECTS 

This film and also Film Loop 3-8 each contain 
one advanced quantitative problem. We rec- 
ommend that you do not work on these loops 
until you have analyzed one of the Events 8 to 
13 of the series, Stroboscopic Still Photo- 
graphs of Two-Dimensional Collisions, or one 
of the examples in the film loops entitled 
"Two-Dimensional Collisions: Part II," or "In- 
elastic Two-Dimensional Collisions." All these 
examples involve two-body collisions, whereas 
the film here described involves seven objects 
and Film Loop 25, five. 

In this film seven balls are suspended from 
long, thin wires. The camera sees only a small 
portion of their motion, so the balls all move 
approximately along straight Unes. The slow- 
motion camera is above the balls. Six balls are 
initially at rest. A hardened steel ball strikes 
the cluster of resting objects. The diagram in 
Fig. 1 shows the masses of each of the balls. 




Fig. 1 



Part of the film is photographed in slow 
motion at 2,000 frames per second. By project- 
ing this section of the film on paper several 
times and making measurements of distances 
and times, you can determine the directions 




and magnitudes of the velocities of each of the 
balls. Distance and time measurements are 
needed. Discussions of how to make such 
measurements are contained in the Film Notes 
for one-dimensional and two-dimensional col- 
hsions. (See Film Loops 18 and 21 .) 

Compare the total momentum of the sys- 
tem both before and after the collision. Re- 
member that momentum has both direction 
and magnitude. You can add momenta after 
collision by representing the momentum of 
each ball by an arrow, and "adding" arrows 
geometrically. What can you say about the ac- 
curacy of your calculations and measure- 
ments? Is momentum conserved? You might 
also wish to consider energy conservation. 

FILM LOOP 25 EXPLOSION 
OF A CLUSTER OF OBJECTS 

Five balls are suspended independently firom 
long thin wires. The balls are initially at rest, 
with a small cylinder containing gunpowder in 
the center of the group of balls. The masses 
and initial positions of the ball are shown in 
Fig. 2. The charge is exploded and each of the 
balls moves off in an independent direction. In 
the slow-motion sequence the camera is 
mounted directly above the resting objects. 
The camera sees only a small part of the mo- 
tion, so that the paths of the balls are almost 
straight lines. 

In your first viewing, you may be interested 



Unit 3/80 



Film Loop Notes 




^ 



"> 



>. 



Fig. 2 



in trying to predict where the "missing" balls 
will emerge. Several of the balls are hidden at 
first by the smoke from the charge of powder. 
All the balls except one are visible for some 
time. What information could you use that 
would help you make a quick decision about 
where this last ball will appear? What physical 



quantity is important? How can you use this 
quantity to make a quick estimate? When you 
see the ball emerge from the cloud, you can 
determine whether or not your prediction was 
correct. The animated elliptical ring identifies 
this final ball toward the end of the film. 

You can also make detailed measure- 
ments, similar to the momentum conservation 
measurements you may have made using other 
Project Physics Film Loops. During the slow- 
motion sequence find the magnitude and direc- 
tion of the velocity of each of the balls after the 
explosion by projecting the film on paper, 
measuring distances and times. The notes on 
previous films in this series. Film Loops 18 
and 21 , wiU provide you with information 
about how to make such measurements if you 
need assistance. 

Determine the total momentum of all the 
balls after the explosion. What was the 
momentum before the explosion? You may find 
these results sHghtly puzzling. Can you ac- 
count for any discrepancy that you find? Watch 
the film again and pay close attention to what 
happens during the explosion. 



B.C. 






by 


John 


Hart 


«|i^ 


^ 


^ 


1 




i 


f 


^ 




n^t- 



By permission of John Hart and Field Enterprises, Inc. 



Film Loop Notes 



Unit 3/81 



KINETIC ENERGY 
CALCULATIONS 

You may have used one or more of Film Loops 
18 through 25 in your study of momentum. 
You will find it helpful to view these slow- 
motion films of one and two-dimensional colli- 
sions again, but this time in the context of the 
study of energy. The data you collected previ- 
ously will be sufficient for you to calculate the 
kinetic energy of each ball before and after the 
colhsion. Remember that kinetic energy imi;^ is 
not a vector quantity, and hence, you need only 
use the magnitude of the velocities in your cal- 
culations. 

On the basis of your analysis you may wish 
to try to answer such questions as these: Is 
kinetic energy consumed in such interactions? 
If not, what happened to it? Is the loss in ki- 
netic energy related to such factors as relative 
speed, angle of impact, or relative masses of 
the colhding balls? Is there a difference in the 
kinetic energy lost in elastic and inelastic col- 
lisions? 

FILM LOOP 26 FINDING THE 
SPEED OF A RIFLE BULLET I 

In this film a rifle bullet of 13.9 grams is fired 
into an 8.44 kg log. The log is initially at rest, 
and the bullet imbeds itself in the log. The two 
bodies move together after this violent colh- 
sion. The height of the log is 15.0 centimeters. 
You can use this information to convert dis- 
tances to centimeters. The setup is illustrated 
in Fig. 1 and 2. 



BALLISTIC 
PENDULUM 



.f?^ 





^■^^^^M^ 


WxCjKC^ ^J.^-X-^^ 





io ^ect 



^3-- 




Fig. 1 



Fig. 2 Schematic diagram of ballistic pendulum (not to 
scale). 

You can make measurements in this film 
using the extreme slow-motion sequence. The 
high-speed camera used to film this sequence 
operated at an average rate of 2850 frames per 
second; if your projector runs at 18 frames per 
second, the slow-motion factor is 158. Al- 
though there was some variation in the speed 
of this camera, the average frame rate of 2850 
is quite accurate. For velocity measurements 
in centimeters per second, a convenient unit to 
use in considering a rifle bullet, convert the 
apparent time of the film to seconds. Find the 
exact duration with a timer or a stop-watch by 
timing the interval from the yellow circle at the 
beginning to the one at the end of the film. 
There are 3490 frames in the film, so you can 
determine the precise speed of the projector. 

Project the film onto a piece of white paper 
or graph paper to make your measurements of 
distance and time. View the film before mak- 
ing decisions about which measuring instru- 
ments to use. As suggested above, you can 
convert your distance and time measurements 
to centimeters and seconds. 

After measuring the speed of the log after 



Unit 3/82 



Film Loop Notes 



impact, calculate the bullet speed at the mo- 
ment when it entered the log. What physical 
laws do you need for the calculation? Calculate 
the kinetic energy given to the bullet, and also 
calculate the kinetic energy of the log after the 
bullet enters it. Compare these two energies 
and discuss any differences that you might 
find. Is kinetic energy conserved? 

A final sequence in the film allows you to 
find a lower limit for the bullet's speed. Three 
successive frames are shown, so the time be- 
tween each is 1/2850 of a second. The frames 
are each printed many times, so each is held on 
the screen. How does this lower limit compare 
with your measured velocity? 



FILM LOOP 27 FINDING THE 
SPEED OF A RIFLE BULLET II 

The problem proposed by this film is that of 
determining the speed of the bullet just before 
it hits a log. The wooden log with a mass of 
4.05 kilograms is initially at rest. A bullet fired 
from a rifle enters the log. (Fig. 1.) The mass of 
the bullet is 7. 12 grams. The bullet is imbedded 
in the thick log and the two move together after 
the impact. The extreme slow-motion se- 
quence is intended for taking measurements. 

The log is suspended from thin wires, so 
that it behaves like a pendulum that is free to 
swing. As the bullet strikes the log it starts to 
rise. When the log reaches its highest point, it 




Fig. 1 



momentarily stops, and then begins to swing 
back down. This point of zero velocity is visible 
in the slow-motion sequence in the film. 

The bullet plus the log after impact forms a 
closed system, so you would expect the total 
amount of mechanical energy of such a system 
to be conserved. The total mechanical energ> 
is the sum of kinetic energy plus potential 
energy. If you conveniently take the potential 
energy as zero at the moment of impact for the 
lowest position of the log, then the energy at 
that time is all kinetic energy. As the log begins 
to move, the potential energy is proportional to 
the vertical distance above its lowest point, 
and it increases while the kinetic energy, de- 
pending upon the speed, decreases. The kinetic 
energy becomes zero at the point where the log 
reverses its direction, because the log's speed is 
zero at that point. All the mechanical energy at 
the reversal point is potential energy. Because 
energy is conserved, the initial kinetic energy 
at the lowest point should equal the potential 
energy at the top of the swing. On the basis of 
this result, write an equation that relates the 
initial log speed to the final height of rise. You 
might check this result with your teacher or 
with other students in the class. 

If you measure the vertical height of the 
rise of the log, you can calculate the log's initial 
speed, using the equation just derived. What is 
the initial speed that you find for the log? If you 
wish to convert distance measurements to cen- 
timeters, it is useful to know that the vertical 
dimension of the log is 9.0 centimeters. 

Find the speed of the rifle bullet at the 
moment it hits the log, using conservation of 
momentum. 

Calculate the kinetic energy of the rifle bul- 
let before it strikes and the kinetic energy of the 
log plus bullet after impact. Compare the two 
kinetic energies, and discuss any difference. 



FILM LOOP 28 RECOIL 

Conservation laws can be used to determine 
recoil velocity of a gun, given the experimental 
information that this film provides. 

The preliminary scene shows the recoil of a 
cannon firing at the fort on Ste. Helene Island, 



Film Loop Notes Unit 3/83 




Fig. 1 



near Montreal, Canada. (Fig. 1.) The small 
brass laboratory "cannon" in the rest of the 
film is suspended by long wires. It has a mass 
of 350 grams. The projectile has a mass of 3.50 
grams. When the firing is photographed in slow 
motion, you can see a time lapse between the 
time the fuse is Ughted and the time when the 
bullet emerges from the cannon. Why is this 
delay observed? The camera used here exposes 
8000 frames per second. 

