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Full text of "Project Physics Resource Book"

PROJECT 
PHYSICS 



RESOURCE BOOK 




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PROJECT 
PHYSICS 

RESOURCE BOOK 




Digitized by the Internet Archive 

in 2010 with funding from 

F. James Rutherford 



http://www.archive.org/details/projectphysicsreOOfjam 



PROJECT 
PHYSICS 

m RESOURCE BOOK 




Diret;lorfi of Harvard Project Physics 

F. James Kutherford 

Department ot Science Education, New York University 

Gerald llulton 

U(;|)at1in('iit of fMivsics, Haivard Universitv' 

Fletcher ii. IV'atson 

Harvaixi Graduate School of Education 



Edilorinl Dcvcloiynw.nt: William \. Moore, Roland ComiitM , Lorraine Smith-Phelan 

Editorial Processing,: Maigaiet M. Byrne, Regina Chiicoat, HolK' Massey 

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

Annette Sessa, Beverly SiKei', Anita Dickhuth, I3orina \'irxlo 
Product Manager: Laura Zuckennan 
Advisory Board: John Tiiggatl, Maurice E. Fey, Xorman Hughes, Da\id J Miller, John W. Griffiths, 

William L. Paul 
Consultant: John Matejowsky 
Researchers: Pamela Floch, Gerard LeV'an 



(opvnuhl © 198L 1!)75. 1!I70 In l»n)jr«l Pinsics 

All Rights Hcsenrd 

l»i-iiUc(l in Ihp linitrd Stales of Atncnrii 

ISBN D-03-QSS1S1-X 
567-014-9876543 

l'n)ii'( I Physics is a regist»>ifci tr^uliMD.ir k 



IntroduGtioii 



General Background 

The Project Phxsics Course is based on the ideas 
and research results of the Han ard Project Phxsics 
curriculum de\elopment group. This national course 
Lmpix3\ement effort fomialh' began in the spring of 
1964. At that time Gerald Holton, James Rutherford, 
and Fletcher W atson of Har\ aixl L'ni\ei-sif\ recei\ed 
support from the United States Office of Education 
and the National Science Foundation, which ena- 
bled them to bring together professional people 
firom all parts of the nation to work on the im- 
provement of physics education. 

Informalh', the Project had started se\'eral vears 
earlier, \%hen Rutherford \\ as a phxsics teacher and 
science department head in a public high school. 
Holton and Watson agreed to collaborate with him 
in testing the feasibility' of designing a new ph\sics 
course. With the stor\'-line and aims in Gerald Hol- 
ton s college text, Introduction to Concepts and 
Theories in Physical Science, as a general guide, 
preparation of a course outline and instructional 
materials was begun. In 1962, the founders ob- 
tained initial support from the Carnegie Corpora- 
tion in New York, which allowed them to test their 
materials. The success of these tests, coupled with 
the increasing national awareness that something 
needed to be done about decreasing high school 
physics enrollments, led to the formation of Har- 
vard Project Physics. The decision to expand to a 
national program was stimulated b\ a request from 
the National Science Foundation late in 1963. 

The general purposes of Project Phxsics re- 
mained constant from the beginning, when thi^e 
indi\iduals worked without support, through the 
time of peak developmental acti\it\' inxolving 
hundreds of scientists, teachers, psychologists, art- 
ists, and other professional participants from 
throughout the United States and Canada, as well 
as thousands of students in trial classes. To some 
degree, the purposes reflected the fact that the di- 
rectors of the Project were, respectixeh', a uni\er- 
sit\' physicist, a professor of science education, and 
an experienced high-school phxsics teacher. The 
chief purposes \%ere: 

1. To design a hun^anistically oriented physics 
course. Hanard Project Ph\sics would show the 
science of physics in its proper light as a broadlx 
based intellectual acti\it\' that has firm historical 
roots and that profoundly influences our whole 
culture. 

2. To develop a course that would attract a large 
number of high school students to the study of in- 
troductory physics. Such a course must be mean- 



ingful not only to those who are already intent on 
a scientific career, but also to those who may not 
go on to college and to those who, while in college, 
will concentrate on the humanities or social 
sciences. 

3. To contribute to the knowledge of the factors 
that influence science learning. In addition to its 
long-term \alue. this extensi\e educational re- 
search should suppK' information needed by 
teachers and administrators in deciding whether 
to introduce the course and, if so, in what way and 
for which students. The research results ha\e been 
reported in professional journals and dissertations, 
and in the book A Case Stud\- in Curriculum Eval- 
uation: Harvard Project Physics. 



Specific Goals of the 
Project Pht'sics Course 

The first two general aims of Han ard Project Phys- 
ics I to de\elop a humanisticalK' oriented ph\sics 
course, and to help increase high school ph\sics 
enrollments! can be restated in somewhat more 
specific terms. The Project Physics Course and 
course materials were designed to accomplish the 
following goals: 

1. To help students to increase their knowledge 
of the physical world b\' concentrating on the ideas 
that chaiacteiize ph\sics as a science at its best 
ifor example, the consenation lawsi, rather than 
concentrating on isolated bits of information isuch 
as the lens formula i. 

2. To help students see physics as the mam- 
sided human acti\it\' that it really is. This means 
presenting the subject in historical and cultural 
perspecti\ e, and showing that the ideas of physics 
ha\e not onlv a tradition but methods of adapta- 
tion and change. 

3. To increase the opportunitx' for each student 
to ha\e immediate rewarding experiences in sci- 
ence while gaining knowledge and skill that \NiLl be 
useful throughout life. 

4. To make it possible for teachers to adapt the 
physics course to the wide range of interests and 
abilities among their students. 

5. To recognize the importance of the teacher in 
the educational process, and the \ast spectrum of 
teaching situations that prexail. 



LNTRODUCnON 



Table of Gontenls 



Unit 1 / Concepts of Motion 



Organisation of InNtruction 1 

Multi-media Systems Approach 
Suggested Schedule Blocks and 

Timetable 5 
Resource Charts 6 



Background and Development 10 

(AeiAJevv ot I'nit 1 10 

Chapter 1 10 
Chapter 2 13 
Chapter 3 16 
Chapter 4 18 

Brief Descriptions of Learning 
Materials 20 

Summary List 20 
Film Loops 18 mml 21 
Sound Films 116 mm) 21 
Film Sources 25 
Transparencies 26 

Demonstration I\otes 26 

Dl Kecognizing Simple Motions 26 
D2 Uniform Motion, Using Accelerometer 

and Dvnamics Cart 26 
D3 Instantaneous Speed 26 
D4 Uniform Acceleration, Using Liquid 

Acceleit)meter 30 
D5 Comparative Fall Rates of Light and 

Heaxy Objects 31 
D6 Coin and Feather 31 
D7 'I\vo Ways to Demonstrate the Addition 

ofV'ectors 31 
ns Direction of Acceleration and 

X'elocity 32 
DO Diitution of Acceleration and Velocity: 

an Air Track Demonstiation 32 
DIO N'oncommutative Rotations 33 
Dll Neulons Fii-st I>a\v 33 
D12 \evv1on's Law Kxperiment lair 

track I 33 
D13 Lffect of Friction on Acceh'ration 33 
D14 Demonstrations with Rockets 34 
D15 Making an Ineilial Balance 38 
D16 Action-n'a('tion Forces in I»ulling a 

Rope. 1 38 

D17 Action-reaction Fones in Pulling a 

Rope. II 38 

D18 Reaction Foice of a VX'all 38 
D19 \ew1on s Third l-iuv 38 
D20 Action-ii'action Foix'es Between c:ar 

and Road 30 
D21 Action-reaction Foi-ces in llammeiing 

a Nail 30 
1)22 Action-n'action I'oix-rs in .lumping 

Upwartl 30 
\)2A Tramcs of ReftMt'ncc M) 



D24 Inertial vs. Xoninertial Reference 

Frames 40 
D25 Uniform Circular Motion 40 
D26 Simple Harmonic .Motion 40 
D27 Simple Harmonic Motion; Air 

Track 40 



Experiment .\otes 41 



41 



50 



53 



Ll-1 .\aked-tye Astronomy 
El-2 RegulariK' and Time 43 
El-3 Variations in Data 44 
El-4 Measuring Uniform Motion 
El-5 A Seventeenth-Centupi' 

Experiment 46 
El-6 Twentieth-Century \'er-sion of 

Galileo's E.xperiment 47 
El -7 Measuring the Acceleration of 

Gra\ir\' a^ 49 
El-8 Newton's Second Law 
El -9 Mass and Weight 52 
El-10 Curves of Trajectories 
El-11 Prediction of Trajectories 
El-12 Centripetal Force 54 
El-13 Centripetal Force on a 

Turntable 55 



Film Loop i\otes 56 

Ll Acceleration Caused by Gra\it\ . I 56 
IJi Acceleration Caused by GravitA'. II 56 
L3 \'ector Addition: \'elocit>' of a Boat 57 
L4 A Matter of Relati\e Motion 57 
L5 Galilean Relati\it\': Ball Drxjpped from 

Mast of Ship 58 
L6 Galilean Relativity: Object Dropped from 

Ain;raft 58 
L7 Galilean Relati\ir\ : Projectile Fired 

VerticalK' 58 
L8 Analysis of a Hurtile Race I 58 
L9 AnaK'sis of a Hur-dle Race. II 59 

Equipment !\otes 59 

I'olarxiid I'hotographv 59 

Str-oboscopic F'hotograjihx 64 

Calibration of Strx)boscopes 67 

The Blinky 72 

Air Tracks 73 

(iuantitatixe Wor-k uith Liquid-surface 

•Acceler-onieter- 73 
A \er"satile Clannon 74 

C;athode-ra\ Oscilloscope 75 

Suggested Nolutions to Htudy Guide 
l^robirms 84 

t haptcr 1 84 

Chapter 2 88 
C^hapter- 3 95 

{"hapt»M ■» 101 



vi 



TAHLi; oi (;o^m^\TS 



Unit 2 / Motion in the Heavens 



Organization of Instruction 107 

Multi-media Sx'stems Approach 109 
Suggested Schedule Blocks and 

Timetable 110 
Resource Charts 111 

Background and Development 115 

CheiAiew of Unit 2 115 

Chapters 115 

Chapters 117 
Chapter 7 119 
Chapter 8 121 
Concept Flow Chart 124 

Additional Background Articles 125 

Background Information on 

Calendars 125 
Armillan' Sphere 126 
Notes on the Sizes and Distances to the Sun 

£md Moon, by Aristarchus 126 
Epicycles 127 

Note on the Chase Problem ' 127 
Atmospheric Refraction 128 
Relations in an Ellipse 129 
About Mass 129 

The Moons Irregular Motion 130 
Measuring G 130 
Theories 131 

Brief Descriptions of Learning 
Materials 132 

Summary' List 132 
Film Loops i8 mmi 133 

Film Strip 133 
Sound Films il6 mmi 133 
Transparencies 135 

Demonstration \otes 135 

D28 Phases of the Moon 135 

D29 Geocentric Epicycle Model 135 

D30 Heliocentric Model 135 

D31 Plane Motions 136 

D32 Conic-sections Model 136 



Experiment \otes 



136 



E2-1 Naked-Eye Astronom\' 136 
E2-2 Size of the Earth 139 
E2-3 The Distance to the Moon 139 
E2-4 The Height of Piton, A Mountain on 

the Moon 140 
E2-5 Retrograde Motion 141 
E2-6 The Shape of the Earth's 

Orbit 141 
E2-7 Using Lenses to Make a 

Telescope 143 
E2-8 The Orbit of Mars 143 
E2-9 Inclination of Mars' Orbit 145 
E2-10 The Orbit of Mercury 145 
E2-11 Stepwise Approximation to an 

Orbit 146 

E2-12 Model of the Orbit of Halley s 

Comet 147 



Film Loop and Film Strip \otes 149 

Retrograde Motion of Mars 
' (Filmstripi 149 

Retrograde Motion: Geocentric 

Model 149 

Retrograde Motion: Heliocentric 

Model 149 
L12 Jupiter Satellite Orbit 150 
L13 Program Orbit. I 151 
L14 Program Orbit. II 152 
L15 Central Forces: Iterated Blows 
LIB Kepler's Laws 152 
L17 Unusual Orbits 153 



LIO 



Lll 



152 



Efjuipment .Notes 153 

Epic\cle Machine 153 
Planetarium Use for Project Physics 

Suggested Solutions to Study Guide 
Problems 155 

Chapter 5 155 

Chapter 6 156 
Chapter 7 158 
Chapter 8 161 



154 



Unit 3 / The Triumph of Mechanics 



Organization of Instruction 166 

Multi-media Svstems Approach 169 
Suggested Schedule Blocks and 

Timetable 170 
Resource Charts 171 



Background and Development 175 

0\ er\ie\v of Unit 3 175 
Chapter 9 175 
Chapter 10 178 
Chapter 11 183 

Chapter 12 186 
Concept Flow Chart 189 



Additional Background .Articles 190 

The State of Ph\ sics as a Science at the 

Beginning of the Nineteenth 

Century 190 

Conservation Laws in Physics 190 
Elastic and Inelastic Collisions 191 
The Equivalence of the Definition of "EJeistic' 

Collision 191 
Energv Reference Levels 191 
Food Energ\- 191 
Classifications of Energv' 192 
Watt 192 

Discussion of Conservation Laws 193 
Feedback 194 



TABLE OF COVTENTS 



A Method for Calculating the Pressure of the 
Atmosphere 197 

Brief DeMf;ripdonM of Learning 
MalerialN 1 9K 

Suininaiy List 198 
Film Ix)ops 18 mm) 199 
Sound Films 116 mmi 200 
Transparencies 200 



E3-15 Wave Properties 236 

E3-16 Waves in a Ripple Tank 

E3-17 Measuring Wa\elength 

E3-18 Sound 238 

E3-19 Ultrasound 238 



237 
237 



UemoDHtradon i\oteH 201 

D33 An Inelastic Collision 201 
Predicting the Range of a 
Slingshot 202 
Diffusion of Gases 204 
Biovvnian Motion 204 

A Note Concerning Demonstrations and 
Experiments in Chapter 12 205 

D37 Wave Propagation 206 
Energy Transpoil 206 
Superposition 206 
Reflection 207 
Wave Trains 207 
Refraction 207 
Inteifei-ence Patterns 207 
Diffraction 208 
Standing Waves 208 
Two Turntable Oscillators 
(beats) 210 



D34 



D35 
D36 



D38 
D39 
D40 
D41 
D42 
D43 
D44 
D45 
D46 



240 
241 



242 

242 



Experiment IMotes 2 1 1 



E3-1 

E3-2 

E3-3 

E3-4 

E3-5 
E3-6 
E3-7 

E3-8 

E3-9 
E3-10 

E3-11 
E3-12 
E3-13 

E3-14 



Collisions in One Dimension. 

I 211 

Collisions in One Dimension. 

II 212 

Collisions in 'I\vo Dimen- 
sions. I 218 
Collisions in Two Dimen- 
sions. II 221 
Conservation of Energ\'. I 
Conservation of Energ\'. II 
Measuring the Speed of a 
Bullet 227 

Energy Analysis of a Pendulum 
Swing 227 
Least Energy 228 
Temperature and 
Therinometei-s 228 
Calorimetiy 230 
Ice C^aloiimetiy 231 
Monte Carlo K.xpeiiment on 
Molecular Collisions 232 
H(?ha\'i()r of Cases 233 



224 
227 



Film LfMip .\ote» 239 

L18 One-Dimensional Collisions. I 
L19 One-Dimensional Collisions. I 
L20 Inelastic One-Dimensional 

Collisions 242 
L21 Two-Dimensional Collisions. I 
L22 Two-Dimensional Collisions, i 
L23 Inelastic Two-Dimensional 

Collisions 242 
IJi4 Scattering of a Cluster of 

Objects 242 
L25 Explosion of a Cluster of 

Objects 242 
L26 Finding the Speed of a Rifle 

Bullet. I 243 
L27 Finding the Speed of a Rifle 

Bullet. II 243 
L28 Recoil 244 
IJJ9 Colliding Freight Cars 245 
L30 D>'namics of a Billiard Ball 245 
L31 A Method of Measuring Energ\': 

Nails Dri\en into Wood 246 
L32 Gra\itational Potential Energ\' 246 
L33 Kinetic Energv' 247 
L34 Conservation of Enei^: Pole 

\ault 247 
L35 Conservation of Enei^: Aircraft 

Take-off 248 
L36 Re\ ersibilitx of Time 248 
L37 Supeqjosition 248 
L38 Standing \Va\es on a String 248 
L39 Standing Wa\es in a Gas 248 
L40 Vibrations of a \\ ire 249 
L41 Vibrations of a Rubber Hose 249 
L42 \'ibrations of a Drum 250 
L43 Vibrations of a Metal Plate 250 



Equipment IXotes 250 

Turntable Oscillator 
Thermistor 255 



150 



Su^^ested SoiutionM to Studi C^uide 
l*robiem(i 256 



c:hapler 9 


256 


Chapter 10 
Chapter 11 
Chapter 12 


259 
265 
269 



Unit 4 / Li^ht and Electronia^iietisiii 



Or^ani^.ation of InNtrut^tion 274 

Mulli-incdia Systems .\ppn)a(h 
Suggested ScIkhIuIc Blocks and 

'Tim(>table 277 
Resourc-e Charls 278 

llaiTk^riiiind and Deielopment J 

(hj'iMCW ot I 'nit 4 282 

Chapt(M- 13 282 
C;hapl(«r 14 28 J 



176 



Chapter 15 287 
Chapter 16 288 
Concept Flow c:hart 290 



Additional Backpground /lirticlriM 

\()tc^ oil I it'lil.s J;ii 
RomtM 292 

Tlie C'ost of an Electrical Motor 
in 1850 29;i 



291 



tiii 



rAHi.i: oi coviKvis 



Brief Descriptions of Learning 
•Materials 294 

Sunimarv' List 294 
Film Loops i8 mmi 295 
Sound Films (16 mmi 295 
Transparencies 295 



Demonstration \otes 



296 



D47 Some Electrostatic 

Demonstrations 296 

D48 The Electrophorous 298 

D49 Currents and Forces 299 

D50 Currents, Magnets, and Forces 

D51 Electric Fields 301 

D52 Microwaves 301 



300 



Experiment i\otes 303 

E4-1 Refraction of a Light Beam 303 

Young s Experiment: The V\'a\'elength 
of Light 305 
Electric Forces. I 305 
Electric Forces. II: Coulomb's 
Law 306 



E4-2 



E4-3 
E4-4 



E4-5 Forces on Currents 308 

FA-6 Currents, Magnets, and Forces 311 

E>l-7 Electron Beam Tube. I 313 

E4-8 Electron Beam Tube. II 314 

E4-9 Waves and Communication 314 



Film Loop \ote 320 

L44 Standing Electromagnetic 
Waves 320 



Ec|uipment \otes 320 

Light Source 320 
Current Balance 320 
Microua\e Appeiratus 323 
Electron Beam Tube 324 

Suggested Solutions to Studi Guide 
Problems 324 

Chapter 13 324 
Chapter 14 328 
Chapter 15 333 
Chapter 16 335 



Unit 5 / Models of the Atom 



Organization of Instruction 337 

Multi-media Systems Approach 338 
Suggested Schedule Blocks and 

Timetable 339 
Resource Charts 340 



Background and Development 344 

CXeniew of Unit 5 344 
Chapter 17 344 
Chapter 18 347 
Chapter 19 351 
Chapter 20 356 
Concept Flow Chart 360 



Additional Background Articles 361 

Comments on the Determination of Relative 

Atomic Masses 361 
Spectroscopy 363 
Rutherford Scattering 364 
Angular Momentum 365 
Nagaoka s Theory of the "Satumian " 

Atom 365 



D54 



Demonstration IVotes 368 

D53 ElectroKsis of Water 368 

Charge-to-Mass Ratio for Cathode 
Ra\s 369 
Photoelectric Effect 
Blackbodv Radiation 
.Absorption 372 
Ionization Potential 



D55 
D56 
D57 
D58 



371 
372 



373 



374 



Experiment IVotes 

E5-1 Electrolysis 374 

The Charge-to-Mass Ratio for an 

Electron 376 

The Measurement of Elementan' 

Charge 376 

The Photoelectric Effect 379 

Spectroscopy 382 



E5-2 

E5-3 

E5-4 
E5-5 



Film Loop \'otes 385 

L45 Production of Sodium by 

Electrolysis 385 
L46 Thomson Model of the Atom 
L47 Rutherford Scattering 385 



385 



Brief Descriptions of Learning 
.Materials 366 

Summary List 366 
Film Loops i8 mmi 367 
Sound Films I16 mmi 367 
Transparencies 368 



Suggested Solutions to Study Guide 
Problems 386 

Chapter 17 386 
Chapter 18 389 
Chapter 19 391 
Chapter 20 396 



TABLE OF CO.VTEXTS 



Unit 6 / The Nucleus 



Organization of Instruction 400 

Multi-media Systems Approach 
Suggested Schedule Blocks and 

Timetable 403 
Resource Charts 404 

Background and Development 

Overview of Unit 6 408 
Chapter 21 408 
Chapter 22 410 
Chapter 23 412 
Chapter 24 415 

Brief Description of Learning 
Materials 421 

Summary List 421 
Film Loops (8 mm) 422 
Sound Films (16 mm) 422 
Transparencies 423 

Demonstration IVotes 423 

D59 Mineral Autoradiograph 
D60 Naturally Occurring 
Radioactivity 424 
D61 Mass Spectrograph 424 
D62 Aston Analogue 425 



Experiment IVotes 425 

E6-1 Random Events 425 

E6-2 Range of a and /3 Particles 

E6-3 Half-Life. I 429 

E6-4 Half-Life. II 432 

E6-5 Radioactive Tracers 435 



E6-6 



402 



Measuring the Energy of /3 
Radiation 436 



408 



Film Loop IXotes 436 

L48 Collisions with an Object of Unknown 
Mass 436 

Suggested Solutions to Study Guide 
Problems 436 

Chapter 21 436 

Chapter 22 438 
Chapter 23 440 
Chapter 24 442 



445 



423 



428 



Appendix 


445 


Radiation 


Safety 


Tests 455 




Unit 1 


455 


Unit 2 


472 


Unit 3 


487 


Unit 4 


502 


Unit 5 


516 


Unit 6 


531 


Suggested Ansners 


Unit 1 


547 


Unit 2 


549 


Unit 3 


553 


Unit 4 


555 


Unit 5 


558 


Unit 6 


561 



547 



TABLE (JF COVTEMTS 





f 




^ - 'J 


1 




RJI 




^ . J • " 



Goncepis of Moiion 



Organization of Instruction 



THE MULTI-MEDIA SYSTExMS APPROACH 



The Multi-media Systems approach is just one of 
many possible stales of classroom management. 
Here the teacher is a manipulator of emironment 
and a tutor. The manipulation affords the control 
of the program b\ the teacher. At the same time, 
the students experience a measure of freedom in 
styles of learning. Much of the time the teacher 
tutors by answering and asking specific questions 
of small groups of indi\iduals. The stvle is informal 
and nonauthoritative. However, on occasion the 
teacher makes a presentation to the entire class. 



For example, in the Chapter 1 daily plan the 
teacher presents graphs, velocitv', and acceleration, 
on the 6th da\'. The students can request addi- 
tional presentations on specific topics. 

These are styles of teaching as good as this one. 
There are many different organizations of work 
within the framework of Multi-media Systems. 
Howe\er, this plan is offered so that a new teacher 
may see one organization of a program for a unit 
of Project Physics. Teachers are imited to modify 
this plan or invent their own st)'le. 



THE MULTI-MEDIA SCHEDULE 



Day 1 

Devote the time necessarv for the opening of 
school. Take the class through a tour of six or seven 
media of instruction. Mention that the first reading 
assignment is not about ph\sics but about what a 
physicist does and the materials a\ailable to learn 
physics. 

Day 2 

This day is used to explain the Multi-media System 
and to charge students with the responsibUit\' of 
self-directed instruction 



Day 3 

.After the Film Loop, dixide the class at random into 
small groups. Pass out three or four open-ended 
questions about the Film. Loop. Be a listener. 

Take a minute to comment on how to use the 
Text most effecti\ely. 

Day 4 

Lab Stations: Uniform Motion 

Students are to make qualitative observations of 
objects undergoing uniform motion. Students spend 



ORGANIZATION OF INSTRUCTION 



8 to 10 minutes at each station. Brief instruction of 
what to look for at each station will be helpful. 

1. balloon pucks on glass tray 

2. pucks on plastic beads 

3. D2 (dynamics cart with accelerometer) 

4. Polaroid photograph of tractor, blinky 

5. LS and L9 {Film Loops) 

6. TO or Tl [Transparencies! 

Take a minute to describe the Handbook. 

Day 5 

Lab Stations: Accelerated Motion 

1. D4 (dynamics cart with accelerometer) 

2. st[X)be photo of free fall 

3. L4 (a matter of relative motion) 

4. D3 (analysis of strobe photo) 

5. L9 (analysis of hurdle race) 

Take a minute to mention your presentation to- 
morrow. Encourage the recording of data. 

Day 6 

Although the time is overdue for explanations, you 
have the students where you want them. Each has 
a head full of questions and a fist full of data. Var- 
ious demonstrations, transparencies, and exam- 
ples may be used to clarify the concepts and their 
measurements. Comment on Study Guide ques- 
tions. 

Day 7 

Post answers. Let students who have many correct 
answers go on to other activities you have set up. 

Design problem-solving group procedures care- 
ftjUy. 

Allow individual problem solving. Drap in on all 
groups. 

Take a minute to comment on the assignment 
with the first student evaluation in mind. 

Day 8 

Divide the class into small discussion groups. Have 
each group read and discuss the quoted dialogues 
in Galileo's Two New Sciences. Give some open- 
ended questions to each group. 

Day 9 

Explain El-5 "A Seventeenth Century Experiment' 
in detail. Concentrate on the stnjcture of scientific 
thought, including definitions, assumptions, and 
Galileo's difficulty in testing his notion of acceler- 
ation. 

Day 10 

Students perform "A Seventeenth Centurv Experi- 
ment" (El-5). Near the end of the period disturti 
each group with man\ questions about this exper- 
iment. 

Day 11 

Assign a few problems on five-fall acceleration and 
"A Sirventeenth Clentuiy Experiment to each gnuip 
You can work toward dexelopiiig mathematical 
skills lor {hv. five-fall laboratoiy loinorixnv. 

lake a minute to mention to students that the\ 



should survey the Activities section of the Hand- 
book with the idea of choosing their own activity 
tomorrow. 

Day 12 

Students can elect to do a detailed study on one 
of the following. 

1. LI or L2 

2. a^ by any of the methods in El-7 

3. any activity 

Day 13 

This lecture should touch upon the life and times 
of Galileo and also on the need to test theories by 
performing experiments. 

Discuss: Is free fall the same for different m<isses? 
Is it the same at all positions in space? 

Day 14 

El-1 "Naked-Eye Astronomy" requires the taking 
of data systematically o\er a period of weeks. As- 
sign students and or groups specific objects on 
which to gather data: sun, moon, specific stars, 
planets, etc. These observations will be ver\' useful 
if they are carried out before Unit 2. 

Day 15 

Lab Stations: Vectors 

1. D7 I two ways to demonstrate addition of vectors) 

2. D8 (direction of acceleration and velocity) 

3. L3 (vector addition! 

4. PSSC film, "Vectors" 

Day 16 

Explain vectors. Use Film Loop 3. 

Day 17 

Lab Stations: Force, Mass, and Acceleration 

1. Dll (inertiai 

2. reSC Exp. 21 (dependence of acceleration upon 
force and massi 

3. PSSC Exp. 20 (changes in velocity' with a constemt 
force) 

4. D12 (Newton's laws — air tracki 

5. T8 (tractor-log problem) 

6. El-8 (Newlons second law) 

Day 18 

Teacher-led discussion: \ew1ons Laws 

Clarify points still unclear about Newlons three 
laws of motion. This should be the most crucial 
class so far this year. 

Day 19 

Small-group problem-soKing session 

Post answers. Let students who have many of the 
convct answers go on to other activities vou have 
set up. Design pixjblem-soUing group procedure 
carefulK .Allow individual problem solving Drop 
in on all activities to help. 

Da\ 20 

Invite student leadei-s from differtMit research groups 
to prvsent obseiAations about the sun moon and 
planets Also encourage individual pivsentations 



UNIT I / CONCEPTS OF IVIOTION 



It is very' important for the teacher to summarize 
the findings and to make clear what should be 
learned from this session. Suggest that students 
work on one more naked-eye observation of the 
hea\ens during the next fixe school da\'s. 

Day 21 

Student Evaluation 

This evaluation may be an examination. Or, the 
teacher can use more imaginative devices, such as 
laboratory reports, poetry, science fiction, addi- 
tional problem sets, etc. 

Day 22 

Lab Stations: Complex Motion 

1. El-10 icurves of trajectories) 

2. El-11 (prediction of trajectories) 

3. El-12 (centripetal forcei 

4. El-13 (centripetal force on a turntable) 

5. L6 (Galilean relativity i 

Students are to use the apparatus at each station, 
making qualitative observations. 

Day 23 

Lab Stations 

Same stations as Day 22 but students are to pick 
one experiment and do it quantitatively. 



Day 24 

Small-group problem solving 

Have students discuss problems in small groups. 
Be sure an outstanding student is in each group. 
Circulate among groups. 

Day 25 

Students report to the rest of the class on results 
of experiments on Day 23. Urge that presentations 
be verv' short and clear to allow plenty' of discus- 
sion time. 

Day 26 

Show the first 13 minutes of the film "Frames of 
Reference." Divide class at random into small 
groups, pass out three or four open-ended ques- 
tions related to film, and use the rest of the period 
for discussion. 

Day 27 

Review projectile motion and uniform circular mo- 
tion. Discuss satellite motion in detciil. 

Days 28-30 

Evaluation 

One method of evaluation is to review, test, and 
discuss the test. Take one day for each activity. 

Another method is to ev aluate each student in- 
dividually during three davs of conferences. 



ORGAMZATIOX OF LVSTRUCTIOX 



Unit 1 SAMPLE MULTI-MEDIA SYSTEMS APPROACH 

Note: This is just one path of many that a teacher may take through Unit 1. 



Survey: 

Multi-media 



Introduce; Multi-media 
procedures 



Text: Introduction 
Handbook: Introduction 



5-min film loop 
Small-group 
discussion 



Text: 1.1-1.4 



Lab stations: 
Uniform motion 



Handbook: 
E1-2, E1-3, 
E1-4 



5 


6 


7 


8 


Lab stations: 


Teacher presentations: 


Small-group 


Small group 


Acceleration 


graphing 


problem solving 


discussion of 






velocity 






Sec. 2.3 






acceleration 
















Selected Study Guide 






Text: 




Text: 1.5-1.8 




questions 


Write up lab 




2.1-2.4 



9 


10 


11 


12 


Teacher presentation: 


Lab stations: 


Small-group 


Lab. Stations: 


17th century Exp. 


17th century Exp. 


problem solving 


Free Fall 
















Write up 












Handbook: Survey 




observa- 




Text: 2.5-2.10 




Handbook: E1-5 




Chapter 2 




tions 



13 

Teacher presentation: 

Galileo & free fall 

Quiz 



14 

Organize E1-1: 

Naked-Eye 

Astronomy 



Handbook: Survey E1-1 



15 

Lab stations: 

Vectors 



Observe sky 
Text: 3.1-3.4 



16 

Teacher 

presentation: 

Vectors 

Observe sky 

Text: 

3.5-3.9 



17 
Lab stations: 

force 

mass 
acceleration 



18 

Discussion: 

Newton's laws 



Ob«erve sky 
Text: 3.10-3.11 



19 

Small-group 

problem solving 



Observed sky 

Selected Study Guide 
questions 



20 
Student presentation: 
Naked-Eye Astronomy 
Summary by teacher 



Observe sky 

Write up EM & E1-8 



Review 
Ch. 3 



21 

Student 

evaluation 



22 
Lab stations: 



Text: 4.1-4.3 
Handbook: Survey Ch. 4 



Text: 4.4-4.6 



23 
Complex motion 



Write up lab 



24 
Student 
presentations 
Selected 
Study Guide 
questions 



25 

Small-group 

problem solving 

session 



Text: Reread 4.4 



26 

PSSC Film: 

Frames of Reference 

Small-group 

discussion of film 

Text: 4.7-4.8 
Epilogue 



27 

Teacher presentation: 

Satellites 



Review Unit 1 



28 

Review 



Individual student 
evaluation 



29 


30 




Test 


Discuss test 




or 


or 




Individual student 


Individual student 




evali 


jation 


evali 


jation 





UNIT 1 / CONCEPTS OF MOTION 



Unit 1 SUGGESTED SCHEDULE BLOCKS AND TIMETABLE 

Each block represents one day of classroom activity and innplies approximately a 50-minute period. The words in each block 
indicate only the basic material under consideration or the main activity of the day. The suggested homework (listed above 
each block) refers mainly to the Text and Handbook, but is not meant to preclude the use of other learning resources. 



Open school 

Show multi-media 

samples 



CHAPTER 1 THE LANGUAGE OF MOTION 



Text: Introduction 
HB: Introduction 



Introduce 
multi-media 
procedures 



Any film loop 

on motion (5 min) 

Small-group 

discussion 



Text 1.1-1.4 



Lab Stations: 

Uniform Motion 

(See day 4.) 



HB: Use E1-4for 
Lab Analysis 



Lab stations: 
Acceleration 
(See day 5.) 



Text: 1.5-1.8 



Teacher presentation: 

Graphing, velocity, 

acceleration, 

instantaneous 

acceleration 



Text: selected 
S G questions 



CHAPTER 2 FREE-FALL: GALILEO DESCRIBES MOTION 

Write up E 1-4 Text: 2.1-2.4 Text: 2.5-2.10 



Small-group 


problem solving 


or 


Special activity 



Small-group 
discussion: 
Section 2.3 



Teacher presentation 
El -5: A Seventeenth 
Century Experiment 



A Seventeenth 

Century Experiment 

(cont'd.) 



Handbook: E1-5 
Write up lab 



Small-group 
problem solving 



Handbook: Survey 
Chapter 2 



Lab stations: 
Free Fall 



CHAPTER 3 THE BIRTH OF DYNAMICS: 



Write up lab 



Teacher presentation: 

Galileo 

Free Fall 

Quiz 



HB: EM Survey 
Naked-Eye 
Astronomy 



Organize E1-1 : 
Naked-Eye 
Astronomy 



Observe sky 
Text: 3.1-3.4 



Lab stations: 

Vectors 
(See day 15.) 



Observe sky 



Teacher presentation: 

Vectors 

Film Loop 3 



Observe sky 
Text: 3.5-3.9 



Lab stations: 

Newton's 2nd 

Law 



NEWTON EXPLAINS MOTION 

Observe sky 
Text: 3.10-3.11 



Teacher-led 

discussion: 

Newton's Laws 



Observe sky 

Text: selected SG 

questions 



Small-group 
problem solving 



Observe sky 

Write up E1-1 
and E1-8 



Student 

presentations: 

Naked-Eye 

Astronomy 



Review Ch. 3 



Student evaluation 



Write up labs 
Text: Reread 4.4 



PSSC Film: 

Frames of Reference 

Small-group 

discussion 



CHAPTER 4 UNDERSTANDING MOTION 



Text: 4.1-4.3 
HB: Survey Ch. 4 



Lab stations: 

Projectile 

Motion 



Text: selected 
S G questions 



Teacher-led 

discussion: 

Uniform Circular 

Motion 
S G Questions 



Observe sky 



Teacher 
discusses projectile 

motion 



Unit 1 Epilogue 



Review 

or 

Individual student 

evaluation 



Observe sky 
Text: 4.4-4.8 



Lab stations: 
Circular Motion 



Review Unit 1 



Test 



Individual student 
evaluation 



Observe sky 
Write up conclusions 



Student 

presentations: 

Circular Motion, 

Naked-Eye 

Astronomy 



Discuss Test 



Individual student 
evaluation 



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ORGAMZATIO\ OF INSTRUCTIOX 



Background and Development 



OVERVIEW OF UNIT 1 



How do things move? Why do things move? The 
principal task of Unit 1 is to provide answers to 
these questions. A secondary task is to provide in- 
sight into the way scientists go ahout their work. 

The first question, How do things move?" is an- 
swered gradually, starting with a very simple mo- 
tion and proceeding to more complex motions. 
One of the main reasons for starting the course 
vvath kinematics is that it provides an immediate 
opportunity for student lahoratoiy activity. Fur- 
thermore, these activities, though easily carried 
out, are both interesting and significant. I he stu- 
dents usually like kinematics experiments and 
gain confidence in their ability to do physics. 

Most of Chapter 1 is spent developing the intel- 
lectual tools to describe straight-line motion. The 
key concepts are average speed and instantaneous 
speed. The chapter concludes by making an anal- 
ogy between the change in position with time 
(speed) and the change in speed v\ith time Ian ex- 
ample of acceleration). 

Chapter 2 extends the description of motion to 
accelerating objects, specifically an object in free 
fall. We follow Galileo thitjugh his ov\ti analysis as 



he seeks to confirm that the speed of a freely falling 
object is proportional to the elapsed time of this 
fall. By using Galileo as an example. Chapter 2 also 
serves to provide the student with some under- 
standing of the scientist as a person working 
v\ithin a social milieu. 

The second question, "Why do things move?" is 
the fundamental question of dynamics, \ew1on 
provided the answer to this question with his three 
laws of motion. These three laws are developed in 
Chapter 3. Vector concepts are introduced and are 
used throughout the remainder of the unit. 

The final chapter in Unit 1 brings together the 
concepts learned in the first three chapters and 
applies them to projectile motion and uniform cir- 
cular motion. The chief example used to develop 
these ideas is that of a journey from the surface of 
the earth to the surface of the moon. 

Chapter 1 begins by citing an old maxim: "To be 
ignorant of motion is to be ignorant of nature. ' 
Indeed, kinematics and dynamics are to phvsics 
what grammar is to language or what scales are to 
music. The techniques and ideas learned in Unit 
1 are used thixjughout the course. 



CHAPTER 1 / THE LANGUAGE OF MOTION 



1.1 I The Motion of lliings 

The most significant case of motion in the devel- 
opment of science is not mentioned in this section, 
although it foims the main topic of Unit 2. The fii-st 
scientific problem facing humanity' dealt with mo- 
tion in the heavens. From earliest times, people 
questioned the nature and causes of the motions 
of the various astixjiiomical bodies. Galileo's studv 
of the motion of objects at the earth's surface iter- 
restiial motion i led to an understanding of the 
motions of th(^ inaccessible heavenlv bodies. 

I bus, an understanding of the basic concepts of 
motion as formulated in seventeenth centuiy phys- 
ics is taken up at the beginning of this course. 
These conccfjts an* still useful in explaining and 
understanding much of the physical world that 
surn)unds us. Moreover, the concepts have histor- 
ical impoilance. 

Do not devote much class time to the justifica- 
tion of starting the coui'se with the studv of motion. 
The students will not yet know enough phvsics to 
know what alternatives there ar^^ It is nunv crucial 
to get the coui-se going (juicklv, raise interesting 
questions, and encourage studcMit partici|)ation 
ISome teachei-s, after having I.uight the course for 
one year, preftM- to stai1 with (:ha|)lri-s 5 and (? and 
part of C^haptei- 7 to establish motivation for studv 
n\ motion I 



xMotion goes on about us all the time. It mav be 
complex and confusing, or it mav have regularities 
that make it simpler to under-stand and classifv To 
help students begin to think about the motions 
around them, ask them to classifv several motions 
into those that are regular and those that are ir- 
regular. Examples might be a pendulum, a sewing 
machine, a leaf blowing on a tree, and a bird in 
fiight. Some motions may contain both regular and 
ini^gular features. 

Students will certainlv recognize on an intuitive 
level that while events such as falling leaves or 
flying birds mav be commonplace, thev are not 
necessarilv simple. .As a first a[)proximation. the 
motion of one object is moi-e complicated than that 
of another if it is undei^going moiv eiratic changes 
of direction and or speed. In the long run. however, 
the distinction depends upon experience and our 
ability to find functional ivlationships with which 
to describe events or statements. Generallv . when 
we luidertake investigations in a relativelv new 
field, we look aniund for what appear to be simple, 
straightfonvaixl examples of the pin'nomenon Ix'ing 
investigated Ihe simplic itv mav later on tuni out 
to be deceptive, but at least we have made a start 
We can cont'd oui-selves later 

The Greeks look unifomi cireular motion, rather 
than unifomi straight-line motion as the simplest 



I'MT 1 / COXCIEITS OF .VUXriDM 



Both their ph\sics and their metaphysics helped 
to direct attention to circular motion as primars'. 
It is pointless to debate which is really simpler — 
but progress in physics was greatly helped by 
adoption of the Galilean \iew in the earl\' and mid- 
dle part of the se\enteenth centurA'. 

1^ I A Motion Experiment that 

Does ^at Quite Work 

Section 1.1 ends with a suggestion that we can 
leam from experiment. Section 1.3 suggests an ex- 
perimental means for establishing regular time in- 
ter\als and measuring distance as a function of 
time, which leads to the definition of speed. 

Section 1.2 is a bridge to help the students bring 
intuiti\e feelings about motion and speed into an 
experimental en\ironment. 

1^ I A Better Experiment 

The main burden of teaching the student how to 
interpret and use strobe photographs rests on the 
laborator\' and audio-\isual aids under the direc- 
tion of the teacher. The treatment in the Text is 
not sufficient b\' itself. 

Quickl\' get the students started on an anal\sis 
of motion and on laboratory' work associated with 
it. Don't start the course with protracted philo- 
sophical discussions about the role of e.xperiment 
or the nature of simplicity. .After thev understand 
more ph\'sics, there is time enough to come back 
to some of these questions. 

Experiments are done so that events can be ma- 
nipulated and made "simple." Furthermore, the\' 
can be reproduced and done o\er and o\er again 
while measurements and observations are made. 

It is in the laboratorv' that students should leam 
about the role of laboratory- in science. Laboratory 
experiments, which sometimes may seem \er\' ar- 
tificial and almost trixial, do lead to undei-stand- 
ings that better explain the complex and interest- 
ing events seen in the world outside of a lab. If, 
howe\er, we begin with the study of complex mo- 
tions, such as falling lea\es, we may never find the 
regularities for which we search. 

If you use E3-2, Method B Stroboscopic Photo- 
graphs I Unit 3, page 104 in the Handbooks de\elop 
the idea of "freezing " motion. However, this need 
not be done rigorously: the idea that regular mo- 
tion \iewed at regular intervals can cause the mo- 
tion to seem to stop or move slowK' will be enough. 

1.4 I Leslie's Snim and the Meaning 

of Average Speed 

This section introduces the cnjcial definitions of 
average speed and internal and applies them quan- 
titatively to a kind of real motion that may pre\i- 
oush have interested the student. 

The preceding section hinted rather casually at 
the profound and essential notion that all meas- 
urements are approximations. This statement is in- 
complete, of course, until it specifies what meas- 



urements are approximations lo. In this section, 
there is a clear e.xample of one measurement ithe 
o\erall axerage speed i that is an appi-oximation to 
each of se\'eral othei" measurements ithe average 
speeds over the intenalsi. Likewise, each of these 
a\erage speeds is presented as an approximation 
to what really happened " at e\er\' moment of the 
motion. Ibis ma\ well be the student s first expo- 
sure to the idea of an experimentally unreachable 
concept that can ne\ertheless be approached, one 
step at a time, as nearh as one wishes, until one s 
measuring instruments are no longer good enough 
to improxe the picture further. 

1.5 I Graphing Motion and Finding 

the Slope 

This section presents no more than a bare outline 
of constructing and interpreting graphs related to 
speed. For a good man\' students this will be ex- 
tremely elementarv, and for them this section may 
be enough. 

In addition to helping students to use graphs 
correctly, an effort should be made to get them into 
the habit of tiding to interpret them ph\sicall\'. For 
e.xample, you would like students to be able to look 
at a graph and to describe in words the physical 
beha\ior illustrated. 

PSSC Lab 1-4 is an excellent exercise in graphing 
that can be done either under classroom supervi- 
sion or as a homework assignment. 

Below are some general rules that should be ob- 
served in plotting a graph. This list is far from com- 
plete and is used as a minimum standard. The fol- 
lowing four ideas should be stressed to \our 
students. 

1. Proper graph format. Each graph prepared 
should include a title, e.xperiment name or num- 
ber, and student name and date, presented in 
block form near the top of the graph. 

Each axis should be labeled clearly with the 
quantit\' plotted and the unit of measure used. The 
scale \'alues should also be clearK given. .All of this 
information should be easih readable from the 
lower righthand comer of the graph sheet without 
rotating the page. 

2. Size. .All graph presentations should be large 
enough to show clearK the behaxior of the quan- 
tities being plotted. The number of points included 
in the graph should also affect the choice of size. 
Poor choices include numerous points shown on 
too small a graph as well as \er\' few points pre- 
sented on too large a graph. 

3. Scales. The choice of scale on any graph is 
arbitrary- but should be made to maximize clarit\'. 
The range of \alues to be plotted should detennine 
the placement of the origin and the maximum 
scale \'alue for each axis, \\hene\er possible the 
scale should be chosen that allows decimal mul- 
tiples isuch as 100, 10, 1, or 0.1 1 of the units being 
graphed to be located easily along the axis. 

4. Plotting techniques. Experimental points 



BACKGROL.VD A.VD DE\TiLOP.ME,\T 



11 



should be plotted with small, shaqj dots. To avoid 
"losing" a data point, a small circle should be 
drawn around each point. 

The unceitainty in the values on the cooixlinate 
axes can be indicated by the size of the circle used, 
or perhaps by the length of horizontal and vertical 
bars drawn through the point as a cross. 

Seldom will data points fall on a totally smooth 
curve. Whenever there is some reason to believe 
that whatever- is being giaphed actuailly does change 
smoothly, a smooth curve should be drawn as 
close as possible to the data points plotted. En- 
courage students to consider what is implied in a 
broken-line graph connecting data points in con- 
trast to the implications of the smooth cuive. 

When two or more curves are plotted on the 
same graph, students should use either different 
colors for each curve, or dotted and broken lines. 
In either of these cases, a key or legend should 
appear on the graph defining the use of each line. 



students, and it makes the point that not all graphs 
are linear. 

Note: The particular selection of weights will de- 
pend on the size of the rubber band available. 




2 3 * i (, 

Tfrne (se:~> 

.3^(6') 

Fig. 1 From the graph, we can see that at time 0.25 sec the 
distance traversed is about 2.8 cm. 



1.6 I Time Out for a Warning 

The process of estimating \alues between data 
points is called interpolation. 

The pi-ocess of predicting values that extend be- 
yond the range of data points is called extrapola- 
tion. 

A discussion of the unceilainties imoKed in in- 
teiijolation and (\\trap()lation might be warranted. 
Stress the fact that interpolation is usually more 
reliabh; than e.xtraiiolation. Both are risky and 
sliould be undertaken with care, and the \alues 
should not b(' ascribed greatei- certainty than is 
warranted. 

The danger of e,\tra|)olation can be illustrated 
with a iiibbtM- band and a set of weights. Suspend 
the iTjbber- band and load it successi\ely with hea\ - 
ier and heaxier weights, ivcording and graphing 
the amount of extension for ('ach weight. ,\fter sus- 
pending son g ask the students to predict by e,\- 
liapolation the e.xtension of the iiibber band when 
the l()()()-g weight is suspended. I he actual exten- 
sion will tuni out to l)e imich less than the e.xtrap- 
olatiul \alu(^ because the elastic chaiacteristics of 
lh(* laibbei- band change. I his will a|)peal to youi 



iCO 


" 


400 


' \ 


300 


\ 




V 




N 


200 


V 




\ 




\ 


100 


\ 




\ 




N 


O 


1 i 1 \ 1 1 



t)00 a 20 6>«0 « 00 9 iO 9 40 
TIME (Pm) 

Fig. 2 From the graph we can see that the plane will be 
about 96 km from San Francisco at 9:00 pm if it does not 
change speed and direction. 

1.7 I Instantaneous Speed 

The concepts of instantaneous speed and limit are 
intix)duced hei-e, although the latter concept is not 
given its customaiy name. I he now-familiar notion 
of average speed is associated with its graphical 
representation as the slope of a choixl joining the 
endpoints of an interval on a distance-time curve. 
The student should alreadx know about tangents 
from geometiA', and this section should help to 
identify the slope of a tangent as a graphical rep- 
resentation of an instantaneous speed. 

It may be puzzling to some students land e\en 
disturbing to others i to learn that their common- 
sense notion of instantaneous speed corresponds 
to nothing that can be specified exactly b\' the 
"exact " science of physics. They may e\en resist 
the idea and thus miss the conceptual leap in- 
volved. Instantaneous speed is a conceptual in- 
vention." It is justified in physics b\ its usefulness 
in describing and explaining motion and its con- 
sistency with other ph\sics concepts. I he point is 
not that there really is or isn t such a thing, but 
that the idea is fruitful. 

.Although the idea of speed is introduced with a 
car speedometer, a car speedometer does not gi\e 
instanteous speed an\' more accurately than this 
method. A speedometer also a\erages o\er a time 
inteival. (Note the lag of a speedometer in register- 
ing as you begin with lai-ge acceleration.! 

1.8 I Acceleration bv Comparison 

The scalai- definitions of average and instantaneous 
acceleration are ptvsented by analogx- with the 
speed definitions. Ihe common-sense basis of Gal- 
ilean ivlati\it\ also appears, but only casualK 

Since (iaiilean ivlatixitx will pla\ a major role in 
later chaptei-s. it would he well to pause herv and 
make a point of the fact that thetv is a ival quali- 
tatixi' ditTeivni'e between sjieed and acceleration 



12 



UNIT 1 / CONCEPTS OF MOTION 



that the equations do not show lat least at this 
level). Ever\' student's experience with carnival 
rides and automobiles is wide enought to recall 
incidents that will make the distinction real, if each 
is stimulated to search his or her memor\'. If stu- 
dents come out of this section cominced that you 
can t tell when you re mo\ing uniformh', but \'ou 
can tell when you're accelerating,' they will be bet- 
ter equipped to tackle the physics of Galileo and 
Newton. 

Beginning physics students are sometimes con- 



fused by the units for accelerations, m sec". It may 

be helpful to show that the units come from the 

definition of acceleration, Av/Af, and that change 

m/sec 

in velocitv per unit time, , is written just for 

sec 

convenience as m/sec^. If your students can't get 

comfortable with the 'square time, ' stick with the 

more obxious expression: msec sec. Conxentions 

of notation are the least important aspects of ph\s- 

ics you can teach, and ought not to be purchased 

at the cost of understamding the ideas. 



CHAPTER 2 / FREE FALL: 
GALILEO DESCRIBES MOTION 



Many discussions of Galileo and his stud\' of me- 
chanics are quite critical of .-Vristotle. It is, perhaps, 
as unfair to condemn .-Vi'istotle for not accepting 
what the \acuum pump would pro\e as it would 
be unfair to criticize Galileo for not disco\ ering ra- 
dio astronomy. 

It should be pointed out that the ph\'sics inher- 
ited by Galileo is really a \er\' different and ad- 
xanced kind of physics compared to the original 
work of Aristotle. 

A fi^quentK o\erlooked contribution to .Aristo- 
telian physics was made by the .Arabs. After the 
decline of the .Alexandrian period of Greek science 
(about 200 .AD I, the knowledge of the Greeks was 
not lost to the West. During the so-called Dark Ages 
in Europe, there was great actixity in the Arab 
\%orid. From the eighth centuiy through the tweUth, 
considerable scientific and scholarly work was 
done b\' the Muslims. Working in Damascus, Bagh- 
dad, Cairo, and ultimateh' in sexeral centers in 
Spain, the Muslims modified the work of Aristotle 
and other Greeks in man\' wa\s. Furthermore, the 
Muslims were influenced by the studies of the Per- 
sians, Hindus, Chinese, and others in the East, and 
by certain Christians fix)m the West. 

Muslim science flourished in Toledo, Cordoba, 
and other Spanish cities. As these cities were grad- 
ually reconquered by the Christians during the 
eleventh to fifteenth centuries, Muslim and ancient 
Greek knowledge filtered into Europe. 

Before the time of Galileo, Aristotelian science 
had been blended with Christian philosophy, par- 
ticularK' by Thomas Aquinas. There were, howe\er, 
\arious criticisms and interpretations made during 
the later Middle Ages at Oxford, Paris, Padua, and 
other centers of intellectual acti\it\'. Vou should be 
aware of these points, not necessarily to bring 
them up in class for discussion, but to avoid 
overemphasizing the conflict between Aristotle 
and Galileo, which would thereby seem to impK' 
that nothing happened during the 2,000 years sep- 
arating these t\vo great men. 

For a more complete ibut by no means extensive i 
treatment of Aristotle s physics the student may be 
referred to any of the following: 



C. B. Boyer, "Aristotle's Physics, " Scientific Amer- 
ican, May 1950. 

M. R. Cohen and I. E. Drabkin, A Sourcebook in 
Greek Science, New York: McGraw-Hill, 1948. See 
pp. 200-203 on natural and unnatural motions, and 
especially pp. 207-212 on falling bodies. 

Alexandre Ko\'re, Galileo, pp. 147-175, in Philip 
P. W'einer and Aaron Noland i editors i. Roots of Sci- 
entific Thought, New York: Basic Books, 1957. Pp. 
153-158 clearly describe the Aristotelian theory of 
motion. 

There are other papers in the Weiner and No- 
land anthology that will help you. We recommend 
the book for your librarv amd for the school library. 

O. L. 0'Lear\', How Greek Science Passed to the 
Arabs, London: Routledge and Keegan Paul, 1948. 

S. F. Mason, Main Currents of Scientific Thought, 
New York: Henr\' Schuman, 1953. 

The influence of China, India, and the craft tra- 
dition in mediexal Europe, as well as the influence 
of the Arabic world, is outlined in Chapters 7 
through 11 I pp. 53-981. 

A. C. Crombie, Medieval and Early Modern Sci- 
ence, Garden City: Doubleday-Anchor, 1959. \'ol- 
ume I deals with the fifth through the thirteenth 
centuries; Volume II treats the thirteenth to the 
sexenteenth centuries. 

2.1 I The Aristotelian Theori' of Modon 

The Aristotelian scheme is a complex and highly 
successful one. For approximateK' 2,000 years this 
scheme dominated intelligent thinking. 

Aristotle was. perhaps, the first to realize that an 
explanation of the universe must be based on care- 
ful descriptions and classifications of what is in it. 
He was primaril\ an encxclopedist, and his writ- 
ings were authoritative accounts of what was 
known at the time in such wideh- dixerse fields as 
logic, mechanics, physics, astronomx , meteorologs', 
botany, zoologx', psycholog\', ethics, economics, ge- 
ography, politics, metaphysics, music, literature, 
and mathematics. He was among the first to un- 
derstand and to discuss such things as the prin- 
ciple of the lexer, the concept of the center of grav- 
it\'. and the concept of densitx'. 



BACKGROL.VD A.\D DEMiLOP.ME.VT 



13 



Aristotle's notion that the motion of an object 
moving with constant speed i-equin^s a force pro- 
poitional to the speed is not true for an object 
falling in a vacuum. It is tiue, however, for an object 
moving in a viscous medium, and most terrestrial 
motion is in the air, a viscous medium. Remember 
that vacuum pumps wei-e not invented until nearly 
2,000 years after A/istotle. 

Students certainly should not be required to 
learn the details of the Aristotelian or medieval 
physics of motion. There are, however, some gen- 
eral points that might well be emphasized. These 
are: 

1. The ideas on motion appear as logiccil parts 
of a larger- theory about the nature and structure 
of the universe. In a sense, the "giand stiucture " 
existed first and the various aspects of it could be 
learned by deduction from this grand structure. 
This contrasts with the modern approach in which 
individual topics and disciplines are studied and 
only gr-adually merge into a larger, mor-e compr-e- 
hensive structure. 

2. The rules governing the motion of bodies on 
or near the earlh wert; differ'ent fr-om the riiles gov- 
erning the motion of nonterrestriai objects. Thus 
it was the nature of objects near the earth to be 
stationary once they reached their "natural place." 
There was no conflict in saying at the same time 
that the natural behavior of stars and planets was 
to move continuously in cir'cles. 

3. The Aristotelian scheme was essentially qual- 
itative and nonexperimental. 

The main Aristotelian ideas about motion sur- 
vived for a long time for many reasons. One of the 
reasons was that they did not seem to violate 
"common sense." Even today the instinctive phys- 
ics of most people is pr-obably Aristotelian. For ex- 
ample, r-ecall how hard it is to convince students 
that a 10-kg mass and a 1-kg mass will fall at es- 
sentially the same speed. Recall also the difficulties 
of teaching Newton's fir'st and third laws, not to 
mention special relativity theory or quantum me- 
chanics. 



23, I Galileo and His limes 

The roots of (ialileo s thinking extend far back to 
the drt^ek tradition. He was able to applv the tra- 
ditions of I*v thagoras and Plato to a new context 
and to give them new vitalitv. (Ialileo contribirted 
gr-eatly to shaping the new science, but he did not 
do it alone, and, indeed, he never entirvlv escaped 
from the |)ast. 

A thumbnail biography of Galileo cannot do jus- 
tice* to his colorful life and carver. Stirtlents who 
would like to know morr al)out (ialileo should i)e 
refen'ed to one of the following: 

!,aur-a Fer-mi and (J. hernaclini, ChUIco and the 
Srirnlific licvolulion, New York: Basic Books. Inc., 
19H1. Shori and rvailable. 

I. B. Clohen, "("ialileo," Scicniitic Anwrican, Au- 
gust 1949. 



t. J. Gr-eene, One Hundred Great Scientists, Lean- 
der, Texas: Washington Press, 1964. 

The time-line chart on page 42 is one of a series 
of similar charts that appear throughout the Text. 
These chains are to help students place the indi- 
vidual and the events into the lar-ger context of his- 
torA'. Most students know something about Shake- 
speai-e and ,\Iarv' Stuail and Galileo, but frequently 
they are not aware that these people were contem- 
poraries. We hope that the students will gain more 
perspective by being able roughly to r^elate Galileo 
to his contemporaries: to Elizabeth I, Kepler, and 
to the founding of Jamestov\Ti, the first American 
settlement. 

Under no circumstances should students be re- 
quired to memorize the names or dates appearing 
on these charls. The names and events here rep- 
resent only a sampling. Students might wish to add 
additional names to the chart. 

2^ I Galileo's Two Xew Sciences 

The mention of the Inquisition and Galileo's con- 
finement may stimulate students to riiise questions 
about this whole affair. Most of the controversy had 
to do with the concept of the solar system, that is, 
with Galileo's astronomy. More about this v\ill be 
encountered in Unit 2. 

Since the focus in this chapter is on one aspect 
of Galileo s study of motion at the earth s surface, 
it may be well to defer the more dramatic aspects 
of Galileo's carreer until Unit 2. However, for stu- 
dents who might like to prepare themselves for the 
issue, you can recommend: 

Georgio de Santillana, The Crime of Galileo. Chi- 
cago: The L'niver^ity of Chicago Press, 1955. 

F. Sherwood Taylor, Galileo and the Freedom of 
Thought, London: C. A. Watts, 1938. 

The dialogue of Sagredo, Simplicio, and Salviati 
is a discussion of a new book on mechanics b\ an 
unnamed author who is a friend of theirs. The 
eminent academician" who wrote the book is, of 
course, Galileo, whose views are presented through 
Salviati. 

Several copies of the Dover paperback edition of 
the Crew and de Salvio translation of 'A\o ,Vew Sci- 
ences should be on hand for students who wish to 
locate these ({notations and follow them in greater 
length. 

A technique useil in pr-esent-dav experiments 
can be mentioned in discussing the ar-gument be- 
tween Simplicio and Salviati It is easier to deter- 
mine the dnycrcncc between the outcome of rv\x) 
events when Ihev are simultaneous. For e.xample. 
it is easv to tell which riinner has won a 200-m 
race, cveMi when one leads the other In onlv 1 m. 
It would be morv difficult to make this detenni- 
nation by timing two rimner-s in separate races. 

2.4 I Uln Study thv .\Io(ion of 

Freeh Falling; Bodies? 

This brief section einph.»M/c.s th.Jt our main inter- 
est is in studving th«' approach used bv (ialileo 



14 



UNIT 1 / COXrUTS Ol MOl lOV 



The quotations ftx)m Two A'en Sciences show that 
Galileo himself realized that his work was of sig- 
nificance and that it would lead to a new science 
of physics. 

2^ i Galileo Chooses a Definition 

of Uniform Acceleration 

This section and the three sections that follow it 
deal with Galileo s free-fall experiment. There is 
some danger that the student will get lost before 
reaching the end. For this reason, the opening par- 
agraph of Sec. 2.5 summarizes the o\erall plan of 
attack. Point out, especially to students who are 
not accustomed to invohed derixations or proofs, 
that it is important to consider the plan of attack 
before beginning to stud\' such a logical argument. 
The summar\' land the marginal commentary' 
accompanxing iti should pro\e useful, yet is is po- 
tentially misleading. This, like all summaries, makes 
the events sound more organized and systematic 
than the\' were. Galileo is, after all, gi\ing an e,\: post 
facto description of work that he did o\ er a period 
of years. Furthermore, he embedded that descrip- 
tion in a contro\ersial document that had wider 
aims than mereK to present some research find- 
ings. Take care, therefore, to see that the student 
does not accept these actixities as a model for all 
scientific endeavor. 

2.6 I Galileo Cannot Test His 
Hypothesis DirectU' 

Frequently' a direct test of a particular hxpothesis 
cannot be made. One or more of the quantities 
in\'ol\ed cannot be measured accurateh' because 
the means for making the measurement has not 
yet been established. 

Suppose that, in an attempt to test directh the 
hypothesis that v t is a constant, Galileo had per- 
mission to use a 20-stor\' building 65 m high. Sup- 
pose that he put marks on the building at 1, 4, 9, 
16, 25, 36, 49, and 64 m from the top. An object 
dropped from the top of the building should pass 
these marks at equ£il time intervals of a little less 
than 0.5 sec. 

But obsen-ing that the mari« are passed at equal 
time inters als is not realK' a direct test that \' t is 
constant. For this, he would ha\ e to determine the 
instantaneous speed as the object passed each 
mark and the distance used for these speed meas- 
urements should be ver\' small. Suppose instead 
that he considered a rather large distance, 1 m. 
The time to co\'er the first space interval, ftxjm 0.5 
to 1.5 m, would be about 0.23 sec. If we assume 
that Galileo could measure to within 0.1 sec, his 
probable error for the first distance would be about 
half the quantity he was trying to measure. The 
time elapsed while the object moved through the 
last interval, ftxjm 63.5 to 64.5, would be less than 
0.03 sec. To measure this time interval to 5% ac- 
curac\ would require a clock that one could read 
to about 0.001 sec, at least 10 and probabK 100 



times better than am'lhing Galileo had a\ailable. 
i\o wonder he resorted to an indirect test. 

Now there are many methods available to us, 
such as stroboscopic pictures or electrically driven 
timers, which allow us to test directh' whether or 
not v/t is constant for a freeh' falling body. But all 
of these methods depend on our abilitv to measure 
small time intervals lO.OOl seci with precision. Such 
methods were simpK' not available to Galileo. 

2.7 I Looking for Logical Consequences 

of Galileo's Hypothesis 

Part of the reason for the scientific breakthixjugh 
begun in the sixteenth and sexenteenth centuries 
was use of a mathematical approach to the study 
of motion. 

The Test derives d — air 1 but you should repeat 
the derivation carefully in class. The main point of 
this section is not to teach the deri\ation: it is to 
emphasize the \alue of mathematics in science. 
Simple algebra allows us to arri\ e at a relationship 
that is self-e\ident at the beginning. While the final 
equation contains no new information, it presents 
the information in a different and useful wa\'. For 
instance, it allows us to make predictions concern- 
ing distances tra\eled b\' accelerating bodies that 
are not exident in the parent equations. 

The word constant is used in sexeral ways in 
physics. In the context of this section, constant 
means: in uniform acceleration, the numerical 
xalue of the ratio dt^ is the same lis constant i for 
each and e\er\' interval for which distance and 
time measurements are made, proxided other pa- 
rameters are fixed. The numerical xalue of that 
constant ratio xxill depend on the xalue of accel- 
eration in a particular case. 

2.8 Galileo Turns to an Indirect Test 

This section contains xvhat is probablx the largest 
conceptual leap in Galileo s argument. He assumed 
that the inclined plane xxas primarilx a dexice for 
diluting free fall xvithout changing its fundamental 
nature, and he was able to proceed to e.xperimental 
tests of his hxpothesis. If the students can be made 
to see that this is a reasonable though not neces- 
sarily true assumption, the inclined-plane experi- 
ment should not be difficult. 

Encourage all students to carry out this experi- 
ment. With reasonable effort the students xvill find 
that for anx distance along the incline, the ratio 
dt^ xxill be constant for a gix en angle of incline. For 
practical reasons the experiment is limited to rel- 
atixelx' small angles. 

In summaiy , the main purpose of this section is 
to make the association betxx een the incUned plane 
experiment and the oxerall problem of fi'ee fall. An 
understanding of the actual experiment itself should 
come from the laboratorx'. 

2.9 I Doubts about Galileo's Procedure 

Healthy skepticism is one of the characteristics of 
scientists. Students fi-om the beginning should be 



BACKGROL.VD .4.\D DEX'ELOPMEXT 



15 



encouraged to be critical of scientific claims and 
experiments. In a textbook it is difficult not to 
sound authoritarian fi-om time to time. A section 
such as this, which lists several reasons for ques- 
tioning (ialileo's results, is intended to counteract 
that tendency. 



2.10 I Consequences of Galileo's 
Work on Motion 

The purpose of this section is to show that Galileo's 
work on motion had consequences far beyond the 
particular issues at hand. His contributions to the 
advancement of science were both substantive and 
methodological. 



CHAPTER 3 / THE BIRTH OF DYNAMICS: 
NEWTON EXPLAINS iMOTION 



3.1 I ''Explanation" and the 

Lait's of Motion 

The purpose here is to place the kinds of motion 
we studied in kinematics into per-spective. These 
are motions that N'ewlon's laws will explain. 

Dynamics is introduced by contrasting it to 
kinematics. The distinction between these was 
mentioned in Chapter 2 when Salviati (Galileo) said 
that the time to talk about causes of motion was 
after accurate descriptions existed. 

One of the pr-eoccupations of science is to pro- 
vide systematic explanations of observable phe- 
nomena. Newtonian mechanics iNevvlon's laws of 
motion, the law of universal gravitation, and var- 
ious force functions) represent one such explana- 
tory system. As the students progr'ess through this 
chapter, studying the three laws individually, they 
should be r-eminded from time to time of this over- 
all theme of explanation. 

3J3 I The Aristotelian Explanation 

of Motion 

Aristotelian ideas concerning motion should be 
presented for better appreciation of the Newtonian 
development. Contr'ast between the two will and 
should be made. Aristotelian and so-called "com- 
mon-sense" observations shoirld be connected. 

3.3 I Forres in Equilibrium 

Deveioj) the ideas of uiihalaiiced forx:e and equilib- 
rium for- the condition of rt\st. This will set the 
stage for the e(|uilibrium condition of constant ve- 
locity with no unbalanced forces to be considered, 
in Sec. AH. 

3.4 I About W'ctors 

The concept of a vector is developed briefly in this 
section. The 7r,\/ states, birt does not show, that 
acc('l(Mation can he ti-eattui as a vector. Actuallv the 
section will not stand alone as a way of teaching 
vectoi-s. It is necessary that all students under-stand 
what vector cjuantities art* and whv they ar-e im- 
portant. Ihev should also be able to do vector ad- 
dition and subtraction graphically. 

3.5 I Xeulon's First Lau of .\Iotion 

i'ln^ signili(an((> of the liist law ot motion cannot 
be overstated. Most physics textbooks point out 



that the first law is reallv a special case of the sec- 
ond one, since by the second law the acceleration 
is zero if the force is zero. While this is true, it 
misses the point. 

The law of inertia is fundamental to modem 
mechanics, for it states what is to be the starling 
place of the entire theory of motion. The first law 
makes it perfectly clear from the beginning what 
is to be basic, what requires further explanation, 
and what does not. In so doing, the law of inertia 
dramatically exposes the difference between the 
Newtonian system and the Aristotelian system. 

The main points to be emphasized about the 
first law of motion are these: 

1. Fundamentally, the law is a definition. It 
states the convention to be followed in studving 
forces. Forces are not to be considered as the 
causes of motion but rather as whatever creates 
acceler-ation. 

2. The law of inertia cannot be proven bv obser- 
vation or experiment. One reason is that the ordi- 
narv method for deciding whether or not there are 
unbalanced forx"es operating is to observe w+iether 
or not there is acceleration. 

Many teachers demonstrate the plausibility of 
the first law bv showing that, as the retarding fric- 
tion on a moving bodv is reduced, the object ap- 
pear-s to behave more and more in accortiance with 
the fir-st law. The teacher should be extremelv care- 
ful not to pass off a dassrxjom demonstration as 
prxjof of the fir-st law: Definitions cannot be prxjven 

3.6 I The Significance of the First Law 

This per-sents one of the fir"st opporlirnities in the 
cour-se to delve into deep philosophical issues. Stu- 
dents are usuallv fascinated by the fir-st law land 
somewhat skeptical aboirt it', and are more than 
willing to discuss several of the issues listed in this 
section, especially frame of reference and univer- 
sality . 

3.7 I Xeulon's Second Lau° of .\Iocioii 

Forre and mass are ver^ difticult conct'iits to mas- 
ter, rhis section postpones a definition of those 
terms and avoids consideration of the empirical 
content of the second law Vhc section does not 
explain the equation 



Hi 



I'MT 1 / COXCHmS OF IVI()TI()\ 



The student should understand that for a single 
object, a is proportional to F. For different objects 
acted on by a constant force, a is in\ersely pro- 
portional to m. We want the student to realize that 
if the second law is true, then certain mathematical 
relationships must exist between force and accel- 
eration and between mass and acceleration. 

The neulon is the only force unit mentioned in 
this chapter. Perhaps the student should know 
that there are other force units depending upon 
the system of units being employed. However, 
there is little to be gained from comparing them or 
being able to conxert from one to the other. 

Perhaps it is worth noting that implicit in 
the equation F = ma is a proportionality constant 
that does not appear because it was set equal to 
1 (F = kma and k = li. An alternatixe approach 
would ha\ e been to define a standard unit of force 
as well as standard units of mass and acceleration, 
and then do experiments measuring force, mass, 
and acceleration from which the \ alue of the con- 
stant could be computed. Examples of this ap- 
proach include G in the unixersal gra\itation equa- 
tion, and the spring constant in Hooke's law 
equations. 

The first law of motion is mathematicalK' a nec- 
essary' consequence of the second law, while the 
re\erse is not true. One can conceive that the sec- 
ond law of motion might ha\ e been \'er\' much 
more difficult to formulate. 

The first law does not really take on meaning 
and is not at all useful in the real world of physics 
until certain additional operational definitions ha\e 
been given for such terms as rectilinear motion, 
equal time intervals and constant speed. Also, a 
frame of reference must be established for all the 
measurements. 

The main point to be emphasized is that, while 
the first law provides a general explanation of an 
event, the second law provides a quantitative and 
therefore more useful explanation. For example, 
when we can say that an object slows dov\Ti be- 
cause there is a retarding force of 4.0 \ acting on 
it, we know a great deal more than when we mere- 
ly say that it slows down because there is a retard- 
ing force. 

3.8 I Mass, Weight, and Free Fall 

Why do all objects in free fall at a given location 
fall with the same acceleration a^? The answer is 
in the proportionalitv' bervseen weight F^ and mass 
m. It is imperative, therefore, that students under- 
stand the distinction between weight and mass. 
The only thing that really makes this difficult is 
that they are accustomed to using the terms inter- 
changeably and usually think of weight as a meas- 
ure of mass. iThere is some dispute among physics 
educators about how weight ought to be defined. 
Here we are using weight as synonymous with 
gravitational force. i Perhaps the relationship be- 
tween mass and weight is easier to understand if 
the second law equation is put in the form 



'^ = m 
Then it is clear that the force is proportional to the 
mass: no matter what the value of the mass, the 
acceleration will remain constant. 

Why in a given location do objects fall with uni- 
foiTH rather than nonuniform acceleration? That 
the acceleration is constant is an experimental fact 
described by the second law. 

3.9 I Neftton's Third Law of Motion 

The purpose of this section is to enable the stu- 
dents to understand what the third law says. 

Once students grasp the notion that forces al- 
ways appear or disappear in pairs due to the in- 
teraction of real objects and that the two forces act 
on different objects, then the rest is not difficult. 
However, these ideas are contrarv to evervday ex- 
perience and are not easily accepted. The tremen- 
dous inertia of the earth, the ever-present forces of 
friction, and the imperceptible distortion of rigid 
objects I like floors and walls i all help conceal ac- 
tion-reaction. 

The third law allows one to examine a small part 
of a complex chain of events. Students find it hard 
to believe that the earth can exert a force on a 
runner. One wav of demonstrating the value of the 
third law at this point is to ask students to invent 
an explanation for the acceleration of a runner that 
is not like the third law in form. 

If forces are equal and opposite, how can an ob- 
ject accelerate? The point to be emphasized in 
going over this example is that the opposite forces 
act on different objects. In discussing the forces 
between two objects in a system, the third law is 
needed. It describes the location and magnitude 
of various force pairs. On the other hand, when 
one becomes interested in the motion of a partic- 
ular object, then one must ask about the net un- 
balanced force acting on that object and apply the 
second law to determine its acceleration. Distin- 
guishing which law to use is not easv and students 
should be furnished with other examples. 

3.10 I Using Xenton's Lans of Motion 

In many real situations the laws of motion are used 
together. The first and third law help keep the 
qualitative situation clear and the second law per- 
mits a quantitative analvsis. Two examples of the 
usefulness of the three laws in dealing with real 
physical situations are presented in this section. 
The main purpose is to demonstrate the applica- 
tion of the laws of motion and not to make all stu- 
dents highlv skilled at numerical problem solving. 

3.11 I Nature's Basic Forces 

Treat this section as a reading assignment. It gen- 
ereilizes and extends the laws of motion and helps 
introduce Chapter 4. The four basic interactions in 
nature are mentioned to reduce the complexity the 
world so far seems to present and to point to what 
lies ahead in the studv of physics. 



BACKGROL.VD AND DE\T;L0P,ME\T 



17 



CHAPTER 4 / UNDERSTANDING MOTION 



4.1 I A Trip to the Moon 

What is simple and what is complex is not alto- 
gether easy to decide. Ceilainly one s puiposes for 
making such distinctions will have something to 
do with the ciiteria. In general, there are three cri- 
teria heing used in this unit. The fir-st two are uni- 
fonnity and symmetry. If the parameters that de- 
scrihe the motion are uniform or constant in value, 
or if the path of motion is symmetrical, the motion 
is regarded as simple. ' 

The thiid criterion has to do with dimensional- 
ity. Motion hecomes more complex as you go from 
one or two to three dimensions. By this standard, 
projectile motion and uniform circular motion can 
he considered as heing more complex than recti- 
linear motion and less complex than, say, helical 
motion. 

4.2 I Projectile iVf otion 

Not much is gained by emphasizing the definition 
of projectile motion. Students should under-stand 
that a projectile is an object moving through space 
without the aid of any self-contained motKe power. 

The historical significance of the problem of pro- 
jectile motion does not receive as much emphasis 
as it could in this section or in the chapter as a 
whole. Many historians of science feel that it was 
one of the key issues in the whole controversy o\er 
the nature of motion. Aristotle s theory was least 
able to explain projectile motion. 

The concept of independent horizontal and ver- 
tical motion may be difficult for- students to accept 
because it conflicts with their common-sense no- 
tion that horizontal speed affects the r-ate of fall. 
The stirdenls should car-ry through the analysis. 
Demonstrate the appar^atus that prt)jects one spher-e 
and dr-ops another at the same instant. iSee Hand- 
bonk page 28.1 Also, students should make their 
own measirrtMiients, or at least see measurements 
mad(^ on pholografjlis or- transparencies similar- to 
the one on 7e,\f page 105. 

■I\vo major and (juite separate points need to be 
made: 

1. It is an undeniable experimental fact that a 
shor-t-range prt)jectile launched horizontally will 
reach the ground at the same time as a similar- 
object dixjpped at the same instant from the same 
height. I his fact, that the giavitationai acceleration 
of a prxjjectile is e.xactlv the same as the gravita- 
tional acceleration of an objei:t falling frvel\' from 
r-est, comes fr-om observation, not from deduction 
from fir-st principles. 

2. This experimental fact can be explained or 
rationalized by assirriiing that the ohser\-ed motion 
of a projectile is the vector sum of two other mo- 
lions that are completely independent of each 
other-: uniforrn horizontal motion and accelerated 
vjM-tical motion 

Students who ar-t' interested in pr-ojectile motion 
and who uruici-stand some irigonomctrA ought to 



be encouraged to analyze the general case of pro- 
jectile motion; the case in which the projectile is 
launched at any angle. Once they have derived 
what they believe to be general equations, they 
should show that the equations used in this and 
the next section can be deduced from the general 
equations. 

4.3 I Ulial Is the Path of a Projectile? 

rhe purpose of this section is \o establish and 
demonstrate the power of mathematics in science 
and to justifv the need for continued scholarly 
work in pure mathematics. In order to do this ef- 
fectively, it is important that prior to the derivation 
of the equation of the parabola, two other points 
be made. 

1. There is no a priori r^eason to favor one curve 
over another. In fact, there is no r^eason even to 
suppose that the trajectory of a projectile will al- 
ways have the same mathematical shape. 

2. The question cannot be settled simply bv ob- 
serving the paths of projectiles with the unaided 
eye. For one thing, the angle of observation and 
problems of perspective make obserAation difficult. 
Secondlv , manv mathematical curves look very 
much alike and can be distinguished onlv by anal- 
ysis. Finally, many objects that are thrown do not 
follow a parabolic path because of the lai^e and 
changing air r-esistance they encounter along the 
flight path. 

The difficulties in determining the shape of the 
projectile s trajectories can be easilv demonstrated 
by throwing objects inside the classroom, or. pref- 
erably, out on the pla\ ing field. See Project Physics 
Transparenc\' 110 Path of a Prxjjectile. 

The su|)por1 that experiment gives to the purely 
mathematical combination of motions can provide 
the student with evidence that mathematical ma- 
nipulation of svmbols that express known princi- 
ples can lead to new r^elations among the svm- 
bols — r-elations that are also tr\re. 

\\ ith a particularlv able class, the teacher might 
wish to develo|) the r-elationship berween the range 
of a projectile and its velocitv and angle of tire. This 
pr-oblem could also be assigned as a project for 
better stirdents who ar-e familiar with trigonometry, 
rhev should trA to derive the general projectile 
ecjuation: 



(i. = tan H (i. -t- ' 



'^' \v;cos2e/ 



See "Projectile Motion, in Foundations of Mmicm 
Physical Science bv Ci. Holton and D. Roller i.Addi- 
son-W'eslev , 1958i 

4.4 I Motin^ Frames of Reference 

rhert' ,uv two i-cl.itcd p()iiit> to be made in this 
section fir-st, therv is the Cialrlean r-elativitv prin- 
(iple which is merviv a formal statement or gen- 
eiali/.ation of the oliserAable fact tliat mecfianiral 



IN 



I'MT 1 / COMCKITS (IF IVI()ri()\ 



experiments gi\e the same results no matter what 
the I constant I \elocit\' of the laborator\ may be. 
The second point is that the laws of motion are the 
same for all reference frames mo\ing uniformly 
with respect to each other. 

Make certain, ho\ve\er, that students realize that 
the appearance of am motion the\ see does de- 
pend on the relative motion of the \iewer. The 
Perception of Motion, b\' H. Wallach, in Scientific 
American, July 1959, may be of interest. It concerns 
the fact that people \ie\v relative motion as if it 
were absolute. 

4^ I Circular Motion 

This is an introduction to the terminolog\ of cir- 
cular motion and does not desen e great emphasis 
in class. .AJthough it is not taken up in the Te\t, it 
is probably worthwhile to demonstrate the diffi- 
cult\' of deciding whether or not an object is in 
circular motion when observing it from a frame of 
reference that is mo\ing with respect to the object. 
For some interesting results of being located on 
the earth in a noninertial frame of reference, see 
the cirticle b\' J. McDonald, The Coriolis Effect," in 
Scientific American, Ma\', 1952. 

4.6 I Centripetal Acceleration 
and Centripetal Force 

The difficultx in this section is to show that the 
acceleration of an object mo\ing uniformly in a cfr- 
cle is truK' centripetal. The Test gi\es onl\' a plau- 
sibilit\' argument; and, while this v\ill convince 
some students, there ma\' well be skeptics. In most 
classes it ma\' be worthwhile for \'ou to go through 
the derivation on the chalk-board, or use Trans- 
parency 11. 

Material is included to provide the students with 
practice in thinking in terms of vectors. It also pro- 
vides an opportunity' to review and compare the 
three kinds of motion considered: rectilinear, pro- 
jectile, and circular. 

This might also be ^m appropriate time to sug- 
gest that circular motion is really a special case of 
projectile motion. This can be done in one of two 
ways. One wav is to compare the vector relation- 
ship of an object moving with uniform circular 
motion with the v ector relationship of a projectile 
at the top of its trajectorv . The second wav is to 
approach it through the use of a diagram such as 
that used bv Newton ithe figure on page 112'. 

The relationship a = v^ fl is used to carrv out an 
arithmetic solution to the same uniform circular 
motion problem solved graphicallv in the te.xt. This 
should provide some of the brighter or more math- 
ematicallv inclined students with an opportunity 
to check it out for themselv es. B\ no means should 
all students be held responsible for studying the 
derivation. 

If a sample problem is worked in class, an inter- 
esting e.xample might be to find the acceleration of 
a point on the earth s equator due to the rotation 
of the earth ifi = 6,400 km, f = 24 hr = 8 64 x lo^ 



sec). The result of this calculation can be compared 
with the value of the acceleration of gravitv . The 
question can then be asked: What would happen 
if the earth were rotating at a speed such that the 
centripetal acceleration were equal to the accel- 
eration of gravitv?' This idea will be taken up again 
in this chapter. 

4.7 The Motion of Earth Satellites 

There is no new phvsics in this section. What the 
student has learned about circular motion up to 
this point is almost entirely theoretical or, at least, 
deals with cases, such as a blinkv on a turntable, 
about which most students care ven little. 

The satellite Alouette is used because it has a 
nearly circular orbit and because it has some his- 
torical significance. 

The trouble with such a list as the one in Table 
4.2 is that it becomes obsolete almost as fast as it 
is printed. .\n effort was made to select satellites 
of continuing interest. Perhaps some students 
should be assigned the task of finding the addi- 
tional entries needed to bring the list up to date. 
See Sk\' and Telescope magazine for frequent arti- 
cles on satellites. 

The important questions that will reallv show 
whether the student has learned what has been 
covered up to this point are Whv does the satellite 
not fall back to earth?" and Why does a satellite 
not fly off into outer space? 

After most of the students in the class hav e been 
able to answer these questions successfully in 
terms of the kinematics of circular motion, you 
mav impress them with the progress the\ have 
made bv asking them to respond to those ques- 
tions as the .Aristotelians probablv would have. 

The section ends bv suggesting that speed, dis- 
tance above the earth, and period of rotation are 
not independent v ariables. This is probablv not the 
time to take up questions about how satellites get 
fTX)m one orbit to another, and what effect this has 
on their speed. However, it mav not be possible to 
avoid the issue altogether, especiallv if some dra- 
matic event has recently happened. 

.Although satellite orbits in particular planetan- 
orbits I will be taken up in greater detail in Unit 2, 
the statement is made in this section that at a par- 
ticular height a satellite must hav e a certain veloc- 
ity' in order to maintain a circular orbit. The ques- 
tion Whv have all of the satellite launchings to 
date been in an easterly direction? is useful. The 
answer involves having the students think about 
vector addition of v elocities in terms of a frame of 
reference related to the center of the earth rather 
than the more familiar frame of reference of the 
individual on the surt'ace of the earth. 

4.8 I Whaat About Other Motions? 

This section should be treated merely as a reading 
assignment. Its only purpose is to remind the stu- 
dent that there are many interesting kinds of mo- 
tion that we have not dealt with. 



BACKGROLND AVD DEX'ELOPMEVr 



19 



Brief Descriptions of Learning Materials 

SUMMARY LIST OF UNIT 1 MATERIALS 



Experiments 

El-1 i\aked-Eye Astronomy 

El -2 Regularity and Time 

El-3 Variations in Data 

El-4 Measuring Uniform Motion 

El -5 A Seventeenth Centuiy Experiment 

El-6 Twentieth Century Version of Galileo's 

Experiment 

El -7 Measuiing the Acceleration of Gravity a^ 

El -8 Neu'lon's Second Law 

El -9 Mass and Weight 

El-10 Cuives of Trajectories 

El-11 Prediction of Trajectories 

El-12 Centripetal Force 

El-13 Centripetal Force on a Turntable 

Demonstrations 

Dl Recognizing simple motions 

D2 Uniform motion, using accelerometer and 
dynamics cari 

D3 Instantaneous speed 

D4 UnifonTj acceleration, using liquid acceler- 
ometer 

D5 Comparative fall rates of light and heavy 
objects 

D6 Coin and feather 

D7 Two ways to demonstrate the addition of 
vectors 

D8 Direction of acceleration and \elocity 

D9 Direction of acceleration and velocity: an air- 
track demonstration 

DIO Noncommutative rotations 

Oil Newton's first law 

D12 Newton's law experiment lair track) 

D13 Effect of friction on acceleration 

D14 Demonstrations with rockets 

D15 Making an ineitial balance 

D16 Action-r(;action forces in pulling a rope. I 

D17 Action-it;action forces in pulling a ixjpe. II 

D18 Reaction foixu^ of a wall 

rJ19 Newlon's thiixl law 

D20 Action-rtuiction forces bet\\'een car and r-oad 

D21 Action-ieaction forties in hammering a nail 

D22 Action-it'action forces in jumping upwaixls 

D23 Frames of refei-ence 

D24 Inertial vei-sus noninertial roference frames 

D25 Unifonii cii-cular motion 

D26 Simple hamionic motion 

D27 Simple hannonir motion: aii- tiack 

Film Loops 

I-l Acceleration Due to (iraxity 1 
E2 Acceleration Due to (ira\ity. il 
1,3 \'ector Addition: \'el()cit\' of a Boat 
L4 A Matter of Relative Motion 
I>5 (ialilean Relali\it\': Rail Di-op|)ed from Mast of 
Ship 



L6 Galilean RelatKity: Object Dropped from Air- 
craft 
L7 Galilean Relativitx': Projectile Fired Vertically 
L8 Analysis of a Hurdle Race. I 
L9 Analysis of a Hurdle Race. II 

Reader Articles 

Rl The Value of Science 

by Richard P. Fevnman 
R2 Close Reasoning 

by Fred Hoyle 
R3 On Scientific Method 

by P. V\'. Bridgman 
R4 How to Solve It 

by G. Polya 
R5 Four Pieces ofAch'ice to Young People 

by V\'arren W. VVea\'er 
R6 On Being the Right Size 

by J. B. S. Haldane 
R7 Motion in Words 

by J. B. Gerhart and R. H. .\ussbaum 
R8 Motion 

by R. P. FexTiman. R. B. l^ighton, and 

M. Sands 
R9 The Representation of Movement 

by Gyorg\' Kepes 
RIO Introducing Vectors 

by Banesh Hoffmann 
Rll Galileos Discussion of Projectile Motion 

by G. Holton and D. H. D, Roller 
R12 Newton's Laws of Dynamics 

by R. P. Fevnman, R. B. Leighton, and 

M. Sands 
R13 The Dynamics of a Golf Club 

by C. L. Stong 
R14 Bad Physics in Athletic Measurements 

by P. Kirkpatrick 
R15 777e Scientific Revolution 

by Heri)ert Butteriield 
R16 How the Scientific Revolution of the Seven- 
teenth Century Affected other Branches of 
Thought 

by Basil Willey 
R17 Report on Tails Lecture on Force, at British 
.\ssociation, 1676 

b\' James Clei-k Ma.xwell 
R18 Fun in Space 

by Lee A. Du Bridge 
R19 The Vision of Our Age 

by J. BiDtiowski 
R20 Becoming a Physicist. b\ .Anne Roe 
R21 Chart of the Future. b\ Arthur C. CMarke 

Sound Films (16 mm) 

PeopU' and I'ariicles 

Elect n)n S\ tichi-otitin 

The WoiUi of Eniico lemii 



ZO 



UNIT 1 / (X)\(:i:i»TS oi- ivurnox 



Fl Straight-Line Kinematics 

F2 Inertia iPSSLi 

F3 Free Fall and Projectile Motion iPSSCi 

F4 Frames of Reference iPSSCi 

F5 X'ector Kinematics iPSSC) 

Transparencies 

TO Using Sti"oboscopic Photographs 
Tl Stroboscopic Measurements 



T2 Graphs of Various Motions 

T3 Instantaneous Speed 

T4 Instantaneous Rate of Change 

T6 Derivation of d = v,f + '/a at' 

T8 Tractor-Log Problem 

T9 Pi-ojectile Motion 

TIO Path of a Projectile 

Til Centripetal Acceleration: Graphical 
Treatment 



FILM LOOPS 



Quantitative measurements can be made with Film 
Loops marked (Lab), but these loops can also be 
used qualitatively. 

LI ACCELERATION DLT TO GRAMTY. I 

Slow-motion photograph\ in one continuous se- 
quence allows measurement of average speed of a 
fcilling bowling ball during two 50-cm intenals sep- 
arated by lo m. I Lab I 

L2 ACCELERATION DLT TO GRA\TT1'. H 

Slow-motion photographx' allows measurement of 
average speed of a falling bowling belli as it passes 
through four 20-cm intenals spaced 1 m apart. 
iLabi 

L3 \TCTOR ADDITION: 
\TLOCIT\' OF A BOAT 

A motorboat is \iewed from abo\e as it moxes up- 
stream, downstream, acixjss stream, and at an an- 
gle upstream, \ector triangles can be drawn for the 
\arious velocities. (Lab) 



L4 A MATTER OF RELATRT MOTION 

A collision between two equalh' massixe carts is 
\iewed from \arious stationary eind moving frames 
of reference. 



Lo GALILEAN RELATRITI': BALL 

DROPPED FROM MAST OF SHIP 

A realization of the experiment digested in Galileo's 
Dialogue on the Two Great World Systems; the ball 
lands at the base of the mast of the moving ship. 

L6 GALILEAN RELATmT\': OBJECT 
DROPPED FROM AIRCRAFT 

A flare is dropped from an aircraft that is flying 
horizontally. The parabolic path of the flare is 
shown, and freeze frames are provided for meas- 
urement of the position at ten equallv spaced in- 
tervals. (Labi 

L7 GALILEAN RELATHTTi': PROJECTILE 
FIRED \TRTICALLY 

A flare is fired \ ertically from a Ski-doo that moves 
along a snow-co\ered path. Events are shown in 
which the Ski-doo s speed remains constant and 
in which the speed changes after firing. 

L8 ANALYSIS OF A HLTiDLE RACE. I 

Slow-motion photography allows measurement of 
speed variations during a hurdle race. I Labi 



L9 ANALYSIS OF A HLTU3LE RACE. II 

A continuation of the preceding loop. (Lab) 
Note: A fuller discussion of each Film Loop and suggestions 
for its use wiU be found in the section of this Resource 
Book entitled "Film Lxjop .Notes." 



SOUND FILMS (16 mm) 



FILM NOTES ON PEOPLE AXD PARTICLES 



In planning this course and preparing all the nec- 
essarv texts, laboratory equipment, film loops, 
teacher guides, and so forth, we felt that we should 
also make a film of what it is like to be working on 
a real physics problem at the research frontier. We 
did not want to fUm a set-up interview or a pre- 
pared lecture; we wanted to show people who are 
working in science. 

Of course, we could choose onlv one out of the 
great varietx' of physics research problems of inter- 
est today. Our preference was to film a group of 
moderate size to show the variety of people in- 
x'olved in experimental work. -Also, we had to select 
a problem that was not too difficult to understand, 



since the first showing of the film will be near the 
beginning of the course. 

We decided to focus the camera on a group of 
Harvard Universitv' students and professors strug- 
gling with their work at the Cambridge Electron 
-Accelerator iCEAi. For over 2 years eventhing was 
filmed as it happened. iWith regret we report that 
the Cambridge Electron Accelerator has been dis- 
mantled. As the cost of building and operating even 
more powerful machines increased, money was 
not available to continue operating many older 
machines, so they were dismantled. i 

The film traces the work of the participants in 
the experiment as they design construct, and as- 



LKARMNG MATERIALS 



21 



semble the equipment to a point where the physi- 
cists are pi-epai-ed to tai<e actual data. The experi- 
ment itself took several more months. As this 
course is pailicularly concerned with the human 
element in the making of science, it is appropriate 
that the film should not emphasize advanced 
physical theory. Instead, it concentrates more on 
the style of work in a lab, on the men and women 
who are working together, and on some of the joys 
and pains of doing original scientific work. It raises 
a number of themes, from the international char- 
acter of science to the fact that work on this scale 
requires a great range of skilled people, including 
shop machinists, scientists, engineers, secretaries, 
and so forth. 

Most people have no way of being directly in- 
volved in any scientific work and cannot look o\er 
the shoulders of scientists. If they could, even 
through such a film, there might be fewer strange 
and false notions about work in a laboratoiy; false 
notions of exaggerated glamor, just as much as of 
dark doings. We hope our film shows that work on 
a real research problem, whether in physics, in 
other sciences, or in any field, can be a truly hu- 
man enterprise. 

An extensive guide to the themes and physics of 
the film has been prepared, which can be studied 
in connection with a second showing of the film 
late in the coui*se. Refer to the separate document 
entitled People and Particles. Furthemiore, the 15- 
minute film Electron Synchrotron gives additional 
technical infonnation about the Cambridge Klec- 
tron Accelerator, liven before the first showing, 
however, the following brief notes will be helpful 

The terni pair production lefeis to the pioduc- 
tion of a pair of elementaiy particles, an electi-on 
and an anti-electron (a positron). Under the right 
conditions such a pair of particles may be pro- 
duced when a very energetic packet of light, a pho- 
ton Iwhich may also be considered an elementarv 
particle), passes near a massive object, such as an 
atomic nucleus. The electn)n beam fi-om the Cam- 
bridge Klectron Acceleratoi' is used to produce the 
particle pairs by a several-step process. The motion 
of the particles is d{;t(;cted with spark chambers in 
which electric sparks jumjj along the paths of the 
particles. 

The Stix)uch-VV'alker group (named for the two 
physicists who head the group. Professors Karl 
Strouch and James W'alkeri is using the 
electitjn-positron pail's to look into a newK sus- 
pecteul flaw in one of the most solidly established 
theories of physics. They are trying to establish 
whether the present theoiy of the influence of elec- 
trical (rharges on one another is correct. Some 
physicists think the present theoiy is bound to 
show its limitations when it bc^comes possible to 
expeiiment with charges that ai-e e.\tirmel\ near 
each other 

Ibis proccdur-j' has often occ iiin'd in science. 
B\' testing the limits of a theoiy. bv looking into the 
conlr adictioMs between the exper inuMilal result 



and the theoretical predictions, the scientists are 
led to new theoretical structures. Of the several 
themes that run through the film, the relation of 
research to education, the international character 
of science, communication among scientists, and 
the relation of science to technologv', the last is 
particularly evident on the first showing. Pure 
physics of the last 100 years or so has made pos- 
sible the development of such practical de\ices as 
the oscilloscope, the electronic computer, the scin- 
tillation detector, the high-\oltage generator, and 
the electron accelerator itself. These devices are 
being put to use in this experiment to produce 
more advances in pure physics; without these 
technological devices the performance of this ex- 
periment would be impossible. Conversely, with- 
out the development of physics itself, these devices 
might never have been invented. It is a ne\'er-end- 
ing interplay. 

Most of the pieces of equipment seen in the film 
are technological devices that are not based on the 
physical laws being tested. Such equipment is 
often refen-ed to as "hardware." It is the construc- 
tion of the hardware that is so costiv, and yet it 
would be quite impossible to do experiments like 
this v\ithout it. 

In a sense, the film does not and cannot show 
new physics "being done." It shows construction 
of new equipment, on the basis of known laws. 
Fr'om the operation of this equipment the new 
physics ' will be fashioned in the minds of the ex- 
perimenters. 

FILM NOTES OX ELECTROX 
SY\CHROTROX 

Electron Synchrotron is a 15-minute film showing 
one of the world's great accelerators, the 6-billion- 
electron-volt Cambridge Electrxjn Accelei-ator iCLA'. 
This accelerator in Ciambridge, Massachusetts, was 
operated jointiv b\ the Massachusetts Institute of 
Technologx' and Harvard I'niversitv . 

Dr. William A. ShurxMifT. Senior Researx'h .Associ- 
ate at the CEA, is the narrator I'sing simple dia- 
grams, he explains the principle of operation of the 
accelerator. Then he conducts a guided tour of 
the accelerator itself, showing and explaining how 
electrons arv injected into a GO-m-dlameter oi4)it 
how the\ art' accelerated almost to the sfieed of 
light while makiiig 10.000 tirnis arx)und the or-liit 
and how they are then dir-ected into a 90-m-long 
Kxperimental Hall wherx' they are used in high- 
eneigv physics experiments. 

The enormous size of the acceler-ator and its 
clean, functional design are dr-amatically evident: 
and the phvsical principles of its operation are in- 
dicated. 

I he film ends with some glimpses of the t>i)e of 
mammoth experimental ecjuipment that is neces- 
sary for anaUzing the new kinds of fundamental 
particles that arv crvated wlien th«* (i-billion-volt 
(•leitix)ns collide with a small targel 



Z2 



I'lviT 1 / (:()\c;i:pt8 (if motion 



The mo\ie is designed for high school or college 
students, and ma\' be used at the beginning of or 
during their first course in ph\sics. No prior un- 
derstanding of accelerators or of ph\sics is as- 
sumed. The mo\ie Electron Synchrotron was de- 
signed as a companion to the 28-minute black-and- 
white mo\ie People and Particles, also prepared b\' 
Project Physics. The latter film concentrates on one 
actual experiment performed b\' a team of ph\ si- 
cists at the CEA, but does not go into the details 
of operation of the accelerator. Thus, the two films 
complement one another. Together, the\ gi\e the 
\iewer a direct feel for how particles ai-e acceler- 
ated to high energv and how the\- are then used 
in explorations of the fundamental particles of na- 
ture. 