Project the film on paper. It is convenient 
to use a horizontal distance scale in centime- 
ters. Find the bullet's velocity by timing the 
bullet over a large fraction of its motion. (Only 
relative values are needed, so it is not neces- 
sary to convert this velocity into cm/sec.) 

Use momentum conservation to predict 
the gun's recoil velocity. The system (gun plus 
bullet) is one dimensional; all motion is along 
one straight hne. The momentum before the 
gun is fired is zero in the coordinate system in 
which the gun is at rest. So the momentum of 
the cannon after colHsion should be equal and 
opposite to the momentum of the buUet. 

Test your prediction of the recoil velocity 
by running the film again and timing the gun 



to find its recoil velocity experimentally. What 
margin of error might you expect? Do the pre- 
dicted and observed values agree? Give 
reasons for any difference you observe. Is ki- 
netic energy conserved? Explain your answer. 

FILM LOOP 29 COLLIDING 
FREIGHT CARS 

This film shows a test of freight-car coupling. 
The colhsions, in some cases, were violent 
enough to break the couplings. The "hammer 
car" coasting down a ramp, reaches a speed of 
about 6 miles per hour. The momentary force 
between the cars is about 1,000,000 pounds. 
The photograph below (Fig. 1) shows cou- 




Fig. 1 Broken coupling pins from colliding freight cars. 



Unit 3/84 



Film Loop Notes 



pling pins that were sheared off by the force 
of the coUision. The slow-motion collision al- 
lows you to measure speeds before and after 
impact, and thus to test conservation of mo- 
memtum. The collisions are partially elastic, 
as the cars separate to some extent after col- 
lision. 

The masses of the cars are: 

Hammer car: w, = 95,000 kg (210,000 lb) 

Target car: m^ = 120,000 kg (264,000 lb) 

To find velocities, measure the film time for the 
car to move through a given distance. (You 
may need to run the film several times.) Use 
any convenient units for velocities. 

Simple timing will give Vi and V2. The film 
was made on a cold winter day and friction was 
appreciable for the hammer car after collision. 
One way to allow for friction is to make a veloc- 
ity time graph, assume a uniform negative ac- 
celeration, and extrapolate to the instant after 
impact. 

An example might help. Suppose the 
hammer car coasts 3 squares on graph paper in 
5 seconds after collision, and it also coasts 6 
squares in 12 seconds after colhsion. The aver- 
age velocity during the first 5 seconds was v^ = 
(3 squares)/(5 sec) = 0.60 squares/sec. The av- 
erage velocity during any short interval ap- 
.70 . 
1 
->'N 



.bO '- 



V* .so 



\~ 



Fig. 1 Extrapolation backwards in tinne to allow for fric- 
tion in estimating the value of v, immediately after the 
collision. 



proximately equals the instantaneous velocity 
at the mid-time of that interval, so the car's 
velocity was about i;, = 0.60 squares/sec att - 
2.5 sec. For the interval 0-12 seconds, the ve- 
locity was Vi = 0.50 squares/sec att = 6.0 sec. 
Now plot a graph like that shown in Fig. 1. 
This graph shows by extrapolation that v^ = 
0.67 squares/sec at t = 0, just after the colh- 
sion. 

Compare the total momentum of the sys- 
tem before collision with the total momentum 
after colhsion. Calculate the kinetic energy of 
the freight cars before and after colhsion. What 
fraction of the hammer car's original kinetic 
energy has been "lost"? Can you account for 
this loss? 

FILM LOOP 30 DYNAMICS OF 
A BILLIARD BALL 

The event pictured in this film is one you have 
probably seen many times — the striking of a 
ball, in this case a bilhard ball, by a second 
ball. Here, the camera is used to "slow down" 
time so that you can see details in this event 
which you probably have never observed. The 
ability of the camera to alter space and time is 
important in both science and art. The slow- 
motion scenes were shot at 3000 frames per 
second. 

The "world" of your physics course often 
has some simphfications in it. Thus, in your 
textbook, much of the discussion of mechanics 
of bodies probably assumes that the objects are 
point objects, with no size. But clearly these 
massive billiard balls have size, as do all the 
things you encounter. For a point particle we 
can speak in a simple, meaningful way of its 
position, its velocity, and so on. 

But the particles photographed here are 
billiard balls and not points. What information 
might be needed to describe their positions and 
velocities? Looking at the film may suggest 
possibilities. What motions can you see besides 
simply the linear forward motion? Watch each 
ball carefully, just before and just after the 
collision, watching not only the overall motion 
of the ball, but also "internal" motions. Can 
any of these motions be appropriately de- 



Film Loop Notes 



Unit 3/85 




Billiard bails near impact. The two cameras tool< side 
views of the collision, which are not shown in this 
film loop. 

scribed by the word "spin"? Can you distin- 
guish the cases where the ball is rolling along 
the table, so that there is no slippage between 
the ball and the table, from the situations 
where the ball is skidding along the table with- 
out rolling? Does the first ball move im- 
mediately after the colUsion? You can see that 
even this simple phenomenon is a good bit 
more complex than you might have expected. 

Can you write a careful verbal description 
of the event? How might you go about giving a 
more careful mathematical description? 

Using the slow-motion sequence you can 
make a momentum analysis, at least partially, 
of this collision. Measure the velocity of the cue 
ball before impact and the velocity of both balls 
after impact. Remember that there is friction 
between the ball and the table, so velocity is 
not constant. The balls have the same mass, so 
conservation of momentum predicts that 



velocity of cue 
ball just before 
collision 



sum of velocities 
of the balls just 
after collision 



How closely do the results of your measure- 
ments agree with this principle? What reasons, 
considering the complexity of the phenome- 
non, might you suggest to account for any dis- 
agreement? What motions are you neglecting 
in your analysis? 



FILM LOOP 31 A METHOD OF 
MEASURING ENERGY— NAILS 
DRIVEN INTO WOOD 

Some physical quantities, such as distance, 
can be measured directly in simple ways. 
Other concepts can be connected with the 
world of experience only through a long series 
of measurements and calculations. One quan- 
tity that we often would like to measure is 
energy. In certain situations, simple and reli- 
able methods of determining energy are possi- 
ble. Here, you are concerned with the energy of 
a moving object. 

This film allows you to check the validity of 
one way of measuring mechanical energy. If a 
moving object strikes a naO, the object will lose 
all of its energy. This energy has some effect, in 
that the nail is driven into the wood. The 
energy of the object becomes work done on the 
nail, driving it into the block of wood. 

The first scenes in the film show a con- 
struction site. A pile driver strikes a pile over 
and over again, "planting" it in the ground. 
The laboratory situation duphcates this situa- 
tion under more controlled circumstances. 
Each of the blows is the same as any other be- 
cause the massive object is always raised to 
the same height above the nail. The nail is hit 
ten times. Because the conditions are kept the 
same, you expect the energy by the impact to 
be the same for each blow. Hence, the work 
from each blow is the same. Use the film to find 
if the distance the nail is driven into the wood 
is proportional to the energy or work. Or, bet- 
ter, you want to know how you can find the 
energy if you know the depth of penetration of 
the nail. 

The simplest way to display the measure- 
ments made with this film may be to plot the 
depth of nail penetration versus the number of 
blows. Do the experimental points that you ob- 
tain he approximately along a straight Une? If 
the line is a good approximation, then the 
energy is about proportional to the depth of 
penetration of the nail. Thus, depth of penetra- 
tion can be used in the analysis of other films to 
measure the energy of the striking object. 



Film Loop Notes 




fliyinie.y df i>ioL^3 



Fig. 1 



If the graph is not a straight Une, you can 
still use these results to calibrate your 
energy-measuring device. By use of penetra- 
tion versus the number of blows, an observed 
penetration (in centimeters, as measured on 
the screen), can be converted into a number of 
blows, and therefore an amount proportional to 
the work done on the nail, or the energy trans- 
ferred to the nail. Thus in Fig. 1, a penetra- 
tion of 3 cm signifies 5.6 units of energy. 

FILM LOOP 32 GRAVITATIONAL 
POTENTIAL ENERGY 

Introductory physics courses usually do not 
give a complete definition of potential energy, 
because of the mathematics involved. Only 
particular kinds of potential energy, such as 
gravitational potential energy, are considered. 

You may know the expression for the gravi- 
tational potential energy of an object near the 
earth— the product of the weight of the object 
and its height. The height is measured firom a 
location chosen arbitrarily as the zero level for 
potential energy. It is almost impossible to 
"test" a formula without other physics con- 
cepts. Here we require a method of measuring 
energy. The previous Film Loop 31 "A 
Method of Measuring Energy," demonstrated 
that the depth of penetration of a nail into 
wood, due to a blow, is a good measure of the 
energy at the moment of impact of the object. 

Although you are concerned with potential 
energy you will calculate it by first finding 



kinetic energy. Where there is no loss of energy 
through heat, the sum of the kinetic energy 
and potential energy is constant. If you mea- 
sure potential energy from the point at which 
the weight strikes the nail, at the moment of 
striking all the energy will be kinetic energy. 
On the other hand, at the moment an object is 
released, the kinetic energy is zero, and all the 
energy is potential energy. These two must, by 
conservation of energy, be equal. 

Since energy is conserved, you can figure 
the initial potential energy that the object had 
from the depth of penetration of the nail by 
using the results of the measurement connect- 
ing energy and nail penetration. 

Two types of measurements are possible 
with this film. The numbered scenes are all 
photographed from the same position. In the 
first scenes (Fig. 1) you can determine how 
gravitational potential energy depends upon 
weight. Objects of different mass fall from the 
same distance. Project the film on paper and 
measure the positions of the nailheads before 
and after the impact of the falling objects. 




Fig. 1 

Make a graph relating the penetration depth 
and the weight mag. Use the results of the pre- 
vious experiment to convert this relation into a 
relation between gravitational potential energy 
and weight. What can you learn from this 
graph? What factors are you holding constant? 
What conclusions can you reach from your 
data? 