Further data on the operation of the CEA ai^ to 
be found in the Film Guide to the documentar\' 
film People and Particles. 

The following questions and answei"s will serve 
as a discussion guide: 

1. Why accelerate the electron? 

To gi\'e it great energ\-. An electron accelerated 
to almost the speed of light has an energ\- con- 
centration exceeding anything known to us 30 
years ago. 

2. Why choose the electron to accelerate? 

It is the easiest kind of particle to accelerate 
since it has an electric charge and extremeK lit- 
tle mass. We don t \et know how to accelerate 
neutral peirticles, and to accelerate a much more 
massi\'e particle, such as the proton, requires far 
more time and effort. 

3. Is the great concentration of energ\' resulting 
irom particle acceleration of any special use? 

A particle ha\ing a concentration of se\eral 
billion electron-\ olts of energy' can break up any 
existing atomic nucleus. More importanth', it 
can transform an\' of the smallest parts of the 
nucleus ithe so-called fundamental particlesi 
into particles of new and e\ en more puzzling 
t\pes. B\ producing and anahzing such new 
particles, high-energ\' phxsicists are beginning 
to find out what matter is and does on the small- 
est scale known to us. Also, by means of this 
de\ice physicists are beginning to understand 
these subnuclear particles in terms of charge, 
spin, and angular momentum. 

4. In an electron s\'nchrotron, what keeps the elec- 
trons traveling in a curved path? 

A long series of magnets arranged accurateK' 
in a circle supply vertical magnetic fields that do 
this with no friction at all. The magnets neither 
speed up nor slow down the electrons. The\ 
control only the shape of the path, which is a 
circle in this accelerator. 

5. What makes the electrons speed up; that is, what 
accelerates them? 

Intense electric fields are fluctuating loscillat- 
ingi; and the timing of the oscillations is con- 
trolled accurateK' so that, as the electrons come 



by, the electric forces are in a forward direction. 
The electric fields are prxDduced at certain spe- 
cial locations along the circular orbit, in copper 
chambers called radio frequency ca\ities, or rf 
ca\ities for short. 

6. Wh\' bother to make the electrons traxel in a cir- 
cular orbit? Wh\' not let them travel straight 
ahead, along a straight line? 

If each electron tra\ els in a circular oit)it and 
travels around this oit)it several thousand times, 
each of the rf cavities pushes it forward several 
thousand times. But if the electrons alwavs trav- 
eled straight ahead, thev would pass through 
each rf cavitv' onlv once and each rf cavitv would 
give onlv one push. To get enough acceleration, 
vou would hav e to provide thousands of rf cav- 
ities, instead of only 16, as at the CEA. Thou- 
sands of rf cavities would cost manv millions of 
dollars, and pi"oviding power to thousands of rf 
cavities would also be verv expensive. iNote: The 
electron accelerator at Stanford L'niversitv in 
California, which uses straight-ahead accelera- 
tion and thousands of rf cavities, costs SIOO- 
000,000, eight times the cost of the CEA. It has 
of course, advantages of its own, too. For exam- 
ple, the flow of electrons is continuous and 
strong instead of in pulses and relatively weak.' 

7. Is there any limit on how fast the electrons can 
go? Can thev be accelerated to speeds exceeding 
the speed of light? 

According to Einstein's principle of special 
relativity', no object can travel as fast as light does 
in a vacuum. Electrons in electron accelerators 
reach speeds almost but not quite equal to the 
speed of light. Electrons accelerated at the CEA 
reach a speed of 0.999,999,99 c, where c stands 
for the speed of light labout 3 x lo** msecL 

8. As the CEA svnchrotron is so huge 1 72 m in di- 
ameter' and so complicated does it not use very 
complicated phvsical principles? 

No. The principles are exactlv the same simple 
ones that can be demonstrated in a laboratory 
with crude, ine.xpensive apparatus. The people 
who designed the CEA accelerator made mag- 
nificent use of simplv phvsics. 

9. How about the e.xperiments in which the high- 
energy electrons strike a target made of hvdro- 
gen atoms and produce new kinds of particles? 
Are the results exactlv predictable here too? 

\o. This is just the pointi Phvsicists do not vet 
know all the basic principles of particles. They 
don t know vet how manv kinds of particles 
there are. They don't know vet how manv dif- 
ferent families of particles there are. Thev don't 
know yet how manv kinds of properties the par- 
ticles can displav . Thev don t know vet how 
mam kinds of forces there are. These are among 
the most exciting current problems in science, 
and manv phvsicists are working on them todav . 
Further data on the operation of the CEA are to 

be found in the Film Guide to the documentarv 

film 'People and Particles 



LEARMXG \UTERIALS 



23 



HLM NOTES ON THE WORLD 
OF ENRICO FERMI 

The film Ihe VVoikl of Knrico Fermi" is a docu- 
mentary film, primarily for pedagogical purposes, 
produced by Haivaid Pi-oject Physics with a grant 
from the Fotd Foundation. Its main aim is to give 
the viewer- an appreciation of the life and contri- 
bution of a nearly contemporary physicist, one 
who was widely honored and loved, and whose 
work helped to transform not only physics and the 
style of doing science, but even the course of his- 
tory itself. 

The film serves several purposes. Viewers meet 
a number of the for-emost scientists in documen- 
tary film footage or stills. Ihey see some of the 
equipment Fermi used and glimpse the way Fermi 
made teams work so well. They see the locations 
at which FerTiii was active including footage of lab- 
oratories in Rome, Columbia, Chicago, and Los 
Alamos). But in following this work from the dis- 
covery of slow-neutron-induced radioactivity to 
the nuclear reactor and the A-bomb, they also find 
the kinds of questions raised that are much on the 
minds of students and the public today. These 
questions concern the r-elations between physics, 
technology, and social concer-ns. 

(Note: In discussions with students, it is essential 
to keep certain distinctions in mind. Enrico Fermi 
and his grxjup contributed to the basic physics of 
nuclear reactions between 1934 and 1938. Then, in 
his work on r-eactor-s and the nuclear bomb during 
the war- years, Feimi became concerned primarily 
with applied and developmental research rather 
than with the pur-suit of pure physics. i 

In focusing on a major historical per-son, this film 
provides an opportunity for comparisons and con- 
trasts with the other major documentary' film of 
Project Physics, People and Particles, which should 
be shown in class for the fir-st time pr-eferably not 
too long before or after- the Fermi film showing. 
People and Particles is trying to captur-e what goes 
on in a tvfjical basic resear'ch laboratory in which 
a grxjup of younger and older scientists collaborate 
on a problem in pure physics. The Fermi film 
makes another "cut" through the grtnvlh of mod- 
ern physics, and presents a coherent life history of 
one physicist. We ha\e carefully selected a per-son 
who was at all times dirxH'tly accessible, and not 
one who was completely out of rt*ach of orclinan 
appr-tuiation as wei-<> \e\\1on and Kinstein. Fermi 
had many students, some of whom ai-e shown in 
this film, and of whom a i-emarkabl\ large numbei- 
ha\e Ixu'omc tlistinguished pinsic ists themseUcs. 
Also, I-(M-mi was always closely associated with 
()th(M- pix)minent plnsicists of his own generation. 
man\' of whom iuv shown in this lilm, too. In tliis 
wa\ the i-errni film can be regarded as a kind of 
guided tour that intrixluces some of the most im- 
portant pinsicists of todaN' and of the r«»cent past * 
rhcy iuv all identified in document.uA footage and 
photographs oi- in inter-views that wen- conducted 
s|)C(-ili(-all\ lor the film In a sense the lilm in its 



own right is itself a contribution to the histor\' of 
science. 

In its final version, the 46-minute-long film di- 
vides into two halves; the two parts should be 
shown together if possible but can be shown sep- 
arately. The first half brings the stor\' up to about 
1938, when Fermi obtained his Noble Prize and fled 
to America. The second half includes the war \ears 
and after. 

In order to make the film useful to the early 
stages of an introductorA' physics course, and to 
wider audiences, the amount of detailed physics 
instruction is kept to a minimum. The ideas nec- 
essary for understanding slow neutron fission are 
the only ones from F"ermi's \oluminous and im- 
portant wor-k that ar-e treated at any length. If the 
film is used late in an introductorA' physics course 
(either for the first or for the second time', the dis- 
cussion of nuclear physics can be elaborated in 
class by referring to the significant portions in Unit 
6, The \'ucleus. The unit develops a theme first 
introduced in the Prologue of Unit 1. 

The film p»ro\ides opportunit\- for a good dis- 
cussion among students. Among some of the 
points that in\ite treatment are the following: 

• the simplicity and directness of Fermi's 
methods; 

• the abilitN' of his group to do outstanding work, 
evidently v\ithout sacrificing the kind of fun that 
students rarelv associate with the image of the 
physicist; 

• Fermi's own remarkable abilit\' to differentiate 
between what was merel\- fascinating and what 
was tmlv inter-esting and relevant in phvsics — the 
mark of genius; 

• the fact that his career coincided with that of 
the arrival of the Nuclear Age, and contributed to 
it; 

• that the possibility- of an .A-bomb effort bv the 
Nazis during World War II necessitated a relation- 
ship between basic research scientists and the VS. 
militarA', a relationship that many of the scientists 
wished to modifv or discontinue immediately after 
the war; 

• Enrico Fermi's own isolation from political bat- 
tles, and the fact that not irntil the last year> of his 
life did he see the need for scientists tt) make clear 
their views to jjolicv maker's; 

• the fact that the nuclear r-eactor, initiallv a bv- 
prxxluct of the [)rt)giam to make the .\-bomb in war 
time, now has become one of the chief hopes for 
cheap ener-g\'; 

• the gr-eat variety that exists in the personal 
characteiistis of phvsicists, within this counlrk- and 
among dilTervnt coirntties: 

• the maturing of America during the past 40-50 



'I'tir list ot ptM-soiis stiowii inrliicips Kemii -Xllison Amaldi 
Aiuh'rson Hohr liiamlM'riain ('hanclraM>k.har Chwv Complnn 
.Marir Ciiiir Khivnfr.st Kin.sliMn (i<nulsnut liriM>nlx>r){ l^»v- 
ii-ncf NtfitiHT Mi<-tj«'lMin Millikaii McimMiii (>p|H'nhoim«T 
K.il)i K.isftti S«Ml>oif{ S«'f{iv S/ilaul I it\ Van*; 



llIVrT 1 / CONCEITS OF MOTION 



years in top-level research in most branches of 
phxsics, and the contributions of immigrant sci- 
entists to this pixjcess; 

• the great fondness with which some of the 
most eminent persons look back upon their asso- 
ciations with Fermi, and particularly Fermi as 
teacher; 



• his abilitv' to excel both in theoretical v\ork and 
in experiments. 

After the film, the interested student will enjo\' 
reading further. We suggest the paperback biog- 
raphy, Atoms in the Family, written by his wife, 
Laura Fermi. 



FILM SOURCES 



Combined film catalog and order forms are avail- 
able on request ftxim: 

U.S. Department of Energy' 

Office of Public Affairs 

C-460 

Washington, DC 20545 

NATIONAL AERONALTICS ANT) 
SPACE ADMLNISTRATION FILMS 

Requests for the free loan of X.AS.A films should be 
addressed to the library assigned responsibilitv' for 
your area, as indicated b\' list below. .An up-to-date 
catalog of films available from NASA can be ob- 
tained free on request from NASA Headquarters, 
Audio-\isual Branch, Code FAM, Washington, DC 
20545. 

V\'HO x\IAY BORROW FILMS FROM NASA 

Residents of the United States and Canada, who 
are bona fide representatives of educational, ci\ic, 
industrial, professional, youth activitv, and go\em- 
ment organizations are invited to borrow films 
ftx)m the NASA Film Library- that services their area. 
There is no film rental charge, but the requestor 
must pa\' return postage and insurance costs. In 
view of the wear and tear that results from re- 
peated projection, films are loaned for group show- 
ing and not for screenings before individuals or in 
homes. Because custody of the films involv es both 
legal and financial responsibility, films cannot be 
loaned to minors. 

To expedite shipment of film, requestor should 
give name and address of person and organization, 
specifying showing date and alternate date. It is 
also advisable to indicate a substitute film. 

Unless othenvise noted, television stations may 
order films for unsponsored public service or sus- 
taining telecasts. 

PSSC HLM SOLTICES 

PSSC films are available from Modern Talking Pic- 
ture Service. To order prints or for additional in- 
formation, contact the appropriate film rental of- 
fice listed below. 

412 West Peachtree Street X.W. 
Atlanta, Georgia 30308 

230 Boviston Street 

Chestnut Hill, Massachusetts 02167 

1687 Elmhurst Road 

Elk Grove \ illage, Illinois 60007 



1145 North McCadden Place 
Los Angeles, California 90038 

315 Springfield Avenue 
Summit, New Jersey 07901 
2323 New Hyde Park Road 
New Hvde Park, New York 11040 

ADDITIONAL FILM SOLTICES FROM 
WTIICH PmSICS HLMS ARE AA'AILABLE 

Phvsics films ma\ be obtained on a rental basis 
ft-om the distributors listed below. It is advisable to 
review films vou may select before class use. 

Contemporary Films, Inc. 
McGraw-Hill Book Companv 
Princeton Road 

Heightstovvn, New Jersey 08520 
Attention: Film Rental 

Encyclopedia Britannica FUms 

180 East Post Road 

White Plains, New York 10601 

General Dynamics 

Convair Division M.Z. 251-30 

P.O. Box 80847 

San Diego, California 92138 

General Electric Company 
Educational Films 
705 Corporation Park 
Scotia, New York 12302 

International Film Bureau, Inc. 
332 South Michigan Avenue 
Chicago, Illinois 60604 

National Film Boaixi of Canada 

1251 6th Avenue 

New York, New York 10020 

NET Film Service 

Audio-visual Center, Indiana University 

Bloomington, Indiana, 47401 

Shell Film Library 

1433 Sadlier Circle, West Drive 

Indianapolis. Indiana 46329 

U.S. NavA', Commanding Officer 
Naval Education <k. Support Center 
Atlemtic Naval Station, Building Z 86 
Norfolk, Virginia 23571 

Western Electric Company 
Motion Picture Bureau 
195 Broadwav, Room 1626 
New York, New \ ork 10007 



LEARVLVG MATERIALS 



25 



TRANSPARENCIES 



TO USING STROBOSCOPIC 
PHOTOGRfVPHS 

I'his tians[)ai(!n(.v pra\ides an opportunity to ana- 
lyze th(! inoti(jn of a f^oU club durinf^ the full suing. 

Tl STROBOSCOPIC MEASLTtEiMENTS 

Stroboseopie facsimiles of uniform speed and uni- 
form acceleration are shoun. Measui-ements may 
he taken directly, data recorded on tables and 
graphs plotted on grids. 

T2 GRAPHS OF V^ARIOUS MOTION 

Multiple examples of distance-time, speed-time 
and acceleration-time graphs. Useful for slope con- 
cept, area-under-the-curve concept, and review. 

T3 INSTANTANEOUS SPEED 

Stroboscopic facsimilie of body-on-spring oscilla- 
tion, data table, and grid. Find approximate instan- 
taneous speed by appr-oaching the limit and graph- 
ical estimation. 

T4 INSTANTANEOUS RATE OF CHANGE 

Ueteirnines v^^^ from enlarged portion of a dis- 
tance-time curve as time inter-vals ar-e deer-eased. 
Shows slopes of chords approach the slope of the 
tangent as the slope of the tangent appr-oaches 
zero. 



T6 DERIVATION OF d = i',f + V2 at" 
Colored overlays illustrate graphical procedures 
using area-under-the-cur\e technique. Space is 
pr-o\ided for teacher-directed derivation. 

T8 TRACTOR-LOG PROBLEM 

Classic third law hor-se-and-uagon paradox is up- 
dated with this tr-actor-log version. 

T9 PROJECTILE MOTION 

Str-oboscopic facsimile of objects projected hori- 
zontally and falling freely are analyzed graphically. 
Space is prxjvided for derivation of equation of the 
trajectory'. 

TIO PATH OF A PROJECTILE 

This demonstration transpar-enc^i' suggests that 
students approximate porlion of circles, h\perbo- 
las, parabolas, and ellipses by throwing objects. 
Leads to determination of actual path of projectile. 
Use with 79. 

Til CENTRIPETAL ACCELERATION- 
GRAPHICAL TREATMENT 

Stroboscopic facsimiles allow derivation of \'fl and 
graphical measurement of a^. 



Demonstration Notes 



Dl RECOGNIZING SIMPLE MOTIONS 

As an introfluction to the section, perform the 
events listed below. Ask the students to select the 
event which would be the best starling point for a 
study of motion. Ask them to give reasons for their 
selection. 

(a) roll a football 

(b) rxill a marl)l(! 

(c) drxjp a sheet of paper 
(dl bounce a ball 

lei swing an object around in a circle 

D2 UNIFORM MOTION, USING 
ACCELEROMETER ANT) 
DVN/VMICS C/VRT 

Tape a large litiuid-suriace accelerometer- to a d\- 
namics cail and show the stirdents that the water 
sur lace is horizontal w hen the cart men es with uni- 
form motion. Do the same with the coi'k-in-l)ottle 
accelerometer- is(»e HnndhookK stressing the fact 
that the cor-k i-t*mains ver1i(-al when the motion is 
uniform. Contrast uniform motion with the rest 
condition 

D3a INSTANTANi;OUS SPEED Wsing 
Strolu' Photos of" Boch on Spring) 

in this (Icnion.sti.ition-.icliv il\ Ihc class analw.es a 
coinpiex motion ithat of a luuK on a springs that 



is definitely nonuniform. Simple equipment is used 
to develop step b\' step the quite sophisticated 
concept of instantaneous speed, introduced in Sec. 
1.7 of the Test. A stroboscopic n^corxl is made of 
one-half oscillation of the bod\ -spring assembU 
and this rxH'ord is used to estimate the instanta- 
neous speed of the mass at one point of the ob- 
served oscillation. 

Equipment 

HocK -ami-spring assembly, hung so as to oscillate 
fr-eelw with a light soirrx'e taped to the oscillating 
bod\' and a sliding pointer arranged so as to indi- 
cate the point of interest. iSee Fig. 1.) 

Polarxjid camer-a 
Motor stix)be and disk 

or xenon strobe 
(hcrheail projector for projection of print. You 

ma\ find 7'r.i/i.s-;>.ireno- 7"3 usefirl for nn-ord- 

ing and anaUzing the data 

Proceiiure 

rhe bod\-spring assembU is shown to the class, 
extended, released and allowed to oscillate brielK. 
A |)r-obU>m is [losed orally to the students Mow 
fast was th«* boil\ mo\ing.' Stateil in tin* teriiii- 
nol()g\ of Se(- 17. this {jueslion is What is the 
instantaneous s|)eed \ of tin* hotly' Ihe students 
will r-et-ognizc that the body had ditVer"«Mit speeds 



UNIT 1 / (;o\(;ki»T8 of moiiox 



at different instants and that they can't even begin 
to answer \'our question until \ou mai<e it more 
specific. \o\v choose a point fairK near ibut not ati 
the end of the oscillation and ask for the speed at 
that point. Attach the pointer to mark the point of 
interest. If we could tie a speedometer to the body, 
we could watch and record its reading as it passes 
the pointer. Since we cannot do this, we ha\e to 
estimate the \alue of v from distance and time 
measui-ements. Two possible appixjaches are sug- 
gested in the Test. 

Method 1 Measure the axerage speed \\^ — 
AdAf over some internal that includes the point 
of interest. Begin with long time intenals and then 
progressi\ely shorten the interA'al until there is no 
longer any trend in the \alues of \\^ as the inten al 
is reduced still further. This final value of v_^^ is, 
within the experimental unceitaint\', equal to the 
value of the instantaneous speed \-. iXote that as 
the distances and time intenals measured become 
smaller, the percentage uncertainty' in v^^ increases. 
Therefore, for small enough A t, the calculated \al- 
ues of v^^ will ha\'e random \ariations due to ex- 
perimental uncertainties in Ad and Af.i 

Method 2 Make a graph of displacement against 
time. Draw a tangent to the cune through the cho- 
sen point P and compute its slope, which is ap- 
proximateK- the instantaneous speed at P. This is 
a straightforward exercise in graphical anahsis of 
a complex straight-line motion. Since the drawing 
of tangents to cun es is not a \er\' precise opera- 
tion, many students will get more satisfying results 
from Method 1. Method 1 also has the adxantage 
of emphasizing the concept of approaching a limit 




Fig. 1 

Method 1 Approaching the limit 

Set up the body-spring assembly and camera as 
shown in Fig. 1. The best strobe rate to use will 
depend on the characteristics of your spring and 
the mass of the body used. Tiy a rate of 30 sec i6- 
slot disk, 300-ipm moton. Vou will want at least 15 
or 20 intenals to measure. 

Altematixely, if you ha\'e a polished steel ball that 
can be attached to the spring, a xenon strobe light 
gives good results. 

With the apparatus aligned and the lights out, 



extend the spring by pulling straight down on the 
body. Open the camera shutter just before releas- 
ing the bod\', and then close the shutter again just 
as the body reaches its highest point and starts 
down again (to avoid the confusion of overlapping 
traces!. 

Hints on photographv and techniques for mak- 
ing the infomiation on a single photograph quickly 
a\ailable to the whole class are discussed in the 
notes on photography in the Equipment Xotes sec- 
tion of this Resource Book. 

Calculate \'^^ = A d/ A ( for se\'eral asxTOmetric in- 
ter\'als containing point P and ha\ing one endpoint 
in common isee Fig. 2i. Start with a Af of 20 or so 
time internals and work down to two intervals. 
Have some of your faster students repeat the proc- 
ess for a different interval. Others might tr\' sxtii- 
metric intenals isee Fig. 2i. 




[■;]] 



Fig. 2 A facsimile of a typical strobe photograph of the mo- 
tion of a mass on a spring, showing two possible ways of 
choosing a set of decreasing time intervals. 

Sample results are shown in Table 1, as meas- 
ured from a stroboscopic photo like that shown in 
Fig. 2. Precision of this order is obtained b\- using 
the 0.1 mm scale and magnifier. lOne millimeter in 
Fig. 2 represents 1 cm in real space.) 





TABLE 1 




Asymmetric Intervals 


t 




d 


(in intervals) 


(mrr 


on photo) 



1 




0.0 
0.5 


2 




1.6 


3 




3.1 


4 




4.8 


5 




7.0 


6 




9.5 


7 




12.2 


8 




15.0 


9 




17.8 


10 




20.7 


11 




23.6 


12 




26.3 


13 




28.9 


14 




31.1 


15 




33.0 


16 




34.5 


17 




35.6 


18 




36.5 



DEMONSTRATION NOTES 







TABLE 2 






Symmetric Intervals 






A<y 


Ad 




Af 






v.. 


one interval 


mm 


mm/Vso sec 


cm/sec 


is Vjo sec 




(on photo) 


(real space) 


18 intervals 


36.5 


2.03 


60.9 


16 intervals 


35.1 


2.19 


65.7 


14 intervals 


32.9 


2.35 


70.5 


12 intervals 


29.9 


2.49 


74.7 


10 intervals 


26.3 


2.63 


78.9 


8 intervals 


21.9 


2.74 


82.2 


6 intervals 


16.8 


2.80 


84.0 


4 intervals 


11.4 


2.85 


85.5 


2 intervals 


5.7 


2.85 


85.5 



We see in this example that the value of \'_,^ does 
not change as A/ is deci-eased below six intervals. 
This value of v,^ is equal, within the precision of 
this experiment, to the value of the instantaneous 
speed V at the point P at the center of each of the 
inteivals tabulated above. 

Have students graph the results tabulated in Ta- 
ble 2: average speed \er-sus size of time interval iFig. 
3). Ask students: 

"What would we find if we could make meas- 
urements over even smaller- time inteivals? Is there 
a point on the cuiac whose value represents the 
instantaneous speed?" You may be able to suggest 
that it is reasonable (since the body doesn't sud- 
denly speed up or slow down at Pi that v is the 
point where the cui-ve v\'ould cut the v,^ axis, and 
that in this case ithe case of a limiting processi it 
is legitimate to extenil the cuiAe (extrapolatei to 
that axis. The important idea of extrapolation must 
be introduced with some care and a variety of ex- 
amples. 



Fig. 3 



M 



Method 2 Estimating v sr«/>/iir-«//r 

(Irajih (/versus t dirtutK IVom rai)le 1. IJraw chords 
(■entcMcd on a chos(Mi point P (corresponding to 
the intenals of Method 1 aboxei and compute their 
slopes, which are the various values of v^^. C.on- 
stiaict a tangent to the cuiac at point P, and com- 
pute its sl()|)e as the value of \ at P. I'lie slope of 
the tangent at anv point gives the value of \ at that 
|)()itit 

i-.stimate the slope of the tangent at each of the 
(lata points, and plot a giajjli of \ vci-sus t. Repeat 
the prdcess, estimating the slope's of the \-/ cuim* 
at the data points, and plotting acceleration versus 
lime iSvv Fig. 4 i 




Fig. 4 Typical plots of d, v, and a against t. The lowest point 
of the body's motion is taken as d = 0, t = 0. 



Ask students, "At what point was the mass mov- 
ing fastest? Was it at one of the data points? How 
accurately do vou know where and VNhen the max- 
imum speed occurred?" Ask similar questions 
about the time (and positioni of zero acceleration. 

D3b IXSTANTAXEOl'S SPEED (I'sing 

Strobe Photos of Pendulum Swing) 
As an alternative to the body-and-swing demon- 
stration, a pendulum swing can be analvzed. In 
El-4, Measuring i'niform Motion, students meas- 
ured the average speed of a I'nifonn Motion Device 
iL'MDi over long and short time inteiAals, and prob- 
ably concluded that the Iniform Motion Device 
was moving with nearlv constant speed. In this 
demonstration-experiment the class looks at and 
analyzes a more complex motion ithe swing of a 
pendulum I. which is definitelv nonunifomv Simple 
ecjuipment is used to develop step bv step the 
(|uite sophisticateil concept of instantanetnis speeti 
intixjduced in Sec. 1.7 of the lest. A value for the 
instantaneous speed of the pendulum bob at the 
bottom icenteri of its swing is estimated experi- 
mentallv. 

Equipment 

i'ciululum about 50 cm long, hung ftx)m rigid 

supi^ort 
Polai-oid camera 
\Iotoi' strobe disk and light source taped to 

pendulum bob. 
or ac blinkv 
or xenon striibe 
(Aerhead pixijector. for pnijection of print 
II(>\ible scale for measuring projection of 

print 



I 'NIT I (;()\c;Km"s of motion 



Procedure 

A pendulum is shown to the class, drawn back, 
released, and allowed to describe a full arc. A prob- 
lem is posed for the students: "How fast was the 
pendulum bob mo\ing at the \ er\' bottom of its 
suing?' Stated in the terminolog\' of Sec. 1.7, this 
question is: "What is the instantaneous speed v of 
the bob at the lowest point P? ' This is the reading 
we might get from a speedometer at the moment 
of passing through the bottom position if we could 
possibly attach one to the pendulum. Since we 
cannot do this, we ha\e to estimate the \alue of v 
from distance and time measurements. Two pos- 
sible approaches are suggested in the Te\t. 



Method 1 Measure the average speed, v^ 



Ad 
Af 



, over some interval centered on the point P. 



The pendulum clearly mo\es more slowly the far- 
ther away it is from the bottom point. Therefore, 
the longer the intenal o\er which v^^ is measured 
the lower v^^ will be. All xalues of v^^ will be less 
than the speed right at the bottom. To get an es- 
timate of the instantaneous speed we must pro- 
gressively shorten the time intenal until there is 
no trend in the \alues of \;^ as the time intenal is 
further reduced. This \alue of v^^ is, within the ex- 
perimental uncertainty', equal to the \'alue of the 
instantaneous speed v at P. i\ote that as the dis- 
tances and time intervals measured become smaller, 
the percentage uncertainty in v^^ increases. There- 
fore, for small enough A t, the calculated \alues of 
v'^^ will ha\ e random variations due to experimental 
uncertainties in Ad and Af. 

Method 2 Make a graph of displacement against 
time, and draw a tangent to the curve through the 
points for the highest observed velocities. The 
slope of the tangent is approximately the instan- 
taneous speed at P. 

The drawing of tangents to curves is not a verv' 
precise operation. For this reason, and because it 
emphasizes the idea of the approach towards a 
limit, Method 1 is recommended. 

Method 1 Approaching the limit 

There are several alternative experimental proce- 
dures here. The one described first is the simplest 
experimentally. 

Set up the pendulum, light source, and camera 
as shown in Fig. 5. Use a strobe rate of 60 sec il2- 
slot disk, 300-rpm motor i. Alternatively, you may 
use the ac blinkv, with some added mass, as the 
pendulum, iln this case, of course, you do not need 
the strobe disk in fixjnt of the camera. i It is impor- 
tant to set up a marker to indicate the bottom point 
of the swing, and to have a rigid stop so that the 
bob can be drawn back to the same position for 
each release. Obviously the instantaneous speed at 
the bottom point depends on the amplitude of the 
swing. Be careful not to pull down on the string 
prior to release; the stretch will disturb the motion 
of the bob. Photograph as much of the swing as 



possible. Be sure to close the camera shutter before 
the bob begins the return swing to avoid the con- 
fusion of overlapping traces. 

Hints on photography and techniques for mak- 
ing the information on a single photograph quickly 
available to the whole class are discussed in the 
Equipment \'otes on photography. 




\L 



>JHlTe POINTiJR TO ' 

iiyoiCf^re sorropi of 

SiAjt(i6 (POINT P) 



Fig. 5 




Calculate v„ 



Ad 
Af 



for several intervals centered 



on the bottommost point or the bottommost inter- 
val, depending on the particular photograph iFig. 
61. Start with a Af of between 30 and 40 time in- 
tervals and work down to 2 or 3 intervals. The dis- 
tance intervals Ad are measured along the arc, 
which requires use of a flexible scale. 



* • • 



-at 



Fig. 6 (a) Trace is symmetrical about bottommost point, (b) 
Trace is symmetrical about bottommost interval. 

Sample results are shown in Table 3. lOne mm 
on the photograph represents 1 cm in real space.) 





TABLE 3 






Id 


\d 




Af 






w„ 


one interval 


mm 


(mm/Vso sec) 


cm/sec 


is Vso sec 


on 


photograph 


real space 


26 intervals 


84.0 


3.23 


194 


22 intervals 


76.0 


3.46 


208 


18 intervals 


66.0 


3.66 


220 


14 intervals 


53.5 


3.82 


229 


10 intervals 


39.5 


3.95 


237 


6 intervals 


24.0 


4.0 


240 


2 intervals 


8.0 


4.0 


240 



DEMO.VSTRATION VOTES 



In this example, the value of v^^ does not change 
as A f is decf-eased below G intervals. I his \alue of 
v,^ is, within the precision (jf this ex|)eriment, equal 
to the value of thf! instantaneous speed v at the 
point I'. 

Method 2 Alternative procedure 

A slight variation perhajis emphasizes more clearly 
that what we aitj doing here is measuring v'_,^ o\er 
successively shorter time intenals. A series of 
strobe photographs taken, each one at a higher 
stobe rate i smaller time intenal between images i 
than the pit!vious one. This is most convenientK 
done with the light soun:e and disk strobe method 
by |)i-ogressi\(;ly opening up more slots. In this 
m(Mh()d one measuit^s v_^ over the lowest interval 
only on each trace. At the lowest strobe rateisi it 
may be impossible to find an internal that is ade- 
quately centered on the bottom point. The change 
between the value of v,^ over the longest A/ and its 
value over the shortest Af will be less in this 
method, because the range of time intei\als over 
which the measurement is made is less. 

A calibrated xenon sti-obe and steel-ball pendu- 
lum bob could be used for this method. S'et an- 
other possibility is to feed the ac blinkv v\ith \ar- 
ious known fi-equencies from a icalibiatedi audio 
oscillator via an amplifier and transformer, i Re- 
member that the neon lamp does not glow below 
about 70 \ peak voltage.) 

Possible extensions 

1. In Method 2 above, plot d against t, draw 
chords centered on P to find various values of v,^, 
and draw the tangent at P to find the v alue of \ at 
P. The slope of the tangent at any point gives the 
\alue of v at that point. 

2. Plot a graph of the nisults obtained above av- 
erage; that is, speed against time intei"val I'ig. 7'. 



/! 



--o^ 



■o^ 



Fig. 7 



^r 



.Ask students: What would we find if we could 
make measuit'ments over even smaller time inter- 
vals?" Can lliev (ind a point on the cune whose 
value represents the instantaneous speed? \ou 
may be able to suggest that it is ivasonable i since 
the bob doesn't suddenlv speed up or slow down 
at P) that V is the point vv Iumv the cune would cut 
thi- v,^ axis, and that in this case it is legitimate to 
extend the cune lextrapolatei to that a.\is Ihe im- 
|)()i1anl idea of extrapolation must be intixuluced 
with some care and a variet\ of examples. 

A rheii' are man\ other mea.sun-ments that can 



be made with this simple experimental setup. It 
could be instructi\e for students to make graphs 
of d against /, v,^ 'measured over one interval! 
against t, and, if possible, of acceleration, a, against 
t. Measure / and d frxjm the point P, Pig. 8.i 




Rg. 8 Plots of s, v^, a against t. The bottommost point of 
the pendulum swing is taken as s = 0, f = 0. 

4. The concept of instantaneous speed will come 
up again in I'nit 3. The kinetic ener^' of a hod\ at 
a given instant depends upon its speed at that in- 
stant. The inteix-hange of potential and kinetic en- 
erg\' in a pendulum will be referred to specificalK 
It is wor'thwile to make jjliotogr-aphs that encom- 
pass both the topmost point, the bottom point P 
and some scale to give absolute measures of dis- 
tance. The photographs should be kejit for use 
later in I'nit 3. 

Questions for discussion 

1. Could one ever measure v experimentally? 
How? 

2. .A car speedometer appears to measure in- 
stantaneous speed. Does an\ student know how it 
works? How is it calibrated? i This is done at con- 
stant speed, that is, bv measuring A/ for a known 
Ad while the speedometer reading is unchanging. 
So reallv all that we know is that the speedometer 
tells us instantaneous speed for the special case of 
uniform motion, that is when v = v^^ at even' 
point. I 

D4 I^IFORM ACCELER/VTIOX, I'SING 
LIQllD ACCELEROMETER 

This demonstration allows vou to show that when 
a carl moves with constant acceleration a, the sur- 
face of the li(|uid is a straight line tilted in the di- 
ivction of the acceleration. 

Cive the carl a uniform acceleration by suspend- 
ing an object over a pullev as in Fig. 9. 

It is best to us«' objects w hose masses r-ange from 
100 to 400 g It is imporlant to keep the string as 
long as possible so that voir use the entirv length 
of the table B\ changing the mass of the sus- 
pended object, \<)u can vaix the acceleration of the 
carl. Notice that the slojie of the li(|irid increases 
with givater acceleration. I lie slope is thus a meas- 
urv of the acceler-ation. It can be sIh)wti that tan ^ 
= n g so that a = g tan W \ ou will find detailed 
comments on qirantitative work with the li(|uid- 
surface ai'celei-onuMer in the Equipment .\otrs foi- 
I'nit 1. The fan cart, Fig. :i5. can be usj'd as a non- 
gravitational source of irniform acceleration A 
small acci'leralor mav be mounted on the fan cait 
()!• on another cart pulh'd In the fan can 



.'lO 



I'MT 1 COVCI.ITS ()l MOlfOV 




Fig. 9 Arrangement to demonstrate uniform acceleration. 

Do COMPARATHT FALL RATES OF 
LIGHT AND HEA\T OBJECTS 

Drop se\eral paii-s ot objects such as a marble and 
a lead shot, simultaneousK from the same height. 
Decide \vhether the theory of Aristotle or that of 
Galileo agrees best with the observations. Account 
for an\' discrepancies. 

On a large book place sexeral objects, such as a 
small piece of paper, a marble, and a paper clip. 
Drop the book. Do the objects fall at the same rate 
and stay on the book? 

IDG COIN AND FEATHER 

If the equipment is a\ailable, do the coin and 
feather e.xperiment. Failures are usualh due to a 
poor \acuum pump or to a defectixe seal on the 
apparatus. Check ahead of time to see how long it 
takes to evacuate the apparatus sufficiently to 
show that the coin and feather fall together. 

D7 T\^'0 \^'AYS TO DEMONSTRATE 
THE ADDITION OF \TCTORS 

Method 1 

Apparatus: 

50 cm X 50 cm board 

Two dynamics carts 

Two Uniform Motion Devices UMD' 

Two sheets of clear plastic i Kodak Safet\- 3, for 
overhead transparencies 

Three marking pens of diflfei-ent colors 

Clamps, stands, etc., to support pens 

Stopwatch 

Three people to operate Uniform Motion De- 
vices I UMD I and stopwatch 






'^?":ij^i;r^VsHHeTAT.p.o 



TO Platform 



Fasten the two carts underneath the board to 
fonn a i-olling platform, as shown in Pig. 10. Hook 
up UMD 1 to push the platform along the table. 
Attach one plastic Sheet A to one corner of the 
platform, as in Fig. 11. 




^t 



Fig. 10 The rollowing platform and the arrangement of the 
plastic sheets. 



PCS A 



Fig. 11 The apparatus as seen from above. 

.Attach plastic Sheet B to UMD 2 as shown, so 
that the UMD tows the sheet along smoothlx be- 
hind it. .Adjust the tow rope so that \\^ ithe velocity' 
of I'MD 2 1 is parallel to the long edges of Sheet B. 

Choose a direction for \,,. and aim UMD 2 in that 
direction, laving Sheet B across Sheet .A as shown. 
Draw a guide line on Sheet A, using the edge of 
Sheet B as a ruler. This is vour onlv record of the 
direction of v^. 

Attach one marker iPen Ai by means of a ring- 
stand and stiff wires so that it makes a line on 
Sheet .A as the platform rolls along. From the length 
and direction of this line, you will be able to figure 
out the magnitude and direction of \\. assuming 
that v_j is constant. 

.Attach Pen B to the rolling platform. It makes a 
line on Sheet B that indicates the motion of UMD 
2 relative to the platform. 

Pen .AB also marks on Sheet B, but it is fastened 
to the stationaiA- ringstand on the table. The mo- 
tion of Sheet B with respect to the table is made 
up of the nvo simple motions added together vec- 
toriallv . Fixjm the line that Pen .AB makes, you can 
deduce the vector w^ + \,,i. 

.Adjust Pen B and Pen ,AB so that they begin at 
the same point P on Sheet B. 

With the pens in place, set the Uniform Motion 
Devices in motion at the same time. This will take 
a little practice.! Shut them off, again simultane- 
ousK-, when the longest line that has been drawn 
is 10-15 cm long. Use the stopwatch to time the 
motion. 

Vou now have three lines of different lengths, 
colors, and directions. If you make certain as- 
sumptions, vou can treat these lines as direct rep- 
resentations of v;,, v^. and iv^ -I- v^i. ,Add an arrow- 
head to each line to indicate the actual direction 
of the velocity that it represents. Remove both plas- 
tic sheets from the apparatus and slide Sheet B 
ov er Sheet .A until the head of v^ is at point P. Be 



DEMONSTRATION NOTES 



31 



sure to keep the edge of Sheet B parallel to the 
guideline. 

Ask the students if Iv^ + v,,i seems to be the vec- 
tor sum of v^ and v,,, using the parallelogram lule. 
Convince them that if these velocities have added 
as vectors, the three vectoi-s should form a triangle. 
Is this the case isee Fig. 12i? 




Gotoe 

S57- 
FAfiAU£L 

723 



Fig. 12 Try the same procedure for a few other directions 
of v^ (v/3 and v^ parallel, opposite, at right angles, etc). 



Method 2 

Apparatus: 



dc blinkv, set to about 1 flash per second 

The same rolling platform as in Method 1, 
painted black 

'I\vo Unifonn Motion Ue\ices 

Polaroid camera mounted on tripod I for 3000 
speed film, the lens setting is about E\' I61 

Bench stand and pointer to indicate the stall- 
ing point of the blinkv 

Three people to operate l?niform Motion De- 
vices and camera 




UMD2-i^ 



Fig. 13 Arrangement of apparatus when ^, and v, are par- 
allel. 

Set u|) the ix)lling platioiiu, pushed In I'MIJ 1. 
as in Method 1. '\'\\v vciocitv of the platlonn as it 
m()\(!s past the camera is again v, Place I MI) I on 
the platfoini: its xclocitx with r-<'sp('ct to the plat- 
lonn is \, Point the cameia downw.ird, so that it 



takes a picture of the apparatus from directly oxer- 
head. Mount the blinkv on L'.MU 2 and position the 
pointer so that you will be able to put the blinkv 
back in its original position after taking the first 
picture. 

Turn out the lights, open the camera shutter and 
set UMD 1 in motion. Let it tow the platform across 
a good part of the camera's field of \iew. ,Ask the 
students which \elocit\' is obtained from the strobe 
record of this motion. 

Replace the platform so that e\erything is the 
same as it was before the last step. Repeat the proc- 
ess without advancing the film iwhen you are 
through, you will have a triple e,\posurei but this 
time ha\'e only UMD l in motion. Which velocity' 
can you calculate from this stixjbe record' 

Return to the starting point again, and take a 
third picture (on the same filmi of the motion of 
the blinkx', this time with both Uniform Motion 
De\'cies moving. 

Develop the print and calculate the three veloc- 
ities (speeds and directionsi v^, C,, and iv, -1- \\. Skip 
the first interval in each strobe record: It takes the 
Unifonn Motion Devices a little time to get up to 
speed. Draw arrows representing the three veloci- 
ties, and check as in Method 1 to see if the paral- 
lelogram law of vector addition holds for the mo- 
tion you have observed. Do the three vectors form 
a triangle? Should the\? 

Again, trv the cases where \\ and v^ are parallel, 
at right angles, and at several other angles of your 
choosing. 

D8 DIRECTION OF ACCELERATION 
AND \TLOCIT\ 

L'sing the same airangement as D4, demonstrate 
that acceleration and velocitA can have different 
directions. Hang an object of 100- or 200-g mass 
over the pullev and give the cai1 a push to the left 
so that it goes nearly to the end of the table before 
it stops and reverses direction. Vou should trv to 
give a short, smooth push so that the liquid 
reaches its steady state quickly. 

Once the water has reached its steady state, the 
surface is a straight line whose slope does not 
change, even when the velocit\- iwei^es direction 
The explanation, of coui-se. is that the acceleration 
is constant and independent of the velocity. Onlv 
the weight of the object over the pullev detemiines 
the acceleration of the cart. 

D9 DIRECTION OF ACCELERrVTIOX AXD 
VELOCITi— A\ AIR-TR^VCK 
DEMONSTRATION 

Mount the small acceleiDmeter on an air-track cart. 
\\ hen the track is hoiizontal and the cai1 is at rest 
or moving with unifonn s|)t'ed the surface of the 
liquid is also hoiizontal. ()nl\ when a horizontal 
force causes the cart to accelerate ifor e.vample, 
when lh«' cai1 stalls or stops or collides with some- 
thing elsei is tin* slope of the surface not horizon- 
tal 



A'i. 



I'MT I / COXCEmS OF MOTIOX 



Next, place the track at a slight incline. When the 
cart slides freely on the air track, the surface is 
parallel to the track. These interesting facts are ex- 
plained in the Equipment Xotes. 

Again, you can show that \elocit\' and accelera- 
tion can have different directions. Gi\e the cart a 
push up the incline. If friction is negligible, the 
slope of the liquid remains the same while the cart 
slows down, i-exerees direction, and mo\es dowTi 
the incline. If frictional forces are increased by add- 
ing mass to the cart, the slope will decrease when 
the cart begins to moxe dowTihill. 

DIO NONCOMMUTATRT ROTATIONS 

One of the points frequenth made about \ector 
addition is that it is commutatixe: that is, the order 
of addition does not affect the sum. Students are 
frequenth conxinced fixjm their experience with 
arithmetic that this is true of all operations. It is 
useful to be able to show them an example of an 
operation that is not commutatixe. 

If a closed book is placed on the desk in front of 
a student, rotated 90° about an axis along the spine 
of the book, and then rotated another 90" about an 
cixis parallel to the near edge of the desk, the final 
orientation of the book is different than if the op- 
posite order of the two operations is followed. 

Dll NEWTO.VS FIRST LAW 

There is an aesthetic appreciation in science for 
simple statements that describe \'er\' complex phe- 
nomena. £ = mc' is an example of such a state- 
ment. Newton's first two laws and the equation 
F — ma, which follows from them, are early 
examples. For the teacher, these simple statements 
often create difficulties because the students fail to 
realize their importance. There is a tendency to 
feel that what is not complex and filled with math- 
ematical sxTnbols cannot be \ers' important. Noth- 
ing could be further from the truth. Man\- people 
contributed to the e\entual de\elopment of the 
three laws, and Newton's own work was a per- 
ple.xing amalgam of intuition, definition, and ex- 
periment. While one cannot say preciseh' how 
Newton came to his conclusions, he was deeply 
familiar with the related phenomena. Therefore, 
we suspect that the students introduction to the 
laws of motion should a\oid the didactic and fa\or 
direct experience and an intuitixe approach. 

The demonstration described below may seem 
trivial, but firsthand experience with \er\' low- 
friction motion is \aluable for understanding 
Newtonian ph\ sics. While this is listed as a dem- 
onstration, it should be conducted as an informal 
experiment. This has alwaxs been a very enjo\ able 
experiment for the students, who frequently men- 
tion it as their fa\orite. 
Equipment 

Se\ eral pucks with balloons or plastic beads 

Puck table 

Large rubber band 

Air track (optional 



Procedure 

Student lab groups are given single pucks without 
the balloons or plastic beads, with the instiuctions 
that they are to pla\ with them for several minutes 
so as to be able to describe how the pucks move. 
Then a brief discussion is held to establish what 
happens to the pucks' motions under \arious cir- 
cumstances, for example, just resting on the table, 
being pushed briefly, being pushed steadily, when 
the table is tilted, etc. Friction may be mentioned; 
perhaps someone will suggest w'hat the motion 
would be like without friction. 

Immediately demonstrate the low-friction capa- 
bilitx' of the pucks, and supply students with bal- 
loons and or plastic beads for another short period 
of imestigation. iHalf the class could use balloons 
and half plastic beads to make the results more 
general. I The instructions are, as before, to be able 
to describe the motion of the pucks. Fences made 
from the large rubber bands are excellent as re- 
flectors because the\' allow long runs. 

The leveling of the surface may be a problem, 
especiaUy with the balloon pucks. The concluding 
discussion might become heated on the how-do- 
\'OU-know-wtien-there s-no-force paradox, but that's 
fine. If students argue about these things, they are 
aware of the issue. 

The disk magnets or ciir track can be used for a 
further extension of frictionless motion. 

D12 XE\\'TON'S-LAW EXPEREVIExXT 
(AIR TRACK) 

With the calibrated accelerometer \'ou can perform 
experiments to define forces in terms of the accel- 
erations of objects whose masses are known. The 
accelerometer would enable you to determine the 
accelerations directK . See Equipment !\'otes for in- 
formation on how to use the liquid-surface accel- 
erometer quantitatively. 

D13 EFFECT OF FRICTION 
ON ACCELERATION 

Demonstration 12 works onl\ if fiiction is negligi- 
ble. Since the direction of the frictional force F^^, 
is always opposite to the \elocit\', sou can show 
the effect of friction on acceleration b\' attaching 
tape with adhesi\ e on both sides to the wheels of 
the cart. 

When the cart mo\es to the right, the horizontal 
forces acting on it are illustrated, as in Fig. 14. The 
acceleration is then 

a — 

M 

where M is the mass of the cart plus the acceler- 
ometer. When the cart mo\es to the left, however, 
the forces act as in Fig. 15. The acceleration is now 
T + F 

M 
The tension T is simply the weight of the object 
hanging o\ er the pullex' and is independent of the 
velocit\ . 



DEMONSTRATION NOTES 



33 




Fig. 14 Force diagram when cart is moving to right. 




Fig. 15 Force diagram when cart is moving to left. The 
change in is exaggerated. 

Since the acceleration is less when the cart 
moves to the right than when it mo\es to the left, 
the slope of the watei- when the cail nio\(!s to the 
right will also he less. This difference in slopes is 
slight, but noticoahie. 

D14 DEMOXSTR/VTIOXS UlTH ROCKETS 
Introduc^tion 

The demonstration experiments suggested heiv 
can accomplish two things. They are e.xciting. 
which makes them ideal as motivating experiments 
at the hc'ginning of the coui-se. Rockets and space 
(light air maltei-s of gi-eat public inteivst toda\', and 
exp(Mim(*nts like these could do much to amuse 
interest land perhaps increase enrollment i in a 
physics cour-s(». The experiements can also he used 
to teach (|uite a lot of pinsics: fn'c fall, foi-ce. im- 
pulse, (-onsenation of (>nerg\ , application of trig- 
onometiA', etc. 

We cannot stit^ss too sti-ongl\- the need for strict 
supeiAJsion b\- the teacher at all times. (Jet pei- 
mission and suppoil from loc.il officials and school 
administrators befoix' starting model rockeliA 

Small solid-fuel rocket engines. iightwjMght rt)ck- 
els, and a considerable bod\ ot supplcmentaiA' in- 



formation can be purchased from Estes Industries. 
Inc., Box 227, Penrose, Colorado 81240. Their cat- 
alog is available on request from the address gixen. 
We have tested the "Scout," the "Corporal, and 
the "V-2.' Assembly for these models ranges from 
1 to 2 hours and could be done by students. 

VMien used with some care under strict super- 
vision of the teacher, these rockets are probably 
considerably safer than a good number of other 
experiments that are performed in the classroom. 
However, students should not be permitted to take 
home lockets from the schools suppK or to use 
the school's rockets during school hours without 
careful supervision. Although quantitati\e experi- 
ments of real pi-ecision are probably mathemati- 
cally too involved, students can learn much from 
a series of demonstrations that permit some stu- 
dent participation. 

Rocket engines come in a variety' of sizes with 
maximum thrusts of either 6\ or 39\ and thrust 
durations from 1.7 sec to 2.0 sec. In addition, a 
special-pui^jose engine (B.8-O1P1I for use in static 
tests is available. 

Experiments nith rockets in free flight 

If a large, open space is accessible to the class, a 
number of experiments can be performed with 
fi-ee-flight rockets. For example, one may use suc- 
cessively more powerful engines in se\eral other- 
wise identical rockets. .Another set of experiments 
would make use of rockets of identical exterior de- 
sign but of difTerent mass: in fact, one might make 
one of the ixjckets so heavy that it will not lift off. 
V\'e all get a thrill from firing the small rocket and 
seeing it rise rapidly. Students should stand at 
known distances, at least 30 m from the launching 
pad, each with a simple altimeter, consisting of a 
protractor with a small plumbline and a viewing 
tube, made, for example, from a large soda straw 
iFig. 161. 



ZiZ: 



^ 



i> // .V TT 



:2Z=^^ 



Fig. 16 

Kach student should try to measuiv the angle of 
elevation of the rxxket at the .same moment pref- 
erably when the ix)cket has rvached its maximum 
height rhe teacher can call (uit the time for this 
measurement I'sing sim|)l«' trigonometr^v stu- 
dents can calculate the height of the nu'ket if then* 
is little wind and the rocket rises vi^rlicallv . the\ 
can calculate the In'ight knowing the liistance and 
elevation angle lach studiMit will lind a value for 



I'MT 1 / (;0\(:KI»T8 of M()TItl\ 



H. A comparison of the results will pro\ide an op- 
portunit\' to discuss errors of measurement. 



f^at/trr ffi^ OiAi /Foc/fiJ 





^Oi-CBB. 




"TuRNTABLe 



Fig. 19 



Fig. 17 



Fig. 18 



In most cases, the rocket will not rise verticalK . 
The computations will become fairh in\ol\ed un- 
less students can measui^ both the angle of ele- 
vation at points of ma.ximum H and the angle 
through which they must turn from a fi.xed line 
when measuring H. For e.xample. if two students 
A and B, stand at a fixed distance D. and each has 
to turn from the line connecting their position by 
angles of 6 and (b. respectix el\', we can then at once 
find the point abo\e which the rocket was at its 
highest point and determine the distcinces ,v and 
y. Knouing ,v and y, each student can calculate H 
isee Fig. 18'. Write to Estes Industries for copies of 
their Technical Report TR3 Altitude Tracking, 
which gives detailed instructions. This e.xercise 
and excursion into trigonometry \ although not di- 
rectly part of a ph\sics course, is useful in showing 
the need for mathematics as a tool. 

When firing ixackets, all possible saferv precau- 
tions should be followed. Estes Industries will sup- 
pl\' an outline '.Attachment #3i of how to handle 
the rockets and what methods to emplo\' to pre- 
\ent accidents. In fact, the safet\ code as supplied 
by Estes has an educational \aJue in showing stu- 
dent show to handle potentially dangerous situa- 
tions. 

Experiments with test stands 

The design of a simple test stand for rocket engines 
requires knowledge of fundamental phxsics prin- 
ciples. Basically, one wants to measure as accu- 
rately as possible the force i thrust i a rocket exerts 
as a function of time. Since the burning times of 
these rockets are short ifrom a minimum of 1.7 sec 
to a maximum of 2.0 seci, one needs to use a re- 
cording de\ice. In order to measure thioist cor- 
rectly, the apparatus should be ti-uh static, that is, 
there should be as little motion as possible while 
the engine fires. If a spring is used to pro\ide the 
balancing force, precautions must be taken to 
avoid oscillations: in fact the damping should be 
critical and furthermore should be \elocity- 
dependent so that the recording pen will always 
return to the s£ime zeixa position. 

Test stcmds can be designed in a \ariet\' of ways. 
Two designs ha\e been tested. 

A. The first test stand consists of an engine 
holder iFigs. 19 and 20i, made from a rocket-body 
tube lEstes Cat. #651-BT-+0, 0.765" I.D., 0.028" wall 
thickness I connected to an aluminum rod R that 
is free to mo\e in two bearing blocks B. 




Fig. 20 

Attached to the far end of the rod R is a tube P 
into which a marking pen or some similar eas\ and 
light writing de\ice can be inserted. The rod R car- 
ries a pin that serves two purposes: It compresses 
a spring as the engine is fii-ed and it pre\ents the 
rod from turning about its axis by riding in a key 
slot attached to one of the bearing blocks. The 
spring constant should be chosen so that a steady 

1 static I force of 20 \ will gi\'e a compression of ap- 
proximateh' 7.5 cm. Friction in the bearings may 
just provide the necessary damping force: other- 
wise, one can add some damping b\' pressing a 
cloth strip against the rod. The test stand is set up 
radialK' near a turntable so that when no force is 
applied to the spring, the pen will leave a circular 
trace near the edge of a circular sheet of paper 
attached to the rotating turntable. When the rocket 
engine is fired, the pen is pushed toward the cen- 
ter of the turntable and plots a graph that can be 
analyzed for a measure of the force applied to the 
spring. With the turntable rotating at 33 rpm, a 
"firing" of an Estes ,A8-0 iPi rocket engine will leave 
a polar coordinate record that covers almost a 
complete re\olution, indicating that the force was 
applied for appro.ximately 1 33 min or just under 

2 sec I Fig. 211. If a linear chart drixe is available that 
will move the paper at a high enough speed to 
spread out the graph over a reasonable distance 
I at least 25 cm sec i, you can substitute this for the 
turntable. Howe\er, there is merit in using a polar 
graph, if only to show students a different method 
of recording and analyzing data. 



r//i/A/6 




Fig. 21 



-NO OF Ti^fi rr 



DE.VIONSTRATIO.V NOTES 



35 



To translate the curve drawn by the recording 
device during the firing into a force-versus-time 
plot, one needs to calibrate the test stand. This can 
be done by applying known forces; for instance, 
weights applied via a pulley to the spring, while 
the turntable is moved by hand through sections 
for each applied force. A calibration curve, relating 
displacement to (static) force can then be drawn. 

Note that the spring used was nonlinear. The 
reason for this is that the initial large force acts for 
a short time only, and thus the impulse due to this 
force is fairly small. To measure accurately the 
much smaller sustained force, a spring is needed 
which will give reasonably large deflections for the 
small force acting over the longest part of the firing. 
Again, there is additional educational benefit to be 
derived from the fact that another illusion is shat- 
tered for most students land many teachersi: 
namely, that springs by natuitj are linear and that 
Hooke's law can be applied without thought. 

It might be worthwhile to point out that there is 
another problem invovied in this analysis; namely, 
that the force applied by the rocket engine is an 
"impulsive" force, acting for a short time only, 
whereas the calibration of the test stand is done 
statically. 



If this impulse is assumed to occur in a short 
time, compared with the total flight, a first approx- 
imation gi\es jFdt = mv^„^, - mv',„„^, \%hen m is 
the mass of the rocket plus engine and v is the 
speed of the rocket after the impulse has been ap- 
plied. If we neglect all external forces except gra\- 
ity, we can find the maximum height to which it 
would rise from simple kinematic considerations 
ivf - v^ = 2g/j). The actual height to which the 
rocket will rise is much less than the comput- 
ed one. 

B. A second type of test stfind (Figs. 24 and 25 1 
can be assembled easily in most schools fiDm odds 
and ends. It in\ol\'es a 37.5-cm wooden ruler in 
which a vertical shaft has been placed at the 30- 
cm mark. The ruler can turn freely about this shaft 
in a horizontal plane. ilnexpensi\e steel shafts with 
bearings are available from radio supply houses, 
for example. Allied #44Z094, panel-bearing assem- 
bly with 7.o-cm shaft. At the 35-cm mark a rocket 
motor holder is fastened secureU' by gluing it with 
a good contact cement, then tying it with string 
iFig. 261. Finally, paint the string and motor holder 
v\ith glue, coil dope, shellac, or some other mate- 
rial that will bind to the ruler, string, and rocket 
motor holder. 




Fig. 22 

Students would benefit fitjm transferring the po- 
lar-coordinate graph to a Cartesian-coordinate 
graph. They can then compute the total impulse 
of the engine i jFdt) by finding the area under the 
curve. 




Fig. 23 



wWj^ 



ROCrpT MOTOB. ^ «>eH MOLDS* 



« 



.-4*^-^ 



■^ — I- 






>'''■' 



Fig. 24 




Fig. 25 




Fig 26 



At the S-cm mark a spring is fastened that will 
extend not more than 7.5-cm when a force of 4 N 
is applied to it. .\ nonlinear spring would ha\e the 
same advantage.s explaim-d earlier 



UNIT 1 / CONCKPTS (IF MOTIOX 



Note: There are \'cirious ways to make nonlinear 
springs. In this particular case one could, for ex- 
ample. ha\e t\vo springs attached iFig. 27i, such 
that for small forces Spring 1 will stretch, but 
Spring 2 will not be under any tension. As Spring 
1 stretches, eventually the string that connects 
Spring 2 to the ruler will become tight and the 
force constant of the combination uill become the 
sum of the force consteints of both springs. 






Sn\lN(r 







Fig. 27 



A second method to obtain a nonlinear spring 
uses one single spring and a thin string looseK' tied 
between some of the coils of the spring iFig. 28 1. As 
the spring is stretched, all coils uill open up at 
first, until the string becomes taut. From then on, 
only those coils can extend that are outside the 
tied-down section of the spring. It is easy to adjust 
the relati\e spring constants simply b\- shifting the 
position of the string, holding back more or fewer 
of the coils. 




/«7/?//y6 



Fig. 29 

Damping can be applied in a \'ariet\ of wa\s and 
will pro\ide a \er\ interesting exercise in applied 
physics. The most ob\ious wa\' to decrease oscil- 
lations is to apply a frictional force. A bottle brush 
held perpendicular to the ruler near the 7.o-cm 
mark and pushed against the flat side of the ruler 
IFig. 30 1 will help dampen out the \ibrations iFig. 
31), but the damping is not critical. In addition the 
fiiction will introduce sizeable shifts in the zero 
position. Idealh' the damping force should be \"e- 
locit\'-dependent. We ha\ e tested a \iscous de\ice, 
consisting of a metal \ ane being pushed through 
oil and find that it is also noncritical iFig. 32 1 but 
does not have a zero correction, .\nother method 
would be to use a metal plate mo\ing in a strong 
magnetic field i eddy-brake i. 



\. 



Fig. 30 




-^nm^i^mi)— 



STHtrjG 



Fig. 28 



A light tube which can hold a marker pen is at- 
tached to the far end of the ruler, near the 2.5-cm 
mark. We again use a turntable so that the pen can Fig- 31 
trace a graph of its excursion as a function of time 
on a paper disk fastened to the turntable. 

The reason for using the unequal lever arms in 
this design is to ha\e the rocket engine move 
through as small a distance as feasible, thus ap- 
proaching a true static test, and also to ha\e the 
mo\ing parts of the de\ice be as light as feasible 
while still gi\ing a reasonably large trace on the 
graph paper. 

Figure 29 shows the result of a firing using a 
linear spring and no damping force. A number lat 
least foun of oscillations following the initial ex- 
cursion of the pen can be seen. Fig. 32 




2£f(.0 SHIFT 



*- /7/p/A'6 




DEMONSTRATION NOTES 



37 



This part of the project is completely open- 
ended. Students can undertake a systematic ex- 
perimental study of damping forces and hegin to 
appreciate the prohlems of the scientist or the en- 
gineer. They will also begin to realize that through 
systematic study of a problem one uill slowly be 
able to approach better and better solutions. 

This demonstration can teach a good deal about 
free fall, propelled flight, the operational meaning 
of force, momentum, conservation of energy, the 
use of trigonometry, experimental unceriainty, and 
the scattering of data; but it can also be justified as 
a motivating experiment that is interesting and 
exciting. Rockets and space flight today hold a 
unique position in the public eye. It seems reason- 
able to make use of this interest in attempting to 
attract students to the physics couj-se. There is no 
question that the news of such firings in a course 
will spread rapidly through a school. As a conse- 
quence, students who otherwise might not have 
found out about the excitement and challenges of 
physics may become interested. 

D15 MAKING AN INTRTIAL BALANCE 

An ineriial balance may be an aid to help the stu- 
dents distinguish between mass and weight. One 
end of a hacksaw blade is clamped to a bench so 
that it can vibrate in a horizontal plane. Various 
masses are attached to it, but their weight is sup- 
ported by suspending the masses from a string. 
The hacksaw blade is pulled to one side and then 
released so that it swings. 

D16 ACTION-REACTION FORCES 
IN PLTLLING A ROPE. I 

Attach a hea\y spring balance to a wall and find 
two students whose maximum pull is about the 
same. Then place the spring balance between the 
two students and have them pull against each 
other with their m£iximum foree. The balance will 
read the same in each case. This should help bring 
home the point that a "pushed or pulled object, 
such as a wall, will exert an opposing foree when- 
ever a foree is applied to it. 



D17 ACTION-REACTION FORCES 
IN PITLLING A ROPE. II 

Place a student on each of two carts and pass a 
rope between them. First have one student pull 
alone, then the other, and finally both. Start the 
carts frx)m the same position each time and note 
the place where the\ mei't. ,\sk the class whether 
an obsener, watching the carts alone, could tell 
which student was actively pulling in each case. 



D19 NEWTOxVS THIRD LAW 

The following simple demonstrations dramatically 
illustrate Xewton s third law. Their simplicity, 
moreover, gives some indication of the elegance 
and profundity' of this remarkable law. 

To show that forces exist in pairs on different 
objects, and that the paired forces act in opposite 
directions, set up a linear equal-mass explosion 
between two dynamics carts. Propel the carts apart 
with a steel hoop, magnets, streams of water, or 
any other forces you can think of. See Fig. 33 for 
some suggestions. Stress that this concept of force- 
opposite-force is N'alid for all types of forces. 




Fig. 33 

The experiment on conservation of momentum, 
E3-1, gives detailed instructions about the explo- 
sion using the steel hoop. Vou can take a strobe 
photograph of the explosion, and show that if the 
carts have equal masses. the\' mo\e apart at equal 
speeds. If the carts ha\e equal speeds, the accel- 
erations they receixed during the explosion were 
equal in magnitude. Since the carts have equal 
masses and since the duration of the interaction 
is the same for each cart, \e\\lon s second law 
implies that they experienced equal forces during 
the explosion. 

A more direct method to show that the forces 
are equal in magnitude is to modifv the demon- 
stration by prT)pelling the two dxnamics carts with 
large magnetron magnets. Mo\e the magnets back 
about 3 cm on the carts. Place a pencil or dowel 
in the hole at the front of each cart and loop an 8- 
cm rubber band around the pencils. When \ ou re- 
lease the carts. the\' will separate, stretch the rub- 
ber band, oscillate, and finalK come to rest 

When the carts are at tvst. the forces acting on 
each cart are those shown in Fig. 34. The tension 
in a rubber band is unifoniv so 7' = 7''. Since each 
cart is at rest, then T = F and 7'' = F' Thus F = 
F' , and the magnetic forces on the carts are equal 
Note that in this demonstration the carts can haw 
difleivnt masses. 



U18 RL/VCTION FORCE OF A WALL 

When \()u U'an on a wall dcH's it e.xert a foix'e on 
you.' Stand on a cait or rollci- skates and lean 
against the wall 




Fig 34 



i'\iT 1 / c:oM:iii»Ts OF \un'i()\ 



Another exciting way to illustrate \eu1on's third 
law is to mount a sail on the fan cart that was used 
to illustrate uniform acceleration D4, and let the 
propeller blow against the sail. Since the sail bends 
forwai'd, clearly there is a force on it. But the cart 
does not mo\e because when the pix)peller pushes 
against the air, the air exerts a reaction force 
against the propeller. Thus, the net force on the 
glider is zero, ilf the sail does not catch all the air 
ftx)m the pixapeller, the cart ma\' mo\e slightly. i If 
you remo\'e the sail, the onl\' force on the glider is 
the reaction force exerted by the air on the pro- 
peller. This force causes the glider to move back- 
wards. 

The fan cart rigged for uniform acceleration is 
sketched in Fig. 35. The placement of the sail to 
show action and reaction is sketched in Fig. 36. 




Fig. 35 




Fig. 36 



D20 



ACTION-REACTIOX FORCES 
BETWTEEX CAR AND ROAD 

Demonstrate the coupling of forces between a cai' 
and the road. Obtain a motorized tov car. Place a 
piece of cardboard on top of some plastic beads or 
an upside-down skate-wheel cart. Then place the 
wound-up car on the cardboard roadway. The op- 
posing forces will cause the roadway to moxe back- 
ward when the car moves forward. 



One familiar example is the situation of two 
trains in a station on parallel tracks. .An observer 
in one train cannot tell which train is mo\ing, or 
whether both trains are moving, unless he or she 
watches the station. 

In the following demonstration, a camera pho- 
tographs a blinkv, with either the camera or the 
blink\' mo\ing at constant velocits'. From the pho- 
tograph, one cannot tell which object was mo\ing. 
The photos in the two cases are identical, unless 
pail of the laboraton' also appears in them. 

This idea that an observer's \iew of a motion will 
depend on ones frame of I'eference will be a major 
theme in Unit 2. To an observer on the earth, the 
sun seems to move dail\' around the earth. But the 
same apparent motions would be seen if the sun 
were stationary' and the earth rotated on an axis. 
The impossibility of distinguishing between the 
two motions caused much intellectual contixaversy 
in the sixteenth and seventeenth centuries. 

Equipment 

Polaroid camera, cable release, and tripod. 
With 3000-speed film, use the E\' 15 setting. 
Two dvTiamics carts 
Two Uniform Motion Devices lUMDl 
dc blinkv' 
Black screen 
Turntable 

Straight-line motion 

Mount the blinkv on one cart and the camera on 
the other. Use the UMD to push the carts. It may 
be necessarv' to increase the mass of the cart with 
the blink\', so that both carts are driven at the same 
speed. Arrange the apparatus as shown in Fig. 37. 



>^^: 



■ /// , ■ // \ — _^ — _ 



Black 
scReeN 



DC SLIKiKV 



D21 ACTIOX-REACTIOX FORCES 
IX HAMMERING A XAIL 

Hammer a nail into a plank while the plank is first 
on a bench, then on a soft pillow. The force e.xerted 
on the nail depends not only on the hammer but 
also on the opposing force of the plank. 

D22 ACTIOX-REACTIOxX FORCES 
IX JUMPIXG LTW'ARD 

When you jump off the floor, does the floor push 
harder on you in order to cause the upwaixl ac- 
celeration? Jump up from. a bathroom scale and 
watch the scale. 

D23 FRAMES OF REFEREXCE 

The following demonstration illustrates the idea 
that different motions can appear the same when 
observed from different reference frames. 



CAMe(«.A ^ 






UMD 




Fig. 37 Apparatus for linear motion. 

Take two photographs, one with the blinkv' mov- 
ing and the camera stationarv. and the other with 
the cart moving and the blinky stationarv. I'se the 
cable release and be careful not to jar the camera 
when you open the shutter. 

Circular motion 

Mount the camera on the tripod and attach the 
blinkv to a turntable. .Aim the camera straight 
down. Figure 38 shows this arrangement. 



DEMONSTRATION NOTES 









Fig. 38 Apparatus for circular motion. 



Take a time exposure with the camera at rest 
and the blinky moving one i-evolution in a circle. 
If you do not use the turntable, move the blinkv by 
hand around a circle drawn faintly on the back- 
ground. Then take a second print, with the blinkv 
at rest and the camera moved steadily by hand 
about the axes of the tripod. Tiy to have the camera 
move at the same rotational speed as the blinky 
moved in the first photo. 

Extension 

Observers in the train cannot tell which train is 
moving if there is a relative velocity between the 
trains. If there were a relative acceleration, how- 
ever, they could tell which train was accelerating. 
They could detect the acceleration of the train, for 
example, uith a liquid-surface accelerometer. If the 
acceleration were gi-eat enough, they uould also 
feel themselves being pushed back or thrown for- 
ward. An object cannot accelerate unless a force 
acts on it. 

Strictly speaking, ourobseivers could not be sure 
they were accelerating. According to Einstein's 
principle of equivalence, the effects of a unifomi 
acceleration a are indistinguishable from those of 
a uniform gravitational field -a. In the train, how- 
ever, the observers can be reasonably confident 
that the acceleixjmeter detects an acceleration, not 
some bizarre gravitational field. 

D24 IXERTLVL XTRSl^S NOXIXERTIAL 
REFERENCE FR/VMES 

Have a student toss a ball straight upwards and 
catch it again while walking at a constant speed. 
Ask for d(;scr-iptions of the path of the ball as seen 
by the ball tosser and by a seated student. How do 
the accelerations compar'e as measured by the 
walker and by the seated student? i rhe\ ar-e the 
same.) How would the path appear if the ball tosser 
had stood still and the student had nuned side- 
ways with the original speed of the walker'' (It 
would appear- to he the same as before to both 
viewer's. I 

Now toss the iiall as you acceler-ate, walking 
faster- and faster, and again as \()u slow down .Also 
toss the ball as voir walk in a (-ir-cie Show that, in 



these cases, the two frames of reference give two 
different accelerations. 

You might want to discuss this idea again in 
Chapter 4, where the idea is de\'eloped that accel- 
eration is caused by an unbalanced force. An ac- 
celerated frame of reference requires apparent lor 
fictitious) forces to explain accelerations that are 
not pr-esent when viewed from a fixed frame of ref- 
erence. 

D25 UNIFORM CIRCLTAR MOTION 

To demonstrate the acceleration in uniforTn cir- 
cular motion, place the accelerometer along the 
diameter of a phonograph turntable. When the 
turntable rotates, the liquid surface is parabolic. 
Figure 39 shows this situation. The acceler-ation in- 
creases with the distance from the center and is 
always directed inward. B\- changing the speed of 
the turntable, you can show that the acceleration 
is greater for higher speeds of rotation. This is also 
discussed in the Equipment \otes on the liquid- 
surface accelerometer, page 73. 




Fig. 39 Accelerometer on rotating turntable. The surface of 
the liquid is parabolic. 

D26 SIMPLE HARMONIC MOTION 

Harmonic motion can be demonstrated as an ex- 
ample of a more complex motion. To show that 
harmonic motion can be discussed in terms of cir- 
cular motion, set an object such as a peg on a 
phonograph turntable, mo\ing in uniform circular 
motion. Then illuminate this motion from the side 
and project its shadow onto a screen so that all 
that can be seen is a back-and-forth motion. Har- 
monic motior-r can be developed furlher. but it is 
prx)babl\- enough just to gi\e several examples of 
objects that ha\e this motion, such as a \ibi"ating 
timing for-k, a pendulum, and an object suspended 
on a sjiring. 

D27 SEVIPLE ILVRMONIC MOTION: 
AIR TR/\CK 

By attaching a long rubber l)and or string to each 
end of the carl and pulling back and forlh. \ou can 
make the carl mo\e in apprx)\imatel\ simjile har- 
monic motion. The class can see qualitati\el\ that 
the acceler-ation is directed opposite to the \el(»cit\ 
and is at maximirm when the carl is tanhest awa\ 
fn)m tin' ('({uilibrium [position 



HIVIT 1 / C;C1\CKPTS OF MfXTIOIV 



Experiment Notes 



El-l NAKED-E^T ASTROXOiVn' 

Useful equipment: 

SC-1 Constellation chart 
Star and satellite pathfinder 
Celestial calendar 

The subject of motion in the heavens is not taken 
up until Unit 2. Houe\er, it is advisable to have 
students carefully observe the skv in advance of 
studying that unit. This is because the motions of 
the heaxenh' bodies appear to be ver\' slow. The 
acti\ir\ is unusual in that it continues over several 
weeks. However, the time i"equired for each obser- 
vation can be quite short. Start earlv in the vear. 

There are no substitutes for the students own 
e.xperiences in making astronomical observations 
for themselves. For some students this mav well be 
the first time that their attention has been guided 
to the beautv of the night skv. At least they will 
come to appreciate the skill and patience of earlv 
astiT)nomers working with the same sort of primi- 
tive instruments. Some students may be excited 
enough to continue their observations beyond the 
outlines suggested. 

Suggest that each student or pair concentrate on 
only one of the observ ations. Later they can share 
their observations and make comparisons. \o stu- 
dent should feel compelled to attempt all the ob- 
servations, although am one mav do so. 

Conditions will varv greatlv, from areas where 
useful observation is nearlv impossible, as in smoggv 
cities, to places where the skv is ideally clear. Even 
in good areas, there will be bad nights. 

A planetarium visit can be used as a supplement 
to, or, in poor viewing areas as a substitute for, 
personal observation. Contact the nearest plane- 
tarium and explain brieflv your need for a special 
program. Most planetarium directors will be will- 
ing to put on a special show for your class that 
emphasizes the celestial motions important in Unit 
2. A suggested program is given at the end of these 
notes. 

A. Sun 

Warn vour students never to look directly at the 
sun since this can cause permanent eve damage. 
They should make all their obsei-v ations of the sun 
by indirect methods. 

The sun s azimuth, its direction measured from 
north through east for 360°, changes continuallv. 
In vour location it is not likelv to be at its highest 
point in the skv' at 12 o clock noon. One i-eason is 
that you may be on daylight saving time, in which 
case noon is about 1 p.m. But even on standard 
time you may not be located in the center of your 
time zone. Places near the center of each time zone 
are given in Table 1. If you are east of the center 
for your time zone, the sun will cross your local 
meridian 4 minutes earlier for each 1° eastAvard. 



Similarly, if you are west of the center of your time 
zone, the sun will transit 4 minutes later for each 
1° of longitude westward. 

TABLE 1 
Some Places Near the Centers of Time Zones 





Mid- 


Places Near 


Zone 


Longitude 


Mid-Longitude 


Eastern 


75=W 


Philadelphia 


Central 


90°W 


Memphis, St. Louis, New 
Orleans 


Mountain 


105^W 


Denver 


Pacific 


120^W 


Lake Tahoe 



Even if you were exactlv on the central meridian 
for your time zone, onlv rarely would noon occur 
at 12 o'clock. Each day the sun moves east among 
the stars, but not at a constant rate because the 
earth's orbit is elliptical rather than circular. Vour 
students will understand this when Kepler's sec- 
ond law is discussed in Chapter 7. Even a uniform 
motion of the sun along the ecliptic would i-esult 
in uneven da\s because the sun s annual path also 
has a north-south component. So, our clocks run 
on a fictitious average dav iMean Solar Time i based 
on the length of a vear. .Actuallv the sun gains and 
loses on Mean Solar Time. The difference is called 
the Equation of Time and may amount to ov er 16 
minutes. 

B. Moon 

The moon appears to move eastward among the 
stars approximatelv 360° per month. Bv plotting 
the position and shape of the moon on the con- 
stellation chart, the students ma\' be able to con- 
firm how the moon's phase depends on its position 
relative to the sun. iThe sun's position at 10-day 
intervals mav be given along the ecliptic on the 
chart. I 

Students could use the astrolabe described in 
the Handbook to measure the altitude and azimuth 
of the moon. During a winter night, the full moon 
reaches a higher altitude than the sun did at noon. 
During a summer night, the full moon reaches a 
lower altitude at night than did the sun at noon. 

The new moon is close to the sun, full moon is 
180° from the sun, and quarter moon is 90° from 
the sun. Xote that the new moon to first quarter 
moon can only be seen in the late afternoon and 
evening while the third quarter to new moon can 
only be seen in the morning. lYes, the moon can 
often be seen while the sun is up.i 

C. Stars 

The "Star and Satellite Pathfinder shows which 
stars are above the horizon at latitude 40°i\ at a 
particular date and time. The Constellation Chart 
shows the stars in a band 60°\ and S around the 
celestial equator. This includes all the stars high in 



EXPERIME.XT VOTES 



the sicy at middJe latitudes. The curved line across 
the middle of the chart is the ecliptic. The suns 
path throughout the year and its position at 10-day 
intervals is marked on the ecliptic. 

Relative to the sun, the stars move about 30^ 
westward per month. Diffei-ent stai-s appear in the 
sky as the seasons change. For example, Orion is 
prominent in winter but is not seen in summer 
when the sun is in that part of the sky. See the 
"Star and Satellite Pathfinder" for information on 
what stai-s are \'isible month by month. 

D. Planets 

Because the sun, moon, and planets stay in the 
same narrow band around the sky, we can con- 
clude that they ail move in nearly the same plane; 
that is, the planetary system is essentially "flat." 

In "normal" motion planets mo\'e eastward among 
the stars; in retrograde motion they move west- 
ward. Consult the Celestial Calendar to find out 
when the different planets are in retrograde mo- 
tion. Post the month's Celestial Calendar with 
eclipses, conjunctions, and similar data marked 
on it. 

Supplementary note on coordinate systems 

Although coordinate systems for locating objects 
in the sky are not an important aspect of this study, 
teachers may wish an explanation of the various 
systems used. 

Coordinates on the Earth: The latitude-longitude 
system is used to locate objects on the earth's sur- 
face. The equator of the earth is established as a 
great circle along the earth s surface halfvvay be- 
tween the north and south poles and perpendic- 
ular to the earth s polar axis. Meridians are a set 
of great circles passing through the poles and are 
perpendicular to the equator. The local meridian 
lyour norlh-south linei establishes your east-west 
location. The meridian passing through Green- 
wich, Kngland, is called the prime meridian and 
has an assigned longitude of 0°. Places west of the 
prime meridian up to halfv\a\' arf)und the earlh ito 
the International Date Linei ha\e longitudes west. 
Places east from Cr-eenwich up to the International 
Date Line ha\e longitudes east. Ma.ximum longi- 
tutles art; thert^foit? 180°K and 180°\\'. 

Latitudes art; angular distances measirivd norlh 
or south frxjm the equator to the poles, a total dis- 
tance of 90°. Thus, the latitude of a place is the 
angular distaiKre betv\'een the place and the equa- 
tor as one might see it fi-om the earth's center. 
Coordinates in the Sk\r One convenient wa\' to es- 
tablish the position of a star oi- other hea\enl\ ob- 
ject is to use the aUitiide-ay.innith system. The co- 
ordinates in this syst(Mii ar-e: 

,'\ltitu(l(v tin; angle of the object above the ob- 

s(M-ver's local hoiizon. 
/Xzimuth: the direction aroirnd the hoiizontal 

plane measurt^d eastwarxl lixjiu triie norlh. 

Sucli a system is local. No two obseiAer-s \e\-en a 
few miles apart i have at the sarin- moment the 



same coordinates for the same star. Also, as the 
earth turns, a star's position on this system con- 
stantly changes. 

For this reason, astrxjnomer-s long ago devised a 
coordinate system attached to the so-called celes- 
tial sphere. This is sometimes referred to as the 
equatorial system and the elements measured are 
right ascension and declination. Imagine that we 
extended the earlh s axis to the celestial sphere. 
Also, extend the plane of the equator until it inter- 
sects the celestial sphere. Great circles passing 
through the \orlh Celestial Pole and crossing the 
celestial equator- at right angles ar« called hour cir- 
cles. These are similar to meridians on the earth's 
surface. 



COORDINATES IN THE SKY 
AfMud^-AzimuITt Syslani 




Equatorial SysMm 



F H W CTW W 0« ••rt*' 




The hour cirxMe that passes through the vernal 
equino.x is the refeivnce ciivle fr-om which r/g/if 
asrensioi} is measui-ed The right ascension of a 
star is the angle measuivd easrwanl along the ce- 
lestial e(|uator frx)m the venial ecjuinox to the hour 
ciri'le passing tliix)ugli the botlv Ihe angle is meas- 
urvd in hoiri"s Since it takes 2-4 li()ur> tor the re- 



42 



UNIT 1 / COMCKPIS OF MUl lfl\ 



lestial sphere to rotate through 360°, 1 hour is 
equKalent to 15°. 

Declination establishes the distance of a star 
along an hour circle north or south of the celestial 
equator. Declinations are like latitudes on the 
earths surface. A star ha\ing a declination of 40°\ 
passes overhead at places having a latitude of 40°\. 

Stars remain ver\' nearly fixed with respect to 
their coordinates in the right ascension-declination 
system. 

SUGGESTED LVTRODUCTORY 
PLAXETARILTVl PROGRAM 

This is an outline of the major phenomena that 
would be most useful and appropriate to Project 
Phvsics students. A program used at the Morrison 
Planetarium in San Francisco is used as a model. 
Other planetarium programs ha\ e been outlined in 
this Resource Book. 

1. The current night sk\ 

lai Set slcv at predaun and at 9:00 P.M. 
lb) Locate Polaris 

(c) Point out a few constellations 
111 Ursa Major 

(21 Cassiopeia 

(31 C\gnus I including the binar\ .Albireoi 

(41 Sagittarius ion the ecliptic i 

(d) Show off the planets 

le) Display sun and moon against starr\ field 

2. Motions in the heavens 

la) Circumpolar stars for 24 hours 

(b) Sun for 

(1) 12 hours demonstrating westward motion 
(21 1 month demonstrating eastward motion 
(3) 6 months demonstrating north-south mo- 
tion 

(c) Moon for 
111 6 hours 
I2i 1 month 

(d) Planets emphasizing 
111 retrograde motion 

l2i maximum angle of elongation for Mercurv 
or \'enus 

3. Celestial coordinate s\stems 
la) Altitude and azimuth 

lb) Right ascension and declination 

Ic) Constellations used to locate planets 

El-2 REGLTuARITi AND TEVIE 

Equipment needed: 

Dragstrip i chart recorder) 

Blinky 

Pendulum 

Metronome 

The first part of this experiment is designed to 
show the regularity' of a few natural events. Stu- 
dents compare a variety' of recurrent phenomena 
with a standard clock,' such as the blink\ or met- 
ronome. The recurrent phenomena might include 
another blinkv', pendulum, object on a spring drip- 
ping burette, the human pulse, or tape-recorded 
crickets. 



The mention of "time ' should be avoided in this 
part, because the students notion of absolute time 
will confuse the problem of regularitv . However, do 
not force the issue. The point is to investigate first 
the regularitv' found naturalh' in the world and 
then move on to contrived measurement stand- 
ards. 

Caution the students not to bear too heavily on 
the recording tape with their pens, because the 
increased drag might affect their results. When stu- 
dents have completed the measurements from the 
dragstrip recorder on their own tapes, the infor- 
mation can be pooled on a master graph. It might 
look like the Fig. 1 below. 



T/cks 























( 1 




. 


,^ ^-^H 


^,^JI 


f-^- ..-..._- 


;;— -;:..-..-3 


















1 







pu/j e ^ 



/KsJb\0 



'sS ^si % To ^o^^^^'^y rictus 

Fig. 1 The regular events are those that show similar curves 
on a graph. 

If a light, even that which comes in the window, 
is allowed to fall on the blinkv bulb the rate will 
change bv 4'^o or more. This is especiallv true for 
some of the earl\ experimental models. .All recent 
models have had a radioactiv e gas added to trigger 
the bulb in total darkness and to maintain stabilitv. 
It might be a good idea to intentionalI\' cause the 
rate to change during the run. The relativitv of reg- 
ularitv' would thus be emphasized, since the good 
clocks" will all show common cuned records on 
the graph, i However, since we want the students 
to accept the blinkv as a reasonably good clock, 
the disturbance should be accounted for after- 
wards. If one or two blinkies are used as unaltered 
"controls, " the explanation will be more convinc- 
ing, i 

Answers to questions 

1. .Answer depends upon results. In general, it is 
not possible when comparing two isolated sets 
of events to state which is more regular. 

2. Two events, here B and C, are compared to a 
third one, .A: therefore .A is taken as being the 
standard. If one event is defined as being reg- 
ular then all other events can be compared to 
it. Thus, for example, if C is more consistent than 
B in the number of recurrences in equal time 
periods as marked bv A, then C is the more reg- 
ular. 

3. There is no measure of absolute regularitv. Tim- 
ing is alwa\s a matter of comparison. Whatever 
is taken as the standard of comparison is as- 
sumed to be regular for the purposes at hand 
whether it be rotation of the hands on a wall 



EXPERIMENT NOTES 



clock, the apparent annual north-south move- 
ment of the sun in the noonday sky, the \abra- 
tions of a crystal, or any other seemingly peri- 
odic phenomenon. 



El-3 V^ARIATION IN DATA 

Equipment needed: 

A wide variety of objects to count, measure, 
and weigh 

The student should become familiar with differ- 
ent kinds of variation in measurement by doing 
this experiment. V\'hile it is possible to introduce 
significant figures, the intention here is only to 
make students comfortable with variation in its 
simplest tenns. 

The general plan outlined below is intended to 
start the student on familiar ground, where it is 
firmly believed the variation is in the things meas- 
ured. Then the experiment progresses through sit- 
uations in which variation is in the measuring 
process to those in which the source of variation 
is uncertain. 

You may wish to give these classifications of \'ar- 
iation to the students after they have finished mak- 
ing the measurements but before the discussion; 
or you may feel it u'ill be more valuable if these or 
similar categories are discovered through discus- 
sion. A few examples follow. 

1. Situations where the variation is unquestionably 
due to differences among the things being meas- 
ured. 

(a) Students' heights or weights 

(b) Family size 

(c) Number of pieces of candy, raisins, or other 
objects in different boxes 



2. Variation unquestionably due to changes in the 
thing being measured. 

(a) Temperature of a beaker of warm water 

(bl Weight of a chunk of dry ice 

(c) Weight or length of a burning candle 

3. Variations unquestionably due to the process of 
measuring. 

(a) Separation of blinkv dots on a photograph 
using a ruler 

(b) Separation of blinkv dots on a photograph 
using magnifier 

(c) Diameter of a piece of wire measured with a 
ruler 

(di Diameter of wire using micrometer or mag- 
nifier 
le) Diameter of a puck 

4. Sources of variation uncertain. 

lal Rotation rates of students' phonograph turn- 
tables 

ibi Height as measured in the morning com- 
pared to height at night. 

These classifications are not the only ones pos- 
sible. One important class of variation not really 
coxered here is the statistical variation of random 
events (such as background radiation count i. This 
classification will be considered in more detail in 
Unit 6, The \'ucleus. For the lab work in this course. 
Class 3, which includes variations due to the proc- 
ess of measurement, is the most important, and is 
emphasized in some of the experiments. 

Station suggestions 

More ideas are listed hei'e than can probabh- be 
used and you may have other ideas that \ou wish 
to substitute. \'arier\' is the ke\Tiote, of course. 

All students should \isit e\er\' station but they 
do not have to begin at the same point in the c\'cle. 



Object 



Measuring Instrument 



Quanttty 



Marble 

Large steel ball 

Beaker of water colder than room temperature 

Beaker of water at room temperature 

Beaker of water warmer than room temperature 

Empty beaker 

Metal cylinder 

Puck 

Puck 

Blinky dots on photo 

Blinky dots on photo 

Wire 

Any object 

Bottle of water 

Dry ice 

Rotating wheel (slow) 

Burning candle 

Line circuit 

Dry cell 

'\()l«v Stmlcnls iiia\ rvali/.c Itial all thcnnomiMi 
ilifiTnoniflci-s lo ii'cliici' oi (>liiniiiali< this 



Vernier caliper 

Vernier caliper 

Thermometer* 

Thermometer 

Thermometer 

Thermometer 

Ruler 

Common calipers 

Ruler 

Ruler 

Magnifier 

Ruler 

Stopwatch 

Graduated cylinder 

Balance 

Stopwatch 

Balance or ruler 

Voltmeter 

Voltmeter 



Diameter 

Diameter 

Temperature 

Temperature 

Temperature 

Temperature 

Length 

Diameter 

Diameter 

Distance between 

Distance between 

Diameter 

Time of fall from indicated height 

Volume 

Weight 

Rotation rate 

Weight or length 

Line voltage 

Voltage 



i> (1(1 nol IT. Ill r\.ict!\ alike iinilcr like ciniinTitanccs ( ai-rfiilK sclcrt thr 



I'NiT 1 / co\c:ept8 of M()TI0\ 



The purpose of this laboratorv' is not to achieve 
unanimous agreement on the sources of variation 
in the measurements, but to make students aware 
of the issue and how critical the issue can be in 
experiments. It is important for students to realize 
that variation problems are not confined to school 
laboratories. The best of scientists with the most 
expensive equipment are often faced with the in- 
terpretation of variation. 

Advance preparation 

The stations around the room must be set up be- 
fore class starts. Suggestions for stations are listed 
below. About 10 or 12 stations will be needed if 
students are to gain a varietv' of experiences. 

If each student makes every measurement, one 
50-minute period will be needed. From one-half to 
a full period will be needed to vviite the results on 
the board and discuss them. 

Answers to questions 

1. Differences in use of instruments and especialh 
in estimating between marks on scales. Also, 
similar objects, including instruments, for ex- 
ample, a set of meter sticks, are not exact du- 
plicates. Sometimes the objects being measured 
can change between measurements in response 
to changing conditions. 

2. No, there are no absolutely correct measure- 
ments (aside from the trivial case of counting a 
few discrete objects . 

3. \o, since the av erage will reflect all of the meas- 
ured V alues, some of which could be far from 

the correct one, whereas some one of the in- 
dividual measurements could be quite close. 
The trouble is that there is no wav of knowing 
for sure which one of those values is most cor- 
rect. Therefore, one uses the average on the as- 
sumption that "on the average ' it will be closer 
than any other v alue. This assumption is rea- 
sonable if it seems likelv, in a given set of meas- 
urements, that errors are as apt to be in one 
direction as another, that is, are comparablv dis- 
tributed on both sides of the true value. 

4. B\' indicating the range of variation. One might 
indicate the average v alue and add a statement 
indicating the range of values that includes 
some proportion isay two-thirds i of all the meas- 
ured values. Statistics textbooks provide meth- 
ods for expressing distributions. 



Encourage students to study uniform motion in 
a variety of ways. For example, two students woi-k- 
ing together can 

1. photograph a puck sliding on bead-covered 
glass ripple tank or some other smooth enclosed 
surface 

2. photograph a glider coasting on a level air track 

3. photograph a tov tractor pushing a blinkv 

4. measure motion of an object in a film loop pix)- 
jected on the chalkboard 

One student alone can measure 

5. a transparency showing what is asserted to be 
uniform motion 

6. a stiT)be photograph, such as the momentum- 
conservation collision photos or the photo on 
page 12 of the Test 

If not enough apparatus is available for the 
whole class to do the same experiment, perhaps 
the class can be broken up into small groups, each 
of which will use a different method. 

InstiTJCtions for operating the Polaroid camera 
and for using the rotating disk stroboscope are 
found in the Equipment \'otes section of this Re- 
source Book. 

In the Handbook we describe the experiment as 
done by Method 1 abov e. Other methods differ onlv 
slightlv and in obvious wa\s. The procedure for 
using the data from anv of the methods is identical. 

If an air puck or a puck sliding on beads, is used, 
it should have a large white X or a rubber stopper 
painted white for easv reference in the photograph. 
Since the puck will probablv rotate, the white in- 
dicator must be at the center, not on the edge, of 
the puck. 

We assume students have studied the Te^t 
through Sec. 1.4, in which case thev will end their 
write-ups after the section entitled Graphing mo- 
tion and finding the slope." 

If they have studied graphs iSec. 1.5 1, however, 
it may be desirable to have them go on to subse- 
quent sections of this experiment in which they 
graph their data. In this case it mav be worthwhile 
to take X\vo runs at different speeds in order to 
show how the two resulting graphed lines differ in 
slope. 



El-4 MEASLTUNG UNIFORM MOTION 

Major equipment for v ersion described: 

Flat smooth surface 

Plastic beads 

Puck or other smooth-bottomed disk 

Polaroid camera 

Rotating disk strobe 

Light source 

Millimeter ruler for measuring picture 

Blinky 

Uniform Motion Device 



Answers to questions 

1. Yes: because it is straight. 

2. .Answer depends on student graphs. 

3. Yes, the same general method can be used, but 
the technique will varv' with circumstances. 

4. Ia)± 2.5 km/hr. 

lb) \o. not reliably. The changes are smaller than 
the uncertaintv'. For the 2-km hr change the 
reliabilitv is greater than for the others. 



EXPERIME.VT .VOTES 



El-5 A SEVENTEENTH CENTURY 
EXPERIMENT 

Major equipment for seventeenth century experi- 
ment: 

Grooved incline about 2 m long 

Supporting ringstands 

Ball to coll in groove 

Water clock 
The next three experiments deal with the accel- 
eration of gravity. Since El-7 is di\'ided into six 
parts, there are eight possible attacks upon a^. 
Some thought should be given to which experi- 
ments should be selected. 

Only one phase of Galileo's investigation has 
been selected for this experiment. A full descrip- 
tion of it can be found in Dialogues Concerning Two 
New Sciences, the "Third Day." See also the i-efer- 
ences to the Crew-de Salvio translation partially 
reproduced in Chapter 2 of Unit 1. A Dover Pub- 
lishing Company i-eprint of this enjoyable book is 
available. Also a careful modern repetition of this 
experiment is described by Thomas B. Settle, "An 
Experiment in the History of Science," in Science, 
Vol. 133, January 6, 1961. Historians and philoso- 
phers of science are still hotly debating whether or 
not Galileo took actual experimentation very seri- 
ously, and whether he actually did some of the 
experiments he described in such graphic detail. 

At least two students are needed for each setup, 
one to handle the rolling ball and the other to op- 
erate the water clock and record data. By dixiding 
the jobs further, as many as four can be usefully 
employed. 

Apparatus 

An inclined plane about 2 m long is needed. It 
should have a groove or channel dowii one edge 
in which a ball njns very smoothly. 

If only one inclined plane is axailable, it can be 
oper'ated by one or two students while the rest of 
the class, individually or in pairs, ar-e oper-ating 
water clocks. 

Distances mar-ked on the incline are arbitr'ary, 
but should be chosen to work well with the rate of 
How of the water clock. We find that 12 marks l.'i 
cm apai1 serve well. Students should not con\er1 
to present-day standards of length, but should 
merely record the distances as units of length, 1, 
2, 3, .... 

The right size of tube and of collecting vessel for 
the water clock must be found by trial and eiTt)r. 
The flow should last at least thrt'e or four seconds 
without oxeiilowing the collector-. The water clock 
does not work as well if it is started and stopped 
with a pinchcock on a nrbber exit tube below the 
funnel. 

It is itHX)mmended that the students peribriii a 
minimirm of foirr trials for each distance and that 
the a\erage time \alire r-ecorded be used in the 
calcirlations. For- longer distanc('s fewer trials ma\ 
be used if the lim(\s seem to be in close agrT'enrent 
lour difleirnl distance's, for- example, 3, 6, 5), and 



12 units should be sufficient for each angle of the 
ramp. It is prxjbably not practical for any one gnaup 
to attempt to take measur-ements at more than two 
differ-ent angles of inclination. An exact judgment 
of the slope of the channel is not critical. The re- 
sults for- heights oxer 30 cm may show considerable 
scatter, depending upon the skill of the students 
and quirks of the equipment. 

Recording data 

It is a good idea at this early stage in the course to 
firmly insist on a neat data table. If students cire 
always quick to r-ecord all their data in ink directly 
into their final r-eport, it does wonder's to de\elop 
clear careful thinking as the year goes on. Mistakes 
will be made, of course, but should be crossed out 
neatly. 

It is useful to plot d versus t first to show that 
the graph is not a straight line. Point out that there 
is no way of recognizing with the unaided e\'e any 
cune except a circle and a straight line. Only by 
plotting in such a way as to generate one of these 
shapes can we identih' the relationship between d 
and t. 

In graphing the results, plot itimei" along the 
horizontal axis. Not only is this comentional, but 
also, when d is plotted along the vertical axis, the 
resulting slope is equal to twice the acceleration. 

If students suggest that the d ^ t cur^e does look 
like a parabola and is ther-efore a d ^ t' relation- 
ship, challenge them to show that it isn't a d ^ t^ 
relationship, which may ha\e the same general 
form las can be \erified by trial i. 

It may be useful to have a pair of students do 
the twentieth century' version of Galileo's experi- 
ment iEl-6) by photographing a glider sliding 
down a tilted air track. This \ ields more precise 
data, which may be r^eassuring when the final con- 
clusion of the experiment is discussed. 