Film Loop Notes 



Unit 3/87 



Later scenes (Fig. 2), provide information 
for studying the relationship between gravita- 
tional potential energy and position. Bodies of 
equal mass are raised to different heights and 
allowed to fall. Study the relationship between 
the distance of fall and the gravitational poten- 
tial energy. What graphs might be useful? 
What conclusion can you reach from your 
measurements? 




Fig. 2 

Can you relate the results of these mea- 
surements with statements in your text con- 
cerning gravitational potential energy? 



FILM LOOP 33 
ENERGY 



KINETIC 



In this film you can test how kinetic energy 
(KE) depends on speed (v). You measure both 
KE and v, keeping the mass m constant. 

Penetration of a nail driven into wood is a 
good measure of the work done on the nail, and 
hence is a measure of the energy lost by what- 
ever object strikes the nail. The speed of the 
moving object can be measured in several 
ways. 

The preliminary scenes show that the ob- 
ject falls on the nail. Only the speed just before 
the object strikes the nail is important. The 
scenes intended for measurement were photo- 
graphed with the camera on its side, so the 
body appears to move horizontally toward the 
nail. 

The speeds can be measured by timing the 
motion of the leading edge of the object as it 
moves from one reference mark to the other. 



The clock in the film (Fig. 1) is a disk that ro- 
tates at 3000 revolutions per minute. Project 
the film on paper and mark the positions of the 
clock pointer when the body crosses each ref- 
erence mark. The time is proportional to the 
angle through which the pointer turns. The 
speeds are proportional to the reciprocals of the 
times, since the distance is the same in each 
case. Since you are testing only the form of the 
kinetic energy dependence on speed, any con- 
venient unit can be used. Measure the speed 
for each of the five trials. 




Fig. 1 

The kinetic energy of the moving object is 
transformed into the work required to drive 
the nail into the wood. In Film Loop 31, "A 
Method of Measuring Energy," you relate the 
work to the distance of penetration. Measure 
the nail penetration for each trial, and use your 
results from the previous film. 

How does KE depend on t;? The conserva- 
tion law derived from Newton's laws indicates 
that KE is proportional to v'^, the square of the 
speed, not to v. Test this by making two graphs. 
In one graph, plot KE vertically and plot v'^ 
horizontally. For comparison, plot KE versus i'. 
What can you conclude? Do you have any as- 
surance that a similar relation will hold, if the 
speeds or masses are very different from those 
found here? How might you go about determin- 
ing this? 

FILM LOOP 34 CONSERVATION 
OF ENERGY— POLE VAULT 

This quantitative film can help you study con- 
servation of energy. A pole vaulter (mass 68 kg, 
height 6 ft) is shown, first at normal speed 



Unit 3/88 



Film Loop Notes 



and then in slow motion, clearing a bar at 11.5 
ft. You can measure the total energy of the sys- 
tem at two points in time just before the 
jumper starts to rise and part way up, when the 
pole has a distorted shape. The total energy of 
the system is constant, although it is divided 
differently at different times. Since it takes 
work to bend the pole, the pole has elastic po- 
tential energy when bent. This elastic energy 
comes from some of the kinetic energy the 
vaulter has as he runs horizontally before in- 
serting the pole in the socket. Later, the elastic 
potential energy of the bent pole is transformed 
into some of the jumper's gravitational po- 
tential energy when he is at the top of the 
jump. 



0. 




^^^B^^l 



Position 1 The energy is entirely kinetic 
energy, ^mv'^. To help you measure the runner's 
speed, successive frames are held as the run- 
ner moves past two markers 1 meter apart. 
Each "freeze frame" represents a time interval 
of 1/250 sec, the camera speed. Find the 
runner's average speed over this meter, and 
then find the kinetic energy. If m is in kg and v 
is in m/sec, E will be in joules. 

Position 2 The jumper's center of gravity is 
about 1.02 meters above the soles of his feet. 
Three types of energy are involved at the in- 
termediate positions. Use the stop-frame se- 
quence to obtain the speed of the jumper. The 
seat of his pants can be used as a reference. 
Calculate the kinetic energy and gravitational 
potential energy as already described. 

The work done in deforming the pole is 
stored as elastic potential energy. In the final 
scene, a chain windlass bends the pole to a 
shape similar to that which it assumes during 
the jump in position 2. When the chain is short- 
ened, work is done on the pole: work = 
(average force) x (displacement). During the 
cranking sequence, the force varied. The aver- 
age force can be approximated by adding the 
initial and final values, found from the scale 
and dividing by two. Convert this force to new- 
tons. The displacement can be estimated from 
the number of times the crank handle is pulled. 
A close-up shows how far the chain moves dur- 



ing a single stroke. Calculate the work done to 
crank the pole into its distorted shape. 

You now can add and find the total energy. 
How does this compare with the original ki- 
netic energy? 

Position 3 Gravitational potential energy is the 
work done to raise the jumper's center of grav- 
ity. From the given data, estimate the vertical 
rise of the center of gravity as the jumper 
moves from position (1) to position (3). (His 
center of gravity clears the bar by about a foot, 
or 0.3m.) Multiply this height of rise by the 
jumper's weight to get potential energy. If 
weight is in newtons and height is in meters, 
the potential energy will be in joules. A small 
additional source of energy is in the jumper's 
muscles: judge for yourself how far he hfts his 
body by using his arm muscles as he nears the 
highest point. This is a small correction, so a 
relatively crude estimate will suffice. Perhaps 
he pulls with a force equal to his own weight 
through a vertical distance of | of a meter. 

How does the initial kinetic energy, plus 
the muscular energy expended in the pull-up, 
compare with the final gravitational potential 
energy? (An agreement to within about 10 per 
cent is about as good as you can expect from a 
measurement of this type.) 

As a general reference see "Mechanics of 
the Pole Vault," 16th ed., by Dr. R. V. Ganslen; 
John Swift & Co., St. Louis, Mo. (1965). 



Film Loop Notes 



Unit 3/89 



FILM LOOP 35 CONSERVATION 
OF ENERGY— AIRCRAFT TAKEOFF 

The pilot of a Cessna 150 holds the plane at 
constant speed in level flight, just above the 
surface of the runway. Then, keeping the throt- 
tle fixed, he puUs back on the stick, and the 
plane begins to rise. With the same throttle set- 
ting, he levels off at several hundred feet. At 
this altitude the aircraft's speed is less than at 
ground level. You can use this film to make a 
crude test of energy conservation. The plane's 
initial speed was constant, indicating that the 
net force on it was zero. In terms of an approx- 
imation, air resistance remained the same 




after lift-off. How good is this approximation? 
What would you expect air resistance to de- 
pend on? When the plane rose, its gravitational 
potential energy increased, at the expense of 
the initial kinetic energy of the plane. At the 
upper level, the plane's kinetic energy is less, 
but its potential energy is greater. According to 
the principle of conservation of energy, the 
total energy (KE + PE) remained constant, as- 
suming that air resistance and any other simi- 
lar factors are neglected. But are these negligi- 
ble? Here is the data concerning the film and 
the airplane: 

Length of plane: 7.5 m (23 ft) 

Mass of plane: 550 kg 

Weight of plane: 

550 kg X 9.8 m/sec- = 5400 newtons 

(1200 lb) 

Camera speed: 45 frames/sec 



Project the film on paper. Mark the length 
of the plane to calibrate distances. 

Stop-frame photography helps you mea- 



sure the speed of 45 frames per second. In 
printing the measurement section of the film 
only every third frame was used. Each of these 
frames was repeated ("stopped") a number of 
times, enough to allow time to mark a position 
on the screen. The effect is one of "holding" 
time, and then jumping a fifteenth of a second. 

Measure the speeds in all three situations, 
and also the heights above the ground. You 
have the data needed for calculating kinetic 
energy (^v^) and gravitational potential 
energy (maji) at each of the three levels. Cal- 
culate the total energy at each of the three 
levels. 

Can you make any comments concerning 
air resistance? Make a table showing (for each 
level) KE, PE, and £ total. Do your results sub- 
stantiate the law of conservation of energy 
within experimental error? 




Steve Aacker of Wheat Ridge High School, Wheat 
Ridge, Colorado, seems a bit skeptical about elastic 
potential energy. 



Unit 3/90 



Film Loop Notes 



FILM LOOP 36 REVERSIBILITY 
OF TIME 

It may sound strange to speak of "reversing 
time." In the world of common experience we 
have no control over time direction, in contrast 
to the many aspects of the world that we can 
modify. Yet physicists are much concerned 
with the reversibility of time; perhaps no other 
issue so clearly illustrates the imaginative and 
speculative nature of modem physics. 

The camera gives us a way to manipulate 
time. If you project film backward, the events 
pictured happen in reverse time order. This 
film has sequences in both directions, some 
shown in their "natural" time order and some 
in reverse order. 

The film concentrates on the motion of ob- 
jects. Consider each scene from the standpoint 
of time direction: Is the scene being shown as it 
was taken, or is it being reversed and shown 
backward? Many sequences are paired, the 
same film being used in both time senses. Is it 
always clear which one is forward in time and 
which is backward? With what types of events 
is it difficult to tell the "natural" direction? 

The Newtonian laws of motion do not 
depend on time direction. Any filmed motion of 
particles following strict Newtonian laws 
should look completely "natural" whether seen 
forward or backward. Since Newtonian laws 
are "invariant" under time reversal, changing 
the direction of time, you could not tell by ex- 
amining a motion obeying these laws whether 
the sequence is forward or backward. Any mo- 
tion which could occur forward in time can 
also occur, under suitable conditions, with the 
events in the opposite order. 

With more complicated physical systems, 
with extremely large number of particles, the 
situation changes. If ink were dropped into 
water, you would have no difficulty in deter- 
mining which sequence was photographed 
forward in time and which backward. So cer- 
tain physical phenomena at least appear to be 
irreversible, taking place in only one time di- 
rection. Are these processes fundamentally 
irreversible, or is this only some limitation on 
human powers? This is not an easy question to 



answer. It could still be considered, in spite of a 
fifty-year history, a frontier problem. 

Reversibility of time has been used in 
many ways in twentieth -century physics. For 
example, an interesting way of viewing the two 
kinds of charge in the universe, positive and 
negative, is to think of some particles as "mov- 
ing" backward in time. Thus, if the electron is 
viewed as moving forward in time, the positron 
can be considered as exactly the same particle 
moving backward in time. This backward mo- 
tion is equivalent to the forward-moving parti- 
cle having the opposite charge! This was one 
of the keys to the development of the space- 
time view of quantum electrodynamics which 
R. P. Feynman described in his Nobel Prize 
lecture. 

For a general introduction to time reversi- 
bility, see the Martin Gardner article, "Can 
Time Go Backward?" originally published in 
Scientific American January, 1967. 

FILM LOOP 37 SUPERPOSITION 

Using this film, you study an important physi- 
cal idea — superposition. The film was made by 
photographing patterns displayed on the face 
of the cathode ray tube (CRT) of an oscillo- 
scope, similar to a television set. You may have 
such an oscilloscope in your laboratory. 

Still photographs of some of these patterns 
appearing on the CRT screen are shown in 
Figs. 1 to 2. The two patterns at the top of the 
screen are called sinusoidal. They are not just 
any wavy lines, but lines generated in a precise 
fashion. If you are famihar with the sine and 
cosine functions, you will recognize them here. 
The sine function is the special case where the 
origin of the coordinate system is located 
where the function is zero and starting to rise. 
No origin is shown, so it is arbitrary as to 
whether one calls these sine curves, cosine 
curves, or some other sinusoidal type. What 
physical situations might lead to curves of this 
type? (You might want to consult books of 
someone else about simple harmonic oscil- 
lators.) Here the curves are produced by elec- 
tronic circuits which generate an electrical 
voltage changing in time so as to cause the 
curve to be displayed on the cathode ray 



Film Loop Notes 



Unit 3/91 




Fig. 1 




Fig. 2 




Fig. 3 



tube. The oscilloscope operator can adjust the 
magnitudes and phases of the two top func- 
tions. 

The bottom curve is obtained by a point- 
by-point adding of the top curves. Imagine a 
horizontal axis going through each of the two 
top curves, and positive and negative distances 
measured vertically from this axis. The bottom 
curve is at each point the algebraic sum of the 
two points above it on the top curves, as mea- 
sured from their respective axes. This point- 
by-point algebraic addition, when appHed to 
actual waves, is called superposition. 

Two cautions are necessary. First, you are 
not seeing waves, hut models of waves. A wave 
is a disturbance that propagates in time, but, at 
least in some of the cases shown, there is no 
propagation. A model always has some limita- 
tions. Second, you should not think that all 
waves are sinusoidal. The form of whatever 
is propagating can be any shape. Sinusoidal 
waves constitute only one important class of 
waves. Another common wave is the pulse, 
such as a sound wave produced by a sharp 
blow on a table. The pulse is not a sinusoidal 
wave. 

Several examples of superposition are 
shown in the film. If, as approximated in Fig. 
1, two sinusoidal curves of equal period and 
amplitude are in phase, both having zeroes at 
the same places, the result is a double-sized 
function of the same shape. On the other hand, 
if the curves are combined out of phase, where 
one has a positive displacement while the other 
one has a negative displacement, the result is 
zero at each point (Fig. 2). If functions of dif- 
ferent periods are combined (Figs. 3, 4, and 5), 
the result of the superposition is not sinusoidal, 
but more complex in shape. You are asked to 
interpret both verbally and quantitatively, the 
superpositions shown in the film. 

FILM LOOP 38 STANDING 
WAVES ON A STRING 

Tension determines the speed of a wave travel- 
ing down a string. When a wave reaches a fixed 
end of a string, it is reflected back again. The 
reflected wave and the original wave are 



Unit 3/92 



Film Loop Notes 



WWW 




Fig. 4 



Fig. 5 



superimposed or added together. If the tension 
(and therefore the speed) is just right, the re- 
sulting wave will be a "standing wave." Cer- 
tain nodes will always stand still on the string. 
Other points on the string will continue to 
move in accordance with superposition. When 
the tension in a vibrating string is adjusted, 
standing waves can be set up when the tension 
has one of a set of "right" values. 

In the film, one end of a string is attached 
to a tuning fork with a frequency of 72 vibra- 
tions per second. The other end is attached to a 
cylinder. The tension of the string is adjusted 
by sliding the cylinder back and forth. 

Several standing wave patterns are shown. 
For example, in the third mode the string vi- 
brates in 3 segments with 2 nodes (points of 
no motion) between the nodes at each end. The 
nodes are half a wavelength apart. Between 
the nodes are points of maximum possible vi- 
bration called antinodes. 

You tune the strings of a violin or guitar by 
changing the tension on a string of fixed 
length, higher tension corresponding to higher 
pitch. Different notes are produced by placing 
a finger on the string to shorten the vibrating 
part. In this film the frequency of vibration of a 
string is fixed, because the string is always 
driven at 72 vib/sec. When the frequency re- 
mains constant, the wavelength changes as 
the tension is adjusted because velocity de- 
pends on tension. 



A high-speed snapshot of the string at any 
time would show its instantaneous shape. Sec- 
tions of the string move, except at the nodes. 
The eye sees a blurred or "time exposure" 
superposition of string shapes because of the 
frequency of the string. In the film, this blurred 
effect is reproduced by photographing at a slow 
rate: Each frame is exposed for about 1/15 sec. 

Some of the vibration modes are photo- 
graphed by a stroboscopic method. If the string 
vibrates at 72 vib/sec and frames are exposed 
in the camera at the rate of 70 times per sec, 
the string seems to go through its complete 
cycle of vibration at a slower frequency when 
projected at a normal speed. In this way, a 
slow-motion effect is obtained. 

FILM LOOP 39 STANDING 
WAVES IN A GAS 

Standing waves are set up in air in a large 
glass tube. (Fig. 1.) The tube is closed at one 
end by an adjustable piston. A loudspeaker at 
the other end supphes the sound wave. The 
speaker is driven by a variable-frequency oscil- 
lator and amphfier. About 20 watts of au- 
diopower are used, giving notice to everyone in 
a large building that filming is in progress! The 
waves are reflected from the piston. 

A standing wave is formed when the fre- 
quency of the oscillator is adjusted to one of 
several discrete values. Most frequencies do 
not give standing waves. Resonance is indi- 



Film Loop Notes 



Unit 3/93 



^^^angmtMe^^hiunm i mf 




Fig. 1 

cated in each mode of vibration by nodes and 
antinodes. There is always a node at the fixed 
end (where air molecules cannot move) and an 
antinode at the speaker (where air is set into 
motion). Between the fixed end and the 
speaker there may be additional nodes and an- 
tinodes. 

The patterns can be observed in several 
ways, two of which are used in the film. One 
method of making visible the presence of a 
stationary acoustic wave in the gas in the tube 
is to place cork dust along the tube. At reso- 
nance the dust is blown into a cloud by the 
movement of air at the antinodes; the dust re- 
mains stationary at the nodes where the air is 
not moving. In the first part of the film, the 
dust shows standing wave patterns for these 
firequencies: 

Frequency Number of half 

(vib/sec) wavelengths 



230 


1.5 


370 


2.5 


530 


3.5 


670 


4.5 


1900 


12.5 



The pattern for/= 530 is shown in Fig. 2. From 
node to node is iX, and the length of the pipe is 
3X -I- iX (the extra iX is from the speaker an- 
tinode to the first node). There are, generally, 
(n -I- i) half- wavelengths in the fixed length, so 
X oc l/(n + i). Since/ oc l/x, f ex (n + i). Divide 
each frequency in the table by (n -i- i) to find 
whether the result is reasonably constant. 

In all modes the dust remains motionless 
near the stationary piston which is a node. 

In the second part of the film nodes and 
antinodes are made visible by a different 




Fig. 2 

method. A wire is placed in the tube near the 
top. This wire is heated electrically to a dull 
red. When a standing wave is set up, the wire is 
cooled at the antinodes, because the air carries 
heat away firom the wire when it is in vigorous 
motion. So the wire is cooled at antinodes and 
glows less. The bright regions correspond to 
nodes where there are no air currents. The os- 
cillator frequency is adjusted to give several 
standing wave patterns with successively 
smaller wavelengths. How many nodes and 
antinodes are there in each case? Can you find 
the frequency used in each case? 

FILM LOOP 40 VIBRATIONS 
OF A WIRE 

This film shows standing-wave patterns in thin 
but stiff wires. The wave speed is determined 
by the wire's cross section and by the elastic 
constants of the metal. There is no external 
tension. Two shapes of wire, straight and cir- 
cular, are used. 

The wire passes between the poles of a 
strong magnet. When a switch is closed, a 
steady electric current from a battery is set up 
in one direction through the wire. The interac- 
tion of this current and the magnetic field leads 
to a downward force on the wire. When the di- 
rection of the current is reversed, the force on 
the wire is upward. Repeated rapid reversal of 
the current direction can make the wire vibrate 
up and down. 

The battery is replaced by a source of vari- 
able frequency alternating current whose fre- 
quency can be changed. When the firequency is 
adjusted to match one of the natural frequen- 
cies of the wire, a standing wave builds up. 
Several modes are shown, each excited by a 
different frequency. 

The first scenes show a straight brass wire, 
2.4 mm in diameter (Fig. 1). The "boundary 



Unit 3/94 



Film Loop Notes 




Fig. 1 

conditions" for motion require that, in any 
mode, the fixed end of the wire is a node and 
the free end is an antinode. (A horizontal plas- 
tic rod is used to support the wire at another 
node.) The wire is photographed in two ways: 
in a blurred "time exposure," as the eye 
sees it, and in "slow motion," simulated 
through stroboscopic photography. 