Possible extensions 

After each grxjup has completed its investigation, 
one of two possible prx)cedures is recommended: 

(a I Each grxiup ma\' r^eporl its findings orally and 
comparisons ma\ be made during a discussion 
period. 

Ibi Composite findings ma\ be tallied on the 
chalkboarxl, and, using these combined results, all 
students ma\- plot the entir-e famih of curves for 
the difler-ent inclinations. 

When romjiaiing ivsirlts make the point that the 
linear r-elation lu'tween d and t' appear-s to hold 
for the rxilling ball, at least for- small angles of in- 
clination of the channel iwithin the \ariation ex- 
pected'. .As an aside, mention that for an\ gi\en 
angle of inclination the distance internals rolled 
down the incline, in successive units of time, will 
follow the pattern l:3u=i:7. . . . 

The value of a^ foirnd by extr-apolating data from 
this experiment, will be too low because onl\ part 
of the balls inci-easing kinetic energ\ as it de- 
scends is in the form of eneigv of motion along the 
plane I he rtMuainder is in the form of ener-gv of 



4« 



ii\iT I / (;()\(:epts or iviotkix 



rotation. Secondly, friction is a \er\' large factor in 
reducing the acceleration. Both these important 
effects are reduced or eliminated b\' photograph- 
ing a glider descending an air track isee EI-Gk 

Derivation of a ^ 

Some of the better students may want to use their 
data to calculate a^, the acceleration of graxity. This 
is not easy. 

The potential energ\' of the motionless bcill at the 
top of the track equals its kinetic energ\' at the 
bottom if friction is so small as to be ignored. But 
the kinetic energ\' is not mereK energ\' of linear 
motion, mv' 2: some energ\' is also in the form of 
rotational motion, described by the expression 
III)' 2. Thus the conversion of energ\' ftx)m potential 
to kinetic is described b\' the equation 

ma„n = H 

2 2 

The angular \elocit\' of the ball co at the bottom 
of the ramp is defined lin radians seci b\' o) — vr 
where r is the radius of the ball and \' is its \ elocitv 
along the track. / is the moment of inertia of a ro- 
tating object. For a rolling solid sphere of mass m 
and radius r, I = 2 /5(mr"i. 

Putting these expressions for oj and / into the 

loi- 
rotational energy term, — , the energv' equation 

becomes 



ma„h 



mv 


9 


2 


■+■ —mr 
5 


mv" 


mv~ 

-i 



2r- 



^ 10 

But v~ — lad where d is the length of the plane 
and a is the acceleration parallel to the incline, so 

7 2ad 

a„ = — X 

^ 10 h 

7 ad 

where h is the height of the plane and d is its 
length. The acceleration a is measured by means 
of the water clock or b\' other more precise meth- 
ods. 

Notice that m and r cancel out of the final 
expression for a^. Hence the acceleration does not 
depend upon the size of the ball, which refutes 
Aristotle s assertion. 

It would be a mistake to present this analysis to 
a class at this stage. Only a very special student 
might be able to follow this. 

Answers to questions 

1. The graph, d \ersus t'. should be a straight line. 

2. Student answer. 



Going further 

1.-2. Student answei-s. 

3. A student probably cannot do as well as Galileo. 
Howe\er, estimating is a skill that can be im- 
pro\'ed with practice, and it is not out of the 
question that Galileo could ha\e attained the 
claimed accuracy. 

4. Ratio should be 1:3:5:7 .... 

El-6 A T\\TNTIETH CE.VTLTIV \TRSIOX 
OF GALILEO'S EXPERIMENT 

Major equipment for r^ventieth century \ersion: 

Air track and glider 
Polaroid camera 
Rotating disk strobe 
Light source 
Blower for air track 

The modem version of Galileo's inclined-plane 
experiment with an impro\ed clock and an air- 
track glider gi\es the same results, namely, that 
dr is a constant. However, the precision is im- 
pro\ed. The idea behind the impro\ed experiment 
is still Galileo s. It is a test of a logical consequence 
from the assertion that things accelerate in the 
physical world. 

Measurement of a^ 

.A simple procedure for students who ha\e not 
taken trigonometn' is to ha\e them calculate the 
accleration laj, a,, a^, etc.i for different angles of 
inclination between the air track and the horizon- 
tal. Next plot a graph of acceleration \ersus angle 
of inclination. Finally extrapolate this graph to 90° 
in order to estimate the free-fall acceleration, a^. 
Students who ha\e had trigonometn- max- cal- 

a 
culate a by the following relationship: a = 



sin 6 



or a„ = 3( ~ I Refer to the diagram below for an 

explanation of the algebraic s\Tnbols in these re- 
lationships. 



sin d - 



sin 6 



Fig. 2 




EXPERIMENT NOTES 



47 



6 = the angle between the horizontid and air tiack 
/ - length of air track 
h — The height of the aii- track 
a, = the acceleration parallel to air track at angle 6 

d = the distance through which the sled is allowed 
to slide 



Sample Data and Results 



DISTANCE (crr^ V5 T/Me*^»«c)* 






.5° 


1.0° 


2.5° 


5.0° 


10° 


^ 


d{cm) 


fisec) 


fisec) 


t(sec) 


f (sec) 


t* (sec) 


si 

V 


20 


2.7 


1.9 


1.1 




.4 


30 


3.3 


2.5 


1.4 




.5 


:i 


40 


3.7 


2.8 


1.6 


0.9 


.6 


t 


50 


4.2 


3.1 


1.8 


1.0 


.7 




60 


4.5 


3.5 


2.0 


1.1 


.7 




70 


4.9 


3.8 


2.1 


1.2 


.8 





'Data for 10° was obtained from a strobe photo. All others 
were obtained using a stopwatch. 



Calculations for 6 = 5° 



t« 




{—<^\ 


(5) 


0.8 


slope 


1.0 


from 


1.2 


Graph B 


1.4 


^ = 50 



151 



MK^' 



100 cm sec^ 
100 cm m (0,087) 



&»»-»>PH A 
DISTANCC V». TIME 
2 5* 




time (sec^ 



Extension ol' laboratory 

The above calculation indicates that the free-fall 
acceleration was 1 1 in sec" rather than 9.8 m sec*. 
Consider the following points regarding this e.\- 
pcrimcntal result 




5676910 1112131*1916 If » tt to 

1. Graph B shows that the acceleration down the 
incline is directly proportional to the square of the 
time. This supporis Galileo's notion that a body 
does accelerate. 

2. This is in disagreement with Aristotelian 
physics, which has no wa\' of talking about accel- 
eration. This result is, therefore, re\olutionarA'. 

3. One more remarkable aspect of either \ersion 
of the se\enteenth centutA' e.xperiment is that Gal- 
ileo mentalK eliminated friction and that he thought 
fotAvaixl to the possibility' that there might be such 
a constant as free-fall acceleration near the surface 
of the earth. There is no evidence that he at- 
tempted to calculate this value. 

4. The teaching of some mathematics from Graph 
B might be more impoilant than ani\ing at 9.8 m 
sec". That is, a x t' and a - kt' where the constant 
of pft)poi1ionalit\' k is the slope of a line and rep- 
resents the acceleration of the sled at a certain an- 
gle of inclination. FinalK . if the slopes are plotted 
against the angles, an extrapolation to 90° gives a 
new idea, namel\ , fiT*e-fall acceleration. 

5. The fact that 1 1 m sec" was calculated in com- 
parison with 9.8 m sec" is not had. It is not bad 
because Project Physics is moiv than the \erifica- 
tion of the data in the Handhook nf Chrrmstn and 
Physics. 

Theiv is an oppoi1unit\ here to guide the stu- 
dents toward operiitionalism. ,Ask them to desciibe 
in detail the measuring instalments and the scales 
on these instiaiments. Ha\e them calculate the un- 
certainties of measurements Ha\-e them calculate 
the uncertaint\- that can be e.xpected in the result 
1 1 m sec" . 

Refer students who want to l)e more precise to 
1:1-7 



48 



I'MT I / CONCEPTS (II MOTION 



Answers to questions 

1. Student answer. 

2. If the students claim that the graph is a straight 
line within the limits of uncertainty of their 
measurements, then the significance is that the 
cdr-track glider is accelerating. 

3. Student answer. 

error 

4.% of error = ; — ; — x lOO 

accepted value 

5. (a) Human reaction time. In measuring small 
time intervals the reaction time in operating 
a stopwatch becomes an increasingly' signifi- 
cant factor. Notice the absence of data for the 
shorter distances on the 5° slope. Strobe pho- 
tography is recommended for the steeper 
slopes. 

(bi In spite of the \'er\- low friction with the air 
track, friction is not eliminated completely. 

(ci The technique of i-eleasing the glider is \er\' 
critical to a \alid measurement. Se\eral dr\' 
runs are needed to achie\e successful oper- 
ation of the equipment. 

Idi The air blasts may exert an impulse upon the 
glider. 

El-7 MEASURING THE ACCELERATIOX 

OF GRA\Tn' Og 
Introduction 

Acceleration due to gra\it\' is a crucial topic that 
has been illustrated indirecth' in the two pre\ious 
experiments. Since there are six methods of finding 
a in this experiment, a decision must be made 
regarding which of these should be done. Ideally 
different groups of students should use different 
methods and then compare results in class dis- 
cussion. This will pro\ide an opportunit\ to raise 
questions about \'ariations and error and about 
how "standard \'alues" are arrixed at. 

Method A: ag by Direct Fall 

In any direct measurement of a , a falling object 
has to be timed accurately as it falls through se\- 
eral preciseK' measured distances. Ordinarily the 
distance of fall must be kept small in order to a\oid 
the appreciable air resistance encountered at high 
speeds. But a short fall is usualK too brief to time 
accurately without elaborate equipment. In this 
experiment with \er\' simple equipment these two 
limitations cause an error of less than Z'^o . 

If a recording timer is available, it may be more 
convenient than a tuning fork for marking the mo\- 
ing tape. The experiment is otherv\ise the same. 

Clamp the timer at the edge of a table in such 
a way that the paper passes freeK' through it \er- 
tically and clears the edge of the table. Since it is 
difficult to measure the frequency of the clapper 
accurately when operated by 1.5 \' dc you might 
short-circuit the breaker gap inside the timer and 
operate the timer on 60 Hz ac. Use a short length 
of wire with a small battery clip on each end. Be 
careful that your short circuit connections do not 
interfere with the free motion of the clapper. 



You can pro\ide the necessan' low \'oltage from 
a bell-ringing transformer, or in some cases from 
the 6-\' ac tap of your power suppK'. Use a small 
rheostat, such as the one used to control a ripple 
tank wa\e generator, in series with the power sup- 
ply. It is important that the current be adjusted 
until the \ibrator action is loud. firm, and regular. 
A skipped beat or two can completeh' spoil your 
results and occasionally does. Since you are oxer- 
loading the coils of the timer, you should leaxe the 
current on as briefly as possible. 

The clapper is now \ibrating at either 60 or 120 
Hz. To discoxer which, you need merely pull 1-2 
m of tape through the timer by hand at a speed 
sufficient to resohe the dots being made by the 
clapper, and count the number of dots made in 
approximately one second. The choice between 60 
and 120 Hz will be obxious and no other frequen- 
cies are possible. 

To measure a^ hold the weighted tape in the 
timer, start the timer, and release the tape. The 
series of carbon-paper dots on the tape can then 
be analyzed in the same way as the waxes formed 
by the tuning fork. 

Answers to questions 

1. Student ansxxer. 

error 
1% of error 



X 100) 



accepted x alue 

Method B: a^ from a Pendulum 

■Although this is an indirect method for measuring 
a , it is probably the simplest method that can be 
considered accurate. 

The derixation of the equation for T. the period 
of a pendulum, draxxs upon concepts of simple 
harmonic motion that students at this stage are 
unable to follow. Most first-xear college texts in 
general phxsics gixe the derixation. The practical 
considerations, hoxxexer, are xery simple. 

The clamp that holds the top of the pendulum 
suspension must not haxe rounded edges to its 
jaxx's, for if it does, the suspension xxill, in effect, be 
shortened slighth as its sidexxays motion xxTaps 
the top fexv millimeters around the rounded edges. 
The clamp must also be very rigid: anx back-and- 
forth wobble xvill increase the period. 

Since the formula is onl\' correct for xery small 
amplitudes of sxxing icertainlx' no more than 10°i, 
the timing should be done with the smallest 
sxxings that can still be seen after 20 trips. 

If 20 round trips lasting 12.0 sec are timed xxith 
starting and stopping errors of 0.2 sec each, the 
total timing error is 0.4 sec. Since this error is 
shared among 20 swings, the timing error per 
sxxing is onlx' 0.02 sec. Because each sxxing takes 
12.0/20 = 0.60 sec = T. the uncertaintx' in 7 due 
to timing is 3% . This is xeiy large indeed compared 
xxith other possible sources of error. To reduce it, 
time a larger number of sxxings, say 50, xxhereupon 
the same error in timing leads to onlx- about 1.3% 
uncertainU' in T. 



EXPERIMEVr .NOTES 



49 



The length of a pendulum whose period T is 1 
sec is 24.8 cm. Remember that 7' is the time for a 
round trip. The pendulum that takes 1 sec to suing 
one way only will be 99.4 cm long. 

More values are in Table 1. 



TABLE 1 




Period of Various Pendulums 


/ 


T 


20 cm 


0.75 sec 


40 


1.26 


60 


1.54 


80 


1.78 


100 


1.98 



Answers to questions 

1. Student answer. 

2. Answer should agree with accepted value within 
1%. 

3. Student answer. 

Method C: a^ with Slow-Motion Photography 
(Film Loop) 

A successftjl motion pictui-e uall depend upon the 
control of light. The black backdrop should be in 
shadow. It is worlhwhile to arrange the lights so 
that they will cast a shadow on the backdrop but 
bathe the meter stick and falling ball in the fore- 
ground. It is best to always shine the light fixjm the 
side, that is, about 90° from the direction of the 
movie camera. This will result in excellent contrast. 
Most teachers will want to use the film loop. The 
film loop projector is one of the most effective 
pieces of laboratory equipment. Try it. 

Method D: a^ from Falling Water Drops 

The simplicity of the apparatus and the clever 
manner of finding t recommends this accurate 
method for detemiining a^. Be sure that students 
maintain the water soui-ce at a constant le\'el. 



Answers to questions 

1. Student answer. 

error 

2. (% of error = 



X 1001 



accepted value 

3. Student answer. 

4. Student answer. 

Method E: a^ with Falling Ball and Turntable 

Vhc clistanc(! bcMweeii the marks made b\ the fall- 
ing balls indicates the diffcrcncr in fall times. The 
expii'ssion for a^ is managtuible onK if the lower 
ball is hung veiy close to the turntable. 

born here 



il 



hall#l 



n 



L 



I turntable 



Use a turntable frequency of 33V3 rpm. At this 
speed, if the difference in heights of the balls is 
about 20 cm, the turntable will turn through about 
40°. 

Answers to questions 

1. Student answer. 

error 

2. (% of error = : : — x lOOi 



accepted value 
3. Student answer. 

Method F: a^ with Strobe Photography 

It is helpful to illuminate the ball from the side 
while a black cloth is draped in the background. 
Be sure to photograph a meter stick. For this wor-k 
a meter stick with white calibrations on a dark 
background is best. 

With 12 slots open and a 300-rpm strobe disk 
motor, the period between consecutive peaks at 
the free-falling object is 1'60 sec. 

If a xenon strobe flash is used, refer to the Equip- 
ment Notes for calibrating information. 

Answers to questions 

1. Student answer. 

entar 

2. (% of error = 



accepted vcilue 
3. Student answer. 



X lOOi 



Fig 3 



El^ NEWTOiVS SECOND LAW 

Major equipment: 
Pynamics cart 
BJinky 

Spring scale taped to cart 
Table-corner pulley 
Weights I hooked I and string 
Polai-oid camera 

Either rotating disk stroboscope and light 
sourx:e or acceleixameter 

It is assumed that students have recently com- 
pleted Sec. 3.7 in the Test on Newlons second law. 

Purpose 1 

In this experiment students familiarize themseh-es 
with the relationship betAveen F,,^,, m. and a. In no 
sense do the\' prtnt' or e\en \erifv the law 

Using stroboscope photographs will certainly 
not aft'ord enough time for a single student lor 
group of students I to take a series of data on a 
versus F,„., and also on a \ersus m. If the task is 
distributed among several students or groups. 
howe\er. two graphs can be tirawn and each stu- 
dent can contribute a point or two to one of the 
gi'aphs. 

The graph of a \ersus F^, with m held constant 
will be a straight line thniugh the origin 

The graph of .j \ er-sus ni with F,,^, held constant 
will be a h\pei4)ola. The graph cannot bo recog- 
nized as a h\j)eH)()la. ho\Ne\er Studt'iits should be 
challenged on this point to show that it is not part 
of a cinle, an ellipse, or a parabola ()nl\ In tinding 
liow to (•on\«'rt it into a sliaighl lin«' <oi a ( in^lei, 



50 



iiiviT 1 / (xiNCEiTS or Mtniox 



which are identifiable by simple inspection, can 
one then work backwards to discover the original 
shape. 

Thus in the case of a versus m. a graph of 1 a 
versus m yields a straight line through the origin. 
All such lines must ha\e the equation y = k,\, or, 
in this particular case, 



Thus, 



- = km 
a 

ma — constant 



which is the graph of a h\perbola. 

Moreover this beha\ior is consistent with New- 
ton's second law. Using a calibrated liquid-surface 
accelerometer on the dynamics cart a single stu- 
dent lor group of students i may be able to gather 
all the data alone, if this seems desirable. Certainh 
the work can be done faster than it can from pho- 
tographs. 

The action of the accelerometer is described un- 
der /\cf/\'/nes in the Handbook. 



Purpose 2 

A second major purpose of this experiment is the 
study of experimental erroi^. 

It may be desirable to pursue this treatment in 
a subsequent laboratory' or class period. The ideas 
developed here will be assumed in future discus- 
sions of experimental error. 

The question is asked: Does \'our measured 
value of F„g, really equal \'Our measured \ alue of 
ma?" Since all three quantities are the results of 
measurements that haxe inherent uncertainties, 
the measurement of F^^, will almost certainly not 
equal ma. This discrepancy- does not necessarily 
mean disagreement with Newton's second law. It 
does mean that experimenters must consider un- 
certainties of measurement and the propagation of 
error. 

If students do find that F^^, is equal to ma. within 
the e.xperimental uncertaintv, then all is well. If the 
uncertainty' is large, they may justifiabK' point out 
that it is a poor experiment. If the difference be- 
tween the measured \alue of F^^., and the calcu- 
lated \ alue of ma is greater than the experimental 
uncertaintv, the most likely explanations i apart 
from miscalculation, or the use of inconsistent 
units) tire: 

(ai the force measured by the spring scale is not 
the only force acting i friction i, and 

(bi the spring scale has a significant error, or is 
inaccurateK' calibrated. 



Discussion of error propagation 

The Handbook points out that the uncertaintA- in 
the difference between two measured quantities is 
the sum of the uncertainties in the two measure- 
ments. The same is true for the sum of the two 
measurements. Students may ask about the uncer- 
tainty in a product. This is a general rule: The per- 



centage uncertainty in a product is equal to the 
sum of the percentage uncertainties in each meas- 
urement, howe\er man\' terms there are in the 
pix)duct. Similarly, for a quotient , the percentage 
uncertaintv' is equal to the sum of the pereentage 
uncertainties of all the temis. 

While the simplifications outlined abo\e are use- 
ful for an intixaductoiy e.vercise, a much more gen- 
eral approach to uncertaintx' and its analysis is re- 
quired for most experimental situations. This 
generalits' is needed because lai there ai"e a \ariet\' 
of mechanisms responsible for introducing uncer- 
tainties, and ibi there are, of course, other kinds of 
functions thixjugh which uncertainties are to be 
propagated during the course of the year. A brief 
outline summars' of these factors follows. 



Mechanisms responsible for uncertainties 

1. Scale-reading uncertainties. Finite space be- 
tween marks on scales. 

2. Object irregularities 

lai Obxious variations that can be identified and 
have predictable effects. 

lb I Perturbations requiring a statistical treatment 
of the final results (for example, population 
surveys, radioactive disintegrations!. 

3. Systematic discrepancies introduced by 
lai Bias, due to poor experimental design. 
lb) Use of ox'ersimplified theoiy. 

Propagation rules to calculate maximum 
uncertainties 

1. Sums and differences 

Add absolute uncertainties to obtain absolute 

uncertainty' in result. 

Elxample: 

If A = 2.51 = 0.01 

and B = 3.33 rt 0.02 

then A + B = 5.84 ± 0.03 

2. Products and quotients 

Add % uncertainties to obtciin % uncertainty in 

result. 

Example : 

If A = 2.51 ± 0.01 lor ± 0.4%) 

and B = 3.33 ± 0.02 lor ± 0.3%), 

then AB = 8.36 ± 0.06 lor ± 0.7%) 

3. Power and roots 

Multiply % uncertaintv' by power or root le.xpo- 

nenti to obtain % uncertaintv in the result. 

Example: 

If A = 2.51 :!: 0.01 lor ± 0.4%) 

then A" = 629 ± 0.05 lor ± 0.8%) 

Exercises could be invented to provide drill and 
practice on any of the items listed in the tables. 
But it is probablv more appropriate to call atten- 
tion to them as thev are needed. 



EXPERIMENT NOTES 



51 



Remember not to give the whole bottle of med- 
icine in one sitting: parcel it out in gentle doses 
over the whole year. The therapy takes time! 

Answers to questions 

1. F^ las measured I ^ rn^^, 'as computed). 

2. Observations may support \ev\1on's second law 
if the uncertainties of measurements are taken 
into consideration. 

'i-4. Student answers. Refer to sample results be- 
low. 

Sample calculations: 

1. Predicted acceleration 

F = 1.6 \ lAvof 5 runs I 
F = 1.6 \ ± 0.2 \ = 1.6 N ± 12% 
m = 1.032 kg ± 0.050 kg = 1.032 kg ± 5% 



_ _F _ 1.6 N ± 12% 
m 1.032 kg ± 5% 



= 1.55 m/sec ±17'! 



or 1.55 m/sec ± 0.25 
Range: 1.30 to 1.80 m/sec" 



"8 
2L 











040 


- 


unknoivn m*90 




aas 


- 


X 




OJO 


- 


/ 




ois 


- 


/ 




010 


/ 


1 1 1 1 1 


1 1 I 



50 lOO ISO lOO 150 500 &50 400 
m*&S on pan (groma) 



Fig. 4 



2. Actual acceleration 
projected distance = 
2.0 m ± .05 = 2.0 m ± 2.5% 
actual distance = 
12.0 m ± 2.5%) X 4 = 8.0 ± 2.5% 

d 8 m ± 2.5% 
actual speedy = - = ■; = 

'" z '/^ sec ± 2.5% 

1.6 m/sec ± 5.0% = 1.6 m/sec ± 5" 

V, - \', 



actual acceleration = 



t, - t. 



1.6 m/sec ± 5% - 
"/,„ sec ± 2.5% - 



= 1.4 m/sec ± 7.5'! 







Sample data 








Time for 50 








oscillations 




Mass on balance 


(Av of 4-5 trials) 


T 


9 




sec 


sac 


50 




11.3 


0.23 


100 




13.6 


0.27 


150 




14.6 


0.29 


200 




16.0 


0.32 


250 




17.1 


0.34 


300 




18.6 


0.37 


350 




19.9 


0.40 


300 




21.0 


0.42 


unknown mass 


(a) 


18.4 


0.37 




(b) 


18.4 


0.37 



It is logical that actual a < predicted a because 
of friction. 

Newton's second law — » F^,., - ma 



F = F 



- F^ 



tension ' fhrlion 

'hiciion m^V be too small to measure accurately. 



El-9 MASS AND WTEIGHT 

This is a \eiy subtle demonstration. The impor- 
tance of it may be lost on all e.xcept the most per- 
cepti\e students. Students should be gi\ en the idea 
that the calrulations an* not the piimai> puqxjse 
of the e.xperiment Ihe principle that inertial mass 
is distinctK diflerent fnjm gravitational mass is the 
significant concept. Furthemiort". the diflference is 
inherent in the difjcrcnt opcrntions used to find 
gravitational and inertial mass. Vhv fact that differ- 
ent optM-ations based on quite difleivnt concepts 
of mass give equivalent ivsults ithat is. 3 times the 
mass accoixling to one concept is e.xactly 3 times 
the mass accoixling to lln> ollien is an e.xtraoixli- 
iiai> and signilicant ide.i 



lai unknown mass resting on the pan 
ibi unknown mass supported b\ string, inde- 
pendent of balance 

Answers to questions 

1. rhe\' are the same to within the errors of meas- 
urement. 

2. Compare the masses in the same wa\' as before. 
Compare the magnetic forces by supporting 
each mass on beads or on a practically friction- 
less puck or other bearing, and use a spring 
scale to measure the pull of the large magnet 
acting horizontalK' on each one in turn. 

El-10 Cm\TS OF TR/\JECTORIES 

Major ec]uipment: 

1 rajectoiA -plotting equipment 
Onion skin paper 
Cariion paper 
Steel ball 
Graph paper 

In the short wr-sion of this e.xperiment students 
c^n slop after ix'coixling the path of the ball, Ix'fore 
the section entitletl \nal\ zing \ our tiat.i lU this 



52 



IMI 1 COXCKPTS ()l MtniOV 



point they have plotted the trajectory' for them- 
selves, which may be sufficient. 

However, another important result of this exper- 
iment is an understanding of the principle of su- 
perposition, and for this the students must go on 
to anal\ze their data. The principle can be made 
particularh' clear if the \ertical displacements are 
graphed against time squared . This graph and the 
graph of horizontal motion against time should 
both be straight lines, as would be expected of the 
two motions if they took place separateh'. 

Notice that there are two tracks that can be fixed 
to the pegboard. One track will launch a sphere 
about 45° upwards ftxjm the horizontal. If the first 
track is used, the experiment will be easier. How- 
ever, it is worthwhile to ha\e some students use 
both tracks so that projectile motion can be dealt 
uith more generalh' in a postlab discussion. It is 
v\ise to be sure that the students understand Test 
Sees. 4.2 and 4.3 before doing this experiment. 




(time intCTMal* Arc made oefj^') 
dii>tAnce — ►• 



Fig. 5 



Answers to questions 

1. The graph of horizontal distance against time is 
a straight line beginning at the origin. 

2. The vertical motion can be described as uniform 
acceleration. One way to show this is to plot the 
vertical distance fallen against the square of the 
time. If the plot is a straight line, it demonstrates 
uniform acceleration. Another method is to 
show that Ad, the change in distance traversed, 
in consecutive equal time intervals, is constant. 

3. The horizontal and vertical motions are inde- 
pendent of one another. 

4. A;< = vAf 

5. Av = VzajAfi" 



Try these yourself 

1. You can expect nearK' the same results uith the 
glass marble as uith the steel sphere of the same 
size. \'er\' lightweight balls are slowed down by 
the roughness of the track and air resistance. 

2. The horizontal component of its \elocit\' uill be 
different. The curve of the trajector\' uill be par- 
abolic, but it uill be a different parabola. 

3. To the degree that friction can be ignored, dif- 
ferent-size balls uill ha\e the same trajectories 
if started from the same positions on the ramp. 

4. The descending half of the curve is similar to 
the first trajectorv'. 

El-11 PREDICTIOxV OF TRAJECTORIES 

Major equipment: 

Steel ball 

Meter stick 

Clock uith sweep second hand (preferably 

stopwatch I 
Support stand 
Ramp 

Students should understand Sees. 4.2 and 4.3 of 
the Te.xf before doing this experiment. 

In a postlab discussion of this experiment, it is 
worthwhile to point out to the students the power 
of logic and the drama of prediction. If one as- 
sumes the equations in the Handbook to be correct, 

2v 
then ,x = V' -=- is true, for it is a logical conse- 
\a^ 

quence of the previous assumptions. Here it is in- 
teresting to have students ai^e whether this log- 
ical consequence gives us anything new. The value 

'2v 
of the equation, K —v^ ^ lies in the fact that the 

prediction of the landing point can be made with- 
out knowing the time of flight. 



Answers to questions 

1. This is the same situation as that examined in 
the laboratorv". If the slingshot is held at a dis- 
tance y abov e the ground, the range .x will be 

" ~- '° k 

But what is v^,? To find this, shoot the same pro- 
jectile verticallv upward and time its flight. Since 
v,j = at for each half of the flight, the time T for 
the round trip will be twice t, or 

r = 2r = — 



when V(, 
,x becomes 



a„T 



and our expression for the range 



aT 



= V /2y ^ /v: 



\ a 



EXPERIMENT .VOTES 



53 



2. We assume; that the hall is launched with the 
same horizontal velocity, v^, and vertical velocity, 
v;,, as on the earth. 

Consider the first half of the ball's flight on 
earth, in which it rises to the top of its trajectory. 
It will reach this high point in a time t defined 
by v^ = a^l; therefore. 

During this same time the ball is also traveling 
horizontally with a velocitv' v,, and will therefore 
have covered a horizontal distance d = v,,f, 
which becomes on substitution 



If you use a glass medicine-dropper tube for the 
bearing, be careful to tape it completely so that if 
it cracks it will not shatter. You may also use a 
plastic or metal tube. 

A student with a watch counting out the time 
aloud may replace the metronome. 

An assumption 

As the stoppers are swxing in a circle at low speed 
the string is by no means horizontal, and the stop- 
pers' distance frrjm the \ertical stick, R' . grows less 
than H. the length of the string. Students ma\' won- 
der whether the centripetal force is determined by 
H or by fl'. The answer is that R is still the correct 
length to measure, as the following anaUsis shows. 



"-"t 



The ball will cover an additional equal dis- 
tance during the descending half of its trajec- 
tory, so its total range R on earth v\ill be 

. 2v,v, 
R = 2d = 

On the moon a^ is only one-sixth as great as 
on the earth, and hence R must be six times as 
great. 
3. The assumptions hold as long as we can ignore 
the effect of air resistance and as long as we 
assume the force of gravity is constant in mag- 
nitude and dii'ection. 

If the earth had no atmosphere, therefore, the 
answer to the questions would be "yes, ' but in 
fact air resistance will reduce both the horizon- 
tal and the vertical distances traveled in a time 
/. The quickest way to appreciate this is to play 
"catch" with a ping-pong ball. 

El-12 CENTRIPETAL FORCE 

This experiment assumes that students have not 
studied 'I'c,\t Sees. 4.6 and 4.7, in which the formula 
for centripetal force is derived. Instead the exper- 
iment leads the students to discover that F is pro- 
portional to m, f^ , and R. 

Apparatus 

The e(iui[}ment is (;asy to assemble if no ready- 
made device is available. One needs: 

A spring scale calibrated in newains or dvnes 

String 

Rubber stoppers for weights 

A stick (meter sticki an)und which the weighted 

string can be ixjtated 
An audible liming device imeti-onomei 
Me(li( ine dit)pper loi plastic or metal tuh(>i 

A scale calibrat(ui in giams can be converted to 
a fotx-e scale l)v placing a piece of tape along one 
edge and marking the corresponding force units 
on it in newlons. 

1 newlon = 102 grams weight 
1 kg weight = 1 X «) S newtoiis 




Fig. 6 

When the string sags, the mass moves in a 
smaller circle whose radius is 



R' = R cos 6 



Its velocity becomes 



ZttR 2ttH cos e 

V = = = \- cos d 

T T 

and the centripetal force is reduced to 

F' = F cos d. 

Substituting these expressions for fl', v', and F' 
into 



F' = 



R' 



(1) 



gives us 



F cos - m 



v' cos" 6 
R cos d 



which simplifies to 



F - 



R 



121 



Thus fonnula ili which describes the cvntripelal 
force when the string sags is rvallv the same as 
fonnula i2i. which students have been seeking to 
verify on the assumfition that the string was hori- 
zontal. 

Students who have not had trifionometrv should 
be cautioiKHl to ineasuie R 



:i4 



ll.MT 1 / (X)lVt]lil»TS OF MOTION 



Discussion 

Groups of students can be assigned diJBFerent sets 
of conditions: then the data can be pooled. The 
students can then proceed to compare the data 
collected. 

Because the error terms associated uith the \ ar- 
iables in this experiment range from \er\' small ifor 
the mass I to \er\' large ifor the period i, a discussion 
of error and its estimation would be appropriate 
here. 

As a second topic for discussion present the stu- 
dents uith the height of a satellite orbit and its 
velocit\' ifor.Alouette l.h ^ 1.040 km and \ = 26,400 
km hri, and ask them either how much faster it 
would ha\e to go to boost its orbit 100 km, or how 
much its oi+)it would be increased if it added 100 
km hr to its \elocit\'. This is a simplified problem 
quite similar to those which astronauts sohe dur- 
ing maneuvering. 

To soKe the problem, one can assume that the 
gra\itational force does not change appreciably 
o\er relatively small distances such as 100 km. Set 



F, = for the initial orbit, and 

mvl 
F, = for the final orbit. 

When any three of the \alues are known, the fourth 
can be calculated. The radius of the earth is 6,334 
km. So fl, = 6,334 + h km. 

Answers to questions 

1. F ^ m 

Z.F^f 

3. F ^ R, if/ and m are kept constant. 

\ote that if/ is kept constant and the radius of 
orbit is not decreased, the speed decreases be- 
cause the stopper tra\els a smaller distance in 
the same time. Remember that according to the 



relation f — 



R 



-, the decrease of radius will 



increase the force, but this will be counteracted 
by the decrease in speed, which is a second 
power dependent \ariable. The net effect of de- 
creasing R and decreasing v' will be a decrease 
in force. .All this is a consequence of the stu- 
dents' keeping the frequency' the same. 
4. F X Rmf- 

F — kRmf'. where k = 4-17" 
lactuall> F = 477-flm/-| 

El-13 CENTRIPETAL FORCE 
OX A TLTtXTABLE 

Major equipment: 

Turntable with lai^e masonite top 

Weights 

Spring scale and string 

The instructions for this experiment assume that 
the students ha\e already studied the subject of 
circular motion thix)ugh Sec. 4.7 



The previous experiment. "Centripetal Force" 
i£3-32i. assumes that the students ha\e not yet 
studied circular motion, and they discover F = 
4TT~mRf' for themsehes in the lab. 

Whichexer circular motion e.xperiment is used, 
the teacher should notice that it uses insights de- 
rived from the preceding work on Newton s second 
law: that is, an acceleration, \'' R, results from a 
force, F. It is also very important to notice that the 
stud\' of circular motion is central to the study of 
planetary- motion in Unit 2. 

The object of this experiment is to predict the 
maximum radius at which an object can be located 
on a rotating platform as a function of the period 
and the friction force. If student predictions are 
within 10% of the experimental results, you can 
consider them a success. 

The friction force needed to get the object 
started differs from the sliding friction. Vou might 
want some students to imestigate this curious dif- 
ference. 

Remember that R must be measured to the cen- 
ter of the mass on the turntable. Ha\ e the students 
mark the inner and the outer edges of the mass in 
the position where it begins to slip, and then later 
measure R to the midpoint between them. Also 
when we measure R to the center of mass we as- 
sume that R is se\ eral times larger than the radius 
of the weight. Txpical data are in the table below. 

Typical data for brass weight on masonite turntable 

Force to Radius 

33V3 rpm 45 rpm 78 rpm 
16 rpm cm cm cm 



Mass start 
g slipping 



1000 
500 
300 
200 
100 



1.5to2N 
0.9 to 1.1 
0.5 to 0.6 
0.4 to 0.5 
0.2 to 0.3 



no slip 
no slip 
no slip 
no slip 
no slip 



21.5 
20.9 
14.5 
21.0 
19.2 



9.7 
9.3 
7.8 
9.7 
10.0 



3.0 
2.5 
2.2 
2.5 
3.2 



Note: .Measurements of radius include the radius of the 
weight. 

The students can be asked to determine the fre- 
quenc\- of the turntable. The table ma\' not be turn- 
ing at 33V3, 45, or 78 rpm. Vou ma\ ha\e to re\iew 
the difference between frequenc\' and period and 
also make sure that periods are expressed in sec- 
onds not minutes. 



Answers to questions 

1. The percentage difference should not be more 
than 5%. Have the students use the frequencx' 
found experimentally. 

2. When the mass is made smaller it might seem 
that the radius R would have to be smaller, since 
R appears to be proportional to m in the expres- 
sion F = mv'R. 

But F is also a function of i', and \' is a function 
of R. so the answer to the question is not ob- 
vious. 

In particular, the centripetal force F is equal 



EXPERIMENT NOTES 



55 



to the force of friction at the moment of slipping, 
which means that 

F = kma^ 

where k is the coefficient of friction and nia^ is 
the weight of the object on the turntable, as- 
sumed to be horizontal. 



Also v^ = 



^j 



Putting these expressions for F and x'^ into the 
centripetal force equation, we get 



kma., - m 






and simplifying 



ka..T 



« = 



Since m has now vanished from the expression 
for H, it follows that the \alue of fl is independent 
of m. This means that for a given solution for B 
you can use any mass. 

3. Changing the mass of the object should have no 
effect because the friction force should increase 
uath the mass as will the centripetal force, and 
the two effects will cancel. 

F. = V^ma^ = 



3v' 
B = — , vdiich does not contain m 



100 km/hr = 27 m/sec 

3 X 127 m/'seci" 



B = 



477" 



9.8 msec" 
= 200 m (wide arc) 



Film Loop Notes 



Ll ACCELERATION DUE TO GRAVITY'. I 

Slow-motion photography in one continuous se- 
quence allows measurement of the average speed 
of a falling bowling ball during two 50-cm inteivals 
separated by 1.5 m. In case the question comes up, 
an iron plug was inserted into the top of the bowl- 
ing ball to allow for magnetic release of the ball. 

The key operating assumption in this measure- 
ment loop is that for uniformly accelerated motion 
the average speed v equals the instantaneous 
speed at the midtime of the intenal. A fomial proof 
of this statement is as follows: 

If the acceleration is constant during a time in- 
terval of duration 7', the speed at the midtime is 



V... = V. + a I - 



But the average speed is 



V, -I- V, 



V, + (V, + aT) 



= V, + a 



Hence v;„ = v. 

The simplifying assum|)tion that a\erage speed 
equal instantaneous speed at the midtime is valid 
in unifoimU' accelerated motion for any size time 
interval. The statement is tme for an aHiilraiy mo- 
tion only as the time inteival appixiaches zem. 

Dexaations of pit)jector speed fit)m the nominal 
18 frames/sec are usualK' no greater than ± 1 
frame/sec but this e.xceetls 5"<'i. This is why it is 
suggested to students that the\' calibrate their pn)- 
jector. 

An ernir of ±0.04 in the \alue of .i^ icompai-etl 
to the accepted \'aliu» at Montival. Clanadai wouki 
still gi\'e some signilicance to a final digit in a 



result such as 9.76 or 9.81 m sec". This would re- 
quire a student's measurements to be within half 
of one percent, which is very unlikely. A more rea- 
sonable expectation would be to obtain a to within 
±0.1 m/sec' I to within l%i. 

L2 ACCELERATION DUE TO GRA\1T\'. II 

Slow-motion photograph\- allows measurement of 
the average speed of a falling bowling ball as it 
passes through four 20-cm intervals spaced 1 m 
apart. 

Remind students of the need to calibrate projec- 
tors if they wish pi-ecise results. The Technicolor 
pixijectors are unlikely to ha\e speeds in error by 
more than ± 1 frame sec. This is, howe\er. more 
than 5%. 

When Unit 3 has fieen studied, the student will 

see that the equation a, = — '- can also l>e derived 
" 2d 

fixjm the law of conservation of energv". If the initial 

speed is \\ and the final speed is Vp then 

£p + £k = £p + £k 

ma^d + Vzmvf = -t- Vzmv^ 
2a ^d = y- - v- 

and for the case of i'^ - 

2a. d = vf 



" 2d 

.At the vui\. the students are asked a wry difficult 
question, nameU Is tlieiv an\' evidence for a svs- 
tJMiiatic InMid in the values.'' In theorv, a svstem- 
atic trentl exists liecause of the approximation 



IIIVIT 1 / COMCKPTS 0|- MOl'IO.V 



made. The ball speeds up as it passes thixjugh any 
interval; at the midtime it is slightK abo\'e the mid- 
point of the intenal. Hence each value of d should 
be decreased \er\ slightK-, and the effect is largest 
for small \ alues of d where the speed changes b\ 
a larger fraction during the inten al. In practice, the 
error is negligible and will not be obsened. 

For the worst case, consider the motion of the 
ball from d = 0.90 m to d = 1.10 m. 

From d — VzaA'. the time to fall 1.10 m is 



f = 



2d 



2(1.10 mi 



9.80 msec 



-; = 0.473 sec 



and the time to fall 0.90 m is 
210.90 mi 



- = 0.43 sec 
9.80 m sec" 

The midtime is thus at t^ = 0.45 sec and the dis- 
placement at midtime is 

d 



So we see that the \ alue of d that corresponds 
to the measured v is 0.98 m, only 0.02 m or 20 mm 
abo\e the midpoint of the inten al. The en-or in d 
is 2°b and therefore the error in a is also onl\' 2%. 
The percent error is even less for the measure- 
ments at d = 2 m, 3 m, and 4 m. 



Some 


tvpical results: 








d 


It 


V, 




3g 


1 m 


4.5 sec 


4.45 m sec 




9.85 m see 


2m 


3.2 


6.25 




9.75 


3 m 


2.5 


8.00 




10.5 


4 m 


2.3 


8.70 


Av: 


9.5 
9.9 - 0.4 m sec^ 



L3 \TECTOR ADDITION: 
\TLOCI'n' OF A BOAT 

A motorboat is \iewed from abo\e as it mo\es up- 
stream and downstream and as it heads across 
stream and at an angle upstream. \ ector triangles 
can be drawn for the \arious \elocities. 

The student notes are somewhat more detailed 
than usual, because vector addition as such is not 
discussed \er\- fully in the te.xt. 

At the time this film was made the ph\sical con- 
ditions were not ideal. You can easil\' verify- that 
the ri\er's speed is not as uniform as would be 
desirable. There is about a 25% \ariation between 
the speeds at the extreme left and the e.xtreme 
right of the picture. The average speed at the mid- 
dle of the frame should be used. Careful observa- 
tion also shows that the direction of the ri\er flow 
is a few degrees off from being perpendicular to 
the line connecting the markers: in Scene 4 the 
flow is 265° and in Scene 5 it is 268°. Scene 1 is 
really superfluous, since the river speed can be 
measured well enough in the other scenes using 
patches of foam floating on the surface. Howe\er, 
the pieces of wood ma\' be easier to see. 



As with all measurements of speed using film 
loops, it is essential to repeat each time measure- 
ment se\eral times to average out erroi"S lor to al- 
low one to discard an obxioush' wrong value). If 
this is done, surprisingly good results can be ob- 
tained. 

As an indication of the consistenc\ of results ob- 
tainable with this loop, we gi\e some tvpical i-esults 
of measurements b\ the techniques described. In 
the four scenes i using foam as reference points) 
the water speed was 2.0, 2.0, 2.1, and 2.0 units. On 
the same scale, the values of Tg^ were 4.0, 3.9, 4.3 
and 4.5 units. The agreement between calculated 
and observed boat headings was = 1" for Scene 4, 
and ±5° for Scene 5. Probable reasons for varia- 
tions in v'm^ were the inabilitv' of the operator to 
maintain exactly constant motor power, and some 
erixji-s in steering. 

L4 A MATTER OF RELATHT MOTION 

.A collision between two equallv massive cars is 
viewed fix)m v arious stationarv and moving frames 
of reference. 

This is a qualitative demonstration loop for re- 
peated classroom use bv the teacher. The concepts 
used are lai relative velocitv and Galilean relativity 
I Unit H: and ibi principles of conservation of mo- 
mentum and conservation of energv' in elastic col- 
lisions I Unit 3 1. It is suggested that the teacher stop 
the projector near the beginning of the loop when 
a message on the screen asks. How did these 
events differ? Encourage the students to describe 
the events thev have just seen, without attempting 
to speculate on the ways in which the ev ents were 
photographed. Then project the rest of the loop 
and initiate a discussion of relative motion and 
frames of reference. Come back to the loop when 
the conservation laws are studied in Unit 3. 

In a technical sense, the word event implies 
knowledge of both places and times. A student 
walks fix)m home to school between 8:00 and 820 
on .Mondax', and again between 8:00 and 820 on 
Tuesdav. These are tv\o different events. Thev are 
similar events, one being a repetition of the other. 
In. the loop, three events not onlv occur during 
different time interv als. but also appeal' to be phv s- 
icallv different. Ihe student should be encouraged 
to describe what he or she sees , and the events do 
seem to requii-e different descriptions. 

The principle of Galilean relativitv is discussed 
in Sec. 4.4 of the Test: Any mechanical experiment 
will \ield the same results when pei-formed in a 
frame of reference mo\ing with uniform v elocitv as 
in a stationaiA frame of reference. In other words, 
the form of any law of mechanics is independent 
of the uniform motion of the frame of reference of 
the observer. Einstein broadened the principle to 
include all laws of phvsics, not just the laws of 
mechanics. Thus, Einstein relativitv includes the 
laws of electromagnetism, which describe the 
propagation of light, as well as the mechanical laws 
of conseixation of momentum and consen ation of 



FIL.M LOOP NOTES 



57 



energy, which are sufficient for our study of collid- 
ing carts. 

The student may he ahle to get some clues as to 
what is "really" happening by closely observing the 
rolling wheels of the carts, and will perhaps see 
the apparent motion of urinides in the pale blue 
background cloth. The teacher should ask the stu- 
dent what he or she means by "really ' happening; 
it should become clear that one fiame of inference 
(the earthi is being subconsciously identified as the 
"real" frame. The point of the film is that the other 
two lmo\ang) frames are just as "i-eal, ' and e\'ents 
taking place in them are described by the same 
laws of mechanics. 

When using this loop in Unit 3, a discussion of 
the "laws" can be given. We are dealing with a col- 
lision. This is governed by the "law of mechanics," 
which we call the "law of conservation of momen- 
tum." In each event, momentum is conserved. 



event 



before 
collision 



after 
collision 



total 
momen- 
tum 



A + (-mv) (-mv) + -mv 

B i + mv) + -H i + mv) + mv 

C { + Vimv) + ( - Vimv) ( - Vimv) + ( + Vimv) 



The total momentum of the pair of carts has a 
different magnitude and dii-ection in each of the 
three frames of reference, but the "law" or "prin- 
ciple" of conservation of momentum is equally 
valid in each frame of i-eference. 

The collisions of the carts are, moreover, of the 
ty|3e called "perfectly elastic." In this t\'pe of col- 
lision, the "law of mechanics" that also applies is: 
Kinetic energy is conserved. Again we find that this 
"law" is equally valid in the thr-ee frames of refer- 
ence. The total kinetic energ\' in each case is in- 
deed the same before and after collision. But its 
value needn't be the same in each fr-ame of refer- 
ence. It is V>mv^ in Event A, v>mv~ in E\ent B. and 
V*mv~ in Event C. 

L5 GALILEAN RELATmTA': BALL 

DROPPED FROM \L\ST OF SHIP 

This film loop is a i-ealization of the expiMimerit 
sirggosted in (ialileo's Dinlogiic on the 'l\\o Great 
World Systems. A ball dr-opped frx)m the mast of a 
moving ship lands at the l)ase of the mast, just as 
it would if th(^ shi|i wer^e not moxing. 

Descriptions of E\ents 2 and 3 in the frames of 
reference of the boat and the earth might run 
something like these: 

Event 2, bout frnnie: .A ball is nun ing horizorUalK 
toward the right and. at the moment it is opposite 
an obsei\er', it is allowed to mo\e frvel\' as a pr-o- 
jtH'tile. The path of the ball is a parabola, and the 
ball mo\'es downwaid and to the right." 

Event 2, earth frame: "A ball is allowed to fall \er- 
licallx frx)m ivst, and it strikes a point dirvctU be- 
low lh(! point of rt'lease 



Event 3, boat frame: "A ball, initially moving toward 
the right, is stopped by the muscular action of a 
student who is stationary on the mast. The student 
lets go of the ball, which then falls vertically down- 
ward." 

Event 3, earth frame: "A stationary ball is gi\'en a 
forward velocirv' to the left by the muscular action 
of a student. The ball is then released, and its mo- 
tion, that of a prxjjectile. takes the ball to a point 
dowTiwarxl and to the left of the starling point. 

L6 GALILEAN RELATHTn': OBJECT 
DROPPED FROM AIRCRAFT 

A flar~e is drxjpjjed fr-om an aircraft which is fl>ing 
horizontally. The parabolic path of the flare is 
shown, and freeze frames are provided for meas- 
urement of the position at 10 equallv spaced in- 
tervals. 

For the student who likes the challenge of care- 
ful quantitative wor-k, the most interesting question 
raised is whether the effect of air resistance is no- 
ticeable in the horizontal and vertical components 
of the flar^ s motion. If air resistance were negli- 
gible, the horizontal displacement graph would be 
a straight line passing thrxjugh the origin, ,v = v^t, 
assuming that the correction B is 0. The vertical 
displacement graph would be a parabola, d - 
Vzat' . In fact, careful measurement and plotting 
show that the horizontal motion e.xperiences a de- 
cided "droop" due to air resistance. The vertical 
motion is surprisinglv good, for the graph of Vs 
versus / remains almost str-aight for the whole mo- 
tion, even in Scene 2, which is the longer of the 
two. To explain this, note that air resistance de- 
pends on speed. The flare is moving at lai^e hori- 
zontal speed fr-om the instant it is released, but it 
has large vertical speed only for the latter part of 
the trajectorv'. 

L7 GALILEAN RELATIMTA : 

PROJECTILE nRED \TRTICALLV 

A flar-e is lir-ed verlicallv frx)m a Ski-doo that moves 
along a snow-covered path. Events art* shown in 
which the Ski-doo s speed remains constant, and 
also in which the speed changes after firing. 

Accorxling to (lalilean ivlativit\ . the tlarx> retains 
the velocitx lif anyi of the Ski-doo. Relative to the 
earlh, each event is the usual parabola of prx)jertile 
motion. Relative to the Ski-doo Event 2 is a vertical 
motion: the flarv falls down again into the Ski-doo. 
In Event 3. the Ski-doo comes to a halt after the 
flare is fir-ed. so the Hart' lands aheail of the Ski- 
doo. In Event 4, the Ski-doo accelerates in the for- 
warxl dirvction after the flare is fired, so the flare 
lands behind it 

L8 ANALYSIS OF A HlTtDLE RACE. I 

Slovv-mt)tion photographv allows measurvment of 
speed variations dirring a hirrxlle race This loop, 
along with Film Untp 9. is intendeil to give students 
a feeling for' the power' of carvfirl mt'asirrvment to 



r>N 



UNIT I / coivc:ei»ts of ivionox 



reveal "structure" in motion that seems to a casual 
observer to be nearl\ uniform. It also encourages 
them to speculate on the causes of the changes in 
motion that the\ observe. 

A student ma\' suggest that a systematic error 
has occurred because of perspective. This was 
taken into account when the meter marks were 
located on the wall behind the runner. The camera 
was positioned opposite the middle of the 6-m in- 
terval, and the markers were "spread somewhat 
so that the runners positions are correctly indi- 
cated when his image coincides with those of the 
markers. 

The front of the runners shorts would be un- 
reliable because he straightens up after the start. 
Using the forward edge of the \ertical meter marker 
is helpful because it gi\ es the observer time to an- 
ticipate the moment of tangency. 

Careful measurements gi\e a speed graph for the 
first 6 m similar to the one below. The droop" at 
5 m can be related to Newton s second law, as sug- 
gested in the Handbook. A good student should be 
encouraged to stud\ the film closely perhaps plot- 
ting regions of push as in the graph shown here 
for your information. It is e\ident that the runner 
is practically coasting as his hip mo\es from 5 to 




pushe* by feet 
of njnrtcr 



d i stance ^meters) 



"6." The initial acceleration can be found from d 
= '/2af'; for d = 1.4 m and t = 38 80 sec. a is 
calculated to be 12.5 m sec'. The average acceler- 
ation during the first 1.4 m is about 1.3 times the 
acceleration due to gravity. Thus, the ground 
pushes on the runner, and the runner on the 
ground, with a force about 1.3 times his own 
weight about 100 kg . If this seems unreasonable, 
note that the world record for weight-lifting using 
arm muscles onlv, is about 180 kg. The acceleration 
is even greater at the verv- start, during the first 0.1 
m of motion. .A frame-b\-frame analvsis of the film 
gave an acceleration of about 40 m sec' during this 
interval, corresponding to a momentarv force of 
more than 2,600 N. 

L9 ANALYSIS OF A HLTU3LE RACE. II 

A continuation of the anal\ sis of motion begun in 
Film Loop 8. 

The graph from Physical Review Letters is shown 
to reassure the student who ma\ be unhappy with 
a graph whose plotted points show considerable 
scatter. It is not necessarv to go into the details of 
the experiment summarized b\' that graph, except 
to point out that this graph is a real-life e.xample 
of published work b\ a team of fi\ e highly capable 
physicists. 

In Scene 1, the speed increases just after the run- 
ner clears the hurdle, while he is stiU in the air. 
This paradoxical result is explained b\ the fact that 
the runner straightens up after clearing the hurdle. 
If his center of mass maintains constant speed 
then his hip must come forward as his knee and 
torso come back relati\e to the center of mass. This 
unexpected result is clearlv shown in a tvpical stu- 
dent measurement and should provoke a valuable 
discussion. A similar effect e.xplains the continued 
rise of speed in the 2-m to 3-m interval of Film Loop 
8: the runner is stUl straightening his torso follow- 
ing the start of the run. In Scene 2 the measure- 
ments are less precise than in Scene 1 because the 
magnification is less. There is a modest rise in 
speed as the runner approaches the finish line at 
50 m. 



Equipment Notes 



POLAROID PHOTOGRAPH\' 



CAMERAS 

.Almost am Polaroid Land camera can be used in 
classroom demonstrations and experiments in 
ph\sics. The notes refer in detail to k\' the modi- 
fied model 320 camera, and iB' the older models 
95, 150. and 800 that can be bought relati\el\ in- 
expensi\el\' and cire used in man\' classrooms. .A 
third section of these notes iCi on photographic 
techniques refers to all models. 



A. The modified model 320 
Polaroid Land Camera 

This camera is a modified version of the model 210 
camera, in which e.xposure time is controlled au- 
tomatically by the electric e\ e. The manufacturer's 
instruction booklet describes the normal use of the 
320 loading the film pack, processing, etc.i. 

The modifications consist of: 

1 a cover for the electric eve that makes it pos- 



EQl IPMEVT NOTES 



59 



sible to take bulb exposures. When the eye is cov- 
ei-ed the camera shutter r-emains open as lonf^ as 
the shutter release lor cablei is held depressed 
There are veiy few, if any, experiments for which 
you will use the eye to control exposure time au- 
tomatically. Always keep the eve covered when the 
camera is not in use to prevent rapid fxindown of 
the internal battery. 

2. a cable release clamped semipermanently on 
the shutter release button. 

3. a base plate with locking thumb screw. For 
most classroom work the camera is used as a fixed- 
focus camera. It is convenient to use the camera 
at a distance that gives a 10:1 photographic reduc- 
tion. The locking scr-ew is used to fix the camera 
bellows at the correct extension. 1 he base plate 
also has a sci-ew hole that takes a standard '/i"-20 
screw for mounting it on a camera tripod or motoi- 
strobe disk unit. 

4. a close-up accessory lens, which clips onto 
the camera lens to give an apptx)ximately 1:1 re- 
duction for photographing traces on an oscillo- 
scope sci'een, etc. 

5. a clip-on slit, to be used in conjunction with 
the motor strobe unit isee notes on strobe photog- 
raphy) 

6. a focusing screen of ground glass mounted in 
a frame that has the same dimensions as the film. 

FOCUSING 

The camera has a nonautomatic range finder. Look 
through the finder uandow: the position of the ar- 
ixnv on the scale at the left of the window indicates 
the focal distance in feet. Focus is adjusted by 
pushing the buttons marked "1" back and foilh. 

For most classroom use, it is con\enient to \\'ork 
at a standard distance from the e\ent being re- 
corded. A distance of about 1.2 m gives a 10:1 pho- 
tographic reduction. We recommend that one of 
the fii-st things students do with the camera is es- 
tablish precisely what the 10:1 distance is. 

Camera model 320 provides a focusing screen. 
If you have another model, such a screen can be 
made \eiy simply as follows. Take a discarded film 
pack apail into its three component pieces. One of 
these pieces is a frame that encloses an area the 
size of the pi(■tult^ Fix a piece of gixjund glass in 
the frame, so that the ground surface faces towaixJ 
the l(Mis wIkmi the fram(> is put into the camera. If 
ground glass is not a\ailai)l(?, a satisfactorA' scrven 
can be imprxnised by sticking tape (not the clear 
varietyl on a piece of flat glass, or using tracing 
paper. 

Insert the frame in the cameia. just as if it were 
a film pa(;k, grxnind glass surface toward the lens. 
Leave the camera back open. Set up a well-illumi- 
nated meter stick about 12 m in fr-ont of the cam- 
era. Cover the electric eye and set the speed selec- 
tor to 75. ()|)en the shutter and keep it open, by 
keeping the "2" button or the cabl(> release de- 
pressed. I The cable release can be locked In tight- 
ening the set screw. I Look at the image on the fo- 



cusing screen and adjust the range finder until the 
image is sharply focused. Measure the image of the 
meter stick on the screen. Adjust the camera-stick 
distance and focus until the sharply focused image 
of the meter stick is 10 cm long.* 

Once the 10:1 distance has been found and the 
camera focused, use the thumb screw provided to 
lock the camera bellows in this position. Measure 
the lens-to-object distance. It will now be easy to 
set up and photograph an object or event at 10:1 
reduction. Do not lefocus the camera or loosen the 
locking thumb screw unnecessarily. 

This preliminary exercise can be extended to es- 
tablish two imporlant points about using the cam- 
era to record events at the 10:1 distance. 

(a) What is the field of view at this distance? lit 

should be just under 1 m.i 
lb) Is the photographic reduction uniform over 
the print? Is it the same near the edge as at 
the center, or is there some distortion? iThere 
is in fact very little distortion: the 10:1 factor 
can be used on all parts of the print.) 

EXPOSURE 

\a> Aperture. Students attention may also be di- 
rected at this time to the effect of the "Film Selec- 
tor" (manufacturers instruction bookleti. Remove 
the screen, open the camera shutter, and look 
through the lens with camera back open. .At the 
3000 setting the lens aperture is small: at 75 the 
aperture is 40 times lar-ger in area. 

For most strtjbe photographv, use the 75 setting, 
even though the camera is loaded with 3000-speed 
film. The numbers refer to the ASA "speeds" of the 
two types of film. For nonnal outdoor use the se- 
lector is set to 300 for 3000-speed black-and-white 
film and to 75 for 75-speed color film. But this does 
not apply to our special classroom use of the cam- 
er*a. Although 3000-speed film will be used in our 
experiments, in manv instances the 75 setting is 
needed. The lighten darken control imanufac- 
turer's instruction bookleti is effective onlv when 
the electric eve is open. 

Ibi Time. If the electric eye is open, the exposure 
time is contrx)lled automaticallv If the electric eye 
is closed, the shutter- will remain open as long as 
the cable release lor shutter release' is held de- 
pressed. For strobe wor-k, cover the electric eve and 
control the exposure time manuallv . TrA' to keep 
the shutter open for the minimum time necessary 
to record the event. The longer the camera is open 
the poorer will be the contrast in the picture. 

The electric eve will not woi-k when the battery 
has lost most of its char"ge. 

UGHTIX'G 

The strx)he photographv experiments and dem- 
onstrations that are described in detail in the Re- 

•I'nlutlunalrK . lln" st-iwii i> lu.sl i«'s» than 10 cm loiiji llipre- 
fore the meter stirk must Ik* sel up ol>liqueK Alteniatiwlv. ad- 
just until a ;W-rm-long part (if thr nictrr stirk ijhi's an ima^r f» 
nn l(inn 



«>o 



IIIVIT 1 / COIVCKITS ()l ,\1()TI()\ 



source Book and Handbook do not require a dark- 
room. In man\' cases it is not even necessan' to 
turn off the room lights, unless there is a light di- 
rectly o\er the lab table. 

Since a dai-k backgixjund is essential in sti-obe 
photograph\', a black cloth screen works well isee 
Fig. II. 




Fig. 1 Blinky photograph taken with modified model 210 
Polaroid Camera. Room lights were on and a black cloth 
screen was used. 

It is often useful to record both the strobe e\"ent 
and a scale imeter stick' in the same picture. Table 
1 summarizes conditions for the \arious strobe 
techniques. 

When working at 75 aperture, a small decrease 
in exposure can be effected b\- adding the clip-on 
slit o\er the camera lens. 

TABLE 1 

Suggested exposure conditions using modified 

model 320 Polaroid Land Camera 



Strobe 
Technique 



Lighting 



Film 
Selector 
(aperture) Procedure 



light source 
and disk 
strobe 



xenon strobe 



blinky 



normal — but 
not directly 
overhead 



normal — but 
not directly 
overhead 



darkened 
room — but not 
dark room 



75 Single-bulb 

exposure 
records both 
event and 
scale 

3000 Single bulb 

exposure 
records both 
event and 
scale 

75 Single bulb 

exposure 
records both 
event and 
scale 



CLOSE-UP ACCESSORY LEXS 

With the accessorv' lens clipped in place over the 
regular camera lens the camera can be used for 
close-up work. Focusing with this lens is quite crit- 
ical and must be done with a focusing screen. The 
object should be between 12 and 14 cm from the 
front surface of the accesson' lens, depending on 



the magnification \ ou w ant. With the camera focus 
set to infinit\ the ratio of image size to object size 
is about 0.85, and with the bellows fully extended, 
the ratio is about 1.2. 

For most classixjom work the camera is used at 
a bellows extension that gi\es a 10:1 reduction 
I without the clip-on lensi, and it is comenient to 
keep the bellows fixed in this position. Vou can use 
the camera for close-up work without changing the 
bellows extension. Add the accessorv lens, insert 
a focusing sci'een in the camera back, and focus 
on the object by mo\ing the whole camera toward 
or away from it. The magnification will be approx- 
imateh' 1 x . 

PHOTOGKWH} OF TRACES 0\ 
THE OSCILLOSCOPE SCREEM 
Remove an\' colored plastic v\indow that may be 
in front of the screen. Clip on the close-up lens 
and focus the camera as described aboxe. For sta- 
tionan patterns set the film selector to 3000 and 
gi\e a bulb exposure of about 1 sec duration. It is 
not necessarv' to darken the room. For single-trace 
work it ma\ be necessar\' to set to 75 and darken 
the room or add a light shield ai-ound the oscillo- 
scope face, long enough to reach to the camera. 
Keep the shutter open for the minimum time nec- 
essary to record the trace. 

B. Models 9o, 150, 160, 800 
Polaroid Land Camera 

These cameras all use roll film and gi\e a picture 
size just under 7.5 cm x lo cm. Models 150, 160, 
and 800 have a range finder: model 95 does not. 
On these cameras one adjustment determines both 
the lens opening and the time the shutter stays 
open. Speed and aperture combinations corre- 
sponding to the E\ numbers of the various cam- 
eras are gi\en in Table 2. 

Notice that to convert E\ numbers given for a 
model 95B, 150, or 800 to values for a model 95 or 
95A, one must subtract 9, and vice versa. A setting 
of 15 on one series gives the same e.xposure as a 
setting of 6 on the other series. In this note and 
others in this Resource Book, we will give both set- 
tings, for example, E\' 15l6i. 



TABLE 2 



Models 95A. 95B, The 700, 150, 
160. and 800 Cameras 
Shutter 



Model 95 Camera 



Shutter 

No. 
Models 

95A, 
The 700 



No. 

(EV 
Scale) 
Models 

95B, 
150.800 



Lens 
Open- 
ing 



Shutter 
Speed 



Lens 
Open- 



Shutter 
Speed 



10 



f 8.8 
f 8.8 
f 8.8 
f 8.8 
f 12.5 
f 17.5 
f25 
f35 



1 12 sec 
1 25 sec 
1 50 sec 
1 100 sec 
1 100 sec 
1 100 sec 
1 100 sec 
1 100 sec 



f 11 
f/11 
f/11 
f 16 
f?2 
f32 
f45 



1 8 sec 
1 15 sec 
1 30 sec 
1 60 sec 
1 60 sec 
1 60 sec 
1 60 sec 
1 60 sec 



Reprinted from "Polaniid Pointers" with permission of Pola- 
roid Corporation. 



EQLIP.ME.VT NOTES 



61 



A deci-ease of one unit in EV number means that 
twice as much light reaches the film. This is true 
for "instantaneous" photographs, but not neces- 
sarily so for time exposui'es. Note frT)m the table 
that at all settings below EV 13i4) the camera lens 
is wide open. For time exposures any further de- 
crease in EV will not affect the amount of light 
reaching the film. 

All these roll cameras have a little knob on the 
camera face close to the lens. This can be set to 
either "I" for "instantaneous" exposures (exposure 
times as given in Table 2i, or to "B ' for "bulb" ex- 
posures (shutter remains open as long as the shut- 
ter release or cable release is held depressed). This 
knob i-eturns to the "I" position automatically after 
eveiy bulb exposure, and must be reset to "B ' for 
each time exposure. Failure to reset it is the most 
common cause of unsuccessful exposures. (Possi- 
bly the second most common cause is forgetting 
to check that there is film in the camera.) 

FILM 

The most useful type of film for classroom use is 
the 3000-speed, type 47. It is the most sensitive and 
has the shortest development time (10 sec). The 
two transparency films ai-e useful occasionally but 
are less sensitive. One of them 146-Ll also needs 
longer development time. 







TABLE 3 






ASA 


Development 




Film 


Speed Value 


Time 


Format 


47 


3000 


10 sec 


prints 


46-L 


800 


2 min 


half-tone 
transparency, for 
slides 


Polaline 


120 


10 sec 


black-and-white 


146-L 






transparency (high 
contrast for line 
drawings) 



If prints are to be kept more than a few days, 
they should be coated soon after exposure with 
the squeegee supplied with each ixjU of film. Prints 
are normally somewhat curled; flatten prints b\' 
pulling ()\er a straight edge, picture side up, before 
coating thcMii. Iranspai-encies iwv. pIt!sel^(?d by im- 
nuMsion in Dippit ' liquid for at least 20 sec, and 
can then be mounted in easil\' assembled frames 
for projection. Head the instiiutions su[)plieci with 
film, with 'Dippit, and with slide frames for moit* 
details. 

EXPOSURE 

It is im|)ossible to gi\e hard-and-fast mles about 
exposures, as these will \iuy according to local 
conditions. Exposuif \alues gi\en in the notes on 
pariiculai' experiments and demonstrations must 
be regai'ded as suggestions only. In all kinds of 
multiple-exposuiv photography lblink\', sti-obei. it 
is imporlant to incit-ase contrast as much as pos- 
sible It is not necessaiy to ha\(> a com[iletel\ 



blacked-out room. Regular opaque shades are 
quite adequate; some Venetian blinds are satisfac- 
tory. A black background such as the black cloth 
screen mentioned earlier in these notes will im- 
prove the contrast enormously. In the particular 
conditions of the laboratory at Harvard, the follow- 
ing values were found to be useful starting points. 

Photography of moving blinlt\': EV 15(6) 

Photography of moving light source i pen-light 
cell and bulbi with 300 rpm disk strobe: EV 14(5) 

Xenon strobe photograph\': falling steel ball: E\' 
16(7) 

Xenon strobe photography: white mast on dy- 
namics cart: E\' 15i6i 

C. Photographic techniques (all modelsi 

All blinkv or strobe photogi-aphs could be called 
multiple-exposure. By multiple-exposure we mean 
the recording of more than one image lof the same 
or different bodiesi on one photograph. An exam- 
ple is the Unit 1 Demonstration 7. "Two Ways to 
Demonstrate the Addition of Vectors. ' Usually it is 
necessary to move either the object or the camera 
slightly between each exposure or to tilt the cam- 
era a bit, to prex'ent successive images fhjm over- 
lapping. The shutter must be recocked (model 2001 
or the knob returned to B ' (model 95, etc.i each 
time. The background light level is more important 
in this sort of work, but up to 20 blinkv traces have 
been recorded on a single print 




Fig. 2 A "multiple-trace-" exposure blinky photograph. 
Shutter setting for 3000 film was EV 16(7) on model 210 and 
75 on the experimental camera. 



liou' to use the pictures 

Ideallv. each student team will be able to make and 
analyze its own photographs. 

Students can [)n)babl\ best make measurements 
using a 10 X magnifier and a liansparent scale. 
Even quite dark prints can be measuivd with the 
magnifier in good light. Hold the print against a 
window pane, put it on the stage of an oxertiead 
prx)jector, or use a reading lamp close to the print. 
The scale is made much more visible b\ backing 
it with white sticky tape (ACS tapei. 

Also satisfactory- is a technique using dividers 
and millimeter scale 



«2 



UNIT I / COMCEinS ()l Mtniov 



Because the protective coating takes se\eral min- 
utes to dr\', it will sa\e time to measure photo- 
graphs before coating: ho\ve\er, the uncoated 
emulsion is soft and easily scratched. 

Students can also use the negatixe to take 
measurements that halves the number of expo- 
sures needed. To preseive the negati\e wash it 
with a damp sponge and coat it in the usual way. 

If it is impossible for each team to produce its 
own print, the information on one print can 
quickly be passed on to the class by projection. 
Carefully make a pin prick at each dot on the print 
and use the o\erhead projector to project onto a 
sheet or pad of paper pinned to the wall. Vou may 
need a sheet of glass or comer weights to keep the 
print flat. A trial ma\ be needed to find the best 
pinhole size for \our projector. Each student 
makes a separate cop\ of the print b\ making a 
mark at each point of light on the projected image. 
He or she then takes down or tears off the sheet, 
and the ne.\t student in turn makes an enlarged 
cop\ of the print. These enlarged copies can be 
measured with rulers. 

It is possible to make projection transparencies 
from black-and-white prints on a cop\ing machine. 
Do not coat the print before make a cop\ . A high- 
contrast print and careful adjustment of the lighter 
dariter knob on the copier are important. 

For demonstrations it may be useful to project 
at high-projection magnification directh' onto a 
meter stick and read off the positions of the dots. 

When using projection techniques, make sure 
that the projected image is not distorted. The pro- 
jector must be set perpendicular to the wall so that 
no 'ke\stoning ■ exists. A quick wa\ to check \our 
projector for ke\'stoning is to place a transparent 
ruler in the position later to be occupied b\ the 
photograph and to see if the scale in the projected 
image remains similai' to the original one. Measure 
distances between equixalent points, for example, 
cm marks. Most projectors introduce some distor- 
tion near the edge of the picture area. 

Opaque projection of prints is onl\- marginally 
successful. Most opaque projectors do not have a 
lamp that is bright enough. 

Polaroid transparency film t\pes 46-L and 146-Li 
can be projected, using either an o\erhead projec- 
tor or a slide projector. This is more successful 
than opaque projection of prints, but in general 
has been found less useful than the projection of 
pricked-through prints. Transparencx' film is not 
available in pack form and so you cannot use the 
technique with the modified model 320 camera. 
Transparencies can be made from Polaroid nega- 
tives. Buford L. Williams of Kimball Countv High 
School, Nebraska, describes the procedures in The 
Science Teacher, March 1974. p. 41. 

Scale 

For many experiments and demonstrations, dis- 
tance measurements can be made in arbitrary 
units. Millimeters on the film is most con\enient. 



Similarly, time intervals can often be expressed in 
multiples of an arbitran unit, flashes of the blink)' 
or the strobe. But there are instances in which it 
is necessarv to know actual distance and time val- 
ues in conventional units. In the determination of 
the acceleration of a freeK falling bod\ , for e.xam- 
ple, you must conxert distances and times into 
some familiar units to compare xour result with 
known \alues. 

It is quite easy to take a picture that shows both 
a mo\ing object and a scale, such as a meter stick. 
For the scale image to be useful, the scale must be 
in the same plane as the motion being photo- 
graphed. 

With the modified model 320, both the mo\ing 
object and the scale can be photographed in a sin- 
gle exposure iFig. 3i. See Table 1 on page 61 for 
recommended exposures and lighting conditions. 
If you are using one of the older cameras i95, 150, 
etc. I a double exposure may be necessaiy iFig. 4i. 

It ma\' be worth bearing in mind that if the 
10 X magnifier is used to measure photographs- 
taken at 10:1 reduction, each millimeter on the 
print is 1 cm in real space. 




Fig. 3 Single exposure, 3000 film, model 210 camera. Fall- 
ing light source, disk strobe. Selector at 75, room lights on, 
electric eye covered. 




Fig. 4 Double exposure photograph, 

3000 speed film, (a) To photograph 
scale: EV 13(4), instantaneous expo- 
sure, with room lights on. This type of 
scale, with 1-cm wide bars is easier to 
photograph than a scale with millimeter 
divisions, (b) To photograph falling light: 
EV 13(4), bulb exposure, darkened room. 
Disk strobe with 18 slots, 300 rpm. For 
explanation of the elongation of the im- 
ages see notes on stroboscopic photog- 
raphy. 



EQLIPME.NT .NOTES 



63 



If two points in real space are known to be a 
certain distance apart when photographed, it is 
possible to reestablish the real scale by projection. 
Move the projector to or from the screen until the 
images of the two points are the same distance 
apart as the objects were. 

Checklists of the actucil operations involved in 
using the two types of camera appear in the Hand- 
hnnk. 

Illuminated scale for Polaroid photography 

Often you want to include a scale (meter sticki in 
a strobe photograph so that you can convert meas- 
urements taken on the photo into real units. Al- 
though you can do this by double-exposing a real 
meter stick, the special illuminated scale described 
here makes this sort of photography easier and 
produces very impi-essive pictures. 

lake a piece of 0.5-cm thick lucite, about 3.75 cm 
wide and 1 m long. Use an engraving tool to in- 
scribe lines at exactlv 1-cm intervals. The lines 



should be about 1 mm deep. Make evei>' tenth line 
the full width of the rule Number the 10. 20, 30 
. . . cm marks IN REVtRSL, scribing the numbers 
carefully and being careful not to scratch adjoining 
surfaces of the lucite. The scale (engraved side 
down) should now look like this: 



-T-r i'"i 'iri M iii ii i i T 
10 



JC. ^^ ^ 



it' lo p 



T I|>l 



Fig. 5 

To use the scale, shine light into the stick from the 
two E.\DS. Light is scattered in all directions wher- 
ever there is a scribed line on the stick, causing the 
numbers and lines to show brightK' against a dark 
background. Set up the stick so that you view it 
from the UNSCRATCHED face. Vou will then see 
the numbers in proper position, ilf the numbers 
were on the frxjnt of the stick, some light would be 
scattered from them back into the ruler, and be 
reflected fixjm the back, causing a double image. i 



STROBOSCOPIC PHOTOGRAPH\^ 



BVTRODUCTIOX 

Many of the experiments and demonstrations de- 
scribed in this Resource Book require stroboscopic 
photographs. There are several reasons why we 
use this technique so often. 

1. The strobe photograph can sometimes give at a 
glance a qualitative idea of the time-displace- 
ment relationship in a particular motion. L'ni- 
form circular motion, free fall, and trajectories 
are examples. 

2. The strobe photograph is a permanent record. 
Measurements made on a permanent record 
can be more precise and unambiguous than 
those made during the fleeting moment while 
the event is occurring. The measurements can 
be checked several times if necessary. Strobe 
photographs ar-e by no means the only penna- 
nent records that will be used in this course. 
See, for cxarTipie, the experiment on unifor-m 
motion, the photography of spectra, and the use 
of a strip char1 r-ecorxler. This corresponds to a 
very modern tendency in the r-esearch lab — 
nanuiiy, to Ivi the event "record itself" on an ,vv 
plotter or on-line computer. 

3. Measur-ements can be made oxer rather shoii 
time internals, so that rapidK* moving objects 
and events of shor1 duration can be anahzed. 

Someone familiar- with strobe techniques can 
often \ery (juickly take a photograph to illirstr'ate 
a point discussed in class. 1 he morv familiar one 
is with the camer-a and stroboscope equipment 
and their- irse in the particirlar' local conditions of 
backgr-ound illumination, etc , ihv more easily 
these dcnionstrations can be perlonned and the 
mor-e (^lT(u-li\c the\ become 



TECHXIftt^S 

It is con\enierTt to classif\' three kinds of strxjbo- 
scopic photography. Most of the experiments and 
demonstrations described in this Resource Book 
can be done by any of the three methods. 

(ai The moving object is illuminated intermit- 
tently by an intense light such as a .\enon 
strobe. 
(b) The moving object carries a flashing light 
souree, for example, a blinkv (relaxation os- 
cillator with neon bulbi. 
Ic) The moving object carries a steady light 
sourx^e, and light from this source to the cam- 
er-a is intenxrpted bv a chopper in frxjnt of the 
camera lens la motor-driven disk strobe. 

Xenon strobe 

Xenon stitjbe photogr-aph\- has the advantage that 
often nothing needs to be added to the moving 
object. Xenon strx)boscopes iStansi model 1812\\. 
Stansi Scientific Co., 1231 \. Honor-e Str-eet, Chi- 
cago, Illinois; or Strobotac by General Radio Co.. 
West Cloncord, Massachusetts', ar-e r-eadilv avail- 
able. The Str-obotac is calibratt'd and can be set to 
rates between 100 and 25.000 flashes min. The 
Stansi strobe gives much morx? light, but is uncal- 
ibrated (see notes on "Calibration of Strobo- 
scopes"). Of course, once a .xenon strT)boscope is 
available, much i-nore can be done with it than sim- 
ple str-obe photography. For example, the meas- 
ur-ements of rates of rx)tation and some very effec- 
tive visiral (h-monstrations that de[)end upon the 
■freezing of various motions mav be canied out 

,\s in all str-obe photogr-a|>hv. a suitable back- 
grx)irnd is verA imporlant: Black cloth or- a surface 
painted flat black aw good A clean chalkboartl or 
cheap paper used in r-ooting .md flooring can be 



64 



UNIT 1 / CONCEPTS OF MOl IO.\ 



used. But e\'en these surfaces will give a surpris- 
ingly bright and troublesome reflection if the stro- 
boscope is not carefully placed. It should light the 
background at a glancing angle, if at all. 



ri^Himim^^i 


Jt^rfrVA ^^^H 


■■■■fli 



Fig. 1 Xenon strobe light reflected from black cloth back- 
ground. 

The mo\ing object is illuminated fi'om the side 
lor occasionally from abo\e or below). The back- 
ground should, if possible, be some distance be- 
hind the e\ent or screened to make sure that the 
background is in shadow while the object remains 
well lighted. The black cloth in Fig. 1 should ha\'e 
been mo\ed backwards to pre\ent the poor con- 
trast on the light. The stroboscope must not be in 
front of the object near to the camera. Make sure 
that the object is illuminated by strobe light 
throughout the motion that you want to photo- 
graph. Sometimes it is helpful to ha\'e a student 
hold the stroboscope and follow the mo\ing object. 
Figures 2 and 3 show tvpical arrangements. 

Objects to be photographed using xenon stro- 
boscope: 

lal A golf belli will look more like a ball than any 

other object due to its surface texture. 
Ibi A ping pong ball, if clean, will give a white 

disk. 
(cl A steel ball proxides a sharp, bright point of 
light due to the con\ex mirror effect of the 
spherical surface. These points are ideal for 
taking measurements from a photograph, but 
the focusing effect may introduce a small er- 
ror. The camera "sees" the \irtual image of 
the light source reflected in the polished sur- 
face of the sphere. As the relati\e positions of 
light source and ball change, the \irtual im- 
age will shift. The maximum possible error 
that can be introduced in this way is one ball 
radius. For most setups the error is less, and 
for any ball of less than 2.5-cm diameter it 
can usually be ignored. The size of the \irtual 
image also depends on the radius of the ball. 
For very' small balls the image may be so 
small that is is hard to photograph. 
Idi D\Tiamics carts can be strobed. It is impor- 
tant, however, to ha\e some bright object to 



ser\'e as a reflector. A pencil painted black, 
except for the sharpened end which is painted 
white, can be fixed to the cart in a vertical 
position. Reflectixe tape iScotchlike silvan, 
knitting needles, and metallized drinking 
straws are good also. 




Fig. 2 In this set-up for a free-fall demonstration, the xenon 
strobe on the floor illuminates the falling steel ball, but not 
the background. 



fiLMJ< CLOTH SCKt^'J 




6^ Ce=3, Qf^^ 



'2./n 



Fig. 3 Xenon strobe photography of dynamics carts. Note 
position of strobe and cloth screen, which is not immediately 
behind event to be photographed. 

As always, optimum camera setting vxill depend 
upon local conditions. The photographs shown 
here iFigs. 4 and 5i were taken using a Stansi 
strobe, v\ith a black cloth background behind the 
moving object. 




Fig. 4 Xenon strobe photo of dynamics carts, in the partic- 
ular conditions of our laboratory a setting of EV 5 on model 
800 camera was used. For model 320, set film selector to 
3000. Strobe rate about 60 sec. 



EQL'lPMEVr VOTES 



65 




Fig. 5 Xenon strobe photograph of trajectory of steel ball. 
Strobe rate about 20 sec. Aperture setting of EV 16 on 
models 800 and 150; EV 7 on model 95. Film selector to 3000 
on model 320. 



Strobe-difik photography 

Ihe small light sources supplied by Damon have 
a mass of about 25 g, and so their mass can often 
be ignoi-ed if they ai-e used on the 1-kg dNTiamics 
carts. But their mass can \erv definiteK' not be ig- 
nored if they ar-e added to air-track gliders, the 
smallest of which has a mass of about 30 g. 

The heart of the motor strobe kit is a 300 rpm 
synchronous motor. If the disk supplied by Damon 
has 12 equally spaced slots, this gives a mcLximum 
strobe rate of 3,600/min or 60/sec. By taping o\'er 
some of the slots iso that the open slots are equally 
spaced I, the rate can be reduced to as low as 300 
min (5/seci when only one slot is open. iThe re- 
quirement that the open slots be equally spaced 
limits the possible rates to submultiples of the 
maximum frequency.) 



Blinky 

The batteiy-power-ed blinky can be made to flash 
at rates between about 20 and 200/min and it is 
fairly massive. But the principle of strobe photog- 
raphy is probably most easily explained using the 
blinky. The so-called "ac blinky" is certainly light- 
weight and it flashes at a known frequency iline 
frequency I. However-, it is not self-contained and it 
must always be attached to an ac outlet of at least 
90 V. Because of its higher flash rate the ac blinky 
is suitable for faster mo\ing objects, such as pen- 
dulums. It is possible to make a simple \ariable- 
fr-equency blinky (20-2000/ second i using an audio- 
oscillator, amplifier-, transformer, and neon bulb. 

Although the blinky is not always the most con- 
venient of the three strobe methods discussed 
here, students will probably find a photograph 
taken with the blinky technique the easiest to un- 
derstand. The blinky is the first choice for dem- 
onstrations early in the cour-se I uniform motion, 
vector addition of \'elocities, etc.i. Because the light 
outpirt of the blinky is rather low, it is imporlant 
to keep the backgr-ound illumination low so that 
fairly wide aper1ur-es (low E\ number-si can be used 
without losing contrast. The data given in Fig. 6 
should be rcgar-ded as onl\ a starling point tmm 
which to establish o|)timum conditions for your 
own local situation. 





Fig. 7 Disk-strobe photograph of dynamics carts; 1.5-V light 
source on each cart. Six slots, 300 rpm (30 sec). Shuner set- 
ting EV 14(5) on old cameras; film selector to 75 on modified 
model 320. 




Fig. 8 Disk-strobe photograph of uniform acceleration. One- 
slot disk, 300 rpm (5 sec). 1 .5-V light source; EV 14(5). or film 
selector to 75 on the Polaroid camera. 



Of course, by changing the motor or the disk, the 
r-ange can be e.xtended. SxtichrxHious motor-s of \ar- 
ious speeds ar-e a\ailable fri)m most radio-supply 
houses lUifayette, .Allied, Radio Shack, etc.i. t.\tra 
disks can easih be made of (-arxlboaixl. 



Strobe rates for 300-rpm motor 



Fig. 6 Blinky photograph: three traces. Model 150 camera 
was set to EV 17 (model 95 setting would be 8) With model 
320, set film selector to 75 







Time between 


Number of 




Successive 


Slots Open 


Rat* 


ImagM 


12 


3,600 min 


V«o sec 


6 


1.800 min 


'/io sec 


4 


1,200 min 


Vw sec 


3 


900 min 


%»sec 


2 


600 mm 


v.osec 


1 


300 mm 


' ' sec 



(i« 



I'M I I CO.XCKITS OF IVIOTIO.V 




Fig. 9 Free-fall, disk-strobe 
technique, showing elonga- 
tion of images. 



Rg. 10 Free-fall, disk-strobe 
technique. Slit on camera 
lens reduces elongation of 
images. 



An important point to remember when using the 
disk-strobe technique is illustrated by the pair of 



prints shown in Figs. 9 and 10. A slot 0.5 cm wide 
in a disk of 10-cm radius rotating at 300 ipm takes 
about 0.005 sec to pass in fi-ont of the camera lens 
diameter about 1.5 cmi. The camera lens will be 
open' for this time. In 0.005 sec a bod\' mo\ing at 
4 m/sec (the speed of a freeK' falling body 80 cm 
below release! will move about 2 cm. This explains 
the elongation of the images, which increases with 
the speed of the object, in Fig. 9. This elongation 
is reduced, but not completely eliminated, by tap- 
ing a fixed slit (supplied with the kiti to the camera 
lens iFig. lOi. This slit should be parallel to the slot 
in the rotating disk as it passes in fhint of the lens. 
The length of the streak could be further reduced 
by using a narrower slit on the lens, but image 
brightness will be reduced b\' lens slots narrower 
than the disk slot. 

The duration of a blinky flash is about 0.010 sec, 
but since the blink\' is unlikeK to be used for fast- 
mo\ing objects, the problem of image elongation 
is unlikeh to occur. The duration of a .xenon stro- 
boscope flash can be se\ eral orders of magnitude 
less, ranging from about 5 /xsec in more e.xpensixe 
strobes to around 10 fxsec or more in others. In all 
experiments that one is likel\- to do in the class- 
room, the object will be efifectively "frozen' by stro- 
boscopic illumination. 



CALIBRATION OF STROBOSCOPES 



Inexpensive stroboscopes are usuall\- uncalibrated; 
that is, the numbers on the frequency-control dials 
don't correspond to actual frequencies. Below aie 
se\'eral methods for finding the dial readings that 
correspond to a set of known frequencies. .A cali- 
bration graph is constructed b\' plotting the dial 
values against the known frequencies and drawing 
a smooth curve through the plotted points. Dial 
readings Ccin then be con\erted to frequencies b\ 
referring to the calibration curve. 

OSCILLOSCOPE METHOD 

1. "Linear trace" on oscilloscope 

Connect a phototube such as the IP39 tube, which 
is part of the phototube module recommended by 
Project Physics' to the \ertical input terminals of 
the oscilloscope. Notice that no \oltage source is 
needed in this circuit. 




J-P3^ 



Fig. 1 

Set the horizontal sweep rate to about 10 sec. 
Adjust the vertical gain until you see a 60-cycle 



trace on the oscilloscope ithe phototube has a ven 
high impedance, and the wires to it act as an an- 
tenna picking up 60-cycle noisei. .Adjust "sync " 
control of oscilloscope until this 60-c\de pattern 
is stable. 

Position the stroboscope so that light from it falls 
on the phototube. Each flash will produce a sharp 
\ertical line on the trace iFig. 2k Adjust the flash 
rate until there is one flash per c\ cle of the 60- 
CN'cle pattern. 

With the flash rate slightK- abo\ e 60 sec the lines 
will be slightly less than one wavelength apart, 
and will mo\e to the left, and \ice \ersa. Only when 
the strobe rate is exactly 60 sec will the vertical 
lines be stationarv on the 60-c\ cle trace. 



VyVsywv 



Rg. 2 

Now reduce the strobe rate. The next simple fre- 
quencies to recognize are 30 sec iFig. 3' and 20/ 
sec. See Fig. 4 on the next page 



EQIIPMEVT NOTES 



67 





^^^-/Ny\/^ 



Fig. 5 

In Fig. 5 there are two flashes for eveiy three 60- 
cycle periods. The time between flashes is there- 
fore Vz X 3 X Vfeo = V40 sec. So the frequency is 40 
flashes/sec. 

Patterns for other fractional frequencies of 60/sec 
can also be it;(X)gnized and interpreted. 

On the low range of the Stansi strobe the sta- 
tionary patterns sh{)wn in Figs. 6 and 7 werv ob- 
ser-v'ed 




2. "Circular trace" on strobe 

Connect the phototube to the verlical input as de- 
scribed abo\e. 

Establish a circular or elliptical trace on the os- 
cilloscope face either b\' lai setting the horizontal 
fr-equency selector to line sweep or ibi setting to 
external input and connecting the horizontal input 
terminal to the 60 \ibrations/sec calibration signal 
available on the scope lor simply attach a short 
wire to the horizontal input that will act as an an- 
tenna to pick up 60-cycle noise 1. .Adjust horizontal 
and verticed gain as necessarv' to obtain an open 
figure. 

The electron beam is now tracing out one re\'o- 
lution of this figure in Vbo of a sec. Turn on the 
stroboscope. Ever^' flash will cause a sharp verlical 
peak. Adjust the flash rate until this peak is sta- 
tionary'. The simplest figure to interpret is one 
flash/cvcle. 




Fig. 8 

As the flash rate is reduced other stationar^■ pat- 
terns will be produced and can be interpreted For 
instance, 30 flashes sec will produce a peak e\er> 
second r^exolution of the spot. 




Fig. 6 t 



15 sec 



Fig 9 



UMT 1 / CONCKITS OF MOl'ION 



Notice the subtle difference between this pattern 
and the pre\1ous one. Here the vertical spike is 
superimposed on a closed ellipse. 

As the flash rate is reduced further this pattern 

fiO 
will recur at f = — ^/sec where n is an integer, that 
n 

is, 

60 60 60 ^^ 60 

f = — = 30; — = 20: — = 15; — = 12; 
•^ 2 3 4 5 

60 60 60 ^^ 60 

— = 10; — = 8.6; — = 7.5: — = 6.^: . . . 
6 7 8 9 

Other series of stationary patterns can be pro- 
duced. 




Fig. 10 Two spikes in — sec indicate a frequency of 120 sec 
but few strobes can flash at this rate— the Stansi strobe can- 
not. 

The pattern in Fig. 11 will occur if the strobe flashes 
twice in every 3, 5, 7, . . . cycles, corresponding to 

60 60 60 „^ , 

flash rates of — = 20: — - 12; — = 8.6, . . Jsec. 

3 5 / 



the circular trace than on the linear trace. On the 
other hand the linear trace is much easier to in- 
terpret. A combination of the two methods is use- 
ful. Use the circular trace to establish a stationary 
pattern. Then at the same flash rate switch to linear 
trace for interpretation. 

ROTATING-DISK METHOD 

Any rotating object with a known rate of rotation 
can be used. A synchronous motor with a suitable 
disk is the most reliable. Some electric fans and 
other rotating machines that have speed ratings 
given may also be used. Rotation rates of less than 
about 360/min are not very satisfactory, as ex- 
plained below. 

The method will be described here in terms of 
a specific example. Be quite careful about gener- 
alizing to other disks rotating at different rates. 

Mount a disk v\ith 12 equally spaced marks on 
the shaft of a 300-rpm synchronous motor ithe 
motor strobe kit supplied by Damon 1. Add another 
single mark, such as a white star or a piece of mask- 
ing tape, between two of the slots iFig. 12 1. Start the 
motor, darken the room, and turn on the strobe. 
As the strobe rate is changed, different stationary 
patterns will appear. 





Fig. 11 

Patterns containing 3 and more spikes per cycle 
can also be obtained. 

Linear versus circular trace metliod 

Clearly the circular technique needs more careful 
interpretation than the "linear trace" method de- 
scribed abo\e. However, it is particularly useful at 
low flash rates. It may not be possible to get more 
than 6 cycles of 60-cycle signal on the oscUloscope 
face, and this puts a lower limit on the frequency 
at which the 'linear trace" method can be used. 
One spike in 6 cycles means / = ^% = 10/sec. The 
circular trace method can be used down to the 
lowest frequencies. 

The "circular trace" method has the advantage 
that it is easier to obtain a stationaiy pattern on 



Fig. 12 

The simplest pattern to interpret is one that 
shows 12 slots and 12 stars iFig. 13i. The strobe is 
flashing 12 times for each re\olution of the disk 
and the strobe rate is 12 times the rotation rate of 
the disk: 12 x 300 = 3,600 flashes/min. 




Fig. 13 



EQUIPMENT NOTES 



69 





Fig. 14 

Reduce the stixjbe rate slowly until a stationary 
pattern showing 12 slots and 6 stars is obsen'ed. 
The strobe rate is now six flashes/re\olution, or 
1,800 flashes/min. 

Other patterns that are easy to interpret are shown 
in Figs, lo-17. 



Fig. 15 



Fig. 16 

Flasli rat«' - ;J i-cxolution 
I - ;)«)() mill 



Fig. 17 

Flash rate — 2/revoiution 
I = 600/mini 

Figures 16 and 17 correspond to flash rates of 10/ 
sec and 5/sec, respecti\'ely, which bring us down 
to rates slow enough to be counted directK'. 

This really completes the simple calibration of 
a stroboscope by this method. However, it is prob- 
ably wodhwhile mentioning some of the other sta- 
tionary patterns that can be obsened, and their 
inteipretation 





Fig. 18 

Flash rate - 1 i-exolution 
I = :U)0 mini 

Figure 18 is the pattern obseiM'd if the lamp 
Hashes once per revolution. The same pattern 
would also be seen if the lamp flashed once for 
r\ei>' rwo revolutions of the disk and once for es- 
v\y thive ivxolutions. and so on But in fact there 
need be no confusion, for tAvo rvasons First, if one 
woi-ks fit)m high to low flash rates in this calibra- 
tion, the fii-st time that Fig 18 is obsened it will 
conx'spond to one flash iv\olution the next time 
to one Hash per two iv\olutions and so on. Second, 
in the pailicular case of a JOO-rjjm motor the flash 
lates conciMiieil are low enough .j, 2'j, 1'4 
tlaslies seci to be identified b\ diivct counting 



70 



|i\l'I' I COXCHPTS OF IVKH'IO.N 



The same sort of thing will happen at other flash 

rates too. Consider, for instance, Fig. 16. The strobe 

flashes 3 times for each revolution of the disk. If it 

flashed once for e\ery IV3 lor V3I re\olutions of the 

disk the same pattern would be obtained. SimilarK', 

Fig. 15 would be obtained with one flash for I'u 

(or %! re\'olutions of the disk as well as at four 

flashes/re\olution. And in general a figure with n 

stars (which is obtained at a flash rate of n flashes/ 

revolution I is also obtained when the rate is one 

n + 1 

flash for e\erv revolutions. 

n 

Other stationary' patterns can be observed in 

which more than 12 slots are seen. For instance, 

a flash rate of 8/ revolution 12,400 min with 300-ipm 

motori will give a pattern showing 24 slots and 8 

stars (Fig. 19). 




Fig. 19 

Flash rates of more than 12/re\'olution will give 
more than 12 slots, of course. At a flash rate of 16/ 
revolution, 16 stars and 48 slots are seen iFig. 20i. 





1 


mm. 


t 



Fig. 20 

Disks that rotate at 78 rpm 1 called phonograph 
tumtablesi are easy to obtain, but their usefulness 
for strobe calibration is limited. They can only be 
used for slow flash rates. 

With a turntable rotating at 78 rpm carrv'ing a 
disk with 6 symmetrical radii, a stationarv image 



of the disk is observed for flash rates of 156 1 = 2 
X 781; 234 1 = 3 X 78); 468 ( = 6 X 78)/min. At 936 
flashes/min a disk with 12 radii is seen iFig. 211. At 
higher flash rates the number of radii grows and 
counting them becomes difficult. 





Fig. 21 



Sample results 

Figures A and B ion the next pagei illustrate caU- 
bration curves for a Stansi strobe obtained b\' the 
oscilloscope method. The \emier adjustment was 
kept at its upper limit. The plot of flash rate against 
scale reading is quite nonlinear as in Fig. B. 

A plot of- I the period, or time interval between 

flashes! against scale reading is linear. iThis is be- 
cause the period is determined by the time con- 
stant of an RC circuit in the stroboscope, and turn- 
ing the knob adjusts the resistance, evidently in a 

linear manner.i The plot of- against scale reading 

makes interpolation and particularly extrapolation 
easier, and fewer points are needed to complete 
the graph (see Fig. Ai. 

\ote that a plot off against ; is not 

scale readmg 

linear. This is because there is a fixed resistor in 

the circuit, as well as the variable one controUed 

bv the knob. 



/ 



1 



.+ r 



— so / is not proportional to 



blc ' ' ll.xed 



EQUIPMENT NOTES 



71 




to zo JO 40 so 

FlaSm iNreRVAL (ncllisetorvl* ) 



Fig. A 



Fig. B 



THE BLI\KY 



A simplified circuit diagram of the blink\' is shown 
in Fig. 1. 




Rg. 1 The blinky 

Since it goes through a certain sequence of actions 
periodically on its own initiati\e. the blink\' is an 
oscillator. It is one of a class known as rela.xation 
oscillators. 

Ihe three 30-\ batteries E change the capacitor 
C through the i-esistance R. 

riie neon lamp ivmains nonconducting as long 
as the potential diffeivnce acn)ss it ivmains below 
the bivak-down voltage which is about 70 \ Ihe 
xoltage acix)ss the neon lamp is. of coui-se equal 
to the Noltage across tin* capacitor Ihe capacitor 



continues to charge up until the neon lamp be- 
comes conducting at 70 V. 

Once the neon lamp becomes conducting, the 
capacitor begins to discharge through it. The neon 
lamp continues to discharge e\en when the poten- 
tial difference across it has fallen below the break- 
down voltage. In fact, it continues to conduct and 
the capacitor continues to discharge through it 
until the potential difference across them reaches 
about 53 \ . rhis all happens verv quickl\-: the whole 
process just described takes on the order of 10 
milliseconds lO.Ol seci. 