Study the location of the nodes and an- 
tinodes in one of the higher modes of vibration. 
They are not equally spaced along the wire, as 
for vibrating string (see Film Loop 38). This 
is because the wire is stiff whereas the string is 
perfectly flexible. 

In the second sequence, the wire is bent 
into a horizontal loop, supported at one point 
(Fig. 2). The boundary conditions require a 
node at this point; there can be additional 
nodes, equally spaced around the ring. Several 
modes are shown, both in "time exposure" and 
in "slow motion." To some extent the vibrating 
circular wire is a helpful model for the wave 
behavior of an electron orbit in an atom such as 
hydrogen; the discrete modes correspond to 
discrete energy states for the atom. 




FILM LOOP 41 VIBRATIONS 
OF A RUBBER HOSE 

You can generate standing waves in many 
physical systems. When a wave is set up in a 
medium, it is usually reflected at the bound- 
aries. Characteristic patterns will be formed, 
depending on the shape of the medium, the 
frequency of the wave, and the material. At 
certain points or lines in these patterns there 
is no vibration, because all the partial waves 
passing through these points just manage to 
cancel each other through superposition. 

Standing-wave patterns only occur for cer- 
tain frequencies. The physical process selects a 
spectrum of frequencies from all the possible 
ones. Often there are an infinite number of 
such discrete frequencies. Sometimes there 
are simple mathematical relations between the 
selected frequencies, but for other bodies the 
relations are more complex. Several films in 
this series show vibrating systems with such 
patterns. 

This film uses a rubber hose, clamped at 
the top. Such a stationary point is called a 
node. The bottom of the stretched hose is at- 
tached to a motor whose speed is increased 
during the film. An eccentric arm attached to 
the motor shakes the bottom end of the hose. 
Thus this end moves slightly, but this motion is 
so small that the bottom end also is a node. 

The motor begins at a frequency below that 
for the first standing-wave pattern. As the 
motor is gradually speeded up, the amphtude 
of the vibrations increase until a well-defined 




Fig. 1 



Film Loop Notes 



Unit 3/95 



steady loop is formed between the nodes. This 
loop has its maximum motion at the center. 
The pattern is half a wavelength long. Increas- 
ing the speed of the motor leads to other har- 
monics, each one being a standing-wave pat- 
tern with both nodes and antinodes, points of 
maximum vibration. These resonances can be 
seen in the film to occur only at certain sharp 
frequencies. For other motor frequencies, no 
such simple pattern is seen. You can count as 
many as eleven loops with the highest fre- 
quency case shown. 

It would be interesting to have a sound 
track for this film. The sound of the motor is by 
no means constant during the process of in- 
creasing the frequency. The stationary reso- 
nance pattern corresponds to points where the 
motor is running much more quietly, because 
the motor does not need to "fight" against the 
hose. This sound distinction is particularly 
noticeable in the higher harmonics. 

If you play a viohn cello, or other stringed 
instrument, you might ask how the harmonies 
observed in this film are related to musical 
properties of vibrating strings. What can be 
done with a violin string to change the fre- 
quency of vibration? What musical relation ex- 
ists between two notes if one of them is twice 
the frequency of the other? 

What would happen if you kept increasing 
the frequency of the motor? Would you expect 
to get arbitrarily high resonances, or would 
something "give?" 

FILM LOOP 42 VIBRATIONS 
OF A DRUM 

The standing-wave patterns in this film are 
formed in a stretched circular rubber mem- 
brane driven by a loudspeaker. The loud- 
speaker is fed large amounts of power, about 
30 watts, more power than you would want to 
use with your Uving room television set or 
phonograph. The frequency of the sound can 
be changed electronically. The lines drawn on 
the membrane make it easier for you to see the 
patterns. 

The rim of the drum cannot move, so in all 
cases it must be a nodal circle, a circle that 
does not move as the waves bounce back and 



forth on the drum. By operating the camera at 
a frequency only shghtly different from the res- 
onant frequency, a stroboscopic effect enables 
you to see the rapid vibrations as if in slow mo- 
tion. 

In the first part of the film, the loudspeaker 
is directly under the membrane, and the vi- 
bratory patterns are symmetrical. In the fun- 
damental harmonic, the membrane rises and 
falls as a whole. At a higher frequency, a sec- 
ond circular node shows up between the center 
and the rim. 

In the second part of the film, the speaker 
is placed to one side, so that a different set of 
modes, asymmetrical modes, are generated in 
the membrane. You can see an antisymmetri- 
cal mode where there is a node along the 
diameter, with a hill on one side and a valley on 
the other. 

Various symmetric and antisymmetric vi- 
bration modes are shown. Describe each mode, 
identifying the nodal lines and circles. 

In contrast to the one-dimensional hose in 
Film Loop 41 there is no simple relation of 
the resonant frequencies for this two-dimen- 
sional system. The frequencies are not integral 
multiples of any basic frequency. There is a 
relation between values in the frequency 
spectrum, but it is more complex than that 
for the hose. 

FILM LOOP 43 VIBRATIONS 
OF A METAL PLATE 

The physical system in this film is a square 
metal plate. The various vibrational modes 
are produced by a loudspeaker, as with the 
vibrating membrane in Film Loop 42. The 
metal plate is clamped at the center, so that 
point is always a node for each of the standing- 
wave patterns. Because this is a stiff metal 
plate, the vibrations are too shght in ampli- 
tude to be seen directly. The trick used to make 
the patterns visible is to sprinkle sand along 
the plates. This sand is jiggled away firom the 
parts of the plates which are in rapid motion, 
and tends to fall along the nodal Unes, which 
are not moving. The beautiful patterns of 
sand are known as Chladni figures. These 
patterns have often been much admired by 



Unit 3/96 



Film Loop Notes 






artists. These and similar patterns are also 
formed when a metal plate is caused to vibrate 
by means of a violin bow, as seen at the end of 
this film, and in the Activity, "Standing Waves 
on a Drum and a Viohn." 

Not all frequencies will lead to stable pat- 
terns. As in the case of the drum, these har- 



monic frequencies for the metal plate obey 
complex mathematical relations, rather than 
the simple arithmetic progression seen in a 
one-dimensional string. But again they are 
discrete events. As the frequency spectrum is 
scanned, only at certain sharp well-defined fre- 
quencies are these elegant patterns produced. 




Answers to End-of-Section Questions (Continued) 



Q10 Both will increase. 

Q11 Answer (c) 

Q12 Answer (a) 

Q13 Answers a, b. c are correct 

Q14 (a) unbroken egg 

(b) a glass of ice and warm water 
Q15 (a) True 

(b) False 

(c) False 
Q16 Answer (b) 

Chapter 12 

Q1 Transverse, longitudinal and torsional 
Q2 Longitudinal. Fluids can be connpressed but 
they are not stiff enough to be bent or twisted. 
Q3 Transverse 

Q4 No. The movement of the bump in the rug 
depends on the movement of the mouse; it does not 
go on by itself. 

Q5 Energy (Particles of the medium are not trans- 
ferred along the direction of the wave motion.) 
Q6 The stiffness and the density 
Q7 (1) Wavelength, amplitude, polarization 

(2) Frequency, period 

Q8 The distance between any two successive points 
that have identical positions in the wave pattern. 
Q9 (1) 100 cps 

1 ^ 1 

/ ~ 100 cps 

i'i\ X -^ 10 m/sec - ^ 

(3) ^ - 7 = -rp^ = 0-1 meter 

f 100 cps 

Q10 Answer (b) 

Q11 A, + A, 

Q12 Yes. The resulting displacement would be 

5 + (-6) = -1 cm 

Q13 Cancellation 

Q14 Antinodal lines are formed by a series of 

antinodal points. Antinodal points are places where 

waves arrive in phase and maximum reinforcement 

occurs. (The amplitude there is greatest.) 

Q15 Answer (a) 

Q16 When the difference in path lengths to the two 

sources is an odd number of half wavelengths (^\, 



(2) T 



0.01 sec. 



3 5 
Q17 



(1) No motion at the nodes 

(2) Oscillates with maximum amplitude 
\ 
2 
2L, so that one-half wavelength just fits on 

the string. 

Q20 No, only frequencies which are whole number 

multiples of the fundamental freqi>ency are possible. 



Q18 
Q19 



Q21 All points on a wave front have the same phase; 
that is, they all correspond to crests or troughs (or 
any other set of similar parts of the wavelength 
pattern). 

Q22 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, 
Q23 If the opening is less than one-half a wave- 
length wide the difference in distance to a point P 
from the two edges of the opening cannot be 
equal to x/2. 

Q24 As the wavelength increases, the diffraction 
pattern becomes more spread out and the number of 
nodal lines decreases until pattern resembles one 
half of that produced by a point source oscillator. 
Q25 Yes to both (final photograph shows diffraction 
without interference; interference occurs whenever 
waves pass each other). 

Q26 A ray is a line drawn perpendicular to a wave 
front and indicates the direction of propagation of 
the wave. 

Q27 The angles are equal. 
Q28 Parabolic 

Q29 The reflected wave fronts are parallel wave 
fronts. 
Q30 (1) Stays the same 

(2) Becomes smaller 

(3) Changes so that the wave fronts are more 
nearly parallel to the boundary. (Or its 
direction of propagation becomes closer to 
the perpendicular between the media.) 

Q31 




Q32 



033 



(1) /\ V relationship 

(2) Reflection 

(3) Refraction 

(4) Diffraction 

(5) Interference 

Sound waves are longitudinal. 



A1 



Brief Answers to Study Guide Questions 



Chapter 9 

9.1 Information 

9.2 Discussion 

9.3 (a) Yes 

(b) The solar system 

9.4 Discussion 

9.5 No 

9.6 Discussion 

9.7 (a) 220.2 g 

(b) 20.2 g 

9.8 Derivation 

9.9 (a) All except v^a' (which = Vu') 

m.K + mn 

(c) 0.8 m/sec 

9.10 Dictionary comment 

9.11 3.3 X 10 « kg 

9.12 Discussion 

9.13 Derivation 

9.14 Discussion 

9.15 (a) 0.2 sec 

(b) About 0.05 m 

(c) 5 X 10-1' m/sec 
(d)2.5 X 10-15 m 

(e) About 15 X 1 0*^ m2 or a square of about 
40 km on a side 

9.16 Yes 

9.17 Derivation 

9.18 Discussion 

9.19 1.2 X 10^ kg m/sec; 4 x 10^ newtons; 
30 meters 

9.20 (a) about 100 m/sec 

(b) about 4.6 kg m/sec 

(c) less than 0.003 sec 

(d) at least 1.5 x 10' newtons 

9.21 Yes 

9.22 Derivation 

9.23 (a) M = — ^—= •' 

(b) m {v., — v) 

9.24 Derivation 

9.25 10 m/sec 

9.26 10.5 x 10** kg m/sec 

9.27 Discussion 

9.28 Discussion 

9.29 Discussion 

9.30 Discussion 

9.31 Discussion 

9.32 (a) 0.8 x mass of ball 

(b) -0.8 X mass of bail 

(c) 1.6 X mass of ball 

(d) Depends on system considered 

9.33 Discussion 

9.34 Derivation 

9.35 Table 

9.36 Derivation 



9.37 Both speeds = - but in opposite directions 



9.38 


Discussion 


9.39 


Discussion 


Chapter 10 


10.1 


Information 


10.2 


Discussion 


10.3 


(a) V, - u and v^- u 




(b)No 




(c) No 




(d)No 




(e) Yes 




(f) iii 




(g) Discussion 


10.4 


5 X 10 1-^ joules, 2 X 101' electrons 


10.5 


(a) 67.5 joules 




(b)4.5 X 100 joules 




(c) 3.75 X 103 joules 




(d)2.7 X 1033 joules 


10.6 


(a) 2 m/sec2, 30 sec, 60 m/sec 




(b) 60 m/sec 


10.7 


(a) -90 joules 




(b) 90 joules 




(c) 18 X 10- newtons 


10.8 


2.3 X 102 joules 


10.9 


(a) 2.2 X 10-3 joules 




(b) 5.4 X 10-2 joules 


10.10 


(a) 0.2 meter 




(b) 7 X 109 joules 


10.11 


Discussion 


10.12 


Discussion 


10.13 


(a) 1.1 X 10'2 seconds (about 3 x 10* years) 




(b) 1.6 X 10--'s meters 


10.14 


Discussion 


10.15 


Discussion 


10.16 


Derivation 


10.17 


Discussion 


10.18 


Sketch 


10.19 


Proof 


10.20 


(a) 96 X 10^ joules 




(b) 8.8 X 102 meters 




(c) 48 X 10^ newtons 




(d) Discussion 




(e) Discussion 


10.21 


Discussion 


10.22 


Discussion 


10.23 


Discussion 


10.24 


Discussion 


10.25 


(b) 


10.26 


No 


10.27 


(a) >1000 




(b) Discussion 


10.28 


1/8" C; no 


10.29 


Rowing 1375 watts or 1 .8 H.P. 


10.30 


1/4 kg 


10.31 


21.5 days 



A2 



10.32 Discussion 

10.33 Discussion 

10.34 Discussion 

10.35 Discussion 

10.36 Discussion 

10.37 (a) Discussion 

(b) Greater in lower orbit 

(c) Less 

(d) Less 

(e) Discussion 

10.38 (a) Discussion 

(b) i: all three, ii: all three, iii: AH, iv: ah, 
v: all three, vi: AH 

10.39 Discussion 

10.40 Discussion 



Chapter 11 

11.1 Information 

11.2 Discussion 

11.3 Discussion 

11.4 Discussion 

11.5 No 

11.6 Discussion 

11.7 (a)10-"m 
(b) 10 9 m 

11.8 (a) 102' 
(b) IQis 

11.9 Zero meters 

11.10 10.5 kilometers 

11.11 Shoes— about 1/7 atm 
Skis — about 1/60 atm 
Skates — about 3 atm 

11.12 Derivation 

11.13 Discussion 

11.14 Discussion 

11.15 Discussion 

11.16 Derivation 

11.17 No change 

11.18 Pressure, mass, volume, temperature 

11.19 Discussion 

11.20 Discussion 

11.21 Discussion 

11.22 Discussion 

11.23 Discussion 

11.24 Discussion 

11.25 Discussion 

11.26 Temperature will rise 

11.27 No 

11.28 Discussion 

11.29 Discussion 

11.30 Discussion 

11.31 Discussion 

11.32 Discussion 

11.33 Discussion 

11.34 Discussion 

11.35 Discussion 



Chapter 12 

12.1 Information 

12.2 Discussion 

12.3 Discussion 

12.4 Construction 

12.5 Construction 

12.6 Discussion 

12.7 Discussion 

12.8 Construction 

12.9 Discussion 

12.10 Discussion 

12.11 Construction 

12.12 Derivation 

12.13 No; discussion 

12.14 (a) 3/4/. 
{b)2/3L 
(c) 1/2 L 

12.15 (a) \ = 4L 

(b)X= '' 



4L 
n + l 



12.16 
12.17 
12.18 
12.19 
12.20 
12.21 
12.22 

12.23 
12.24 

12.25 
12.26 
12.27 



2n + / 

(c) X = 2L, \ = 

Discussion 

Maximum 

100 and 1000 cps; yes 

Discussion 

Discussion 

Construction 

Straight line 

R 

2 

Construction 

_1_ 

4/c 

Discussion 



(n = 0, 1, 2, 3, etc.) 



12.29 



Two straight-line waves inclined toward 
each other. 
12.28 Discussion 

(a) ^A = Z BAD 

(b) Ok = ZCDA 

(c) Xa = BD 

(d) Xn = AC 

(e) Derivation 

(f) Derivation 
Xp = 0.035 m 
Xs = 0.025 m 
Discussion 

(a) 1.27 X 10" watts 

(b) 8 X 10'- mosquitoes 

(c) subway train 
2d = vt 

I air 1.125 ft 
1000 cps: I sea water 4.8 ft 
( steel 16 ft 
One tenth of each of these values for 10,000 cps 
Discussion 
12.35 3 X 10'' cps 
2.5 X 10" cps 



12.30 

12.31 
12.32 



12.33 
12.34 



A3 



INDEX/TEXT 



Absolute temperature scale, 81 

Acoustics, 132-133 

ADP (adenosine diphosphate), 52 

Aeolipile, 40 

Air pressure, and mercury barometer, 79 

Anechoic chambers, 132 

Antinodal lines, definition, 113 

ATP (adenosine triphosphate), 52 

Average values, 73 

Bernoulli, Daniel, and kinetic theory of 

gases, 74 
Biological systems, energy in, 51-55 
Biophysics, 54 
Boltzmann, Ludwig, and irreversibility, 

89, 94 
Boyle, Robert, mechanistic viewpoint, 2 
Boyle's law, and kinetic theory of gases, 

74, 80 
British Thermal Unit (BTU), 51 
Brown, Robert, and Brownian 

movement, 84 
Brownian movement, 84, 94 

Caloric theory, 50 

Calorie, definition, 53 

Carbohydrates, 52 

Camot, Sadi, and engine efficiency, 

86-87 
Celestial equator, definition, 10 
Chemical energy in food, 52-53 
Circular reflectors, 124-125 
Clausius, Rudolf, and entropy, 87 

gas molecule size and, 76, 78 
Collision(s), and conservation of 
momentum, 9-16 

elastic, 19-20 

gas molecule, 71-72, 76, 78 

in two dimensions, 23 
Conservation of energy, 56-63 

neutrino and, 62 
Conservation of kinetic energy, 20 
Conservation of mass, 5-8 
Conservation of mechanical energy, 

34-36 
Conservation of momentum, 10-16, 61 

Newton's laws and, 15-17 
Conservation laws in science, 62-63 

Descartes, Rene, 21 
Diffraction, definition, 120 

wave, 120-122, 132 
Disordered motion, definition, 72 
Displacement, waves of, 102-103 
Doppler effect, 130 
Drum, vibration of 119 
Duty, of electric motors, 56 

Earth, mass of 8 
as open system, 8 



energy conservation on, 59 
Efficiency, definition, 51, 86 
Einstein, Albert, and kinetic theory, 94 
Elastic collision(s), 19-20 
Elastic potential energy, 33 
Electric battery, 56 
Electric current(s), transfer of, 56 
Electric motor, duty of, 56 
Electric potential energy, 33 
Electromagnetic induction, 56 
Energy, in biological systems, 51-55 

chemical, 52-53 

conservation of 34-36, 56-63 

conversion of 39-43 

dissipation of, 85-88 

heat, 39-42 

internal, 60 

mechanical, 34-36 

potential, 31, 32-33 

rates for human use of, 53 

transfer of 105 

transformation of 52-53, 56-58 

work and, 29-31 
Engine(s), efficiency of, 86-87 

see also Steam engine(s) 
Entropy, concept of, 87 
Experimental Philosophy (Henry 
Power), 134 

Faraday, Michael, and electromagnetic 

induction, 56 
Fat(s), as energy source, 52 
Feedback, definition, 42 
First Law of Thermodynamics, 61 
Food(s), as energy source, 52-53 

world supply of 53-55 
Foot pound, definition, 45 
Force(s), gas pressure and, 79 

work and, 37-38 

see also Collision(s), Motion 
Fourier, Jean-Baptiste, and 

superposition principle, 110 
Frequency, see Wavelength 

Gas(es), density of 79-80 

expansion of, 80 

kinetic theory of 76, 78 

nature of 71-72 

predicting beha\aor of, 79-85 

volume of 79-80 
Gas molecule(s), motions of, 71-72 

size of, 76, 78 

speeds of, 74-77 
Gas particle(s), see Gas molecule(s) 
Gas pressure, 79- 85 

effect of temperature on, 80-81 

kinetic explanation of 81-85 
Gaseous state, model for, 71-72 
Gay-Lussac, Joseph- Louis, and gas 
volume, 80 