The capacitor now begins to charge up again 
from the batteries, and the neon remains noncon- 
ducting until the breakdown' \oltage is reached 
again. 1 hen the neon bulb glowTi briefly as the \Tjlt- 
age drops down to about 53 \'. 

I he knob on the front of the blinky box adjusts 
the \ariable resistance that controls the rate at 
which the capacitor charges between tlischanges. 

Do not worn- alwut the batteries running dovNn 
The current drawTi from them is \-er\ small It is 
the shelf life of the batteries that detennines how 
long the\ will last Ihis can be extended b\ keeping 
the blink\ cool as in the refriuerator' dnrini; the 



summer 



72 



I MT 1 CONCKPTS OF MOTIOV 



The most likeh- reason for a blinks' not to blink 
is poor contact between one of the 30-\ batteries 
and its holder. 

AC BLEVKY 

This is eas\ to make and is a useful piece of equip- 
ment for motion studies and photographs. 

An ac blink\' is a neon glow lamp circuit that 
operates directK' from the 100-120 \' ac line. The 
intensity- and duration of the flashes can be \aried, 
but the flash rate ifrequencyi cannot; It is fixed at 
60 sec. 
Two factors make the ac blink\' especially useful: 
(a) The flash rate is accurately' known since line 
frequenc\- is usually maintained \erv' pre- 
cisel\' at 60 sec. 
ibi The flash rate is high, making it useful for 
rapidJ\ mo\ing objects. 

But, unlike the regular idci blink^-, the ac blink\ 
is not a self-contained unit. It must always be 
plugged into the line. 



- f/£5 
//i '-. eo C/CLES 



O^fXhT/0/. 



^r- 



- aX\ 



OK 



Ci' Ifi'UC^ti' 



fL-HZi^ 



r^ds/ 



f2.oYfr-^ 



^'7A 



-vw- 



Fig.2 



Schematic of ac blinky circuit 




Fig. 3 Physical layout of ac blinky 



AIR TRACKS 



The linear air track recommended b\' Project Phys- 
ics is an inexpensixe model. Although it is quite 
adequate for many demonstrations and experi- 
ments, it is not a high-precision de\ice. 

Any medium-to-lai^e household \acuum cleaner 
that can be used as a blower should be adequate. 
The air flow will be increased if \ou remo\ e the 
dust-bag from the cleaner. If \ou use a large in- 
dustrial-t\pe cleaner ifor instance, one borrowed 
from the school shop or from the janitori, you ma\' 
find that it helps to plug it into a variac. Too strong 
an air flow will cause the gliders to float too high. 
We ha\e found that the compressed air suppK 
sometimes a\ ailable in laboratories is generall\- not 
enough to operate the air track. 

To test the track, raise one end a few centimeters 
and release a glider from the top. The glider is run- 
ning satisfactorily if it rebounds from the rubber 
band at the lower end of the track to within 25 cm 
of its starting point. 

Use the le\eling screws to adjust the track so thai 
a glider, released from rest, has no tendency to 
move toward one end or the other. Because of the 
slight drop in air pressure along the track, this bal- 
ance will not necessarih' be achiexed when the 
track is perfectK' horizontal. 



The two small gliders supplied b\ the manufac- 
turer ha\'e equal masses. The one large glider has 
twice the mass of either small one i±2%i. The 
three gliders allow you to perform equal-mass elas- 
tic collisions, unequal-mass elastic collisions, and 
unequal-mass inelastic collisions. Note that if the 
gliders are earn ing light sources for strobe pho- 
tography, the mass ratios will not be 1:1 or 2:1. 

The range of mass ratios can be extended by 
taping extra mass to the gliders. Be sure that the 
added mass is distributed s\Tnmetricall\'. It is im- 
portant to keep the center of mass low and there- 
fore it is better to add mass equalh' to the two sides 
of the glider than to the top. Check the glider for 
free running after \'ou ha\e added extra mass b\' 
doing the rebound test described above. The large 
glider should support an extra load of at least 
40 g. 

The following setup can be used to impart the 
same initial \elocit\' to a glider on consecutive 
trials: .Attach a small block to the glider. Draw the 
pendulum bob back and let it strike the block. If 
the pendulum is always released from the same 
point and the glider is in the same position iso that 
the bob hits it at the bottom of its swingi, the glider 
will alwax's acquire the same initial \elocity. 



QUANTITATIVE WORK WITH LIQUID-SURFACE ACCELEROMETER 

Theorv' predicts that the slope of the liquid surface 
is gi\en b\' 

tan 6 - a/a^ 

Figure 1 shows an accelerometer mo\ing hori- 
zontalK' with constant acceleration a. Fig. 1 




EQUIPMENT VOTES 



73 



If the cell has length 2/ and the liquid rises to a 
height h above its i-est position at the end of the 
cell, then the angle that the surface makes v\ith 
the horizontal is given by 

h 
tan 61 = - 

So 

h a 



and 



/ 



3 = 7^^ 



That is, the ratio of the two lengths h to / gives 
the acceleration in a^. 

The mattei- can be simplified further. Since a^ is 
almost 10 m/sec", if we make the length / - 10 cm, 
then 



h cm 
10 cm 



X 10 m/sec" - h m/sec" 



The height h, in centimeters, is equal to the accel- 
eration in meter's/sec". 

To i-ead h it is convenient to stick some centi- 
meter tape to the front surface of the cell, with the 
scale vertical, and exactly 10 cm from the center of 
the cell. The zero mark of the scale should be at 
the height of the undisturbed horizontal level of 
the liquid, usually about halfway up the cell. It also 
helps to stick a slightly wider piece of white paper 
or tape on the back of the cell, opposite the scale. 
This gives a definite background against which to 
observe the liquid level (Fig. 21. 




CALIBKATIOiX OF THE 
ACCELEROIVIETER 

Vhv. tlKHjiftical dcrixation des(-ribed aboxe can be 
conlirmed exp(Minu*ntall\ In the following [)n)re- 



dure. Use a conventional string, pulley, and mass 
setup to produce uniform acceleration of a dvnam- 
ics cart carrying the accelerometer. The actual ac- 
celeration can be measured from a strobe photo- 
graph. I'ltie strobe rate and photographic reduction 
must be known, of course. The calculation is much 
simplified if the strobe rate is 10 sec, and the re- 
duction is 10:1.1 From the same photograph the 
height h can be measured on successive images 

and the a\erage value of -- calculated. lA variation 

of less than 10% was found. i 

This is repeated with several different falling 

weights lor masses on the carts i to produce a range 

h 
of values of a. The average value of -- is plotted 

against the average value of a for each photograph. 
A typical result is shown in Fig. 3. 



f.^ 



ijoV- 




■r JO 



Fig. 3 



,-\s an altemati\e iwhich is less precise, but in- 
volves more studentsi, hav-e several students sta- 
tioned along the cart's path and let each one ob- 
serve the \ alue of h as the cart passes b\'. 

For further details and theoretical derivation of 
the formula mentioned abo\e. see the article by J 
Harris and .\. ,\hlgren, Physics Teacher, \ol. 4, 
pages 314-315 (October 1966i. 



A VERSATILE "CANNON 



A very versatile and inexpensive nibher-banti-pow- 
ered "cannon " can be built, either as an indi\'idual 
arti\'ity, or as a mass-|)rt)(lu(ti()ii class acti\it\ . Four 
of the immediate uses we ha\i' tried are: 



1 a launchei for range of projectile dcinon.stra- 
tion.s 



2 a launcher for- the Monkey in the Tree" dem- 
onstration 

3 a de\ice for rvprodiirihle forees for acceler- 
ating carls, air-track gliders etc. 

4 a sighting tube for astmnom\- tmade more ac- 
curate in tailing a plastic soda straw along the lop 
of the' bair-<*l since a paper- straw gets soggx and 
IxMids in (tani|i night air 



UNIT 1 / conc:ki»ts oi Mcrriox 










S^/i55 Tu^J/i'O- 



P/<?077i/:C7Zs/^ 



ST/^/A^ MA/0 OJfiS^I- 






Fig. 1 










T7rv .^Tf^/^P 



Fig. 2 



Our model isee Fig. H consisted of a 20-cm 
length of 8-mm bore brass tubing, with a piece of 
solid brass brazed to the middle. To a\oid brazing 
an altemati\ e would be a length of aluminum tub- 
ing with a wooden dowel fastened to it with epoxy 
cement and a small metal strap around the tube 
Isee Fig. 2i. The plunger consists of a wooden 
dowel uith a larger piece of wood screwed to the 
end. A slot cut across the end of the wooden piece 
keeps the loibber band from slipping off the end of 
the plunger. A plastic protractor is glued to the 
side of the tube. A short pin is glued in the refer- 
ence hole in the protractor, and a thread and 
washer are attached to it for determining a plumb 
line. For use as a sighting instrument, the handle 
can be put through a hole in a piece of wood that 
is pixoted on a flat board marked off in degrees. 

RANGE PREDICTION EXPERIMENT 

Procedure 1 

(ai Determine the muzzle velocitv' by firing the 
canno n \ er ticaih , measuring h. and substituting in 

V = \ 2gh. 

(bi Then estimate the horizontal range, knowing 

V fhjm the above calculation, and h, height above 
the floor, from the relations: 



n = Vza t', t — — , and range - W 
^ \ a 

g 

Mark the expected range on the floor and ta- to hit 
the mark. 

Procedure 2 

For more advanced students, develop lor have 

them derivei the general range formula, B = 

v' sin 20 

, and then trv the expenment. 



Procedure 3 

In Unit 3 the energy concept can be used for the 
same situation: 

(ai Make a graph of force versus length for the 
rubber band. 

Ibi From the graph, find maximum F and mini- 
mum F when the rubber band is used for a 
particular shot. 

(ci Find the mass of the plunger and cannon 
ball. 

(d) Find the estimated velocitv', using 

F^^ >: d — kinetic enei^ (Vzmv^l 

Sample results 

Using Procedure 2, a measured value of 4.34 m was 
obtained for an estimate of 4.20 m. 

The discrepancy was slightlv larger for Proce- 
dure 3. The predicted muzzle velocitv- i&x)m force- 
e.xtension curve i was 6.9 msec: the measured ve- 
locitv' ifixjm the height to which the ball rises when 
fired verticallyi was 6.4 nx'sec. A direct check with 
a strobe photo or two photocells and an oscillo- 
scope would be excellent. 

\ote that the force versus extension curve for 
Procedure 3 iFig. 3i does not pass through the or- 
igin because the rubber band is alreadv stretched 
before the plunger is pulled back at all. There is a 
finite force for a zero extension ion our scalei. The 
energv' given to ball and plunger is the total area" 
under the graph. 




Uork Done - I F- ds 



E'ttns.c 



Fig. 3 



CATHODE-RAY OSCILLOSCOPE 



The cathode-ray oscilloscope iCROi is one of the 
most versatile laboratory instruments. This note 
can only summarize its different capabilities and 
functions in the lab and some of its uses as a teach- 



ing aid. The approximate numerical values given 
in this note refer to a tvpical inexpensive scope 
such as the Heathkit Model IO-12 (wired from 
Heath Benton Harljor .Michigan^. 



EQl IPME.VT NOTES 



75 



FUNCTIONS OF THE OSCILLOSCOPE 

Ihe cm) is a voltmeter. It can measure voltages 
down to about 10 ■ V (depending on the amplifier). 
It can measure short voltage pulses idovvn to about 
10 *■ sec). Because it has a high input impedance, 
it draws little current from the voltage source being 
measured. 

As well as being used to measure voltages, the 
CRO is useful as a null detector. Kxamples are: 
(al a phototube (illuminated by a pulsed light 
source) is connected through an amplifier to 
a CRO and the reverse voltage across the pho- 
totube is increased until the pulses on the 
CRO trace disappear, indicating that the 
"stopping voltage" for the photoelectrons has 
been reached. 
(b) an ultrasound detector or microphone is 
connected to the CRO and moved through an 
interference or standing-wave pattern until 
the signal falls to zero, indicating that the 
detector is at a node (point of zero intensit>'). 
The CRO can be used to show the wave form of 
a voltage signal (sinusoidal, square, saw-tooth, etc.) 
and to measure the phase difference between Uvo 
signals. It can be used to measure time inten^als 
(10 ' to lO"** sec) and fi-equency (10 to lO*" cycles/ 
sec). 

These and other functions make the CRO a val- 
uable "trouble shooting" tool in the lab, in the re- 
pair of radio and TV sets, and in electronics work 
genercilly. 

THE OSCILLOSCOPE AS A TEACHING AID 

rhe CRO also has many applications in the teach- 
ing of physics, some of which are listed here. 

Electricity: demonstration of the effect of ca- 
pacitance and inductance in a circuit; phase 
relationships between xoltages across differ- 
ent elements in an LCR circuit: oscillations 
in tuned cirouits. 

Sound: demonstrations of the wave fomis of 
pure and impuit? tones; beats. 

Simple hamionic motion: addition of two sine 
curves to show amplitude modulation (beats) 
and Lissajous figures, measuivment of phase 
and frequency; Fouiier synthesis. 

Electn)nics; display of the function and char- 
acteristics of devices, such as diodes, tran- 
sistor's, and vacuum tubes. 

Time measuivments: time-of-flight measurv- 
ment of |)it)jectiles; pulse of sound; displa\ 
of pulses fix)m (ieiger counter. 

rhe CRO can also be used to set up some xer^ 
effective attention-getting displays, corridor dem- 
onstrations, science fair- projects, and so on. E.\- 
amples of intei-c'sting traces ait" gixcn in Figs. 1, 2. 
and 3. 

OPER/\TIO\ 

These notes ar(> neccssariK' of a \('i> gctu'ral na- 
tui-e Refer to IIk* maruilactunM s in.striidion man- 




Fig. 1 




Fig. 2 




Fig. 3 

ual for detailed notes on the operation of a partic- 
ular- oscilloscope. 

The ON OFF switch is often combined with the 
INTKXSrn' control Wait a minute or two after 
turning on before tiAing to get a tr-ace on the 
screen .Adjust intensit\ and the FOCT'S knob until 
a blight, shaip spot or line is obtained Its position 
on the screen can be \aried b\ means of the 
NFRliKAI.i I'()S)rHON' and HORIZOMAL) 
RlSdllOX) contrx)ls It is bad practii-e to lpa\-e the 
scope turned on with a high-intiMisit\ stationary 
spot since it ma\ bum a boh* in tlie phosjihor coat- 
ing of the scrven 



7« 



I'Ml 1 CdNCKITS OF MCmOlV 



Vertical deflection: \'oltage measurement 

The cathode-ray tube itself is sensitive to both dc 
and ac voltages, but its sensiti\it\' i displacement of 
the spot per volt of potential difference between 
the deflecting plates) is low — tApically of the order 
of 0.2 min/\'. Amplifiers are therefore added to in- 
crease the sensitixitv'. In most simple oscilloscopes 
these are ac amplifiers, and so these oscilloscopes 
cannot be used for dc signals.* A dc oscilloscope 
has a dc ac switch that must be set to the appro- 
priate position. 

A \oltage signal to be measured is applied be- 
tween the \ER'I IXPLT terminal and GROUND ter- 
minal. I Make sure that the connection to ground 
is consistent with the circuit or de\ice providing 
the signal: that is, beware of crossed grounds, i 
X'oltages applied here deflect the beam up and 
down on the screen. This terminal is sometimes 
referred to as the V IXPLT, and the deflection as 
V DEFLECTION'. The amplification of this signal is 
controlled by two knobs that ma\ be called X'ERT 
INPUT, X'ERT GAIN, \ERT ATTENUATOR, \ERT AM- 
PLIFIER, etc. Usually one knob provides coarse 
control in three or more steps 1 1 x , 10 x , lOO x 
and the other gives fine control. In more expensive 
oscilloscopes these controls are calibrated in volts 
per centimeter deflection of the spot on the screen. 
With simpler scopes it is necessarv' to calibrate the 
sensitivitv' at a given setting by applving a signal of 
known voltage and measuring the deflection. Such 
a calibrating signal mav be provided at one of the 
terminals on the scope itself. On the Heath IO-12, 
for example, the l-\ P-P terminal provides a 60- 
cycle signal with a peak-to-pe£ik amplitude of 1 \ . 
(Note that the legends 100 x , 10 x, ix may refer 
to how much the input signal is attenuated rather 
thcin to how much it is amplified, so the 1 x is the 
range of highest sensitivitv'. i 

Pickup 

Sometimes you may find that a signal is seen on 
the oscilloscope face even if no obvious voltage is 
applied to the oscilloscope input. To see some of 
the characteristics of this pickup,' try the follow- 
ing procedure. Set the FREQ SELECTOR to about 
10/sec, and turn up the \ERT G.AIN to the maxi- 
mum setting. .Attach one end of a short length of 
wire to the X'ERT INPUT, leaving the other end un- 
connected. .An approximately 60-cycle sinusoidal 
trace will appear on the oscilloscope. Its amplitude 
increases if you touch the end of the wire, or if you 
use a longer piece of wire. 

The v\ire is acting as an antenna and is picking 
up the 60-c\'cle electromagnetic field that e.vists, to 

"In ac oscilloscopes it is sometimes possible to bvpass the 
amplifier and applv a signal directly to the tube, thus getting a 
deflection for a dc input. This usujilly in\olves removing a panel 
at the back or side of the instrument to expose the appropriate 
terminals. PROCEED WITH GRE.AT CALTION: THESE TERMI- 
NALS MAY BE .AT VOLTAGES AS HIGH .AS 1300 V Be sure to 
unplug the instrument before you e.xpose the terminals Be- 
cause of the \aw sensiti\it\' of the cathode-ra\' tube itself, the 
deflection will probably be small. 



a greater or lesser extent, in the vicinitv' of anv 60- 
cycle current. The field is pailicularly strong near 
transformers, fluorescent lamps, etc. .Although the 
ac voltage due to the varving field is small, the lai-ge 
amplification and high input impedance of the os- 
cilloscope can result in an appreciable trace am- 
plitude. 

Connect a resistor ifi ~ 1 megohml between the 

antenna ■ and the ground terminal of the CRO. 

The amplitude of the signal decreases, but is still 

appreciable. If the value of R is decreased, the pick 

up becomes smaller. 

Because of this spurious 'pick-up ' signal, shielded 
cable must be used to connect the CRO to high- 
impedance, low-voltage sources. The same consid- 
erations apply to all high-gain amplifiers. Note that 
the phototube supplied by Damon ifl ~ 5 meg- 
ohms i is mounted in a grounded metal box, and 
shielded cable is used to connect it to the ampli- 
fier. 

Horizontal deflection: measurement 
of time and frequency 

With the HOR FREQ SELECTOR (or SU'EEP SELEC- 
TORi set to EXIiERNALi, a signal applied to the 
HORIZ INPUT iX input! terminals causes the beam 
to move left or right across the tube face i horizontal 
or X deflection!. .As with the X'ERT INPUT, the signal 
is amplified and except on dc oscilloscopes 
a steady idci voltage does not produce a deflec- 
tion. The amplification is controlled by 
the HORiIZONTALi GAIN or HORIZON- 
TAL! .AMPiLI-R^DEi knob. 

When the FREQ SELECTOR or S\\ EEP SELEC- 
TOR is in the LINE SWEEP position, a 60 cvcle-per- 
sec sinusoidal voltage is applied to the horizontal 
deflection plates. If there is no vertical deflection, 
the spot will move back and forth across the screen 
in simple harmonic motion. Note that the deflec- 
tion is not linear with time in this setting. If another 
sinusoidaJ voltage is applied to the vertical deflec- 
tion plates, the resultant motion of the spot will be 
the combination of two perpendicular SHM's, i.e., 
straight line, circle, ellipse, Lissajous figure, de- 
pending on the relative amplitude, phase, and fre- 
quency of the two signals. 

VX'ith the SXX'EEP SELECTOR at UNE SWEEP, the 
PHASE knob is used to shift the phase of the sweep 
voltage with respect to the input signal. The traces 
shown in Figs. 4 and 5 were both made with the 
selector on LINE SWEEP and 60-cps signal on the 
vertical plate. The phase is shifted 90° between Fig. 
4 and Fig. 5. 

For other settings of the HOR FREQ SELECTOR 
or SWEEP SELECTOR control, an internal circuit 
applies a var\ ing voltage to the plates that control 
the horizontal position of the spot. This voltage has 
a saw-tooth wave form:/VV\A- ^^^ ^P°^ moves 
across the screen from left to right at a uniform 
rate while the voltage is increasing, and ver\' rap- 
idly flies back to its starting position when the volt- 
age drops to its minimum value. In this setting, 
deflection is linear with time. lAutomatic retrace 



EQUIPME.NT VOTES 



77 



blanking" reduces the intensity of the spot so that 
it is not seen as it flies back to the left of the screen. i 
The sweep ft-equency is controlled by two knobs. 
The HOR/FRKQ SELECTOR, or SWEEP SELECTOR, 
provides coarse control in steps. Typically, one set- 
ting will cover a "decade" of frequencies, for ex- 
ample 10-100, 100-1000, etc., cycles/sec. The FREQ 
VERNIER or SWEEP VERNIER gives fine control 
within these ranges. 




Fig. 4 




Fig. 5 

On expensive oscilloscopes, these contrcjis may 
alivady be calibrated. Othenvise, they can be cali- 
brat(!d by the following procedure. A signal of 
known fitniuencv is applied to the VERT IN'Pl'T. \i\ 
counting the number of cycles of the known-fiv- 
quency signal on the trace, one can establish the 
sweep fretiuencv. Ihat is, if there are exactl\- n 
cvt:les of a 60 cycle-per-second signal on the trace, 

then the sweep frequency is - 60/sec (see Fig. 6.1. 

n 

For low sweep rates, a 60-cycle signal can be used 
On the Meath oscilloscope IO-12, changing the 
I RI.O, SELECTOR one step will change the sweep 
rate In apiii-oximateK a factoi- of 10, for instance 
fn)m 20 sec to ZOO sec. For moiv accurate calibra- 
tion at higher swe(»p ratios, use a calibrated audio 
oscillator- isignal geneiatori to pixnide a signal of 
known fr('(|U(Mic\ Som(> expensive oscilloscopes 
have a built-in oscillator that can be used to apply 
known IrtMiuencx' signals to the xcrlical dellection 
plates Other's ha\e an output tcrininal tlial gi\»'s 
a 1-V (iO-c\(le signal 



The length of the trace is controlled by the HOR 
GAIN knob; this does not affect the sweep fre- 
quency (number of sweeps per second i — one full 
sweep still represents the same time interval — but 
it does, of course, change the sweep rate 'cmseci, 
and 1 cm will represent a different time interval. 

Synchronization of the horizontal sweep fre- 
quency wTth the signal applied to the verticiil input 
is important. If the two are svnchronized. then the 
same pattern will be repeated for successh'e sweeps, 
and what appears to be a stationaiy trace will be 
obtained on the screen las in Fig. 6i. 




If the signal and sweep frequencies are not sntj- 
chr-onized, then the traces obtained for successh-e 
sweeps of the screen will not coincide Fig 7 




FSg. 7 

Svnchronization is achieved bv fine adjustment 
of the FREQ \ ERN'IER contml until the sweep fre- 
(|uencv is an exact fraction of the signal fn*quenc>'. 
By setting the SVNC: SELECIOR to IVI^ or INT- 
the start of the sweep can be sv iichrx)nized with 
either the positive or negative slope of the input 
signal I Figs. 8a, bi. 

The sw«'ep can also be svnchninized with a sig- 
nal applied to the EXIiERNM. SNNC! temiinal b\' 
setting the SVNC: SELECIOR knob to EX I SN NC. 
.Adjust th«' E.\r SVNC: AMPLIil I)E contml until the 
sweep is sv ruhitmi/.ed with the signal Ihe EXI" 
SNNC" amplitude .setting has no elT«Mt unless Ihe 
S\NC SELFCrOR is set to EM S\ \( 



78 



I MI 1 CONCKITS OF MOTKIN 




Fig. 8a 




Fig. 8b 

There may also be a "LINE" setting of the SYNC 
SELECTOR. In this position, the horizontal sweep 
is svnchronized with the i60-c\'clei line frequency'. 

A useful feature present on some oscilloscopes 
is a "trigger." iThe Heath IO-12 does not ha\e this 
feature. I The horizontal sweep can be triggered by 
a signal applied from an external circuit to the trig- 
ger input. Until the triggering signal is applied, the 
spot remains stationary'. This is particulariy useful, 
for example, in time-of-flight measurements. If one 
wants to measure the time interval between two 
signals I such as, the interruption of two beams of 
light to two photocells I, it is desirable i though not 
essential! that the two signal pulses occur on the 
same horizontal sweep. This can be achiexed by 
triggering the sweep on the rise of the first signal 
pulse. If the CRO has no trigger facUitv', then it ma\' 
happen that the first signal will occur toward the 
end of one sweep and the second signal will occur 
on the next sweep. This makes measurement of 
the time interval between the signals difficult. 

Some, but by no means all, oscUloscopes ha\e a 
two-beam display: there are two Y inputs and it is 
possible to apply different signals to the two 
beams. This makes it \'er\' easy to compare the am- 
plitudes and frequencies of two different signals. 
In realitv', there is only one electron beam that is 
switched up and down so rapidly that two appar- 
ently continuous beams are seen iFig. 9i. The 
sweep rate for the two beams must be the same, 
but the amplifications of the two signals can be 
adjusted independently. 

On a "two-beam oscilloscope" the switching is 
done internally. External sv\itching circuits are 
a\ailable that enable one to make a two-beam dis- 
play of two independent signals on a regular os- 
cilloscope iHeathkit Electronic Switch ID-22. un- 
assembled!. 




Fig. 9 

Intensity' modulation (Z modulation) 

Some oscilloscopes ha\e an input terminal that is 
connected i through a capacitori to the intensity- 
control grid of the cathode-ray tube. This terminal 
is commonly marked Z-AXIS. On some oscillo- 
scopes it is necessan' to remo\e the back panel to 
uncover this terminal. Always unplug the oscillo- 
scope before remo\ing the panel. 

The potential of the grid iwith respect to the 
cathode! controls the intensity of the electron 
beam, and hence the brightness of the spot or trace 
on the screen. It is this grid potential that is ad- 
justed by the I\TiEXSIT\'! control knob. If the grid 
is made more positive, the spot becomes brighter. 
If a \'ar\ing \oltage is applied to the grid, the beam 
intensity will be modulated at the frequency' of the 
applied signal. Tvpically, about 10 V is required for 
complete blanking of the trace. 

Intensity modulation may be used to provide 
accurate time markers on the trace. The same in- 
tensity modulation, by the way, creates the light 
and dark areas in the picture on a TV screen. 

PHOTOGRAPH!' OF CRO TRACES 

The use of fast i3000-speedi Polaroid film makes it 
possible to photograph the trace. A close-up aux- 
iliary lens to gi\'e an appix)ximately 1:1 object-to- 
image ratio is necessar\'. If possible, remove the 
camera back and insert a ground-glass or other 
focusing screen in the plane of the film. With the 
sutter open, adjust the camera position to sharp 
focus. A rigid support for the camera is needed, of 
course. Turn up the oscilloscope intensity' control 
until a bright trace is obtained. Howe\er, if the in- 
tensitv' is increased too far on some oscilloscopes, 
the whole screen may begin to glow faintly and 
there will be a loss of contrast. If there is a colored 
screen or filter mounted in front of the oscilloscope 
face, remoxing the screen or filter ma\' improxe the 
image. Background illumination should be low, but 
it is certainly not necessary' to work in a darkroom. 
The appropriate aperture and time settings can 
quickly be found by trial and error. Don't forget 
that if the shutter speed is faster than the sweep 



EQUIPMENT NOTES 



79 



rate, only pai1 ol tJie tiace will be photographed: 
lor example, at 1/50 sec exposure, you cannot pho- 
tograph a complete 1/30-sec trace. 

The photographs used to illustrate this note 
were taken with the modified model 210 Polaroid 
Land camera, using the clip-on auxiliary lens. 

EX/\MPLES OF THE USE OF A CRO 
IN TEACHING PROJECT PHYSICS 

1. The ultrasound transduceis used in the wave 
experiments in Unit 3 have a \ery sharp r-esonance 
at 40 kilocycles. Before attempting any experiment 
with them, the oscillator driving the source trans- 
ducer must be carefully tuned to the resonant fre- 
quency. Set up the equipment as shown in Fig. 10, 
with the receiver transducer a few centimeters in 
front of the source. Set the CRO to a sweep rate of 
about 10 kilocycles/sec. Slowly adjust the fre- 
quency control on the audio oscillator until the 
trace on the oscilloscope screen "peaks" to a max- 
imum signal. 




Fig. 10 

In the experiments themselves, the amplitude of 
the trace on the oscilloscope screen is used to es- 
timate the efTecti\eness of various materials as re- 
flector's and absorbers of ultrasound, and to locate 
the positions of nodes Izero amplitudei and anti- 
nodes (maximum amplitude) in \arious interfer- 
ence and standing-wa\e patterns. 

2. The oscilloscope is used as a curr-ent meter- 
(or-, mor-e accurateh', as a null detectori in the in- 
vestigation of the photoelectric effect, one of the 
I 'nil 5 experiments. The oirt|iiit of the phototirbe 
is fetl (via an external amplilieri to the oscilloscope. 
As the couriterpotential acrxjss the pliotolube is in- 
creased, the photocunt'nt, and thus tiie amplitirde 
of the tract" on the CHO, (Uu-r-eases. 1 he experiment 
consists of finding what "slopping xollage is 
ne(»d('d to r-educe the photocun-ent to zeix) for light 
of dilTertMit fi-e(|iren(nes. 

3. Although these firnctions art? not included in 
Project I'hysicti, the C]RO can be used to make 
(|uanlitative measurements of ac voltages and cur-- 
mits. Clonijiarison of th(^ peak-lo-peak \t)llage with 
the r-eading given b\ an ac xoltmeter-, which is the 



To measure ac current, connect the oscilloscope 
across a known resistance (noninductixei and use 
/ = V/H to calculate current. 

Wave-form display 

The CRO can be used to show the difference be- 
tween sinusoidal and square waxes: to show half- 
and full-wave rectification of ac and the effect of a 
smoothing capacitor. It can also be used to show 
the waxe forms of the sounds produced by \arious 
musical instruments, or by students voices. (Use 
a micrt)|)hone as a detector. A small speaker can 
also be used as a microphone; amplification ma\ 
be necessary.) 




Fig. 11 Recorder plays a high C 




r"(H)l-m('an-s(|uan' voltage 
po.s.sihlc. 






Fig. 12 Harmonica plays a high C 

■I\vo or- more audio oscillators can be set to fun- 
damental and har-monic fr-equencies to sxnthesize 
tones approaching those of various mirsical instiu- 
ments. Ihe higher- frinjuencies must be set to e.vact 
multiples of the fundamental to get a stable trace. 

To demonstrate the for-mation of beats, set the 
two os(-illalor-s to frv(|uen(-ies that are only slightly 
differ-ent 

Spetilic examples of vvave-fonn displav in Proj- 
ect Physics wor-k: 

1. To show that the electron-beam tulnv like a 
vacuum diode, is a rv(-tilier. Kven if an ac voltage 
is applied b«'lv\een tilament and plate the cum'nt 
is d(- ihalf-vvavi" rvclitied i-onvsponding to elec- 
trons moving fiom tilamiMit to plate. i.-\ two-l>eam 
displav woultl be useful berv.i 

1 lo show the aitiori of the transistor switch 



HO 



I Ml I COIVCEPl'S OF MO-liOV 




Fig. 13 



used in \arious standing-vvaxe demonstrations in 
Units 4 and 5. 

3. To show the damped oscillations in an LCR 
circuit (Demonstration on Induction, Resonance — 
Unit 4). 

4. To "see" the signal bix^adcast by a radio sta- 
tion (Demonstration on Induction, Resonance — 
Unit 41. 

Time measurements 

(al Timing moxing objects 

Use two photocells in series and U\o light beams. 
The phototube units (PV'lOOi and light sources ftDm 
the Millikan equipment supplied b\ Damon can be 
used. Notice that no \oltage suppK is needed for 
the phototubes. This arrangement can be used to 
time falling objects, buUets, etc. Sweep rate must 
be known, of course. A rough idea of the speed of 
the object will make it easier to choose a suitable 
sweep rate and distance beUveen photocells. 




Fig. 14 

In some situations, the photocell can be re- 
placed b\' a simple switch that is momentarily 
closed b\ the object as it mo\es past. iFor instance, 
a steel ball making contact between tvvo pieces of 
eiluminum foil as it passes. i 

Some care is needed in interpreting the trace. 
The signals fi-om the two phototubes lor suitchesi 



will have slighth different shapes due to differ- 
ences in illumination. Establish which signal is 
from which tube by intercepting first one light 
beam, then the other. Xow examine carefully the 
trace record obtained when the mo\ing object 
crosses both light beams or switches. If the signal 
due to the interruption of the first light beam oc- 
curs toward the beginning of the sweep and is fol- 
lowed b\ the second signal, then there is no special 
problem. In this case the distance between the two 
signals represents the time bet\veen the two e\ents. 
But it can happen that the first signal occurs to- 
ward the end of one sweep and the second signal 
occurs on the ne\t trace. In this case the sum of 
the distance from the first signal to the end of the 
trace, plus the distance from the beginning of the 
trace to the second signal, represents the time in- 
tenal between the exents. 



Fig. 15a 




Fig. 15b 

If a triggered sweep circuit is a\ aUable, this com- 
plication need not occur. The triggering cii"cuit can 
be used to start the sweep just as the object crosses 
the first beam, or closes the first switch. 

lb I Stroboscope calibration 

A 60-cycle signal is used as a reference. See the 
notes on calibration of a .\enon stroboscope in this 
Resource Book. 

Frequency measurements 

The precision of a frequency measurement de- 
pends upon the accuracy of the reference source 
a\ailable. If the unknown frequency is a simple 
multiple or submultiple of 60 c\cles, then 60-c\'cle 
line frequencx', which is usually \er\ .closeh con- 
trolled, can be used. 

Set the HORIZ SELECTOR to LINE SW'iEEPt and 
the SVXC SELECTOR to LI.\E, or apply a 60-cycle 
signal from a stepdown transformer to the HORIZ 
I.NPLT. 

.Apply the signal whose frequence' is to be meas- 
ured to the \ ERT INPUT. Adjust the HORIZ and 
XTRT gains if necessar>'. Figures 16 through 19 are 
tvpical of the patterns that can be obtained. The\ 
ai-e called Lissajous figures. 

Only if there is a simple whole number ratio be- 
tween Xen ^"d .^oriz ^^"i'l stationary figures of this 
t\pe be obtained. 



EQL'IPMENT VOTES 



81 




1 



Fig. 16 

The pattern obseived depends on the relative 
phase of the two signals as well as their frequency 
ratio. The circle shown in Fig. 16 is obtained from 
two perpendiculai' sinusoidal signals 90° itt/Z ra- 
diansi out of |)hase. If the phase difference between 
the rvvo signals is 0° or 180° Itt radiansi, the result- 
ant trace will bv. a straight line. Intemiediate values 
of phase difference will gi\e ellipses. The PHASE 
knob can be used to vaiy the phase difference be- 
tween the signal applied to the V plates and the 
line frequency sweep. 

If the two frequencies are not equal, then the 
phase difference will vaiy continuously. The trace 
will change from straight line I0°i to ellipse I45°i to 
cirele (90°i to ellipse I135°i to a straight line per- 
pendicular to the original one I180°i, and thixjugh 
ellipse, circle, ellipse, back to the original straight 
line. Ihe frequency at which this change occui-s is 
equal to the frequency diffeience between the two 
signals. For example, if the rvvo signals are 60 and 
61 cycles, the trace pattern will go through one 
complete cycle of transformations in 1 sec. 

This techni(|ue can be used to calibrate an os- 
cillator against a 60-cvcle signal, at frequencies that 
are, or are \'eiy close to, multiples or submultiples 
of 60'sec. Hut if the problem is to measure a fn>- 
quen(\v that does not happen to be ecjual or close 
to a multiple or sui)multiple of 60 cycles, then this 
method cannot be used. Instead, it is necessan- to 
use a variable-trecjuency oscillator whose calibra- 
tion is accurately known as a reference. 




Fig. 19 



Demonstrations of complex motions 

In the pi-e\'ious section on fi-equencN' measurement 
by means of Lissajous figures, two independent 
signals were applied: one to the V and one to the 
X input. 

It is also possible to produce circular and ellip- 
tical traces using onl\' one ac \oltage In making 
use of the fact that in an RC circuit there is a 90° 
phase difference between the voltage across the 
resistor and the \oltage across the capacitor. iSee 
Fig. 20.1 

\ote that the mid|)()int of the HC circuit is con- 
nected to the gitJund temiinal of the oscilloscope; 




Fig 17 



Fig. 20 The HORIZ SELECTOR is set to EXT 



82 



IIIVIT 1 / tXJMIIiPTS ()l MOTION 



It is important that neither output terminal of the 
oscillator be grounded. If both the oscilloscope and 
the oscillator are connected to the line b\ a three- 
wire cable and three-pin plug, \ ou max ha\ e to use 
a three-to-tvvo adapter plug to isolate the oscillator 
&x)m the ground. 

The trace xnll be a circle or an ellipse, depending 
on the two \ oltages and the horizontal and \ ertical 
gains. 

Suitable \alues of fi and C for 1000 cycles sec are 
1000 ohms and 0.1 microfarads. Note that as the 
firequencN is increased the impedance of the ca- 
pacitor drops, the \oltage-drop across it drops, and 
what was a circle becomes elliptical. 

More complex patterns can be made if two os- 
cillators are a\ailable. For e.xample, the trace shown 
in Fig. 3 at the beginning of this article was pro- 
duced by the circuit shown in Fig. 21. iThe HORIZ 
SELECTOR is set to EXT.i 




Rg. 21 



Set up a circular or elliptical trace as described 
abo\'e, at, sav , 60 c\cles. Then apph a higher fre- 
quency', sa\', 1,200 CNcles, sine- or square-wa\e volt- 
age to'the\TRT Ix'PLT. 

Intensity* modulation 

1. Time markei"s. Set up circular, elliptical and epi- 
cycle traces, as described in the section Demon- 
strations of Complex Motions. .AppK a sinusoidal 
signal to the Z-axis to provide intensitx' modulation 
at a frequency that is at least 10 times higher than 
the frequencies applied to the X and V inputs in 
order to provide at least ten time markers per c\'cle. 
This modulation frequenc\ must be adjusted care- 
full\'. Onl\ when it is an e.xact multiple of the trace 
frequency will a stationarv pattern be obtained. 
These time markers show that the spot mo\es 
around the circle with constant speed. In the el- 
lipse it moves most quickl\- when it is close to the 
center; but note that this motion, unlike planetar\ 



motion, is s\Tnmetrical about the center of the el- 
lipse. iSo in this case the equal areas' law can be 
applied to the motion about the center, not to mo- 
tion relative to a focus. The same is true for the 
motion of a conical pendulum. i 




Rg. 22 




Rg. 23 

2. Television. ' Set the HORIZ FREQ to about 15 
kilocycles. Applv a 60-c\de sinusoidal signal of a 
few volts peak to peak to the V input. This can be 
from an audio oscillator, a step-down transformer, 
or the 60-c\cle calibration signal provided b\ the 
oscilloscope itself. .Adjust the horizontal and ver- 
tical gain to obtain a square or rectangular area 
that fills most of the tube face. 

■AppK an ac voltage of about 10 \ peak to peak 
I for instance, from the Project Phvsics oscillator 
unit' to the Z axis. If the frequencv of this signal is 
a few times greater than the sweep frequencv a 
sweep pattern of vertical bars will be formed as the 
trace is blanked out several times in each sweep. 
If the modulation frequency is several times less 
than the sweep frequencv then a pattern of hori- 
zontal stripes is formed. Stabilize the pattern by 
setting the S\'\C SELECTOR to EXT, connecting the 
EXT S\'\C terminal to the Z-axis oscillator, and 
turning up the EXT SYNC AMP control until the 
pattern "freezes.' 

V\ith two oscillators at different frequencies, it is 
possible to combine the vertical bars and the hor- 
izontal stripes to form a checkerljoard pattern. 



EQUIPMENT NOTES 



83 



Suggested Solutions to Study Guide Problems 



CHAFI ER 1 



2. Speed is the ratio of a change in distance to the 
time taken by the moving body to effect the 
change. Symbolically, if: 



d„ = Ad 
/„ = A/ 



- speed 



Ad 

then V - —— 

A/ 

Uniform motion: If the ratio of change in dis- 
tance to time taken is constant for some suc- 
cessive inteivals, regardless of how close the 
intervals are, the moving body is said to move 
with uniform motion. 



Average speed = 



to tal distance covered 
total time taken 



change in vertical distance (Av) 

Slope = — ; : r~r T — " 

change in horizontal distance ( A^ 

Let change in vertical distance = y, - y, - Ay 
Change in horizontid distance = ,v, - ,v, - 
A,K 

V — V A V 
Then slope is denoted by m = ' 



?^2 - ^1 



Ax 



Instantaneous speed is the rate of change of 
distance at a particular instant. 

V . = — , as Ar becomes verv small. 
'""' A/ 



average aceieration = 



change in velocit>' 
time taken for change 
Av 





" ^t 




t 


«iop«' "— 




tr* 




^-g'^X- 


IS' 


>/ j CkX' -Ki.- O., 





•Jt-i 



OC 



distance = d = 30 m 
time = r = 1-5 sec 

d,„,^ 30 m 

average speed = v' ^ = = 

',o.aj 1-5 sec 

20 m/sec 



4- (a) speed = 



distance tra\eled 72 cm 



time elapsed 12 sec 

= 6.0 cm/sec 

. km 20 min 

'") distance traveled = 60 — x 



hr 60 min hr 



= 20 km 



(c) time elapsed =- 



9.0 m 



36 m min 



= 025 min 



(d) The data gi\en indicate that the speed was 
probably uniform at 3 cm sec: so at 8 sec we 
e.xpect the speed to be 3 cm sec and the 
position to be 24 cm. 



(el V, 



Ad 240 km 



It 



6hr 



= 40 km hr 



If) On a trip as long as this it would no doubt, 
be impossible to maintain a constant speed, 
so the best we can do is estimate the posi- 
tion to be about 120 km on the assumption 
that the entire trip was made at 40 km hr. 



Ad 418 cm 

(gi Af = — = 

\' . 76 cm/sec 



= 5-5 sec 



(hi Ad = X A/ = 44 X 0.20 sec = 8.8 m 

sec 



5. If the ili\er falls uith unifonii speed, it means 
that the effect of gra\it\ has been countertial- 
anced by air resistance. Iheivfoiv, acceleration 
due to gra\it\' will not be considered wliile the 
di\er falls. 

\' = 12 m sec 

distance of fall = d, = 228 m 

d, 228 m 

time of fall = f, = — = = 19 sec 

I' 12 m sec 

Adilitional lime of fiiil = / ^ = 25 sec 
Additional distance fallen = d ^ = U ^ = il2m 
sec I (25 sec I - 300 m 

Total distance fallen = d,.,,,, = d, + d^ 

= (228 m + 300 ml 

= yifi Ml 



H4 



UMIT 1 / CONCKinS OF MOTION 



6. la) First, we need to find the total time taken, 
It 



It 



Ar, + It, 
Ad, Ad, 

V, v, 

100 m 



100 m 



\'.. 



5.0 m sec 1.0 msec 
Ad _ 200 m 
A^ 120 sec 



120 sec 



— 1.7 m^ sec 



Ad = Ad, + Ad, 

= 5 m,'sec x 100 sec +1.0 m^secx lOO sec 
= 500 m + 100 m = 600 m 
_ Ad _ 600 m 

"'" ~ a7 ~ 



= 3 m/sec 



200 sec 

\ote:In one case, the a\erage speed is the av- 
erage of the individual speeds; in the 
other case it is not. 



7. d. 



30 m 



practice time for rahbit = 'rabbit 
practice time for turtle = t,^^^ 

d 
practice \elocit\' for rabbit = — ^- 

' rabbi 

6 m/sec 

practice velocity for turtle = 

"turtle 

30 m 



o sec 
120 sec 
30 m 



o sec 



120 sec 

Total distance for race = d^ = 96 m 
.A\erage \elocit\' of rabbit = 6 m sec 

Therefore, total time used b\- rabbit 

96 m 



— 0.25 m sec 



— lO SKU 

6 msec 




a 
Total time used bv turtle = — 




96 m 




— — J54 sec 
0.25 m sec 


Difference in time - 


= 1384 sec - 16 seci = 368 



sec 

8. Af = 4 hr 34 min = 4.57 hr 

Ad = v^\t = 790 kmhr x 4.57 hr = 3.6 x 
10^ km 

9. (ai Compare the speed of the ball with the 

speed of a runner from third base. iThe 
times are about the same.i A runner does 90 
m in 10 sec or 9 m/sec, so it will take about 
3 sec to run the 27 m from third base to 
home plate. The distance to the outfield is 
about 90 m so the speed of the ball will be 
about 30 msec. 
lb) 1) Determine the speed of a leaf, for exam- 
ple, being blown by the wind. 
2) Hang a plumb line ftxjm the center of a 
protractor and calibrate this b\' holding 
it out the window of a car t^n a calm da\'. 



Notice how far away from the vertical it 
is displaced as you travel at various 
speeds. If, then, you find the same dis- 
placement on a windy da\' when the in- 
strument is held fi.xed, the wind must 
have the corresponding speed. 
(ci Sight on an edge of the cloud and detemiine 
the time requii-ed to mo\e thi-ough a certain 
angle. .Assuming the cloud to be 3-5 km 
abo\e the gix)und, you can estimate the ac- 
tual distance mo\ed from a scale diagram 
in\'ol\ing the height and the measured an- 
gle. 

10. (a) It is necessan' to determine first the number 
of seconds in a year: 

\t = 365 day/Vr x 24 hr day x 60 miahr 

X 60 " 

sec/min — 3.18 x lO' sec/yr 
Ad = 3.0 X 10* m/sec x 3.18 x lo" sec = 

9.5 X lO'" m 

(bl Total disteince to and frtim Alpha Centauri 
= 8.12 X lO'*' m 

8.12 X lO'^ m . 

Af = 1 = 2.7 X 10 sec 



3.0 X 10 m/sec 



X 10 sec 



3.18 X 10 sec>T 
8.5 \T 



ici One major problem is the short time inter- 
\als because of the high speed. Use a light 
pulse reflected from a known distance to 
excite a photocell. .Amplih' output of cell to 
excite the lamp giving light pulse. Measure 
high frequency' of pulses. The light beam 
couples the lamp to the cell. If the action of 
lamp and cell takes appreciable time, note 
change in frequency as reflecting mirror is 
mo\ed a known distance. 

(di Confine the ant to a definite path. 

le) Analx'ze the length of the trace on film left 
by a fast-mo\ing bright light. i\'ou need to 
determine the scale of the picture. i 

(f ) A high-speed motion picture camera could 
give the duration of the blink. .Another way 
would be to bounce a beam of light off the 
eyeball into a photocell whose output is am- 
plified and connected to an oscilloscope. 

Igi Determine how man\ hours it takes the 
whisker to grow to a measurable length. 

11. d,„,_,, = 500 m 

Let fj, be the time taken b\- the blue bic\'cle and 
f r the time for the red bic\cle to finish the race. 
t^ = (Tb + 20 sec) 
speed of red bicycle = a; = 10 msec 

d.„.., 500 m 



t, = Ifv, + 20 seel = 



10 m/sec 

50 sec 



SUGGESTED SOLLTIOXS TO STLT»V GLIDE PROBLEMS 



85 



Therefore, /,. = (50 sec - 20 sec) = 30 sec 



average speed oi blue bicycle = v,, = — 



300 m 
30 sec 



= 16.7 nVsec 



12. (al Initial speed = v, = 

Speed after 5 sec = v^ = 30 m/sec 
Time taken = t = 5 sec 

V, — V, 
Average acceleration = a^^ = 



130 m/sec - m/sec) 



= 6 m/sec 



D sec 

(b) If V, = 

a^^. = 6 m/sec^ 

total time taken = f = 10 sec 

V, - V 
Usmg a ., — ; V, = la„ t) + v, 

V, = (6 m/sec^)(10 sec) + = 60 m/sec 



13. la) 11) Average speed from starting line 10 seci 

Ad 
to 6 sec IS given by \\,^ - t~ = 

A t 



(30 m - m) 
(6 sec — sec) 



— 5 m/sec 



(2) Average speed from starting line to 10 

Ad 
sec is given by v_,^ = 



It 



ISO m - m) 
(10 sec — sec) 



= 5 m/sec 



(31 Average speed from 6 sec to 10 sec is 

Ad 150 m - 30 mi 

given by v„, = - — = 

' " It 110 sec - 6 seci 

= 5 m/sec 

(4) A\eiage speed fixim 5 sec to 8 sec is 
gi\en by 

Kv = 

Ad (40 m - 25 m) 

— — = = 5 m/sec 

at (8 sec - 5 sec) 

lb) A sliaighl line oljtaiiH'd fn)m a distanctMime' 
graph iiiilicates iinilonn speed. Heme, no 
matter how small the intenal chosen, the 
same speed will be found. 

(c) Since the speed is unifoini, the instanta- 
neous s[)eed will be 5 m/sec. The speed line 
(distance \t'i-sus time) has the same slope at 
all times 



14. lai The slop«* ol the line i.s pi()|)()rti(iiial to the 
s|)eed IK looking at the f^raph one ohseiM's 



that the ball traveled fastest in section CD 
and slowest in section BC. 
lb) Average speed between A and B 

175 m - m 

v.„ = = 44 nvsec 

4 sec - sec 

Average speed between B and C 



200 m - 175 m 



25 m 



1 1 sec - 4 sec 7 sec 
Average speed between C and D 



= 3.6 m/sec 



500 m - 200 m 



300 m 



15 sec - 11 sec 4 sec 

= 75 m/sec 

(bl Average speed between .A and D 

500 m - m 

V.,, = = 33.3 m/sec 

^' 15 sec - sec 

(c) Instantaneous speed at point fin section CD 
= 75 m/sec as for the section CD. 



15. lai V. 



- ^ 

Af 
_ 30 m - 25 m 

15 sec — 5 sec 
= 0.5 m/sec (at 10-sec mark) 
50 m - 35 m 

30 sec - 20 sec 
= 1.5 m/sec (at 25-sec mark) 



lb) a„ = -— 
A/ 



1.5 m/sec - 0.5 msec 
25 sec - 10 sec 
= 0.06 my sec' 



16. Randall wins over U'eissmuUer b\' 19 sec. Weiss- 

400.0 m 
mullers speed is so in 19 sec he would 



still ha\'e 



297 sec 
400 X 19.0 



297 



25.6 m to go. 



In making the graph for puiposes of e.\trap- 
olation, plot the number of seconds above 4 
mill xereus dates. 



17. 



sec 


2.0 m sec 


10 


0.75 


20 


0.57 


30 


0.85 


40 


0.63 


SO 


0.57 



She was swimming fastest at the beginning 
then slowing down but with a spuii at about 
30 .sec Note: The actual values for \ obtained 
b\ each student will show wide \anation ilue 
Id diHicultx in estimating iiitei\als on the graph 



N« 



I'Mi 1 (;()\c:ki»ts of motion 



..or* 

'.A 

E 



^x-. 



20 3D AO 

tir->« (sec ) 



50 



18. The error, which Mark TA\ain was fulh- aware 
of, was in assuming that ph\ sicaJ processes re- 
main unchanged. In this case, as in so many, 
one effect of a phenomenon 'shortening of the 
ri\eri is to change the rate at which that phe- 
nomenon takes place. .Also, it often happens 
that other e\ents le.g., geologic uplifting! can 
alter the circumstances enough to radicallx al- 
ter the phenomenon of interest. Since 176 \ ears 
is miniscule compared to geologic time, man\ 
such e\ents are likeh' to ha\e occurred, making 
Twain s extrapolation as erroneous as it is hu- 
morous. 

19. (ai The speed was greatest in the interval from 

1 to 4.5 sec. 

_ Ad _ 5.4 m — 1.0 m 
Af 4.5 sec — 1 sec 

4.4 m 



20. 



3.5 sec 



1.3 m sec 



Ibi The speed was least in the interval from 6 
to 10 sec. 

6.7 m - 62 m 0.5 m 

V = — = 0.13 m sec 

10 sec - 6 sec 4.0 sec 

ici A tangent drawn to the curve a\ t — 5 sec 
can be made the hypotenuse of a triangle 
with legs Ad = 6 m: Af = 8 sec. thus, v — 
0.75 m sec. 

idi A similarly drawn triangle at t = 0.5 sec 
gi\es 

Ad _ 8 m 

It 8 sec 



1 m/sec 



lei Reading directly from the graph, distance 
= 6.6 m - 6.2 m = 0.4 m. Using the \alue 
for the speed determined in ib , the distance 
- 0.13 msec x 2.5 sec = 0.33 m. 

Wliich method do \ ou think is more pre- 
cise? 

lai DE was co\ered fastest. 
BC was co\ered slowest. 

ibi EF was supposed to be a resting interval: 
therefore, it should ha\e been drawn par- 
allel to the horizontal axis. 



(cu- = 



_ 600 km - km 
.„.«■ * v\eeks — week 

= 75 km w eek 

Id) Instantaneous speed at P = — — at P 

It 



200 km - km 
2.5 weeks - week 



= 80 km\veek 



Ad 



Instantaneous speed at Q = — at Q 

It 



21. la) 



40r 



50- 



"8 



20 



10 



500 km - 300 km 
6 weeks — 5 weeks 



SPEED vs. TIME 



200 km week 



,x' 



/' (a) vat 2.5 see. = i4-.<nT^3cc. 



ok I I I \ I I I I I I 



ibi 






3- 



J 1- 
<0 



li34-5ft7»9iO 

time (*ec^ 



X ACC&t-PKATION VS. Tl>/1E 
\ plot 5-t for each into-Yal 



(b) m&f^imom acceleration 
occured during first 
second; S.3 m/6cc/^«c. 



J I I I I I I 



ai4.5C7ft9 10 

time C^>cc.) 



22. Total distance — 525 lines picture x 50 cm 
line — 26.250 cm picture 

26.250 cm picture 

Speed = = 875,000 cm sec 

0.03 sec picture 



SUGGESTED SOLITIONS TO STLDV Gl'IDE PROBLE.MS 



S7 



23. 



Data for graphs taken from photographs 



Position 










read on scale 


d 


Id 


Ar 


V 


1.5 


cm 


9.0 cm 


0.20 sec 


45 cm sec 


10.5 


9 


13 


0.20 


65 


23.5 


22 


17.5 


0.20 


87.5 


41 


39.5 


21 


0.20 


105 


62 


60.5 


25.5 


0.20 


127 


87,5 


86 









24. Fire bullets through two rotating thin paper 
discs spaced a shorn distance apart and rotat- 
ing at a high, knov\Ti speed. The bullet hole in 
the second disc will be displaced a certain an- 
gle relative to that in the first. The fraction this 



angle is of a whole re\'olution, times the period 
of one revolution, gives the time for the bullet 
to travel between discs, from which the speed 
may be computed by v = d/t. 

An optional method is to use a ballistic pen- 
dulum, using the law of conservation of mo- 
mentum. 

25. In line with the ideas de\'eloped in this chapter, 
it would seem appix)priate to start by making 
a graph of distance versus time, plotting the 
location of the horses nose, for example, at 
each instant photographed. Inspection of the 
varying slope of the graph will gi\e information 
of the speed that can be correlated with the 
motions being made by the horse s body and 
legs. 



CHAPTER 2 



2. A stone released at the surface of water will go 
to its natural position below the water. Rain- 
drops fall through the air seeking their natural 
position. (Accept any other imaginative exam- 
ples from students.) 

3. Very heaw body with no resistance: Aristotle 
would predict an infinite speed idue to dividing 
by zeroi; Philoponus would have the speed de- 
pend on weight alone. i\ote, however, that 
Philoponus has a basic difficulty for all weights 
and resistances in that he did not indicate how 
these could be expressed quantitatively in the 
same units; that is essential for the subtraction 
process.) Veiy light body with great resistance: 
Aristotle would expect it to fall very slowly; for 
Philoponus, the difficulty described above is 
paramount with at least the mathematical pos- 
sibility that the resistance be greater than the 
weight, giving a negative value to the speed 
(that is, it would travel upwardli. 

4. (a) Simp: Both pieces slow down to half-speed 

and fall togethi'r, taking twice the time that 
the 1-kg i-ock would have taken to fall the 
remaining distance. 

Salv: Both pieces continue to fall at the 
same rate as before fracture and strike the 
ground at the same time as the 1-kg rock 
would have. 
(bl Simp: I'he v5-kg rock would fall at a faster 
rate than the 4 5-kg r-ock. 

Salv: i'hey would fall at the same rate. 
(c) Simp: The sack containing the 100 rocks 
would speed up to fall the it'iiiaining dis- 
tance in ' uKi the time recjuired In uncap- 
tui-ed rocks. 

.S',»/\ . 1 he sack wouid irach the bottom in 
the same time as that taken In the separate 
tx)cks 



If the objects are of comparable densirv, they 
will fall with the same acceleration, and the 
string will hang limp between them. It v\ill be 
similar to the limp umbilical cord which at- 
taches an astronaut to a space capsule for the 
same reason: both have the same acceleration. 
In an extended situation where there is appre- 
ciable air resistance, an object vsith a greater 
cross-sectional area per unit mass will be re- 
taixied more, accelerate more slowly, and cause 
the string to become taut. 

(a) a = 2 m/sec^ for 6 sec 

d = Vzat^ (initial velocit>' = Oi 

d = V2 (2 m/sec^ii6 seer 

= 36 m 

IV, - v. I 
v_ = 





t 




36 m - m 




6 sec 


= 


6 m/sec 


= 


\\j 


= 


(6 m/sec) (6 sed 


= 


36 m 



(a 



ibi The equations used assume that the accel- 
eration is unifomi as with gravirv i .\ vaiAing 
acceleration would ir^quire a different imore 
complex! mathematical model. 

(Constant speed on a speed-lime graph is 
repivsented bv a horizontal line at a height 
abcne the /-axis coiresponding to the jiar- 
ticular speed \ We know that algebraicaliv 
the distance traveled in the time ( is given 
by d = \t. rhe rectangle shown on the 
graph has length = / iseci and width - x 



88 



I 'NIT 1 / (:o\c:ki»ts of moiioiv 



(m/seci, so its area — vt, which is simply the 
distance tra\eled measured in meters. 




time (^Cc) "^ 

(bi and (ci v^ = final speed after t seconds 
v^ = average speed = Vziv^ + Ol 

Area of triangle = Vz base x height 

= >/2 f X Vf 

or Vz Vft 

Area of the rectangle formed by 
\;, andf = v^j 

- Vz Vft 




time, (sec^ 

8. lal, (b), Id are correct answers. 

9. Students may see that each successive trape- 
zoid may be broken up into a number of con- 
gruent triangles that ma\' then be added up as 
1:3:5:7. They may also notice that the total area 
is in the ratio 1:4:9:16, or is proportional to r. 

This problem may also be done b\' comput- 

B \h, + h,i 
mg the area of each trapezoid as — 

using arbitraiy units, but the purely geometric 
approach seems preferable. 




10. Photograph a falling object illuminated by a 
stroboscopic light. The distance between the 
last two images di\ided by the time between 
flashes of the light will gi\e the average speed 
in the final interval. The shorter the time be- 
tween light flashes, the closer this average 
speed will be to the instantaneous speed. 

11. The plan for showing the equixalence of the 
gi\en expression with the Merton theorem is to 
logically deduce the expression from the rule. 

1. Distance traveled at average speed = dis- 
tance traveled when speed is changing uni- 
formly. 

2. Distance traveled is represented by the area 
under the curve of a speed-time graph. 

3. The area of the rectangle above determined 
by v^^ and t — the area of the crosshatched 
trapezoid, i Refer to the argument in the an- 
swer to 7C.I 

4. The algebraic statement of this relationship 
is 



.t = 



+ V. 



+ V. 




12. 



Average 



time 



15 -I- 16 -^ 17 -h 18 -^ 19 



85 

— = 17 years 



Average earning power 



S8,000 -I- 512,000 



S20,000 



SIO.OOO 



13. 


lai a^^ 




t 

V = 284 m sec 
t = 5.0 sec 




a^ 


= 


284 m/sec 
5.0 sec 




^^ 


— 


57 rri/sec" 



SUGGESTED SOLLTIONS TO STUDY GUIDE PROBLEMS 



89 



(bid 


= '/za/' 
a = 57 m/sec^ 


(b) 




f = 5.0 sec 


100 


d 


= Vz X 57 nVsec^ x 15.0 sec)^ 




d 


= 710 m 


80 


(c) a„. 


V 

t 


'•>. 

S " 




V = V, - Vj = m/sec - 284 m/sec 




^av 


f = 1.5 sec 
— 284 m/sec 
1.5 sec 


a«v 


= - 190 m/sec^ 


20 



Since 10 m/sec^ is approximately the accelera- 
tion of gravity, this means Col. Stapp was sub- 
ject to an average acceleration 19 times gravi- 
tational acceleration. iThe fact that he was 
subjected to a maximum of 22 g indicates that 
the actual acceleration was not constant.! 



14. (a) true 

(b) true 

(c) true 

(d) true 
le) true 



15. (a) Given: —, = K 
t 

Show: "... that the distances traversed, dur- 
ing equal intervals of time, by a body falling 
fiom rest, stand to one another in the same 
ratio as the odd numbers beginning with 
unity (namely 1:3:5:7 . . .1. . ." 

d = Kt' 

Substitute equal time internals of any arbi- 
trary but ecjual units of time as follows: 

/„ = units, /, = 1 unit, f, = 2 units, 
^3 — 3 units, and /, = 4 imits. 

Calculations of total distunrcs truvcrscd 
d^, - KO' = OK' units of length 
d, = K\' = IK 
d, = K2^ = 4K 
d, = K3' = 9K 
d\ = K4- = 16K 
*K is constant of pi-oportionalit\' 

Calculation of thr difference between 
consecutive distances traversed 

d, - d„ - IK units of length 

d^ - d^ = 3K 

d, - d, = 5K 

(/, - d\ = 7K 



OU 



X 



^ 



40- 



I 

5 20 

u 

V 
"" 10 



2 4 6 e to 

time (o.36s«c.; 



>>ir 



/* 



30.9cm/^>«c 






9x056 see- 



I I 1 I I I I I I L 



I 234-5ft709 to 
time (p. 35 Ae.c) 



Av 30.9 cm/sec 

acceleration = -— = 

Af 9 X 0.35 sec 



= 9.8 cm/sec 



16. lal 



Position 



(bi When the ball i.s in its u|)\\ard trajecton 
tmm A to C theiv is a decivase in velocity 
in the upwaixl or positive diivction that in 
effect, is an inciTase in velocity in the earth- 
ward or negative ditvction so A\\„ is neg- 
ative When the ball is in its downwanl tni- 
jectoiy fomi C to K. its \elocit\' is increasing 
in the eailhwaixl or negative diivction so 
that AV|„ is negatiw It should be noted that 
lime is not taken to ha\e a dinienMon ot 



BO 



I'lVIT 1 / CONCKPTS CIF IVKmOIV 



direction; time is scalar. Therefore, the ac- 
celeration is negati\'e, as is indicated by the 
definition a = v St. There is within this 
question the notion that when a \ector 
quantitv' is di\ided by a scalar, the scalar 
quantity' has no effect upon direction. 

\ote that a \ ector analysis of this question 
could be equalK' instructive, 
(c) The acceleration due to gra\ity would be 
( + I in all instances. 

17. The points would ha\e a pattern correspond- 
ing to the photo in 16 turned upside down. .A 
strong magnet held o\er a small nail would 
produce an upward acceleration. Also, a piece 
of wood released below the surface of water 
would accelerate upward. 

18. lai d = '/,ar" 

a = — 10 m/sec^ 

t = 1.0 sec 
d = Vz X -10 msec" 11.0 seci" 
d = -5.0 m 
lb) V = at 

a = - 10 m sec" 

t = 1.0 sec 
\' - —10 msec" X 1.0 sec 
\' = — 10 m sec 
Id d = Vzatl - Vzat] 
d = Vza (fj - f^i 

a = - 10 m sec" 

r, = 2.0 sec 

f, = 1.0 sec 
d = 1/2 X -10 m/sec"i4.0 sec" - 1.0 sec'i 
d = — 15 m 



19. By definition, 
acceleration = 
or 

„ final initial 



change in \elocity 
time elapsed 



\ote a, = 10 msec 



lai v'f = 


Vj 


- at 




Vj 




- 20 m^sec" 




a 




= - 10 mysec^ 




f 




= 1.0 sec 




V = 


20 


m/sec - 10 m/sec" 


X 1 sec 


V = 


10 


m/sec 




Ibid = 


vt 








V, + Vf 







d=l^^^|X, 



V, = 20 m/sec 
Vf — 10 m/sec 

f = 1 sec 

/ 20 my sec + 10rasec\ 
d = ( X 1 sec 



d — 15 m 

Id Vf = V, + at 
at = \'f - V, 
V, - \' 



t = 

a 

Vf = m/sec 
\\ = 20 m/sec 

a = — 10 m sec" 

— 20 m sec 

t = ; 

— 10 ni/sec 

r = 2 sec 

Idi d ^ \t 



Vf = m/sec 
Vj = 20 m/sec 

f = 2 sec 

/o m/sec + 20 ni/sec\ 
d = I I X 2 sec 

d = 20 m 



lei Since the object falls the same distance it 
rises and undergoes the same acceleration, 
we can immediateK sa\' that the downward 
trip will be similar to the upward one. Then 
it will take the same time and \v will be the 
same. We can also show this mathemati- 
cally: 

Since d — Vzat', the time to fall a distance 
d is 



a 



The \elocity to which the object will ac- 
celerate in this time is 



V - at - a 



= i2adl' 



2d 
a 

a — - 10 m sec" 
d = - 20 m 

id is negati\'e because the object is going in 
a negati\e direction i 



SUGGESTED SOLITIONS TO STL'D\ GLIDE PROBLE.MS 



91 



V = 12 X -lOm/sec" x -20 m) 

V = + 20 m/sec 

V = - 20 m/sec 

(The negative root is the one v\ith meaning 
in this situation.! 

20. (a) Vf = Vj + at 

V, — 40 m/sec 
a - —W m/sec^ 
f = 2 sec 

Vf = 40 m/sec - 10 m/sec^ x 2 sec 
Vf = +20 m/sec 

(b) Vf = Vj + at 

t = 6 sec 
Vf = 40 m/sec - 10 m/sec" x 6 sec 
Vf = — 20 m/sec 

(c) The ball reaches its highest point when 
V, = 0. 

Vf = m/sec 

Vj = 40 m/sec 

a = - 10 m/sec^ 

m/sec - 40 m/sec 

' = ; 

- 10 m/sec' 

f = 4 sec 

(d) d = Vj/ + Vzat^ 

= 40 m/sec X 4 sec + VA -10 m/sec *I16 sec* 
= 160 m - 80 m 
= 80 m 

(e) The speed is zerxj, since by s\Tnmetrv' the 
ball must reach the ground land hence 
come to a stop) in just 8 sec. 

(f I We can say immediateh' fix)m the s\Tnmetr\' 
of the pn)blem that the speed will be of the 
same magnitude when it gets back to the 
gix)und as it was when it left. This may be 
pro\ed in the following manner: 

V, = V, + at 

The time f is the time it takes the distance 
d to return to zero. 

d = v,f + Vaaf^ 

Letting d = and soMng for t, 

f(v, + Vzar) = (The root t = de- 

scribes the initial con- 
ditions! 

V, + Vzat = 



Substituting, 



-2v 

Vf = V, -f- a = - \', 

a 

Since v, = 40 m/sec, v, = - 40 m/sec 



21. (ai a = — 



(biv„ = 



A V = Vf - Vj = m/sec - 4 m/sec 
Av = -4 m/sec 
A / = 2 sec 

- 4 m/sec 

a = 

2 sec 

a = - 2 m/sec^ 

+ V. 



22 



4 m/sec + nvsec 

v„ = 

2 

\'„ = 2 m/sec 
(c) Vf = V, -I- at 

V. = 4 m/sec 
a = - 2 m/sec^ 
t = 1 sec 
V, = 4 m/sec - (2 m/sec^ x i sec) 
V, = 2 m/sec 
(dl d = v,f + Vzat- 

V| = 4 nxsec 
a - -2 msec^ 
t = 2 sec 
d = 4 nVsec x 2 sec + Vji -2 m./sec*)4 sec* 
d=8m-4m=4m 

or 
d = v„ t 

= 2 m sec x 2 sec = 4 m 
(el V, = V, + at 

t = 3 sec 
V, = 4 m/sec - 2 m/sec* x 3 sec 
V, = - 2 m/sec 

(f I The total time will be 4 sec. just twice the 
length of time needed to reach the highest 
point. 

(a! The proposal is to detemiine whether dif- 
feix»nt masses fall with the same accelera- 
tion, rhis experiment in\ol\es a direct 
measurement of the ratio d /"'. Howe\-er. var- 
iations in t for the same e\ml will pn)babl\ 
be ivIaliwK large: thus, there will Ih» varia- 
tions in a^. 

/\ stuilent might pmpose to detennine the 
disin'paiuies among the falling times and 



I'NIT 1 / CO.XCIKI'l'S C)K VUmOlM 



then compare the small discrepancies v%ith 
the total time to fall. This would imolve a 
change in the design of the experiment to 
a null method, ' which is more sensiti\e. 

(b) This project looks interesting and will allow 
us to determine whether d r is constant for 
different \ alues of d. Howe\ er, the pixjposal 
states that each student will obtain instan- 
taneous speed of the ball as it passes the 
window. The procedure is not outlined and 
ma\' pro\e difficult or impossible to do. 

(c) The effect of air resistance on the cotton 
balls is going to be appreciable. If this is the 
case, we cannot expect the balls to ha\e a 
constant acceleration. The experiment might 
pro\ide interesting inlbrmation, but it would 
not be pertinent to Galileo s problem. How- 
ever, a student might justifv accepting the 
proposal on the basis that the effect of air 
resistance might not be known until some- 
one had performed this experiment. 

23. (a) We can find the acceleration due to gra\it\' 
on Arret by constructing a graph of speed 
versus time for the fi^eh' falling bod\ and 
finding its slope. To do this we could first 
construct a distance-time graph and meas- 
ure the slope of the cune at \arious points: 
or, we can approximate the \ elocits at dif- 
ferent points b\' finding the axerage \elocit\' 
between tAvo points on either side of the 
desired points. This can be done with a ta- 
ble. Xote that the a\erage \elocit\' between 
points A and B approximates the instanta- 
neous speed at the time 0.25 sec. -Also note 
that the time inter\al in the gi\en data 
changes from 0.5 sec to 0.2 sec. 



Time 
Position (surgs) 



d 

(welfs) 



Interval 



/ welfs \ 
\ surg ' 



0.0 
0.5 
1.0 
1.5 
2.0 
2.2 
2.4 
2.6 
2.8 
3.0 



0.00 

0.54 

2.15 

4.84 

8.60 

10.41 

12.39 

14.54 

16.86 

19.33 



AB 
BC 
CD 
DE 
EF 
FG 
GH 
HI 
IJ 



0.54 
1.61 
2.69 
3.76 
1.81 
1.98 
2.15 
2.32 
2.47 



acceleration - 




Av 
Af 
12.3 



1.1 

3.2 

5.4 

7.5 

9.0 

9.9 

10.7 

11.6 

12.3 







2.9 
= 4.3 welfs surg" 



(b) 1 welf = 0.633 m 
1 surg = 0.167 sec 

1 welf/surg^ = 

1 welf 10.633 niAvelfi 
1 surg^ 10.167 sec' surg" 



= 2.27 m/sec" 



227 m/sec 

4.3 welf/surg" x = 9.8 m/sec" 

1 welf/surg" 

The accleration caused b\' gra\it\' on 
Arret is 9.8 m sec", about the same as that 
on earth. 

24. Special conditions implied in gi\'en equations 
II object starts from rest 
21 acceleration is uniform 

(al Derivation: 
Given: \' = at 



t = - 
a 

Given: d = VzaV 



d = Vza I - 



d = — 
2a 

v" = 2ad 

(bl As the ball returns to earth it will ha\e the 
same speed downward that it was gi\en up- 
ward initiall\'. Thus, the height h bxtm wtiich 
it must fall to attain the speed \' is deter- 
mined as follows: 

^r = la^h 

\~ 
h ^ — 



25. The equations we ha\e a\ailable are as follows: 



Da 

n - ^\ 



t 



- from which we can sa\' t = 



21 d = v.t + Vzar 

31 d = v^J 

4>v^^ = 1/2 (Vj -I- Vfl 

Combining equations 1, 3, and 4 we get: 

V, - V. 



time (sorgs) 



d = V2(Vj + Vfl — 

2ad = Vf - vf 
vf = \'f + lad 

The same result is obtained if we combine 
equations 1 and 2. 



d = \-, ' + '-2 a 



SUGGESTED SOLLTIOXS TO STLTIV GUIDE PROBLEMS 



93 



ad = v,v, - vf + Vzlvf - 2v.v, + vfl 
ad - v,v, - vf + '/av^ - v,Vf + Vzvf 

= '/2vf - '/2Vf 

2ac/ = vf - vf 
vf = vf + 2ac/ 

26. The area under the graph line is composed of 
the rectangle of sides corresponding to v, and 
t plus the tiiangle of base t and height v, - v^ 



Total ai-ea - v, / + '/zlv, - v^t 

or V, - V. = at 



V, — V: 

but a = -^ ' 



t 

Thus, total area - distance traveled 
= vj + Viat^. Thus, 




time. 



27. The steps in Galileo's investigation may be 
identified as follows: 

1) Definition: define uniform acceleration as 
constant increase in \elocit\' with time. 

2) Hypothesis: Freely falling bodies are uni- 
formly accelerated, as defined abo\e. 

d a 

3) Deduction: -r = - = constant for balls fall- 

t^ 2 

ing from rest. 

d a 

4) Deduction: -r = - = constant foi- balls 

t^ 2 

rolling dowii an inclined plane. 

5) Observation: 41 is verified by experiment. 

6) Conclusion: 21 is xerified by the above 
process. 

The argument was limited by Galileo's abilit\' 
to mcasiiif time int(MAals accurately and by his 
idealization that rolling motioti was simpl\ a 
slowed-down falling motion. Me ignoitid the ro- 
tational motion of the ball about its center. 

28. (a) The average speed is equal to the distance 

intenal tiaxclcd (ii\ided by the time int(M"\al 
ovv.v which tln" distance was measunul 

The a\iMag(' acceleration is ('{jual to a 
change in xclocitN dixided by the time in- 
teival ()\ei- which the velocity change was 
measured. 

The distance tra\eled by a unifonnh ac- 
celerating object is e(iual to one-half the ac- 
celeration measured Ironi the sl.iil multi 



plied by the square of the time since the 
start. 

(b) A wide variety of equally good problems can 
be expected here. 

(c) For the problem suggested, the answer is: 

-^^ 

^t = — 

3,200 km 
1,000 km/hr 
= 32 hr 

29. Although his followers may ha\e relied too 
hea\il\ on banded-down information, Aristotle 
himself did obser\e nature and successfully 
classified many plants and animals in addition 
to recognizing that stones, for e.xample, do fall 
faster than lea\es. Galileo showed that it is nec- 
essary to question authorit\ in science, espe- 
cially when contradictions are observed in na- 
ture. He was not able to show everv'one directl\', 
but was able to convince man\' b\ indirect and 
mathematical arguments that Aristotle s anal- 
ysis of free fall was wTong. If by science we 
mean the stud\' of nature as it can be observed, 
coupled with an attempt to correlate separate 
events into a coherent pattern, then Galileo was 
certainlv not the first to do this. His principal 
contribution to our modem scientific method 
was the recognition of the kev role played by 
mathematics in describing nature. 



30. lai V, = 5 m/sec a = 



(V, - v.t 



/ 



V, — 30 m sec 



30 msec - 5 msec 



10 sec 

/ = 10 sec =2.5 nvsec^ 

(bi t = 9.0 sec d = ^/zat' 

V, = = V2(9.8 m/sec '119.0 sec) ^ 

= 397 m 

(for a = 10 m/sec^. d = 405 mi 
Id a = 2 m/sec' d = ^zat' 



d = 20 m t' 


a 


t' 


2(20 m) 




2 m sec* 


I' 


= 20 sec' 


t 


= 4.5 sec 


li\\ = 0. V, = alt 




= (2 m sec^ l4-5 seci 


= 9 m/sec 


(di \', = 8 m sec norih 




a = 5 m sec* noilli 




t = 10 SCI 





»4 



I'MIT I / (XI\C:KI»TS of IVIOTIf 1\ 



d = v^t + Vzar 

— (8 m/seci llO seci + Vz I5 nvsec^ 1 10 seel' 

= 80 m + 250 m 

= 330 m north 
Vf = v, + at 

= 8 m/sec + 15 m/sec") (10 sec) 

= 58 m/sec north 

(e) a = 2 m/sec" d = \\t + Vz ar 

d = 4 ni 4 m = + '/2 

1 2 m/sec'i r 
Vj = 4 m = 1 nvsec" t' 

4 m 



t' 



1 m,sec" 
4 sec" — r 

2 sec ^ t 

Vf = at 

= (2 m/sec"l 12 sec) 
— 4 ra sec 

(f ) Vj = 6 m/sec 
a — 1 m^sec" 
d = 2 m 

v^ = vf + 2ad 

Vf = 16 m/seO" + 2(1 m/sec"i I2 m) 

vf = 136 + 4) m'/sec" 

Vf = V40 m/sec 

Vf = 6.3 m/sec 

(g) V, = 5 m.sec d — Vz iv, + vj t 
Vf — 55 m/Sec 100 m = ViiSS m. sec 

+ 5 nvsec If 
d = 100 m 100 m = Vzieo nvsec) t 

100 m =130 msecir 
3.3 sec = t 



t 
55 m/sec — 5 m/sec 

3.3 sec 
50 m/sec 



3.3 sec 
= 15.2 m/sec" 
(hi d =^ 45 m 

f = 4 sec 
d = v.t + Vzat^ 

45 m = Vj (4 sec) + Vz ( — 9.8 m/sec"i 14 seer 
45 m = \', 14 seel + Vz I -9.8 m/sec^ (16 seel 
45 m = V, 14 seci - 78 m 
45 m + 78 m 



4 sec 




' 


31 msec = Vj 


31. (al v^^ 


= 


Ad 30 m - 15 m 
A f ^2 sec 




= 


10 m 15 m 
4 sec 2 sec 




= 


2.5 m/sec lABi = 7.5 m/sec iCDi 


'b' v.. 


= 


15 m - 10 m 
2 sec 




= 


2.5 m/sec (BCI 


^a\ 


_ 


7.5 m/sec - 2.5 m/sec 






2 sec 




- 


2.5 m/sec' 


(c) Discussion 



32. (a I graph : constant \elocit\' 
graph B: acceleration 
graph C; acceleration 

graph D: negati\e acceleration (decelera- 
tion! 
(bl graph A: backward 
graph B: fon\ard 
graph C: backward 
graph D: forward 



CHAPTER 3 



2. V\'e do in fact always obser\e that a force is 
required to keep an object mo\ing with con- 
stant speed across a table, and, if a massive rock 
and a ping pong ball are dropped from the 
same height abo\e the ground, the rock will 
reach the ground first. We tend to remain Ar- 
istotelian since Xeutonian force analysis re- 
quires us to consider ideal situations or to deal 
with in\isible forces like friction. 

3. (ai Mechanics is the branch of physics that 

deals with the study of forces on objects. 
Dynamics is the study of forces that pro- 



duce motion in objects (cause objects to 
mo\ei. Kinematics is the study of motions 
without reference to the causes of the mo- 
tions. 
(b) (1) d (scalar) 

(2) /(vector) 

(3) V (scalari 
(41 /(scalari 

(5) m (scalar) 

(6) a (vector) 

(7) t I scalari 

(8) a (vector) 
191 V ivectori 



SL'GGESTED SOLLTIOXS TO STl D\ Gl IDE PROBLEMS 



95 



4. 



a) Three blocks east of the starting |)oiiit, 

(b) (3 + 4 + 5 + 1 + 2) blocks = 15 blocks 

(c) Part (a) is a vector problem; part lb) is a sca- 
lar pixjblem. 

la) The forces do not balance. The tip of F^ does 
not coincide with the tail of F,. 

Note that the direction system and the 
force scale are needed before attempting a 
solution. 

lb) Net force = 2.4 units west lshov\Ti just be- 
low the diagram). 






Force scale (units) 

?'3-12 units 
30° S of West 




Fj=6 units, 
North 



net? 



=8 units, East 



6. A + B has the same magnitude and direction 
whether it is obtained as the diagonal of the 
parallelogram having /\ and B as adjacentsides 
or as the third side of the triangle ha\ing A and 
B as the other two sides. The essential pmce- 
dure in both methods is to make sure that the 
given magnitudes and directions of the original 
vectoi-s are car-efully preserved before attempt- 
ing to complete either the parallelogram or the 
triangle. 




7. The parachutist s weight is 750 N down. Air re- 
sistance is therefore 750 N up, since the para- 
chutist is falling with uniform speed. lAir re- 
sistance just balances the effect of graxity.l 

8. Another example is a Lunar Module appix)ach- 
ing the surface of the moon at a constant speed 
of 24 m sec. In this example, the downward 
gravitational fotte is etiual to the upward thiiist 
of the module in magnitude but opposite to it 
in diit'ction Uotli ot iIm> forces increase as the 
module appmach(!s the surface of the moon. 

9. lal lujuilibrium is th(! state in which the net 

force acting on a s\st(>m is zeii). 
lb) Two p()ssil)le states of motion lor an oijject 
in e(|uilibrium are: i|i the oi)iect is at i-est: 
i2i the oliject is mo\ing unilormK 



10. lai We would say that the force is not large 

enough to overcome the force of starting 
friction. An Aristotelian would say that rest 
is a natural state and that any motion re- 
quires a force; the force is not enough to 
change the box from its natural state. 

lb) We would sav that the applied force is now- 
greater than the force of friction, resulting 
in an unbalanced force. Consequently, the 
box accelerated according to .\ewlon's sec- 
ond law. An Aristotelian probabU would 
have maintained that the force producing 
the motion was now great enough to dis- 
place the box from its natur-al state. How- 
ever, the Aristotelians had no clear concept 
of acceleration. 

(c) V\'e would say that the frictional force be- 
tween the box and the table is directed op- 
posite to the motion, accelerating it in this 
reverse direction until it comes to rest. Ar- 
istotle would say that the box was returning 
to its natural state of rest. 

11. la) The ice puck will move with a uniform ve- 

locity. 

If the laboratorv itself is moving with a 
uniform velocity', the puck's motion will ap- 
pear unaffected to an observer inside the 
laboratoiy. An observer outside the labora- 
torv will see the puck moving with a velocity' 
equal to the sum of the laboratoiA' s velocitv 
and the velocity- imparled bv the push. 

In a laboratory' under-going uniform linear 
acceleration, an observer inside could not 
detemiine bv obserAing only the puck s mo- 
tion whether the puck was being acceler- 
ated in one direction or- the laboratory- was 
being accelerated in the opposite direction. 

Motion of a puck in a curxed path can be 
explained either bv assuming a forve acting 
on the puck in a direction that makes an 
angle with the velocitv it was originallv 
given, or bv- assuming the laboratorv- to lie 
accelerating at the supplementarA' angle, 
(bl The man will see the puck curve awav from 
him. Reiving on \ew1on s laws, he will think 
that the puck is being subjected to a foive 
that accelerates it in the cirr"\ed path. He 
will be wning, of course, because he is in an 
accelerated frame of referxMice. i This ficti- 
tious forx^e is called a Coriolis force.) 

12. lal The brakes slow the car but not the passen- 

ger's, since they are not rigidiv attached to 
the car. Their inertia causes their for-warxl 
motion to continue unchanged momentar- 
ily while that of the car is rvduced 
Ibi \ elocitx is a v ector (|uantit> W hen the force 
of the r-oad against the tiivs changes the 
dir-ection of the car it fails to change the 
dirvction of the passenger-s immediatelv 
They continue in the original tlirvction until 
the loi-ce of the seat and tlie side of the tar 



9« 



ii\i I I / c:c)\(:kpts oi- mutioiv 



on their bodies changes their direction of 
motion. 
(c) The centripetal force needed to hold the 
coin in "orbit" increases as the rotation rate 
of the turntable increases. The frictional 
force that links the coin to the turntable re- 
mains constant. When the frequency of the 
turntable ixjtation has increased so that the 
centripetal force equals the frictional force, 
a further increase in the centripetal force 
required to keep the coin in its orbit" can- 
not be pro\ided by friction. The coin will 
then slip toward the rim of the turntable. 
Note: This anticipates the discussion in the 
next chapter. 

13. One way would be to use a rubber band lor a 
springi stretched to the same extent and attach 
it to each of the different masses successively. 

14. (al Newton's second law says that acceleration 

is 111 directly proportional to the magnitude 
of the force, I2i in the same direction as the 
force, and I3i imersely pixjpoitional to the 
mass. 

ibl a = 10 m/sec" north 
F — ma 
f 



Therefore, a is directly pixjportional to F and 
imersely proportional to m. If the force be- 
comes V2F and the mass becomes Vbm; 

a a: VzF :. 10 m sec" X V2 = 5 m/sec' 
a ^ 3m .". 5 m sec" x 3 = 15 m/sec" 

The new acceleration is 15 m sec" east 
(since the new force is to the easti. 

15. That k must have the dimension hr/sec is seen 
as follows: 

Ad = kv'lt 
imii = k imi/hr)(sec) 

We need to compensate for the sec/hr found 
in the \' and Ar terms. The value of k is then 
the number of hours in a second, which is the 
fraction 1/3,600 or 2.78 x lO^^ Thus, k = 2.78 
X 10"* hr/sec. 

16. As explained in Sec. 3.10, students should find 
that their weight seems to increase as the ele- 
vator accelerates upward. The new weight will 
be equal to the original weight plus the added 
force caused by the upward acceleration iF = 
ma). As the elevator slows down, a student's 
weight will gradually seem to decrease to the 
original weight. If the ele\ator mo\es up and 
down at constant speed, the student's weight 
will also appear to remain constant. Although 
your weight does not really change during an 
elevator ride, the scede shows a difference when 
the elevator is accelerating because the scale 
measures the net force acting on your bodv. 



This foix;e increases when the elevator is accel- 
erating and decreases when it is decelerating. 
In a space vehicle, your weight would seem 
to decrease as you got farther from earth. 

17. Provide yourself with suitable standard masses 
and measure the accelerations associated with 
particular \alues of the extension of the spring. 
For each case, F can be detemiined by multi- 
pKing the mass by the acceleration and mai^ked 
at the place on a scale indicating the extent to 
which the spring was stretched. To actualh' do 
this it would be difficult ill to maintain a uni- 
form stretch of the spring, 12 1 to eliminate fric- 
tional forces, and (3) to measure the accelera- 
tions precisely. 

18. lal A simple experiment could be set up by 

hanging different masses from a spring and 
noting the extension of the spring for each 
mass. Since the force of gravity on each 
mass can be calculated, we can plot force 
versus extension. If Hooke is correct, the 
points will lie along a straight line. This law 
does not hold when the spring is stretched 
beyond its elastic limit; that is, when the 
spring fails to resume its original length 
when the mass is removed. This perma- 
nently damages the spring and should be 
avoided. iNote: Modern usage requires the 
substitution of the word "force " where Hooke 
used 'power." Today, power has a different 
meaning.i 
Ibl A static method of calibration may now be 
used. When the spring is stretched by a 
known mass, the force on the spring is just 
equal to the gravitational force on the mass 
I its weight I and is equal to the mass times 
the \alue of a^ at the particular location. 

19. (cl 24 N out 
(dj 15 N left 

(e) 0.86 N north 

(f) 9.0 kg 

(g) 0.30 kg 
(h) 0.20 kg 

(i) 3.00 m/sec" east 
(j) 2.5 m/sec^ left 
(k) 2.50 m/sec" down 

F 

20. la) a^,, = — 

m 

F == 8.9 X 10" N 
m = 4.44 X 10' kg 
_ 8.9 X 10" kg m sec" 
4.44 X 10^ kg 
a 3, = 2.0 X 10" m/sec^ 
V = at 

t ^ 3.9 sec 
\' = 2.0 X 10" m/sec" x 3.9 sec 
\ = 7.8 X 10" m sec 



SUGGESTED SOLL'TIOXS TO STLT)V GLIDE PROBLEMS 



97 



21 



(b) 2.0 X 10'' m/sec^ is about 20 g. Since the 
maximum acceleration is 30 g, the acceler- 
ation varied. Note that 20 t^ is the average. 

(cl v^ = 2ad 

= 2:1 

^" ~ 2d 

V = 860 m/sec 
d = 1530 m 
_ I860 m/seo' 
"^ ~ 2 X 1530 m 
a = 2.4 X 10^ m/sec^ 

The average acceleration and the maxi- 
mum speed turn out to be higher than that 
obtained by using the equation for Xeulon's 
second law of motion, as in lai. The discrep- 
ancy is explained by the fact that the rocket 
mass is constantly decreasing, and hence it 
is incorrect to use the initial mass for the 
whole r-un. 

A good student might like to try to cal- 
culate the mass lost during the 3.9-sec run. 

To detennine the unknown mass, we first cal- 
ibr-ate the spring balance. This ma\' be done by 
acceler-ating the 1-kg standard with a constant 
force indicated on the spring balance. The time 
to cover a measur-ed distance from r-est can be 
determined and the acceler-ation calculated; 



d = Vzat' 



2d 



From the knov\Ti values of m and a, F can be 
calculated using Xewlon's second law, F = ma. 

The unknowii mass can then be accelerated 
with this same force and its acceleration meas- 
ured. If \alues for F and a ar-e substituted into 
F - ma, the unknowii mass can be calculated. 

It may be noted that since the same forx:e is 
used each time, it is not necessary to compute 
the value of the force to (iiul the mass: 

F = F 

m^a^ = m.2.^2 

m,a. 



m, = 



a.. 



22. Since the balance r-eading is 0.40 \ when the 
block is dragged at any constant \el()(it\', this 
must be the trictional force. Ihe net force is the 
applied forxe less the fractional force. 

F = F - F 

nrl applliHl ' Inrliiin 

F,,^, = 2.1 N - 40 N = 1.7 N 
F...., = rria 



' nol 

a 



85 m sec 



1.7 kg m/sec^ 

m = r 

0.8o nvsec 

m = 2.0 kg 

23. This question is intentionally phrased in per- 
sonal terms of "you" and "your. ' The students 
may propose a variety of explanations, such as 
the following: 

(1) Because all parts of the body are accel- 
erating downward at the same rate, those 
below do not support those abo\e them 
as they normally do. There are no upward 
forces being exerted that compensate for 
the downward gra\itational forces. 

(2) This is only an apparent" weightless- 
ness, because gra\ilational forces cer- 
tainly are acting on the bodv, making it 
fall. " 

131 "True" weightlessness, as in deep space, 
we can only imagine: but we understand, 
in \ew1onian terms, that there would be 
no appr-eciable forces among the \arious 
parts of the body. 

24. la) 111 The mass will be 1 kg in both places. 

liilF = ma^ 

m = 1 .000 kg 

a^ iParisi = 9.81 msec^ 

F = 1.000 kg X 9.81 msec^ 

F iParis = 9.81 \ 
a^ lV\'ashingtoni = 9.80 nvsec^ 
F = 1.000 kg X 9.80 m/sec* 
F iVVashingtoni = 9.80 N 

ibi The change in any student's weight ma^ can 
be calculated as follows: 

AF = FiParisi - F' iWashingtom 

AF = ma, - mal 

AF = m la^ - a^i 

where a^ = acceleration at Paris 

a^ = acceleration at Washington 

25. lai Since the pound is a unit of force iweighti 

and the kilogram is a unit of mass, they can- 
not be dircctU con\ cried Weight is a meas- 
ure of the earlh s gravitational attraction at 
its surface and then'forc comparisons can 
onl\' be made on earlh 

ibi Student answer-s will \ar\\ 

(el Stirdent answers will \arA' iFor each 1 kg of 
mass lifted. 9 8 \' of force arv reqiriivd.i 

26. Ibis (juestion anticipates the discussion on cir- 
cular- motion in later- chapters, but it maN ha\-e 
aInMiK been raised b\ stirdenls in connection 
with 23 When in odnX a few hundrx'd kilome- 
tj'rs aboNC the earth. Werghtlessness cannot 
lie due to a \»m"\ small \alire of .».. it can hv 



UNIT I / CXJMCKPTS OF MOTIUN 



Grrif 
shouTi that a„ — — r^ where G is the Ca\ endish 
« R- 

constant, aMj the mass of the eailh, and R the 
distance to the center of the earth. Since the 
radius of the earth is about 6 400 km, a feu- 
hundred more kilometei-s will not make a large 
change. The correct e.xplanation lies in the fact 
that the astronauts, their capsule, and all its 
contents are in a constant state of centripetal 
acceleration = a^: a kind of fi'ee fall. 

27. lai F„ = -Ff 

(601(51 = -i60 X lO'^i Oj.) 

a^ = -5 X lO'^msec" 

(bi v„ = a^t 

\-„ = I5II2) 

\ „ = 10 m sec 

(ci \\ =^ 3,^t 

Vf = i5 X 10"^)I2) 

Vp = 10 X 10"" msec 

28. F = ma 

_ _F_ 

_ 80 kg • m sec' 

40 kg 
= 2 m sec" 

According to Newton's third law, the force on 
the bo\ ^F^^J is 80 \ ikg • m sec"'. Therefore, 

_ 80 kg • m sec" 
^"^ " 70kg 

= 1.14 m^ sec" 

29. lai True, but this would also be true without 

the condition of standing perfecth still. 
Ibi True. The propeller e.xerts a force on the air. 

The air exerts an equal and opposite force 

on the propeller, enabling the plane to moxe 

forvvard. 
(CI True. Both are numericalh equal to the 

weight of A. 

30. Think what the tractor must do to bring about 
its motion. As power is applied, the tracks push 
backward against the surface of the earth. Some 
loose earth ma\ be pushed awa\'. The loco- 
motion of objects commonK in\ol\es pushing 
backward, opposite to the direction of motion. 
But according to the third law, if the treads of 
the tractor push backwaixl on the surtace of the 
earth, the earth must simultaneousK push for- 
ward on the treads. Whether or not the tractor 
moves depends soleK on the balance of forces 
impinging on the tractor: the tractor will ac- 



celerate if, and onl\' if, there is an unbalanced 
force on it. The force of the log on the tractor 
opposes the motion of the tractor, as does the 
friction in the mo\ing parts of the tractor and 
between the tractor and the grxjund. It is only 
when the force of the earth on the tractor be- 
comes greater than these retarding forces that 
it will begin to mo\e. 

Another wa\ of answering the question about 
wh\ the tractor mo\es is to sa\ that the force 
it exerts on the ground is greater than that 
which is exerted b\' the log. Therefore, the ac- 
celerating force of the earth is greater than the 
retarding force of the log. 

For a diflferent presentation of the tractor-log 
parado.x refer to Transparency T8. 



31. (al 




F,, = the force with which the earth pulls on 

the ball 
Fp = the force with \Nhich the ball pulls on the 

earth 
Ibi F, = m^a^ 

Ft, = il.OMlOi 

F^ - 10 .\ of force acting on ball 

Fp — 10 X of force acting on the earth 

Fp = rn^,a^ 

10 = 6.0 X lo'^i la^i 

a„ = 1.7 X lO"-"* msec" 



10 



.ci^ = 

a„ 1.7 X 10 



6 X lo'"" 



The ratio of the accelerations is just the in- 
\erse of the ratio of the masses. 




idi 



32. lai lii To accelerate a 75-kg person at 1.5 m sec" 
requires an unbalanced net force that has 
a magnitude equal to the product of mass 
and acceleration. 

m = 75 kg 
a = 1.5 m/sec" 
F„^, = 175 kgi 11.5 msec"! = 112.5 X 

Gra\it\ exerts a constant downward force 
on the person equal to weight F^^. In order 
that the person experience an upward ac- 
celeration, the ele\ator floor must e.xert an 
upward force F^ that is greater than the 



SUGGESTED SOLITIOVS TO STl D^ GUIDE PROBLEMS 



99 



weight. The net foi-ce will equal the excess 
of F^ over F^. 

F.,.., = F - F. 



F„ = ma^ 

F^ = 75 kg X 10 m/sec' = 750 N 
F„ = 112 N + 750 N 
F^ = 862 N upward 

(ii) The net force on any body is zero if it 
moves with constant \'elocity. Therefore, the 
elevator floor must exei1 an upward force F,. 
equal in magnitude to the person s weight 
F^. F^^ has already been found to be 750 \. 
(iii) A person accelerating downward expe- 
riences a net force downward. Again, the 
net force will equal the difference of F^ and 
F^. However, in this case, the person's weight 
F^ must be greater than the upward force 
F,. exerted by the elevator floor. 

F = F - F 

fne, = 112 N 

F^, = 750 N 
F^ = 750 N - 112 N 
F^ = 638 N upward 

(b) According to \'eu1on's third Uiw, for every 
force there is an equal and opposite force. 
When the elevator floor exerts a certain 
force on the pei-son, the person will in turn 
exeii an equal force I in the opposite dii-ec- 
tion) on the floor or scale. The bathroom 
scale would read the values calculated in 
(al for each of the three cases. 

(c) As a it;sult of the diffeitmt forces in the con- 
ditions examined above, it does appear that 
the pei-sons weight changes, since we am 
ac(-ustomed to associating weight with the 
force \\v. e.xeil against the floor lor \ice xei-sa 
according to \e\\1oii s thiid lawi. We should 
r(Mn(Miiber, howe\er, that since we defined 
weight as F„ = nm^, the actual weight does 
not chang(^ The appaitMit change was due 
to the accelerated fiame of i-efeivnce. 

Xi. Srv Test page 99. 

34. (a) F,..., = F„ - F, 

- 20 \ - 5 N 

- 15 ,\ light 



a = 



nel 

m 
_ 15 kg • m/sec" 

5 kg 
= 3 m/sec^ 
d = Vzat^ 

= '/2 (3 m/sec^i (10 sec)' 
= 150 m 

(b) F = F - F 

"' ' net ' 2 ' I 

= 1,200 \ - 200 \ 
= 1,000 \ 

F„, 

a = 

m 

_ 1,000 kg m/sec^ 

" 50Tg 

= 20 m/sec^ 

V = at 

= (2 m/sec") (20 sec) 

= 400 m/sec 

(c) Av = \', - \', 

= 80 m/sec — 40 m sec 
= 40 m/sec 

Av 
/ 

40 nvsec 



a = 



idiF 



a = 



10 sec 
= 4 m/sec^ 

F ^ ma 

= (4 kg) (4 m/sec^i 
= 16 N 
= F,-F, 
= 40 N - 15 \ 
= 25 \ 
A_v- 
t 
_ 55 m/sec 

11 sec 
= 5 ra'sec* 

F - ma 

F 

m = - 
a 

_ 25 kg • nvsec^ 

5 nvsec* 

= 5 kg 



F 

(e) a = — 
m 



_ 80 kg nvsec" 

~ 5kg 

- 16 m sec' 



I on 



IM r 1 / (XJNCKITS ()l ,M()TK),\ 



When m is reduced by one-half (2.5 kgi 
80 kg msec' 
2.5 kg 
- 32 nvsec" 
F...., 



(fia, - 



40 kg nvsec' - 4 kg ■ nVsec' 
= 12 m sec" 

_ ' nel 

m, 

40 kg ■ msec' - 4 kg ■ nvsec' 

= 4 m sec' 



a,„,a, = a, + a, 

= 12 nVsec' + 4 nvsec' 
= 16 nvsec' 
d = '/2 ar 
200 m = Vi (16 nVsec") t^ 
25 sec" = r 
5 sec = ^ 

= 18 N - 115 N + 3 N'l 
= 

F 

a — — 

m 

a = 



CHAPTER 4 



2. F^ _ = ma; when the rocket rises \ertically, 
F , = thmst - weight 

* nel ^ 

F_^^._ 7.37 X 10' - 5.4 X 10' X 9.8 

^ ^ ~i^ ^ 5.4 X lO'' 

a = 3.8 m/sec' = the acceleration at lift-off. 

d ^ Vzar As the fiael bums, the mass m 

,2 ^ rr decreases. Therefore, — 

decreases. 
2 X 50 

r 



4. when t 



10 sec 



t 



2 X 50 

= 26.4 sec' 

3.8 

5.1 sec 



(a)if v; = 4 msec, A.v = 
if V = 3 iTL'sec, Av = 



8 m 
6 m 



20 m 
15 m 



40 111 
30 m 



lb I total distance =V,v- + / = 10 m 25 m 50 m 

{C\ d 5 msec 5 msec 5 msec 

t 



\o, the bullet will not follow the line of sight 
along the barrel. It will start to drop as soon as 
it clears the end of the gun. 

Yes. the bottle will be hit. It and the bullet 
both fall with the same \ertical acceleration, a^. 
In an anahtical argument it will simplifv the 
algebra to assume that the line of sight along 
the gun barrel is horizontal: that is, that the 
gun and the bottle are initially at the same 
height abo\e the gix)und. Then it is quite easy 
to show that for an initial bullet speed \',, in the 
horizontal direction and an initial distance d,, 
beUveen gun and bottle it will take d,, \,, sec- 
onds for the bullet to reach the bottle. Howexer, 
during this time the bullet and the bottle will 
both be accelerating at the same rate xeitically 
and so both will fall a \ertical distance d, = 
V2a^r. Analysis of the moi^ general case when 
the" gun must be pointed at some angle with 
the hoi izontal reciuires lesohing the initial bul- 
let velocity into hoiizontal and \ertical com- 
ponents. Perhaps some of the better students 
max wish to do tiiis. 



5. The general equation for a parabola is y = a.v' 
+ b^ + c when a, b, and c are constants. In 
this case 
X = \\t andy = \\.t + y^a^r 

Since t = - we can substitute - for ( in the 
expression for y 

V = V. 1-) + V^aJ- 



„.^^).. f.,. 



Comparing this with the general equation for 
a parabola, we find that 



IV 



b = -, c = 



Since a^, \\. and \; are all constants, the trajec- 
toiy is indeed a parabola. 

B d, = v;f = 1.0 nvsec x 0.5 sec ^ 0.5 m 
The horizontal displacement therefoit- 0.5 m. 



SUGGESTED SOLITIONS TO STUDY GLIDE PHOBLEMS 



101 



a^ = 10 m/sec 

d^ = '/zl 10)10.5)^ = 125 m 

The veilical displacement is 1.25 m. 

d = Vdl + dl = Vo.25 + 1.56 = vTsT 
= 1.3 m 

The resultant displacement is 1.;} m. 

The direction of d lelative to the horizontal 
can he determined eithef hy measurement of 
the angle formed hv the 1.3-m hypotenuse uith 
the horizontal 0.5-m leg of a right triangle on 
a carefully drawn scale diagram, or by trigo- 
nometry, since 

« ^v 1.25 

tan = — = = 2.5 

d, 0.5 

Either method gives an angle of about 67° below 
the horizontal. 



V = VvfT^ 

v^ = 1.0 m/sec 

v^. = a^t = 5 m/sec 

V = Vl + 25 = Vie = 5.1 m/sec 

Either a scale diagram or trigonometry gi\'es an 
angle of about 79° below the horizontal. 

d 

7. (a) V = - 

/ 

d 
t = - 

V 

_ 25 m 

10 m/sec 
= 2.5 sec 
lb) d = Vzat^ 

= Vz (-9.8 m/sec^)(2.5 sec)^ 
= - 30.6 m 

(c) I he horizontal \elocity \\ imisl he large 
enough so that the xeriical distance d^ does 
not (!xceed 20 m: 

d^ = Vzat^ 
20 m = '/2(9.8 nVsec'l t^ 
2 sec = / 
d, 

_ ^^ "^ 
2 sec 
= 12.5 m/sec 

8. v? = vf + 2ad 

= ( - 1 m/secl^ + 2( -9.« m sec^i I -45 mi 
= - 29.7 nVsec 
V, = V. + at 
V. - V. 



— 29.7 m/sec - i - 1 m/seci 

-9.8 m/sec" 

= 2.93 sec 

9. lai d = Vzat^ 

80 m = '/2(9.8 m/sec n t^ 

4.1 sec = t 

(b) Since time of fall is independent of the hor- 
izontal \elocity, / does not change if \\ is 
doubled. 

Id v^ = at 

= 1-9.8 m/sec^)(4.1 seel 
= —40 m/sec 

(dl The horizontal velocits' v, = 8 m/sec. 

10. They increase at the appro.ximate rate of 1:3:5:7. 

11. (al The ball would mo\e straight down. 

ibi It would seem to travel along a parabola 
curving backward. 



t = 



;i 



(c) It would seem to tra\el along a parabola 
curving forward. 

(dl It would mo\e along a straight line tending 
toward the rear of the \an. the angle de- 
pending upon the magnitude of the \an s 
acceleration i-elative to a 

(el same as (b) 

(f I same as (c) 

12. The condition described could take place if one 
pei-son in a train traveling at a unifonii velocity- 
let an object dn)p to the flooe. lo that [jerson 
the path would be a straight line. .An outside 
obseiAer watching the train go b\ would see 
the object fall along a |)ath which, seen in that 
frame of n'leixMue, is a parabola. 

13. None of the gi\en alternatives describes the 
pilot's obser"vations 

lai The |)ilot will sec the bullets move awa\ 
eastAvaixl at 1,000 km hr 

(bi The pilot will see the bullets move awav 
westwaixl at 1,000 km hr 

(ci The pilot will see the bullets move straight 
down. In each of these cases, the plane s 
actual speed rvlative to the grDiind has no 
effect on the pilot s obser^ations. la'. ibi, and 
(cl would be the obsei-vations made when 
the earth is the frame of reference, 

14 f, = 1//, = 1/16.6 = 6.0 X 10 'min 

/j = l//j = 1/33.3 = 3.0 X 10 ' min 

t, = Vf, = 1/45 = 22 X 10 ■' min 

t . 1 f", 1 7.S 1 .< > 10 iiiiii 



102 



I'MT 



(:o\c:i;i»Ts oi \i(n'i()\ 



15. The passengers tend to mo\'e in a straight line 
at a unifomi speed, while the car is being ac- 
celerated b\' a centripetal foice towaixl the left. 
The door exerts a centripetal foix'e on the pas- 
sengers causing them to mo\e contran to this 
straight-line motion at a constant speed. I'he 
door is "thix)\\n against the passengers." The 
passenger's, of coui-se, e.xert an equal and op- 
posite force against the door. 

16. lai The loose surface may not be able to provide 

the force required to keep the car on the 
road. 
ibi Softer tii'es would gi\e a larger surface-con- 
tact area. Therefoi-e, less frictional force per 



square centimeter of road surface would be 
required. 

A banked road exerts a force on the car as 
a reaction to the cars weight and its speed 
as it travels on a cuned path. The foroe ex- 
erted perpendicular to the road surface 
now has a component directed inward to- 
ward the center of the cur\ e, thus providing 
pail of the required centripetal force. 

Note: A complete discussion of all the forces 
cind their angles relative to each other will 
be found on pp. 246-249, \ol. 1 of the book, 
Physical Science: Its Structure and Develop- 
ment by Edwin C. Kemble iM.I.l. Press, 
Cambridge, Mass. 19661. 



Name of 
Concept 


Syml 


Total distance 


d 


Displacement 


d 


Average speed 
or constant 
speed 


V 


Instantaneous 
speed 


V, 


Velocity 


V 


Acceleration 


a 


Acceleration of 
gravity 


h 


Centripetal 
acceleration 


a. 


Frequency 


f 


Period 


T 



Definition 



Example 



Length of a path between any two points 
as measured along the path 

The straight-line distance and direction 

Time rate of change of total distance 

The value of the average speed taken for a 
very small time interval. If the calculation 
is made for a smaller time, v, will not 
change. 

Time rate of change of displacement 

Time rate of change of velocity 

The acceleration of a freely falling body 



Time rate of change of velocity toward the 
center of a circle 

The number of complete cycles per unit of 
time 

The time it takes to make one complete 
revolution 



The speedometer reading recorded on a 
trip from Los Angeles to San Diego and 
return 

Straight-line distance and direction from 
Detroit to Chicago 

A car drives 8 km through traffic in 20 min: 
V = 26 km/hr 



Measurements from a high-speed strobe 
photograph of a pendulum show that Ad 
= 1.3 cm and Af = 0.10 sec. Thus, v^ = 13 
cm/sec. 

An airplane flying west at 640 km hr at 
constant altitude 

A car accelerates at 3 m/sec^ toward the 
north 

The acceleration of gravity in San 
Francisco is 9.800 m/sec' toward the center 
of the earth. 

A child on a merry-go-round 

The drive shaft of an automobile turns 600 
rpm in low gear. 

The period of a drive shaft turning 600 rpm 
is 0.1 sec. 



Note: Answers to be supplied b\' student are in bold t\pe. 



18. (al a,, = — 
R 



V = 2.5 X 10" m/sec 
fl = 13 X loMightyear) 19.46 x lo" 
m/llght year) 
= 2.84 X 10^" m 
_ 12.5 X 10" m/secr 

2.84 X 10'" m 
= 2.2 X 10 '" m sec" 



ibi F^ = ma^. 

m = 1.98 X 10^ kg 

a, = 2 X 10""'m/sec^ 
F, = 1.98 X 10^ kg X 2 X 10 "'"m/sec 
F^ = 4 X 10-" N 

mv' 



Icl F„ = 



H 



m = 5.98 X 10"^ kg 



SUGGESTED SOLUTIONS TO STUDTi GUIDE PROBLEMS 



103 



H = 1.495 X 10" m 

2TTrt 

T 

Ztt X 1.495 X lO" m 



V — 



V = 



F.. = 



1 yr X 365 day/yr x 24 hr/day X 3600 sec/hr 
V = 2.98 X 10^ m/sec 

5.98 X 10^ kg (2.98 X 10^ m/secl^ 
1.495 X 10" m 
F^ = 3.55 X 10^^ N 
Fp is about 100 times greater than F^. 

19. (a) From the photograph, the radius of the cir- 
cle seems to be about equal to the athlete's 
height, which can be estimated as 1.8 m. 
During the Olympic coverage of the ham- 
mer throw on lY, it appeared that the pe- 
riod of the swing was about 1 sec. 



F.. = 



m 4iT"fl 



7.27 X 4 X 9.86 X 1.8 



1 



= 5.2 X 10^ N 



Because of the estimations imolved, we 
give an order of magnitude \alue at 10"* N. 
lb) There must be an upward component of 
force sufficient to balance the downward 
force of gra\it\' on the hammer. Also, there 
would be some air resistance to overcome. 

20. Rectilinear motion is motion along a straight 
line. Example: a car mo\'ing along a straight 
road. The velocity at any instant will depend 
on the object's initial velocity and the length of 
time it has been subjected to an acceleration. 
In r-ectilinear motion, the only two possible di- 
rections for acceleration ai-e in the direction of 
the original velocitv' or in opposition to it. 
Projectile motion is the motion of a bod\' that 
is not self-pmpelled and that has Ikhmi launched 
with a specific initial \elocity and then comes 
under the influence of a gravitational force (its 
weight). 

Example: any object hurled into the air at 
any angle. If we neglect air resistance, as was 
done in this chapter, the object maintains a 
unifomi horizontal \elocity as long as it is in 
flight while being accelerated downwaixl. 
Uniform circular motion is motion at a con- 
stant speed along a circular path. 

K.xamphv a point on a n)tating turntable. .Al- 
though nuning at unifomi speed, the dii-ection 
of the ()l)je(t's v(»locit\ is continualK changing. 
At any instant the xclocitx' is dii-ected along a 
tangent to th(^ circular path at the location of 
the object. It is sul)jected to an acceleration al- 
ways at right angles to its diitution of motion: 
that is, l\w acceleralion is (lin'clcd touaixl the 
center of the ciniilar path I he acceleiation in 
this case does not spe«'d u|) or slow down the 
object but serMvs oiil\' to change its dii-ection 



21. (a) V = 



2itR 

T 
2ir(2 m) 



2 sec 
= 6.3 m/sec 



(b) a = — 
B 

_ (6.3 m/sec )" 

2 m 

= 19.8 m/sec^ 

ici F — ma 

= (2 kg)(19.8 m/sec^ 

= 36.9 N 



22. F = 



mv 



, _ FR 

m 



v" = 



15 kg • m/sec^)(S mi 



2.5 kg 
v^ = 10 m^/sec^ 
V = 3.2 m/sec 

x-^ 4lT^fl 

23.a = — = ;- 

fl T- 



r = 



47T'fl 



T- = 



a 

4-17' 13 ml 



10 m/sec" 
7'- = 11.8 sec' 
T = 3.4 sec 

The mass of the object is not in\ol\ed since 
force is not considered. 



24. a = 



4TT-fl 



1 



— 


T 








1 




0.1 


sec 


= 


10 


sec 



7- 
_ 4tt-^ (0.5 ml 

(0.1 seer 
= 1,970 nvsec' 

25. (al S\T«com 2 has the most nearl\ circular ori)it 
since the distance from the surface land 
also fixjm the earth s center' varies b\ onl\ 
8 km. If the radius of the earth is taken as 
6,400 km, this is a difference of only about 
02% 18 km 35,520 km x 100% = 6.02%. i 
(bi Without actualK calculating the eccentric- 
ity, it would be reasonable to estimate 
which satellite has the greatest pereentage 
variation in its greatest and least distani^e 
tmm the rvnWr of the earth. This is l.unik 
3. 

Eccentricit\ is explained in detail in I'nit 
2. The actual calculations for the two in()>t 
obviouslv eccentric satellites an* 



I04 



I MI I ' c:c)\c;i':in"s of iviotiox 



Lunik 3 e = c/a = 208,800 263,200 = 0.80 
Luna 4 e' = c'/a' = 303,200 399,200 = 
0.76 

\ote that the percentage of \ ariation is the 
ob\ious method. 

(ci Luna 4 

(di The earth rotates once in 1,440 min and 
Syncom 2 orbits once in 1,460 min. If the 
satellite begins directh- o\erhead, it uill be 
onl\- 5° to the west in 24 hr. That is, it will 
take 20 additional minutes to reach the po- 
sition directly o\erhead. 

The following relationships between de- 
gree and time measurement were used: 

24 hr = 360° 

1 hr = 15° 

20/60 hr = 5° 

It is recommended that star time and 
sun time" not be discussed. The point of 
the problem is to appreciate a near-syn- 
chronous oi-bit. 



29. 



26. a, = 



v^ 



R 

v" = a^R 

v-^ == iS.Tmsec'i i6.8 x 10*^ ml 
\'^ = 59.2 X 10*" m- sec" 
\- = 7,690 m/sec 
The mass of the satellite is not important. 



4TT-R 



— ■ But a^ = a - 9.8 m/sec" at the 



earth s surface. 

(4) (9.9) (6.38 X lO^mi 



9.8 rrLseC = 



T- 



r = \ 26 X 10" sec- 
r = 5.1 X 10^ sec 
T = 85 min 



a, — ~ 
R 



9.8 



6.38 X 10 



\' = V62 X 10 

\ = 7.9 X 10^ m/sec 

30. A satellite is held in its orbit only b\- the pull of 
gra\it\'. ,As problem 29 shows the shortest pos- 
sible period for a satellite is 85 min. A shorter 
period would require a centripetal acceleration 
greater than that of gra\it\ . 
Yes, it is impossible. 

31.fi = 110 km + 1,730 km moon radius = 1,840 
km 
fi = 1.84 X 10** m from the center of the moon 
a^ = 1.43 m sec" is the acceleration 110 km from 
the surface of the moon. 
Then: 

4-'fi 



27. F = 



47T-mfi 



1.43 



R T- 

_ 477- 1500 kgi 118 X 10^ ml 

122,800 seer 
= 683 N 



28. This problem is the same as question 27, e.xcept 
that the center of motion is the moon rather 
than the earth. Therefore, the central i acceler- 
ating' force is one-sLxth that of earth las found 
in question 27i: 

F = -(683 Nl 

6 

= 114 N 
For the same distance R 118,000 kmi a^^^„ — 



(aT-i 



moon 

llil380min)^ 

T- = 

moon -* ,r* 

lb 
r = 931 min 



i4i 19.91 (1.84 X 10 I 
T- 



i4i i9.9i 11.84 X 10 I 
1.43 



r = V 51 X 10 

r = 7.1 X 10^ sec or 7 = 1.2 x lO" min 

32. Given: 

a at moon's surface =^1.5 msec" 

a at 100 km from moon's surface = 1.43 m sec" 

d^ = 1.0 X 10" km = 1.0 X lO" m 

V, = 1.0 X 10" msec 



(a) d, = ^/2a^r 

lio X lo" = IV2 ) (1.51 r 

1.0 X 10" r J 

t = = \ 1.3 X 10' = \ 13 X 10 

\ 0.75 

f = 3.6 X 10" sec 
(bi d^ = v^t 

d^ = il.O X lO'l (3.6 X 10"! 

d^ = 3.6 X lO"* m 

d, = 36 km 
Ici All that can be estimated is that the braking 

must start at a distance greater than 36 km 

from the landinij target. In order to answer 



SUGGESTED SOLLTIOXS TO STID\ GLIDE PROBLE.MS 



105 



the question in more detail, one would have 
to gather the following intbimation: 

(1) lunar preoihital injection speed at 
100 km 

(2) thrust value of the engines in nev\1ons 
13) the mass of Apollo 8 

(4) the desired time of burning in sec- 
onds 

33. Given: 

Vp = speed necessary for orbit 
V = preinjection speed 
F = thmst 
m = mass 

To calculate time, It, for engine to burn 

v., - V 



a = 



At 



At = '-^^^ 
a 

F = ma 

a = F/m 

V — V 

At = ^ 

F/m 

At = im/Fi (v„ - VI 



34. 1. Simplest motions 

Ic) car going from 50 km/hr to a complete 
stop The car mo\es along a straight line. It 
is not clear, hovve\er, that acceleration is 
constant. V\'e will assume that it is. 

If) rock dropped 3 km It moves along a straight 
line with a constant acceleration. Assume 
that air resistance has little effect. 

(g) person standing on a moving escalator He 
or she mo\es at a constant speed in a 
straight line. 

2. More complex motions 

Ibi "human cannon ball' in flight This is an ex- 
ample of pr-ojectile motion. IchuilK , the path 
is a parabola and th(> \elocity changes in 
magnitude and diivction. Ibis motion as- 
sumes that the horizontal component of the 
\(!locity is constant, which ma\ not be e\- 
actU true due to air friction. 

(e) child riding a ferris wheel We assume that 
the child tra\(>ls in a ciicle at a constant 



speed. The magnitude of the acceleration is 
constant, while its direction changes uni- 
formly. 

(i) person walking The motion may ha\e a reg- 
ular rh\1hm and may be in a plane parallel 
to the earth. Howe\er, the direction and 
speed components of velocity are contin- 
ually changing. 

(hi climber ascending Mt. Everest The \elocit>' 
will undergo man\' complicated changes. 

3. Ver\' complex motions 

la) helicopter landing This is a complex motion 
when one considei-s; the motion of each ro- 
tor; the motion of the helicopter as a whole. 
The rotors exhibit unifomi circular motions 
at right angles to one another. Each rotor 
exerts a controlled force on the vehicle. The 
velocity of the whole helicopter is onl\ at 
times constant and in straight lines. 

id) tree growing If this motion is considered 
for a short time period, such as 1 sec. it is 
a simple motion. Howe\er. if the motion is 
considered o\er a long time period of 25 \t, 
the motion is complicated. 

iji leaf falling from a tree This is the most com- 
plicated motion. The mass is so small that 
frictional and gravitational forces will pro- 
duce large and \aried accelerations. .An ad- 
ditional complication lies in the three- 
dimensional nature of the motion due to 
wind and tumbling effects. 

35. Some ideas that might well be included in the 
essay ave. 

1. Identification of pi-obable details of how 
the photograph was made. For example, 
how was the camera shutter controlled? 
Was the shutter open for a long or a short 
time? Was the shutter opened more than 
once to pix)duce the final photo' If more 
than once, w hat was the prxibable orxler of 
magnitude of time between exposures.' 

2. How can each motion be identified as ex- 
amples of unifomi velocity and or- uniform 
acceleration? 

3. What forces seem to be acting in each 
case? 

4. Could this picture be inter^Jivted in morv 
than one wav? 



I (Mi 



i!\iT 1 / coxcKms oi \i()rn)\ 




lUlolion in the Heauens 



Organization of Instruction 



THE ML1.TI-MEDIA SCHEDULE 



Day 1 

Make assignments for student debates to be held 
on Da\ 6. Vou will need debater's, timekeepers, and 
judges. This is a good opportunit\ to work with the 
English department. 

Daj 2 

Small groups plot data from El-1 and discuss 
questions from the e,\periment. If data were not 
available, use the data pro\ided in £2-1. 

A \isit to a planetarium or an evening star-gazing 
session would he useful. 

Daj 3 

Teacher presentation on Aristotle and Plato. Pre- 
sent and discuss the scientific and philosophical 
\iewpoints of the ancients. 

Oi-ganize debate acti\it\- for Da\ 6. 

Day 4 

Lab stations: Ptolem\ 

1. EpicNcle machine Handbook, Activities section 

2. D28 I phases of the moon 

3. Film stiip, Retrograde Motion of Mai's" 

4. LlO 'retrograde motion — geocentric model) 

5. Celestial sphere ^Handbook. Activities section' 

6. Making angular measurements ^Handbook, Ac- 
tivities section! 



Design activities so that students either mov e from 
station to station or select one station. 

Day 3 

Class discussion: Ptolemv and Copernicus 

.Answer questions that will arise with regaixi to 
geocentric and heliocentric celestial mechanics. 
T13 and T16 should be helpail. 

Daj 6 

Student debate: The natui-e of the universe as de- 
scribed bv Ptolemv and Copernicus. Students should 
present both viewpoints in standard debate form. 

Day 7 

Students coUectivelv do E2-6. "The Shape of the 
Earth s Orbit. Several students read measure- 
ments of solar diameters from projected photo- 
graphs and ever\ student makes an orbit plot. 
Large sheets of graph paper are v eiA helpful. 

Day 8 

Some students can assist the teacher in running 
help sessions to clear up all questions regai-ding 
E2-6. 

Some students can work in small problem-solv- 
ing groups 



ORGA.VIZATION OF INSTRLCTIOV 



107 



I3ay 9 

Divide class into small groups to discuss Study 
Guide questions. Circulate among these groups. 

Take about 20 min to summarize Chapters 5 and 
6. Explain evaluation procedure. 

Day 10 

Give a quiz and then discuss it. 'Some other eval- 
uation procedure as indicated in notes for Days 
22-24 may be used instead.) 

Day 11 

Lab stations: Kepler 

1. T17 loHiit parameters) 

2. L11 (retixjgrade motion) 

3. E2-7 (using lenses) 

4. rhi-ee-dimensional model 

5. U30 (heliocentric model) 

6. Drawing ellipses luse pins and string) 

Day 12 

Teacher presentation or class discussion 
Possible discussion topics: 

1. models of the universe (Aristotle to Kepleri 

2. the changing nature of physical laws 

3. separation of celestial physics and terrestrial 
physics in history 

4. Kepler's law 

Day 13 

E2-8 (orbit of Mars) 

Students plot Mars' orbit on the graph the\' 
made on Day 7. 

Day 14 

Teacher can discuss with the class the details of 
E2-8. In addition to answering questions and giv- 
ing indi\idual help, point out some possible choices 
for student activities on Days 20 and 21. Refer to 
Days 20 and 21 for ideas. 

Day 15 

Divide the class into small pi-oblem-sol\inggixJups. 
They might discuss the assigned pnjblems, work 
on othei-s of their ouii choice, help one another-, 
or- work independently. Gi\e concr-ete help to each 
group as you cir-culate. Teach to the point of a spe- 
cific (|irestion. 

Day 16 

Lab stations: \e\\1on 

1. T18 (motion under central forces! 

2. L12 (JupitiM- satellite orliit) 

3. Ph()l()mcti> : With a light meter- measur-e inten- 
sity at various distaru-c's from a small light 
source. 

4. Radioacti\'ity: Measure? intensity of radiation at 
various distances. 

5. Soirrul; Micr-ophone and ampliti(<i- ilri\e a \u 
meter-. Measuri; intensities at \arious distances. 



Suggestions 4, 5 and 6 are intended to illustrate the 
inverse-square law. (Consult Unit 3 for details of 
apparatus.) Arrange equipment so that students 
may stay in one group ail period. 

Day 17 

Students demonstrate activities carried on during 
Day 16 and show results to class 

Day 18 

Teacher presentation: The \e\\1onian svTithesis 

At this time Kepler's laws, Galileo s observations, 
and terrestrial physics are combined into one law. 
See Holton and Roller, Foundations of Modern 
Physical Science. Chapter-s 11 and 12: Kemble. Phys- 
ical Science, Its Structure and Development, Chap- 
ter 9; and Andrade, Sir Isaac i\e\\ton. 

Day 19 

By equating the centripetal force to the gravita- 
tional foix^e, show how one can calculate the mass 
of Jupiter. This is a stailling achievement of New- 
ton's work. Refer to Test page 228. 

Organize optional activities for the ne.xt two 
days. 

Day 20 

Student option 

In small groups or individually, students may 
plan their own activity for this day. Possibilities in- 
clude: 

1. E2-2 (size of the earlhi 

2. E2-4 (the height of Pitom 

3. E2-7 (using lenses to make a telescope) 

4. E2-11 (stepwise appro.ximation to an orbit) 

5. E2-9 I Inclination of Mar-s orl)iti 

6. E212 iModel of the or4jit of H alley s comet 

7. L12 iJupiter satellite or-ljiti 

Day 21 

Student option 
Some possibilities: 

1. field trip to a planetariirm 

2. essav about I'nit 2 topics 

3. PSSC Film #0309 'Universal Gravitation fol- 
lowed bv discirssion 

Days 22-24 

One method of evaluation is to rwiew, test, and 
discuss the test. Devote a dav to each acthitv . 

.Another- method of evaluation is thrxiirgh indi- 
vidual stirdenl-tt'acher conferences dirring a pe- 
riod of thrve davs. F.valiration can be based irpon 
laboratory r-eports, essavs poems, equipment de- 
sign, sets oi Study Guide answer-s. etc 

Note that two of these thrve davs of testing could 
be done at other times tlirring the 24 davs. 



I\n 2 / >VIOTI()\ IN THK IUvWIiNS 



Unit 2 SAMPLE MULTI-MEDIA SYSTEMS APPROACH 

Note: This is just one path of many that a teacher may take through Unit 2. In this 
system, the teacher is a manipulator of environment and a tutor. 



1 

Small-group discussion 



Exchange and plot 

data from E2-1 : 

Naked-Eye Astronomy 



Text: Prologue to Unit 2 
Handbook: Survey Ch. 5 



Teacher presentation: 

Plato's 

and 

Aristotle's Views 



Text: 5.1-5.4 



Text: 5.5-5.9 



Lab stations: 
Ptolemy 



Text: 
6.1-6.5 



Class discussion: 

Ptolemy 

and 

Copernicus 



Student debate 



7 

Filmstrip 

demonstration 

E2-6, The Shape of 

the Earth's Orbit 



Prepare for debate 



Handbook: E2-6 



8 

Help session on E2-6 

or 

Problem-solving 

session 



Finish orbit plot 
Text: 6.6-6.8 



Selected 
Study 
Guide 
quest. 



9 

Small-group 

problem solving 


10 
Quiz on Ch. 5 & 6 

or 
Other evaluation 


11 

Lab Stations: 

Kepler 


12 

Teacher presentation: 

Brahe versus Kepler 

Kepler's laws 




Review previo 
in Unit 2 


us work 


Handbook 
Survey Ch. 7 




Text: 7.1-7.4 




Bring earth 
plot to class 
Text: 
7.5-7.9 



13 
E2-8 Orbit of Mars 



14 
Class discussion 

on 
E2-8 Orbit of Mars 



15 

Small-group 

problem solving 



Handbook: E2-8 
Finish Mars plot 



Selected Study Guide 
questions 



Text: 8.1-8.4 



16 

Lab stations: 

Newton 



Handbook: 
Survey Ch. 8 
for options 
days 20 & 
21 



17 

Demonstrations 

from day 16 



18 

Teacher presentation: 

The 

Newton Synthesis 



Text: 8.5-8.7 



19 

Teacher presentation: 

The Mass of Jupiter; 

and Organization 

of Student option 



20 
Student option 



Text: 8.8-8.10 



Prepare for optional 
activity 



Student 
assignment 



21 
Student option 



22 

Review 

or 

Other evaluation 



23 
Test 

or 
Other evaluation 



24 

Discuss test 

or 

Other evaluation 



Text: Unit 2 Epilogue 
Review Unit 2 



Review Unit 2 



ORGAMZATIO.V OF IVSTRLCTIO.V 



109 



Unit 2 SUGGESTED SCHEDULE BLOCKS AND TIMETABLE 

Each block represents one day of classroom activity and implies approximately a 50-min period. 



CHAPTER 5 WHERE IS THE EARTH? THE GREEKS' ANSWERS 

Text: Prologue 
HB: Survey Ch. 5 Text: 5.1-5.4 



Small-group 
discussion 



Lab E2-1: 
Naked-Eye 
Astronomy 

Lab E2-3: 
Distance to 

the Moon 



Teacher 

presentation: 

Plato 

and 

Aristotle 



Text; 5.5-5.9 



Lab stations: 

Ptolemy 
(See day 4.) 



CHAPTER 6 



Text: 6.1-6.5 



Class discussion: 

Ptolemy 

and 

Copernicus 



Prepare debate 



Student debate 

on 

Ptolemaic 

versus 

Copernican 

models 



HB: E2-6 



Lab E2-6: 

The Shape of 

the Earth's Orbit 



DOES THE EARTH MOVE? 

Text: 6.6-6.8 Selected SG Quest. 



Discussion; 
Lab E2-6 



Small-group 
problem solving 



Review 



Quiz 
Ch. 5 and Ch. 6 

or 
Other evaluation 



HB: Survey Ch. 7 



Lab. stations: 

Kepler 
(See day 11.) 



CHAPTER 7 A NEW UNIVERSE APPEARS 



Text: 7.1-7.4 



Teacher 

presentation: 

Brahe versus Kepler 

Kepler's Laws 



Bring earth plot 
Text: 7.5-7.9 



Labs E2-8 and 

E2-9: 
Orbit of Mars 



HB: E 2-10 The Orbit 
of 
Mercury 
Finish Mars Plot 



Class discussion: 
Lab E2-8 



Selected SG Quest. 



Small-group 
problem solving 



Text: 8.1-8.4 



Lab stations: 

Newton 
(See day 16.) 



CHAPTER 8 THE UNITY OF EARTH AND SKY: THE WORK OF NEWTON 

HB; Survey Ch. 8 Text: 8.5-8.7 Text: 8.8-8.10 Student assignment 



Student 
demonstrations 



Teacher 

presentation: 

Newtonian 

Synthesis 



Teacher 

presentation: 

Mass of Jupiter 

and 
student options 



Student option 



Student assignment 



Student option 



Text: Epilogue 
Review Unit 2 



Evaluation 

or 

Review 



Review Unit 2 



Evaluation 

or 

Test 



Evaluation 

or 
Discuss test 



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T 14 Celestial Sphere 

F 6 Universe — NASA (Prologue) 

F 7 Mystery of Stonehenge — McGraw-Hill 

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E 2-2 Size of the earth 

E 2-3 Distance to the Moon 


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114 



UMT 2 / IVUnUlIM IIV THb HEAXliMS 



Back^ound and Dev^elopment 



o\t:r\ie\\' of uxit 2 



Unit 2 is a brief story of the physics that de\ eloped 
as people attempted to account for the motions of 
hea\enl\ bodies. It is not a short coui"se in astixjn- 
omy. The pixjlogue to Unit 2 gi\es a brief o\enie\\ 
of the unit. 

The clima.x of the unit is the work of \e\\1on. 
For the first time in histon . scientific generaliza- 
tions to explain earthh e\ents were found to apply 
to e\ents in the hea\ens as well. This remarkable 
SN'nthesis, summarized in Chapter 8, produced 
echoes in philosophy, poetrv', economics, religion, 
and e\en politics. 

The early chaptei-s are necessaiy to establish the 
nature and magnitude of the pixablem that Newton 
solved. They also show that obsen ational data are 
necessary to the gixDwth of a theory . I bus. Chapters 
5, 6, and 7 are a prelude to Chapter 8 and consti- 
tute a case history in the de\elopment of science. 

Chapter 5 constructs a model of the uni\ erse 
based upon the kinds of obser"\ ations made by the 
ancients, the Greeks, and b\ people in modern 



times. It describes the motions of sun, moon, 
planets and stars as seen tram a fixed-earth frame 
of r-efer-ence. It r-elates Plato s model and Ptolemy's 
geocentric uni\er-se. 

Chapter- 6 describes the work of Coperrricus and 
Tycho Brahe, discusses the arguments that devel- 
oped, and cites the historic consequences of this 
radical \iew of the univer-se. Ihe diligent observa- 
tions and r-ecords kept b\' Tycho show the impor- 
tance of such work to science. 

The work of lycho s successor, Johannes Kepler, 
and that of Galileo are r-elated in Chapter 7, as well 
as the world-shaking consequences of these works, 
which changed the coui-se of r-eligion, philosophy, 
and science. 

Chapter 8 presents a profile of Newton, the in- 
di\idual, and an insight into the tremendous 
power of the syiithesis of earthly and celestial me- 
chanics. The cement of his synthesis, the law of 
universal gravitation, is developed, and several 
tests of the law are discussed. 



CHAPTER 5 / WHERE IS THE EARTH? THE GREEKS' ANSWERS 



5.1 MOTIONS OF THE SL^^ AXD STARS 

\\ hen the change from the Julian to the Gregorian 
calendar was made in England in 1752, September 
2 was follow ed by September 14 for a connection of 
11 davs. Manv peasants are reported to have 
claimed thev Wanted their eleven davs back. 

George \\ ashington was actuallv born on Feb- 
ruary 11, 1732, according to the Julian calendar 
then used bv the British. Scholars must be car-eful 
to distinguish Julian lOld Stvle' dates frx)m Grego- 
rian I New St\lei dates on original documents from 
the latter half of the eighteenth century . 

There are 88 official constellations. Bv interna- 
tional agreement all the boundaries have been de- 
fined along north-south or east-west lines, al- 
though older star maps show curved boundaries. 

In the Test we have avoided r-eferring to the zo- 
diac and to sidereal time. Sidereal time is star time 
that gains on mean solar time by 3 min 56.6 sec 
per day, due to the motion of the earth about the 
sun. 

Students may wish to read Stonehenge " by Jac- 
quetta Hawkes ^Scientific American. June 1953. \ol 
188, No. 6i or Stonehenge Decoded bv Gerald Hawk- 
ins. Doubledav. 1965 



5.2 MOTIOXS OF THE MOOX 

.As recentlv as the Revolutionary War- befor-e ma- 
rine chronometer-s wer-e developed to keep accu- 
rate time at sea, navigators depended upon the 
position of the moon among the star-s to determine 
their longitude. Because the position of the moon 
among the stars changes rapidlv , a pr-ecise obser- 
vation of the moon could be used for this purjjose. 
The procedure, knowrr as lunar-s, is mentioned 
in various historical novels as well as in more of- 
ficial documents. 

Longitudes shown on old maps are often much 
in error. This reveals the difficulties of fi.xing the 
longitude of a place. The need for precise predic- 
tions of the moon s position greatly stimulated as- 
tronomical observations and theorv'. 

The motions of the moon could be predicted 
mor^e accuratelv after Newton s studies of gr-avita- 
tion. Today many of the very small residual varia- 
tions are forecast from cor-rections based on ob- 
servations of eclipses and the orbits of earth 
satellites rather than from theory. 

5.3 THE "WAXDEREVG" STARS 

Sections 5.1 5.2 and 5 3 review the basic observa- 
tions to be explained bv a theory If the students 



BACKGROrVD WD DEXTiLOP.MEVT 



115 



can complete the following table correctly, they 
know the major motions to be explained. 



Motion 



Stars Sun Moon Planets 



Daily motion from eastern 

horizon to western horizon x x x x 

Generally move eastward 

among the stars x x x 

Moves N-S-N while 

moving eastward x x x 

Moves N-S-N in one year x 

Moves N-S-N in one x 

month 

Retrograde motion x 

5.4 I PLATO'S PROBLEM 

In Archimedes in the Middle /\s,cs, Vol. 1, Uni\'. of 
Wisconsin Press, iyei4, Marshall C:lagKtt re\'iews the 
histoiy of the manuscripts axailable on the works 
of Archimedes. Because Archimedes was one of the 
giants of Hellenistic Gr-eece, we might expect that 
many manuscripts would be available. However, 
modem knowledge of Aix;himedes is actually based 
on three Byzantine (ireek maniisciipts, two of 
which were copied in the Middle Ages, but are lost. 

The discovery of the third manuscript in 1906 by 
Heilberg is itself an exciting slo\y. This manuscript 
is of the type known as palimpsest: that is, a parch- 
ment used a second time after the initial writing 
has been erased. The first wiiting can be revealed 
with iiifrartid light. Careful scientific- detecti\e work 
by H(;ilb(!rg rexeahul that a religious tract cover-ed 
the [irecious copy of Ar-chimc^des Oil the Method. 
previously not available. 

Some pirpils might wiah to in\esligate how dates 
are derived for- old manuscripts and what means, 
in addition to infr-ared light, are used to stirdy 
them. 

5.5 I THE GREEK IDEA 

OF "EXPLANATIOIV" 

It may hv. visciul to refcM- back to these thr-ee fea- 
tures of an explanation as the course proceeds. 

In Unit :{, the kinetic-moUuarlar- theory of gases 
is developed in a \ ery similar- wa\ . 1 bat discirssion 
explains th(! natuit; of a gas b\' summarizing the 
laboratory obser-\ations of its prx)perties in a series 
of mathematical statements. Each statement is 
then "(explained ■ by inxenting assirmptions about 
gases that luv. ihv. simplest possibU? ones (-onsist- 
enl with the fac-ts. This (-ollection of assum|Jtions 
finally suggests an imaginary model of a gas which, 
if coir-(MM, should (whibit all tln^ pr-opcities that ar-e 
aciirally ()bser\(Hl. Ihc model siiouid also sugg»\sl 
pr-operties of gas(\s not yvl obs(>r\('d. If liritber- ex- 
pt-rimenl confir-ms thes(' suggested pr-()|)eiti»'s. our- 
iailh in tin* accurac\ of the model is irurx'ased. 

Ihe study of ele(-tricit\' and magnetism in I'nit 
4 leads to the ('lectr-omagiK'tic Ibeor^ in a similar- 
wa\ . in Unit ,■> a lik(> serii^s of ste|)s leatls to an 
understanding of the outer- stiuttuie of tbi< .ilom 



5.6 I THE FIRST EARTH-CE.VFERED 

SOLLTTIOX 

To the ancients the eailh was \'eiy large, immobile, 
and at the center of all motions. It seemed easy to 
explain the motions of the fixed stars with the 
earth at the center. It is mentioned that Eudoxus, 
Plato s pupil, needed only 26 spheres to explain 
the general obser-vations. Aristotle added 29 more 
mainly to pro\ide enough motions to account for 
the \'arious cycles observed. Thus, as more cycles 
wer-e included for gr-eater pr-ecision. more motions 
were needed. 

From our point of \iew, one big drawljack of the 
eai1h-(;enter-ed scheme was its failur-e to preciseK' 
predict the positions of planets in the sk\ . But 
Greek science had dilTer-ent purposes than modem 
science; its theor-ies were, at first, onl\ intended to 
account for the gener-al changes observed. The de- 
sire for gr-eater precision came later. 

Students should under-stand that it is impossible 
to describe the theory' of the Greeks. Ther-e were 
many \ariations. Plato believed that the earth was 
spherical because of the shape of its shadow 
thromi on the moon at lunar eclipses Heraclides 
of Pontus, who, like .Aristotle, was a pupil of Plato, 
believed that the earlh was at the center- and r-o- 
tated while the heavens were at rest. 

Students will prxjbabh be amazed to find that in 
the thir-teenth centirr^ most astr-onomical expla- 
nations were still those of Grvek antiquit\. To elab- 
orate is to trace western civilization: perhaps some 
students will want to pr-esent a capsulated history' 
to the class. Or perhaps a student will want to ex- 
[jlain to the class how Dante in the Divine Comedy 
11300 A.D.I described the spherical eailii in the cen- 
ter of the wor-ld and the planets and star-s moving 
in celestial spher-es. 

The Grvek view of the ar-rangement of the planets 
has come down to us in the names of the days of 
the week. Students might note that manv of the 
names we use ifor example, Thors dav i reefer to 
characteristics of varioirs gods and goddesses in 
the Cireek and leirtonic rmthologies. Langirage 
students will note that the dav names in French. 
Spanish, and Italian art* close to the original GrT^ek 
r-rames. 

5.7 I A SnV.C:EXTERF:iJ SOLITIOX 

rhrx)irgh the Almngesl. which cirvulated among 
scholar-s and students in the Middle .Ages, the itiea 
of a heliocentric system was known. Gopemicus 
tried to defenil a sun-centeriul svstem isee Ghapei- 
6) and to refute the argument of Ptolenn in the 
Ahn.if^est. 

Ihe drawing on Te.\t page 146 illustr-ates retro- 
grade motion. It shows the earth and a planet mov- 
ing in cir-cular or-liits ar-ound the sun Both move 
at constant speed but in this example the earth 
moves faster. When the earlh is at points 1 and 2 
the pr-ojection of the line of sight below shows that 
the planet apfiear-s to bi- moving to the right, ahead 
of the earth or eastwarxl Between points A and o 



UNIT 2 / IVIOTiON l\ THli HK,\\Ti\8 



the \elocir\' of the earth peipendicular to the line 
of sight becomes greater than that of the planet, so 
the planet appears to be nio\ing backwaid iwest- 
ward or retrograde!. At points 6 and 7, the eai-th s 
velocity perpendicular to the line of sight again 
becomes less than that of the planet i although its 
actual speed remains unchanged, of coui-sei and 
hence the planet is seen to be mo\'ing eastward 
again. See Transparency T15. 

See Additional Background Articles for a note on 
the sizes of and distances to the sun and moon b\' 
Aristarchus. 



5.8 I THE GEOCENTRIC SYSTEM 

OF PTOLEMY 

During the 500 \'ears between Plato and Ptolemy 
the Greeks had made great achie\ements in ge- 
ometiA . Ptolemx applied some of these results in 
his attempt to find wa\'s to predict the positions 
of the planets precisely. He was willing to sacrifice 
Plato's assumption of uniform angular motion 
around the centers of circles for greater precision 
in his predictions. Emphasis was upon the longi- 
tudes, or positions along the ecliptic, rather than 



upon the latitudes, or positions peipendicular to 
the ecliptic. The latitudes could be predicted, at 
least roughly, by tilting the planes of the epicycles 
a bit fixjm the plane of the ecliptic. 

The intent is not to sress the details of the \ ar- 
ious geometrical de\ices used b\ Ptolemy, but 
rather to indicate his abilir\' to introduce man\ dif- 
ferent tspes of motions to satisf\ the increasingh' 
more precise obseiAations. To satisfy a \ariety of 
cycles found in planetaiy motions, Ptolenn intro- 
duced a xarietA' of geometrical models: the eccen- 
tric, epicycle, and equant. He used geometry 
to soke problems for which we would now use 
trigonometric equations composed of terms 
containing sines and cosines of \arious angles. 

See Additional Background Articles for notes on 
epicycles. 

5.9 I SUCCESSES AND LLMITATIOXS 

OF THE PTOLEMAIC MODEL 

In summaiA , it appears that the Ptolemaic model 
meets the requirements for successful e.xplanation 
discussed in Sec. 5.5. E\en befoi-e starting the ne.xt 
chapter it may be worthwhile to consider wh\' this 
apparently successful model is no longer accepted. 



CHAPTER 6 / DOES THE EARTH M0\T:? 
THE WORK OF COPERNICUS AND TYCHO 



6.1 1 THE COPERXICAN SYSTEM 

Since both the Copemican and the Ptolemaic s\'s- 
tems had to account for the same obseiAations. the 
two systems had about the same number of mo- 
tions. Copernicus also used Ptolemy's numerical 
constants, which described the magnitude of the 
motions. Consequenth , the Copernican system 
was no more precise than the Ptolemaic system it 
proposed to replace. But increased precision was 
not i^all\ Copernicus piimaiA intention. He wished 
to purifv the model, to describe all the motions on 
the basis of combined unifonn circular motions. 

Copernicus had been requested by Pope Paul III 
to assist in the i"eform of the calendar, a procedure 
which resulted later in the Gi^gorian calendai" in 
current use. But Copernicus declined, claiming 
that a better calendar should be based on an im- 
proved system for pi-edicting celestial events. Some 
idea of the complexit\' of forming a calendar for 
ci\il and religious puiposes is included in the ar- 
ticle Calendar in the Encyclopaedia Britannica. 

About 1512 Copernicus prepared and circulated 
to a few friends the Commentariolus, which was a 
sketch of his pix)posed system. I hixjugh it a num- 
ber of people learned a bit about the ideas he was 
developing. Later in his Bcvolutionibus Copernicus 
made some changes in the argument and added 
other, small cyclic motions, perhaps as a result of 
criticisms ftom his friends 



6.2 I NEW CONCLUSIONS 

The orbital distances for Mercurv and \ enus were 
found from the maximum angular elongations 
from the sun. The orbit of \'enus is almost circular. 
An optional experiment, E2-10, uses such obser- 
vations to \ield an orbit for Mercurv'. 

As the table below suggests, students can work 
out their own approximate values from the dia- 
gram on Text page 163. iThe data are not intended 
to be exact. I 

Derivation of Planetary Orbits 





Diameter 




Radius* 






of 




of 






Epicycle 


Ratio 


Deferent 


Radius 


Object 


cm 


Sun/Planet 


cm 


AU 


Sun 


2,00 


1.00 


1.00 


1.0 


Mars 


3.25 


0.61 


2.62 


1.6 


Jupiter 


2.15 


0.93 


4.40 


5.0 


Saturn 


1.45 


1.38 


7.25 


10.0 



"The diameters of the deferents are not shown in the figure. 

While the details of these calculations will inter- 
est some students, it is more impoilant for all stu- 
dents to realize that the heliocentric model al- 
lowed such results to be obtained for the first time 
in histoid'. 

At the end of this chapter we raise questions 
about the reality of these orbits. CertainK they 
seem much more 'real than the computing de- 



BACKGROLND A.ND DE\TiLt)P.ME\T 



ii: 



vices used bv Ptolemy or the transparent crystal- 
line spher-es proposed earlier. 

6.3 I ARGIT^VIENTS FOR THE 

COPERXICAX SYSTEM 

In stjriie ways, th(! (lojjcrriican s\'st(Mn was rela- 
tively simple, but its details wert; just as complex 
as the Ptolemaic system. The simfilicity is essen- 
tially aesthetic or- philosophical: that is, the basic 
idea as shown in the bottom diagram on Ic'sl page 
1B3 is simple. Vet the ('omputations needed to 
make pr-ecise predictions vverj; just as complex as 
ever". In this "messiness' lay the motivation of Iv- 
cho Brahe and of Kepler- to lind a simpler- model. 
Useful background r-eading would be Chapter- 1 
of The Origins of Modern Science by Her-l)er-t But- 
terfield (Free Press, NY; 1951). 

6.4 I ARGUMENTS AGAEVST THE 

COPERNICAN SYSTEM 

Members of all the r-(!ligious gi-ou|3s attacked Co- 
pernicus; the attacks and ridicule were not limited 
to any one group. As the religious leader-s r-ealized, 
if the sun were the center- of the system, the star's 
must be very far- away and very luminous, perhaps 
even themselves suns. If they were suns, thev 
might have planets, and these planets riiight sup- 
port intelligent life. This idea, that ther-e might be 
other planets around other- stars, was called the 
pluralirv of worlds." It presented the notion that 
the earlli and our- religious experiences her-e might 
not be unique. The possibility of life existing on 
bodies other- than the ear1h was voiced only slow Iv , 
in England by Thomas Digges and on the C:onti- 
nent by Gior-dano Bruno, who was burned at the 
stake for- heresy in !(>()(). 

6.5 I HISTORICLIL COXSEftUEXCES 

Follow Te\t. 

6.6 I TYCHO BRAHE 

Some hackgi-ound in rcgar-d to the "new" star- ob- 
serveil by lycho in 1.172 will bo helpful. A star- is 
"new" only in the sense of becoming mori' obserA- 
able or conspicuous. Cirrrent e\|5lanations de- 
scribe the pr-ocess as one in which the stars hv- 
drogen content has graikrally been consirmed so 
that the oirtwarxl radiation pi-essur-e within the gas- 
eous star- no longer- balances the gravitational at- 
tr-action tovvarxl the center, llien the star- collapses. 
Very high central pressures ai'e then devloped 
and the star flart^s up for a few year-s as a nova. F'ar1 
of the outer envelope is blown off and, for- fairlv 
nearby novaer. mav later- be detected as wispv 
filaments of outwar-d moving gas. .Apparentlv 
novae eventirallv settle down as "white dwarfs, 
still gaseous but having surface tJ'mp«M-atur-es 
ar-ound l(),(){)()°c: and inlerrial sperilic gravities 
of 10' or 10". 

Other- slar-s seem to urulergo much gr-ealer- out- 
bursts and be(-()me suptMriovatv \'oi a lew veais 
their luminosilv oi- aitual output of radiant enei-gv 



may be as great as lO" times that of our- sun. \\ hen 
a sun-sized star becomes a supernova, it blows off 
much of its mass and appears to change into a 
neutr-on star only a few kilometers in diameter. A 
larger star seems to turn into a black hole and 
disappear ' except for its intense gravitational 
field. Much of the information about these strange 
ex-supemovae is obtained with radio telescopes. 

The novae obser-ved in 1572 by Tycho, and in 
1604 by Kepler- and Galileo, wer-e both supemovae. 
The supernova obseiAed in laums in 1054 a.u. has 
r-esulted in the Crab Nebula. Ther-e is evidence that 
the Indians in western America saw this event 
since they cut in a rock face the symbols of a star 
and the crescent moon. What else could have been 
so impr-essive that it produced this record? Prob- 
ably a star like this superiiova was visible during 
the day. From oriental records and recent com- 
putations, it has been found that the nova ap- 
pear-ed in Julv 1054 near the cr-escent moon. 

Tycho's I'ranibor^ might be likened to one of our 
present large r^searx:h centers supported bv the 
government; perhaps the BrookJiaven Laboratory , 
the Lawrence LaboratorA', the Argonne Laboratory , 
or CERN in Switzerland. An article in Scientific 
American, February 1961, page 118, discusses T\- 
cho s obser-vatorA'. 

Interested students might read about comets 
and how thev ar-e studied. Sky and Telescope for 
December- 1965 and for 1973-74 included many ar- 
ticles and photographs of bright comets. 

Perhaps some discussion of the manv super-sti- 
tions surrounding the unexpectedness of comets 
and novae would be useful to suggest the variety 
of ways in which people interpret unexpected 
events. The wiitings of Shakespeart; contain manv 
allusions to astronomical ever"rts as omens. Al- 
though we still have manv rudiments of such su- 
per-stitions with us, we mav be incrvasinglv con- 
scious that we ar-e r-eacting fearfullv on the basis of 
unwarranted assumptions aboirt the world. 

6.7 1 Tl'CHO'S OBSERl'ATIOXS 

■As we saw earlier. Copernicus relied mainly upon 
the r-ecorxls of Ptolemv , which wer-e inaccirrate. 
These old observations had been made bv diflferent 
people at diflervnt times Scholar-s still discirss the 
extent to which the observations in the Alntagest 
wer-e made bv Ptolemv or- vver"<' in part atlopted 
and corr-ected fr-om earlier work bv Mipparchus 
aboirt 150 lu T\(-ho coricluded that new and more 
pr-ecise observations made over a number- of v ears 
werv essential befori' anv new description of pla- 
netarv motions coirld be crrated or- jirdged 

The sections on Tvcho s e(|uipment might stim- 
irlate some stirdents intervsted in mechanics and 
ecjiripment design The inhervnt limitations of our 
eves also could lie investigated thrx)irgh readings, 
or by a project for those interested 

6.8 TYCHO'S C:OM PROMISE SYSTEM 

Tvihos obsc'n.itions wen- planin-d as the ha^is for 
the development ot a new model ot planctaiv mo- 



I IN 



l'\n 2 / \l(n'l(}\ l\ THIi HliAXIiXS 



tions. Although he died before much of the analysis 
could be completed, his general idea of a sxstem 
is that indicated b\' the low er drawing on 7e,vf page 
175. lycho s s\'stem had the phuiets mo\ing aix)und 
the sun as in the Copernican system. Howe\er, be- 
cause he could not obser\e any parallactic shift 
that should ha\e resulted Irom the motion of the 
earth ai"ound the sun, TncIio assumed a fi.xed earth 
with the sun oiiiiting it. In all other wa\s, Ixcho s 
system was essentialK the same as the Copernican 
system. Since no stellar paraiia.x had been ob- 
served, man\' people accepted the 1\ chonic model 
rather than the Copernican. 

At the end of the section we intrxjduce the im- 



portant question: "Are scientific models descrip- 
tions of reality', or only convenient computational 
devices? The Ptolemaic svstem permitted com- 
putations of the positions of the planets: it did not 
attempt to describe realit> . The heavens were vis- 
ualized by the Greeks and bv the medieval world 
in temis of ciAstalline spheres. Ihis vision is de- 
scribed bv Dante in the Divine Comedv i\D 1300i. 
But Copernicus and Tycho were concerned with 
the real motions of the planets. Here, as well as 
elsewhere in the te.xt, we raise the question about 
the realitv' of conclusions based on scientific the- 
ories. The point should be included in class dis- 
cussion. 



CHAPTER 7 / A NEW UMVTLRSE APPEARS: 
THE WORK OF KEPLER AND GALILEO 



7.1 i THE ABAXDOXME^'T OF 

UXIFORAI CIRCIXAR MOTION 

Kepler was a strange blend of my stic and scientist 
v\ith a deep Pvthagorean feeling for the numerical 
perfection in the world and the music of the 
spheres. His earl\ paper on the spacing of planets 
and his later work on the third law suggest that 
sometimes scientists begin with aesthetic or artis- 
tic premises. The recent stress upon svmmetrv 
in particle phvsics is a similar example. 

Due largeK' to the fact that Kepler inherited all 
of Tycho s data on Mars and had access to the 
writings of preceding astronomers, the time was 
ripe for new ideas not pi-ejudiced by the assump- 
tion of uniform circular motion. Vou might remind 
students that in many instances in life one mav be 
forced to reexamine earlv assumptions and per- 
haps to replace them. 

Seen in historical perspective, Tvcho and Kepler 
made an ideal pair. T\ cho stressed the importance 
of improved observations and devoted his life to 
obtaining such observations. W ithout them, Kepler 
would have had the same difficulties as did his 
predecessors. 

After more than 70 unsuccessful trials Kepler 
found that he could not fit the observations with 
any combination of circular motions. Vou might 
wish to dramatize the situation in which Kepler 
found himself. He felt that some satisfactory solu- 
tion could be found. Since Mai"s continued to move 
across the sky. oblivious to Kepler's efforts, the 
trouble must lie with the theory. Therefore, he was 
obliged to look at the problem in a new way. This 
is alwavs difficult for us, but Kepler did it. 

His work, both unsuccessful and eventuallv suc- 
cessful, was laborious because the mathematical 
techniques of his time were cumbei"some. Kepler 
was one of the first scientists to use logarithms. 

Kepler was caught in the religious confiicts of 
the rhiitv' Veai-s War and the struggles between 
the Catholics and Protestants. At best he had a dif- 
ficult time earning a living, despite the promises of 



the king. The trial of his mother for witchcraft 
might be paralleled with similar occurrences a bit 
later in the American colonies. It indicates the cul- 
tural and social context within which Kepler, like 
Galileo, was working. The popular book entitled 
Kepler, 1571-1630 bv Max Caspar might interest 
some students. 

7.2 KEPLER'S LAW OF AREAS 

Kepler noticed that the speed of Mars changed as 
it moved through its orbit. His challenge was to 
find something constant about this changing speed. 
Students should be sure that thev clearly under- 
stand this pr-oblem. 

His first discovery , that a line drawn from the 
sun to Mars sweeps out equal ai-eas during equal 
time intervals, reveals something unchanging or 
constant about this orbital speed. Such an un- 
changing mathematical constnjct is an empirical 
law. It is based upon observations: and it must only 
satisK the constraint of accounting for those ob- 
senations. .As a matter of fact, all of Kepler's laws 
are empirical. 

Note that E2-6, The Shape of the Earths Orbit," 
and E2-S. Orbit of Mars, might be done even be- 
fore reading the text. 

7.3 KEPLER'S LAW OF 

ELLIPTICAL ORBITS 

rhe Te.xt quoted Kepler s comment that, Mai's 
alone enables us to penetrate the seci-ets of astixjn- 
omv which otherwise would remain forever hid- 
den from us.' This statement almost surely refers 
to the sizable eccentricitv of the orbit of Mars le 
= 0.09). Of the outer planets, onl\ Mars is near 
enough to be studied accurately by Kepler s trian- 
gulation method. .Although Mercui^ has a more 
eccentric orijit i — 0.2 li, studies of it were prac- 
ticalK impossible by his method. MercuiA is seen 
only in the twilight when few stars can be observed 
to detemiine accurate positions. Todav telescopic 
observations of the position of .Meix'Uiy and even 



BACKGROl \D AND DEXTiLOP.MENT 



119 



of bright stars can he made in the daytime. Of 
course, Uranus, Neptune, and Pluto were unknown 
at the time of Kepler. 

An excellent background on the mathematics of 
conic sections app(;ai-s in the S.MSd publication In- 
termediate Mathenvttics, Teachers Commentary,', 
Unit 19, Yale University Press, .New Ha\en, 1961. 



universe, he could accurately predict the future of 
the universe. The great success of \e\\1onian me- 
chanics, discussed in Chapter 8, supported such 
a mechanistic \ie\v. Only within the current cen- 
tuiy ha\e physicists been obliged by new t\pes of 
observations to abandon such sweeping general 
assertions (see Units 4, 5, and 6i. 



7.4 I KEPLER'S LAW OF PERIODS 

Previously, Kfjpici discovered b\ trial and en-or 
laws that accounted for the shape of an oi+)it and 
the speed of a single planet. His next problem was 
to determine what is invariant about a set of 
planets that have different elliptical orbits and 
therefor-e different speeds. He disco\ered that the 
ratio of the square of each period to the cube of 
each average distance is constant. This is the law 
of periods. 

Two insights might be stressed in class. One is 
that this last law is general since it deals with all 
of the satellites of the sun. The second insight is 
that the law does not show the relationship among 
satellite systems. That is, each satellite system has 
a different constant of proportional it>' between the 
square of the periods and the cube of the average 
distances. 



* K, 



K 



Jupile 



7.5 I THE IVEW CONCEPT OF 
PHYSICAL LAW 

Kepler's work reflects the change from a mx'stical 
inteipretation of how the world ought to be to a 
reliance upon obsonations as the final basis for 
decisions. He had a growing feeling that some 
mechanism was essential to mo\e the planets. We 
know that he often wrote to (lalileo but that after 
a few lettei-s fn)m Galileo the correspondence was 
one-sided. Why (ialileo did not accept the elliptical 
od)its of Keplei- is difficult to undei-stand. Perhaps 
it was because, as one authority noted, Kepler- 
wrote in a flowerA' style that was often most diffi- 
cult to irnder-stand. Unfortunately, his major con- 
tributions art' buried in masses of worxls. Us therv 
a moral in this for- xoirr- stirdents.-' Ha\e the\- e.\- 
amined some scientific and technii-al writing in 
|irx)fessi()nal joirriials?i 

In the historical stirdy of science, it is often dif- 
ficult to cslablisb who actiralK had an idea fir-sl 
Ideas often grow as various people* c-onsider- them 
and their- conse(|uences I he idea of the irni\er-se 
operating like a (lot-kwor-k or- a giant machine was 
implicit in the se(|uence of invisible celestial spher-es 
pr-oposed In Kudoxus However-, Keplers analogv 
is imporlant because, as (Chapter- S slunvs. this idea 
became firniiK' entrenched. Perhaps the irltimate 
fbr-m of the idea was the statement of one later 
scientist to the (<tTect that, if he knew the initial 
positions and velocities of all the Ixulics in the 



7.6 ' GALILEO AXD KEPLER 

A number of lines cjf evidence, including Galileo's 
work in mechanics and the astr-oncjmical models 
of Copernicus and Kepler, were undermining the 
premises on which the Aristotelians based their 
arguments. Even such figures as the poet and 
wiiter John Milton in England were aware of what 
was happening. Miltcjn visited Galileo in southern 
Europe during the summer of 1638. In the quoted 
section of Paradise Lost, the poet raises the ques- 
tion that had been rejected bv the Rolemaics. 

Galileo was incensed that his contemporaries 
would not e\en use the telescope and try to refute 
his obser\ations. They remained entrenched in 
their own ideas and wouldn t consider either chal- 
lenging them by looking for themselves or accept- 
ing his reporls. We all e\|jerience great difficultv' in 
making a major shift in concepts. Ceriainlv the 
shift ft-om an earlh-center"ed svstem was gigantic 
in its implications. Can students suggest other 
comparable shifts that have caused us to reinter- 
pret the worid and our place in if:* Do not restrict 
the list to those shifts that seemed to appear ab- 
ruptly. It required nearlv 1,800 veal's for the sun- 
centered svstem to be considered seriously. The 
students might list such reinterpretations as de- 
temiination of the age of the earlh. the Darvxinian 
theoiA of evolution, relativitv. and Freud s psy- 
choanalvtic theories. 



7.7 1 THE TELESCOPIC E\1DEXCE 

The photograph on lc.\l page 193 shows two of 
Galileo s telest^opes. with wiiich he saw and inter- 
preted manv new objects. His conclusions are even 
more imporlant than his drawings of what he saw. 
Other-s might have vienved the moon, but not ha\e 
found the mountains he recognized. The differ- 
ence between raw data and interpretation might 
be developed with the students .Although new in- 
stiaiments permit new obsei-vations. insti-uments 
onlv pixnide data that must then be inteipieted. 

7.8 c;alili:o fcm:i'Si:s 

THE COVrRO\'i:RSV 

Follow Test. 

7.9 SCIENC:E /VNU FKEE1KI.\I 

Stiulents max want to repoil on the history of the 
Catholic c;lurr-t-h in the seventeenth centuiA and to 
compare it with the Churvh in the twentieth cen- 
tur-v Others inav want to discuss the rise of the 
l'i-()trstant gt()up> 1> it likclv lliat a (-oinrminity 



1211 



IMP 2 M()TI()\ l\ THK HKAVliNS 



hospitable to the followei's of Martin Luther or students examine the known facts objectivelv and 



John Cahin would be hospitiible to new ideas in 
science? 

Do not attempt to create a "hero and \'illain" 
image of Galileo and the Chuixii. Rather ti^' to ha\e 



conclude that thei-e pi"obabK was en-or and pitjv- 
ocation on both sides. The Crime of Galileo b\ de 
Santillana presents Galileo as a mai-t\i-. Is it a fair 
position? 



CHAPTER 8 / THE UMT\^ OF EARTH AND SK\^ 
THE WORK OF \EVVTON 



8.1 : NEWTON AXD SE\TXTEEXTH- 
CEXTITIY SCIENCE 

The emphasis in this section is upon the growing 
acceptance in noi-thern Europe of the new phi- 
losophy of empirical experimental science. In ad- 
dition to the Royal Philosophical Societ\ of Lon- 
don, there was in France the Academie des Sciences, 
and in Itah', the .Accademia de Lincei iLvTixesi at 
Rome and the .Accademia de Cimento at Floi-ence. 
These scientific societies, first in Itah and then in 
England and France, were important because the\ 
allowed scientists to work and argue together and 
to publish journals that could be sent to their col- 
leagues in other countries. Sti'ess how the work of 
mam people, as illustrated b\ the quote from Lord 
Ruthertbrd on Te\t page 209, is demonstrated b\ 
the achie\ement of Xeuton. 

The origins of the great generalizations of sci- 
ence can be traced in preceding decades. .An\ 
study of Xewton's analysis should emphasize the 
importance of the historical background leading to 
Newton s great sxnthesis of his laws of motion and 
of uni\ ersal gra\itation. The barrier between celes- 
tial and terrestrial motions set up b\' Aristotle was 
gradualh being broken down. T\cho Brahe located 
comets be\ond the moon. Kepler replaced per- 
fect ' circular motions by motions in elliptical or- 
bits. Jeremiah Horrocks. bom in the \ear Kepler s 
thirxl law was published, entei"ed Emmanuel Col- 
lege of the Uni\ ei"sit\ of Cambridge at the age of 13. 
When 19 and curate of Hoole in Lancashire, he 
applied Kepler s first law to the motion of the 
moon around the earth. He e\en showed that the 
eccentricit\' of its oit)it changed periodicalK' and 
the major axis of the ellipse slowly rotated. This 
was 25 years before the youthful Newton concei\ed 
his ideas on uni\ersal gravitation, which were not 
published for xet another 20 yeai"s. 

The persistent question that might be raised is 
whether this great theory iuni\ei"sal gravitation i is 
ever "proved. Students should conclude that a rig- 
orous pitjof is not possible. \eX in spite of this, the 
theoiy seems to work well. It explains much that 
is known and predicts manv other phenomena 
and quantities. 

The arguments in this chapter follow Newton s 
rather closelv, although some have been modified 



or reworded in the spirit of Newlon for- this dis- 
cussion. 

8.2 NEWTON'S PRIXCIPIA 

The fii-st edition of the Priiuipui was published in 
1686. The second edition in 1713 included many 
corrections to the first printing, some new argu- 
ments, and considerablv more data on comets 
based mainlv on Hallev s work. 

Various authoi-s hav e repeatedh pointed out that 
Newton did not attempt to explain gravitation. He 
postulated an inv ei-se squaie force of attraction be- 
tween bodies and it worked. He did not know how 
it worked or whv it seemed to be associated with 
masses. In his famous General Scholium" at the 
end of the Principia. he obsened that he framed 
no hvpotheses" on the nature of gravitv. He was 
concerned, but had no conclusions that he wished 
to publish. 

.At this point \ou mav wish to ask students about 
the usefulness of an undefined concept such as 
gravitv . We can measure its effects, predict the out- 
come of certain experiments, and in general make 
some use of gr^avitv , vet we do not know w hat it is. 
Einstein was working on a unification of several 
aspects of gravirv at the time of his death. People 
irivolved in this r^esearch todav have still not ex- 
plained gravitv . 

8.3 THE INlTERSE-SQrARE LAW 

OF plant;tary force 

Point out the shift in Newton s assumptions from 
the Gr-eek notion of circular motion as pert'ect, to 
the inertial circular motion of Galileo discussed in 
Chapter 4, and then to the definition of inertial 
motion in an opticallv straight line. .Also, Newton s 
idea was that circular motion is caused bv a for-ce 
in action, and he extended this to include the el- 
liptical motion of Kepler s laws. 

A Blend of the Laus of Kepler and .Newton 

1 he synthesis of Kepler s laws and Newton's laws 
to reach basic conclusions about the nature of the 
central force acting on the planets prxjvides an ex- 
cellent example of logical reasoning. The following 
material, taken from earlier editions of the 7"e,v/, is 
included here so that the teacher can gradualh 
develop the argument step bv step with the class 



BACKGROl AD AND DEXTiLOPMEVT 



121 



Newton's laws Kepler's laws 

1. A body continues 1. The planets move 
in a state of rest, or of in orbits that are el- 
uniform motion in a lipses and have the sun 
straight line, unless at one focus. 

acted upon by a net 
force (law of inertia). 

2. The net force act- 2. The line from the 
ing on an object is di- sun to a planet sweeps 
rectly proportional to over ai-eas that are pro- 
and in the same direc- portional to the time in- 
tion as the acceleration, tervals. 

3. To every action 3. The squares of the 
there is an equal and periods of the planets 
opposite reaction. are propoilional to the 

cubes of their mean dis- 
tances from the sun tT' 
= kR'\. 

According to Newton's first law, a change in mo- 
tion, either in direction or amount, requires action 
of a net force. But, according to Kepler, the planets 
move in orbits that are ellipses, that is, curved or- 
bits. Therefore, such a force is acting to change 
their motion. Notice that this conclusion does not 
specify the type or direction of the net force. 

Combination of Newton's second law with the 
first two laws of Kepler clarifies the direction of the 
force. According to Newton's second law, the net 
force is exerted in the direction of the observed 
acceleration. So, what is the direction of the force 
acting on the planets? Neulon showed by a geo- 
metrical analysis, which is developed in the Test. 
that a body moving under a central force will, when 
viewed from the center of the force, mo\e accord- 
ing to Kepler's law of areas. But Kepler s law of 
areas relates the planets to the sun. Therefore, 
Newton could conclude that the sun at one focus 
of each ellipse was the soui"ce of the central force 
acting on the planets. 

Newton then found that motion in an elliptical 
path (or a path defined by any of the conic sections 
mentioned in Chapter 71 would occur only when 
the central force was an inverse-square force, F ~~ 

1 

— . Thus, only an inverse-square force exerted b\ 

the sun would rcsull in the obsened elliptical or- 
bits of the planets described by Kepler. Newton 
then clinched the argument by showing that such 
a force law would also i-esult in Kepler s thiril law 
the law of periods, 7" = kM\ 

Fix)m this anaKsis Neulon concludeci that one 
general law of uni\ei-sal graxitation that applied to 
the earth and an apple also applied to the sun, 
planets, comets, and all other bodies mo\ing in the 
solar system. This is the central ai-gument of New- 
toiis grcat s\nthesis. 

Motion I'nder a Central Force 

Your students may encounter diffi(ult\ in the geo- 
mc'tric dexelopment of Newton s argument be- 
cause thev i\n\ unable to see how the method ot 



measuring the triangular areas changes. To mini- 
mize this difficult^', make use of the drawings on 
Te;ict pages 218 and 219, which demonstrate how 
each side of a triangle may be used as a base and 
how a perpendicular may be dropped from each 
vertex. 

It might also be a good idea to emphasize the 
unexpectedness of the conclusion that the law of 
areas holds exen when no central force is acting. 

It might be useful to see .Newlon s original de- 
velopment of this argument in Book I of the Prin- 
cipia. While the students ma\' not be able to follow 
the text, the wording of the pixjpositions and schol- 
iums and the illustrations will ser\e to demon- 
strate how neatly .Newlon tied his aiigumentati\e 
package together. 

The universal law of gra\itation was a veiy bold 
proposal. Dramatize the audacity' of Newton to 
propose the uni\ersalit\' of physical laws whose 
action could generally only be observed on the 
earth. 7 he people of .Newton s time were still 
bound by the concepts of separate worlds and 
other Aristotelian doctrines. 

8.4 I LAW OF LTVnXRSAL 

GRAnXATIOX 

One of the high points of the text is the philosoph\ 
of the Newtonian synthesis. This asserts that gra\- 
iation applies throughout the unixerse. Thus one 
law explains observations on the earth as well as 
in the heavens. Furthermore, uni\ersal graxitation 
is a synthesis in the sense that it accounts for all 
three of Kepler's laws. 

Students are often fascinated by Descartes' al- 
ternate argument that a fluid causes the planets to 
stay in their orbits. .Also, such basic questions in 
philosophy as the meaning of explanation and 
of "cause can be the topics of discussions. 

The French philosopher Descartes il596-1650i 
proposed an alternate theory' that all space was 
filled with a subtle, invisible fiuid that carried the 
planets around the sun in a huge whirlpool-like 
motion. Descartes' theorv was first published in 
1644 and received wide acceptance on the conti- 
nent. An English edition was finally published in 
Lxjndon in 1682 before the Principia was published. 

This theorv was a popular nonmathematical 
statement read by large numbers of people and 
readiK' accepted as a better explanation than none. 
It sounded good and was not too nidicalK dift'erent 
fn)m the Aristotelian attitudes that the people had 
previously learned. Descartes theoiA was widely 
taught, even at Cambridge long after the publira- 
tion of the Principia'. 

It might be interesting to point out that \oltaire s 
famous essa\ . Flements of Newtonian Philoso- 
ph\', " was banned in France because the man in 
chai-ge of permissions to publish was a Cartesian. 

8J5 .VEWTON /i.V» IIl-POTIIESES 

Ihe tlisrussit)n raises the (|u»'siion ot action at a 
distance Note the ({notation tix)m .Newton on I'e.xt 



IZ2 



I 'MIT 2 / MCmOIV l\ THE HEAVENS 



page 222. Direct the students' attention to the fact 
that, from the observations of T\cho and the em- 
pirical relations of Kepler and of Galileo, Newton 
had been able to fashion an exceedingly general 
and abstract description of hea\enl\ motions. But 
in the process he had been obliged to postulate 
the gra\itational force that he could not explain. In 
much of science, as in mathematics, there are 
some postulates and axioms that cannot be ana- 
lyzed within the problems considered. Occasion- 
cilly someone can interpret one or more of these 
axioms by a more basic proposition. 

8.6 THE .MAGXITITJE OF 

PLAATETARY FORCE 

Note that the discussion of geometric points in the 
case of Kepler s law of areas changes to a discus- 
sion about the masses of stones and planets. The 
idea of mass has already been introduced b\ 
Newton's second law of motion. Note particularK 
the argument on Test pages 224-226 that it is the 
mass of a body that is associated with the notion 
of gra\itational force. This argument realh marks 
Newton s great contribution, a leap in understand- 
ing from a consideration of the direction of the 
force to that of the amount of the force. 

The gra\itational constant G serves the same 
function as any constant that changes a proportion 
into an algebraic equation. In the case of equations 
invoking physical quantities, the constant also 
serves as a balancer of units. It might be worth- 
while to remind the students at this point that 
symbols that stand for ph\sical quantities are not 
sacred cows, and that the\' onl\' mean what \'ou 
want them to mean. It could also be pointed out, 
for example, that this same kind of operation in- 
\ol\ing a constant turns up in the algebraic form 
of Kepler s third law, where T' ~ R^ becomes T' 
= kR\ 

8.7 PLAXETARY MOTIOX AXD THE 

GRA\TTATIOXAE CONSTANT 

The experiments that we can earn out in order to 
determine that G is a universal constant are limited 
in number. The e.xtension to uni\ ersality must be 
carried out in terms of a kind of well-sustained 
faith. Such a conclusion may come as a shock to 
those students who feel that science is a rational 
process that has little or no room for imagination 
or statements based upon re\elation. Vou have 
a ready-made situation for an interesting discus- 
sion. 

The reason that the masses of the sun and Ju- 
piter can be compared is that Jupiter acts like a 
miniature solar s\stem. iGalileo had this thought 
upon identihing the satellites of Jupiter.' The only 
difference between the sun system and the Jupiter 
system, insofar as Kepler s law is concerned, is that 
each k in\ol\es a different central mass. .As long as 
we can measure the R and 7' for a re\ol\ing bod\' 
in each svstem, the two central masses can easih 



be compared. .Note that the power of the law of 
universal gravitation is that one does not have to 
know the value of G in order to make such a com- 
parison! B\' fonning ratios of the equations i refer 
to Text pages 228-229), the constants, including 
undetermined G, cancel. 

8.8 THE \'ALIT OF G AND THE 

ACTUAL CLASSES OF THE PLANETS 

.As long as Ne\%ton had to depend upon the use of 
ratios, in which G did not enter into the quantita- 
tive results, his statement about gravitation was 
really a h\'pothesis. Once G was measured, the hv- 
pothesis could really be called a law. since all quan- 
tities in the statement were now measurable. 

.Although the Test defines mass as the quanti- 
tative measure of the inertia of an object, we should 
distinguish the inertial mass of an object from its 
gravitational mass. The fact that we must exert a 
force F to giv e an object an acceleration a is a prop- 
ertv of the inertial mass m_. Bv Newton s law, we 
know that F — m^a. This law presumably would 
hold whether or not we perform the e.xperiment in 
a gravitational field. The weight W of an object and 
the attractiv e gravitational force F^ between objects, 
however, depend on the gravitational mass m^. 
These phenomena are independent of the inertiaJ 
properties of matter, but thev could not exist with- 
out a gravitational field. In describing them, we 
should wiite 



and 



F, = 



8.9 FtHTHER SUCCESSES 

Tides 

Some of vour students max have difficultv in un- 
derstanding the concept of differential forces in the 
case of the pulls e.xerted on the earth bv the sun 
and the moon. Thev ought to see that this differ- 
ence is reallv a function of distance. Even though 
the gravitational pull of the sun on the earth is 
much greater than that of the moon, the sun is so 
far awav that it does not distinguish between the 
near and far side of the earth. The moon, which is 
much closer, does. 

If one had the data for high and low tides from 
all different parts of the earth, would that infor- 
mation be enough for the formation of a general 
predictive theorv' for the tides? In what wav does 
the principle of universal gravitation become a 
breakthrough here? 

Comets 

Students will be able to find allusions to comets as 
omens in Shakespeare. Chaucer, Julius Caesar, and 
in all kinds of folklore. Remind your students to 
look carefully at the reproduction of the Bayeux 
tapestrx in the Test 



BACKGROUND AND DEMiLOPME.VT 



123 



Students might also be asked to refer back to the 
poilioM of the re;<t where the contribution of ly- 
cho Brahe to the understanding of comets is dis- 
cussed. 

Halley's comet is foi-ecast to i-etum in 1986 and 
pass perihelion on Febiuarv 9 of that year. Probably 
the comet will be detected early in 198o and [jos- 
sibly earlier (when it is as far as Jupiter's distance 
from the sun). This will be a disappointing ap- 
pearance of the comet: Perihelion passage will be 
on the far side of the sun and far south in the sky. 
Bright moonlight before and after perihelion pas- 
sage win also blot out most of the faint tail when 
it is likely to be brightest. The best observations 
may be made in April 1986 when the comet is far- 
ther norih in the sky, at about the ear1h s distance 
from the sun, and seen 90° from the sun isee I'he 
Physics Teacher 15, 260, 19771. 

The model dexeloped in the Unit 2 Activity can 
be used for the 1986 appearance as well as for that 
of 1910. The date of perihelion can be changed to 
February 9, 1986, and the calendar for the comet s 
motion then worked out befoi-e and after perihe- 
lion passage. The calendar for the earth's positions 
remains the same. With a simple model made of 
cardboard, students can see for themselves why 
the 1986 appearance of Hcdley's comet will be dis- 
appointing. 

8.10 I SOME EFFECTTS AND 

Li\ii'iv\TioNs OF xEwnroivs 

WORK 

It is worthwhile to discuss Newtonian physics at 
some length. Appreciation of the achievements of 
universal gra\itation should be one of the foremost 
goals of Project Physics. Students ha\e a right to 
see the "big picture. ' 

Part of the big picture is that unixei-sal giavitation 
accounts for- obsenations on the earth as well as 
in the heavens. Another aspect is that unixersal 
griivitation is a gi-eat theory based upon numeixjus 
laws (such as those of Keplei- and of \'eu1on), a 



priori statements (force acts through a distance), 
observations, and arbitrary definitions. Of course 
the most dramatic result of a great theor>' is its 
predictive power, for example, the e.xpectation that 
Halley's comet is a satellite of the sun that will 
return every 75 years. 

Finally, and perhaps most important, is the re- 
lationship of univer-sal gravitation to small things 
ileading to quantum mechanicsi and to large 
things Ileading to r-elati\ityi. This consideration \nt11 
allow the teacher to give a pr^exiew of coming at- 
tractions in Units 3-6. Applying orbital ideas from 
planets to the tiny electron going around the nu- 
cleus gives rise to the quantum constraint of cer- 
tain specified allowable orbits. .\Io\ing to large- 
scale considerations, Einstein saw the space aiDund 
Jupiter as warped rather than as a simple .\eu1on- 

F 
ian gravitational field (g = — ). Perhaps it is wise to 
m 

only hint at these relationships at this time. 

EPILOGUE 

Some students may want to look up the Ency- 
clopedist movement in France, begun b\ Denis 
Diderot. The influence of \eu1on s work upon the 
great X'oltair-e is also worth some researrh. A good 
encyclopedia will cerlainly ha\ e much to sa\ about 
this particular- period in France and England, when 
ideas like "the rights of man and democracy 
were emerging from the minds of people who were 
beginning to use logic rather than r^evelation as 
their apprxjach to a philosophx' of life. 

The limitations of the Neulonicm analysis of cer- 
tain phenomena will become apparent later, when 
we discuss r-elati\it\' and quantum mechanics. 

The eighteenth centur\' saw the rise of iatrome- 
chanics, that is, the concept that the human bod\ 
is a machine whose par1s n.in accorxling to the laws 
of Newlonian d\namics. .About 1745 a French doc- 
tor. La Mettrie. published a ph\siolog\ te,\t whose 
title was "L'Homme Machine iMan, the Ma- 
chine "). 



CONCEPT FLOW CHARTS 



These charts art* designed to help \oir follow the 
de\(*l()pment and interconn(H:tion of ideas in the* 
Project Physics c^ourse. 

ihriM! kinds of ler-ms appear on each charl, dif- 
ferfMitial(Hl by the for-m of the letter's. Lower- case 
(for (wample, "ol)serAalions ) rt'fer-s to phenomena. 
obsJMA'ations, or e.\perim«Mits. Upper case (for e.\- 
amplc!, MODELS I refer-s to well-delined concepts, 
models, or- theories. Italics ifor- example, I hemes ' 
rvfer-s to swcH'pingly general ideas, themes, or view- 
points, which art* often closely rt^lated to philoso- 
phy. Thert» is sometimes rx)om for ar-girment as to 
which shoirld be irsed, but for the most pari oli- 
ser-valions art* distinct fr"om models, and both arx" 
distinct fr-om general \iewiMiints. 



Some of the conceptiral themes are so general 
that it would be impractical to show all the arrows 
on the char1. For- example the idea of riiathemat- 
ics as the jir-oper- langirage of nalirrv affected al- 
most e\er>- development in pliNsics tmm (iaiileos 
time and perhaps long beforv Ihe ancient cori- 
ceptiral themes at the top of the chart wer^. of 
cour-se. derixed fixim ohserAation and are all part 
of the e\en biDader \ie\\point of nature as know- 
able. 

The arrows do not all mean exactly the same 
thirig. Some impl\- a dirvct deri\atiori, while others 
impl\ onl\- a sirspected irifluence. Ir-r any case. the\ 
r-epresent onl\ those connections that ar-e dis- 
cussed in the Project Phvsics I'rst. 



124 



I'lVIT 2 MOTIOX l\ I'Hi; HIvWIAS 



There is a rough progression in time from top to 
bottom, but this is not ngoi-ous and ought not to 
be taken too literally. 

A word of caution! The interconnections of con- 
cepts, \ie\\points. and experimental data in the 
de\elopment of plnsics are subtle, complex, and 
numerous. For even' connection we know about, 
there are man\' more of which we ma\' be ignorant 
or which we misundei"stand. E\en the necessarily 
simplified stor\ that is sketched in the Project 
Physics Te}it is complicated and to some extent 
speculati\'e. These charts are intended only as one 
way of outlining the connections as they are 
treated in the Te,vf. The charts are, then, an incom- 



plete and somewhat arbitrary' skeleton of a Te}(.t 
that itself is incomplete. Furthennore, hardh' any 
indication is gi\en of the exceedingly important 
relations of physics to philosophy and social de- 
velopments. These i"esei"\ations keep us from call- 
ing the charts The Stoiy of Ph\ sics.' 

Nonetheless, the Test does tell a ston' something 
like that depicted on the chart, and some kind of 
map through the forest seems to be needed. How- 
ever clear the account ma\' be in the Test, it is 
necessarily cut into somewhat arbitrary chunks 
that are presented one after the other. The charts, 
if they do nothing else, show cross-connections 
between concurrent developments. 



CONCEPT FLOW CHART UNIT2 



observations, experiments 

THEORIES, MODELS 

Viewpoints. Themes 



Natural violent 
motion 



Heavenly spheres 

PTOLEMAIC MODEL 



/ 



naked-eye sky 
observations 



projectile 
motion 



GALILEAN 

KINEMATICS 

\ 

NEWTON'S 
LAWS OF 
MOTION \ 

NEWTONIAN 



COPERNICAN MODEL 




KEPLER'S LAWS 

KEPLERIAN MODEL 

Y telescopic sky 

!\.^J.^I^?'1NUNIVERSAL>^ observations 



OF DYNAMICS 



GRAVITATION 



Additional Background Articles 



BACKGROL^T) UVFORMATIOX 
ON CALENDARS 

ISecs. 5.1 and 8.1 1 

In 45 B.C., Julius Caesar decreed a new ci\il cal- 
endar of 365V4 days. .As the Test, Unit 2, indicates, 
this "Julian" year exceeded the actual motion of 
the sun by 11 min 14 sec per year. As a result, the 
Julian calendar v\as slow b\' one day in 128 years. 
By 1582 .A.D., the Julian calendar was in error by 10 
da\s and the sun passed the \ernal equinox on 
March 11 rather than on March 21, as required b\ 
church canons. In 1582, Pope Gregor\' XIII abol- 
ished the old calendar and replaced it with a new 
ci\il calendar now known as the Gregorian or New 



StA'le Calendar. October 4, 1582 was followed by 
October 15th. The new calendar was immediately 
adopted b\' all Catholic countries, but England and 
some other non-Catholic countries would not 
adopt the new calendar because it was established 
by Catholics. \ot until 1752 did the Gregorian cal- 
endar finaU\' become official in England, when Sep- 
tember 2 was followed b\ September 14. 

Some examples of the confusion presented to 
historians by the difference between the Julian and 
Gregorian calendars can be illustrated b\ the birth 
and death dates of Newton. Often Newton is said 
to have been born in the year that Galileo died. 
Galileo died in Itah' on Januarv 8, 1642 i\ew St\lei 



ADDITIONAL BACKGROUND ARTICLES 



125 



and Newton was born in England on December 25, 
1642 lOld Style). VVtien the date of Newton s birth 
is changed to New Style iGregorian) it becomes Jan- 
uary 5, 1643. 

Newton s death is generally fepoiled as occur- 
ring on Maich 20, 1727, yet the year caived on his 
tomb in Westminster Abbey is 1726. VXtien the Cal- 
endar Act of 1750 went into effect in 1752, not only 



were 11 davs dropped from the time record, but 
also the date of New Year's Day was changed from 
March 25 to January 1. Actually Xewlon died on 
March 20, 1726 lOld Style), but on the new calen- 
dar, which was adopted later, this became March 
20, 1727 (New Stylei. 

Schematically the change looked like the dia- 
gram below: 



Julian (o.s.) 



Jan. Feb. Mar 



: 175:Z 

I ! > 



Gregorion(N.s) 



Apr. Moy June ' July j Aug. Sept. Oct, . Nov. Dec- 

53 ^ 



17 



7 S3 




These dates (Jon.llo Mar25) ore in 
different years according to the Julian and 



1 I I i I • ■ I ' ' ■ 

Gregorian Calendars as used in England 



ARMILLARY SPHERE 

(Sees. 5.1 to 5.31 

An aimillaiy sphere is a mechanical device that 
shows the various coordinate systems used in the 
sky. Metal arcs are used to represent the horizon, 
the celestial equator, and the ecliptic, as well as 
noiih-south and east-west coordinates. You will 
find such a device very helpful as you trv to visu- 
alize these imaginaiy lines in the sky. 

Armillaiy spheres, and plastic spheres which 
can serve the same fimction, are available from sev- 
eral scientific equipment companies ifor example, 
the Welch Scientific; Companyi. However, you can 
make a reasonably satisfactoiy substitute from a 
liemispheiical hanging-plant basket puix:hased fixjm 
a garden sufjply store. 

A wire basket 25 or 30 cm in diameter would 
probably by most useful. Two would make a 
sphere. They will have a great cirt:le with ribs going 
toward the bottom (polei. One or two small ciix^les 
of wire paralleling the great circle help support the 
ribs (meridians), ^'ou can add wire circles for other 
coordinates. For example, if the great cirele ivp- 
ixisents the equatcji-, add another gi^eat ciix;le tipped 
at 23V2° to show the ec^liptic. Use bits of |ja|irr to 
locate some of the biighter stai-s. 

NOTE ON THE SIZES AND DISTANCES TO 
THE SUN AND MOON, BY ARISTARCHUS 

I Sec. 5.71 

Ibis summaiA is based on a sc^'lion in,\ Source- 
hook in Creek Science, M. K (lohen and I K Diab- 
kin iMc(iraw-Hill, New York, l!MSi 



Aristarchus assumed that the moon was a sphere 
shining by reflected sunlight. As the figure shows 
in an exaggerated manner, when the moon ap- 
peared to be just half-illuminated, it would be lo- 
cated less than 90° from the sun. Aristarchus meas- 
ured the angle at the earth between the sun and 
moon when the moon appeared to be e.xactlv at 
fii-st quarter i half-illuminated i as 87°. (.Actudlv the 
angle is about 89°50'.i By a complicated geometri- 
cal analysis, he concluded that the sun must be 
between 18 and 20 times farther from the earth 
than the moon. But the distance to the moon was 
known appmximatelv to be several hundred thou- 
sanfls of kilometei-s. Therefore, the distance to the 
sun must be several million kilometers. 



nioot. 




S.r. 



The analysis also provided mlonnation on the 
sizes of the moon and sun. The moon was found 
to have a diameter about one thiixl of that of the 
eai1h ihen the sun. hav ing the same angular size 
at 18 times the moon s distance, must be at least 
18 3. oi- 6 times the diameter of the earth, and 216 
times the volumi" of the earth it) some philoso- 
phers, this raised a (juestion of whether the lai^er 
bodv would move aixuind the smaller one. Note 
that there was no evid«Mi( «• «if concern for the 
masses ol these bodies 



126 



UNIT 2 / MOTION IN THK HEAVENS 



EPICYCLES 

(Sec. 5.81 

The epicycle sketch on Te\t page 149 has a ra- 
dius about half that of the detei-ent. Ptolemy s val- 
ues for the planets are almost the same as those 
used by Copernicus and shown in Tables 6.1 and 
6.2 

The rates of angular motion obtained b\' the use 
of epicycles did not agree well with the obsena- 
tions at certain sections of the orbits. As the bottom 
drawing on Te.xt page 151 indicates, in a series of 
oppositions of Mars no two occurrences were 
identical. To provide a better fit between theoiy 
and obsei^ ations, Ptolemy introduced another geo- 
metrical device called the equant. A planet P 
moved at a constant distance from the center, C. 
(Epicycles around P could also be added. i P mo\ed 
at a uniform angular rate about an off-center point 
C', while the obsener was located on the earth, 
offset equally but oppositely to C. The search for 
stabilitv and uniformity', or predictability, required 
increasingly complex descriptions. 

There is little evidence that anyone believed that 
the planets actuallv moved through space in paths 
described by Ptolemy. His analysis was strictly 
mathematical for the prediction of precise posi- 
tions of each planet separatelv. 

The drawings on Te^t page 151 illustrate a sim- 
plified scale diagram of the Ptolemaic svstem. The 
simplification results from the omission of the ec- 
centrics and equants, and of the several motions 
of the moon. Notice that each planet had only one 
epicycle. All other cyclic motions were represented 
by eccentrics and equants. The very large epicycle 
for\enus occupies about three fourths of the space 
beUveen the earth and sun. To Copernicus this was 
of special interest. V\'ith a protractor, students can 
check the angles subtended at the earth by the 
epicycles of Mercurv and \ enus to see if they agree 
with those shown in the drawing on Text page 141. 
The lower drawing on Te.xf page 151 shows that 
the radii of the epicycles for Mars, Jupiter, and Sat- 
urn had a period of 1 year and were always in line 
with the earth-sun line. This diagram will become 
important in Chapter 6 when we discuss how Co- 
pernicus replaced all these large epicvcles by one 
annual motion for the earth, and derived distances 
to the orbits of the planets. In the Ptolemaic sys- 
tem, each planet was considered to be at a distance 
such that its motion did not quite overlap that of 
the adjacent planets. Actually, to Ptolemy these 
planetarv' distances were not important. 

Our awareness of the adv anced degree of Greek 
mathematics and technical skill was sharply in- 
creased by the discovery of the so-called Antiky- 
thera machine, named for an island near which it 
was found. About the size of a large book, this de- 
vice apparently contained at least 20 gears and a 
crown wheel, as well as pointei-s moving over dials. 
While the use of this complex, but badly corroded, 
machine is still to be unravelled, it is suspected 
that it was used to compute the positions of the 




sun and moon, and possibly of the planets too. The 
machine was recovered from the remains of a ship 
that sank about 65 B.c (See diagram above.) Some 
students may want to read "An Ancient Greek 
Computer" by Derek J. De SoUa Price in Scientific 
American, June, 1959. 

NOTE OX THE 

"CHASE PROBLEM" iSec 6.2) 

The motions of the hands of a clock provide a 
commonplace illustration of the "chase problem" 
described in the Te?ct, Sec. 6.2. Because the hands 
are moving in the same direction and because the 
hour hand continually moves ahead, the minute 
hand must "chase" the hour hand in order to 
overtake and pass it. The questions listed below 
might be used to stimulate class discussion before 
or during a demonstration with a clock. 

1. How many times does the minute hand 
overtake and pass the hour hand during an elapsed 
time of 12 hours? HI timesi 

2. Starting with the hands in the 12 o'clock po- 
sition, at what time will the minute hand overtake 
the hour hand? (l''05"'27'i 

3. Can we derive an expression to show the re- 
lationship between the period of the minute hand, 
the period of the hour hand and the synodic period 
(the interval between successive overtakes i? 

The rate of synodic motion, — , is the difference 

1 1 

■■ . For 

T. T„ 



between the other two rates, or 



T T 
Tj, this becomes T^ = *! r ' 

Observe the hands of a clock or watch through 
12 revolutions of the minute hand. Students may 
predict 11, 12, or 13 "overtakes," but there will be 
only 11. 

Now find the relationship between the period of 
the hands and the synodic period; let T,,, be the 
"sidereal period ' of the minute hand. iThe "side- 
real period" is the time required for the revolving 
object to make one complete revolution, 60 min in 
this case. I Let T^ be the "sidereal period" of the 
hour hand I12 h or 720 min) and let T^ be the syn- 
odic period" of the hands. In 1 min, the minute 



ADDITIONAL BACKGROLND ARTICLES 



127 



hand advances — revolution; its rate of motion is 
— revolution/min. Therefore, in time T^, it must 



make ^ revolutions. In the same time, the hour 

T,„ 

T. 1 

hand must make — revolution at a rate of — rev- 

olution/min. 

Turn the hands of the clock through one synodic 
period. Note that the minute hand makes one com- 
plete revolution, then goes an additional fraction 
of a revolution, or angle, that is the same as that 

T 
traversed hv the hour hand. In svmhols, -^ = 

T T 

— ^ -t- — ^ . This expi-ession can be rearranged to find 

'h 'h 

the value of any one of the quantities. Solving for 
T^ gives the synodic period: 



n = 



T„T 



r„ - T^ 

Substituting 60 min for T^ and 720 min for 7,, 
gives '/; = 65.45 min, l''05"'27^ or 1.019 h. 

This clock analogy is very close to the situation 
for earth and Jupiter. The synodic period is about 
1.092 years, and Jupiter's sidereal period is then 
about 11.8 years. 

A simple de\ice to demonstrate these revolution- 
ary relationships on an overhead projector can be 
constructed fix)m a "dollar" pocket watch, a piece 
of 0.5-cm plastic and a few bits of wire. Remove the 
ciystal from the watch and cement the watch, face 
up, to the center of a 30-cm x 30-cm sheet of plas- 
tic. Cement a 5-cm length of wire to the minute 
hand and a 10-cm length of wire to the hour hand. 
Cement small disks of paper to the ends of the 
wii-es to repi-esent planets. Solder or braze a 17.5- 
cm length of hea\y wire to the winding stem of the 
watch. Place the plastic on the stage of an overhead 
projector and rotate the hands slowly by tvxisting 
the ware soldered to the winding stem. 