Glucose molecule, 52 

Goethe, Johann Wolfgang von, and 

nature philosophy, 57 
Gravitational potential energy, 34 
Guitar string, vibration of, 35 

Harmony, and standing waves, 117-118 
Heat, as energy, 39-42, 60-61, 69-70 

loss of, 86-87 

mechanical equivalent of, 51 

as transfer of energy, 60-61 
Heat death idea, 88 
Heat energy, 39-42 
Helmholtz, Hermann von, and 

conservation of energy, 58 
Herapath, John, and kinetic theory of 

gases, 74 
Horse power, definition, 45- 46 
Huygens, Christian, and elastic 
collisions, 20-21 

wave diffraction and, 120-121 
Huygens' principle, 120-121 

Ideal gas law, 80-81 

Industrial Revolution, and steam 

engine, 46-47 
Interference, sound wave, 132 
Interference pattern, 110—115 

definition, 110 
Irreversible processes, 87-88 

see also Second Law of 
Thermodynamics 
Isolated (closed) systems, 6-8, 18 

energy conservation and, 61 

Joule, definition, 31, 51 

Joule, James Prescott, 49-51 
conser\'ation of energy and, 56 
duty of electric motors and, 56 
heat energy experiments of, 50-51 
kinetic theory of gases and, 74 

Kehin, Lord, 93 

absolute temperature scale and, 
81 

atom size and, 78 

energy loss and, 87 

reversibility paradox and, 91-92 
Kelvin scale, 81 

Kilocalorie(s), daily human requirement 
of 53 

definition, 51 
Kinetic energy, conservation of 20 

definition, 31 

work and, 29-31 
Kinetic theory, criticisms of 94, 96 
Kinetic theory of gases, gas model, 
71-72 

gas pressure and, 81-85 

history of 69-70 



A4 



molecule sizes and, 76, 78 
molecule speeds and, lA-11 
Newtonian mechanics and, 69-70 
predicting gas behavior and, 79-85 

Lavoisier, Antoine, and conservation of 

mass, 6-7 
Law(s), conservation, 63 

see also individual laws 
Law of Conservation of Energy, 20, 

56-63 
Law of Conservation of Mass, 7-8 
Law of Conservation of Mechanical 

Energy, 35-36 
Law of Conservation of Momentum, 

10-14 
Leibnitz, Gottfried Wilhelm, and kinetic 

energy, 21-22 
Lift pump, and air pressure, 79 
Longitudinal wave, definition, 103 
Loschmidt, Josef, and reversibility 

paradox, 91 

Machine, see Engine 
Magnetic energy, transfer of, 56 
Magnetic potential energy, 33 
Mass, conservation of, 5-8 
Matter, gaseous state of, 71-72 

states of, 71 
Maxwell, James Clerk, 91 
molecular velocities in a gas and, 
75-76 
Maxwell velocity distribution, 75-76 
Maxwell's demon, 89-90 
Mayer, Julius Robert, and conservation 

of energy, 57 
Mechanical energy, conservation of, 
34-36 

heat energy and, 40-43 

loss of, 86-87 
Mechanical equivalent of heat, 51 
Mechanical waves, 101 
Mechanics, Newtonian, 113 
Mercury, effect of air pressure on, 79 
Mersenne, Martin, and speed of sound, 

129, 132 
Mitochondria, 52 
Mks system, 51 
Model, gaseous state, 71-72 

types of, 71 
Molecule(s), definition, 69 

gas, 71-72, 74-77 
Momentum, conservation of, 10-16, 61 

definition, 10 

see also Motion 
Motion, disordered, 72 

momentum and, 15-17 

Newton's theory of, 15-17 

quantity of, 9-10 

see also Momentum 



Musical instruments, vibrations of, 35, 
117-119 

Nature Philosophers, and conservation 

of energy, 57-58 
Neutrino, discovery of, 62 
Newcomen steam engine, 41-44 
Newton, Isaac, 1-2 
conservation of momentum and, 

15-17 
Newtonian mechanics, and kinetic 

theory, 69-70 
Newtonian physics, 134-135 
Newtonian world machine, 1-2 
Nietzsche, Friedrich, and recurrence 

paradox, 93 
NQdal Unes, definition, 113 
Normal distribution law, 75 

Oersted, Hans Christian, and energy 

transfer, 56 
Open system, and energy conservation 

61 
Orbit(s), satellite, 37 

Parabolic reflectors, 125 
Parsons turbine, 47-49 
Particle point of view, 101-102 
Pauli, Wolfgang, and neutrino, 62 
Periodic vibration, 106-107 
Periodic wave(s), 106-109 

definition, 106 

snapshot of, 108 

speed of, 107-108 
Photosynthesis, 52 
Poincare, Henri, and recurrence 

paradox, 93-94 
Polarized wave, 104 
Potential energy, forms of, 31, 32-33 
Power, Henry, and gas pressure, 79 

Newtonian physics and, 134 
Pressure, air, 79 

definition, 74, 79 

force and, 79 

gas, 79-84 
Principia (Newton), 3 
Principles of Philosophy (Descartes), 21 
Propagation, of waves, 104-106, 

126-127 
Proteins, as energy source, 52 
Pulse, wave, 102 

Quantity of motion, 9-10 

Recurrence paradox, 91, 93 
Reflection, wave, 122-125, 132-133 
Refraction, definition, 127 

wave, 126-128, 132 
Reversibility paradox, 91-93 



Satellites, in orbit, 37 
Savery steam engine, 40-41 
Schelling, Friedrich von, and nature 

philosophy, 57 
Science, role in food production, 53-55 
Second Law of Thermodynamics, 87-90 

as statistical law, 88-90 
Simple harmonic motion, 107 
Sine wave, 107 
Sonar, 132 

Sonic boom, 130-131 
Sound, loudness of, 129 
Sound waves, definition, 128-129 

snapshot representation of, 103 

speed of, 129, 132 
Speed-time graph, 25-27, 29-30 
Standing (stationary) waves, 115-118 
Star(s), globular clusters of, 1 
Stationary (standing) waves, 115-118 
Steam engine(s), 39-49, 56 

efficiency of, 86 

Newcomen, 41-44 

Savery, 40-41 

Watt, 44-46 
Steam turbine, 47-49 
Superposition principle, 109-110 

Theoretical model, definition, 71 
Thermodynamics, definidon, 61 

First Law of, 61 

Second Law of, 87-90 
Thomson, William, see Kelvin 
Time line. Watt, James, 95 
Torricelll's barometer, 79 
Torsional wave, definition, 103 
Towneley, Richard, and gas pressure, 79 
Transverse wave, definition, 103 

polarization and, 104 

Vibrations, of musical instruments, 
117-119 
periodic, 107-108 
Volta, Alessandro, and electric battery, 
56 

Water, boiling point of, 80 

fireezing point of 80 
Watt, definition, 45 
Watt, James, 43-49 

time line, 95 
Wave(s), calculating wavelength of, 
115-116 

definition, 105 

diffracted, 120-122 

displacement, 102-103 

harmony and, 117-118 

interference pattern of 110-115 

longitudinal, 103 

mechanical, 101 

periodic, 106-109 



A5 



polarized, 104 

propagation of, 104-106, 126-127 

properties of, 102-105 

pulse, 102 

reflection of, 122-125 

refraction of, 126-128 

sine, 107 

snapshots of, 103 

sound, 103, 128-135 

speed of, 126-128 

standing, 115-118 



stationary, 115-118 

superposition principle of, 109-110 

torsional, 103 

transverse, 103-104 

types of, 103 
Wave front, definition, 120 

diffraction patterns and, 120-122 
Wave reflectors, 124-125 
Wavelength, calculation of, 115-116 

of periodic wave, 107 
Work, concept of, 29-31 



definition, 29-31 
energy and, 29-31 
energy transformation and, 56, 60-61 
force and, 37-38 
kinetic energy and, 29-31 
on a sled, 32 

as transfer of energy. 56, 60-61 
Working model, definition, 71 

Zermelo, Ernst, and kinetic theory, 
91-94 



INDEX/HANDBOOK 



Activities 
conservation of mass, 56 
diver in a bottle, 60-62 
exchange of momentum devices, 

56-57 
ice calorimetry, 37-38 
measuring speed of sound, 66-67 
mechanical equivalent of heat, 60 
mechanical wave machines, 67 
moire patterns, 64-66 
music and speech, 66 
perpetual motion machines, 62-63 
predicting the range of an arrow, 59 
problems of scientific and 

technological growth, 58-59 
rockets, 62 
standing waves on a drum and violin, 

63-64 
steam-powered boat, 57-58 
student horsepower, 57 
weighing a car with a tire pressure 
gauge, 62 
Air, as a gas, 44-45 
Air track in bullet speed experiment, 
26-27 
and collisions (experiment), 4-5 
inclined, in energy conservation 
experiment, 24-25 
Aircraft takeoff, and conser\ation of 

energy (film loop), 89 
Alka-Seltzer, in mass conservation 

activity, 56 
Antinodal lines, 50 
Arrow, predicting range of (activity), 59 

Ballistic pendulum, 81 

in bullet speed experiment, 27 
Billiard ball, dynamics of (film loop), 
84-85 



Biography of Physics, The (George 

Gamow), 60 
Bullet, measuring speed of 

(experiment), 26-28 
(film loop), 81-82 

Calorie, defined, 34-35 
Calorimeter, 33-34 
Calorimetry (experiment), 33-38 

of ice, 37-38 
"Can Time Go Backward?" (Martin 

Gardner), 90 
Cannon, recoil velocity of, 82-83 
Car, weighing with tire pressure gauge 
(activity), 62 