With a little more effort, the de\'ice described 
here can be used to show how the retrograde mo- 
tion of the planets occurs. Instead of paper disks 
on the wires, cement thumbtacks with their points 
up. Then cut a verv' thin pointer from balsa wood 
so that it will ride on the two thumbtacks as the 
hands revoke. The pointer will show cleariv the 
apparent backward motion of the outer "planet" as 
seen from the inner one, each time the minute 
hand overtakes the hour hand. 

For the synodic period P., of a planet of heli- 
ocentric period Pp seen from the earth having a 
period P^ , the equation above becomes T^ = 

T T 1 

— . Reformed as — to gK'e the rate of motion, 
'p ~ 'fc ' 

this becomes 



But the svnodic period T^ can be found bv obser- 
vation, while the earth's period 7,^ is known as 1 
year. Then 



which becomes 



T = 



-a 



For Mars, T^ = 780 days, while Tf = 365 days. Then 
365 davs 365 davs 



1 - 



365 
780 



1 - .469 



= 687 da\'S 



If the planet moves inside the earths orbit, the 
planet gains on the earth and the equation be- 
comes 



111 
To n T. "^ 



T, 



1 -t- 



a 



Consider Venus, which comes to maximum elon- 
gations at an average inter\'al of 584 da\s. Then 

365 davs 365 



•^Vcnu., = 



1 + 



1 -t- 0.626 



= 225 da\s 



ATMOSPHERIC REFRACTION 

iSec. 6.71 

Any ray of light entering the earth's atmosphere 
at a slant is bent dowiiward i refracted i, with the 
result that the source appears to us lo be higher 
above the horizon than it rvallv is The farther the 
bodv is fix)ni the observers zenith i straight owr- 
hcad 1. the greater is the length of the air path and 



\'ZH 



VWT 2 / MOriUX l\ THE HEAVENS 



the greater the cingle at which the ra\' enters the 
atmosphere. .As a result the amount of de\iation 
due to refraction increases rapidh near the hori- 
zon. The cuned atmosphere acts like a thin lens. 

Because the amount of the refraction increases 
rapidl\- near the horizon the observed image of the 
setting sun is distorted. The bottom limb of the sun 
is half a degree farther from the zenith than is the 
upper limb of the sun. Light from the lower limb 
is refracted upward more, and the sun takes on an 
elliptical, or o\al. appearance. 

.An interesting consequence of this refractive ef- 
fect is that the actual sun no atmosphere has set 
below the horizon before the lower limb of the ap- 
parent I refracted I sun touches the horizon. Notice 
also that most of the blue and much of the green 
light frx)m sunlight is scattered b\ the atmosphere. 
The onl\ light not strongK- scattered i Ra\leigh scat- 
tering) is red, the color of the setting sun. 

If \'OU wish to expand on this refractive effect 
and the color due to scattering, consider the ap- 
pearance of the moon in total eclipse. Students 
ma\- be surprised to learn and perhaps can con- 
firm from their own observations that the eclipsed 
moon does not go black. Instead it appears cop- 
pery- red, e\en in the middle of the earth s shadow. 
With a bit of suggestion, students can conclude 
that the thin edge of the earth s atmosphere per- 
pendicular to sunlight is acting like a thin lens. 
Thus, some sunlight is refracted into the earths 
shadow. The path of this light through the earth s 
atmosphere is twice as long as the light we see 
from the setting sun. Therefore. onl\ red light re- 
mains in the rays refracted into the earth s shadow. 

ClearK", atmospheric refraction would result in 
errors in star positions unless corrected. Long se- 
quences of careful observations, such as those 
made by Tycho, are needed before the corrections 
can be determined. 

RELATIONS IX AX ELLIPSE 

iSec. 7.31 

a = major axis 

b = minor axis 

c = distance between foci 

e — eccentricit\ 



changes with shapes, or how round' is an ellipse 
of large e. 



_c b _ 

a a 


(1 - 


- e^)-' 


0.1 




0.995 


0.2 




0.98 


0.3 




0.955 


0.4 




0.916 


0.5 




0.866 


0.6 




0.800 


0.7 




0.715 


0.8 




0.600 


0.9 




0.435 


0.95 




0.313 



Gi\e the equation 



(^}- 



1 - e"' and suggest 




ABOLT MASS 

Sec. 8.8' 

In Unit 1, Sec. 3.7, the concept of mass was in- 
troduced as something that is measured by inertia, 
the resistance to a change in motion This sort of 
mass is called inertial mass, and it is used in New- 
ton s second law. F — ma. 

We measured the inertial mass of a body b\ 
seeing how its motion changes under the action of 
a known force. 

In Sec. 3.8 the concept of mass was connected 
with gra\itational forces of attraction. Here the 
mass of a bod\ is a measure of the gravitational 
force that other bodies exert on it, or that it e.verts 
on other bodies. .Assigning a \alue to the mass of 
one particular bod\ we can, in principle find the 
relati\ e mass of am other bod\ b\ measuring the 
gra\itational force between the tvvo. That sort of 
mass is called gra\itational mass. 

The question now arises whether the inertial 
and gra\itational masses of a bod\ are linearh pro- 
portional to each other. If we know how bodies 
accelerate when experiencing onl\ a gravitational 
force, then we can tell whether inertial and gra\i- 
tational masses are proportional. Consider tvvo 
bodies A and B with inertial masses m^ and m^ 
and gravitational masses m^ and m^ . We put bod\ 
A a distance R irom a fixed third bod\ with gravi- 
tational mass A/, and we ask for the acceleration of 
bod\' A when it experiences onl\ the gravitational 
attraction due to A/. Using Newton s second law, 
we have 

Gm ^ A/ 



R- 



"^v\GA/ 



If we do the same with bodv B we get 

'"^B,\GA/ 
.m„ / fi- 



ll) 



(21 



that someone woric out this table and graph it. Stu- 
dents will be surprised to see how rapidh e 



Now if a bod\'s gravitational mass is linearh- pro- 
portional to its inertial mass: that is, m^ = km^ in 
general, where A; is a universal constant then 

m ^ — km ^ 



ADDITIONAL BACKGROLXD ARTICLES 



129 



and 



m„ = km 



Hi 



If we put the first of these expressions into Equa- 
tion 111 and the; second into Kquation I2i, then we 
get 

GM 



3 . — 3„ — K 



R' 



The two accelerations will be equal. To check this 
analysis all we need do is peifonn the experiment 
and see if the accelerations are equal. 

If we use the earth as M, we would conclude 
that, at a given distance from the earth's center, all 
freely falling bodies should have the same accel- 
eration. Therefore, experimental verification that 
this is true would be proof that inertial mass is 
propodional to gravitational mass. Unfortunately, 
the experiment is very difficult to perform with 
high precision. 

Isaac Newton devised an experiment that tested 
the proposition in an indirect way. A pendulum 
bob is not a freely falling object, but in its motion 
to and fro it does accelerate, and the value of its 
acceleration governs the rate of oscillation. i\ev\1on 
was able to show that only if inertial mass is pi-o- 
portional to gravitational mass will the rate of os- 
cillation be the same for pendulum bobs of differ- 
ent masses. Newton made a hollow pendulum bob 
in the form of a thin metal shell into which he put 
different materials, always being careful to see, by 
using an equal arm balance, that the weight of the 
material was the same each time. Since weight is 
a measure of giavitational mass, any difference in 
the rate of oscillation of the pendulum would be 
due to a difference in ineilial mass. No such dif- 
ference appeared, and Nev\'ton concluded that in- 
ertial and gravitational masses are equivalent. 




But 



One can test for each bob separately, then test 
for various materials. 



Professor Dicke and his co-workei-s at Princeton 
Univei-sity believe they have shown the equivalence 
of ineilial and gravitational mass to within 1 pai1 
in lO" pai-ts. 

THE MOON'S IRREGIT^R MOTION 

(Sees. 8.3 and 8.!)i 

The obs(MM!d motion of llic moon contains many 
small variations that cannot be predicted In the 
simple assumption of a gravitational Ibrce ixMvvccn 
two mass points ni, and lu^, 



Newton's investigations accounted for some of 
these discrepancies, but he studied only a few. 
Nevertheless, his theoretical results were reason- 
ably close to the observed values of his time. 

Though the process of applying the law of uni- 
versal gravitation to separate sets of two-mass sys- 
tems may seem to allow relatively easv solutions 
for motions, what happens when a third body gets 
involved? Thus, the sun-earth and earth-moon 
systems appear to be simple gravitational phenom- 
ena, but the reality is a single sun-moon-earth sys- 
tem that becomes so complex that a solution of its 
motions by gravitational theory becomes possible 
only under ver^' limited conditions. 

U'hat if another large bodv ilike Jupiter at 5 AU 
from the sum were on the other side of the comet 
in the figur-e below? The student could then realize 
that the prediction of or+)ital motion would have 
to be painstakingly worked out by adding up all 
the acceleration vectors concerned. This suggests 
the real problem of computing paths or orbits for 
space probes, moon missions, and Mars and \'enus 
fly-bvs, where all the planets are attracting the 
space ship. 

The computations are so lengthv and complex 
that precision orbits, docking and other maneu- 
vers would be impossible without high-speed com- 
puters. 



Sun 




MEASLTUiXG G 

iSec. 8.81 

It might also be easier for some students to un- 
der-stand a method of measuring G that was de- 
signed and canied out by a German physicist. Von 
Jolly, in the mid-nineteenth century . He used an 
equal-arm balance instead of the rather compli- 
cated tor-sion apparatus of Cavendish. On one side, 
\ on Jollv put a spherical (lask filled with mercury 
that he balanced with weights in the other pan. 
rhen he put a large lead sphei-e below and close 
to the flask of mercury . He could determine the 
distance between the two spheres. The gravita- 
tional force bet\veen the two spheres cairsed the 
side of the balance with the flask to dip down 
slightlv rhus, the weights necessary' to rebalance 
the erurivalent were a measure of the F,^ between 
the spherxjs. 



fkt*r 




:io 



UNIT 2 / M()TICI\ l\ THH IILAXIJNS 



Here is a set of figures typical of the \'on Jolly 
experiment that your students can use to calculate 
G for themselves: 

m, Imass of mercur\'i = 5 kg 
m, Imass of lead sphere! = 5775 kg 
R (between sphere centersi = 0.57 m 
Fg^^ = 0.59 mg I added for balance i 

== 5.9 X 10 'kg X 9.8 m/sec" 

= 57.8 X 10"" \ 



Then 



F,„,. = 



find 



Grriim, 
R- 

FR^ 

G = 

/Ti,m, 

57,8 X 10"' X i5.7 X 10 'i" 

G = 3 

5 X 5.775 X 10 

G = about 6.5 X 10"" m^/kg • sec' 

\ ou might ask the students to indicate which of 
the abo\e measurements present probable sources 
of error, and wh\ . For e.vample, does the lead ball 
also attract the weights in the other pan'^ 

Another important point that could be made 
here is that the Ca\endish e.xperiment represents 
an inertial method of measuiing G, while the \ on 
Jolly experiment uses the gravitational method. 
You might wish to refer back to Sec. 3.8, Unit 1. 

THEORIES (AN EXTENSION) 

iSec. 8.101 

Theories often ha\e important practical appli- 
cations. This is less apparent in the astronomical 
context, although the de\elopment of instruments 
and mathematics were influenced. Many other ex- 
amples of more direct practical consequences will 
appear throughout the course. Cun-entl\', rocket 
de\elopment is haxing a major impact upon the 
design of many commerical products: Such e\ents 
often occur without public notice. 

As human creations, theories are produced, de- 
veloped, judged, and applied by people uith per- 
sonal prejudices and frailties. Therefore, the com- 
bined judgment of mam scientists is safer than the 
reaction of one. Vet historv ma\' show that, for a 
time, the one might be right and the majority 
wrong. It is important here to tr\ to replace the all- 
too-common snap-judgment, good-or-bad e\'alua- 



tion of new ideas by a critical interest in theories 
as possibilities. One can be informed about and 
interested in a new theon' without necessarily ac- 
cepting or rejecting the theoiy. Suspended judg- 
ment is often a mark of maturity. 

Can the students suggest other theories in sci- 
ence, government, art, or economics, which at first 
seemed shocking, \et have become commonly ac- 
cepted? Perhaps impressionistic art, now com- 
monly used in advertising, would be an example. 
What was the public reaction to Manet and Pi- 
casso? Or have someone interested in music report 
on the initial reception of the compositions of V\'ag- 
ner, Brahms, or Stravinsky, i The latter s ballet. The 
Rite of Spring, was loudly booed when first per- 
formed in Paris in 19131. Or perhaps consider the 
acceptance of James Joyces L'lysses. 

Contest among ideas, the trial by combat, is es- 
sential in science and everv' other field of human 
creation. Supine acceptance or adulation, or casual 
rejection, of a great individuals creation is naive. 

Theories are changed over time. They are not 
fixed and permanent to be idolized, but rather are 
working tools to be used and i-esharpened. Rarely 
is a theorv completelv abandoned. Most are mod- 
ified, but some are replaced. Scientists, like other 
people, cannot tolerate a complete absence of 
some sort of explanation. They will not completely 
abandon an old theorv', ev en if it is known to have 
serious limitations. ,At least it worked in some 
cases, and still satisfies some phenomena. 

In many ways, scientists are artists. Each is a 
specialist in the study and interpretation of some 
set of phenomena. Each brings to his or her work 
a general sense of what txpes of theories and ex- 
planations are satisfving. That is, scientists have 
personal stvies. Some are mainly concerned about 
the precision of measurement and the design of 
equipment. Others look at theories as the bases for 
predictions. Still others trv' to imagine a varietv' of 
possible explanations, some of which are more 
daring than others. Einstein and Fermi are revered 
because they were v erv' imaginativ e and would play 
with possibilities, turning them this way and that 
to see what consequences might result. In this way, 
the individual characteristics of the scientist are 
more apparent. .At least initiallv, the possible line 
of a theorv' is alwavs qualitative and often pictorial. 
The sort-of-like-this imagery comes first and re- 
veals the basic aesthetic approach of the individ- 
ual s vision of the world in which we live. 



ADDITIO.VAL RACKGROL.VD ARTICLES 



131 



Brief Descriptions of Learning Materials 

SUMMARY LIST OF UNIT 2 MATERIALS 



Experiments 

K2-1 Xaked-Eye Astronomy 

E2-2 Size of the Earth 

E2-3 The Distance to the Moon 

E2-4 The Height of Piton, A Mountain on the 

Moon 

E2-5 Retr-ograde Motion 

E2-6 The Shape of the Earth's Orbit 

E2-7 Using Lenses to Make a Telescope 

E2-8 The Orbit of Mars 

E2-9 Inclination of Mars' Orbit 

E2-10 The Orbit of Mercury 

E2-11 Stepwise Appi-oximation to an Orbit 

E2-12 Model of the Orbit of Halley's Comet 

Demonstrations 

D28 Phases of the moon 

D29 Geocentric epicycle machine 

D30 Heliocentric model 

D31 Plane motions 

D32 Conic-sections model 

Film Loops and Filmstrips 

Retrograde Motion of Mars iFilmstripi 
LIO Retrograde Motion: Geocentric Model 
Lll Retrograde Motion: Heliocentric Model 
L12 Jupiter Satellite Orbit 
L13 Program Orbit. I 
L14 Program Orbit. II 
LIS Central Forces: Iterated Blows 
L16 Kepler's Laws 
LI 7 L'nusual Orbits 

Reader Articles 

Rl The Black Cloud 

by Fred Hoyle 
R2 Roll Call 

by Isaac Asimov 
R3 A Night at the Ohserxatory 

by Henry S. F. Cooper, Jr. 
R4 Preface to De Hevolutionihus 

by Nicolaus Copernicus 
R5 The Starry Messenger 

by Cialileo Galilei 
R(i Kepler's Celestial Xlusic 

by I. Bernard Ciohen 
R7 Kepler 

by (Jerald Molton 
R8 Kepler on Mars 

by Johannes Kepler 
R9 Newton and The Principia 

by C. C. Gillispie 
RIO I'he Liws of Motion and Proposition One 

by Isaac Xewlon 
KI I I'he Carden of Epicurus 

h\ Anatole France 
Hli: / 'i}i\crsal Ctravilation 

in Richard P. Feynman. Robert B I^Mghton. 

aiui Matthew Sands 



R13 An Appreciation of the Earth 

by Stephen H. Dole 
R14 Mariners 6 and 7 Television Pictures Prelim- 
inary Analysis 

by R. B. Leighton and others 
R15 The Boy Who Redeemed His Father's Name 

by Terry Monis 
R16 The Great Comet of 1963 

by Owen Gingerich 
R17 Gravity Experiments 

by R. H Dicke, P G. Roll and J. U'eber 
R18 Space The Unconquerable 

by Arthur C. Clarke 
R19 !s There Intelligent Life Beyond the Earth? 

by IS. Shklovskii and Carl Sagan 
R20 The Stars Within Twenty-Two Light Years That 
Could Have Habitable Planets 

by Stephen Dole 
R21 Scientific Study of Unidentified Flying Objects 
from Condon Report with introduction 

by Walter Sullixan 
R22 The Life-Ston' of a Gala,\y 

by Margaret Burbidge 
R23 Expansion of the Universe 

by Hermann Bondi 
R24 Negative Mass 

by Banesh Hoffman 
R25 Three Poetic Fragments about Astronomv 

by William Shakespeare, Samuel Butler and 

John Ciardi 
R26 The Dyson Sphere 

by I. S. ShkloNskii and Carl Sagan 

Sound Films (16 mm) 

F6 Universe 

F7 MysterA' of Stonehenge 

F8 Frames of Reference 

F9 Planets in Orbit 

FIO Elliptic Or-bits 

Fll Measuring Lar^e Distance 

F12 Of Stars and Men 

F13 Tides of Fundy 

F14 Harlow Shapley 

F15 Uni\ersal Gra\itation 

F16 Forces 

F17 The Invisible Planet 

F18 CMose-up of Mar-s 

F19 Of Stars and .Men 

F20 i\e\\1on s Equal Arr^as 

Transparencies 

ri;j stellar Motion 

I 14 rh(> Celestial Sphere 

115 Retix)giade Motion 

T16 Eccentrics and E(|uants 

117 Oi-t)it Paramet»*rs 

IIS Motion under C'entral Fchtcs 



132 



UMT 2 / MOTION IN THE HEAVEiVS 



FILiVI LOOPS 



Quantitative measurements can be made v\ith F/7f77 
Loops marked (Labi, but these loops can also be 
used qualitatively. 

LIO RETROGRADE MOTION: 
GEOCExNTRIC MODEL 

A machine was constructed in which the planet is 
represented by a lamp bulb on an epicvclic aiTn 
revoKing aixjund a deferent. The camera is at the 
position of the stationary earth, pointing in a fixed 
direction in space. 

11 RETROGRADE MOTION: 
HELIOCENTRIC MODEL 

The epicycle machine is used with the camera on 
an arm re\ol\'ing around the sun. The camera 
points in a fixed direction in space. 

L12 JLTITER SATELLITE ORBIT 

Time-lapse photography, at 1-min interxals, of the 
motion of Jupiter s satellite To. The penod of rev- 
olution can be measured, the sccde is gi\en, and 
hence Jupiter's mass found. iLab) 

L13 PROGRAM ORBIT. I 

A computer is programmed to calculate the same 
orbit that a student calculates in the laboratory' 
when doing E2-11, "Stepwise Approximation to an 
Orbit. " The result is displayed on an X-Y plotter. 
Because of th stepwise approximation used, the 
orbit fails to close up exactly. 

L14 PROGRAM ORBIT. II 

The computer calculates an orbit using memy more 



points than in the preceding loop: This time the 
orbit closes up. The display on the X-Y plotter is 
repeated on the face of a cathode-ray tube iCRT). 
All other computer loops in this series use CRT 
display. 

L15 CENTTIAL FORCES: ITERATED 

BLOVV^S (COMPUTER PROGRAM) 

The computer is programmed to gi\e sharp blows 
to a mass at equal time intervals. The blows are 
directed (at random i tov\'ard and away from a cen- 
ter of force, and the magnitude of the blows is also 
random. The law of areas can be \'erified. iLabi 

L16 KEPLER'S LAWS 

(COMPLHTER PROGRAM) 

Two planetary orbits in an in\'erse-square force 
field are programmed for display on the CRT. The 
positions of the planets are shov\Ti at successive, 
equally spaced time intervals. A^ three of Kepler's 
laws can be verified. (Labi 

L17 L^NX^SUAL ORBITS 

(COMPUTER PROGRAM) 

The computer is programmed to display two mo- 
tions taking place in central fields that are not ex- 
act in\ erse-square fields. One perturbation gi\es an 
advance of perihelion, as for Mercury's orbit. The 
other perturbation gives a catastrophic orbit in 
which the planet spirals into the sun. 

Note: A fuller discussion of each Film Loop and suggestions 
for its use will be found in the section entitled "Kilm Loop 
.Notes." 



FILMSTRIP 



RETROGRADE MOTION OF MARS 

Three sequences of photographs taken at irregular 
intervcds show^ Mars and Jupiter in retrograde mo- 



tion. The angular size and durations of retrograde 
motions can be determined. This filmstrip defines 
retrograde motion for students. 



SOUND FILMS (16mm) 



F6 LTMVERSE 

B & V\', 26 min, National Film Board of Canada, 
available from XASA Films. A triumph of film art, 
creating on the screen a vast, awe-inspiring picture 
of the uni\erse as it would appear to the voyager 
through space. Realistic animation takes one out 
beyond our solar system, into far regions of space 
perceived by the modem astronomer. Beyond the 
reach of the strongest telescope, past moon, sun, 
Milk\' Way, into galaxies yet unfathomed, one tra\- 
els on into the staggering depths of the night, as- 
tonished, spellbound at the sheer immensity of the 
universe. The starting point for this journey is the 
Da\id Dunlap Observatoiy, Toronto. Se\enteen film 



awards, including International Film Festival, 
Cannes, France: International Film Festival, Edin- 
burgh, Scotland: British Film Academy, London, 
England. 

F7 MYSTERY OF STONTHENGE 

B &. VV, 58 min (tivo partsi, a\'ailable from McGraw- 
Hill. This film could be shown to awaken interest 
in the explanation of such structures built long 
ago. It was filmed by The Columbia Broadcasting 
System and shown on television in the United 
States and in Britain. The vigorous confiict of in- 
terpretations between F»rofessor Hawkins and oth- 
ers is notable. 



BRIEF DESCRIPTIOXS OF LEAR\L\G VUTERIALS 



133 



F8 FRAMES OF REFERENCE 

B &. W, 28 min, PSSC, Modern Learning Aids. If you 
haven't shown it jireviously, Sec. 6.4 might be a 
good place. Although it presents much more in- 
formation than is necessaiy, it is an excellent film. 
It does gi\e students the idea that the appearance 
of events may depend upon the fiame of i-efer-ence. 

F9 PLANETS IN ORBIT 

B &, VV, ID min, KBI\ I tils film presents animated 
i-epiesentations of some of the diffei-ences between 
the Ptolemaic and Clopeinican systems. 

FIO ELLIPTIC ORBITS 

PSSC, Dr. A. V. Baez, Cat. #0310, Modem Learning 
Aids. I'his film might be used to make clear to stu- 
dents what ai-ea is being discussed in the law of 
areas. 

Fll MEASURING LARGE DISTANCE 

PSSC, Dr. F. G. Watson, Cat. #0103, Modem Learn- 
ing Aids. With a senes of models, the film stresses 
the use of triangulation as the primaiA' means for 
determining large distances. Toward the end of the 
film other techniques based on photometry are il- 
lustrated as a means of extending the distance 
scale when triangulation is no longer possible. 

F12 OF STARS AND MEN 
(ABOUT GALILEO) 

This biography is axailable from Center for Mass 
Communication, Columbia University Press, New 
York, \'Y, 10025. 

F13 TIDES OF FUNDY 

Color, 14 min, National Film Board of Canada, avail- 
able from NASA Films. A fascinating study of the 
phenomenal tides in the Bay of Fundy on (Canada s 
Atlantic coast and how they affect the life of the 
region. 

Animated pictures explain the forces of moon 
and ocean and the earihs r-otation that together 
create in the Bay of Fundy the highest tides in the 
world. 

Filmed with an eye for the dramatic, this film 
brings to the sci-een scenes that are tiuly amazing. 
It shows, in this tiny pocket of the sea, a sequence 
of cause and effect that inxohes the \eiy forces of 
the univei"se. It is a film that will appeal to eveiy 
audience. 

F14 HARLOU' SILVPLEV 

30 min, KncN cioijcdia Britannica Films, #1«()(S. Ibis 
film discusses major astixjuomicai discoveries and 
how they ha\e influenced philosophx . ix'ligion, and 
our orientation to the uoiid 

F15 irM\TRSAL GRyVMTATION 

PSSC, 31 miti, a\ailai)le lron\ Modern Learning .Aids, 
#0305) In this film, the lau' of uni\ei-sal graxilation 
is derived lor an imaginan' solar ,s\strm of one star 
and one planet 



F16 FORCES 

PSSC, 23 min. Modem Leaming Aids. This film is 
relevant to Unit 2. It introduces mechanics in gen- 
eral and shows a qualitative Cavendish experiment, 
in which the gravitational force bervveen two small 
masses is demonstrated 

F17 THE LWISIBLE PL/VNXT 

NET Film Service. As this film opens, students meet 
Peter Van de Kamp, director of the Sproul Observ- 
atory at Swarlbmor-e College, and learn of his in- 
terest in Barnard s star, a small star near us in the 
solar system. With Dr. \an de Kamp and Mr. Her- 
beri as guides, the student leams about the oper- 
ation of the large refractor telescope, the use of 
photographic plates, the recorxling and analvsis of 
data, and the results of data car-efully recorxled for 
over 25 years. From this data. Dr. \ an de Kamp and 
his colleagues were able to determine the apparent 
presence of a small planet near Barnard s star that 
causes a small perlur+jation or wobbling. Ihe pre- 
cision, time, and car-e in astr-onomical observations 
are portrayed with impact in this film. Recom- 
mended for use in physics or in earlh science. 

F18 CLOSE-LT OF MARS 

NLT Film Ser\ice. Ibis is the story of the dex-el- 
opment of the camer^a system aboard the space- 
craft Mariner I\ that took the historic photographs 
of the surface of the planet Mars in mid-Jul\ , 1965. 
The audience follows Roberl Leighton, professor of 
physics at the Califomia Institute of Technologv , 
as he and the scientist-engineers worWng with him 
tackle the problem of designing, building and using 
a camei-a system that can weigh no more than 11 
pounds and use only 10 watts of electricit\ . In 
viewing this film, students can sense the difficulties 
surrxHinding the assignment and the excitement of 
success as the fir-st filrTis are relayed back to earth 
from 520 million km out in space. Par1icular4\ rec- 
ommended for students of physics or electronics. 

F19 OF STARS ANT) MEN 

Color-, 53 min, available fixjm Brandon Films Inc. 
Prxiduced and adapted b\ John and Faith Hubley 
fr-om the book b\ Hariow Shaplev . Ihe film helps 
the audience to locate our place in the universe of 
atoms, prxjtoplasm, stars, and galaxies. Our rela- 
tionships to space, time, energN'. and matter are 
explored. 

F20 NTWTON'S EQl'AL AREAS 

Cioloi-. 8 min Briice and Kalheiine (dm well, Alfred 
Bork .Available from Interriationa! Film Bureau. 
This animated film is based on Isaac Newlon s sim- 
ple geometrical pn)of of the law of areas for anv 
central forx'e It fir-st established the laws of motion 
in Ihe form needed b\ Newton, goes thrx)Ugh New- 
ton s prx)of for- several difiervr-it cases, including the 
limit considerations, and then shows several ex- 
amples ifir"st simple then complex' of ecfual areas 
being traced out with a cenlral foix »■ 



i:i4 



HIVIT 2 / MfniOIV l\ THK HKAVKNS 



ADDITIONAL FILMS 

Se\eral astix)nomical computer-produced, color 
and sound films, 7 to 8 min, ha\e been pixjduced 
by Dr. M. L. Meeks and are available for sale or 
rental by the Houghton-Mifflin Companx', Depart- 



ment M, 1 Beacon Street, Boston, Massachusetts 
02107 Those related to Unit 2 are. The Motion of 
.Attracting Bodies, and ' Planetary Motions and 
Kepler s Laws." Silent loops ai'e available for both 
Technicolor super-8 and Kodak spool pi-ojectors. 



TRANSPARENCIES 



T13 STELLAR MOTION 

Displax s a Uvo-sphei-e universe explanation of ap- 
parent stellar motion as obsened at mid-northern 
latitudes, the equator and north pole. iSecs. 5.1, 5.5i 

T14 THE CELESTIAL SPHERE 

Illustrates the scheme of the celestial sphere, in- 
dicates the meaning of equino.xes and solstices, 
shows the sun s path in relation to the zodiac, and 
gi\es the meaning of declination, right ascension, 
and celestial longitude and latitude. (Sees. 5.1, 5.5, 
E2-6. E2-8\ 

T13 RETROGRADE MOTION 

Elxplains apparent retrograde i westward i motion of 
an outer planet b\' means of heliocentric model. 
(Sec. 5.61 



T16 ECCENTRICS ANT) EQUANTS 

Displays features of geocentric schemes of Ptolemy 
in accounting for observed planetary motion. iSec. 
5.81 

T17 ORBIT PARAMETERS 

Illustrates the six elements that define any orbit. 
iSecs. 7.2, 7.3, E2-8. E2-10< 

T18 MOTION L^T)ER CENTRAL FORCE 

Illustrates in geometric steps that objects subject 
to a central force obey Kepler's law of areas iSec. 
8.3, E2-11I 



.Note:, A fuller discussion of each Transparency and sugges- 
tions for its use will be found in the \ isu-Book contain- 
ing L'nit 2 Project Physics Transparencies. 



Demonstration Notes 



D28 PHASES OF THE MOON 

The following model will help to clarif\ the phases 
of the moon. Attach a ping pong or tennis ball to 
a thread. Then in a darkened room have students 
watch the phases of this moon as you swing it: 

(a) around a single lamp bulb inot too bright* 

(b) around their heads with the lamp bulb a few- 
meters to one side. 

The latter matches best with our observations of 
the moon s phases. 

D29 GEOCENTRIC-EPICYCLE MODEL 




In this demonstration, students themselves play 
the part of planets. As well as showing that the 
geocentric-epic\'cle mode of the solar system gives 
retrograde motion, it demonstrates the effect that 



an observer s own motion can ha\e on his or her 
view of the motion of another object. 

Student A, representing the earth, stands still 
while the two others, B and C, mo\e around; B in 
a circle, and C. representing a planet, in an epi- 
CA'cle. A length of string i about 5 m* between A and 
B. and a shorter one i about 2.5 mi between B and 
C keep the radii of the circles constant. Student C 
will ha\e to mo\e fairh' fast to make one or more 
re\olutions about B. while B, walking at a steady 
rate, makes one revolution about A. Once the\ have 
established appropriate speeds, the\ must tr\' to 
keep them as constant as possible. In this dem- 
onstration, A is the earth, B is mereh a point in 
space, C is a planet. Ecirth' observes the motion 
of the planet with reference to a distant back- 
ground, such as trees, goalposts, school buildings, 
which represent "the fixed stars. Does the planet 
alwa\'s appear to be mo\ing in the same direction? 
When does it retrograde? How long does the retro- 
grade motion last? 

D30 HELIOCENTRIC MODEL 

In this model, the stationary student represents 
the sun. The earth, now displaced from its position 
at the hub of the uni\ erse, moves around in a circle 
(radius about 4 mi. The third student, representing 
Mars, moves around the sun in the same sense in 
a larger circle i6 mi. If eaxth and Mars Wcilk in step, 
but Mars takes a shorter step earth s period will 



DE.MONSTRATIOV NOTES 



135 




be considerably less than that of Mars and their 
relative motions will approximate fairly vvell the 
actual movement of the two planets. Again, earth 
is asked to describe the relative motion of Mars as 
it appears against the distant background of fixed 
objects. Does the motion appear uniform? Is there 
retrograde motion? When does it occur? Retro- 
grade motion of an inner planet may be more dif- 
ficult to spot. I'ly these parameters: earth's orbit, 



10-m radius; Mercury's orbit, 4-m radius; Mercuiy 
takes two paces for every one taken by earth. 

D31 PLANE MOTIONS 

rhe importance of Kepler s use of motion in a 
plane cannot be overstressed. V\ith Unit 1 in mind, 
have students make fists with their left hands to 
i-epresent the sun. Have them hold pens or pencils 
in their right hands to represent a point in space 
and a velocity vector. You can anticipate Chapter 
8 and ask students what forces are acting on the 
body. iThe only foix;e is the central pull of the sun.i 
What initial motion does the body have? iThe ini- 
tial \elocity vector is represented by the pen or 
pencil.) But one point and a line define a plane. 
What would you infer if the body did .NOT mo\'e 
in a plane? iSome other force is acting from a place 
not in the orbital plane. i This planar assumption 
is applied in the acti\it\' in which the orbital incli- 
nation of Mars is derived from the observations of 
the positions of Mars north or south of the ecliptic, 
E2-9. 

D32 CONIC-SECTIONS MODEL 

If the mathematics department has a model of a 
cone, use it to let the student see the natural oc- 
currence of ellipses and other conic sections. 



Experiment Notes 



E2-1 NAKED-EYE ASTRONOMY 

Equipment: 

Astrolabe 
Constellation Chart 

It is best to have students make their own ob- 
servations rather than use the results in E2-1 ex- 
clusively. It is also most important to observe direct 
motion, that is, eastward motion with respect to 
the fixed background of stars. 



A TViC etin or. t«ptcmbcr Hr-d. 




Answers to questions 

Part A 

The graph was drawn fixjm the data in E2-1 Part 
A. Note that noon occurs after 1:00 P..\i because of: 

1. the equation of time 

2. the observer's location west of the 75th meridian 

3. use of daylight time 

Answers are taken from the graph. 

1. 49 '/2° 

2. Since the sun was on the equator September 
23, the latitude of the observer is 90" minus 
sun s noon altitude, or 90° - 49' 2° = 40V2° 
North. Refer to the drawing on the light. 

3. The sun was highest at about 1:18 p\t EST. 

4. When the sim was highest. 

5. Just after sunrise or just before sunset. 

6. Since the speed of the earth ortiiting around 
the sun is not constant, it happens that on Sep- 
tember 23 the e\ent of smallest shadows, that 
is, so-called noon, will be 8 min before l.-OO iv.\! 
E.S.T. or 12:52. on the 75th meridian. This 8 min 
of time is called the equation of time correction. 

The observer sees noon at 1:18 E.S.T., or 26 min 
later than the per-son on the 75th meridian C"on- 
seqirentK , the observer must be west of the 75th 
mi'iidian. I his is because the 75th meridian rxitates 
under the sun lir-st and then 2li min later the ob- 
s(M"ver- rotates under- the sun and observes that 



:i« 



UNIT 2 / MOTION IN THE HEAVENS 




apparent path 
of Sun 



shadows are shortest. Since 1° of longitude is 
equixalent to 4 min of time, 26 min is equivalent to 
6Vi° of longitude. Thus, the obsener is 75° + 6V2° 
= SlVz" West. The observer s latitude is 4OV2'' N. 
On a map this point is close to Canton, Ohio. 

Part B 

The data in part B are plotted in the figure below. 

7. 43° 

8. Latitude = 90° - 43° = 47° N 

9. Quebec, Canada 

10. 303° - 238° = 65° range 

11. A f = 7.6'' - 4.4'' = 3.2'' = 4*^12'" 

12. Shortest day: s'' 48'" 
Longest day: lo*" 12"^ 



80 



< 

© 20 



0° 



re/K^ or euM o&set2_vATioN'5 



A 



-A. 



A 



^A 



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A 



/J 



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□ 






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□ 



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iio 






H 



170 



I |T] OAL-tttude 050 

I \ 



-h 



\ 



[7] Azimuth 



1 ^ I \ r~^ \ n 

Jan Feb Mat' Apr May Jun July A»>g Sept Get Nov Dec 
TIME (months) 




§ 

O 



Part C 

13. The plot of the positions of the moon follows. 



4o* 






|G9* 



< . o 

OcA26 



0(47° 
oct.fG 



Oct 24- 



Oct 22 



ootao 



57° 



J 
OcX.\e> 



) 62° 
Oct 16 



Ho>^LZ.or> 



-a^ 



70 



90 



n 

1^0° 



\70° 



I" 

I90° 



ZlO 



AzLinoth Y/est — > 

/^Looking -south ~) 



EXPERIMENT .NOTES 



137 



Part C 

13. The; plot of the positions of the moon follows. 
For students who have tioubie; \'isualizin/^, 

the drawing at the right may he helpful in 

accounting for the phases of the moon plot- '^ 

ted on page 137. '^^V\ ^V\ 

14. New moon Oct. 14 \o/Z€> \)\ \\ 
Fii-st quarter-moon Oct. 21 \^-j 
Full moon Oct. 29 , 

15. Sketches aie shown on the plot ' „ 
of positions oi the moon on page "^^ tfT~f 

bright portiori 



•^^ 



of Sun subtends \ 
thi5 ans^<^ 




Barth 



Light 

< 

son-set on the. < 

Above cUtes _ 

Son 



Part E 

The sun will ha\e the same longitude after 364'/2 
moie days. 

Since the perioti of orbit of the earth is constant, 
the date when the question is asked does not mat- 
ter. 

The sun moves about 1° per day. 

The sun moves fastest along the ecliptic in Jan- 
uaiv. 



Mercun' is separated fix)m the sun by no more 
than 28^ 

About e\eiy 88th day Mercur>' will pass in front 
of the sun and then about 44 da\ s later it will pass 
in back of the; sun i superior conjunction i. Student 
answers will differ frxim this because of the uncer- 
tainity of the plot and because of the eccentricitv' 
of the orbit of Mercurv. 



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X'AH 



UNIT a / MU TIUM I.N THIi IIL/WEXS 



E2-2 SIZE OF THE EARTH 

Equipment: 

Astrolabe 

This is an excellent experiment for a few stu- 
dents to do as a special laboiatory, because the 
students must communicate with another school 
at least 300 km a\va\ on the same longitude. Some 
practice with the astrolabe lor a more refined in- 
stalment i is ad\isable in older to measure the al- 
titude of a star or the sun accurateK . All this takes 
time. However, the pa\oft"is monumental, for is not 
the size of the earth a rather dramatic measure- 
ment? 

An entire class ma\ woik on E2-2, but, if so, it 
will be a class effort that takes weeks to arrange 
and peifoim. while i-equiring onl\ a small portion 
of a class period each da\ . Let the students do the 
whole experiment, including contacting the sec- 
ond obseiAing school. If the teacher is enthusiastic 
and is able to catcii the imagination of a gniup of 
students about finding the dimensions of the earth, 
this single activity could inspire a great spirit 
within a class. 

It is best to find the altitude of a hea\'enl\' body 
when it is on the meridian because there is less 
atmospheric refraction at that time. Be sure that 
the star chosen is bright. .A good star to use is 
Sirius. 

An interesting historical note is that Eratos- 
thenes used two points in Egvpt: .Alexandria, on 
the Mediterranean Sea; and S\ene, near the pres- 
ent location of the Aswan Dam. He measured the 
distance between these points in stadia.' Unfor- 
tunateK , we do not know the pixipei- conversion 
from his stadia to our kilometers. According to one 
inteipretation, Eratosthenes \alue for the circum- 
ference of the earth was about 20% larger than the 
presenth' accepted \alue. Howe\er, according to 
another interpretation, his \alue agi^ees v\ith the 
modem \alue to about 1%. 

Some additional points of interest might be men- 
tioned before or after this experiment. It is often 
said that the \o\'age of Columbus to the New World 
was a daring feat because it was not knov\Ti in 1492 
that the earth was spherical. However, by 250 b c , 
Greeks such as Plato and Aristotle had concluded 
that the earth was a sphere. \\ hat obseivations ma\' 
have caused the Greeks to make that conclusion? 
How might you explain the popular concept at the 
time of Columbus that the earth was flat and that 
one could fall off the edge? 

Since about 1673, scientists ha\e known that the 
earth is slightly oblate: that is, the polar diameter 
is slightly less than the equatorial diameter. Near 
the pole there are about 111 km along the earth's 
surface per degree of latitude: while there are onK 
about 110 km per degree at the equator. 

Ansivers to questions 

1. The uncertaintN of the distance of 300 km is per- 
haps 1 part in 200 or 0.5"b. The uncertaintx in 



measuring the angle of a star is considerable. At 
300 km apart the angular dif^eitmce of a star's 
altitude would be 2.9°. A 1° uncertainty would 
be maximum: a percentage error of 30% can be 
expected. Consec|uentl\ , the uncertaint\ of 
could be 00 times the uncertainty of the distance 
bet\veen the two obser\ei-s. 

Of cour-se, one way to impro\e the accuracy 
of the experiment is to increase the baseline and 
imjjitne the technique of measuring angles. If 
one location is 800 km iioith or south of the 
other, the difference in altitude of a given star 
will be 7Vi° instead of 2.9°. 
2. At best, students will be able to calculate the 
circumference of the earth to be about 40,000 
km. The uncertainty will probably be no less 
than 30% . 



3. % of error = 



student s eri-or 



X 100 



40,000 km 
4. A 30% error is acceptable. 

Students should not be discouraged about a 
30% error-. This is a difficult measurement to 
make with crirde instrximents, so that to be able 
to obtain the order of magnitude of the cir-cum- 
fer-ence of the ear-th is an achie\ement. However-, 
encourage them to refine their angle-measuring 
procedur-e. 

E2-3 THE DISTANCE TO THE MOOxX 

This is a stiaightfoi-ward triangulation. By obser- 
vations of their thumb with fir-st one exe and then 
the other-, students notice the parallax angle. 

Ansivers to questions 

1. If the object is mo\ed farther away, the parallax 
angle becomes smaller. 

2. If the baseline is made longer, the parallax angle 
increases; if shorter, the par-allax angle de- 
er-eases. 

3. Since the moon mo\es rapidK' among the stars, 
about 13° per daw or one moon s diameter per- 
hour, the photographs should be nearh simul- 
taneous. Earthshine is sunlight reflected fr-om 
the earth to the moon and back to the earth. 
Since the moon s displacement on these en- 
larged photographs is less than 1 cm, some care 
is needed in the plotting and measuring. The 
images of the moon, \enus. and Jupiter are 
o\erexposed and distorted by light scattered 
within the camera lenses and the films. Ihe dis- 
placement of corresponding points should be 
measur-ed — at least both cusps. Our par-alUix an- 
gle a\'eraged about 0.26°. 

4. The positions of \ enus and Jupiter do not shift 
compared to the star-s. 

Our results with the technique described av- 
erage around 408,000 km. How triangulation can 
be used to get any answer is much mor-e impor- 
tant than agreement with a "correct answer. 

5. The linear diameter of the moon, just less than 
0.5° in the sky, comes out to be close to 3,500 
km. Inter-comparisons of results among students 



EXPERIMENT NOTES 



139 



would show the scatter of values and raise ques- 
tions about experimental errors. 

6. Only two significant figures seem to be justified. 

7. If the obseived parallax from widely separated 
places on the eaiih is very small, or zero, the 
objects must be much farlhei- than the moon. 
This conclusion was reached by I'ycho Brahe 
about the comet of 1577. See Chapter 6. 

8. Every observation has inherent errors. Every 
analysis involves approximations. Those made 
in this analysis are relatively small, but worth 
the attention of curious students. 

(a) The baseline is about 15° from being perpen- 
dicular to the direction to the moon. Thus, 
the foreshorlening of the effective baseline is 
proporlional to cos 15° 10.9701, an errxir of 3%. 

(b) The difference in the baseline between the 
arc over the surface and the chord through 
the earth is about 13 km; the resulting sys- 
tematic error is less than 1%. 

(c) Over-exposure of the moon's images expands 
them. Therefore the systematic error is to- 
ward measuring extra-large displacements. 
For the diameter of the moon the best meas- 
urements are made between cusps on the 
lower pictur-e. 

(d) The displacement of the moon was nearly 
parallel to the horizontal so that differential 
refraction is not significant. 



E2-4 THE HEIGHT OF PITON, A 
MOUNTAIN ON THE MOON 

Equipment; 

Ruler 

Handbook 

10 X magnifier 

Moon photograph 

Piece of string 1 m long 

This experiment also uses indirect measure- 
ment. Our ability to make measurements by direct 
methods (dir-ect application of our sensesi is se- 
verely limited. We can't even see very small things, 
and a moimtain is just too large to measur'e by 
direct means. Some objects, such as a cloud, ar-e 
too inaccessible to measure. In each of these cases 
and in countless other-s, we use indir-ect methods 
to "estimate" dimensions. In this exercise students 
combine observations with a geometrical model to 
measure a \'ery large, very inaccessible object. 

The photographs at the bt^ginning of the exper- 
iment show the large ci'ater tlopeniicus, which is 
near the ecjuator in the moon s oasteni hemisphere 
during the third (juailer. 

1 he discussion of Figs. 2-11 and 2-12 in the 
llnncibook concerns a change in \iew})oint in ob- 
ser\ing the shadow of Piton. This change of view 
can be made ("lear with a basketball Mark the 
noi1h pok" and the central meridian itenninatori. 
Use a golf t('(' to rcpit^sent Piton Show the basket- 
ball to the class so that the\ sec it as in tiic pho- 



tograph and Fig. 2-11. Now rx)tate it b\' bringing the 
north pole up toward you laway from the classi 
until the class sees Piton on the edge of the moon's 
disk. This is the view seen in Fig. 2-12. 

Answers to questions 

Students may measure distances either on a mo- 
saic photograph of the moon or on the photo- 
graphs in the Handbook. The following solutions 
are from data taken from the large photograph in 
the Handbook. 

1. Data: / = 1.0 cm 

d = 2.6 cm 
2r = 81.0 cm 
Calculations: 
r — 40.5 cm 

/ X d (1.01(2.6) 

h = = r = 6.4 X 10 ■ cm 

r 4.05 X 10 

Actual diameter of moon in km 

2. Scale = 

Measur-ed diameter of moon 

on photograph in km 

Change 81 cm to 8.1 x lO ' km 

3.476 X 10^ km , 

Scale = -. = 4.3 x lO 

8.1 X 10 ^ km 

A linear dimension of the moon is 4.3 million 
times as gr'eat as a linear dimension of the pho- 
tograph. 

3. Ther-efore, the estimate of the actual height of 
Piton is 4.3 x lo" i6.4 x lO"'i km = 2.7 km 

4. and 5. Questions 4 and 5 are not easy ones for 

students to answer but they should agree 
that the error in their measurements is 
less than 25% isee discussion belowi. The 
method itself has been somewhat simpli- 
fied, resulting in an additional possible 
error of about 10%. 

Encour-age students to use a magnifier. 
Some may discover for the first time that 
a halftone reproduction is made up of 
dots. 

A question that each student must an- 
swer indi\iduall\' is: "From which point 
in the illuminated part of the mountain 
should I measure: the center or the west- 
ern (left I edge?*' It is probabh- correct to 
measure from the left-hand edge on the 
assumption that it is that par-t of the 
mountain that casts the longest shadow. 

L'ncer-taint\' in / is about 10%. Uncer- 
tainty' in r is comparativeh' insignificant. 
Maximum uncer1aint\' of the final result 
is therefore 15% + 10% = 25% 

Percentage enx)r of our result is 

27 - 2 1 04 

X 100 = — X 100 = 15% 

2.7 2.7 

As the example abo\e shows it is iin- 
likel\' that rrsults obtained b\ the students 
will difler ti-oiii 2 7 km b\ moiv than the 
experimental inuei1aint\ The convct an- 
swer to C!an \(ni suggest win ' is ptx)b- 



I'MT 2 / MOTION IN THE HE/WTiNS 



ably "I underestimated the uncertainty in 
locating the terminator, etc." 

Students should not be disappointed 
v\ith the rather high percentage error of 
their results. To ha\e measured the height 
of Piton e\en to within 30% is an impres- 
sive achiexement. 

This is a good time to emphasize the 
importance and \alue of order of magni- 
tude measurements in physics. Estima- 
tion seems to contradict the cliche Phys- 
ics is an exact science, but an order of 
magnitude value is often all that can be 
obtained, particularly if the quantity' being 
measured is \er\' large or very small. Of 
course, it is essential to ha\e an estimate 
of uncertaintA'. 

E2-5 RETROGRADE MOTION 

The notes that accompam' the Film Loop might 
well be inserted here in your Resource Book. The 
filmstrip of slc\' photographs on the same scale and 
carefully positioned to show the same star field in 
each frame allows students to readil\' see the mo- 
tion of Mars in retrograde. The table belou- gi\es 
the dates that students ma\ estimate as the turning 
points in Mars' motion and the approximate du- 
ration of the retrograde motion in days. At best, 
these are crude estimates because the photo- 
graphs were taken at irregular intervals and often 
fail to show the critical times. In addition to seeing 
the actual retrograde motion, students should re- 
cdize that the durations and displacements differ 
between recurrences of the retrograde motion. 

Retrograde Motions of Mars from the Filmstrip 

Begins Ends Duration 



1941 


Sept. 1 


1943-44 


Oct. 29 


1945-46 


Dec. 8 



Nov. 16 


76 days 


Dec. 20 ? 


53 days 


Feb. 23 


77 days 



The retrograde motion of Jupiter is much longer, 
about 120 days, and extends beyond the intervals 
covered by the 1943-44 and 1945-46 photographs. 
Of interest is the relative motions of Mars and Ju- 
piter and the difference in duration of the retro- 
grade motions 

E2-6 THE SHAPE OF THE 
EARTH'S ORBIT 

Equipment: 

Filmstrip projector i35mmi 

Screen 

Meter stick 

Graph paper — 20 x 20 size i desirable i, or 

smaller sheets 
Filmstrip of sun photos 
Ruler 
Compass 
Protractor 
Handbook 



E2-6, "The Shape of the Earth's Orbit, " should 
be done before E2-8. 9, and 10. in which the stu- 
dents determine the oi-bits of Mars and Mercury. 

You may decide to conduct this experiment as 
a group acti\ir\' in which the whole class partici- 
pates in collecting the data. Each student should 
make a separate plot, howe\er. Once the data ha\'e 
been collected the plotting could be done at home 
since the only tools needed are protractor and 
ruler. 

The Changing Size of the Sun 

This experiment is based on the assumption that 
the angular size of the sun changes because our 
distance from it changes. Before beginning, you 
may want to challenge students to oflfer other ex- 
planations that fit the facts. 

For example, perhaps the sun periodically ex- 
pands and contracts, as many stai-s do. iThe most 
famous tApe of periodic "xariable stars" are called 
cepheid variables. One famous cepheid is Polaris, 
which has a period of just less than four days.i You 
might ask, if the sun does \ ar\' in size, does it also 
\ar\' in other ways, for example, in brightness? 
Could this be an explanation for the seasons? Does 
it seem reasonable that the suns period of \aria- 
tion would coincide exactly with the earth s period 
of revolution about the sun? V\ hat other effects 
would we observe on earth and on other planets 
if the sun does varv in size or brightness? 

There are other possible sources of systematic 
or cyclic error that could account for the obser- 
vations. The effective focal length land therefore 
amount of magnification) of the telescope might 
\ar\' with temperature, and therefore with the sea- 
sons. To test this idea a stud\ of solar diameter 
versus air temperature could be made. A more con- 
clusixe study would in\olve similar photos from 
the southern hemisphere where the seasons differ 
from ours. 

After a variation in the suns physical size has 
been ruled out as a probable explanation, return 
to the idea that this is an apparent \ ariation onh' 
and that probably it results from a \ariation in the 
distance from earth to sun. 

Students ma\ be surprised to learn that the sun 
is actualh' closer to the earth in the northern hem- 
isphere during winter than it is in summer. This 
contributes little to seasonal differences, which cire 
mainly caused by the inclination of the eeirth's axis 
of rotation to its plane of revolution. 

The Coordinate System 

In locating the 0° direction on the graph paper re- 
member that the point where the ecliptic crosses 
the equator on March 21 is called the vernal equi- 
nox. iSee 72 71. The sun is then close to Pisces. A 
line from the earth to the vernal equinox is the 
reference line ftxjm which celestial longitudes are 
measured along the ecliptic. The angles are meas- 
ured eastward from this 0° line. 

This system is used in E2-8, 9, and 10. as well as 
in this experiment, and students should become 



EXPERIMEXT NOTES 



141 




Line 



familiar with it. Part of Transparency T14 can be 
used to help explain the system. 

Gathering Data 

Project the filmstrip onto a wall or other flat surface 
to give an image of the sun with a diameter be- 
tween 50 and 100 cm. Measure the diameters with 
a meter stick. Do not move or refocus the projector 
after the measurements start. 

If you have moi-e filmstrips than projectors, some 
students can measure on the film directly, using 
a lOX magnifier. 

In frames 7 through 18 the directions marked \' 
and E refer to the directions as seen in the sky and 
not to directions on the sun itself. \ote that the 
direction marked "north" varies from frame to 
frame, principally because the photographs were 
made at difterent times of the day. The frames al- 
ways have the same alignment uith respect to the 
horizon. On plates taken early in the morning, the 
north direction is tipped toward the left. We rec- 
ommend measuring cdong the horizontal diame- 
ters, which are parallel to the bottoms of the frames 
since this lessens the effect of atmospheric I'efrac- 
tion. RKM EMBER: Do not move the projector or 
refocus. 




observer 



A 10-cm radius gives an ()i4)it tiiat is convenient 
to use as the starting point for plotting tlie od)it of 
Mai-s EZ-a. 

In discussing the ciiange fix)in an eailh-c(Miten>il 



to a sun-centered plot, remember that if the plot 
is turned thixjugh 180' it now represents the orbit 
of the earth aixjund the sun. Some students may 
see this; many prxibably won t. Rather than tr\' to 
convince them by intellectual arguments that the 
same plot can be used for either orbit we suggest 
that they begin to plot the motion of the earth 
around the sun. If they are encouraged to compare 
the new plot uith their first one as they go along, 
they should soon i-ealize that they do not need to 
make a whole new plot. Rotating the first plot is 
sufficient. 

Here are examples of the two plots. 



US' 












A 




J?iV 


















..--ir; . 
















i.^ 




D^ ■ 


*' 





—i 


^ 
















i 




















y*i 


K*t 
















sV^ 


-"-■ 


••• 





Ansu'ers to questions 

1. The orbit is best drawn as a slightly off-renter 
ciixle. In fact it is an ellipse whose eccentricit\' 
is 01 7 

2. Perihelion is JanuaiA 3. .Aphelion is Jul\ 5 The 
ratio of a|)helion distance to perihelion distance 
is about 1 (M. 

:\ See Fig 2-20 in the Hnndhook 



142 



U\IT 2 / MOTIOM IN THK HEAVENS 



E2-7 USING LENSES TO MAKE 
A TELESCOPE 

Equipment: 

Large lens 
10 X magnifier 

Small lens mounted in wooden cylinder lop- 
tics kit I 
2 telescoping cardboaixl tubes 
Plastic cap for mounting large lens 
Handbook 

Optional equipment: 

Wooden saddle 
Rubber bands 
Camera tripod 
Telescope kits 

Through some simple experiments students can 
learn enough optics to understand how a tele- 
scope works. 

Use your own stock of assorted lenses. Include 
lenses that ha\e the same diameter but different 
powers, some negati\e lenses, and some plane 
glass disks if you have them. If necessar\', use 
lenses from the telescope kits; other\\ise, keep the 
kits in the background. Caution students to handle 
lenses by the edges and to keep their fingers off 
the surfaces. 

The Simple Magnifier 

If the object distance is held constant at its \'alue 
for maximum magnification, the angular size of the 
image will remain constant as the eye is mo\ed 
away irom the lens. 

Answers to questions 

1. Magnifers are thicker in the middle than at 
edges. The lens thickest in the middle has the 
highest magnifving power. 

2. Curvature is the important feature. The more 
convex (fatten a lens is, the greater the magni- 
fication it is capable of, given an appropriate 
choice of object distance and image distance. 



n 



u 

\ncrea.5\n3 M-b.9rnf\ cation 



3. Only lenses thicker in the middle than at the 
edges (convex lenses i produce an image that can 
be projected on a screen. Such an image is called 
a real image. Real images are alwavs inverted. 

4. Uliether the real image is larger or smaller than 
the object depends on the relative distances of 
the object and the image from the lens. If the 
object is farther from the lens than the image is, 
the image will be smaller than the object and 
vice versa. 

5. If you want to look at a real image without using 
the paper screen, you have to put your eye be- 



hind the image and at least 25 cm from it. To a 
student who has trouble finding the image ask: 
"How far would your eye have to be fham a real 
object to see it clearly?" 

6. The image seen in this way may be quite small. 
In order to inspect it more closely use a second 
lens as a magnifier held at the same distance 
from the image as it would be held from a 
printed page for clear viewing. If students have 
difficulty in placing the magnifier in the right 
place, let them focus the image formed by the 
first lens on a sheet of paper. Having thus lo- 
cated the real image, they can adjust the dis- 
tance of the magnifier from it accordinglv and 
then remove the paper. 

7. The telescope produces the largest images if the 
lowest-power lens is used as the objective (the 
fii-st lens the light goes through! and the highest- 
power lens is used as the eyepiece (the magni- 
fieri. 

E2-8 ORBIT OF MARS 

Equipment: 

Booklet of Mars photographs 

Transparent overlays 

Graph of earth's orbit E2-6 

Protractor 

Straightedge 

Handbook 

The Photographs 

None of the photos used in this experiment are 
retouched. They may show lint or dust marks or 
the edges of dried watermarks that mar the origi- 
nals. The center is often heavily exposed while the 
edge is barely exposed. This is due to vignetting 
(shadowingi at the edges of the field caused by 
camera design. The student is as close to the 
"data " of observational astronomy as are research 
astronomers themseKes. 

Some photographs show fewer or fainter stars 
than others because: 

1. In some parts of the sky, well away from the 
MUky Way, star density is low. 

2. Some of the photographs mav have been made 
through thin clouds, smoke, or haze, and will 
not show the fainter stars. 

The size of a focused image of Mars on the orig- 
inal plate is about '/jo mm, as are the diffraction 
disks of star images. Consequently, the images of 
planets are theoretically indistinguishable from 
those of stars. Actually, light scattered within the 
photographic emulsion and "twinkling"' causes 
brighter images to grow larger than fainter images. 

The best image formed by a lens is on the optical 
axis (in the center of the picturei. Distortion be- 
comes more pronounced toward the edges of the 
field. One kind of distortion, which makes the 
shape of images seem triangular, is clearlv evident 
in some of the frames taken from the edge of a 
plate. Nothing can be concluded about real shapes 
and sizes from these photos: They ai-e valuable be- 



EXPERIMEiVT NOTES 



143 



cause they recortl i-elative positions of Mai's and 
the stars. 

Almost surely some questions will arise about 
the "halo" around the bright object iMarsi in plates 
F and O. I his occurs because the photograph 
emulsion is on a glass plate about 1 mm thick. As 
the sketch below indicates, light from a veiy bright 
object will penetrate through the thin emulsion, be 
reflected by the back of the glass, and strike the 
emulsion from below in a ring around the initial 
image. 







The Coordinate System 

Before beginning this experiment be sure students 
have the dates and angles properly recorded upon 
the earih plot from E2-6. If 0° is on the right, this 
should be labeled September 23. Ihat is, on this 
date the earth has a heliocentric longitude of 0° as 
seen from the sun. On March 22 the earih has a 
heliocentric longitude of 180° and the sun has a 
geocentric longitude of 0°. 

Refer to the tvpical Mai-s orbit on page 67 of the 
Handbook I Fig. 2-271. Some students ha\'e trouble 
visualizing that a line drawn horizontally to the 
right fiom the earih takes a dii-ection of 0°. It does 
not matter where the earth is in its orbit; to the 
right on the typical diagram is 0°, to the left is 180°, 
and directly downward is 270°. For further clarifi- 
rntinn refer to the diagram below. 




>0 



t 270 



Because of the changing attractions of the moon 
and sun on the oblate eaith, the earth s I'otation 
has se\eral wobbles. .All except one aii* too small 
to be important in this analxsis. Ihe one large 
change is called f)rr(('ssion and has a period of 



26,000 years. During this time the direction in 
which the north pole of the earth points moves 
through a large circle on the sk\'. The result of this 
motion is to cause the location of the vernal equi- 
nox to slide westward about 50 sec of arc each 
year. Since our coordinate grids are based upon 
the vernal equinox as the zeix) point, the longitudes 
we assign to the stars change verv- slowK' with the 
years. For this study, we have adopted the coor- 
dinate system of 1950.0 

Longitude of Mars, seen from Earth 
Plate Date Longitude 



A 


Mar 21, 


1931 


118.6' 


B 


Feb 5, 


1933 


169.0° 


C 


Apr 20, 


1933 


151.4'- 


D 


Mar 8, 


1935 


204.4° 


E 


May 26, 


1935 


186.7° 


F 


Apr 12, 


1937 


245.7° 


G 


Sep 26, 


1939 


297.5° 


H 


Aug 4, 


1941 


016.5° 


1 


Nov 22, 


1943 


012.1° 


J 


Oct 11, 


1943 


080.1° 


K 


Jan 21, 


1944 


065.6° 


L 


Dec 9, 


1945 


123.r 


M 


Mar 19, 


1946 


107.6° 


N 


Feb 3, 


1948 


153.4° 





Apr 4, 


1948 


138.3° 


P 


Feb 21, 


1950 


190.7° 



A Typical Mars Orbit 

In this plot, we ha\e drawn a circle of radius 15.5 
cm 11.55 AUi that passes through or close to most 
of the positions of Mars. The center of the circle is 
above and to the left of the sun's position. 



.:.j-i 



rjr 



I 



Data for Mars' oiiiit: 



Mean distance a = 1.52 .\\J 
Eccentricit>' e = 0.09 

'I\vo articles on measuring aivas with planime- 
ters ha\e appeared in Scirntific .XnuTiran. .August. 
1958 and F-el)i\iatA 1959 MechanicalK minded stu- 
dents might wish to make and calibrate theii own 
planimetei-s and \eiit\ Kc'pler s law of aiva?> 



144 



IflVIT 2 / MOTION IN THE HEAVENS 



Answers to questions 

1. Student answers will \aiy. 

2. The sun-to-earth distance is often referred to as 
"one asti-ononiical unit' lAl'i. The distance trom 
sun to Mars in AU: 

Maximum distance =^1.7 AU 

Minimum distance = 1.4 AU 

Mean Distance - 1.55 .AU 

3. As seen from the sun, the directions (longitude^ 
of perihelion and of aphelion of Mars are ap- 
pro.\imatel\ 340° and 160", respecti\ely. 

4. The earth is about 0.4 .AU from the orbit of Mars 
in September. This is the closest that the two 
planets can approach each other. 

5. The eccentricirv of the Mars orbit is e = 0.09. 
&-10. Student answers will \ajA . 



E2-9 IXCLIXATIOX OF MARS' ORBIT 

Equipment: 

Booklet of Mars photographs 
Transparent o\erla\ s 
Graph paper 
Handbook 

If some students are interested in earning fur- 
ther the analysis of Mars orbit. the\ can use the 
same star-field photographs and cooixdinate oxer- 
lays to deri\e the inclination of the orbit. 

The elements of an orbit are discussed again in 
E2-12 model of the orbit of Halle\ s Comet. ' 

Transparency 18 shows the details of Fig. 2-37 
more clearK than a single drawing can. 

Data for the completion of Fig. 2-31 i£2-Si follow: 







Position of W 


lars 












As seen 


from 


As seen 


from 








earth 


sun 


Plate 


Date 


long. 


lat. 


long. 


lat. 


A 


Mar 21, 


1931 


118.6" 


3.2N 


150= 


1.8N 


B 


Feb 5, 


1933 


169.0 


4.0N 


150 


1.8N 


C 


Apr 20, 


1933 


151.4 


2.5N 


183 


1.4N 


D 


Mar 8, 


1935 


204.4 


3.0N 


183 


1.4N 


E 


May 26, 


1935 


186.7 


0.6N 


220 


0.3N 


F 


Apr 12, 


1937 


245.7 


0.7N 


220 


0.3N 


G 


Sep 16, 


1939 


297.5 


4.6S 


335 


1.7S 


H 


Aug 4, 


1941 


016.5 


4.1S 


335 


1.7S 


1 


Nov 22, 


1941 


012.1 


0.6S 


043 


0.2S 


J 


Oct 10, 


1943 


080.1 


0.5S 


043 


0.2S 


K 


Jan 21, 


1944 


065.6 


2.7N 


096 


1.3N 


L 


Dec 9, 


1945 


123.2 


2.9N 


096 


1.3N 


M 


Mar 19, 


1946 


107.6 


3.0N 


142 


1.9N 


N 


Feb 3, 


1948 


153.4 


4.4N 


142 


1.8N 





Apr 4, 


1948 


138.3 


3.1N 


169 


1.6N 


P 


Feb 20, 


1950 


190.7 


3.5N 


169 


1.6N 



The plot of latitudes > north or south of the eclip- 
tici is a sine wave. The ascending node, where the 
path crosses frxjm south to north as the longitude 
of Mars increases, is near 050", and the descending 
node is near 230". The maximum latitude as seen 
frtjm the sun is 1.8°. halfvvax bervveen the nodes: 
this is the angle of ori)ital inclination. 



You may wish to discuss the way in which a 
small number of data points can be interpreted, 
i.e., when \ou know that the curve will be a sine 
wa\e. 



E2-10 ORBIT OF MERCIHY 

Lquipment: 

Graph of earth's orbit [E2-6) 

Table of planet positions 

Handbook 

Protractor 

Straightedge 
This simple exercise provides additional e.xpe- 
rience with the concepts of orbit theorv. Orbital 
eccentricit\- and Kepler s second law can both be 
studied. .Although this is a relativeh brief acti\it\' 
that some pupils ha\e completed at home in less 
than 20 min, the results can be surprisingly accu- 
rate. 

Because the oit)it of Mercun- is not a circle, the 
tangent to the ori^it is not perpendicular to the line 
joining sun to planet: that is, the assumption sug- 
gested here is only an approximation. Students 
may find that the\ cannot draw a smooth curve to 
join all the points located in this wa\ . In this case 
it is quite legitimate to mo\ e some of the points 
slightK . The final orbit should be represented b\' 
a smooth curve that touches, without crossing, all 
the sight lines. 



Calculating R^^ and the Orbital Eccentricitj- 

MercuiA s perihelion point occurs at a longitude 
of about 78° lin about the direction of the earth s 
position on December 10, and the major axis of 
the orbit lies along this line. From the data gi\en 
in the experiment, the major axis of Mercuiy s orbit 
is about 7.8 cm (0.78 AUi long. 

2a = 0.78 AU 
a = 0.39 AU 

The accepted v alue for a is 0.387 .AU. 

From the plot. R^ = 4.60 cm and a — 3.9 cm. 
Thus, 



4.60 

e ^ 1 

3.9 

e = 1.18 - 1 

e = 0.18 

The accepted value of e is 0.206. 

Calculating the accepted value should not be a 
major concern. The emphasis should be on deter- 
mining if the orbits of the earth and Mercurv- have 
the same shape. If the\ do not, then how can one 
describe the difference? Eccentricity', a mathemat- 
ical device, describes this difference in shape. How- 
ever, it is moi^ important for students to be able 
to describe the difference between elliptical shapes 
in their own way than to feel compelled to calcu- 
late e = 0.206 fi-om their data. 



EXPERIMENT NOTES 



145 



Kepler'M Second Law 

Labol the sun S and the positions of Mercurv' as 
tbilows: 

Jan. 4, 1963 A 

Feb. 14, 1963 B 

June 13, 1963 C 

Aug. 24, 1963 13 

Count the number of squares in the areas SAB 
and SCU. Note that each of these areas is more 
than half the orbit area. 



s.< »»"• 






I-J1 






, 




o.*& 




\« 






IW^-' 








.,>i 









With uui data: 

SAB 274 squares 

f, 41 days 

SCD 525 squares 
I. 



Average = 



72 days 
274 + 525 799 



= 6.7 sq/day 
= 7.3 sq/day 



41 + 72 113 



= = 7.07 sq/day 



If the law of areas holds, the two ratios should 
be equal. Our experimentally determined ratios 
agree uithin 10%. Perhaps your students, using a 
smoother orbit, will obtain greater accuracy. 

E2-11 STEPWISE APPROXIMATION TO 
AN ORBIT 

EquipuKMit: 

20 X 20 graph paper 

Straightedge 

45° oi' 60° triangle 

Dixidei-s or c(jmpass 

Handbnak 

This e.xpjMimenl has two results. The students 
will come to unth'istand that if a force is central 
toward tin; sun and if it is in\(M-sel\ propoi-tional 
to the stjuare of the distance from the sun, the 
n^sulting orbit is an ellipse K(iiiall\ im|)()i1ant, after 
the |)lot is compietj'd. the student will ha\t' a 
griMtly impnned undiM standing of the infall of a 
body toward the sun. 

rlu' \'elocit\' of a body in a gravitational lield is 
coiitinuousK changing, wlicthri- the bocK is a pi-o- 



jectile near the earth's surface, a satellite in orbit 
around a planet, or a comet in orbit about the sun. 
For a projectile near the earth s surface the gravi- 
tational force is constant and the trajectory can be 
found fairly easily. iSee Unit 1, Chapter 4. i Howe\'er, 
for a body in space the gravitational forx:e changes 
with distance frxjm the sun, earth, and other bod- 
ies. Precise prediction of the orbit of a body in a 
varying force field is complicated. However, if one 
assumes that the force acts intermittently at equal 
time intervals, like hammer blows, an orbit can be 
approximated rather quickly. 

See Fevnman's Lectures on Physics, Vol. I, Chap- 
ter 7 for an algebraic equivalent of the geometric 
method used in this experiment for computing or- 
bits. Newton used this method to prove that Kep- 
ler's second law follows ft-om a central -force hy- 
pothesis. I See Principia, page 40 in the paperback 
(!dition and also Te;ct Sec. 8.4. i The experiment de- 
scribed here is based on one developed bv Dr. Leo 
Lavatelli, l'niversit\' of Illinois, and printed in the 
American Journal of Physics, 1965, 33. p. 605. 

In the Project Physics Film Loops 13 and 14 en- 
titled Program Or+jit. I and Program Orbit. II a 
computer works out the same ortiit bv iteration. In 
the first loop the time interval between blows is 60 
days and the result is close to what the students 
should obtain. In the second loop a 3-day iteration 
interval is used; the oit)it is smoother. Both loops 
should probably be used after the students have 
made their plots. 

The "thought experiment " suggested in E2-11 
("Imagine a ball rolling . . . "i can form the basis of 
a demonstration in which the teacher applies re- 
peated lateral isidev\isei blows, all directed toward 
the same point, to a heavy ball or air puck. It is 
helpful to trv a demonstration involving all the stu- 
dents. Each one gives a centrally directed blow to 
the ball or puck as it passes. Ihe balls initial ve- 
locity must be fairlv high and the blows not too 
strong. 

Answers to questions 

1. The ball will continue to move in a straight line 
with the same velocitv'. 

2. The path direction would change. 

3. Its speed mav change depending on initial 
speed and acceleration imparted bv the blow 

I In cireular motion onlv direction changes, not 
speed. I 

The ball will move in a path made up of a series 
of straight-line segments. 

Effect of the Central Force 

An8»»'ers to questions 

4 Ihe foixi> is gii'atei if Ihe comet is nearer to the 
sun: 

1 

F y- — 

ir 

5 Ihe gixMter the foive of the blow tlic greater 
th«' velo( itv change lAx" ^ Fk 



14B 



UNIT 2 / MCmOX IN THE HlvUTJNS 



Because all blows ha\e the same duiation 
(Af) and m is constant, \e\\1on s second law 
F — m Av/Ar can be simplified to Av » p. 

Because the time intei-val bet^\^ee^ blows is 
constant I60 daysi the comet s displacement 
along its orbit during a 60-day interval is pro- 
portional to its velocity. 

Ad = V X 60 days, becomes Ad ^c v. 

Scale of the Plot 

The particular orbit chosen is similar to that of the 
short-period comets, like Encke's comet, which 
stay entirely within the orbit of Jupiter. These pa- 
rameters, and the 60-day intenai, gi\e an oiijit that 
is completed in about 25 steps. Half of the orbit 
can be obtained in 12 steps. 

The earth's average orbital speed is about 96,000 
km/hr. 

Computing \v 

The following altematixe may be preferable to us- 
ing Fig. 2-50 as a computer by the method de- 
scribed at the end of this section. Plot the graph of 
Av vei"sus R on transparent paper. With the orljit 
plot on a drawing board, stick a pin thiough the 
origin of the graph and the sun point of the orbit 
plot. It is also a good idea to reinforce the graph 
with a piece of masking tape in the area of origin. 
Swing the graph around the pin as a pixot to es- 
tablish the \alue of fi and point d,. 




In either method of making the plot, 12 or 13 
steps should bring the comet to perihelion. If time 
is short, students can complete the orbit by assum- 
ing that the hal\es are symmetrical. 

The Film Loops, Program Orbit. I" and "Program 
Orbit. II, illustrate that a shorter iteration intenai 
results in an orbit that is more nearh' a smooth 
ellipse. A student can draw a smooth ellipse on the 
plot by using the two-pin-and-loop-of-string tech- 
nique. 

Ansivers to questions 

Answers are based on a sample plot. 

6. Perihelion distance = 1.1 AU 

7. e ^ 54 



8. Period of re\olution = 24 x 60 days 

= 1,440 days 
= 4 \eai-s 

9. The closer the comet is to the sun the greater 
the speed. 

10. The path has the approximate shape of an el- 
lipse with the sun occupying one of the two 
foci. If A J were decreased from 60 days, the 
cui"\e would be smoother, and the curve would 
tend to close. This provides a straightforward 
application of calculus, for as we imagine A/ 
approaching zero we model a continuously 
changing force rather than a single blow every 
60 days. This question can lead to much class 
discussion and to Film Loops 13 through 17. 

11. The area is approximately 27 cm". Students can 
find this by calculating A = Vzafa or by counting 
squares. Consider the students' inaccuracies 
due to the iteration process. Discuss what or- 
der of deviation might be considered a constant 
area in view of this approximation of iteration. 

More things to do 

1. The student who completed a superior graphi- 
cal representation of the orbit might be encour- 
aged to repeat the experiment using a different 
initial speed but retaining the same direction. 
Another student might change the initial direc- 
tion and i-etain the same speed, that is, keep the 
length of the vector the same. 

Have these students present their findings to 
the class. Some interesting results include the 
sensitivity of orbits to small changes in the initial 
velocitv'. 

2. An example of such a repulsive force is that be- 
tween two like charges. The path will be hvper- 
bolic and this will help the students appreciate 
alpha-pariicle scattering in Unit 6. A special kind 
of student will be interested in this and benefit 
from it. 



E2-12 MODEL OF THE ORBIT 
OF HALLEY'S COMET 

Equipment: 

Cardboard or stiff paper, rvvo sheets 

Ruler 

Protractor 

Data for Halley's comet {Handbook) 

Compass 

Scissors 

In constructing the orbits of Halley's comet and 
the earth it mav be useful to refer to Transparency 
17, which illustrates the elements of an orbit more 
clearly than a single drawing can. 

The orbit of Halley's comet is an ellipse with e 
= 0.967; the portion close to the sun is very nearlv 
a parabola. All parabolas have an eccentricitv of 1. 

When fitting the rvvo oHiits together, remember 
that the ascending node is the point at which the 
comet s ori)it ri-osses the ecliptic plane going from 



EXPERIMENT iNCXFES 



147 



south to north. The descending node is the point 
at which the comet citjsses the ecliptic hnm noith 
to south. 

Halley's comet mo\es in the opposite sense to 



the earth and other planets. The earth and planets 
mo\'e counterclockwise when \iewed from abo\e 
inoilh ofi the ecliptic: Halley's comet mo\es clock- 
wise 




Pictures showing how the two planes go together. 



Answers to questions 

1. Refer to the sketch below. As the comet mo\'es 
fiom C, to C,, the earth mo\es from E, to E,. Due 
to these relative motions, the comet appears to 
move westward among the stars. 

2. When the earth is at E^ the comet is at C,. Fur- 
thermore, the motion of the comet is in a direc- 
tion directly down the sight line C,E,, a collision 
course! 

3. Refer to the sketch. As the comet moved from C, 
to C, and as the earth moved from E^ to E^, the 
comet moved between the earth and sun fairly 
near the earth, so its angular motion was great. 



4. On May 19 the comet crossed the earth's orbital 
plane between the sun and the earth. 

5. Yes, the earth passed through the tail. Refer to 
the sketch and \isualize the tail sweeping from 
C. and then cutting through the plane of the 
ecliptic at the line of nodes. The earth passed 
through this tenuous tail. 

6. Research question. 

7. Nothing unusual happened. The tail is ven,' ten- 
uous. The comet's tail is much less dense than 
the air near the surface of the earth. Therefore, 
one could not expect anxthing to ha\e hap- 
pened. 



HAULEy'5 COMET 19IO 




closest Approach 
of comef to Earth 



[4H 



I 'MIT 2 / MOTION I!V THE Hli/\\'li.\S 



Film Loop and Filmstrip Notes 



Filmstrip RETROGRADE MOTION 
OF MARS 

Because photographs are the most honest e\i- 
dence we ha\ e of the actual retixagrade motions of 
Mars and Jupiter, the filmstrip should be shown as 
soon as motions of the planets are mentioned. 

The Photographs 

The frames were made from unretouched contact 
prints of sections of the original photographs. The 
photographs were taken with the short-focus cemi- 
era i focal length 15 cmi sho\\Ti in one of the tii-st 
frames. Because Mai's was ne\er in the center of 
the field, but sometimes almost at the edge, the 
star images show distortions from limitations of 
the camera lens. During each e.xposure the camera 
was dri\ en b\ clockwoi4< to follow the western mo- 
tion of the stars and hold their images fixed on the 
photographic plate. Because the sky was less clear 
on some nights and the exposures \aried some- 
what in duration, the images of the stars and 
planets are not of equal brightness on all pictures. 
Howe\er, some of the frames show beautiful pic- 
tures of the Milkx Way in Taurus 1943 and Gemini 
11945). 

These three series of photographs were selected 
as the most e.xtensi\e a\ailable for recent opposi- 
tions of Mars. The photographs were taken as part 
of the routine Harvard Sky Patrol, and were not 
made especialh to show Mars. 1 he planet just hap- 
pened to be in the star fields being photographed. 
I See Notes on E2-3.' 

LIO RETROGRADE MOTION: 
GEOCENTRIC MODEL 

Using a large epic\cle machine as a model of the 
Ptolemaic system this film illustrates the motion 
of a planet, such as Mars, as seen from the earth. 

First, the machine is \iewed hxim abo\e with the 
characteristic retrograde motion during the loop 
when the planet is closest to the earth. When the 
studio lights go up it becomes discernible that the 
motion is due to the combination of two circular 
motions. One arm of the model rotates as an epi- 
c\'cle at the end of the other the deferent. 

The earth, at the center of the model is then 
replaced b\ a camera that points in a fixed direc- 
tion in space. The camera sees the motion of the 
planet relati\e to the fixed stars so we ai'e ignoring 
the rotation of the earth on its a.xis. For an obserx er 
\iewing the stars and planets from the earth, this 
is the same as alwa\s looking toward one constel- 
lation of the zodiac, such as Sagittarius. 

Imagine that you are facing south looking up- 
ward toward the selected constellation. East is on 
your left and west is on your right. The direct mo- 
tion of the planet, relative to the fixed stars is east- 
ward toward the left. The planet, represented b\ 
a white globe, is seen along the plane of motion. 



A planet s retrograde motion does not always 
occur at the same place in the sk\' so some 
retrograde motions are not \isible in the chosen 
direction. 

Three retrograde motions use smaller bulbs and 
slower speeds to simulate greater distances. 

Note the changes in apparent brightness and 
angular size of the globe as it sweeps close to the 
camera. Actual planets appear onl\- as points of 
light to the eye, but a marked change in brightness 
can be observed. This was not considered in the 
Ptolemaic system, which concentrated only upon 
positions in the sk\-. 

Retrograde Motion: Heliocentric Model 'LllJ 
shows a similar model based on a heliocentric 
theorv . 



Lll RETROGRADE MOTION: 
HELIOCENTRIC MODEL 

This film is based on a lai^e heliocentric model. 
Globes representing the earth and a planet mo\e 
in concentric circles around the sun. The earth 
represented by a light blue globe passes inside a 
slower-mo\ing outer planet, such as Mars repre- 
sented b\ an orange globe . .A yellow globe indi- 
cates the sun. The orbits of both planets are as- 
sumed to be circular. 

The earth is replaced b\- a camera ha\ing a 25" 
field. The camera points in a fixed direction in 
space, indicated b\ an arrow, thus ignoring the 
dail\- rotation of the earth and concentrating on 
the motion of the planet relative to the sun. 

Scenes are \iewed from above and along the 
plane of motion. Retixagrade motion occurs when- 
ever Mars is in opposition; that is, when Mars is 
opposite the sun as viewed from the eailh. But not 
all these oppositions take place when Mars is in 
the sector the camera sees. The time between op- 
positions averages about 2.1 \ears. The fUm shows 
that the earth moves about 2.1 times around its 
orbit between oppositions. 

It is possible to calculate this value. The length 
of a year is 365 da\s for the earth and 687 da\ s for 
Mars. In 1 da\- the earth moves ' 6«7 of 360" and the 
motion of the earth relative to .Mars is iVses — Vssri 
of 360'. Thus, it will take 780 da\s for the earth to 
catch up to Mai-s again. The average phase period 
of Mars is 780 days, or 2.14 vears. The view from 
the moving earth is shown for more than 1 vecir. 
First Mars is seen in direct motion, then it comes 
to opposition and undergoes a retrograde motion 
loop, and finally we see Mars again in direct mo- 
tion. 

Note the increase in apparent size and bright- 
ness of the globe representing Mars when it is 
nearest the earth. Mewed with the neiked e\e Mars 
shows a large variation in brightness but always 
appears to be onK a point of light With the tele- 



FILM LOOPS AND FILMSTRIP VOTES 



149 



scope, we can see that the angular size also varies 
as pr-edicted by the model. The heliocentric model 
is in some ways simplei- than th(! geocentric model 
of Ptolemy and gives the general features observed 
for the planets: angular position, retrograde mo- 
tion, and variation in brightness. However, detailed 
numerical agreement between theory and obser- 
vation cannot be obtained using circular orbits. 
Kepler's elliptical ort)its gave better agreement with 
observation. 

Hctrograde Motion: Geocentric Model ILIOJ shows 
a siriiilar model lepr-esenting the geocentric theory 
of Ptolemy. 

Some of the finer details of the motion of Mars 
are related to the planet's rather strongly elliptical 
ortjit I eccentricity 0.093 compared with 0.017 for 
the earth's ort)iti. Some oppositions ar-e mor-e "fa- 
vorable " (closer) than ar-e others. The drawing be- 
low shows that the closest oppositions occur if the 
earth is at A lin late August). A little over 2 years 
later the next opposition is not so close i earth at 
B, in November). The follov\ing distances illustrate 
this point: 

most favorable 

opposition, AA' = 56,000 km 

least favorable 

opposition, CC = 100,800,000 km 

least favorable superior conjunction, 

AC = 398,400,000 km 

The model used in the film is based on the ap- 
proximation of circular orbits. 

OPPOSITIONS OF M/APS 
ROM FA\^\H 

1,1967 

\9SO 




I'^^e 1943 



L12 JIJI'ITER SATELLITE ORBIT 

Ihe piri-|)ose of tin- loop is to give students a feeling 
for the motion of a celestial body: in this case, a 
satellite of .Iirpiter- mo\ ing under- the inflirence of 
gravitational for-ce. ihe loop is primariK intended 
for- (jualitatixc usiv Howexei', some simple meas- 
ui-em(Mits of period and size of oibil can help a 
student appreciate the naturx* of astix)riomi(-al ob- 
sei>ati()ris in a r-eal situation 



Jupiter was in opposition on Jan. 20, 1967, and 
was therefore closest to the earth lat about 42 AUi 
and maintaining a relatK'ely constant size as viewed 
from day to day. The entir-e satellite orbit could not 
be photographed, because on Feb. 8, 1967 ivvhen 
the missing part of the orbit was being traversed 
by loi, the image was blurred because of high-aJ- 
titude jet streams in the earths atmospherie. The 
next return of lo to this par't of its or-bit during 
dar-kness in Arizona was on Feb. 24, and bv then 
Jupiter would have been farther away and its image 
would have been smaller. Also, use of a lai^e tele- 
scope must be tightly scheduled, and our project 
had alr-eady used major amounts of telescope time 
on seven nights. For all these r-easons, we settled 
for a film showing 84% of a complete orljit, includ- 
ing all the portions needed for calculations. 

Exposures were for 4 sec on 35 mm black and 
white film of the tvpe used for aerial mapping and 
reconnaissance. A green filter iWratten 58i was 
used to give maximum sharpness of the images. A 
decision was made to use the best exposure to 
show the satellites, thus overexposing the image of 
the disk of Jupiter. For this reason, the surface 
markings due to atmospheric storms on Jupiter are 
only glimpsed occasionally, during moments of 
haze or cloudiness. 

First, the film shows a segment of the orbit as 
photographed at the telescope: a clock shows the 
passage of time. Due to small erTors in guiding the 
telescope, and atmospheric turbulence, the highlv 
magnified images of Jupiter and its satellites dance 
about. To remove this unsteadiness, each image 
was optically centered in the frame. Thus, the sta- 
bilized images were joined to give a continuous 
r'ecord of the motion of lo. Some variation in bright- 
ness was caused by haze or cloudiness. 

The satellites move nearly in a plane that we 
view almost edge-on. Thus, they seem to move back 
and for-th along a line. The position of lo in the last 
frame of the January 29 segment matches the po- 
sition in the fir-st frame of the February 7 segment. 
However, since these were photographed nine 
days apart, the other thr-ee satellites had moved 
varving distances, so thev pop in and out while the 
image of lo is continuous. 

Jupiter is not seen as a perfect cirxMe because its 
rxjtation causes it to tlatten at the poles and bulge 
at the ecjuator-. Ihe effect is (juite noticeable, for 
Jupiter is large (equatorial diameter of 142,720 km 
and polar diameter of 133,440 kmi and rotates rap- 
idly (period of 9*'55"'i. 

rhe fiattening of Jupiter is about Vis, compared 
with the tlattening of ' jjo for the earth. The cen- 
tripetal forx-e at the equator is mv' r = mrw', so 
the effect depends on r as well as on lo. the angular 
velocitv of rx)tation of the planet For Jirpiter- r is 
11.2 times that of the earth and cu is 2 42 times that 
of the earlh. Ihe centrifugal field larliticial gravit\ 
tending to lift a mass off the surface' is. then'forv, 
i11.2m2.42i' as much ie 64 times as gr-eat on Jir- 
piter as on the earth 



ISO 



DNIT a / MOTION IM THE HEAVKMS 



Before the development of marine chixjnometers 
about 1750, the motions of the satellites of Jupiter 
were used as a clock for the determination of 
longitudes. Fixjm man\' obsenations and theoiA'. 
the times at which the satellites would transit Ju- 
piter or be eclipsed could be pi-edicted for some 
standaixJ place like Greenwich. A distant obsener 
seeing the e\ent would then know the time at 
Greenwich and could compare it with the local 
time, from the time of observed noon or sunset. 
The differences in the two times was then the ob- 
server s longitude east or west of Greenv\ich. 

Measurements 

1. Period of orbit 

(Refer to pages 96 and 97 of the Unit 2 Handbook.' 
In measurement of T and R. a circular orbit is as- 
sumed, lo s orbit is perhaps the most nearh cir- 
cular in all of astronomv isee Table 2-12'. .A student 
ma\' raise the point that the earth is not infiniteh 
far awav . hence the points B and D lin Fig. 2-87 1 
are not on parallel lines tangent to Jupiter. This 
makes the observed time intenal slightK less than 
half a revolution. The effect is negligibK small, onlv 
11 sec of real time. In the film, this corresponds to 
onh 0.01 sec of apparent time. 

2. Radius of orbit 

The simple method of orbit radius described in the 
Handbook gives results even more precise than the 
measurement of the period. 

3. Mass of Jupiter 

As discussed in the Test, the mass of Jupiter and 
the mass of the sun are related as follows: 



( 1 \ / 1.390.000 
1 1048/ \ 139.000 



0.9 = 



m. 



earth s orijit 



d Jupiter 



T„ 



The student knows the values for the earth s orbit 
and has measured the values for lo s orbit. Hence, 
the ratio of the mass of Jupiter to that of the sun 
can be calculated. Using values similar to the ones 
the student will obtain bv measurement of the film, 
we have 



m^ 
m. 



422 X 10 km 



X 



365 X 24 hr 



150 X 10 km/ \ 42.5 hr 
= 12.813 X 10"^i X 1206.11" 
= 94.9 X 10"" 
_ 1 
1059 



If a student wishes to go still further, the density 
of Jupiter relative to that of the sun or that of the 
earth can now be calculated. The respective vol- 
umes are proportional to the cubes of the diame- 
ters, and the ratio of masses has been found. The 
average diameter of Jupiter is 139,000 km: that of 
the sun is 1,390,000 km. The result is that Jupiter s 
density relative to that of the sun is 



Thus, Jupiter is only slightly less dense than the 
sun. The actual densities in gm cm^, based on 
knowledge of the gravitational constant G are: 
earth, 5.52: sun, 1.42: Jupiter, 1.34. 

L13 PROGRAM ORBIT. I 

In this film, a student is plotting the orbit of a 
planet, using a stepwise appix).\imation. Then, the 
computer is instructed to solve the same problem. 
The computer and the student follow a similar pro- 
cedure with 60-dav intenals. 

The computer language used was kortr.a.\. 
The FORTRAN pixjgram i contained in the stack of 
punched cardsi consists of the rules of the game : 
the laws of motion and of gravitation. These de- 
scribe preciseK how the calculation is to be done. 
The program is translated from koktr.a.\ and is 
stoi'ed in the computer memorv before it is exe- 
cuted. 

The calculation begins with the choice of initial 
position and velocitv'. The initial vaues of A and V 
are selected and also the initial components of 
X\'EL and VA'EL. iA\'EL is the name of a single for- 
TRA.\ variable, not a product of four variables. i 

The program instructs the computer to calculate 
the force of the sun on the planet from the inv erse- 
square law of gravitation. The calculational proce- 
dure can be thought of as a blow' applied toward 
the sun. Newton's laws of motion are used to de- 
termine how far and in what direction the planet 
mov es after each blow. 

The computer output, the result of the calcula- 
tion, can be presented in manv wavs. .A table of X 
and Y position values can be tvped or printed. An 
X-Y plotter can draw a graph trom the v alues sim- 
ilar to the hand-constiTJCted graph made bv the 
student. The computer results can also be shown 
on a cathode-ray tube iCRT, similar to that in \ our 
television set. 

The numerical values for initial position and ve- 
locity' are entered at the computer tvpewriter by 
the operator after the computer t\pes messages 
requesting the values. The dialogue for trial 1 is as 
follows: 

GI\E ME IMTLAL POSITION IN AU .... 
X = 4 
V = 

GI\ E ME INITIAL \T:L0CIT\' IN AU VR .... 
XVEL = 
WEL = 2 

GIVE ME CALCULATION STEP IN DAYS .... 
6 

GI\E ME NUMBER OF STEPS FOR EACH POINT 
PLOTTED .... 

1 

GIVE ME DISPLAY \10DE .... 

X-Y PLOITER 



FIL.M LOOPS A,\D FILMSTRIP NOTES 



151 



Studtjnts sec that tin; oi1)it displayed (jn the X-Y 
plotter is like their own graphs. Both oHiits fail to 
close exactly. This result is surprising, as it is 
known that the orbits of planets ar-e closed. Dis- 
cussion should bring out that perhaps the step- 
wise appr'oxiniation is too coar-se. Ihe blows may 
be too infr-(;(|uent n(;ai- perihelion, whert; the foice 
is largest, to be a good approximation to a contin- 
uously acting for-ct;. In the Film Loop Program (Jr- 
bit. I!" the student can see if this explanation is 
r-easonable. 

LI 4 PROGRAM ORBIT. II 

Ibis film is a continuation of Program Orijit. I" A 
computer is used to plot a |)lanetary ori)it with a 
force inversely proporlional to the distance. The 
computer program uses Newton's laws of motion. 
IJlows act on the body at equal time intervals. 

Presumably the orljit calculated in the pr"e\ious 
film failed to close because the blows wer-e spaced 
too far apar1. To test this assumption we need only 
specify a smaller time intei-val between the calcu- 
lated points, and use the program f)re\'iously 
stort;d in the computer memory. 

A portion of the "dialogue" between the com- 
puter- and the operator for trial 2 is as follows: 

GIVE ME CALCULA riON STEP IN DAYS .... 
3 

GIVE ME NUMBER OF STEPS FOR EACH POIM 
PLO TIED .... 

7 

GIVE ME DISPLAY MODE .... 

X-Y PLO TIER 

Points are now calculated every 3 days 120 times as 
many calculations as for- trial 1 in "Program Orbit. 
I "I birt only one out of every seven of the calculated 
points is plotted to avoid a graph with too many 
points. 

rh(! computer outpirt in this film can also be 
displayed on the face of a cathode-ray tube ICRTi. 
Vhv c;r I display has the acKantage of speed and 
flexibility. On the other hand, the pernnanent rtu- 
or'd afforded by \hv. X-Y plotter- is sometimes very 
conxenient. 

VV(! will use the GR I display in the other Filn\ 
Loops in this series, /Jo, Lid, and LIT. 

L15 CENTRfVL FORCES: ITERATED 
BLOWS (computer projjriini) 

,\ bod\' acted on b\' a central loi-ce, one always 
diixu-l(Hl towar-d or away fr-om one point, mo\es so 
that (Mjual ar-eas art; swept oirt in ecjual times. Ihe 
for-c(; can be constant or- xariable, attractixe or rv- 
|)ulsi\c The law of ar-eas is a consetjirence of tin* 
laws of motion and of fbrx'e dir-t>(-ted alwaxs towai-d 
or- aua\ Ir-om a fixed point 

Ibis film was made In photographing the face 
of a computer--(ii-i\en (-athode-i-a\' tirbe Ihe (-om- 
pirter- pr-ogram uses ,\ewton s laws of motion to 
predict the ifsult of ap|)l\ing blows I be basic- pr-o- 



gram remains the same for all parts of the film, but 
we can program the computer to \ary the force. 

The computer generates a random number that 
determines the magnitude of the blow. Blows are 
applied at equal time intervals. The direction, to- 
ward or away fr-om the center, is also selected at 
random, with a slight preference for attractive 
blows to keep the pattern on the screen. The dots 
show the par-tide "s position after equal intervals of 
time. The intensity and dir-ection of each blow is 
represented by a line at the point of impact. 

Have students project the film on paper and 
mark the center, the points where blows wer-e ap- 
plied, and the direction of each blow. They should 
then measur-e the ar-eas of several triangles to de- 
ter-mine if the law of ar-eas applies to this motion. 

If a weight on a string is pulled back and released 
with a sideways shove, it mo\es in an elliptical or- 
bit with the force center i lowest point i at the center 
of the ellipse. The force is proporlional to the dis- 
tance from the center. The computer appro.ximates 
a smooth orbit for such a forc^e b\' delivering the 
blows at shorter time inter-vals. In scene 2a, four 
blows ar-e used for a full orbit; in scene 2b there 
ar-e nine blows. In scene 2c, 20 blows give a good 
approximation to the ellipse that is observed v\ith 
this force. 

A similar prxigram uses two planets with a force 
on each that varies inver-selv as the square of the 
distance from a forc;e center. It is assumed that no 
force acts from one planet to the other. For the 
r-esulting ellipses, the force center is at one focus 
(Kepler's first law), not at the center of the ellipse 
as in the previous case. 

L16 KEPLER'S L/\U'S (computer program) 

A computer program similar to the one used in the 
L15, "Central Forx-es: Iterated Blows, ' causes the 
computer to displav the motion of two planets Im- 
pulsive blows dir-ected towar-tl a center ithe suni 
act on each planet at eqiral time intervals. The for-ce 
exerted by the [ilanets on each other is ignored in 
the program. Each [)lanet is acted upon onlv by 
gravitational for-ce of the sun, which varies in- 
versely as the squar-e of the distance from the sun. 

Initial positions and initial velocities for the 
planets wei-e selected. The positions of the planets 
ar-e shown as dots on the face of the cathode-ray 
tirbe at r-egirlar interAals. iMany morv points were 
calculated between those displayed. i This film is 
a true "loop, " since the motion is continuous: 
Therv is no beginning and no vnd'. 

Students tan (heck Kepler s tbrve laws b\ pro- 
jetting the film on paper- and mar-king the succes- 
sive positions of the planets I he law of arvas ran 
be verili«»d in this sitiration In drawing triangles 
anil measuring areas Students should find the 
ar-eas swept oirt in at least thi'ee places: near 
perihelion, near- apbeliori. and at a point approxi- 
matelv midwav between perihelion and aphelion 

lo (-he(-k Keplers thirxl law stirtients measurv 
the distani-es of perihelion and aphelion lor each 



UNIT a / M()TI()\ l\ THK HKA\T-:,N8 



body and measure the periods of ("evolution. They 
detemiine whether the orbit is an ellipse with the 
sun at a focus by using a string and thumbtacks to 
draw an ellipse, as showii on page 100 of the Hand- 
book. The empts' focus should be s\iiimetrical with 
respect to the sun s position. 

Another method students ma\' use to test whether 
their plot approximates an ellipse is to place the 
plot on an inclined plane and shine a flashlight on 
it. Adjustment of the distance of the light source 
and the inclination of the plane should be possible 
so that the boundaiy of the light matches that of 
the elliptical plot closely. 

Ad\anced students might seek an algebraic 
method. 

LI 7 UNUSUAL ORBITS 
(computer program) 

In this film we use a modification of the computer 
program described in the notes for L15 "Central 
Forces: Iterated Blows." The forces are still central, 
always directed toward or awa\' from one point, 
but the\' are no longer onl\' in\erse-square forc'es. 