Cartesian diver (activity), 60-62 
Celsius (centigrade) scale, 31-32 
Centigrade scale, 31-32 
Chain, in least energy experiment, 

29-30 
Colliding disk magnets (experiment), 14 
Colliding freight cars (film loop), 83-84 
Colliding pucks (experiment), 13 
Collisions, elastic, 4 

inelastic. 4, 8-9 

in one dimension (experiment), 4-12 

in two dimensions (experiment), 
13-21 

perfectly inelastic. 8-9 
Collision rules of a particle. 40-41 
Collision squares, mean free 

path between (experiment), 40-41 
"Computer Music." Scientific American, 

66 
Conservation of energy (experiment). 
22-26 

and iurcraft takeoff (film loop). 89 

and pole vault (film loop), 87-88 
Conservation of mass (activity), 56 



Conservation of momentum, 13 
Constant pressure gas thermometer. 
44-46 

Descartes, Rene, 60 
Diffraction, wave, 49, 52 
Disk magnets, and conservation of 
energy experiment, 23-24 
and two-dimensional collision 
experiment, 14 
Diver in a bottle (activity). 60-62 
Drinking duck (activity), 59-60 
Drum, vibrations of (film loop), 95 
Dynamics carts, and conservation of 
energ>' experiment. 22-23 
and "explosions'" experiment, 4 
Dynamics of a billiard ball (film loop), 
84-85 

Elastic coUision. 4 
Elastic potential energy, 59 
Energy, conservation of (experiment), 
22-26 

(film loops). 87-89 

elastic potential. 59 

gravitational potential. 86-87 

heat, 34-35 

see also Calorie 

kinetic, 81 

(film loop), 81-87 

mechanical, measurement of. 85-86 

potential, 29-30. 82 
Exchange of momentum de\ices 

(activity), 56-57 
Experiments 

behavior of gases. 43-46 

calorimetry, 33-37 

collisions in one dimension, 4-12 

collisions in two dimensions. 13-21 



A6 



conservation of energy, 22-26 
energy analysis of a pendulum swing, 

28-29 
ice calorimetr>', 37-38 
least energy, 29-30 
measuring the speed of a bullet, 

26-28 
measuring wavelength, 49-51 
Monte Carlo experiment on molecular 

gases, 38-43 
sound, 51-53 
temperature and thermometers, 

31-32 
ultrasound. 53-55 
wave properties, 47-48 
waves in a ripple tank, 48-49 
Explosion of a cluster of objects (film 

loop), 79-80 

Film loops 
colliding freight cars, 83-84 
conservation of energy - aircraft 

takeoff, 89 
conservation of energy - pole vault, 

87-88 
dynamics of a billiard ball, 84-85 
explosion of a cluster of objects, 

79-80 
finding the speed of a rifle bullet 1 and 

II, 81-82 
gravitational potential energy, 86-87 
inelastic two-dimensional coUisions, 

77, 78-79 
kinetic energy, 81, 87 
method of measuring energy - nails 

driven into wood, 85-86 
one-dimensional coUisions I and II, 

76-77 
recoil, 82-83 
reversibility of time, 90 
scattering of a cluster of objects, 79 
standing waves in a gas, 92-93 
standing waves on a string, 91-92 
superposition, 90-91 
two-dimensional collisions 1 and II, 

77-78 
vibrations of a drum, 95 
vibrations of a metal plate, 95-96 
vibrations of a rubber hose, 94-95 
vibrations of a wire, 93-94 
Frequencies, 94-95 

Gas(es), behavior of (experiment), 
43-46 
temperature of 44 
volume and pressure (experiment), 

43-44 
volume and temperature 
(experiment), 44 
Gas thermometer, 44-46 



Gay-Lussac's Law, relation between 
temperature and volume, 43 

Gravitational potential energy (film 
loop), 86-87 

Hanging chain, 29-30 
Heat, conduction of 38 

exchange and transfer, 38 

of fusion of ice, 37-38 

latent, of melting, 36 

mechanical equivalent of (activity), 60 

radiation, 38 
Heat capacity, measurement of 35-36 
Hooke's law, 59 
Horsepower, student (activity), 57 

Ice, calorimetry (experiment), 37-38 

heat of fusion of, 37-38 
Incbned air track in energy conservation 

experiment, 24-25 
Inelastic collision, 8-9 
Inelastic two-dimensional collisions 

(film loop), 77-79 
Interference, wave, 48-49 
Interference pattern, wave, 49-50 

Kinetic energy (film loop), 81-87 
see also Energy 

Latent heat of melting, 37-38 
Liquids, mixing hot and cold 
(experiment), 34-35 
Longitudinal wave, 47 

Magnesium flash bulb, in mass 

conservation activity, 56 
Marbles, collision probability for a gas of 

(experiment), 38-39 
inferring size of, 39-40 
Mass, computation of force from, 43 

conservation of (activity), 56 
Mean free path between coUision 

squares (experiment), 40-41 
Measurement(s), speed of a buUet 

(experiment), 26-28 
Mechanical energy, measurement of 

(film loop), 85-86 
Mechanics of the Pole Vault (R. V. 

Ganslen), 88 
Metal plate, vibrations of (film loop), 

95-96 
Moire patterns (activity), 64-66 
"Moire Patterns" (G. Oster and Y. 

Nishijima), 66 
Molecular gases, Monte Carlo 

experiment, 38-43 
Momentum, conservation of 13-15, 

26-28, 85 
measurement of 76-77, 78 
Motion, direction of, 14-15 



Music and speech (activity), 66 

Nails, in measurement of kinetic energy, 
87 
in measurement of mechanical 
energy, 85 
Nodal lines, 50 
Nodes, 50 

One-dimensional collisions I and II (film 
loop), 76-77 

Pendulum, ballistic, 81 

Perfectly inelastic collision, 8-9 

Periodic wave, 48 

Perpetual Motion and Modem Research 

for Cheap Power (S. R. SmedUe), 

62 
Perpetual motion machines (activity), 

62-63 
"Physics and Music," Scientific 

American, 64, 66 
"The Physics oiViohns," Scientific 

American, 66 
"The Physics of Woodwinds, "Sc?entiyic 

American, 66 
Pole vaulter, 87-88 
Potential energy, 29-30, 82 
Precipitate, in mass conservation 

activity, 56 
Pressure, of gas, 43-44 
Puck(s), in conservation of energy 

experiment, 23-24 
in two-dimensional collisions 

experiment, 13 
Pulses, 47-48 

Radiation, heat, 38 
Random numbers, 39-40 

table of 42 
Recoil (film loop), 82-83 
Reflection, wave, 48, 51 
Refraction, wave, 48 
Rifle bullet, finding speed of (film loop), 

81-82 
Ripple tank, waves in, 48-49 
Rockets (activity), 62 
Rubber hose, vibrations of (film loop), 

94-95 

Scattering of a cluster of objects (film 

loop), 79 
Science of Moire Patterns (G. Oster). 66 
"Science of Sounds" (Bell Telephone), 

66 
Scientific and technological growth. 

problems of (activity), 58-59 
Seventeenth-centurv' experiment, 

56-57 
Similarities in Wave Behavior (J. N. 



A7 



Shrive), 67 
Sinusoidal wave patterns, 90-91 
Slinky, 67 

Snell's law of refraction, 60 
Sound (experiment), 51-53 

calculating speed of, 53 
Sound waves, diffraction of, 52 

interference, 52 

pressure variations in, 53 

reflection, 51 

refraction, 52 

standing wave, 52 

transmission of 51-52 

see also Waves 
Specific heat capacity, 36 
Spectrum(a), of wave frequencies, 

93-94 
Speech and music (activity), 66 
Speed, of a bullet, 26-28 
(film loop), 81-82 

of sound, 53 

(activity), 66-67 

of ultrasound, 55 

see also Velocity 
Spring, waves in, 47-48 
Standing sound waves, 52, 54 
Standing waves, on a drum and violin, 
63-64 

in a gas (film loop), 92-93 

on a string (film loop), 91-92 
"Stanzas firom Milton" (William Blake), 
58 



Steam-powered boat, construction of 
(activity), 57-58 

Stroboscopic photographs, of one- 
dimensional collisions, 5-6, 
25-26 
of two-dimensional collisions 

(experiment), 16-21, 25-26 

Stroboscopic Still Photographs ofTwo- 
Dimensional Collisions, 79 

Superposition (film loop), 90-91 

Temperature, defined, 31-32 

of a gas, 44 

and thermometers (experiment), 
31-32 
Thermodynamics, laws of, 62 
Thermometer(s), comparison of, 32-33 

constant pressure gas, 44-46 

making of, 31-32 

and temperature (experiment), 31-32 
Thin-film interference (activity), 32 
Time, reversibility of (film loop), 90 
Transmission, of sound waves, 51-52 
Transverse wave, 47 
Two Cultures and the Scientific 

Revolution (C. P. Snow), 58 
Two New Sciences (Galileo), 56 
Two-dimensional collisions I and II (film 
loop), 77-78 

(experiment), 16-21 

stroboscopic photographs of, 16-21, 
25-26 



Ultrasound (experiment), 53-55 
speed of, 55 

Vectors, 17, 76 

quantity, 24 
Velocity, 13-15, 76-77, 84 

recoil, 82-83 

see also Speed 
Violin, vibrations of, 63-64 
Volume, and pressure (experiment), 
43-44 

and temperature (experiment), 44-45 

Water, measuring temperature of 

(experiment), 33-38 
Waves, diffraction, 49 

properties (experiment), 47-48 

reflection, 48 

refraction 48 

in ripple tank (experiment), 48-49 

sinusoidal, 90-91 

sound, see Sound waves 

in spring, 47-48 

standing, 49 

transverse, 47 
Wave machines, mechanical (activity), 

67 
Wavelength, measurement of 

(experiment), 49-51 
Wire, vibrations of (film loop), 93-94 



A8 




ISBN 0-03-089638-x