The planet Xeptune was discoxered because of 
its gra\ itational pull on Uranus. The main force on 
Uranus is the pull of the sun, but the force exerted 
by Neptune changes the orbit of Uranus \'er\' 
slightK . .Astronomers predicted the position of the 
unknown planet from this small eftect on Uranus. 

Tvpicallv, a planet's orbit rotates slowK' because 
of the small pulls of other planets and the retarding 
force of friction due to dust in space. This effect is 
called 'ad\ance of perihelion. Mercuiy's perihe- 
lion advances about 500 sec of arc/century. Most 
of this was explained by perturbations due to the 
other planets. However, about 43 sec/century re- 



mained unaccounted for. When Einstein reexam- 
ined the natui-e of space and time in developing 
the theoiA' of relati\'it\', he developed a new gra\i- 
tational theoiy. Mercuiy s orbit is closest to the 
sun, and relativity' was successful in explciining the 
extra 43 sec/centuiy. 

The first sequence shows the acKance of peri- 
helion due to a foix:e pixjportional to the distance 
R, added to the usual inverse-square force. The 
"dialogue" between operator and computer starts 
as follows: 

PRECESSION PROGRAM W ILL USE 
ACCEL = Gifl . H) -t- P . H 
Ca'E ME PERTURBATION P 
P = 0.66666 
GIVE ME INITIAL POSITION IN AU 

X = 2 
r = 

GIVE ME INITIAL VELOCITY IN AU/YR 

XVEL = 
WEL = 3 

The svmbol • means multiplication in the For- 
tran language used in the pixjgiam. Thus GiH • R) 
is the inverse square force, and P • fi is the per- 
turbing force, proportional to R. 

In the second part of the film, the force is an 
inverse-cube force. The orbit resulting from the in- 
verse-cube attiacti\e force, as with most forces, is 
not closed. The planet spirals into the sun in a 
"catastrophic" orbit. As the planet approaches the 
sun, it speeds up, so the points are separated by 
a large fraction of a re\olution. Different initial po- 
sitions and velocities would lead to quite diffei-ent 
orbits. 



Equipment Notes 



EPICYCLE MACHINE 

Turning the handle of the epicycle machine shown 
in the photograph on the left, causes the lighted 



bulb to mo\ e back upon its own path and describe 
a retrograde loop, as in the photograph on the 
right. iThe batterv can be fixed to the machine with 





EQL'IPME.NT .VOTES 



153 



tape that is sticky on Ijotli sides. i i he ladius ot the 
epicycle looj) can he adjust(;d hy moving the hat- 
teiy and light hulh tovvaitl the center of the; epi- 
cycle. Fuilh(!mion;, the celative speeds along tht; 
deferent and epicycle can he (thangcid hy changing 
the elastic dii\(; to diflenMit positions on the drive 
wheels. The conseqiicMices of these changes are 
nMrograde loops of diff(;r(!nt shapes and durations. 
In short, the e|)icycle machine may he used t(} 
demonstrate how i(;trograde motion may he ex- 
plained hy the Ptolemaic model. 

PLAXETARILTVI USE FOR 
PROJECT PHYSICS 
Aflvantag(;s of the plancMarium 

rh(; op|KJitunit\ to (visiK point to a planet with an 
armw projected on a star field will in itself make 
a trip to the planetarium wodhwhile. In addition, 
time may he speeded up to illustrate the conse- 
quences of the it)tation, revolution, and precession 
of the eai1h. Most planetariums hii\e auxiliary pro- 
jectors for slid(!s, (nerlays, and hinary motions. Per- 
haps th(! greatest value to a class will he to see the 
projection of various coordinate systems on the 
star (ield. 

rhe (Miiotional impa('t of music used during a 
plan(!tarium show has teaching acKantiiges if the 
music is selected appif)i)riateK and us(!d with dis- 
cretion. 

Mofies of operation 

Most |3lan('tarium cuiators are |)rimaril\' lecturers. 
This mod(! of liujching can assist you in prc^senting 
id(!as reflated to I 'nit 2. 

Other modes of t(!aching that \ou ma\ use an; 
\\c)rksh(M!ts, oral c|uestions and answer's, arguing, 
moving hack and forth from the star- doriu; to out- 
(jf-doors, and challenging groups of students uith 
special projects. 

VVor-ksheets can he used during twilight condi- 
tions or- hy r-igging red lights around the cow of 
the planetarium. The red lights are hr-ought up 
only dirring an answer jjer-iod and do not distur-l) 
night adaptation of the eyes. 

The c]iiestion and answer technique is difficult 
when working in tin; dark. However, if pr-f?plannirig 
is done with the stirdents, quc^stioning can he ef- 
fective. A cerlain amount of argirmcnt rt;sults in a 
high degree of student irnoKcment l)ut it mirst he 
done skilltulU 

Moving gr-oirps of students hatk arul forlh fr-om 
the star- dome to the out-ol-doors will r-esirit in aj)- 
plitation of the knowledge gained in the |)ianetar- 
ium to li(!ld ()hser\ati{)ns However', this i-e{|uir-i's a 
good viewing night and a planetarium lotalcd 
away Ir-om citv lights. 

Preparation 

Discuss and plan the pr-ogram with the planetar- 
ium curator-. The lir-st |)r-ogr-am is olten a (-ompro- 
mis(? hetvvjMMi what vou want to teach in I 'nil 2 
and what the curator is e(|irip|)ed to do in the 
|)lan('lar-iirm. Often the curator- will moditv one ot 



several successful programs to fit vour- needs, l^ater 
you can try one of the other modes of teaching 
descr-ihed previously. 

Sample programs 

Kach program should have a clear hut limited re- 
lationship to what is heing taught in L'nit 2. Ihose 
that follow requir-e ahout 30 min in the planetar- 
ium. Ix'ave plenty of time for- discussion, testing, 
and questions and or- [jeriTiit students to ask ques- 
tions that ai-e not on the suhject of the program. 
These additional activities mean that a 30-min 
planned program will require 60 min. More than 
one program could be given during a visit. 

PROGR.VM I THE MOON AND THE Sl^' 



If this can be followed 
up by having students 
observe the actual mo- 
tion of the moon toward 
the east ever>' 24 hr, then 
one can meaningfullv de- 
fine the dir-ect motion of 
planets. 



A. rhe moon 

1. young cr-escent 

2. diurnal motion to 
west 

3. 13° eastward mo- 
tion per 24 

4. phasing 

5. 1 month of obser- 
vations 

6. eclipses 

7. dailv predictions 
of position and 
phase 

B. The sun 

1. diurnal motion 

2. north-south mo- 
tion 

3.eastvvar-d motion 

among stars 
4. the e(|uinoxes 
.■j. the solstices 
(i. looking at the sirn 

at different lati- 

lirdes 
(;. Discussions, questions, arguments, test, or other 
activities. 



.Most planetarium ma- 
chines can change the 
latitude of the viewer to 
demonstrate pari 6. 



FROGR^ViM H THE 

Pro(>rnm 
.A. \'enus lor Mercury i 
1. twilight observa- 
tion 
2.gr-eatest angle of 
eastern elonga- 
tion 
3.gr-eatest angle of 
western elonga- 
tion 
4. dir-cit motion 
.■) ' phase change 
(i. intensitx change 

7 ' inferior and sir- 
perioi conjunc- 
tions 

8 ' transits 



FL/VXETS 

Coninicnts 
Tr\' to schedule this 
planetariirm visit when 
\enus is at its greatest 
angle of elongation (refer 
to Sky ami I'elescope'. 
Then have students make 
independent dailv obser- 
vations. .Also, it is advaii- 
tageous to schedule this 
visit close to doing E-10. 
Or-fiit of Mer-cirrv 



'Supplement with slide 
pr-ojeitioris 



.14 



UIMIT 2 / MOTION l\ Till-: HivWlilVS 



B. Mars (Jupiter or Sat- 
urn I 

1. identification b\' 
lai intensity 

ibi location 
(CI motion among 
stars 

2. direct motion 

3. retrograde mo- 
tion 

4. out-of-doors ob- 
servation thixjugh 
a telescope 

C. Discussion, questions 
acti\ities. 



Refer to the magazine 
Sky and rclcsrofw to de- 
termine when a supeiior 
planet is up in the eve- 
ning sk\ and plan the 
planetarium \ isit accord- 
ingly. 

Most planetariums ha\e 
a telescope and its use 
will inspii-e students. 



arguments, test, or other 



PROGRAM III THE EATXIXG SKY AND 
COORDINATES SYSTEMS 

Program Comments 

A. The Evening sk\' Most planetaiiums have 

l.What can be seen pixigrams planned on this 
at tVNilight? subject. 



2. moon 

3. planets 

4. circumpolai- stai"s 

5. one constellation 

6. outside observ a- 
tion if possible 



This program should 
always be gi\en before 
any of the others on a 
gi\en evening because the 
operator must set the sky 
carefulK . During all other 
programs the sk\' will be 
changed ftxjm the date of 
the program. 

It is helpful to do class- 
work on coordinate sys- 
tems both befoi^ and after 
this planetarium pro- 
gram. L'se '113, T14, and 
T17. 



B. Coordinate systems 

1. altitude and azi- 
muth 

2. local meridian 

3. declination 

4. celestial equator 

5. right ascension 

6. ecliptic 

7. examples of star 
locations 

C. Discussions, questions, arguments, test, or other 
activities. 



Suggested Solutions to Study Guide Problems 

CHAPTER 5 



2. (a) Local noon will occur when the shadow is 

shortest, or midway between a.m and pm 
times of equal length 

(bi The noon shadow on June 21st will be 
shorter than at anv other noon. 

Id The solar year is the number of days be- 
tween da\s. in the same season, when the 
shadow lengths are equal. For e.\ample, this 
could be June to June. The more accurate 
observations would be made close to the 
equino.xes when the suns \-S position is 
changing most rapidly. 

3. lal 365.25000 davs - 365.24220 days = 0.00780 



day = (7.80 x 
day) = 674 sec 



10 ^ days i( 8.64 x 10^ sec 



(bi 



7.80 X 10 days 
3.6525 X 10" dav s 



X 100 



= 0.0021% of a year 



4. lai Observations 



ibi Reason for Impor- 
tance 

Appai-ent motions of the sun: 

(ai daily westwaixl day-night determina- 
tion 
seasonal changes 



motion 
(2) annual 

north-south 
motions 
lei annual east- 
ward motion 
through the 
stars 



length of the year, ba- 
sis of the calendar 



Appai'ent motions of the moon: 
(a( phase changes tie-in with other phys- 
ical phenomena 
(21 continuous basis of the month: re- 

motion east- lated to eclipses of sun 
ward among the and of moon 
stars 
Apparent planetaiA motions: 
(1) retrograde mo- a seemingly contrary 
motion that should be 
explained 



planets are different in 
some wav's 



tions westward 
at opposition 
(Mars, Jupiter, 
Saturn I 
(21 periodic mo- 
tions of \enus 
and Mercury 
near the sun 
Appai'ent fixed positions of the stai-s: 
(II continuous cir- the most unifonn of all 
cumpolar rota- obsened motions 
tion of the ce- 
lestial sphere 

The nightly and annual changes could be ex- 
plained for an obsei-ver at a fixed location How- 
ever, one could not explain the changes in an- 
gular elevation of the North Star w hen trav eling 
in the \-S direction, nor the variation in rising 
and setting times when trayeling in the E-W 
dii-ection. 

In each 24-hour period the moon rises in the 
east and sets in the west, rising latei- on each 



SLGGESTED SOLI TIOVS TO STl DA Gl IDE PROBLL.MS 



155 



successive date. It also appeai-s to be mo\ing 
eastward among the stars and relative to the 
sun. It goes thr-ough the full cycle of phases in 
a month. 

7. After Mercury and Venus move westward past 
the sun, they appear- in the early morning sky 
and soon reach their maximum brightness. At 
maximum elongation, east or west, they will 
appear in quarter phase, Ther-eafter-, they will 
fade slowly and move in toward the sun, pass 
it, reappear in the evening sky, and brighten 
slowly as they approach maximum eastward 
elongation. 

8. Quadrant No. of Degrees 

1 102 

2 78 

3 78 

4 102 

360° 

9. lal — r = 15 per hour 

24'' 

(b) (1) Check latitudes on a map of the U.SA. or 
compare the length of noon shadows of 
posts of equal height. 

(21 The airplane goes 5 x 800 km or 4,000 
km between Washington, D.C. and San Fran- 
cisco. 

(3) From the differ-ence in sunset times we 
know that the longitudes of Washington, 
D.C. and San Francisco differ by 3 hour-s, or 
45°, or one half of the distance around the 
earth. 

Then the distance around the earth at the 
latitude of these cities (about 38°i is 8 x 
4,000 km or about 32,000 km. The value at 



the equator would be larger by about 20%. 

The equatorial circumference would be 

32,000 X 1.2, or about 38,000 km. The di- 

circumference 
ameter of a circle is the . So 



the diameter of the 
38,000 km 



earth is about 



3.14 



= 12,100 km. 



10. Ptolemy assumed: (a),(b),(c),(d),(e),(f) 

11. (a) They both predicted the location of the 

stars and planets reasonabK' well. 

(bi According to Greek science, the geocentric 
system had the ad\antage of conforming to 
the dogma of the perfect planets moving 
only in perfect circles and the earth being 
stationary'. The disadvantage of the Greek 
system was that too many epicycles were 
necessar\' to explain the heavenly motions. 
Although the heliocentric svstem ex- 
plained retrograde motion mor-e simph', it 
did not conform to the Greek dogma, and 
it predicted stellar parallax, which could 
not be confirmed during ancient times. 

Id Ptolemy did provide a practical system 
based on celestial events as viewed from the 
earth , which served quite well for mam cen- 
turies. 

12. The phenomena of interest in ancient astron- 
omy are mostly slowly changing and cyclic. 
Therefore, a few relatively simple assumptions, 
such as circular motions, could lead to a fairiy 
accurate theor^'. 

The phenomena of meteor-ologv and zoology' 
ar-e far mort? numer-ous and rapidlv changing. 



CHAPTER 6 



M, 



M^ 



M. 



3. Copernicus ar^ed that having the sun at the 
center of the planetary- system was more rea- 
sonable,' and that the planets would rnove 
uniformly about itheiri prxiper center" — as the 
rule of absolute motion r^equires. " .Also, the dis- 
tances of the planets frxim the sun incr-eased 

Mj with their periods, which was pleasing to the 

mind. 

4. If you start with the hands together at 12 
o'clock, you will observe that the first pass oc- 
curs a little after 1:05: the second a little more 
after 2:10: the sixth at about K:33 etc I he elev- 
enth pass occur"s as the hands rvturn to 121)0. 
Thus, the frx'cjirencv of passing is " u or 0.92 
cycle hr, which r-e|)resents the apparvnt fre- 
quenin' of the long hand as viewed by an ob- 
serAer riding on the shorl hanil Ihe actual frv- 
quency of the shorl hand is ' u cycle hr, since 
it takes 12 hr to come back to the original nu- 
meral on the clock face. The actiral frequency 
of the long hand is 1 cycle hr- Since " i- * ' i; 

1, we ar'j' led to the statement th.it 



I SB 



I'M'I' 2 / lVIOTI(l\ l\ THK HK/WTiNS 



actual frequency of 

the long hand — apparent frequenc\' of 
the long hand + actual 

frequency of 
the shoi1 hand 

This statement correlates with the discussion about 
the planets as viewed from the earth. (See Te,\t 
page 160.1 Once the frequencx is known, the period 

follows immediatelv as . 

fi^quenc\' 

5. Copernicus calculated the distances of the 
planetarv' oi+)its from the sun and the periods 
of planetary motion aixjund the sun. 1 he Cop- 
ernican s\stem was more simple and haj'mo- 
nious than that of Ptolem\ . .Also, these oit)its 
began to seem like the paths of real planets, 
rather than onl\ mathematical combinations of 
circles that were useful for computing posi- 
tions. 

6. The Copernican svstem imohed a reordering 
of the relati\e importance of the sun and the 
earth. The sun became dominant while the 
earth became "another planet." These philo- 
sophical results were more important than the 
shift in geometiA'. 

7. lai When Mercur\' and \'enus are mo\ing from 

farthest east to farthest west relati\e to the 
sun, the\ are o\ertaking the earth and pass- 
ing between the earth and sun. In the case 
of \'enus, only about one-quarter of its or- 
bital period is required for the planet to 
move from farthest east to farthest west. 

Ibi To find a period for Mereun s motion rela- 
ti\e to the sun, we ha\e a choice of which 
intervals to measure. Probably the most sig- 
nificant would be the times required for 
Mereun' to pass the sun. The three intervals 
for motion from west to east ai'e: 110, 105, 
and 130 days. For motion from east to west, 
the intervals are 127 da\s and 112 da\'s. The 
average of the five intervals is 115 days. The 
variations result fhjm the eccentric orbit of 
Mercurv'. 

(c) The mean cycle compared to the sun is 115 
days, or 0.315 xear. This is T in the equation 
of the chase problem. M is 1. Then 

T 
period of Mercurv = 

0.315 



1.315 
= 0.240 yr 
— 87.5 days 

Id) The major sources of uncertainty' are: 

(II Only three cycles for Mercurv' are shown. 
V\'ith more cycles a better average would 

result. 



12) The orbit of Mercury is not circular, but 
is rather eccentric. Therefore, the ob- 
served intenals depend upon the direc- 
tion from which we on the mov ing earth 
see Mercury' in its oHjit. 
lei Only a little more than half a cycle is shown 
for Venus. We assume that the motion is 
symmetrical. Since a half cycle takes 289 
days, a full cycle would be 578 days. In one 

365 

vear we observe of a cvcle: this is ,V = 

578 

0.632. T is 1 vear. Then 



period of \'enus = 



1 



1 ^ 0.632) 
= 0.613 yr 
= 224 days 

8. The sequence 0.39 AU to 9.5 AV shows no ob- 
vious regularitv. ,A natural step for the student 
to take would be to plot average orbital radius 
V ersus sequential order and then extrapolate to 
n = 7. This \ields a value of about 14 AU. The 
next planet, Uranus, actuallv has an average 
orbital radius of about 19.2 AU, so extrapolation 
as above is not satisfactorv. ,An empirical for- 
mula, known as the Bode-Titus law, is suipris- 
ingly successful not onlv in correlating the 
known data but also in predicting the correct 
orbit for Uranus. However, it breaks down com- 
pletelv for Neptune and Pluto. This law is dis- 
cussed on pages 198-200 of Foundations of 
Modern Physical Science bv Holton and Roller.) 



1 AU 



?i = 



tan - 



= ( I = 14 X 10"^)° 

\2400/ 




- = 12 X 10 I 
2 

Consulting trigonometric tables we find 

1. for small angles the tangent of the angle is 
proportional to the angle 

2. tan 1° = 0.018 = 1.8 X 10"' 
thus 

6 
tan - = tan I2 x lO l = 3.6 X 10 
2 

then 

1 AU 
.V = rr = 2.8 x 10" AU 



3.6 X 10 



10. Each star could have been assigned a very smcdl 
epicycle with a period of 1 vear. This would be 
similar to the larger epicycles with a 1-vear pe- 
riod assigned o Mars, Jupiter, and Saturn. 



SUGGESTED SOLLTIO.VS TO STLD^ GL IDE PROBLE.MS 



157 



11. Some that might be suggested are 

(a) we do not feel the earth's rotation on its axis 
and its revolution around the sun 

(b) it is difficult to imagine the wave-particle 
duality of light and matter 

(c) time dilation from relativity theory 

(d) atomic description of solids 



12, If the eailh is not the center of the universe but 
just one of many planets, we may not be 
unique. Projects that scan the skies for evi- 
dence of some kind of extraterrestrial intelli- 
gence in the form of radio signal patterns have 
been attempted. Our knowledge of the other 
planets of our solar system virtually lules out 
th possibility of life being found on any of 
them. The possible discovery of such extrater- 
restrial life raises such questions as; How does 
their state of civilization (Compare with ours? 
How do we all fit into a cosmic plan, if indeed 
there is one? 



13. The motion is difficult to explain. From \o\'. 1, 
1909, until May 3, 1910, the comet mo\'ed west- 
ward iretrcjgradei. During April 1910, its posi- 
tion changed verv little, although the earth s 
position relative to the sun changed about 30°. 
During May, the comet moved moi-e than 90° 
eastward, with the most rapid motion occur- 
ring around May 19th. 

Stuchmts who do E2-11 will find that they 
can, with the three-dimensional model, explain 
the comet s observed motion. -Also, they will 
notice some other interesting results. 

14. The phrasing of the question, of course, calls 
for a personal reaction. The Copemican system 
did not seem to most people any more 'real 
than he Ptolemaic, but it did la\' the ground- 
woi-k for the later wor-k of Kepler and .\e\\1on. 
So we develop the feeling that a heliocentric 
scheme is not only simpler but lends itself to 
a dvnamic explanation through the law of gra\- 
itational attraction. 



CHAPTER 7 



2. By definition, 1 min of arc is 1/60°. Fherefore, 
8 min is 8/60° or about 1/8°. The moon's appar- 
ent diameter is close to 30 min; 8 min is there- 
fore about 1/4 of the moon's apparent diameter. 

3. (a) Compare an obseivation of Mars at oppo- 

sition to another obsenation 687 days later, 

that is, when Mars is back at the same place 

in its oriiit. 
(b) Use the observed direction toward Mai-s and 

the sun on these two observation dates to 

triangulate the second position of the earth 

i-elative to the fii-st position. 
(cl Repeat this f)i-ocedure for- diffei(Mit sets of 

obseivations 687 days apait. 

4. Since the earth's orbital speed is inversely pm- 
portional to its distance fixjm the sun, the per- 
centage change in speed will be the same as 
the pereentage change in distance. 1.02 AU 0.98 
AU = 1.04, so the distance 1.02 Al' is 4% greater 
than the distance 0.98 AU. The speed at 0.98 Al' 
will he 4% great(M- than the speed at 1.02 AU. 

5. (a) First plot the oibit of the earth. 

(b) (Compare obseivations of Mars mad(> 687 
days apart ithat is, when Mars is at the same 
position in its orbit i and draw sight lines 
from the position of the earth in its orbit. 
The point wheit! the two sight lines to Mai-s 
intei"sect is a point on the orl)il of Mai's. 

(c) Hepi^at this pixxH'ss on paii-s of obsei\ations 
for dilTtMCMil positions of Mai's in its od)it. 

6. 'I'he length / of the string in the loop is most 
easily found by considering tin* extivme case 
ill one (Mid of the ellipse !f.\ is the distance 



from a focus to the nearest point on the ellipse, 
then, 

/ = 2c -I- 2^ 
but 2,v = a — c 

so I = 2c + a — c 
I = a + c 




7. The shape is shown in the figure abo\e. If the 
tacks ifocii are together, r is zeix) and the figure 
is a ciix-le of radius a. ,As the tacks aix* moved 
farther apart, the ellipse becomes thinner and 
the eccentricitA' ici greater. 

H.c - ac 
5 cm 

e = = 056 

9 cm 

9. lai N'our sketih should look like the one in 
question 6 where .v equals the perihelion 
(listance while the distance tmin the right 
locus to the (•lli|)s(' i> the .i|iheli()n 



IHH 



UNIT 2 / MOTION IN THE HE/WENS 



(bi When P is the perihelion distance and A is 14. The ratio of speeds is the inverse of the ratio of 
the aphelion distance, distances. 

c - \A - P\ and a = i/\ + Pi 
a 



P = 



c a + c 

— and A = 



a A + P 

Icifl^, = - = 

av 2 2 

10. The second focus is empty. 

11. (a) apocrvpha, apostasy, apostrophe, apojove, 

periodontal, pericardium, perijove 
Ibi Apogee is the position of the satellite far- 
thest from the earth, perigee the position 
closest to the earth. 
10 apolune and perilune 

12. lai Given: ^ = 5 cm and P — 2 cm 

c = \A — Pi — ^5 cm — 2 cmi = 3 cm 
a — lA + P\ = \5 cm + 2 cmi = 7 cm 
e = a c = 7 cm 3 cm = 0.43 
(b) Given: e = 0.5 and a = 10 cm 

c = a X e = 0.5 X 10 cm = 5 cm 
P = 2/la - cl - 2/110 cm - 5 cm) 

= 2. 5 cm 
A = 2/la 4- ci = 2/110 cm + 5 cm) 
= 7.5 cm 
Ici Given: P = 5 cm and e = 0.8 

e = a/c = i/\ + Pi/IA - P) 
Ail - e) = Pll + e) 
Ail - 0.81 = 5 cmd + 0.81 
0.2 A = 9 cm 
A = 45 cm 
c = (A — Pi = 145 cm — 5 cml 

= 40 cm 
a = lA + Pi = 145 cm + 5 cml 
= 50 cm 



13. Bv definition, e 



a = fi^ + flp = 70.0 X 10* km + 45.8 x 10 km 

= 115.8 X 10'' km 
c = R3 - fip = 70.0 X 10** km - 45.8 x 10*^ km 

= 24.2 X 10* km 
24.2 X 10" km 



115.8 X 10 km 
e = 0209 




«p 


= Vz(a - c) 




fla 


= Vzla + c) 




«P 


a — c 




/}, 


a + c 






c 

but e = - , so c = 

a 


= 0.254a 


«,- 


.i - 0.254a 




R^ 


a + 0.254a 
0.746a 
1.254a 
= 0.594 




Therefore, -^ 


= 0.594 





15. la) From Kepler's law, for bodies orbiting the 
sun 



R 



AU~ 



For Halley's comet, T = 76 \t, so Iwith the 
understanding that T is expressed in vr and 
fl,.inAU) 



(76 vr)' 



yr" 



fij 


-'av^ 


76- 




HJ 


= 1 


RJ 


= 76' 




= ^^W 




= \^'57,700 




= 17.9 AU 


(b)fl^ = 


c 

Vza + Vzc but - ^ e 




a 


so R^ = 


Vzta + 0.97a) 




1.97a a 


~ 


or, since — R 
2 2 ^ 


= 


1.97 fl_,^ = 1.97 X 17.9 AU 


= 


35.3 AU 



e = 0.97 



(c)fl = Vza - Vzc but - = e = 0.97 
a 



sofl. 



V2ia - 0.97ai 
0.03a 



= or, since — = R , 

2 2 "' 

= 0.03 R^^ = 0.03 X 17.9 AU 
= 0.54 AU 



SUGGESTED SOLLTIO.NS TO STLDV GLIDE PROBLEMS 



159 




(d) The ratio ol speeds is the inverse of the ratio 
of the distances. 
«,, 35.4 
H. 



= 66 



0.54 

The greatest speed is about 66 times the 
least speed. 

r- vr 

16. For planets othHini' the sun, — ; = 1 - — - 

Thus, T = VH.„' when 7' is expressed in vrand 

H^^ in AV. 

For Pluto, «^, = 39.6 AU, so 

T,,,^,„ = V39.6' = V62,000 = 249 yv 



17. For- I'ranus 



7-, 



84.013" 



,060 



li. ' 1!).19' 



= 1.00 



7,070 
27,150 



T- 164.783 

For iX'eptune —^^ = :- = 

R- 30.07' 27,200 



^ 1.00 



Foi- Pluto 



7,.' _ 248.42' 



39.52' 



61,800 
61,600 



= 1.00 



18. ! he figures available to Kepler were less accu- 
rate than those used in (luestion 17, but must 



ha\'e given values of k close to 1 .0. He wanted 
such a regularitx' and was willing to o\eriook 
the small variations in k. 

19 i he radius of the satellite orbit is measured 
fnjm the center of the earth: 380 min is equal 
to 22,800 sec and 18,000 km is equal to 18 x 



10 m. Therefore, 


T- 122.800 seci- 




R^ 118 X 10*" mi' 




= 8.9 X 10 "se 


20. R'n] = k = Rini 

Therefore, 

i7',/7'-,r = (fl,/fl,)^ 




rr,rj\r = il/4)' 




iT/I\r = 64 
T,/T^ = V64 




T,/Tj = SIT, is 8 times T,i 



21. The period of the satellite is 28 days (24 x lO^ 
seccT^/H' = 9.9 x 10 '^ sec" m^ 

fl' = 9.9 X 10 '^ sec'/m^/T^ 
(24 X 10^ seer 



R' = 

9.9 X 10 ■' secvm' 

R' = 5.91 X lo" m' 

fl = 3.9 X 10" m 

22. 7^/fl' is equal to 9.9 x 10 " sec m within the 
accuracy of the data gi\en and the calculations. 

23. (a I Note that, in addition to Kast being on the 

right, .\orlh is downwatxl It ma\ be of spj'- 
cial intei-est to a \el^ feu students that the 
sun is northeast of Jupiter: therefore, a 
shadow is cast towaixl the southwest. Con- 
sequenll\-, on days 7 and 28 lo is in a 
shadow. Furthennore, on da\' 28 Calistro is 
just emerging from a shadow. These aiv 
woilhwhile thive-dimensional thoughts 



Day 



West 



Juptter 



East. 



14 



21 



(in '»H«dow) O 



V 1 -f-rom shadow / 



26 



(jr, •9h»c*Ow) 



Kid 



I .Ml 2 .MOTION l\ Tllli HivUlJ.VS 



ibi, ic) 



IV 



Satellite 



Europa 

Ganymede 

Calistro 



/?., 




T 


T 


P 


ff.v' 


PlR" = k 


(mm)* 


days 


hours 


(X 10^ hours) 


(X 10* hours') 


(X 10^ mm') 


Ihr^mm')"* 


3.4 


1 


20 


0.44 


0.193 


0.398 


485 


5.2 


3 


12 


0.84 


0.703 


1.42 


495 


8.2 


7 





1.68 


2.81 


5.56 


505 


14.6 


16 


16 


3.84 


14.7 


31.3 


470 



'The measurements were made bv means of a 10 x view finder with a millimeter scale attached to it Therefore, the scale 
distance of 3 4 mm for lo represents 419.000 km This is perfectly proper and does not prevent one from deteimining the similaritv' 
of the constant of p^opo^tionalit^• between T^ and H' . 

"Students will wonder whether these numbers adequately demonstrate in\ariance They should answer their own question by 
anal\zing the error in\ oK ed in each measurement and the inaccurac> of the drawing from Sky and Telescope. 



24. Guide students to a discussion of science co\- 
erage in various media i newspapers, maga- 
zines, T\', etc. I. 



25. Kepler s persistence lor stubbornness' is re- 
markable. The discussion of this question could 
be quite interesting. 

26. Kepler expected a theor\' to predict new ob- 
sen ations with accurac\'; began to seek ph\'si- 
cal causes for motions; switched from geo- 



metrical to algebraic mathematics; examined 
with care the limits of accurac\ of the obser- 
\ ations of Brahe and accepted the observations 
to that accuracy' ithat is, he tnjsted the instixi- 
ments and the obsenersi. 

27. -An empirical law is a generalization based on 
observations. Empirical laws are inherentK lim- 
ited since phenomena can ne\er be observed 
completely. Their \alue is that 1 1 <■ the\ are di- 
rectly related to experience and i2i the\ serve 
as a foundation for theoretical speculation. 



CHAPTER 8 



2. (ai along a straight line at uniform speed. 

(bi caused the planets to deviate trom motion 
in a straight line. 

(ci directed to a center; and that center was the 
sun. 

idi varies in\ersel\' with the square of the dis- 
tance. 

3. lai The fall of the apple caused Xewton to con- 

template the fall of the moon toward the 

earth. 
lb) Because the moon is 60 times farther than 

the apple fhjm the center of the earth, the 

acceleration on the moon should be il 60i' 

of that on the apple, or 9.8 m sec' x il 60' 

= 2.7 X 10"^ msec-. 
(c) The centripetal acceleration 

a. 



= 


R^^ 








= 


i2-!Tflr 
T- 


1 

X — 

R 








■X-u- 


13.9 


X 10*1 


m 




128 X 


24 X 


36001 


- sec^ 


2. 


71 X 10"^ m/sec' 





4. The real question is "What holds the moon 
down? ' The Newtonian view requires a force 
to prevent the moon from flving off in a straight 
line. 

5. Ever>' object in the universe attracts every other 
object with a gravitational force directed be- 



tween the centers of the objects. F — GiA/atji R' 
where F is the force between objects, A/ and m 
are the masses of the objects, R is the distance 
between centers of the objects, and G = 6.67 
X 10 " \ • in' kg^. .Available evidence shows 
no change in G with time or position. 



T (days) 



D^ PD'fx 10-^) 



1.77 


3.13 


6.04/- 


222 


1.41 


3.55 


12.55 


9.62 


900 


1.396 


7.15 


51.0 


15.3 


3610 


1.407 



IV 



16.7 



278 



27.0 



20000 



1.39 



The results ai'e in agreement to an accuracy of 
about I'^o, which is about the accuracy of the 
data given. The law of periods appears to hold. 

la I Descartes' theory directed further attention 
to the question "Why do planets move ac- 
cording to Kepler's laws?" 

Ibi Having space filled with a fluid avoided the 
pix)blem of action at a distance associated 
with .Newtonian gravitation. 

Id The theorv conceived of planetan motions 
as explainable in terms of phenomena fa- 
miliar to us on earth. 



8. (al Since the designation of which body is m, 
or m, is arbitrary', the general relation ap- 
pears to be F 5: m^m,. 



SL'GGESTED SOLLTIO.VS TO STLDV Gl IDE PROBLEMS 



161 



m. 



•-» 






*- • 


•"-^ 






-•• 


»• ^'^ 


-^ 




^ JL 


9 




A 



" 'I " 't 
2x3 



(b) (1) (m, + m^i ^ F 

If either mass vanished, a force would still 
exist, but this is contrary to the definition of 
a force that must prtjduce an acceleration 
It is absurd. 

m, 

(21 —^xF 
m. 

If m, vanishes, the force F becomes zero, 
which seems to be consistent with the def- 
inition of a force. But either body could be 
identified as m,. If the denominator van- 
ishes, the force becomes infinite, which is 
absurxl. 

This type of analysis uses the limiting 
case" in an imaginarv solution. This is a 
powerful technique in the study of possible 
solutions. 

9. The acceleration due to gra\it>' at the surface 
of a body a^ -^ m/R^ when m is the mass of the 
body and R its radius. 



a^ (moon) m,./fl 



_ '"M'l M 



a^ feailh) 



^t/f^'t. 



m. 



0.012 



R\. 



1.00 
== 0.012 X 



10. 



F. = G 



But F = F„, so 



1 16 
F^ = /Tip a,. = m 
ni.ni 



3.95 X IQ- 
1.08 X 10^ 

15.6 

= 0.162 

4tt'H 



4TT'fl 



= G-^ 
R- 



-d3 



Ihen 7 -' = 



Gm^ 



or r- = — — 



47r" 
G 






1 1 The gravitational force betvv-een the tvvo spheres 
is given by 

Mm 



F = G 

R- 

= 6 X 10 " \ m 

(;(;7 ■ 10 - \ 



, / 1,000 kg X luukg . 

\ 10 m' / 



The acceleration on the lai-ge sphere is 
a„ = M/F 

= 1,000 k^6.67 X 10"* N 



= 16.67 X 10 
= 6.67 X 10" 



kg m/sec^i/kg 
m/sec^ 



The acceleration on the small sphere is 10 
times greater: a^ = 6.67 x lo'^m/sec^. 

12. The predicted positions and discoveries of 
Neptune and later Pluto were dramatic appli- 
cations of the .\e\\1onian gravitational theoiy. 
The predicted position of Neptune was based 
on small unexplained irregularities in the mo- 
tion of L'ranus By a complex anahsis the mag- 
nitude and direction of the disturbing forces 
were calculated Then the direction to the cen- 
ter of those forces i.Neptunei was derived. On 
the basis of Bode s law, the distance of .Neptune 
from the sun had been assumed. Actually, the 
planet was not so distant nor so massive as 
assumed. 

During the vears when Uranus was showing 
the unexpected motions, it was passing be- 
tween .Neptune and the sun Thus, the attrac- 
tions caused bv .Neptune were at a maximum. 

The predicted position of Pluto was based 
on irregularities in the motions of .Neptune, 
L'ranus. and Saturn, which were all at the limits 
of observational accuracv .Although Pluto was 
discovered near one of the predicted positions, 
its mass is probably too small to exert the sus- 
pected forces. 



13.laiilc, „ =—7 

Jup ri3 



16 



dAT/ 



365 dA 



— AU) 
80 / 



14.57 X lO'^i^ 
1125 X 10*1^ 



20.8 X 10 
1.97 X 10 



TT = 1.05 X lO" vi^/AU' 



R^ 



Similarly 



■Mm 



for a planet moving 
, about the sun, S. iSee 
10.1 

for a satellite moving 
about Jupiter, J. 



47r^ Rj 



Then m^ = f- and m, = 

Ij \126 X 10 V \ 



G 7/ 



m^ 
m. 



45.8 X 10 



26 X 10 
in years and .Al'i 

= (4.12 X 10^1^ 13.85 X 
= 712 X lo" X 148 ^ 
= 1 05 X 10' VT'' At 



119 



10' 

10 



-)' 



162 



l'\IT a / M()TIO\ l,\ THK HEAXlilVS 



Note: In this calculation, the \alues used for R^ and 
Tg were those associated with Jupiter. We 
could ha\e used the set associated with anv 
of the suns planets, for example the earth. 
Would this ha\e been an advantage? 



Alternate Solution 

Since T^" = | I — ^ , we ha\e 



4tt" 
G 



but k^ = 



(f) 



Similarly, for Jupiter m, - 



4-n- 
G 



Then 



h 
^ 



1.05 X lO^ as found in 



part b. 
(ci The distance to Jupiter could be found by 
triangulation in the same wa\ that Kepler 
found the distance to Mars. .At that distance, 
each unit of angle corresponds to a specific 
distance. The angular radius of the satel- 
lite's orbit can be observed and then con- 
\erted into kilometers. 



14. From 10 we have 
'4t7^\ /r^ 
G I \m 



T = 



as applied to an earth satellite. 

T = 24 hr = 24 X 3600 sec ^ 8.65 x 10^ sec 
, G m^ 



47r- 


^ r 












16.67 


X 


10" 


"i 


X 15 


98 


X lO^^I 






4 


X 


3.14' 






X (8.65^ 


X 10* 








6.67 


X 


5.98 


X 


8.65 


X 


8.65 



10^' 
4 X 3.14 X 3.14 

= 75.8 X 10-' 

R = "^ 75.8 X lo" m = 42 X 10^ km 

Note: This is the distance from the center of the 

earth, not from the surface of the earth. 
Alternate Solution 
From the motion of the moon we can derive ky-. 

(27.3 davsi" 



k =1- = 



3.80 X 10" kmi' 
743 



55.0 X 10 



13.5 X 10 



For a satellite to sta\' abo\e the same place on 
the earth it must have a period T of 1 dav. 
Then 



R' - 



, or fl = — 



\13.5 X 10 '^/ 



R = 42,400 km 



= 10.0740 X 10 I- = 
0.424 X 10^ km 



15. F. 



/T7,/T1, 

R- 



Let m, = 1 kg and m, = m^ - mass of earth 
fl'^K 6.4 X lO^^m^] X 9.8 [\] 



G m. 



6.67 X 10 



\m- 



(40. 



X 10''i X 9.8 



l[kg] 



6.67 X 10 " 
= 59.8 X 10^ kg, or 5.98 x lo" kg 



16. m. 



4tt' 
G 



R' 

— ; , where R and T refer to the 

T- 



mean distance to moon = 3.84 x lo* m 
mean period of moon — 27V3 davs 

X 186,400 sec/day) 
= 2.36 X lO*" sec 



4tt- 



6.67 X 10 
4 X 9.88 X 



kg 



\m- 



13.84 X 10*i^m'] 



2.36 X lO^i^sec-] 



156.8 X 10 I 



16.67 X 10 ~) X 15.57 X lO'-i 
= 6.04 X 10'"'kg 
This agrees with the answer to 15 to within 1%. 

17. The moon s gra\itational attraction on the fluid 
w aters on the far side of the earth is less than 
its attraction on the solid earth. Therefore, the 
solid earth is pulled away from the water on 
the far side and a high tide results. 

18. Discussion. The moon's inertial motion is uni- 
form motion in a straight line: its displacement 
,v would be ^ = vt. Combined with that moion 
is the gra\itational displacement y towaixl the 
earth gi\'en bv v = ^zat'. 



19. (a) densitv = 



mass 
\olume 



For the earth, 

mass = 5.98 x lO"^ kg 

radius = 635 x 10*^ m 



SUGGESTED SOLLTIO.VS TO STl'DA Gl IDE PROBLE.MS 



163 



volume; = - tt rt ' = \ Xi X :i 14 X (0.635)^ 
:i 

X lo" 

= 1.08 X lo" m' 

5.98 X 10^' 



tl(;ii.silv = 



1.08 X 10- 

= 5.52 g/cm ' 



5.52 X 10' kg/ni^ 



(bl Somnvvhe;re within the earth, not at the 
surface, there must be a large \olume 
with a cl('nsity well above the mean \alu(; 
of 5.52 g/cm '. Because the eai1h is almost a 
symmetrical sph(;i-e, the large, dense mass 
may be a central coie. 

20. Height above moon - 112 km 

Period of oibit around moon = 120.5 min. 

= 7.23 X 10' sec 



22. 




Mass relative 




Actual mass 


to earth = 1 


Sun 


1,980.000 ^ 10'" kg 


330,000. 


Mercury 


0.328 


0.055 


Venus 


4.83 


0.81 


Earth 


5.98 


1.0 


Mars 


0.637 


0.107 


Jupiter 


1900. 


318. 


Saturn 


567. 


95. 


Uranus 


88.0 


14.7 


Neptune 


103. 


17.2 


Pluto 


1.1 


0.185 



rn 



G / T' 

H = (1,740 + 1121km = 1,852km = 1.85 x lo'in 
4 X 3.14 X 3.14 11.85 X lO'V 



23 g"^-^ 

F„ I sun on earth i «- 

F', I moon on earih' m.,rr 



W^,, — distance moon to earth 
fl,. = distance sun to earth 



"K. = 



6.67 X 10 " 17.23 X 10'|- 

6.40 X lO'" 



= 2.7 X 10' X 


( ' 

MOO 


27 X 10* 




16 X 10' 




= 1.69 X lo' 





AH^, = 5.92 X 10 X 
m^, = 7.30 X 10" kg 



5.20 X lo' 



21. (air' = 



47T 



24. 



G / m. 



H = radius of Mai's 

= 3,385 km, or 3,385 X lo" m 

The mass of , Mars is 0.11 of the eailh s mass, 
or 0.11 X 6.0 X 10-' kg = 6.6 X lO"' kg. 

4 X 9.88 (3.385 x 10*')' 

T = TT ^ TT- 

6.67 X 10 6.6 X 10 

= 35.2 X lo'' sec" 

T = 5.9 X 10' sec' 

7' - 1.65 hours of luutb time 



(bl In a cir(;ular oihil, the satellite's \elocit\ 
would be 

Zirfl 



3.385 X 10 m 

= 27T ; = 3,550 m/sec 

5.99 X 10 ' sec 

V = 3.55 km sec 



(c) Mars is not smooth I be slightest giavila- 
tional eflect ofaMothci- sat«'llite, for- e\am|)le, 
would cause the satellite to collide with the 
surface of Mars 



The gravitational force exeiled on the earlh bv 
the sun is about 170 times as great as the foire 
exerted bv the moon. 



k = —, accoixling to Keplei s hantionic law : A; 

= 1, for motion about the sun in vears and .-Xl'. 

Then, fl' = kT^ [ 

fl = f 75^ = x" 5,600 = 17.7 .Al 



The average distance of Halle\ s comet fi-om the 
sun is 17.7 AL'. 




a - major axis 
R^- Average distance 

■froin S».>n to comet 
C - dtstanoe between ^i 
Rp* p«»"tSehL>n distance 
H^= apheliijn di'^tance 



l(*4 



l'\IT 2 , MOTION I.V Till-; IIIvWlvVS 



closest to sun: 
a = 2H. uhei-e a is the major axis of ellipse, so 
a = 35.4 AU 

c 
e = - . where e is eccentricitN' and c is the 

a 

distance between the two foci, so 
c ^ ea ^ 0.967135.4 AUl = 34.2 AU 
a = c + 2R^, \vhere R^ is the perihelion 

distance 



fl., = 



35.4 - 34.2 



1.2 



= 0.60 AU 



The closest approach of Halley's comet is 0.60 
AU from the sun. 

Farthest from the sun: 

Let R^ = the aphelion distance 

fl^ = a - flj, = 35.4 - 0.60 = 34.8 AU 

At the farthest point in its orbit, Halley's comet 

is 34.8 AV from the sun. 

rtt m 

25.laiF^ = m,a^ == G 



R- 

Since m, appears on both sides of the equa- 
tion and R is constant for a gi\en position 
on the earth, nothing changes. Thus, a^ for 
a particular place should be constant. 

Ibi Since the earth is not a perfect sphere, R 
may be different at different places on the 
earth. Then a , uould also be different for 
those places. , 



(ci 



Since F., = \a„m. 



in m, 

G ^— , where la„m,i 

R- " ' 



is the weight of the mass m, then 

Gm, 

R' 



^a m^\ = m, 



Since G, m,, and R are constant at a position 
on the earth's surface, the weight can be 
seen to varv' as the mass. 

(dl The gra\itational force on a bod\' is F^, while 
the force producing centripetal acceleration 
for a circular orbit is F^. These forces must 
be equal. 

v 
F^ — An,a^. — m^ — 



2ttR 4TT-R 

But \' = so F - m, — — 

T ' ' T- 



F. = G 



R' I 



4tt'R m, 

men, since F = F„, — — = G — fand 
« 7- R- 



R\ 



4tt- 
.Gm.,/ 

which has the form T~ 
hamionic law. 



= kR\ of Kepler's 



H' 



— 7 is a constant; m this case k = 

T- Gm, 

(e) At any given time there will be two tides: 
one toward the moon, the other awa\' from 
the moon. Due to the differences in R in the 
gra\'itational formula, the water is pulled 
away from the solid eai1h on the near side; 
the earth is pulled away from the water on 
the far side. ,As the earth rotates under the 
moon an\ gi\en location will experience a 
high tide about e\eiA' 12'/ 2 hours. 

26. Discussion. 

27. Kepler replaced Plato s problem with a new 
way of explaining planetary motions. Plato's 
problem, stated in Chapter 5, was ne\er sohed, 
for no system of uniform circular motions 
could describe the motions of the planets with 
satisfactoiy accuracy. 

28. Today we accept a heliocentric system because 
the obsened motions of planets, comets, etc., 
are accuiateh described b\- the theoi^' of uni- 
\ersal gra\itation applied to a sun-centered sys- 
tem. 

Today's concept of a heliocentric s\stem dif- 
fer considerabK from that of either Coperni- 
cus or Kepler. 

The geocentric system is not dispro\ed, but 
replaced by another system that is simpler, 
more elegant, and more precise. 

29. Newton's basic assumptions and conclusions 
are used every da\' in scientific work. 

30. Discussion. 



SUGGESTED SOLL'TIONS TO STLD\ GUIDE PROBLEMS 



165 




The Triumph of Mechanics 



Organization of Instruction 

THE MUL'n-MEUl/X SCHEDULE 



Day 1 

Lai) stations: C;()iistM\ati()n ol Mass 
Students do onl\' one oi tlic follow ing: 

1. AlchcMiical. L(?ad lilings nrr in-ati'd in an open 
test tuhi'. The s\steni gains \\t!ight. 

2. Boyl(!. Lead lilings arc ht;al(!d in an open tuhc. 
AttcM' h(>ating. the tube is scaled, cooled and 
weighed. 

;{. I.a\()isiiM\ Lead filings ai(; healed in a i loseii 
tube. 

4. Antacid. Weigh a thick 2-L flask, stopper, water, 
and tablets b(;l()re and alter interaction. I'se a 
balance scMisitixe to 0.1 g. 

.■j. I'l-ecipitate. I'lit H) g ol leatl nitrate dissolved in 
water in a 1-1. llask and 11 g ol |)otassinni di- 
chroinate in water in a small test tube placed 
inside the llask in an upiight |)osition Stopper 
and weigh. In\ert to nii\ and reweigh 

Assignment: Prepare to give the class a repoit on 
\()ur expeiiment. 

I)a> 2 

Students reports: ( onser^ation ol Mass 

Alter a lO-min rehear-sai. gri)ups of students ex- 
plain to the class the experiment thi'\ did on the 
|)re\ious da\ Students might ex|)lain wh.it has 
h.ippened as tlie scientists who originallv pci 



formed that t\pe of e\p»'riment might ha\e ex- 
plained it 

IJa\ a 

Teacher' denujnstration: Inelastic- (.'ollisions 'D33> 

I'se lab carts with bricks to show one-dimen- 
sional inelastic collisions. I'se double-sided tape 

on carts 

Day 4 

O.iralitative lab stations: C'onser^ation of Momen- 
tirm 

Stiulents jvxp«Mience one- and two-dimensional 
collisions using balloon pucks Dxlite beads, disc 
magnets, and dxnamics caits In each case. Ihc^' 
ar-e to look for- conser"\ation of momentum. 

I)a\' 5 

I'.'.i-t: Collisions in One Dimension 

.\ (|uanlitati\e measurement is made of momen- 
tum cxch.mge in a collision An air track ov collid- 
ing d\ riamics carts max be used Data aiv ivcoixletl 
with stnilx' and camera. Kach student will need a 
recorxl of a collision for' analx sis. 

Day H 

Student |)r^-sentation of /■,.1-3 or /i3— I.' Collisions in 
1 W(i Dimensions 

Ha\e two belter students do this and an.il\/e 



l(i(> 



iMi A I III: iKii MiMi Ol \ii:(:ii/\M(:s 



consenation ol nioinentiim in two dimensions tor 
the class. 

Day 7 

VX'oik session: Two-Uiiiiensional Cloiiisions 

Fiist discuss E3-3. then sol\e I'ZC). K(|iial Mass 
Two-Diniensiona! Cloilisions, while the students 
uoik on tlie solution ot E\ent 8 in the Handlfook. 
Supplement this aetixity with L22. Iwo-Dimen- 
sional Collisions. II. The best students can proceed 
to another collision e\ent. Gi\e individual help. 

Day 8 

Film: tnerf^ and Work iPSSC #0311 2« min' 

Small group discussions should tollow the lilm. 
Prx)\ide groups with c|uestions to discuss. 

Day 9 

Qualitative lab stations: Kinetic- and I'otential En- 
ergv' 

1. simple pendulum 

2. Galileo s pendulum 

3. ball on inclined plane 

4. energ\ stoi-ed in compressed spring 

5. F/7f7j Loops: L32, L34. or L35 

Students are to look tor changes in PE and K£. 

Day 10 

Discuss results of pitnious dii\ s experiments about 
10 mini. 

Problem soKing: L'se photos taken on Da\ .I to 
check for consenation of kinetics energv' i about 40 
mini. See Handbook, page 160. 

Day 11 

Teacher demonstration: Consenation of Energ\ in 
Inelastic Collisions 

Acti\'ity: Mechanical Ecjuixalent of Heat isee 
Handbook, page 149i, or 

D33: An inelastic collisicjn. Consult the Demon- 
stration \otcs in this Hesourcc Book. This recjuires 
careful preparation. 

Day 12 

E3-10. E3-11. ov E3-12 

Permit students to choose which experiment 
they wish to complete. 

Day 13 

Quiz or other e\ aluation 

Day 14 

Teacher presentation: CarefulK discuss the kinetic 
molecular- theon of gases. Point to the need for 
statistical mathematics and describe the choice 
between the two games in E3-13, Mcjnte Carlo Ex- 
periment on Moleculai- Collisions. 

Day 15 

E3-13: Monte Carlo Experiment on Molecular C^ol- 
lisions 

Note that students complete either Game I or 
Game II. 

Day 16 

Class discussion: Work through the details cjf how 
to estimate the dimensions of a mcjiecule 



Ha\e students discuss the Sliidv Guide cjuestions 
in small groups. Circirlate among grtjups gi\ ing as- 
sistance and making certain that the stirdents arc 
working eflectixcK 

Day 17 

E3-14: Beha\ ior of Gases 

Day 18 

Student acti\it\' day: Students pick acti\'ities from 
Handbook or other sources. Make airangements in 
advance for needed materials. 

Day 19 

I eacher discussion: The Second Law of Ther-mo- 
dynamics (about 35 mini 

Assignment: Encourage students to exercise the 
freedom of reading an\' articles. Ihis takes some 
selling in order to ha\e a successful discussion on 
Da\ 20. 

Daj 20 

Stirdent discussion 

Students should sit in a circle and the teacher 
should be careful to say as little as possible. Ask 
leading questions, howexer, and encourage stu- 
dents to express opinions. 

Days 21 and 22 

Lab stations: VVa\es E3-15, E3-16, or E3-17 

1. pulses on a rope or rxrbber tube 

2. pulses on a slink\ 

3. pulses in a ripple tank 

4. sound wa\es in air- 

5. ultra-sound 

6. microwaxes 

7. continuous wa\es in a ripple tank 

Studerits ar-e asked to look for and make obser- 
vations of \elocit>' of propagation, wavelength, fr-e- 
quency, difftaction, absorption, r-ellection, super- 
position, energy transfer, standing waves, etc. 

Day 23 

Discuss laboratory obser-vations from Da\s 21 and 
22 (15 mini. 

Day 24 

Teacher- presentation: Pr-esent the details of super- 
position and two-source interfer'ence. Note that 
'I'23 through T29 ar-e \erA useful for this purpose. 

Day 25 

E3-l^: Sound and E3-19: L'ltrasound 

A student should do onK one of the various 
parts of this experiment. A quantitative analvsis 
should be completed rather than the tvpe of qiral- 
itative surxey with waves that was done on Davs 21 
and 22. 

Day 26 

Small-group problem solving 

Have each group decide upon an activitv for to- 
moriow 



ORGA\IZr\TIO\ OF l\STRLCT10\ 



167 



Day 27 Days 28-30 

Student activities: Evaluation 

Some of the possible acti\ities are: One method of evaluation is to re\iew, test, and 

, ,, ,. ■ c .• discuss the test. Dex'ote a dav to each actixitv. 

1. h3-5 and hJ-6 Conservation ot hnergy , .u »u i r i \- .u u ■ j- 
„ .. , „ , r „ II . Another method of evaluation is throut^h indi- 

2. t3-7 Measuring' the Speed of a Bullet 11.1..^ c 1 

, ,. ,. ■ ,, lu I ,-0 vidual student-teacher conferences dunne a pe- 

3. Standing' waves on a drum ^Handbook, page lo2i • , r .. 1 i- 1 .- u u 1 
,...,." ,, ,, , ,„,. nod of three davs. Evaluation can be bcised upon 

4. Moire Patterns \}iandho()k, vai'v. loJi 1 u . 1 .1 

„ .. , . , L ,,, JL I laboratorv reports, essavs, poems, equipment de- 

5. Mechanical wave machines tUnnabook, page .' cc-. ^ ^^ j' . 

' ^ sign, sets of Study Guide answers, etc. 

,. '., , „^ , , ^„ Note that two of these three davs of testing could 

6. /•//m Loops 36 through 43 u 1 . .u •.• .u o« j 1 

' " be moved to other positions in the 30-dav plan. 



I6H I'MT A I THK TRIl'MPH OF MECHAMCS 



Unit 3 SAMPLE MULTI-MEDIA SYSTEMS APPROACH 

Note: This is just one path of many that a teacher may take through Unit 3. 



Lab stations: 
Conservation of mass 



Text: Unit 3 
Prologue and 9.1 



Student reports 
from day 1 



Text: 9.2 



Teacher demonstration: 
Inelastic collisions 



Text: 9.3-9.4 



Lab stations: 

Conservation of 

momentum 



Handbook: 
E3-1 and 
E3-2 



Labs E3-1 and E3-2: 

Collisions in one 

dimension 



6 

Student 

demonstrations: 

E3-3 or E3-4 



Work session: 
Two-dimensional collisions 



P.S.S.C. Film: 

Energy and Work 

Small-group 

discussion 



Write up E3-1 or E3-2 



Text: 9.5-9.7 



Handbook: Finish 
analysis of events 8 & 9 



Text: 
10.1-10.4 



Lab stations: 

Kinetic and Potential 

Energy 



10 

Discuss day 9 lab 

Problem solving: 

Energy conservation 



11 

Teacher demonstration: 

Conservation of energy 

in inelastic collisions 



12 

Lab E3-10, E3-11, or 

E3-12 



Selected Study Guide 
questions 



Text: 10.5-10.8 



Handbook: E3-10, E3-11 
or E3-12 



Write up Lab 

Text: 

10.9-10.11 



13 

Quiz or other 

evaluation 



14 

Teacher presentation: 

Gas models 

(D35 and D36) 



15 

Lab E3-13: 

Monte Carlo experiment 

on molecular collision 



16 

Class discussion of E3-13 

Small-group 

problem solving 



Text: 11.1-11.4 



Handbook: E3-13 



Selected Study Guide 
questions 



Handbook: 
E3-14 



17 

Lab E3-14: 

Behavior of gases 



18 

Student activity 

day 



19 

Teacher discussion: 

Second law of 

thermodynamics 



Handbook: Survey Ch. 1 
for activities 



Text: 11.5-11.8 



20 

Student-centered 

discussion 



Text: 
12.1-12.4 



21 
Lab station: 



22 



Waves and wave behavior 



Handbook: Survey Ch. 12 



23 



Write up observations 



Text: 12.5-12.8 



24 

Teacher presentation: 

Interference 



Text: 12.9- 
12.11 



25 
Lab E3-18: Sound 

or 
E3-19: Ultrasound 



26 

Small-group 

problem solving 



Selected Study Guide 
questions 



Handbook: Survey 
and select activity 



27 

Small-group 

activities 



28 

Review 

or 

individual evaluation 



Write up activity 



Review 



29 

Test 

or 

Individual evaluation 



30 

Discuss test 

or 

Individual evaluation 



ORGA.MZATIO.N OF INSTRUCTION 



169 



Unit 3 SUGGESTED SCHEDULE BLOCKS AND TIMETABLE 

The sample schedule outlined on this page is one idea of how to present the chapter. Each block represents one class 
session (approximately 50 min) and the spaces between blocks are used to indicate homework. More specific suggestions 
can be found under Multi-media. 



CHAPTER 9 CONSERVATION OF MASS AND MOMENTUM 
Text; Prologue to 9.1 Text: 9.2 Text 9 3-9 4 



Lab stations: 

Conservation of 

mass 

(See day 1) 



Student reports 



Teacher 

demonstration: 

Inelastic 

collisions 



Lab stations: 


Conservation of 


momentum 


(see day 4.) 



Handbook: 
E3-1 & E3-2 



Labs E3-1 and E3-2: 

Collisions in one 

dimension 



Write up E3-1 & E3-2 



Student 
demonstrations 



CHAPTER 10 ENERGY 



Text: 9.5-9.7 



Work session: 

Collision events 

8 and 9 



Film: 

Energy and Work 

Discussion 



Text: 10.1-10.4 



Lab stations: 

Kinetic and 

potential energy 

(See day 9.) 



Selected Study 
Guide questions 



Problem-solving 
session 



Text: 10.5-10.8 



Teacher 

demonstration: 

Energy 

conservation 



CHAPTER 11 THE KINETIC THEORY OF GASES 



Handbook E3-10, 
E3-11, or E3-12 



LabE3-10, E3-11, 

or E3-12: 

Temperature, 

calorimetry or 

ice calorimetry 



Write up E3-10, 
E3-11, or E3-12 
Text: 10.9-10.11 



Quiz 



Text: 11.1-11.4 



Teacher 
presentation: 
Gas models 



Handbook E3-13 



Lab E3-13: 

Monte Carlo 

experiment on 

molecular collisions 



Selected Study 
Guide questions 



Discussion 

of E3-13 

Study Guide 

questions 



Handbook: E3-14 



Lab E3-14: 

Behavior of 

gases 



Handbook 
survey Ch. 11 



Student activity 
day 



Text: 11.5-11.8 



Discussion: 

2nd law of 

thermodynamics 



Student-centered 
discussion 



Text: 12.1-12.4 



Lab stations: 
Wave properties 

Waves in a 

ripple tank 

(see days 21-22.) 



Handbook: 
Survey Ch. 12 



Lab stations: 
Measuring 
wavelength 



CHAPTER 12 WAVES 

Write up 
observations 



Programmed 
instruction: 
Waves I, II 



Text 125-128 



Teacher 
presentation: 
Interference 



Text: 12.9-12 11 



Lab E3-18: 

Sound 
Lab E3-19: 
Ultrasound 



Selected Study 
Guide questions 



Small-group 
problem solving 



Handbook: Survey 
and select activity 



Small-group 
activities 



Review 



Review or 
Individual 
evaluation 



Study for test 



Test or 
Individual evaluation 



Discuss test or 
Individual 
evaluation 



170 



I'MIT 3 THI-: TKIl'MIMI Ol MDCHWICS 



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l'\l I A I I Hli TKII'MPH OF Mi;CHAM(:S 



Background and Development 



OXTHX lEW OF UNIT 3 



In I'nits 1 and 2 we ha\e de\ eloped the basic prin- 
ciples of .\e\\1onian or classical' plnsics and 
shown their successful application to the astron- 
om\ of the solar s\ stem. HistoricalK this success 
led phxsicists in the eighteenth and nineteenth 
centuries to use these same principles in explain- 
ing mam other natural phenomena. 

In Unit 3 we will focus on the generalization of 
Xewlonian mechanics b\ means of consenation 
laws for mass, momentum, and energ\'; and the 
application of Newtonian mechanics to collisions 
of objects, heat, gas theoiA and wa\es. 

In presenting the consenation laws, we stress 
the metaplnsical and semitheological oiigin of 
these law s in the se\ enteenth centun : the idea that 
the world is like a machine which God has created 
with a fixed amount of matter and motion. In the 
discussion of the generalized law of consenation 
of energ\ , we point out connections with steam- 
engine technolog\ and other historical and philo- 
sophical factors that la\ in the background of the 
simultaneous discoxeiA of this law b\ several sci- 
entists in the middle of the nineteenth centuiA 



Our purpose here is to make students awai-e of the 
tvlationships between physics and othei- human 
activities. 

In selecting cei'tain applications for detailed 
presentation, while ignoring many others that are 
paris of the traditional phvsics course, we have 
been guided b\ two main criteria: ifi to give the 
bare minimum of technical detail that is needed to 
illustrate the main principles and i2i to develop 
material that will be needed in later units. The cli- 
max of the unit is r-eallv in Chapter 11, where the 
concepts of momentum, ener-gv , and heat ar-e com- 
bined in the kinetic theotA of gases to vield the first 
definite infoiniation about molecular speeds and 
sizes las well as an explanation of the macroscopic 
prT)perties of gases'. This is about as far as New- 
tonian mechanics can take us into the atomic 
world; we will have to wait for the quantum con- 
cepts intrf)duced in Unit 5 before we can go fur- 
ther. Chapter 12 begins the studv of wave phenom- 
ena, still in a mechanical framework, but breaking 
the ground for the studv of light and electix^mag- 
netic waves in L nit 4. 



CHAPTER 9 CO\SER\ ATION OF MASS AND MOMEiXTUM 



Sl^OIARY OF CHAPTER 9 

The law of conservation of mass expresses accu- 
rateU the ancient conviction that the amount of 
matter in the univer-se does not change. A sound 
experimental basis for this belief was prxavided in 
the eighteenth century by the experiments of La- 
voisier. It has since been demonstrated with gi"eat 
precision that the mass of am closed svstem al- 
wa\s r-emains the same. 

Seventeenth-centurA philosopher-s also believed 
that the quantitv of motion in the universe al- 
ways remained the same: the piT)blem was to for- 
mulate a definition of quantitx of motion such 
that it would be conser-\ ed. Descartes prxjposed the 
pi-oduct of mass and speed as the correct measure 
of quantitv of motion. Bv considering collisions be- 
tween two carls, however, it is seen that Descartes 
was wTong. It is the pr-oduct of mass and velocity 
that is conserved. Ihis vector quantitv is called 
"momentum. 

The law of conservation of momentum is per- 
fectly general. The resultant momentum of am 
number of bodies exerting am kind of forces on 
one another is conserAed, so long as the net for-ce 
on the entire svstem of bodies is zero. 

The law of conservation of momentum is a log- 
ical consequence of Newlon s second and third 
laws of motion. \e\ it allows us to solve problems 



that could not be solved bv dii-ect application of 
Newlon s laws. In pariicular, use of the law of con- 
servation of momentum enables one to find out 
about the motion of interacting objects ev en when 
the details of the forces of interaction are un- 
known. 

A sx'stem on which the net force is zero land the 
momentum of which therefore is conserved' is 
called an isolated sv stem. It is neailv impossible to 
arrange a svstem that is truly isolated. .\Ianv svs- 
tems can be r-egarded as ver\ nearlv isolated how- 
ever, and the law of conservation of momentum 
can be applied to them with little error. 

A description of a demonstration witnessed bv 
the Roval Society in 1B66 in which two hard pen- 
dulum bobs repeatedlv collided with one another 
leads to the intr-oduction of another quantitv that 
Huygens showed to be conserved in collisions be- 
tween verv' hard objects iperfecth elastic colli- 
sions i. It was called vis viva. The German philoso- 
pher Leibniz was convinced that vis viva was 
always conserved, even in collisions between soft 
objects linelastic collisionsi. 

Just as Descartes quantitv' of motion" imvi de- 
veloped into the modem concept of momentum 
imvi, so vis viva is closelv related to the modern 
concept of kinetic energv 777\" 



BACKGROl \» AM) DEMILOPMLN t 



173 



9.1 I COXSERV^Al ION OF \L\SS 

J lio main point of this section is the faith that nat- 
ural philosophers had in the conservation of mass, 
even in the ahsence of good expeiimental \erifi- 
cation. The work of Lavoisier, which finally gave 
accejjtahle experimental suppoil to the conserva- 
tion of mass, was not a source of surprise or even 
of much information; it was rather somewhat of a 
i-elief. 

There are any number of visually appealing dem- 
onstrations that could be done in a closed flask to 
show conservation of mass. The one using antacid 
tablets is described in some detail in th(! Handbook 
and can be used to make the more subtle |Mjint of 
accuracy limitations to conser\ation laws. 

The discussion on lest page; 250 may seem to 
be inconsistent with the caption under- the figur-e 
illustrating changes in mass during burning. Burn- 
ing leads to an incr-ease in mass if all products are 
included. However-, if burning takes place in an 
open pan, as shown in the photograph, initial heat- 
ing can drive off gases, and smoke can escape, 
which leads to a net loss in mass. If time permits, 
this would be a good time to discuss some chem- 
ical r-eactions that seem to increase mass by ad- 
dition of reactants and some that seem to deer-ease 
mass by subtraction of products when an open 
container is used. 

Although substances seem to disappear when 
dissolved, this does not constitute a violation of the 
law of conser^'ation of matter-. Solution may occur- 
through dispcMsion of molecules of the dissoKed 
substance in the solvent or- thr-ough a chemical r-e- 
action. 

During World V\'ar- II, Niels Bohr-, the physicist 
who developed the atomic theory discussed in 
Chapter 19, had to flee his labor-atory in Denmark. 
Two of his colleagues gave him their gold Xobel 
Prize medals, which he dissolved in a jar of acid 
and left in his office. Because of his experience uith 
the conservation of matter, he was cerlain that the 
reaction between the a(id and the gold would con- 
ver1 tin; gold into a soluble chemical compound. 
He felt that the (ier-man invaders would ne\er sus- 
pect the jar of acid to be valuable. When he re- 
tirrntid after- the war- he irsed chemicuil methods to 
rt!cover tht; gold from the; aiid and had \hv medals 
recast. 

The distinction l»!tw(H;n mass and weight was 
made in I 'nit 1, l)ut it might be well to trt'at it briefh' 
again. Changcis in weight with changes in distance 
from the ear1h werx; tr-eated extensively in I'nit 2. 
Because mass conservation was, until recentlx. 
studied only with analytical balances, the opera- 
tion of balances and what tlicn' measure migiit be 
discirssed. 

rhe ratio of the weights of two bodies is eqiral 
to the ratio of their masses if they aiv weiglu'd at 
the same place? or- at places wberi' n^ has the same 
value. iBeiall (;alil(M)s work, discussed pii'v iousU.i 
rhe prt)porti()rialit\ has been expiMimtuitalK r'stab- 
lisbed to bellei than 1 part in lo'". In general \vl- 



ativity, the prxjportionalitv' is assumed on the basis 
that acceleration and gravitational fields are math- 
ematically indistinguishable. 

After suggestions that ver\' small changes in 
mass might accompany radiation, pr-ecision chem- 
ical experiments, especially those by Hans Landolt, 
were made about the beginning of the twentieth 
century. These experiments established conserva- 
tion of mass to 1 pai1 in 10*'. 

In some cases, 10 ~ ** g can be quite a potent quan- 
tity. iFor example, that amount of LSD has ter-ri^^'- 
ing effects on humans. i A cube with an edge 0.10 
mm long has a volume of 10 *" cm^. With a densitv- 
near- that of water-, such a cube has a mass of about 
10"'' g. On any desk top, one can pr-obably see se\- 
er-al specks of dust v\ith v ery similar masses. 

It has now been obserAed that in some nuclear 
r-eactions the r^st mass of all the panicles after the 
reaction is detectablv different from what it was 
befor-e the reaction. However, if the mass equivalent 
of the ener-gv' involved in the reaction is included 
as calculated by m = E/c~ , the total mass is still 
conserved. 

If 18 g of methane iCH^i are burned v\ith B4 g of 
oxygen lone gram molecular weight of each i, 211 
kilocalories of heat are released. This amounts to 
a change in r-est mass of about 10 " g or about one 
pari in lO'", as the following equation shows. 

^E 

Am — — — 
C 

_ (211 kcali 14.19 X 10'°eiigs/kcali 

3.00 X lo'" cnvsec)- 

= 9.8 X 10 ' g. 

This is a relatively energetic reaction, so that this 
fractional mass change is ver^' much lai^er than for 
the r-eactions Landholt studied. 

Trv' to avoid anv extended discussion of mass- 
ener-gv relationships. The imporlant point to em- 
phasize is that, with incr-easing knowledge, con- 
ser-vation laws may need some adjustment in terms 
of jirst what it is that is being conserved. In Chapter 
17, return to a discussion of £ = rnc' . 

If questions ar^ r-aised about reactions involving 
the annihilation of electrons and positrons, where 
the r-est mass of the par-tides disappears com- 
pletelv and is turned into the ener-gv of two gamma 
rays, it is mor-p accirrate to sa\ that mass disap- 
pear-s" onl\ in the sense that it cannot be measured 
on a balance. What "disappears, of cour-se. is the 
ele(-tr-on and the positron as identifiablp particles 
having rvst mass. 

Sugj^estions for Quiz or Class Discussion 

1. A conser-vation law can be contir-med experi- 
mentallv oriK to within a cerlain mar-gin of er-ri)r 
^'et conser\ation of ener-gv and momentum are 
believed to be e.xact laws, ami werv believed to 
be e.xact even beforv extrvmelv pr-P(-ise e\}>eri- 
ments vver-e ilone Iherv ar-e marn (|uantities 
that ar-i> iumiIv i-onstant isuch as the nirmber- of 



I7« 



I'M I a / THK TKIHIVIPH OF IVllJCH/WICS 



pianos in Boston during a one-week period i. yet 
we do not belie\e that there are exact conser- 
vation laws for these quantities. Can \ou think 
of anything that influences our lielief in conser- 
vation laws besides quantitative experimental 
evidence? 

2. Give a brief explanation of why physicists are 
interested in consenation laws. 

3. A person obsen es a burning tree and makes the 
following two statements: 



(el 



The wood disappears, heat and light are 
gi\'en off, and ashes remain. 
Molecules of wood combine with molecules 
of oxN'gen in the air to fomi molecules of the 
gas cait)on dioxide; the minerals in the wood 
are left in the fomi of solid oxides, and heat 
and light are gi\en oft". 

Which of the statements is the description of 
what happens? 

VX'hich statement depends upon a theor\'? 
Which, if either, would make sense to a per- 
son who did not belie\ e in atoms? 
In the light of further exidence, which state- 
ment is more likeh' to remain unchanged? 
Will both remain unchanged? 
VXliich statement is tnje"? 



9^ COLLISIONS 

This entire section is designed to show how people 
were using incomplete and xagvie concepts in their 
search for order and fimdamental principles. It 
may well represent the sort of thing that always 
goes on when one is tning to find order amid the 
seemingh' chaotic; a sort of private, prephxsics 
stage in which guess work, hunch, and intuition 
are stirred in about equal parts with experimental 
observation, good existing theor\ . and hope. Look- 
ing back from our present \antage point, it seems 
extraordinary' that am-thing came from it all. One 
hesitates to draw current parallels in other ai^eas 
of study for fear of being misunderstood, but to 
some degree the same sort of searching goes on 
today in physics as we Xr\' to understand the role 
of fundamental particles. Of course, now there is 
a good bit less reference to what is on God s mind 
than there was in Descartes time. 

Note that while the text treats inelastic collisions, 
the experiments and demonstrations deal primar- 
ily with elastic collisions. This should present no 
pix)blem if no distinction is made at this time. Sec- 
tion 9.6 deals with this matter. 

Beginning with an equal-mass elastic collision 
(one body initially at resti, students can be led to 
suggest that speed is conserved. Then, to establish 
the generalitx' of a "consenation of speed ' princi- 
ple, the principle must be confiimed in other col- 
lisions. An unequal-mass elastic collision will show- 
that "conservation of speed' does not hold. But, if 
the masses are in simple ratio, like 2:1, for instance, 
it will be fairly obvious that the product of mass 
and speed is conserved. Again, to claim the gen- 



erality of the "conservation of mass x speed" prin- 
ciple, other tvpes of collisions must be tried. Note 
that the product of mass and speed squai-ed is also 
conserved, but not m'v, mv*, m v. m\ v, etc. In the 
two txpes of collisions considered thus far, niv and 
mv^ have both been conseiAed. Students can be 
advised that the simpler relation will be taken up 
first, and that m\'^ will be taken up later. The linear 
"explosion" oi- head-on inelastic collision will show 
that even mass x speed is not conserved. Then 
the conservation principle can be saved onlv if the 
direction of the speed is considered. .A conserva- 
tion of mass X velocity" principle holds in these 
two cases as well as in the othere. The use of the 
vector velocitv' should suggest that nonlinear col- 
lisions might be tried. The collisions of pucks on 
plastic beads or of air-pucks can be photographed 
stroboscopically with very' satisfacloiy results. The 
film loops on two-dimensional collisions give ex- 
cellent results. An inexpensive air-table pix)vides 
excellent two-dimensional analyses. 

9.3 I CONSERVATION OF MOMENTL^M 

A wealth of material, including experiments and 
demonstrations, film loops, stroboscopic photo- 
graphs, and overhead transparencies, is available 
for teaching conservation of momentum. The topic 
is, however, not so important to warrant the use of 
all parts of all these media. What is important is 
that students get some intuitive feel for the kind of 
outcome to be expected in a collision and that they 
get this feel before the corresponding bald claim 
about consenation is made in the te.xt. The collec- 
tion, or examination, of data in the laboratoiy 
could be coordinated to allow the progressive "dis- 
covery of conservation of momentum independ- 
ent of (and preferably before) the reading of the 
text treatment. 

This is an appropriate time to introduce or re- 
view the summation sv mbol S if vou hav e not done 
so. Here 2 means vector addition. The use of dif- 
ferent objects and subscript numbers 1 and 2 to 
mean befoi^e and after collision is not very gener- 
alizable. If we let subscript numbei-s identifv the 
different objects and let unprimed and primed \'s 
mean before and after, respectivelv , then we can 
write in general 

Sm^Vj = Sm^v'j 
or ' 

ASm-v- = 

Notice that the summation is vectorial. Actually, it 
might be more satisfactory' to write algebraic sum- 
mation for each of the three components of mo- 
mentum: 

A2m,v, = 
A2mjV^_ = 
ASm.v- = 



BACKGROUXD A\D DEMJLOPMENT 



177 



These principles are true generally, both intlt;- 
pendently and all together. 

9 A I MOMExVrUM AND XEWTOiVS 

LAWS OF MOTIOiX 

Attention shoulii ho gi\(;n to the question: How 
does momentum entei- or leave a system?" The 
only way we can change the linear momentum of 
a system is to exert a for*ce on part or on all of the 
system from outside. The force that one pari of a 
system exeils on another pai1 cannot change the 
system's total momentum. Such a force might re- 
distribute the momentum among the parts, but it 
cannot change the total. 

The operation of a rocket engine may be dis- 
cussed in terms of conservation of linear momen- 
tum. The analogy of the recoil of a gun suggests 
that the forward momentum of the bullet as it 
leaves the muzzle of the gun is equal to the recoil 
momentum of the gun in the opposite dii-ection. 
If we then picture the rocket as a continuously 
firing gun, we can derive the "ixjcket equation. " 

9.5 I ISOLATED SYSTEMS 

An isolated system need not be completely iso- 
lated, only isolated with respect to the (juantities 
we are interested in. "Entering " or "lea\ing a sys- 
tem implies passing through a spatial boundary 
surrounding the system. It should be made clear 
that something created or destroyed within the 
boundary of a closed system is not entering or leav- 
ing the system. 

A conservation law must hold true for any closed 
system. We do not speak of "the conserAation-of- 
energy-in-an-insulated-calorimeter law." It is not 
enough to conserve something in a particular iso- 
lated system; that something must be conserved in 
every isolated system; othei-wise there is no law. 

It is very difficult indeed to think of good con- 
ser\'ation laws dealing with concepts outside of 
physics. That is one reason it is so remaikahle that 
they occur in physics at all. Since all aixnind us we 
see change, it is surprising and wonderful to find 
that amidst all the turmoil some things sometimes 
do not change. 

Physicists cling to the conservation laws, some- 
times in the face of seemingly contrarv exidence. 
At fir-st the conservation laws of linear momentum 
and energ\' seemed to be violated in the p deca\ 



of nuclei. Physicists then imented the neutrino in 
order to preserve the conservation laws. 

The students have already studied a ver> good 
conservation law: Kepler s second law lUnit 2, Sec. 
7.2). The Tepct states: "The line from the sun to the 
moving planet sweeps over areas that are propor- 
tional to the time intervals." Of course, for this to 
be tr-ue, it must be that the line sweeps over an 
area at the same rate no matter where the planet 
is in its motion. That is, the rate at which the line 
sweeps out an ar-ea is conserved throughout all the 
changes in the planet's velocitv' and distance from 
the sun. 

Since we do not discuss angular momentum in 
the Te;ict, the student will not recognize Keplers 
second law as just a special case of the conserva- 
tion of angular momentum. It does have the ad- 
vantage of being a phvsical example in which the 
quantity being conserved is not a substance. .-Vlso, 
in this illustration the quantity' that is conserved 
has different v^llues for different sun-planet sys- 
tems. 

9.6 I ELASTIC COLLISIONS 

The main point of this section is the necessity' of 
introducing another conservation law in order to 
successfully account for the motion of the bobs. 
Since we have an interaction involving two un- 
known velocities, we must have two equations to 
arrive at their final values. 

9.7 ! LEIBXIZ AXD THE 

COXSER\^ATION LAW 

This section illustrates the evolution of science 
through conti-oversy and the transition from mv^ 
to heat, the subject of the next chapter. 

Here students could be encouraged to ai^e out 
the Leibniz-Descartes controversv regarding what 
it is about motion that is conserved. This is worth- 
while even though the two men were not contem- 
poraries. Fur"thennore, the desire of Leibniz to as- 
sur-e the conservation of rn\'^ is an e.xample of great 
intuition and faith in an idea. Such faith and in- 
tuition is a part of science. 

Demonstration D33 is appropriate for shovving 
the students an example of the transfer of me- 
chanical ener'gv' to heat or internal energv'. This 
activitv' is timely between Chapters 9 and 10 or be- 
tween Sees. 10.4 and 10.5. Refer to the Demonstra- 
tion Xotcs. 



CHAFl ER 10 / ENERGY 



SinVIMARY OF C:iIAnT:K 10 

The concept of work, delincd as the product of the 
force on an object and the distance the object 
moves while the iovcv is excMted on it, is intei- 
pitMc'd as a measurr of energv changed fix)m one 
form to another With this inteipivtation, exprx's- 
sions can be derived foi- the kinetic energv ' Jin'^ 
of an object and ioi- the change in gravitational po- 



tential energv' F^^^d of an object of weight F^^^ that 
moves thixjugh a verlii'al distance d. Other forms 
of |)otential enei-gv aii- mentioned M friction is neg- 
ligible, the sum of the kinetic energV' and the po- 
tential enei-gv does not ihange: this is the law of 
conservation of mechanical ener-gv . 

Woi-k is more accuratelv defined as the prtiduct 
of the i(^n}fu)n('nt of the tone on the ohj<Mt in ihr 



I7H 



UNIT 3 / THE TRIIIMPH OF MECHANICS 



direction of motion of the object and the distance 
the object mo\'es while the force is being applied. 
Thus, if an object moves in a direction perpendic- 
ular to the direction of the force on it, the force 
does no work. 

The present-day \iew of heat as a fomi of energ\ 
was established in the nineteenth centuiy, parth 
because of knowledge of heat and work gained in 
the de\'elopment of the steam engine. The first 
practical steam engine was in\ented b\' Saxerv to 
pump \%ater from flooded mines. A considerab)l\ 
better engine was that of Xewcomen, which was 
widelv used in Britain and other European coun- 
tries in the eighteenth century. 

The invention of the separate condenser by Watt 
in 1765 resulted in a \'astl\' superior steam engine, 
one that could do moi"e work than the Xewcomen 
engine v\ith the same amount of fuel. Watt charged 
a fee for use of his engines that depended on the 
rate at which they could do work: that is, on their 
power. The Watt engine was quickJ\' adapted to a 
varietN' of tasks and was instrumental in pushing 
forward the Industrial Re\olution. 

One of the scientists who helped establish the 
idea that heat is a form of energ\' was Joule, who 
performed a \ariet\' of experiments to show that a 
gi\en amount of mechanical energ\' (measured, for 
example, in joules i is alwaxs transfonned into an 
equi\alent amount of heat (measured, for example, 
in kilocaloriesi. 

Li\ing systems require a supply of energ\' to 
maintain their xital processes and to do work on 
external objects. Plants obtain energ\' from sunlight 
and, by the process of photos\ nthesis, con\ert it 
into chemical energ\' stored in the molecules of 
the plant. .An animal that eats plants, or that eats 
other animals that ha\e eaten plants, releases the 
stored chemical energv' in the process of oxidation, 
and uses the energs' to run the "machiner\ ' of its 
bod\' and to do work on its surroundings. Different 
activities use food energv' at different rates. A 
healthy college-age person needs at least 1,700 
kilocalories of food energ\' a da\' merely to keep 
the body functioning, without doing any work on 
that person s surroundings. Vet there are countries 
where the a\erage indi\idual intake of food amounts 
to less than 1,700 kilocalories a day. 

In the early nineteenth centurv', developments 
in science, engineering, and philosophy contrib- 
uted to the growing conviction that all forms of 
energ\' (including heati could be transformed into 
one another and that the total amount of energ\ 
in the uni\erse was conserved. 

The newh' de\eloping science of electricitv and 
magnetism re\ealed man\' relations between me- 
chanical, chemical, electrical, magnetic, and heat 
phenomena, suggesting that the basic "forces of 
nature were related. 

Since steam engines were compared on the basis 
of how much work they could do for a gi\en supply 
of fuel, the concept of work began to assume con- 
siderable importance. It began to be used in gen- 



eral as a measure of the amount of energ\' trans- 
formed from one forni to another and thus made 
possible quantitative statements about energ\' 
transformations. 

The Geiman ' natui-e philosophers,' concerned 
with disco\'ering through intuition the inner mean- 
ing of nature, stimulated the belief that the \arious 
phenomena of nature were different manifesta- 
tions of one basic entity that came to be called 
energ\'. 

Of the large number of scientists and philoso- 
phers who proposed a law of conservation of en- 
eigv' in some fonn, it was \on Helmholtz who most 
clearly asserted that any machine or engine that 
does work cannot pro\ide more energ\' than it ob- 
tains from some energv' source. If the energ\' input 
to a system (in the fonn of work or heati is different 
from the energ\' output, the difference is ac- 
counted for by a change in the internal energy of 
the system. 

The law of conser\'ation of energy lor the first 
law of thermodynamics! has become one of the 
\'ery cornerstones of physics. It is practically a cer- 
tainty that no exception to the law will ever be 
discovered. 

10.1 I WORK AND KEVETIC ENERGY 

It may seem strange that we ha\ e brought our dis- 
cussion of mechanics so far with hardl\' a mention 
of energx', which now holds such an important 
place among the concepts of physics. We might 
mention that the logical development of mechan- 
ics is quite possible without it. .All the problems of 
classical mechanics can be sohed without refer- 
ence to energy. The "idea ' of energ\' is historically 
much older than the name. It goes back to Galileo's 
work with machines, in which the concept of 
"work " was involved. 

V\'hile there was confusion between the words 
force and work, due to Leibniz's interpretation, 
Leonardo da Vinci had remarked 200 yeai-s before 
that work implied that an object mo\ed in the di- 
rection of the force. In the literature of physics to- 
day, work means just this: force multiplied b\' the 
distance tra\ ersed in the direction of the force. The 
approach taken in the lest is to present the con- 
cept of work in an easil\' digestible form firet, and 
to qualifv it later as necessary. E\en using the Test 
approach, \ou might want to hedge the initial def- 
inition " by pointing out to students that both force 
and displacement are vectoi-s and that it isn t im- 
mediately ob\ious what to do when the force and 
displacement aren't along the same line. As long 
as they are in line, it is correct to take work as Fd. 

Many physicists use the concept of negative 
work because it allows one to treat these problems 
from a more general point of \iew with one equa- 
tion instead of two. We haxe a\oided doing this in 
the Text to avoid excessi\'e abstraction and stay 
closer to familiar concepts. Thus, we say that B 
does work on A, rather than that A does negative 
work on B 



BACKGROL.VD A\D DEVELOPMENT 



179 



As we saw in Sec. 9.6, Huygens made prxjminent 
the concept of vis viva ur living force, a quantity 
vaiying as the mass multiplied by the square of the 
velocity. The teim energy was not attributed to the 
vis viva concept until the nineteenth centuiy. The 
product Fd is called the "work" done by the force 
during the displacement, while the quantity '/2/nv^ 
is one-half that which used to be known as vis viva. 
We now call Vzmv^ the kinetic energy. 

If a body moves a distance d directly against a 
resisting net force F, it loses kinetic energy. The 
amount of kinetic energy lost can be shown as fol- 
lows, stalling from F — ma: 
F = ma 
Fd = ma ■ d 

(Vf - V,) Iv, + V,l 

= m • / 

t 2 

= Vzmiv,- - Vj^) 
= Vzmv,'^ — V^mv^ 

10:2 I POTENTIAL ENERGY 

Expeiiment E3-5 should be done before (or closely 
in conjunction with) the beginning of this section. 
The "dip" in the kinetic energ\' graph for the slow 
collision is a nice way to introduce 1 "discover"! 
potential energy. Some of the kinetic energ\' dis- 
appeais and then reappears. Where was it? The 
"explosion" from the tied steel loop makes a good 
follow-up demonstration; there is no kinetic energ\' 
in the system before and much afteiAvards, but 
there was something related to energy in the tied 
spring. If £3-5 can't be done until after the gravi- 
tational force of PE has been considei-ed, the dip 
in KE can be predicted instead of disco\'ei-ed. 

If we accept the idea that a falling body is con- 
tinually gaining kinetic energ\' due to the previous 
work we did in raising it, we must then accept the 
idea that the "raised" body, before it stalled to fall, 
possessed the energy that is appearing in kinetic 
foim. Energy due to motion is fairly obvious, which 
is why kinetic energy lv/.s v/vai came upon the 
scene relatively early. Energy due to position is less 
obvious. In fact, it eluded both Leibniz and Ues- 
carles. The first mention of "position energv was 
in a book by Camot in 1803 in which he stated: 

Vis viva can figui-e as the pix)duct of a mass and 
the sc|uare of its velocity, or as the product of a 
moving [)ower and a length or a height. In the fii-st 
case it is a vis viva properly called; in the second 
it is a latent vis viva. 

It was not until 1853 that the name potential en- 
ergy was first used by Rankine, and it has been 
used ever since*. 

See the ailicle "Energv Reference Levels in the 
section of this Hesource Hook entitled Additional 
Background AHicles. Note that its content is not 
used in the Te.Kt development. 

10.3 I CONSERVATION OF 

MEc:iiANicLriL i::ni:rc;y 

Ihe nalui-e of energv leads to Ivxo lundamcnt.ii 
conicipts; kinetic and potential energv Kinetic en- 



ergy is possessed by reason of motion while po- 
tential energv' is possessed bv reason of position or 
condition, as in a raised weight or stretched spring. 
It should be clear that either kind of energy mav 
come into being in consequence of the perform- 
ance of work. In the first case, the work is devoted 
to producing speed; in the second case, woi-k in- 
volves some revei-sible process that subsequentlv 
can be made to yield the energv thus stored. 

In using the momentum and enei^' conserva- 
tion principles for collisions it should be noted that 
pi-ediction is possible onlv for rectilinear collisions 
of two bcdls. As manv students may remember, a 
set of equations, all of which are true, can be solved 
for as many unknowns as there are independent 
equations. Energy and momentum conservation 
principles constitute a pair of such independent 
simultaneous equations and so can be solved for 
only two unknowns: the two velocities after colli- 
sion of two balls. If the collision is in two dimen- 
sions, there are two unknov\Tis for each ball after 
collision i,v and y velocitv components or speed 
and anglei giving four unknowns in all. Thus, two 
other values would be needed in order to be able 
to solve for the remaining two velocities. The sit- 
uation of a single ball striking a row of balls in 
contact has, after the collision, as manv parameters 
as there are balls: one velocitv value for each ball. 
Energy and momentum conservation are not suf- 
ficient to find a unique set of after values, and, 
as a matter of fact, all the balls will have finite ve- 
locities after the collision, even if the balls are per- 
fectly elastic. If the balls are not in contact, but 
separated enough so that the collision of the first 
and the second is over bv the time the second 
touches the thiixl, then the traditional all-zero-ex- 
cept-the-last-ball result is predictable, but the 
event is no more than a series of two-ball collisions. 
Historically, the energy conservation generaliza- 
tion was employed bv Leonaixio da \ inci near the 
end of the fifteenth centuiy : Simon Stevin used it 
in 1605 as the basis for his development of the laws 
of statics, while Galileo emploved a similar ail- 
ment in his analvsis of a frictionless inclined plane. 

10.4 I FORCES THAT DO NO IVORK 

The idea of zeix) work being done v\ hen heavA bod- 
ies are held or transpoHecl horizontallv is tn)uble- 
some to students and should pmbablv be cjualified 
as follows. When someone stands and holds a 
heavA' body, no work is being done on the body . 
but this is not to sav that there is not a great deal 
of chemical transfer of ener^- iwork. in the tech- 
nical sense, on the cellular or molecular scalei re- 
(juired to maintain the muscle contractions that 
ivsult in the bodv being held motionless. Woii, can 
be going on inside vou transfening chemical en- 
eigv ultimatelv into heat without any woii, being 
done on anything outsitle the closed svslem you. 
In this scMise. it is (|uite ix»asonable to say that you 
can get tii-ed and hcit without doing any outside 
work 



I HO 



IIIVIT 3 / THK TKiriVII'H Or .MLCH/XMCS 



It does take a force to move something horizon- 
tally, e\'en without ftiction. If it is an "immeasura- 
bly small force, " then it will take an imnieasurabl\ 
long time for- the body to make the change in po- 
sition. A more reasonable treatment would be to 
admit that a finite foix:e is required, and that there- 
fore VNork must be done on the bod\' to get it mov- 
ing. Point out that when the bod\' reaches the new 
position it v\111 still be moving. The work that went 
into getting it moving went into kinetic energv', 
which it still has after reaching the new position. 
No energv' was "used up" in changing position, ilf 
a small spring gun were used to start the body, the 
spring gun could be moved around and used to 
stop the bodv as it arrived at the new position, and 
the gun would be cocked in the process. Everv'- 
thing then would be the same as before except that 
the body had changed position.) 

"No-work" foixjes, then, are those that are per- 
pendicular to the direction of a body's motion and 
so do not contribute to kinetic energv' changes. A 
motion that is restricted to a prescribed path, like 
a track or wire, or to a prescribed plane, is called 
a constrained motion. In such motion, no-work 
forces are present but knowledge of them is not 
necessarv' for the calculation of changes in kinetic 
energv'. 

10.5 I HEAT AS ENERGY 

One must be aware that the development of prin- 
ciples of heat and mechanics progressed along 
quite different paths: the study of motion and the 
mechanical interaction of bodies on the one hand 
and the studv of temperature and the thermal in- 
teraction of bodies on the other. The work of Joule 
joined these two almost entirely independent dis- 
ciplines. This joining of heat and energv', along 
with early developments in chemistrv', brought 
about the kinetic theorv'. This will be developed 
further in Sec. 10.9. We do not describe the long 
controversy over the nature of heat that extended 
from the time of the early Greek philosophers until 
the middle of the nineteenth centurv . In the long 
run, however, the multitude of physical processes 
in which heat seems to be generated by the ex- 
penditure of mechanical work must be reckoned 
with. 

The detailed operation of steam engines is not 
an important part of the storv line of Unit 3. Ac- 
cordingly, it is necessary only that students learn 
enough about engine operation to make sense of 
the extended treatment of steam engine improve- 
ment and its technological and social effects. If stu- 
dents understand well the principle of Savery's first 
engine, no further detail is important. Subsequent 
improvements can be adequately understood in 
terms of the operation of this fairly simple device. 
The Newcomen engine, for example, was different 
from the Savery engine only in that a piston was 
moved instead of water being moved directly. 
There was also a mechanical linkage that spraved 
the water, inside, at just the right time. Watt's prin- 



cipal improvement was to condense the steam in 
a separate container, so that the main cylinder did 
not need to be reheated by the steam for each 
cycle. The development of steam engines was pri- 
marily a commercial rather- than a scientific enter- 
prise. The early stages of development, it should 
be noted, were the work of amateurs: there v\'er-e 
as yet no professionals of engine design. Sec. 10.6 
discusses the steam engine's impact on the Indus- 
trial Revolution. 

10.6 I THE STEA^I EXGEVE AND THE 

INDUSTRIAL REI^OLITTION 

The Industrial Revolution was a vast and complex 
set of phenomena: the bar-e account given her-e can 
be treated as a r-eading assignment. One point that 
should be made is that distaste for mechanistic 
science could now be based on objections to its 
practical consequences as well as on its philo- 
sophical implications. 

'The Steam Engine," from Technics and Civili- 
zation, by Lewis Mumford, Harcourt, Brace Jo- 
vanovich. New York, 1934, provides background 
material that may be useful for a class discussion 
on the social and cultural effects of the develop- 
ment of the steam engine. The following definitions 
may be useful to the reader: 

eotechnic civilization: the pre-machine age 

paleotechnic civilization: the period beginning about 

1750 that followed the Industrial Revolution. 

In the Watt engine, the use of a separate con- 
denser provided a great gain in efficiency. However, 
it is even more efficient to use high-pressure steam. 
Early attempts to do this were frustrated because 
the materials available for constructing the boiler 
could not withstand high pressures and tempera- 
tures. But advances in metallurgv' later in the nine- 
teenth centur\' made it possible to constr-uct prac- 
tical high-pr-essure engines. Eventually, the separate 
condenser became unnecessarv', but by this time 
it had already served its purpose in getting steam 
engines established as the major sour-ce of energy 
in industry. 

Watt s working model of Newcomen s engine 
had a problem of scale, which is a tangential topic 
of considerable interest in itself. If you have time, 
PSSC Chapter 4, "Functions and Scaling," and the 
PSSC film Change of Scale, " could be used. 

An additional historical sidelight: The first L'.S. 
Government research grant was made in the nine- 
teenth century' to the Franklin Institute of Phila- 
delphia to investigate the causes of explosions in 
steam engine boilers. 

The section "The Steam Engine after V\att ipp. 
42-55), in I. T. Sandford's Heat Engines )Doubleday, 
19621, is a nice account of some of the fir-st steps 
toward modem engines. Internal combustion, steam 
turbines and gas tur-bines, turliojets, and rockets 
ar« taken up in Chapter 7 of Sandford's book. The 
discussion includes some ther-modynamics, which 
is not difficult to skip over-. 



BACKGROUND A.\D DEXTSLOPMENT 



181 



10.7 THE EFFICIEXCV OF F:XC;LN'ES 

A siiif^le ex|jeriinent camiot, ol coui-se. eslaljlish 
that theit; is a constant comtMsion lactoi betwciMi 
say, work and heat. (July \vh(;n the same pn)\)()i- 
tionality constant is found in a numher ol ciitfi'ii;nt 
experiments with different substances can it be 
called the con\ei-sion factor. I he student activity 
"Mechanical E(|ui\alent of Heat ^Handbook, paf^e 
1491 is, accoixlingly, a way of finding a pitjpoilion- 
ality constant for internal eneiXN and gross me- 
chanical (Mi(Mg\' of lead shot. I hat this is the con- 
version factor- fj)r work and lu^at is an unsupporled 
gencMalization. 

If lab time is available, you muy wish to pursut; 
the calori(! in Esin^rimcnl Eli- 11. C.alorimetry. 

rhe distinction between calorie and kilocaloiie 
iCaloriei should be emphasized to iixoid confusion 
later-. 

10.8 I ENERGY LX BIOLOGICAL 

SYSTEMS 

The main [Kjint of this section is that a meaningful 
connection can be made betwe^en principles de- 
veloped in the abstract world of phxsics iwh(!re a 
"body" is a "body" whether- it is a molecule of h\ - 
drogen, Caesai 's corpse, or- a red giant stari and 
the familiar world of widely differentiated living 
organisms. The physical principles are no l(?ss tr\ie 
for bacteria and high school students than foi- as- 
terxjids and roller cai1s. The details of the section 
are not imporlant to the storA line of I'nit ;j 

It should be pointed out that the "energx' avail- 
able in food" must be measured; thus, a known 
mass of the food substance is oxidized in a closed 
container of a calorimeter. 

Throughout the section there is mention of inef- 
fic'iencv and energ\' loss. Ihe dissipation of energv 
is discussed in Sec. ll.fi. If Sees. 10.7 and ll.fi art? 
br-cjught together- as a r-eading assignment, students 
can be asked in the sirbse(iirent (lass dis(-irssion to 
(vxplain what is riu^ant b\' the "loss" of energv in 
the food chain. IIk; point, of coirrse, is that onl\ a 
small amount of food ener-g\' is available forgr-owlh 
and movement Most of it is either dissipated 
internally as heat or- excreted in the tbrrn of un- 
digested foods. 

See Additional Background .Articles for- notes on 
"Food Kner-gv " 

10.9 ARRI\ IXG AT A C;EXERy\L 

C:OXSER\'ATIOX I^IW 

See .Additional Background Articles for a note on 
'(!lassi(ications of Knergv 

rhe philosophical material in this section should 
be tr-eated as was done elsewher-e: the details of 
tlie philosoplu or- theii- (-oii-ectness is not im|)()r-- 
tant rhe cNlcrit to which the iihilosophical con- 



cern influenced the dexelopment of physics is im- 
porlant. 

The majority of those people who first comprie- 
hended the full imporl of the generalit\ of ener-g\' 
conservation were young, and mam were outside 
the field of physics at the time they made their 
contributions: Mayer, a German physician, aged 28; 
Carnot, a French engineer who preceded all the 
rest in the discoverv land who will be discussed in 
Sec. 11.61, aged 34; Helmholtz, a German ph\siol- 
ogist, aged 32: Joule, an Knglish phvsicisi, aged 25; 
and Colding inot mentioned in the 7e.x;/i. a Danish 
engineer who made the same discoverv independ- 
ently of the others and almost simultaneously, 
aged 27. 

10.10 THE I^IW'S OF 

THERMOD\7VAMICS 

This section implies that total energv consists of 
something more than mechanical energ\'. heat, 
and potential enei-gv'. E.xamples of other forms of 
energv' ar-e the ener^' of excitation of an atom 'dis- 
cussed in I'nit oi and nuclear- energx tr-eated in 
I'nit fii. Students ma\ ask about these other forms. 
However-, the point to stress her-e is that the state- 
ment of the first lav\ of themiodynamics, A£ = Al\' 
+ A//, includes all special cases discussed in the 
past and will cover energy situations discussed in 
the futur-e. 

A// can r-epr-esent either- the addition of heat or 
the absorption of electrx)magnetic radiation to ex- 
cite an atom. The fact that the sourx-e of A// does 
not matter r-ender-s the law of conservation of en- 
ergv- gener-al and ther-efor-e \erv powerful. 

A discussion of this idea will illustrate the co- 
herence of physics. However, do not begin a full 
discussion of atomic and nuclear- phvsics at this 
time. 

10.11 i FAITH LX THE I^AWS OF 

THERMOD\'XA\IIC:S 

Students mav think it c urious that faith plavs such 
a ciTJcial role in this gr-eat law . Wlienever accept- 
ance of conser-xation of energ\ is endangered, a 
new motion is postulated to [jreserve the law. 
When mechanical ener-g\ tlisappeaivd 033 inter- 
nal energv was invented; when ener^ disappeared 
in a nirclear- r-eaction, the neutrino was postirlated 
to protect this law. Thirs. faith in ener^ conser- 
vation becomes frxiitfirl, for- it prx'diit.s new obser- 
vations. 

In the discirssion of kiru'tii- theorx C hapter- 11'. 
students will obserxe evidence of the f)ostulated 
internal eneig\ of gases. IJo not miss tin* o[)pt)r- 
tirnitv to str-i'ss this triumph of mechanics .New- 
tonian [)hvsics not onlv explains tlu' |)rx)perties of 
gases l)ut al.so pn'tlii t> new pr-o|)«Mlies 



IN2 



IIMT3' Tllh lailMIMI Ol MICIIWKS 



CHAFIER 1 1 THE KIXE I IC THEOHV OF GASES 



Sl^IAIAKY OF CHAPTER 11 

The citnelopmeiit of the kinetic theoiA ol gases in 
the nineteentli rentur> led to tlie last major triiini|ih 
of Newtonian nierhanies. A simple tlieoretical 
model of a gas was adopted: a large numhei of \ei\ 
small particles in rapid disordei-ed motion. Since 
the size of the pailicles is assumed to he \ eiA small 
compared to the space occupied h\' the gas, each 
particle mo\es most of the time at constant veloc- 
ity', occasionalK colliding with other particles or 
with the sides of the container-. H\ appl\ ing .New- 
tonian mechanics to this model, scientists could 
deduce equations that related ohsei\al)le pi-oper- 
ties of gases isuch as pi-essure. densitx , and tem- 
perature' to the sizes and speeds of molecules. 
With these equations, kinetic theorists could: ili 
explain the known relations hetween ohsei-\ahle 
pi-opei-ties of giise's, such as Bo\'ie s law: (2i pi-edict 
new relations, such as the fact that the \ iscositx' of 
a gas inci-eases w ith temperatui-e but is inilepend- 
ent of densitN" and i3i infer- the sizes and sjjceds 
of the molecules. 

The chapter first discusses the model and its 
consequences qiralitati\el\ : then details and deri- 
\ations are collected in an optional section ill.5i, 
which can be cjmitted by most stirdents. The r-est 
of the chapter is devoted to exploring the relation 
between the second law of therriiocK riamics. or 
principle of dissipation of energx-, and the kinetic 
theoiA . It would appear that, if Newtonian me- 
chanics were stricth applicable on the atomic 
le\el. then all molecular- processes would be time 
re\er"sible. and an\ initial arrangement would e\en- 
tuallv recur. Boltzmann s attempts to reconcile 
these re\er-sibilit\ and r^curr-ence paradoxes with 
experience in\ol\ed statistical reasorning and intro- 
duced interesting speculations about the dir-ection 
of time. 

11.1 AX OA'ERATEW OF THE CHAPIER 

This section should be ti-eatinl piimaiiK as a read- 
ing assignment. Discussion i-ould inclirde the fol- 
lowing. rhe kinetic-h\pothesis-of-heat idea is of 
much longer standing than is orxlinariK realized. 
Disr-egarding Cireek speculations, one can find that 
in the scientific era 11706) John Locke said: 

Heat is a \er\- brisk agitation of the insensible par1s 
of the object, which produces in us the sensation 
from whence we denominate the object hot: so 
what in our sensation is heat in the object is noth- 
ing but motion. 

In 1738, Daniel Bernoulli had r-emarked that: 

it is admitted that heat ma\ be considei-ed as an 
increasing interniil motion of particles. 

In 1780. Lavoisier- and Laplace were even mor-e 
explicit. They said in their- Memoire sur La Cha- 
leur": 



. . . heat is the vis viva r-esirlting tr-om the insensible 
movements of the molecules of a bodv . It is the 
sum of the prodircts of the mass of each molecule 
b\ the squar-e of its velocity. 

In 1798, Rumforxl, drawing conclusions frtim an 
experiment, stated that it appeared to him to be: 

extr-emelv difficult if not quite impossible to fonn 
anv distinct idea of anything, capable of being ex- 
cited and communicated, in the manner- the Heat 
was e.xcited and communicated in these Kxperi- 
ments. except it be MOTION. 

All these pr-onoirncements. except Rumfor-d s went 
considerablv bexond anv r-ealK justifiable evidence 
fi-om experimental data. 

Again, most of the central ideas here will be de- 
veloped later. I reat this section as introductory 
material, but alwavs be prepar-ed to add to a dis- 
cussion. The "equal-sharing principle lequijjarli- 
tioni and the problem of iir-ever-sibilitx- ai-e men- 
tioned only to show contradiction between 
.Newtonian mechanics and obserxable prxiperties 
of matter-. 

11.2 I A MODEL FOR THE 

GASEOrS STATE 

Historicallv . suggestions aUjng the lines of a parti- 
cle theory of gases had been matle in 1738 bv Ber- 
rioulli. Detailed theory was developed during the 
latter- half of the nineteenth century bv such men 
as Clausius. .MiLxxvell, Boltzmann, Krxjnig, and Joule. 
Joule is better- known for his experimental confir- 
mations of the principle of conser-vation of energv 
than he is for his work in developing the kinetic 
theory of gases. The theory stands verA much as it 
was left bv these men. but is r-efined primarilv to 
include knowledge of intennolecular forces. 

f he theory can be looked at svstematicallv in 
historical per-spective in the following manner': 1 
1738. suggestions bv Hooke and Bernoulli: 2 1848, 
fresh attack bv Joule in light of his mechanical 
equivalent of heat experiments: i3' 1807, chief dif- 
ficulties encoirntered bv Joule solved bv Clausius: 
4i 18oy, application of statistical mechanics bv 
\Ia.xwell. 

W hat do we mean w hen we sav model? Ibis idea 
should be discussed so that it can be applied firllv 
tcj the kinetic theory of gases. 

11.3 THE SPEEDS OF MOLECIXES 

rhe Newlonian mechanics of collisions can be 
used to derive the r-elation betxveen pr-essui-e and 
V olume iP and \ i for a collection of perfectK elastic, 
vanishingK small particles. Ihe prxjduct of pr-es- 
sur-e and volume is prxjporticjnal to the total kinetic 
energv of the particles. I his idealized model does 
not appear- to match r-eal gases, however-, because 
Newlonian mechanics shows also that the kinetic 



BACKGROl ND AM) DKX ELOP.MEM 



183 



energy, and therefore the product PV, changes 
when the volume is changed. The model might still 
be collect, however, if something happens during 
volume changes that continually acts to it3tujn the 
total kinetic energy of the paiticles to the initial 
value. This something will later be shown to be 
heat exchange with the sun-oundings; keeping the 
kinetic eneigy constant is the same thing as keep- 
ing the temperature constant. 

Although Newton's static 1/r repulsion model 
does account for- the empiric:al relation of P and V, 
and Beinouili's kinetic model aj)i)aiently does not, 
we have dismissed the fomier because it isn't nec- 
essarily cori-ect and pui-sued the latter because it 
is "reasonable." Students may need an optimisti- 
cally sly "now just wait and see " 

The second part of this section touches on the 
very rich field of statistical description. The essen- 
tial |K)int is that normal distributions are likely to 
i-esult when a very large number of independent 
small effects pix>duce a measurable result. 

11.4 I THE SIZES OF MOLECULES 

rhere was no real need to consider- the collisions 
between molecules in developing the gas laws and 
the specific heat of a gas. Howe\er, many effects 
and properties of gases and molecules cannot be 
under-stood on a molecular basis v\ithout a quan- 
titative study of molecular collisions and primar-ily 
the mean free path. With a molecular radius of 
about 10 " cm, we find that the mean free path is 
10 ' cm in a gas at atmospheric pr-essure and at 
a temperature of about 300°K. This is 1,000 times 
laigei- than the assumed molecular- dimensions. lo 
obtain a mean free path of 1 m, the pressui-e must 
be only 1.5 x 10 ' mm Hg. Many effects and prop- 
eilies of gases and molecules, such as heat con- 
duction and viscosity, cannot be understood on a 
molecular basis without a quantitatrxe stuch of 
molecular collisions. 

11.5 i predicting the beha\tor of 

gasf:s from the kevetic: 

THEORY (Special topic— optional) 

Consider the; options a\ailabl(^ at this poirit in the 
text. One option is to delete Sec. 11.5 and therefoit' 
mo\e from the sizes of molecules (Sec. 11.4) to the 
second law of therriiochnamics iSec. 11.61. The 
other- option is to teach kinetic theory by covering 
Sec. 11.5. The teacher- must decide upon an apprx)- 
priate investment of time and effort. Ihe mathe- 
matical derixation is not needed as a foundation 
for- later- work in this coirrse, birt would be most 
us(>lul to students who will contirure the stud\ of 
physics. 

Those who want to inxest a few daws' time can 
\vi\d stud«*nts towar-d an appi^tH-iation of the em- 
pirical ver"sus the theoretical dimensions of ph\'s- 
ics. 'I\vo equations aiv discussed; 

P = kiyr Empirical equation 
/' ^ ' </-^'i rh<'or-eti("al ecjuation 



"Empirical " suggests knowledge found after ex- 
perience. In this case, the experience is that of 
measuring P, D, and 7 and by trial and error finding 
a mathematical r-elationship. Refer to E3-14, The 
Behavior of Gases.' 

The theoretical equation is based upon the a 
priori statement air consists of molecules. ' "A 
pi'ior-i " suggests knowledge found before experi- 
ence. Therefore, the theoretical equation differs 
from the empirical equation since the molecule 
notion" comes fi-om a pei-son s mind. The molecule 
is postulated to pr-eserve the success of Xewlonian 
physics, momentum in particular. 

Note that these equations ha\e P and D in com- 
mon. If the remaining quantities are equated, the 
stailling discoxePk' is that temperature is equal to 
the average kinetic energy of the molecules of a 
gas. 

The statement in the Te,\t, page 332, that the 
height of the mercury column that can be sup- 
ported by air pressure does not depend on the 
diameter of the tube " may need demonstrating. 

You can \'ividly demonstrate the ti-uth of the 
statement by making mercury- bai-ometers out of 
transpar-ent hoses of \ainous diameters and show- 
ing that the column height is the same for ail of 
them. Barometers made with water are even more 
dramatic if \'ou can manage hoses about 10 m long, 
suspended perhaps in a stairwell. 

Emphasize that, as the diameter of the tube 
changes, the weight of the column changes, but so 
does the area upon which the column is sup- 
porled. Thus, the upwai-d for-ce on columns of dif- 
fererit diameters is alwa\s proportional to the 
weight of liquid being supporled. The balance of 
upwai'd and downward forces cannot, therefore, 
be altered b\' changing the column diameter, but 
it can be alter-ed by changing the pr-essure exerted 
on the air at the bottom of the column iweather 
changes) or by changing the densit>' of the liquid 
used iwater instead of mennirA i 

11.6 THE SECOND L/IW OF 

THERAIODIA'AMICS /iXD THE 
DISSIPATION OF ENERGY 

.Note that Carriot s work \vd to principles with 
sweeping generalitv. e\en thougli his theoiA treated 
heat as a fluid. His conclusions went beyond the 
first law of thennod\Tiamics ithe conservation of 
ener-gy). 

The first law states that the total output of an\ 
heat engine must equal the total enei^ input. It 
folUnvs dir-ecth that the useful wor-k cannot exceed 
the total input energx . C^arnot s theory placed a 
furlher rvstriction. stating that the useful wor-k out- 
|)ut can ne\er exceed a ceilain fraction of the total 
input eneig\ . Ihe fraction is 

T, - T„ 

T, 
where 7", is the absolute temperatun' in the wor-k- 
irig part of the engine and 7', is the absolute tem- 



IN4 



I'M T 3 / THI-: TKIl'MPH OF IVILCH/V.MCS 



perature of the exhaust. A heat engine can ap- 
proach 100% efficienc\ onK as the temperature ot 
its exhaust approaches ahsolute zero, or as its 
working temperature becomes intinitel\' great! Be- 
cause of design limitations, heat engines al\va\s fall 
short of even this theoretical fraction of useful work 
output. 

The principle of dissipation of energy can be ap- 
plied to all prT)cesses. mechanical or organic. The 
principle can be stated in man\' different wa\'s and 
each way can be called the second law of ther- 
mod\namics. Because it was first fomiulated with 
regard to engines, man\' of these statements of the 
second law refer to restrictions on the operation 
of heat engines. The statement second law of ther- 
modynamics is not particularU' better than am 
other. Indeed, it leaves unclear what the maxi- 
mum amount of work ' is. lit is the maximum al- 
lowed b\' the first law of thermodynamics an 
amount equal to the input enei-gv.i 

It may not be particularU pleasing to ha\e a ma- 
jor principle stated so negatixely. Ph\sicists ha\e 
in\ented the concept of entropy, defined as AS = 
Ah T, an abstract quantitv that can be calculated 
for am' process of heat transfer. In an\' real process, 
the entrop\' total for all participating s\stems will 
increase . This is a pleasantK positive statement of 
the second law of thennodxTiamics e\ en if it lacks 
much intuitixe meaning. Some intuitive meaning 
can be attached to the entrop\ quantitv by as- 
sociating it with the degree of disorder in a s\stem. 
(This association has e\en led the de\eloper of 
modem information theor\' to refer to the en- 
tropy of a message, a quantitative measure of the 
disorder, and therefore information, in a set of 
SNTnbols.i So another statement of the second law 
of thermodxTiamics could be: In an\' real process, 
the total disorder for all participating s\stems 
tends to increase. The topic of entropy increase is 
taken up again in terms of kinetic molecular the- 
on', and it will be seen then that these simple state- 
ments need to be qualified to allow for brief statis- 
tical fluctuations during which total entropx can 
decrease. This qualification has been attempted b\ 
the use of tends to." It wont be clear yet to stu- 
dents what tends to means, but it should be 
pointed out that it is a qualifier that will be dis- 
cussed further later. 

Reference is sometimes made to a third law of 
thermod\Tiamics, one statement of which is: It is 
impossible to reduce, in a finite number of steps, 
the temperature of any system to absolute zero. 
B\' using a \er\ large number of cooling steps, tem- 
peratures of 0.001°K or less can be produced. iThis 
restriction is not the same as the "zero-point en- 
ergy of quantum theorvM 

The "entropy death, also called the "heat death, 
appears to be a cold death." The principle of dis- 
sipation predicts onl\' that all ordered motion will 
eventualh' disappear in any closed s\stem and 
onl\ random icompleteh- disordered i thermal mo- 
tion will remain. Whether this is "hot" or cold 



depends on how much energy there is to go 
ai-ound in how much space. The statement about 
the possihilit\' of avoiding the enti-opy death refers 
to the recun^ence theoi-em. If we could wait long 
enough, we might find that entropy decreases 
again and we come back to vvhere we started from. 
The principle of dissipation of energ\' is not best 
characterized as a consequence of the second law 
of thermodvnamics. A better expression of the re- 
lation is that the second law is a quantitative but 
somewhat restricted version of the dissipation 
principle. The statement at the end of the para- 
graph suggests that heat performs work while flow- 
ing, without being used up. An alternate statement 
would be as follows: Mechanical work can be de- 
rived from internal energ\' only when there is a 
temperature difference betvveen two parts of a sys- 
tem. As a system approaches a uniform tempera- 
ture, the possibilitv' of producing mechanical work 
from the internal energ\' approaches zero. 

11.7 MAXl^-ELL'S DEMOX AXD THE 
STATISTICAL \1EW' OF THE 
SECOXD LAW OF 

THERMODl-XAAHCS 

The main point of this section is \er\- important to 
the overall goals of the course. It suggests that laws 
of phvsics need not be absolute descriptions of 
what must or must not happen: thev can just as 
well give probabilities for w hat might or might not 
happen. .An aspect of this kind of law, not men- 
tioned explicitlv in the Text, is that the accuracv 
of statistical description increases with the number 
of particles. In other words, the e.xpected percent- 
age of fluctuation from the most probable value 
decreases as the number of particles increases. 
I Even a dust speck contains some billion billion 
atoms, so virtuallv all directlv observable events are 
verv, verv' close to being completelv determined by 
statistical laws.i .Also, the longer the time interval 
over which an average value is taken, the less the 
expected percentage of fluctuation from the statis- 
tically most probable value. For relatively small 
numbers of particles over relativelv short time in- 
tervals ifor example, a thousand particles over a 
billionth of a second, the statistical description 
predicts pronounced fluctuations from the most 
probable value. Note, however that the fluctua- 
tions do not constitute error in the statistical de- 
scription: the statistical description includes the 
expectation of fluctuation. It is the expected fluc- 
tuation that decreases as the number of particles 
and the time interval increase. The Steinberg car- 
toon from The \'e\\ Yorker might be labeled "a verv 
improbable fluctuation." The statistical form of the 
second law of thermodynamics would give an ex- 
ceedingly small I but not zero' probability to so 
great a fluctuation as the boulder rolling up the 
hill as it and the hill cooled off. 

Maxwells demon is important to the develop- 
ment. It is an example of how the second law could 



BACKGROl \D AND DE\'ELOPlVIE\T 



185 



be violated, on the basis of \eu1onian mechanics, 
if only ont! could get infoimation about molecular 
anangements and soil out the molecules. In this 
respect, it illustrates the connection between en- 
tropy and information mentioned in Sec. 11.6. 
Feynman gives the example of a mechanical de- 
mon consisting of a tiny spiing door of so small a 
mass and so loose a spring that it would open un- 
der the impact of fast molecules only. The impacts 
would result in the door becoming increasingly 
hot, so that its own random thermal agitation 
would progit3ssively disrupt its function. 

11.8 I TIME'S ARROW AND THE 
RECURRENCE PARADOX 

Your students will probably be feeling anxious 
about the statistical turn that the course has taken. 
Statistics may be all right for describing the chaotic 
organic world, but the immutable laws of matter 
are something else. They may find some comfoil 
in knowing that many scientists in the nineteenth 
centuiy were not willing to accept as a basic law 
of physics one that gave only probabilities instead 
of certainties. They argued that if kinetic theor\' 
didn't lead to a completely detemiinistic second 
law of thermodynamics, the theoiy must be wrong. 



Another way of stating this objection is the re- 
versibility paradox." The kinetic theory in\olved 
only reversible collisions of molecules so that 
events could just as easily happen backwaixls as 
forwards. For instance, the hill could cool slightly 
and prtjpel the boulder up it as easily as the boul- 
der could roll down and warm the hill: concen- 
trating enei-gy instead of dissipating it, increasing 
order instead of disorder. But all observed e\ents 
in the universe go in just one direction, that of 
energy dissipation and increasing disorder. So it 
seems that the kinetic theory must be v\Tong. The 
sequence of photographs on Te\t page 344 is in- 
tended to suggest this ver>' sticky question. The 
sequence of e\ents in the \eiy nearly perfectly elas- 
tic collision photographed stroboscopically in the 
top picture could be almost equally good ph\sics 
whether beginning at the right or at the left. In the 
perfectly elastic molecular collisions of the kinetic 
model, there would be no way of choosing between 
a left-to-right and a right-to-left time sequence. In 
the other pictur-es there is no doubt about which 
way the process is going; if a motion picture of one 
of them were reversed, it would easii\' be spotted 
as "backwards." Film Loop 36, "Reversibility of 
Time," could be used to develop this point. 



CHAPTER 12 / WAVES 



SUMMARY OF CHAPTER 12 

Chapter 12, peihaps more than any other chapter 
in the unit, depends upon experiments and/or 
demonstrations to give substance to the material 
discussed in the Te^t. The purpose of the dem- 
onstrations and experiments is to provide a general 
understanding of waves. 

We are concerned with wave phenomena in a 
mechanical framework in order to set the stage for 
the study of light and electromagnetic waves in 
Unit 4. It is imporiant that students become famil- 
iar with wave phenomena, particularly interfer- 
ence, so that a vvave-versus-particle model will 
make sense to them. This will also be useful in I'nit 
5 when they stiicK the atom. 

12.1 I EVTRODUCTIOX 

Our pn;vious work has been concerned, on one 
hand, with gross motion of particles and of rigid 
boditjs and, on the other hand, with some of the 
internal pioperties and constituents of matter. 
rhix)ughout our study of rigid bodies, we have as- 
sumed that the various parts of a body itMiiained 
fixed distances apari. That is, we ignoivd the com- 
pressible nature of the solid form of matter. We 
shall now ask how a compii'ssible bod\ it'sponds 
to e,\ternall\ applied forces and see that this leads 
clin!i:tly to the study of waves. In chaptei-s to come, 
we shall see that these phenomena iwavesi aiv par- 
ticularU descriptive of manv events in natuiv. such 
as light, sound, radio, and water-suriace plnMiom- 



ena. We shall see in the units called "Models of the 
Atom" and "The Nucleus" that atoms, electixins, 
and nuclear and subnuclear particles have wave- 
like, as well as pariicle-like, properties. 

12^ I PROPERTIES OF WAIVES 

Wave motion implies the transmission of a state. 
If we stand dominoes on end in a long line and 
then knock over the first one, we start a chain of 
events that leads eventually to all the dominoes 
being knocked down. Iheiv was no net mass trans- 
port along the line of dominoes: rather, it was their 
state of fiilling that traveled. In this simple case, the 
speed at which the state of falling traveled is called 
wnvc sfyccd. 

All wave pulses possess momentum and enei^-, 
which can be detemiined without much diflicultv . 
This could be assigned to better students with 
some guide lines. 

The first serious suggestion that polarization 
might be due to vibrations of light transvei-se to the 
direction of pix)pagation came finm Iliomas \oung 
in 1817. rhe name polarization was fii-st used in 
this connection in 1808 b\ a French investigator 
named Malus. 

12.3 I WAITE PROPAGATION 

Kner((\ can be transmitted ovt'i considerable dis- 
tances In wavi' motion I hi* energv in the waves is 
the kinetic and potential enei>^ of the matter but 
the transmission of the enei-gv comes about bv its 



IHfi 



I'Ml :» THi: THIl'IVIPH OF VIIJCHA.MCS 



being passed along from one pai1 of the matter to 
the next, not b\ anv long-range motion of the mat- 
ter itself. The pi-operties of a medium that deter- 
mine the speed of a mechanical \va\ e thi-ough that 
medium are its inertia and its elasticit^ . C"ii\en the 
characteristics of the medium, it is possible to cal- 
culate the \va\e speed fixjm the basic principles of 
Ne\\'tonian mechanics 

12.4 PERIODIC WA\TS 

It is most important for students to ha\e a thor- 
ough undei-standing of the important definitions 
in \va\e motion. The \va\elength idea" is an ob- 
vious, imperative, and prerequisite concept for an 
understanding of phase. Since the period 7" is the 
time required to travel one wavelength \, it follows 
that \ = vT. This velocitv' is called the phase ve- 
locity and is the onK velocitv' involved in a simple 
harmonic wave. In contrast to this is a group ve- 
locity, which is important when several wave- 
lengths and phase velocities travel thixjugh a me- 
dium. We shall limit our discussion to phase 
velocitv' and simple harmonic waves. The relation- 
ship V = \/is important and applies to all waves 
impartiallv: water waves, waves on springs or 
ropes, sound waves, and electromagnetic waves. 

12.5 miEX WAATS MEET: THE 

SLTERPOSITIOX PRINCIPLE 

The superposition principle appeai-s so obvious 
that it might be worthwhile to mention when it 
does not applv. When the equations of the wave 
motion are not linear, superposition fails. This hap- 
pens phvsicalK when we hav e relativ elv large wave 
disturbances for which the law s of mechanics are 
no longer linear. An example is the linear relation 
in Hooke's law. Bevond the elastic limit, F — —k,x 
no longer applies. For this reason, shock waves in 
sound behave differently than ordinary' sound 
waves and hence superposition does not hold. A 
quadratic equation governs the behavior of shock 
waves. \ erv loud tones will not add linearly thus 
giving rise to a distortion known in high-fidelitv 
jargon as "intermodulation distortion. Ripples in 
water waves can travel independentlv across gentle 
swells but not across breakers. 

It would be instructive to stress the fact that, if 
two waves act independentlv of one another, the 
displacement of am particle at a given time is sim- 
ply the sum of the displacements it would receiv e 
fhjm the individual wa\es aJone. 

12.6 A TWO-SOmCE EVTERFEREVCE 

PATTERN 

It is interesting to note that Thomas Young 
11773-18291 was a London phvsician who became 
interested in the study of light through his medical 
studies, particularly his discoverv' of the mecha- 
nism of accommodation ifocusingi of the eye. He 
had done extensive e.xperiments with sound and 
was impressed v\ith the studv of beats. Two sounds 



can combine so as to pi-oduce silence and this is 
most easiK explained on the basis of a wave pic- 
ture. It isn t surprising then, that he presented re- 
ports to the Rov al Societv' concerning experiments 
on light, namely, his now famous two-slit interfer- 
ence experiment. 

The node and antinode lines are actuallv hv per- 
bolas. The hvperbola, since it is the curve for which 
the difference in distance from two fixed points is 
constant, obviously fits the condition for a given 
fringe, namelv. the constancy of the path differ- 
ence. .Although this deviation from linearitv may 
become important with sound and other waves, it 
is usually negligible when the wavelengths are as 
short as those of light. 

12.7 I STANDING W'A\T;S 

Standing waves are most commonly pnaduced as 
a result of reflections. However, they can also be 
produced bv two independent sources of disturb- 
ance, one of which prxjduces a wave that travels to 
the right and the other a wave that travels to the 
left. In terms of energv', the difference bervveen a 
traveling and a standing wave is that in the trav- 
eling wave there is a net flow of energv', whereas 
in a standing wave there is none. The energv' is 
"trapped in the standing wave. 

Each characteristic frequencv of the one-dimen- 
sional spring, string, or anv other similar elastic 
system, corresponds to a certain characteristic 
mode of oscillation. Similar characteristic frequen- 
cies and modes of oscillation are found also in two- 
dimensional patterns. .An important difference ex- 
ists between the one- and two-dimensional cases. 
In the one-dimensional case, there is a one-to-one 
correspondence between the characteristic fre- 
quencies and the characteristic modes of vibration. 
In the two-dimensional case, however, there mav 
be several modes of oscillation with the same fre- 
quencv. 

Standing waves are of first importance in the 
study of sound, since, without exception, everv' 
musical instrument depends on this principle. In 
1807, Fourier showed that am periodic function 
can be expressed as the sum of a number of sine 
and cosine functions with appix)priate amplitudes. 
This mode of analvsis is of great value in almost 
everv branch of physics. 

Reference to Sec. 12.6 can now show that the 
pattern for two-slit interference between the sources 
is a standing wave pattern. 

12.8 WAAX FRONTS ANTI 
DIFFRACTION 

Huygens theorv simpK assumes that light is a 
wave pulse rather than, say, a stream of particles. 
It says nothing about the nature of the wave, and 
gives no hint of the electrxjmagnetic character of 
light. This theorv' is based on a geometrical con- 
struction, called Huygens' principle, which allows 
us to tell where a given wave front will be at any 
time in the future if we know its present position 



BACKGROl \D AND DEXTILOPMEVT 



187 



Today this piincipln can be stated somewhat suc- 
cinctly as follows: Kveiy point on an advancing 
wave fiont acts as a soui-ce litim which secondaiy 
waves continually spitiad. The [passage of a lipple 
through an apertui-e illustrates this effect, i Refer to 
rip|)le tank experiments such as E3-16 or D44k It 
was a similai' spi-eading of beams of light that ga\e 
rise to the phenomena that (irimaldi had classified 
under the name dilXraction in his publication of 
l(i65. \(!wt(ni not only i-efeiit!d to (irimaldi s ex- 
jjeriments but he r(!peated and impio\ed upon 
them, rhe pnujccupation with one theoiy blindetl 
e\en a scientist of .\e\\1on s caliber to the signifi- 
cance of the (!\'id(;nce |)ointing toward another. It 
was left to f homas Voung to find the most con- 
vincing evidence for the wa\e hy|)othesis. 

12.9 I REFLECTIOX 

I h(! law of icflection was known to lAiclid. It can 
be d(!ri\ed liom Maxwells equations, which means 
it should hold for iill regions of the electromagnetic 
spectiTjm. 

I'he reflection of waves is familiar from such 
events as the echo of a sound wave, the i-eflection 
of a ripple on a water surface, or the reflection of 
waves on a rope. When wa\es are incident on a 
boundary between two media in which the \eIocit\' 
is appr(!ciably different, the incident wa\e train is 
divided into i-eflected or retracted loi- transmittedi 
trains. 1 he amount of energ\' reflected will be rel- 
atively greater the larger the change in velocity. 

12.10 REFR/\CTION 

Obsei\ation of thi! bending of light waves goes back 
to anticiuity. Cleomedes, in the first centua \i) , 
was one of the fii-st to suggest that the sun remains 
visible for a time aftei- it has actually set. Aristotle, 
in his Book of Problems, coirectiv descM ibed the 
appearance of an oar dipped in water. Ptolemy 
(about A.D. 1501 tabulated angles of refr-action for 
air-water-glass media. He concluded that the an- 
gle of incidence and refiaction wei"e the same. Al- 
hazen (965-10391, an Arabian investigator, pointed 
out the error of Ptolemy's generalization. Rolemv's 
tables were extended ithough not cori-ectlvi bv \'i- 
tellio about 1270 and by Kircher il6oi-lH80i. 
K('pler. in addition to his astronomical studies, 
made important contiibutions to optics and inves- 



tigated reflection and refraction angles in media 
with somewhat more success than Ptolemy. It was 
not until 1621 that the legitimate relation was ex- 
perimentally discovered by Willebrod Snell and 
deduced from an early corpuscular theor\' of light 
by Rene Descartes. The law of refraction is known 
as Snell's law or (in France i Descartes law. There 
is reason to believe that Descartes had seen Snell's 
manuscript, though he subsequentlv published 
the law of refraction as his own discover^'. Des- 
cartes' plagiaiism upon Snell points out, perhaps 
unfortunately, that unethical practices in the pur- 
suit of personal ambition are not entirely unheard 
of in the histor\' of science. 

12.11 I SOLWD w'a\t;s 

Sound waves ai-e longitudinal mechanical waves. 
They can be propagated in solids, liquids, and 
gases. Ihe material particles transmitting such a 
wave oscillate in the direction of propagation of 
the wave itself. The distinction between the sub- 
jective and objective attributes of sound has not 
always been recognized, but John Locke, the sev- 
enteenth centuiy philosopher, said 

That which is conveved into the brain bv the ear 
is called sound, though in truth until it comes to 
reach and affect the perceptive part, it is nothing 
but motion. The motion which produces in us the 
perception of sound is a vibration of the air caused 
by an exceedingly short but quick tremulous mo- 
tion of the body fi-om which it is propagated, and 
therefore we consider and denominate them as 
bodies sounding. 

But in any case the origin of sound can be traced 
to motion of some kind. .Again, historically speak- 
ing, Aristotle made this observation when he said 

All sounds arise either from bodies falling on bod- 
ies or from air falling on bodies. It is due to air 
. . . being moved bv expansion contraction and 
compression. 

Robert Boyle in 1660 impi-oved the air pump iv- 
cently invented bv von Guericke and with it per- 
fomied a bell-in-vacuo e.xperiment where it is re- 
ported he said 

U'e silently expected the time when the alarm 
should begin to ring . . . and were satisfied that we 

heaixl the watch not at all. 



IHH 



UNIT 3 / THE THIIMI'II OF MbCHAIVICS 





£ - 



i 



BACKGROL'VD A\D DEXTILOPMENT 



189 



Additional Back^ound Articles 



NOTE ON THE STATE OI I»H\ SICS AS A 
SCIENCE AT THE BEGINNING OF THE 
NINETEENTH CENTURY 

Until the beginning of the nineteenth century, 
practical sources of electrical current had not been 
devised, so the study of electricity was not well 
developed. Heat and electricity wei-e both consid- 
ered to be weightless fluids, in accordance with 
the mechanical point of view. An adequate theory 
of heat was not developed until the middle of the 
nineteenth century. The nature of light was not 
known at the beginning of the nineteenth century: 
the particle hypothesis of Xewton was faxored 
against the wave hvpothesis of Huygens, but both 
were mechanical theories. 

It should be noted that those who contributed 
to the development of mechanics, like N'ewlon and 
others in the following generations, were great 
mathematicians. This put a definite stamp of pr-e- 
cision and logic on the development of mechanics. 
Mathematics seemed to be a science of unlimited 
possibilities. One of the French mathematicians 
once said that, given sufficient time, it will become 
possible to express human thoughts in the form of 
a mathematical formula. Mechanics was consid- 
ered to be an applied aspect of mathematics and 
as such also as a science of unlimited possibilities. 
(Remember the full title of \eu1ons Principia: Phi- 
losophiac Naturalis Principia Malhc.matica. "Math- 
ematical Principles of Natural Philosophy,"! This 
was trxre both for "pure" mechanics ifor instance, 
laws of motion I and technological or engineering 
applications of mechanics. As we will see later in 
this unit. Heron of Alexandria was able to make 
certain mechanical devices utilizing the power of 
steam in about ad. 100. In the seventeenth and 
eighteenth centuries, many sophisticated mechan- 
ical devices and automata wer-e built, such as Be- 
sancon's duck that coirld swallow food, or the me- 
chanical boy who coirld play a musical instrirment. 
The Swiss vvatc;hmakers prodirced ingenious de- 
vices, some of them able to give a whole theatrical 
performance staged by mechanical dolls. One 
needed only to wind the spring and the mecha- 
nism started to operate. The possibilities of these 
mechanical devic'es ifrom the clocks of the thir- 
teenth century oni greatly stimulated human in- 
genuity in the West. (See L. Mumford, Technics and 
Civilization.) 

This development of mechanics and mathemat- 
ics on one side, and the construction of mecha- 
nisms of high complexity on another side, induced 
scientists to think of (lod as the greatest of all sci- 
entists. Clod had built the workl-macbirK*. the irni- 
ver-se in which we live, and had wound it u[i for- ail 
lime. As Kepler- put it, the task of a s( i(>ntist con- 
sisted tbeiefort' in following the tboirghts of (iotl 
at the moment of creation. Of cour-se, not all s( i- 
entists acceplc'd thi.s view 1 his lack of consensus 



is illustrated by the famous anecdote involving the 
French scientist Laplace, who had suggested a 
mechanical account of the origin of the earlh and 
our solar system, .\apoleon I, to whom Laplace ex- 
plained his theory, asked him. Where is God in 
your theory? to which l^place is said to have re- 
plied, "I have no need of that hvpothesis." ( "Je n'ai 
pas besoin de cette hypothese-la.'i 

NOTE ON COIVSER\'ATION 
LAWS IN PHYSICS 

The discovers of conser\ation laws has been one 
of the most important achievements of science. 
These laws, which are perhaps the most pov\erful 
and certainly the most prized tools of analysis in 
physical science, say in essence that, whatever 
happens within a system of interacting bodies, 
thei^e ar-e cerlain measurable quantities that can be 
counted on to remain constant so long as the sys- 
tem r-emains isolated. 

The list of conservation laws has grown in recent 
year's, particularly as a result of work in the area of 
fundamental lor "elementary'i particles. Some of 
the newer laws ar-e imperfectiv and incompletely 
understood. There are others that are on tenuous 
ground and ar-e still being argued. 

A list of conservation laws is given her^e. It would 
be foolhardy to say the list is complete or entirely 
accurate. Only too r-ecently we have had to surren- 
der long-held, cherished conceptions that ap- 
peared almost self-evident. But this list includes 
those conserAation lav\s that comprise the wor-king 
tool kit of phvsicists todav . Those which are starred 
are discussed in the basic units of this course: the 
others are treated in supplementary (optionali 
units. 

1. Linear momentum* 

2. Energv' (including massi* 

3. Angular momentum (including spim 

4. Charge* 

5. Electron-family number 
fi. Muon-familv number 

7. Baryon-familv number 

8. Strangeness number 

9. Isotopic spin 

Those listed as numbei-s 5 thrx)irgh 9 r-esult from 
wor-k in nuclear phvsics, high-ener-g\' phvsics, or 
elementary or fundamental particle phvsics. If 
these laws are urifamiliar, vou will find Kenneth 
Forxls "Conservation Laws inter-esting. enlight- 
ening, and worlh rvading at this stage. The selec- 
tion is a chapter from For-d s bt)ok The World of 
Elcmcnian Parlirirs Blaisdell I\rlj|ishing Co . 19631. 
The book is a well-written intrxiduction that should 
appeal to anvone who wants to leani about one of 
the leading fixMitier-s of cunvnt phvsics Forxl dis- 
cusses the fir"st seven laws in that selection and 
even for- those uninitiattul into the mvsteries of el- 



i»o 



IIMT 3 / THE TKIUIV1PH OF MECHANICS 



ementarv' pai-ticles, he gi\es a clear presentation of 
many ideas basic to contemporan phxsics. 

ELASTIC AND INELASTIC COLLISIOxXS 

The temis elastic and inelastic appear in Chap- 
ter 9. It is difficult to discuss these temis fulK with- 
out using the concept of energx. For an elastic col- 
lision between bodies A and B, B s \elocitA relati\e 
to A after collision is just the negati\e of what it 
was before collision. Ihe distance between A and 
B afterward will increase at the same rate at which 
it decreased before collision. For a completeK or 
perfectly inelastic collision, the relative \elocir\ is 
zero after collision. For a collision somewhere be- 
tween completely elastic and completely inelastic, 
the bodies will separate more slow 1\ than the\' ap- 
proached. 

More usualh'. an elastic collision is defined as 
one for which the kinetic energv' after the collision 
is the same as the kinetic energ\' before. Below is 
a proof that this is equi\alent to the abo\e defini- 
tion. 

In either treatment, an elastic collision is "re- 
versible '; that is, it would be impossible to tell 
whether a motion picture of the collision was lun- 
ning forwards or backwards. 

NOTES ON THE EQLTR ALENCE OF THE 
DEHNITIONS OF "ELASTIC COLLISION" 



Before 



-After 



(Vg — \\^i is the \elocir\' i\'^' — Vy'i is the \elocit\' 
of B relatixe to A before, of B relatixe to A after. 

If kinetic energ\' before and after are equal, then 

Vzm^v^' + Yzm^Vg^' = Vzm^v^'' -I- V2mBVg'^ ll) 

Since the linear momentum is always conserved, 

^A^A + "Ib^'b ^ "Ia^a' + "IbVb'. (2) 

We can rearrange equation (II 

"^b'^'b'' - V' = "^a'^a'" - ^a"I '31 

and equation 12) 

If we divide equation i4) into equation i3), we get 



vJ, 



151 



which says that the velocity of B relative to A after 
collision is equal to the negative of what it was 
before. 

ENTRGY REFERENCE LE\TLS 

The zero le\el for measuring potential energ\' lor 
even for measuring kinetic energvli is ariiitraiy. Be- 
cause the conservation principle really deals only 



with changes, the zero levels can be chosen as 
whatever is most convenient at the moment. This 
is more easilv seen if the general formulation used 
for momentum is used again here; 

2 (V2m,v,- + F,/i,i = 0. 

i 

Lest students conclude that playing with the 
zero levels is onlv "mathematical," some examples 
can be given to show that there is i-eally no mean- 
ing to absolute zero levels for potential or kinetic 
energv'. It is common, for example, to take the low- 
est point of the pendulum s swing as the zero po- 
tential level. But if the string is cut, the bob can fall 
to the floor and decrease its potential energv' still 
more. The potential energv' at the floor level could 
be taken as "negative," or the floor level could be 
chosen as a new zero level. But if a hole were to 
be dug next to the bob, it could deci-ease its grav- 
itational potential still further bv falling into the 
hole all the way to the center of the earth.' (What 
would happen if it went beyond the center?! But 
neither will the center of the earth do as an abso- 
lute zero of potential energv', since there is still the 
possibilirv of falling all the way into the sun . . . and 
so on, ad infinitum. 

The topic of " inert ial" frames of reference is too 
subtle to take up here so it is not easv to discuss 
the zero level" for speed. Thus, the "zero level for 
kinetic energv is not so easv to handle. The change 
in V' in one frame of reference will not be the same 
in some other frame of reference moving with uni- 
form translation relative to the first, but neither will 
the observed distance through which the force 
acts. \'eiy often ifor example, in nuclear physics) 
V elocities are referred to the center-of-mass of the 
interacting bodies to simplift' calculation. 

.After all this, the main point to be made is as 
follows; Intuitively, V2m\-^ is not an obvious choice 
for quantity' of motion," but neither is it particu- 
larlv objectionable. Likewise, weight times height 
is not objectionable as a measure of a potential" 
something. 

However, by making these choices, we can com- 
pute a quantitv', the sum of —mv and Fh, which is 
conserved lat least in frictionless rising and falling 
neai- the surface of the earth i. It will turn out to be 
valid also for the frictionless rising and falling" in 
the solar system and in many other situations 
ivvhen some other distance measure than distance 
fix)m the earth is used:. We have given the sum the 
name energv .' When related to ideas of heat" 
and "work," energv " pr-oves to be one of the most 
powerful ideas of phvsics, making manv connec- 
tions with other sciences as well as within physics. 

NOTE ON FOOD ENTRGY 

Many foods combine with oxygen in the bodv to 
produce carbon dioxide and water. Since, in the 
body, the process of oxidation progresses through 
many steps, the energv is released much more 
slowlv than when the same food is burned in air. 



ADDITIONAL BACKGROUND ARTICLES 



191 



However, the chain of chemical reactions through 
which the process takes place does not alter the 
amount of energy released. 

The energx' value of a food can be ascertained bv 
burning samples in a laboratoiy. I'his process is 
used as the basis for Calorie charts that tell us how 
much energy a particular food is capable of pro- 
viding. 

Measurements of the amount of food actually 
"burned" in the body may also be made. This is 
done by collecting the gas a person exhales for a 
short time and analyzing the sample to determine 
how much oxygen from the inhaled air has been 
replaced by carbon dioxide. Persons vary in the 
extent to which their bodies obtain and use energy 
from food. The minimum energy used to keep alive 
is called the "basal metabolism." 

Different amounts of food energv are needed by 
the body for different types of acti\aties. The more 
a person exercises or performs heaw physical 
work, the more energv is needed. In general, how- 
ever, the body converts a large amount of the en- 
ergy stored in food to heat energv and utilizes a 
much smaller portion of the total food energv for 
other life processes. 

NOTE ON CLASSIFICATIONS OF ENERGY 

For some puiposes, there are advantages to clas- 
sifying energies. But the kind of classification that 
one might best make depends upon the situation 
under consideration and is, in any case, arbitraw. 

Suppose we are looking at a physical system: a 
bowl of soup, a soap bubble, or maybe our solar 
system. Someone asks, "What kinds of energy does 
the system possess?" We might answer that it has 
kinetic and potential energies; but that statement, 
while true, is likely to be too general, too all-inclu- 
sive. 

We might go further. We might say that the sys- 
tem has kinetic energy as a consequence of the 
motion of the system as a whole. We may have 
dropped the bowl of soup; the soap bubble may be 
floating away; or our solar system may be moving 
relativ^e to the center of our galaxy. Fhe magnitude 
of this energy depends upon the reference frame 
in w^hich we choose to measure the system's mo- 
tion. 

In addition, the system as a whole may possess 
potential energy as a consequence of its position 
relative to external sources of force. The soup and 
soap bubble hav^e graxatational potential energy as 
a consequence of experiencing the earth's graxd- 
tational force. Similarly, the solar system may ex- 
perience gravitational forces due to nearby stars. 

For some purposes, these kinds of kinetic and 
potential energies are not v^eiy important. For ex- 
ample, we do not consider them at all when we 
ask about the soup's temperature or about the way 
the planets move relative to the sun. When that is 
the case, we generally say that all the rest of the 
energy is internal energy. For most thermody- 
namics problems, this is the sort of classification 



we make. (In fact, the laws of thermodynamics are 
independent of the detailed nature of the system.) 

How we go about describing the internal energy 
depends upon our purposes. If we are dealing with 
helium gas at moderate temperature and pressure, 
it is reasonable to say that the internal energy is 
just the translational kinetic energy of the atoms, 
where we measure their speeds relative to the con- 
tainer. That is the approach taken in Sees. 12.4 and 
12.6 of Unit 3. This works reasonably well for He, 
because the helium atoms exert only veiy weak 
forces on each other when not colliding, and be- 
cause the He molecule is monatomic (composed 
of only one atom). 

If we are dealing with a gas composed of more 
complex molecules, we need to consider other 
forms of internal energy; for example, the kinetic 
and potential energies associated with the vibra- 
tional motion of the atoms vuithin the molecule 
and the kinetic energv associated vvath the mole- 
cule's rotation. We might even need to include the 
energy associated with the interactions of the mol- 
ecules with each other, often referred to simply as 
chemical energy. 

Even these forms do not exhaust the sources of 
internal energy. There is the energy involved in 
binding an atom's electrons to its nucleus and that 
associated with the forces holding the particles to- 
gether within the nucleus. There are instances for 
which we might need to include the system's ra- 
diation energy (light). 

It should be clear that there are many different 
ways of making energy classifications. There are 
times when such classifications as elastic, electri- 
cal, and chemical energies are helpful, even though 
we know that elastic and chemical energies are 
predominantly electixjmagnetic in character. (Think 
of a charged compressed spring in a jar of sulfuric 
acid.) It is piobably worthwhile to give some 
thought to the sources of internal energv, but it is 
not the sort of thing one wants to bludgeon stu- 
dents with. Detailed classifications can sometimes 
lead to confusion, and aie always somewhat arbi- 
trary. 

SOME NOTES ON WATT 

(from James Watt, Craftsman and Engineer, H. W. 
Dickinson, Cambridge University Press, 1936.) 

"The problem now was to make the apparatus 
into an engine capable of repeating its motion in- 
definitely. Watt started on the construction of a 
model with a cylinder 2 in. diam. While thus en- 
gaged, Robison's stoiy is that he burst into Watt's 
parlour and found him with a little tin cistern' on 
his lap. Robison began to talk engines, as he had 
done previously, but Watt cut him short by saying: 
You need not fash yourself about that, man; I have 
now made an engine that shall not waste a particle 
of steam.' Robison put to Watt a leading question 
as to the nature of his contrivance but he an- 
swered me rather drilv and vouchsafed no expla- 



192 



UNIT 3 / THE TRIUMPH OF MECHAMICS 



nation.' If an artist ever wislies to paint a g(MTPe 
picture of Watt, instead of perpetuating the un- 
founded story of his playing as a boy with the 
steam issuing from the spout of a kettle, he might 
limn the young workman in his leathern apron 
with the separate condenser on his lap and Robi- 
son tiying to quiz him. 

"If Watt lacked experience in the constiuction of 
engines in great,' i.e. of full size, he had the advan- 
tage of being free from preconceived ideas of what 
engines should be like. In fact he had in view two 
engines, one reciprocating and the other rotary, 
entirely different in design from anything that had 
gone before. This veiy fertility of mind, and his re- 
source in expedient, may almost be said to have 
delayed his progress. Of a number of alternatives 
he does not seem to have had the flair of knowing 
which u^as the most practicable, hence he ex- 
pended his energies on many avenues that led to 
dead ends. In truth this is the attitude of mind of 
the scientist rather than that of the craftsman. Still, 
unless he had explored these avenues he could 
not be certain that they led nowhere. " 

DISCUSSION OF CONSERVATION LAWS 

Conseivation laws are important because they en- 
able us to make predictions. This ability has a far 
wider significance than its application to colliding 
carts in the laboratory. For example, much of what 
is known about subatomic particles comes from 
analyzing scattering (collision) expeiiments. Stu- 
dents hear a great deal about bubble chambers, and 
particle accelerators without understanding the 
u^ay in which information fi'om these devices is 
used. 

A detailed account of Chadwick's discovery of 
the neutron and his calculation of its mass is found 
in Chapter 23 and could well be used here as an 
example in which both conservation laws are used 
in making new discoveries. 

We suggest the following procedure as an effec- 
tive way to show the use of conservation laws to 
make pr^edictions. It will also introduce a method 
of using gr^aphs that is v^ery similar to that used by 
engineer's and physicists. Although the computa- 
tions and plotting may be assigned as homewor^k, 
it is important that you go thr^ough the analysis of 
the gr-aph carefully with the class so that they will 
appr-eciate the full significance of the curves they 
have dr^awn. Data from one of the collisioris you 
photogr-aphed could be used. However, if you feel 
that students would get bogged down in arithme- 
tic, you could give them easy r'ound number's, such 
as those used in the example below^. 

First; ask students to assume only that momen- 
tum is conserved in a collision. For example, let m^ 
= 1 kg, m„ - 2 kg, v^ = 0, and v„ = 10 cm/sec. 
Then, 

^aKv + ^B^/,, = m,,v,' + m„v,/ 
+ 20 - v/ + 2v„' 
V,' = 20 - 2v,/ 



Hav^e the students assign values for- v,,' and con\- 
pute v/: 






+ 20 


'5 


+ 10 


10 





15 


-10 


-5 


+ 30 



Plot these points as in the figure below. 




Point out that ther-e is an infinite number- of val- 
ues for v,/ and v^' that would satisfy the above 
equation, and ask if all of them might actually be 
observed if the experiment were done enough 
times. Make sur-e that the significance of positive 
and negative values is clear-. For example, ask 
students to predict from their gr-aphs what the 
velocity of the smaller cart would be if the 
large cart r-ebounded with a speed of 10 cm/sec 
(v/ = 40 cm/sec I. 

Next, ask students to assume only that kinetic 
energy is conserved in this collision. 



+ 200 



V," + 2v 
200 - 2v 



Constr'ucting a table of xalues for- v,,' and \\' is 
now a little more time-consuming because of the 
squar-e roots. However-, a number of shor-tcuts ar-e 
available. Students might plot values of ,v~ as a func- 
tion of ^, dr-aw a smooth curve, and use this as a 
simple computer-. 

Again, any number of values for- x,/ and v^' can 
be calculated, but most of them are physically 
meaningless. The table of values could be con- 
str-ucted as follows: 



2v'' 












200 


±14.1 


± 2 


4 


8 


192 


±13.8 


± 4 


16 


32 


168 


±12.9 


± 6 


36 


72 


128 


±11.3 


± 8 


64 


128 


72 


± 8.5 


±10 


100 


200 





± 8.5 


±12 


144 


288 


-88 


(imaginary) 



ADDITIONAL BACKGROLWD ARTICLES 



193 



Plot these values of v„' and v^' on the same axes 
as the preceding graph. An equation in the form 

results in the curve called an ellipse. The equation 
resulting from the conservation of energy in the 
collision has this fonn. Students may be familiar 
ifrcjm Unit 2) with the mathematics invoked her«, 
but may not appreciate that these cur\'es can be 
used to infer a number of interesting facts about 
real physical events. 

Use the key letters on the graph below to develop 
the follouing ideas: 










Point A: Any point lying on the straight line rep- 
resents values of Vp' and v^' for which momentum 
is conserved. 

Point B: Any point lying on the ellipse represents 
values for which kinetic energy is the same after 
the collision as before. 

Point C: For any point lying outside the ellipse, 
the corresponding kinetic energy of the carts 
would have to be greater than before the collision. 
In other words, kinetic energ\' was added to the 
system. This could not happen in this kind of col- 
lision; it would reciuire that an explosion be set off 
during the collision. 

Point D: For any point inside the ellipse, the total 
kinetic energy of the carts after the collision is less 
than before. In other words, kinetic energ\' was lost 
from the system, ft)r example, by friction between 
the bumpers. 

Point E: This point on the ellifise r-epr-esents the 
case in which the cart with velocity v„ goes past or 
through the other cai1 ha\ing velocity v^ = with- 
out ex«Mling any foree on it. This case may seem 
trivial or physically meaningless, but a comment 
aliout neutrinos may be in order here. 

Point F: This point on the ellipse represents the 
only values for- v,,' and \\ lother than case Ei for 
which both momentum and kinetic energv are 
conserved, as shown by the inter-section of the two 
cur-ves. if students used starting data from one of 
th(Mr photographs, then should see wheri" ihi' val- 
ues for- V,,' and \\ obtained frx)m their photograpli 
plot on th(! graph 

Point (i; Ibis point on the line shows \„' and v / 
sj't for- a collision in which the carls stick togrtbei 



after the collision. In other words, v^' = v^'. Since 
momentum is always conserAed, the point must 
fall on the straight line, and so it falls inside the 
ellipse. Kinetic energy was lost, so this represents 
an inelastic collision. Points along the line segment 
FG represent all possible \alues of \ „' and v^' for 
which the cart with v^ = 10 cm/sec pushes the 
other ahead. The line segment GE represents all 
Vcilues for- which the first cart overtakes or passes 
thr-ough the second one. 

If the values of v„' and v^' obtained from the 
photograph do not plot at point F, ask the students 
for possible reasons, ilt should fall in the region 
FGi. 

FEEDBACK 

An enormous difference often seems to e.xist be- 
tween the practical world of applied science and 
engineering and the idealized world of pure sci- 
ence and theoi-etical engineering. This is because 
it is often difficult to describe commonplace phys- 
ical situations theoretically by e.xact mathematical 
relationships, while simple quantitative theoretical 
ideas, such as fiictionless surfaces, constant speed, 
point sources, pure sine waves, and a host of other 
theoretical concepts may be difficult to produce 
experimentally. Usually, we must be satisfied to 
produce these basic physical or mathematical 
ideas to within acceptable limits of error. To do 
this, we often make use of a system of control 
called /eedbac/c. In attempting a definition of feed- 
back, let us consider a situation that any motorist 
meets and solves daily. 

Suppose the driver of an automobile on a lightly 
traveled superhighway is asked to hold a speed of 
90 km/hr for 5 min. Assuming that the driver co- 
operates as much as possible, the chances are that 
it will take quite a bit of manipulation of the gas 
pedal to keep the car s speed constant to within a 
few kilometers per hour of 90 because of the dif- 
ferent rxjad conditions to be encountered in 5 min. 
For example, the car will tend to slow up when 
climbing hills unless it is given more gas and v\ill 
tend to speed up on going downhill unless the 
driver lets up on the accelerator. \ arious road sur- 
faces also require different accelerator settings. 
Thus, we realize by experience or observation that 
a motorist who attempts to maintain a constant 
speed is continually moving the accelerator by 
small amounts. In engineering terms, we describe 
this sitiration by saving that the motorist fomis a 
feedback svstem for the car and attempts to keep 
the car moving with constant speed under a 
changing load ithe hills and rtiad sirrlace bv ob- 
seiAing the deviation fr-om the desired output sig- 
nal itln' 90 km hr on the sp(>edometeri and then 
corr-ecting the input signal the tlovv of gas con- 
trolled bv the acceler-iitori in sirch a way as to bring 
the output signal back to the desirvd value i90 km 
hn. The pr-ocess just described is the essence of 
what we mean when we use* the term fredbatk in 
a technical sense 



1»4 



t'MI a THE TRIl'MPH CIF MECH/WICS 




lOa (sccocdsj 



Simulated graphs of (I) automobile speed and (II) vacuum 
gauge readings as a function of time. The vacuum gauge, an 
indicator of gasoline flow to the carburetor as governed by 
the accelerator, is a fast response instrument compared with 
the SF)€edometer. This difference in response contributes to 
phase differences in the two curves. Though simulated, the 
values are realistic. 



There are some other things about a feedback 
system that are characterized by thiis situation For 
one it is quite e\ident that the more cioseK the 
motorist tries to hold the car s speed to 90 km hr 
the more tiring it is. We can sa\ that a tighth cou- 
pled feedback system one allowing onl\ small de- 
xiations from a given \alue generally requires a 
greater e.xpenditure of energ;\- than a system in 
vv+iich larger de\iations are allowed. Secondly we 
note that the fluctuations in the speedometer and 
the \ariations in the accelerator position do not 
occur together because it takes the dri\er and car 
a certain amount of time to react to the output 
signal. Thus, we see that the simple mathematical 
statement v — 90 km hr used to describe the 
speed of the car in a theoretical problem ma\ in 
practice require good equipment concentration 
and skill to fulfill. 

In engineering one often wishes to dispense 
with the human element replacing it with some 
ph\ sical de\ice that doesn t tire \et performs its 
function in the feedback loop. To show in some 
detcul how feedback operates in a ph\ sical s\ stem 
we shall choose a simpler and more direct e.xample 
than that of the automobile and driver. Let us con- 
sider how we might meter and control the flow of 
a liquid in a completeh automatic way. In this 
metering svstem. the objective will be to maintain 
a constant flow of liquid under a pressure \ ariation 
stemming from a change in the height of a liquid 
in a tank at the high pressure side of the tube. 

In order to m<ike a liquid metering device, theoiy 
shows that all that is needed is a tube of constant 



length and uniform bore. For this tube and a spec- 
ified liquid the volume of liquid passing through 
the tube in a given time is directlv proportional to 
the pressure difference across the ends of the tube. 
In a common arrangement one end of the tube is 
open to the atmosphere and the other end is fed 
from a column of Uquid of some height H. .As the 
pressure on the side w+iere the liquid flows out of 
the tube is atmospheric pressure while the high- 
pressure side is at atmospheric pressure plus a 
term proportional to the height of liquid the pres- 
sure diflference across the tube is directlv propxir- 
tional to the height H of liquid. From this analvsis 
we see that by maintaining a constant height of 
liquid feeding the tube a constant flow of liquid 
through the tube is assured The problem therefore 
resolves itself into finding a means for keeping H 
constant as liquid flows through the tube. 

The classical wav of solving this problem is 
through use of a weir. This arrangement is shown 
in Fig. 1. The weir is the open tank labeled \\ The 
metering sv stem is the horizontal tube of length D. 
The weir maintains a constant liquid level at hei^t 
H abov e the tube bv means of an ov erflow pip>e the 
curved tube inserted in the tank Liquid flows 
through the supplv pipe entering the svstem at 
the point labeled input and flows out into the weir. 
From the weir the liquid leaves the svstem at the 
output point after having first flowed through the 
metering tube the tube of length D. .-Vfter the sup- 
plv pipe fills the weir to height H this level is main- 
tained bv allowing the excess liquid to run out of 
the overtlow pipe as waste Once this condition is 



ADDITIO.VAL BACKGROL \D ARTICLES 



195 



INPUT 



INPUT 




OUTPUT 




Fig. 1 Liquid metering system 



Fig. 2 Liquid metering system with feedback 



reached, the output thiough the metering tube is 
constant for all variations of input through the sup- 
ply tube, provided there is always some liquid flow- 
ing out the overflow pipe. 

This system for metering a liquid is very precise 
and as an apparatus for research concerning the 
viscosity of liquids, for example, yields definite re- 
sults. For less exacting uses, such as in the meter- 
ing of hw.\ to the carburetor of an internal com- 
bustion engine, it is more comenient to dispense 
with the over-flow pipe and de\ise a system where 
the height of liquid in the tank is controlled auto- 
matically by a valve in the supply line. Two ways 
of accomplishing this are suggested, both of which 
use feedback. One t\'pe is purely mechanical and 
the other is electix)mechanical feedback. The aim 
here is to illustrate the concept of feedback, not to 
design the most practical system possible. 

In Figs. 2 and 3, the height of liquid is deter- 
mined by a float-controlled valve. I'he simplest of 
the two arrangements is shouTi in Fig. 2, where a 
float is suspended at one end of a beam balance 
and valve V is suspended at the other end. The 
float and \'al\e i shown here as a simple plate dam- 
ming th(! li(|uid in the su|)ply pipei an' adjusted at 
a certain le\el M for a fixed pn^ssure on the injiut 
side of the supply line. I'he subsequent action of 
the float and \al\(' is such that when an increase 
in pn\ssuit\ and hence incrt'ase in flow of the licj- 
uid, occui-s in the supjily line, the resulting flow 
through the \alve leads to an increase in height H 
in th(^ tank. 1 his causes the float to rise. Since it is 
connc'cted to th(> opposite end of a beam balance. 
Ih(' rise of the float depi-esses the val\e, decreasing 
the flow of li(|uids and leading to a fall in the litjuid 
level n When this lewl falls the lloat falls and the 



INPUT 




Fig. 3 Liquid metering system with electromechanical feed- 
back 



\al\e opens. This compensating action tends to 
keep le\el H constant, because the float always 
mo\ es the val\e in such a wa\- as to keep the liquid 
suiiace at a fixed height The float and \al\e con- 
stitute a mechanical feedback s\stem 

Figure ;i is an electromechanical ada[)tation of 
Fig 2. In this feedback s\stem lloat F and \a\\v: \' 
are linked electricalK . not m«*chanicall\ Ilie change 
in the height of the float changes tlu' i-cM^t.mcc 



196 



l'\IT 3 / THI': THIl'MPH OF MKCH/WICS 



and hence the current, in the electrical circuit 
shown. A change in current also occurs in the coil 
L, changing the strength of the magnetic field in 
the coil. By controlling the magnetic Held of a coil 
that is part of a motor or relay, one controls the 
position of a valve connected to the motor or relay 
and hence contixjls the flow of liquid in the supply 
line in the same sense as in Fig. 2. Ihe advantage 
here, of course, is that F and \ need not be close 
to one another, which makes this arrangement the 
more flexible feedback s\stem. 

Historically, feedback goes back to the fl\ -ball 
governor on James V\'att s steam engine. In order 
to keep this engine n.mning at constant speed un- 
der differing mechanical loads, it was necessaiy 
to control the amount of steam admitted to the 
cylinder containing the piston. Ihis control was 
achieved b\' linking a \al\e in the steam line la vahe 
acting like the gas pedal of a can to a fl\'-ball gov- 
ernor ixjtated b\' the shaft of the engine. This gov- 
ernor operates on the same principle as a stone on 
a string. That is, when a stone on the end of a 
string is swTjng in a horizontal circle at constant 
speed, the angle between the string and the vertical 
increases as the speed of the stone increases. On 
the flv-ball governor, this change in angle with 
speed can be used to control the action of the v alv e 
in the steam line. 

Since Watts time, and especialK now, feedback 
has become an important and sophisticated part 
of the structure of modern devices, ranging from 
the oscillators in tinv radio sets to the automatic 
pilots of our largest jet airliners. 

A METHOD FOR CALCLTATIXG THE 
PRESSLT^ OF THE ATMOSPHERE 

Assume any cross-sectional area A for a column of 
height h and liquid densitv' p. The volume of the 
column of height h will be Ah, the mass will be p\' 
— pAh, and the weight will be ma^ = (i-\ha„. The 
pressure on the bottom needed to support the col- 
umn will be the weight divided bv the area: 

pAha 



P = 



A 



Pha^ 



Using mks units, p = 1.4 x 10\gm^, h = 0.76 m, 
and a^ = 9.8 m/sec', so 

- kg m 1 - 2 

p = 10" -^— ^ ^ 10 \Vm . 



You may prefer to find numerical values for each 
step instead of waiting until the last step. 

The detailed historv of Bovle s law is not impor- 
tant in itself. It takes on importance in this chapter 
because it can be used to test models of gases. It 



should be made very clear that PV = constant is 
an empirical rule, summarizing the data of many 
experiments. 

You might want to give students a rough idea of 
the range of pressures it is possible to obtain in 
earthly laboratories: fi-om 10 ~ " atmosphei-e or even 
less in the best vacuum pumps to 10' atmospheres 
or more in special high-pressure apparatuses. The 
initial pressure pixjduced by a hvdi-ogen bomb is 
on the order of several billion atmospheres. 

The Newtonian mechanics of collisions can be 
used to derive the relation between P and V for a 
collection of perfectly elastic, vanishingly small 
particles. The product of pressure and volume is 
proportional to the total kinetic energv' of the par- 
ticles. This idealized model does not appear to 
match real gases, however, because Newtonian 
mechanics shows also that the kinetic energv', and 
therefore the product PV, changes when the vol- 
ume is changed. 

It is essential that students distinguish between 
the empirical relation, P\' = constant i Boyle's law), 
and the hypothetical relation derived from a sim- 
plified kinetic model, PV = -/iXiKEi. The latter ap- 
pears not to agree v\1th the former, because a very 
simple analysis of moving a piston to compress a 
sample of gas shows that KE does not remain con- 
stant. 

V\'e can claim that the two agree only if we can 
show why KE should remain constant. The prob- 
lem is resolved when the constant KE ' of the hy- 
pothetical relation is claimed to be equivalent to 
the constant temperature ' of the empirical rela- 
tion. This solution might be spotted by students, 
manv of whom have long been exposed to the 
identification of molecular motion and tempera- 
ture, especially if the conditions are emphasized 
in your presentation: "PV = constant if the tem- 
perature stays constant : PV = -/aiViKE) if the KE of 
the particles stavs constant ." 

The statements about absolute zero ai-e correct 
but they fail to make explicit two qualifications that 
you may wish to present briefly to students. The 
first is sometimes called the third law of thermo- 
dvnamics: It is impossible for anv process to re- 
duce the temperature of a system to absolute zero 
in a finite number of steps. The approach to ab- 
solute zero becomes progressiv ely more difficult as 
the temperature nears zero, so that each succes- 
sive step becomes smaller. Classically then, abso- 
lute zero may be approached as closely as desired, 
but can never be i-eached exactlv. .Another qualifi- 
cation, the result of quantum mechanics, requires 
that a system have a finite least energv', the zero 
point energy ": thus, even at a theoretical temper- 
ature of absolute zero, the particles of the system 
would not have zero kinetic energy. 



ADDITIO.NAL BACKGROUND ARTICLES 



197 



Brief Description of Learning Materials 

SUMiMAHY LIST OF UNIT 3 MATERIALS 






Experiments 

K3-1 Collisions in One Dimension. I 

E3-2 Collisions in One Dimension. II 

E3-3 Collisions in 'I\vo Dimensions. I 

E3-4 Collisions in Two Dimensions. II 

E3-5 Conservation of Energy. I 

E3-6 Conservation of Energy. II 

E3-7 Measuring the Speed of a Bullet 

E3-8 Energy Analysis of a Pendulum Swing 

E3-9 Least Energy 

E3-10 Temperature and Thermometers 

E3-11 Calorimetry 

E3-12 Ice Calorimetry 

E3-13 Monte Carlo Experimerit on Molecular Col- 
lisions 

E3-14 Behavior of Gases 

E3-15 Wave Properties 

E3-16 Waves in a Ripple Tank 

E3-17 Measuring Wavelength 

E3-18 Sound 

E3-19 Ultrasound 

Demonstrations 

D33 An inelastic collision 

D34 Predicting the r-ange of a slingshot 

D35 Diffusion of gases 

D36 Brownian motion 

D37 Wave propagation 

D38 Ener-gy tr'ansport 

D39 Superposition 

D40 Reflection 

D41 Wave trains 

D42 Refraction 

D43 Interference patterns 

D44 Diffraction 

D45 Standing waves 

D46 'I\vo turntahle oscillators (beatsi 

Film Loops 

L18 ()ri(!-Dimensional Collisions. I 

L19 One-DinuMisional Collisions. II 

L20 Inelastic One-Dimensional Collision 

L21 'I\vo-Dimensional Collisions. I 

L22 'I\vo-Dimensional Clollisions. II 

L23 Inelastic IVvo-Diniensional Clollisions 

L24 Scattering of a Cluster of Objects 

L25 Explosion of a CMirster of Objects 

L26 Finding the Speed of a Rille Ikillct I 

L27 Finding the Spe(>d of a Rille Bullet II 

L28 Recoil 

L29 C:olliding Freight Cuis 

L30 Dynamics of a liilliard Ball 

L31 A Method of Mca.siii iiig Enei-gy: Nails Driven 

into Wood 

E32 (iiaxitalional Potential Ener-g\' 

l,.{;} kinetic l'neig\ 



L34 Conservation of Ener^: Pole \'ault 

L35 Conservation of Ener^': Airx:raft Takeoff 

L36 Reversibilitv' of Time 

L37 Superposition 

L38 Standing V\'aves on a String 

L39 Standing Waves in a Gas 

L40 Vibrations of a Wir"e 

lyll Vibrations of a Rubber Hose 

L42 Vibrations of a Drum 

L43 Vibrations of a Metal Plate 

Reader Articles 

Rl Silence Please 

by Arlhur C. Clarke 
R2 The Steam Engine Comes of Age 

by R. J. Forbes and E. J. Dijksterhuis 
R3 The Great Conser\'ation Principles 

by Richard Feynman 
R4 The Barometer Story 

by Alexander Calandra 
R5 The Great Molecular Theory of Gases 

by Eric M. Rogers 
R6 Entropy and the Second Law of 
Thermodynamics 

by Kenneth W. Ford 
R7 The Law of Disorder 

by George Gamovv 
R8 The Law 

by Robert M. Coates 
R9 The Arrow of lime 

by Jacob Bronowski 
RIO James Clerk Ma.xwcll 

by James R. Newman 
Rll Frontiers of Physics Today: Acoustics 

by Leo L. Beranek 
R12 Randomness and the Twentieth Century 

by Alfred M. Bork 
R13 Waves 

by Richard Ste\enson and R B Moore 
RM What is a Wave? 

by ,Alber1 PLinstein and Leopold Infeld 
Rl,") Musical Instruments and Scales 

In HatAe\ E White 
R16 Founding a Family of Fiddles 

bv c;arleen M Mutchins 
R17 The Seven Images of Science 

b\ Gerald Ilolton 
R18 S(icr}tific CranLs 

h\ Martin (laifiner 



Sound Films (IGmm) 

F17 l.leinents ( Onipounds. and Mi.vtures 
F18 Ihe Peilection of Matter 
F19 Elastic Collisions and Stoivd F.ner-gv 
I 20 Energ\ and W oik 



liiH 



lllVIT 3 / THE TRIUMPH OF MECH/VIVICS 



Transparencies T24 

T19 One-Dimensional Collisions T25 

T20 Equal Mass Two-Dimensional Collisions T26 

T21 Unequal Mass Two-Dimensional Collisions T27 

T22 Inelastic Two-Dimensional CoUisons T28 

T23 Slow Collisions T29 



The Watt Engine 
Superposition 
Square \\'a\e Analysis 
Standing Waxes 
'I\vo-Slit Interference 
Interference Pattern Analysis 



FILM LOOPS 



Quantitati\e measurements can be made with film 
loops marked iLabi, but these loops can also be 
used qualitati\'ely. 

L18 ONE-DEVIENSIONAL COLLISIONS. I 

Slow-motion photography of elastic one-dimen- 
sional collisions. iLabi 



L18 OXE-DEVIEXSIOXAL COLLISIONS. 

A continuation of the preceding loop. I Lab) 



L20 



II 



LXELASTIC ONT-DEVIENSIONAL 
COLLISIONS 

Slow-motion photographx of inelastic one-dimen- 
sioniU collisions. iLabi 

L21 T\V'0-DL\IENSIONAL COLLISIONS. I 

Slow-motion photograph\' of elastic collisions in 
which components of momentum along each axis 
can be measured. ILabi 

L22 T\VO-DEVIExNSIONAL COLLISIONS. II 

A continuation of the preceding loop. iLab) 

L23 LXELASTIC TWO-DIMENSIONAL 
COLLISIONS 

A continuation of the preceding two loops; plasti- 
cene is wrapped around one ball. iLabi 

L24 SCATTERING OF A CLUSTER 
OF OBJECTS 

In slow-motion photograph\', a mo\ing ball col- 
lides with a stationary' cluster of six balls of various 
masses. Momentum is conserved. iLabi 

L25 EXPLOSION OF A CLUSTER 
OF OBJECTS 

A powder charge is exploded at the center of a 
cluster of fi\e balls of \arious masses. One ball is 
temporarily hidden in the smoke. The position and 
velocity' of its emergence can be predicted using 
the law of conservation of momentum. (Lab) 

L26 FINDING THE SPEED OF 
A RIFLE BLXLET. I 

A bullet is fired into a block of wood suspended by 
strings. The speed of the block is measured directly 
by timing its motion in slow-motion photography. 
ILabi 



L27 FINDING THE SPEED OF 
A RIFLE BLT.LET. II 

A bullet is fired into a block of wood suspended by 
strings. The speed of the block is found by meas- 
uring its \'ertical rise. iLabi 

L28 RECOIL 

A bullet is fired fitjm a model gun. Direct meas- 
urements can be made of the bullet s speed and 
the speed of recoil of the gun. iLabi 

L29 COLLIDING FREIGHT CARS 

The collision of tvvo freight cai-s is photographed 
in slow motion during a railroad test of the 
strength of couplings. iLabi 

L30 Dl'NAMICS OF A BILLIARD BALL 

Slow-motion photography' of a rx)lling ball striking 
a stationary' ball. The target ball slides, then starts 
to roll. Linear momentum and angular momentum 
are conserved. iLab) 

L31 A METHOD OF MEASITUNG 
ENTRGY: NAILS DRRTN 
EVTO WOOD 

A nail is drixen into wood by repeated identic£il 
blows of a falling weight. A graph of penetration 
depth versus number of blows can be made; the 
result is nearly a straight line. This loop establishes 
a criterion for energy measurement used in the 
next two loops. (Labi 

L32 GRAMTATIONAL POTENTIAL 
ENTRGY 

Dependence of gra\itational potential energy' on 
weight; dependence on height. iLabl 

L33 KINTTIC ENTRGY 

Dependence of kinetic energy' on speed; depend- 
ence on mass. Sloyv-motion photography alloyvs 
direct measurement of speed. iLabi 

L34 CONSER\'ATION OF ENTRGY: 
POLE \'ALT.T 

The total energy' of a pole y aulter can be measured 
at three times. Just before takeoff, the energx' is 
kinetic; during the rise, it is partly kinetic, partly 
grayitational potential, and partly elastic energy of 



BRIEF DESCRIPTIO.V OF LEARNING MATERIALS 



199 



the distoiled pole; at the top, it is gravitational po- 
tential ener^. (Lab) 

L35 CONSERV'ATIOX OF ENERGY: 
AIRCRAFT TAKEOFF 

Flying witli (-onstant ptjwer, an aircraft moves hor- 
izontally at gKHind level, rises, and levels off. Ki- 
netic and potential energy can be measured at 
three levels. I Lab) 

L36 REVERSIBILITY OF TIME 

After some intioductoiy shots of leal-life actions 
that may oi- may not be reversible, the film shows 
events of increasing complexity: a two-ball collision 
on a billiard table; a foui-ball event. Finally, a ball 
rolls to a stop while making some 10"' i invisible i 
collisions vvath the molecules of the table surface. 

L37 SUPERPOSITION 

Amplitudes and wavelengths of two waves are var- 
ied; the resultant is shown. Display is in three 
colors on the face of a cathode-ray tube. 

L38 STANDING WAVES ON A STRING 

Production of standing waves by inteiference of 
oppositely moving equal waves is shown in ani- 
mation. Then a tuning fork sets a string into vibra- 
tion and several modes are shoun as the tension 
is adjusted. The string's motion is also shouTi stro- 
boscopically. 

L39 STANDING WAVES IN A GAS 

A loudspeaker excites standing waves in a glass 



tube containing air. Nodes and antinodes are made 
visible in two ways: by the motion of cork dust, and 
by the cooling of a hot wire inside the tube. 

L40 \TBRATIONS OF A WIRE 

A horizontal stiff vvii-e is set into vibration. The driv- 
ing force is supplied by the interaction of alternat- 
ing current thrxjugh the wire and a fixed mcignetic 
field. Several modes of vibration are shown, both 
for a straight uire lantinode at free end' and for a 
circular wii-e modes equally spaced aitjund the cir- 
cumference). The patterns are shown in real time, 
and also stroboscopically. 

L41 VIBRATIONS OF A Rl^BER HOSE 

A long vertical injbber hose is agitat«'d bv a variable- 
speed motor- at one end. The fi-equencA' is adjusted 
to show a succession of nodal patterns. 

L42 \TBRATIONS OF A DRLTVI 

A loudspeaker is placed beneath a horizontal cir- 
cular rubber dr-um head. Several symmetric and 
antisymmetric modes are shown stroboscopically, 
in apparent slow motion. 

L43 VIBRATIONS OF A METAL PLATE 

Vibration patterns are made visible bv sprinkling 
white sand on a vibrating plate; sand collects at 
the nodal lines. 



.Note: A fuller discussion of each h'llm Loop and suggestions 
for its use will be found in the section of the Hesource 
Book entitled Film Loop Notes 



SOUND FILMS (16mm) 



F17 ELEMENTS, COMPOUNDS, 
AND MIXTURES 

Color, 33 min, Modeiii Learning Aids. A discussion 
of the differ-ence between elements, compounds, 
and mixtur-es, showing how a mixtur^e can be sep- 
arated by physical means. Demonstrates how a 
compound can be made and then taken apar-t by 
chemical methods, with identification of compo- 
nents by means of their physical properties, such 
as melting point, boiling point, solubility, color-, etc. 

F18 THE PERFECTION OF MATTER 

Color-, 25 min, Xuffield Foirndation. This film is 
mainly for- atmosph(!i-(*. A cameo treatment of me- 
ilieval cirlture and science, principallv alchemv . 
Some explicit discirssion of closed systems and 
conservation of mass. 



F19 ELASTIC COLLISIONS AND 
STORED ENTRG\' 

B & V\', 27 min, Modern Learning .Aids. \ arious col- 
lisions between two dr^-ice pucks are demon- 
strated. Cylindrical magnets are mounted on the 
pu(-ks pri)ducing a r-epelling force. Careful meas- 
ur-ements of the kinetic energv' of the pucks during 
an interaction lead to the concept of stored or po- 
tential energv'. 

F20 ENERGY ANT) WORK 

ii \ \\ 2H min Modem [.earning .Aids. Shows that 
woi-k, measuivd as the aiva under- the fotxx»-dLstant^ 
cirrAC, does measuiv the transfer of kinetic enei^' 
to a bodv . calculated frx)m its mass and speed. Sev- 
eral difTeixMit methods establish wor-k as a useful 
measur-e of energV' transfer. 



TRANSPARENCIES 

T19 ONE-DIMENSIONAL COLLISION sion are provitled Measurements mav be made 

laisimiles of strohoscopii- phologiaphs ot two diivctlv lrt)m the transpatvncv to establish the 
events involving two-bodv collisions in one ilimen- piiiK iple of conseiAation of momentum 



zmi 



I'MIT 3 / THE TKIl'IVIPII OF IVtECHAMICS 



T20 EQUAL MASS TWO-DLMEXSIOXAL 
COLLISIONS 

A sti-oboscopic facsimile of an elastic collision he- 
rvveen spheres of equal mass is shown. 0\erla\ s 
show accurately drawn momentum \ectors before 
and after collision, illustrating conservation of mo- 
mentum. 

T21 UNEQUAL MASS Tl^'O- DIMENSIONAL 
COLLISIONS 

A stroboscopic facsimile of an elastic collision be- 
tween spheres of unequal mass, both of which are 
moving before collision. Cherlays show accurateK' 
drawn momentum vectors before and after colli- 
sion, illustrating conservation of momentum. 



T22 LNELASTIC TW'O-DLMENSIONAL 
COLLISIONS 

A stroboscopic facsimile of an inelastic collision 
between two plasticene-covered spheres of equal 
mass, both of which are moving before collision. 
0\erla\' shows accurateK drawn momentum vec- 
tors before and after collision, illustrating conser- 
vation of momentum. 

T23 SLOW COLLISIONS 

A stroboscopic facsimile shows a collision betvv een 
two dv namics carts equipped with spring bumper's 
and light sources. .Anahsis of momentum and ki- 
netic energv' before, during, and after the collision 
may be made directly from the transparency. 



T24 THE W^ATT ENGINT 

Overlav s depict a schematic diagram of the V\'att 
e.xternai condenser engine during the steam-ex- 
pansion and condensation phases of operation. 

T25 SLTERPOSITION 

Shows two pulses crossing at four instants of time, 
fii-st with both pulses above the equilibrium line, 
then with one on either side of the line. The su- 
pei-jjosed wave for each case is also shown. 

T26 SQUARE \\'A\T ANALYSIS 

Shows how first four Fourier terms add to begin to 
produce a square wave. May be used for varietv' of 
superposition problems. 

T27 STANDING WA\TS 

A set of sliding waves pemiits a detailed step-bv- 
step analysis of a standing wave pattern. 

T28 TWO-SLIT LNTERFERENCE 

1 he first tvvo overlavs are concentric circles drawn 
from rvvo sources. The third overlav suggests the 
resulting constructive and destructive interference 
positions. 

T29 LNTERFERENCE PATTERN 
ANALYSIS 

Overlays illustrate crests and troughs for two in- 
dependent sources. Other overlays show nodal 
and antinodal lines and geometry' for deriving the 
wavelength equation. 



Demonstration Notes 



D33 AN INTLASTIC COLLISION 

A perfecth' elastic collision is one in which the total 
amount of kinetic energv present is the same be- 
fore and after the collision. .An inelastic collision, 
then, must be one in which kinetic energy' is lost. 
What happens to it? \"ou can show qualitatively by 
a very simple demonstration that a loss in kinetic 
energy is associated with a rise in temperatui-e of 
the interacting bodies. Pound a nail into a piece of 
wood, and have students touch the nail. Remind 
students that when the\ analvzed a slow elastic 
collision they found that the total kinetic energy of 
the dynamics carts decreased temporarily, then 
went back to near its original value. Ask what 
would happen if the bumpers on the carts had 
instead been made fi-om a soft metal, such as lead. 
Then demonstrate such a collision with the ap- 
paratus described below. 

Bend the lead strip into a ring and tape it to the 
end of a dvnamics cart. 

Do this several minutes before you want to per- 
form the demonstration so that the lead v\ill have 
time to recover fiT)m the heating caused by han- 
dling and bending it 




If two carts, each loaded so that its mass is 2 kg, 
approach each other with speeds of 1 m sec, their 
total kinetic energy is: 

2 X V2 X 2 kg X (1 m/sec)^ = 2 J 

If this enei^' is converted entirelv into heat, it 
amounts to about 0.5 cal. The temperature rise 
in a 50-g strip of lead with a heat capacity of 0.03 
cal/g C° will be 

0.5 cal 



mr 



50 g X 0.03 



= 0.3 C° 



DEMO.NSTRATIO.N NOTES 



201 



Although studenls are not ready for the quantita- 
tive treatment given here, the calculations are 
shown so that you will appreciate the difficulties 
involved in making the experiment quantitative, 

A practical experiment to show the con\ei-sion 
of kinetic energy into heat in a collision requires 
a very sensitive thermometer; the rise in temper-a- 
ture will he only a few tenths of a degree. 

A sensitive thermometer with very low heat ca- 
pacity is made frxim a thermistor la pellet of semi- 
conductor material whose resistance drops mark- 
edly with inci^ease in temperatur-ei and an amplifier. 
A thermistor-, alr-eady embedded in a thin strip, is 
supplied by Damon Educational. An identical ther- 
mistor-, not embedded, is provided for other- dem- 
onstrations and activities. 

Briefly, an incrt;ase in temperature of the ther- 
mistor increases the input current to the amplifier-, 
which increases the output current many times 
mor-e. Small changes in the large output current 
(and, therefore, small changes in temperatur-ei can 
be detected by blocking out most of the output 
current with the OUVPUl OFFSE I control and con- 
necting the output to a sensitive meter. 



Pov 



Sm 



n, 



V9-8 . 




Connect the thermistor between a 1.5-\' cell and 
the input terminals of the amplifier. Turn the ac/ 
dc switch to dc. Set the OLriPLn OFFSEl contr-ol 
to its maximum Ifull counterclockwise) and the 
gain to about half-way up. Connect a meter acr-oss 
the output of the amplifier. It is most convenient 
to use a multimeter with several voltage ranges. 
Start on the least sensitive (highest voltagei dc \olt- 
age scale. The r-eading shoirld be near zerx): if it is 
not, adjust the gain contr-oi until it is. Change to 
progressively mor-e sensitive scales, using the gain 
contr-ol as necessary to keep the r-eading on scale. 

Finally, with the meter on a scale of not more 
than 250 mV or- 100 |JlA firll scale deflection, adjust 
the gain or Oil TPl'T OFFSKT so that the r-eading is 
appr-(),\irnately zerx). Final adjirstment is \ery sen- 
sitive, and it is not necessary to set to zero e\actl\ . 
If you do not have such a sensitive scale on a mul- 
timeter-, sirbstitirte an independent millivoltmeter- 
or microammeter-, but always use the coar-ser 
scales of the multimeter fir-st, so that the sensitive 
meter will not be overloaded. L'se a projection me- 
ter if you can. 

With the meter- r-t'ading apprxjximately zerx), in- 
cr-('as(' the temperatur-<> of the iher-mistor b\ bring- 
ing vour finger-, a hot ir-ori, or- a match dose to it 
(but not a(-lually touchingi. I her-e should be an ap- 
preciabU' iiKr-ease in m(!ter r-eading. Kvcn blowing 
on the tlKM-mistor- mav send tlu« r-t'ading ofT stale 



L'se one of these techniques to show that the sys- 
tem r-esponds to temperature changes and that an 
increase in temperature increases the meter r-ead- 
ing. Arrange the apparatus so that the wires from 
the amplifier-power supply unit to the cart are 
hanging fr-eely. Wait for the meter reading to reach 
a steady value before doing the collision demon- 
stration. Push the carts together so that the lead 
ring absor-bs the kinetic energv' impact. The meter 
deflection shows the increase in temperature of 
the lead. If you plan to calibrate the thermistor for 
quantitative work later, record the meter readings 
from this demonstration for future reference. 




In a trial, we found that when a cari moving at 
about 1 m sec collided with a stationary object ithe 
walli, the temperature rise in the lead caused a 
change of 50 mV in meter reading on a 250-m\ 
meter, or 30 yiA on a 100-^A meter. 

Suggestion for Quiz, or Class Discussion 

In order to show that the meter deflection is not 
simply due to mechanical shock las one might sus- 
pect with such a sensitive instrument i, you ran tiA' 
putting the lead strip on a br-ass block and hitting 
the block with a hammer. In this case, there will 
be no temperatur-e rise. 

D34 PREDICTING THE RANGE 
OF A SLINGSHOT 

In this ex|}erimerit, the imjiact point of a slingshot 
pr-ojectile is pr-edicted fr-om the drawing foire and 
distance. The objectives are to provide Hi an ex- 
erx:ise in energv' conser^ation that both can land 
Willi engage the students intuition, and i2i an ex- 
perience of successfully predicting from the dry 
machinery of theoretical mechanics an tn-ent that 
is inter-esting to the stirdents. Ihe derivation of the 
expression for range rvquir-es some analvsis not 
tr-eated in the text: r-esolirtion of vectors, and the 
work done bv a varying foive. 

I»rocedure 

.\ satistattoiy sling can be made from a lai^^e rub- 
ber- band. I'o insirr-e a rvasonablv small er-rx)r in 
measirring the length of draw the rirbber band 
should allow a tlr-.iw of at least 20 (tii withoirt 



Z02 



UNIT 3 / THE THIIIMPH OF MECH/WICS 



overstraining. The support can be almost amlhing 
that is sufficiently rigid: a pair of ring stands 
clamped to the table will suffice. The rubber band 
should be attached to the supports in a manner 
that will allow a minimum of friction during diaw 
and release. i.Altemati\el\-, we ha\e found that a toy 
shop model slingshot firing a 2-cm steel ball gi\es 
ver\' satisfactory' results.' 

An excellent projectile can be made by twice 
folding a 2-cm x 12-cm piece of 0.2-cni thick lead 
sheet. Its mass should be about 50 g, great enough 
so that only a negligible fraction of the kinetic en- 
erg\' will appear in the njbber band upon release, 
but small enough to gi\e an impressi\el\' long 
range. Be careful to fold the lead in such a way 
that it will not catch on the njbber band when it 
is released. The margin of error is adequateh' rep- 
resented land the drama is increased) by placing 
a wastebasket at the expected impact point. (See 
Fig. 1.1 



^'^ 




Derivation of Range R: 
R = vt 



R = 2\ t , 



R = 



R ^ — 

a„ 



2£. 



R = 



R = 



t = 2t 



tr = 



2Fd^. 



F„.. d„ 



I for 45° launch angle*) 






(for linear force) 



(1) 
12) 



(31 



(4) 



(5) 



(6) 



(7) 



Rg. 1 

The draw and release can be made satisfactoril\- 
with the thumb and forefinger on the edges of the 
projectile, but some skill is required to a\oid fac- 
tional losses on release. .\ thread-burning tech- 
nique such as that suggested b\ Fig. 2 ma\' pro\e 
to be better. The drawing force and distance are 
measured with a spring balance and meter stick, 
as indicated in Fig. 2. iThe meter stick should be 
remo\ed for launching.! 



Po5iTlOf\l) 




The deri\ation is lengthy; but, even so, if vou 
know each input principle and \our math is cor- 
rect, \'ou should ha\'e confidence in your predic- 
tion. 

With the exception of equations i4i and i5". these 
relations all represent idealizations. .As it happens, 
all the actual de\iations from the ideal are in a 
direction that reduces the actual range. Onl\ the 
step represented by equation i7i allows the de\ia- 
tion from the ideal to be accounted for: students 
can plot F against d and find £p^^ lor Fi from the 
area under the somewhat nonlinear curve. But re- 
member that the work done b\' a \ar\ing force is 
not considered in the Te.vf. ProbabK' the best wax- 
to treat a nonlinear force— extension cur\e is to re- 
place it by an equivalent bar graph. iSee Fig. 3.i 



i 





/^•a,*(l3->Qj 



Hg. 3 



Fig. 2 



•Vector resolution is not treated in the te.xt. but should be 
familiar ftx)m problems. 



DEMONSTRATION NOTES 



203 



The most a|j|)r()|)iial(; approach to the cierivation 
f()|- a pailicular student must he (letrMininecl h\ the 
t(!a(h(!r. Helow are listed some approaches to sug- 
f^est th(! hroad rarif^(! ol possihilities. Anv one ap- 
proach would n(Jt ha\e to he used lor a uliole 
class. For example, a|)|)roach "C" miglit he planned 
foi- most of the class while several faster students 
could hegin as in the apfjroach "A.' hi any case, 
the lationale lor hotheiing with pi-ediction instead 
oljust tiyin^ the; ex|)erim(;nt should he clear in this 
era ol prohing space with (;\tremeK' expensive ma- 
chineiy. 

i'osMble Approaches 

A. Students are retjuc^steid to derixe an exjires- 
sion for the range of a proj(Htile as a homework 
assignment, pcMhaps o\er a week or longer. J he 
teach(!r is availahle to discuss prohlems. 

li. As in A, hut tin; prohl(!m is gi\f!n hetter initial 
direction hv a class discussion of Wliat would you 
have to know in order to liguie out iiow far it will 
go?" 

C. A discussion is hegun as in B, hut it is pur- 
sued thmugh the entire derivation, with sugges- 
tions for steps coming almost always from the stu- 
dents. I'You say you want to know the; initial 
speed. How could you find it?' i 

D. As in C, hut with the teacher- prxniding the 
structur-e by means of a concrete leading c]irestion. 
( "We can find tiie speed if we know the kinetii- 
energ\'. V\'hat is the r(!lation of \ and K7J? 

E. L(!Cture presentation of a derixation. 

The more that students are ahle to ('ome up with 
on their- own, the; more \alual)le the experiment is 
likely to hv. for- thjMii. (Jn the oIIkm hand, |)ainfull\ 
heavy demands on the students w ill spoil the (effect 
of tlie demonstration and the time invohenl must 
always he weighed. In any appr-oach, it should 
eventually he pointed out that the procedure is one 
very common to science. iSee Fig. 4.i 









Fig. 4 



An inlei-esting supplenxMitaiA discussion might 
center- on the (|uestion. What do vou do when the 
actual event doesn I match th(> pr-eiliction.' .\nv 
discussion of the uncertaintv in the predicted lo- 
cation of the impact |)oirit should precede the 
ailual launching \n addition to uncertainties in 



measuiTement, the effects of the idealizations shcjuld 
be considei-ed. 

i he amtjunt of derivation required of the stu- 
dents mav vary a gr-eat deal; whatever treatment of 
the experiment is used, the intent is that the ex- 
periment he predictive and interesting. I o this end. 
it is important that the actual launching be the \ei\ 
last step and that the anticipated range be as long 
as is prac^ticable. 

I Courageous teachei-s who choose approaches 
U or- K might want to tiy a dramatic tack wherein 
an erroneous step is surreptitiously included in 
the derivation or- calculation, causing the pr-ojectile 
to over-shoot the target hv a factor of two or more, 
per-ha|)s pr-cjci^eding out a door or windovN'. The 
immediate drop in the stock of science must, of 
course, he quicklv recouped, ideallv hv the stu- 
dents di.scovering the error themselves. i 

D35 DIFR^SIOX OF GASES 

1 he forniation of a (loud of .\H,Cl vividlv demon- 
strates the diffusion of HC;l and .Ml, molecules 
through the air. 



/y^^ct CUU 



iitifftf 




^' "rrhfif'i*. ^' 



The dcMiionstration also shows: 

lai rhe lighter- .\H, molecules diffirse more tiuicklv 
than the heavier- lii'A molecules: The cloud of 
\H,C1 is fomied farther frtjm the \H, sourre. 
ihi Diffusion is comparativelv slow at atmos- 
pheric pressure. 

D36 BROU'XIAX MOTION 

The molecirlar-kinetic- theorA of matter- developed 
in Chapter 11 as a model for- a gas is consistent 
with manv experimental ohserAations igas laws 
s|)ecilic luvit, etc.". Br-ownian motion is the most 
dir-ect ev idence we have been able to pr-esent so far- 
lor- the molecular-kinetic- theory of matter-. Ihe 
phenomenon is called iirxnvriian becairse the bot- 
anist Kobert Br-owri. althoirgh not the lii-st to ob- 
serve it. showed icirx-a 1827i that it was found for 
a wide variet\' of particles, both or>;anii- arui inor- 
ganic. 



liisluriral Si^iiifiraiice of Urouiiiaii Motion 

Brown himself had no theory to account lor- Ihe 
motion, lie fomul that it existed for- all kinds of 
inorganic" particles, antl var'ious suggestions in- 
cluding the irregular- mol(>cirlar- bombartlmcMit of 
the parlicle wer-i* made to arcoirnt for- it. iln lact 
iirown used the per-sistence of the phenomena in 
isolated dr-ops to prove that it could not be due 
tc molecular activilv Birt until work bv Linstein 



204 



DMT 3 Till-: THIl'MI'ii Ol' MIICIIAMCS 



and Smoluchowski and In Feriin eaii\ in the twen- ti\e pit;dirtions iibout the ohsetAed motion of the 

tieth centuiA', se\eral eminent scientists still dis- particles that wei-e closeK' confiinieti In Penin s 

puted the existence of atoms. measurement. One of the achie\ ements of the the- 

Einstein and Smoluchowski. using the mathe- or\ was the first accurate determination of A\o- 

matics of probabiiit\\ were able to make quantita- gadro's number. 



A i\OTE CO\CER\Ii\G DEMOXSTRATIOXS 
AND EXPERIMENTS IN CHAFI ER 12 



Chapter 12, perhaps more than an\' other chapter 
in the unit, depends upon demonstrations and e.\- 
periments to gi\'e substance to the material dis- 
cussed in the 7"e.vr If vou e.xamine the chart that 
follows, \'ou will notice that the wa\e properties 
are carried entireh' b\' either the e.xperiments or 
the demonstrations. If you ha\e the time and wish 
to do Experiments E3-15 through E3-19, it is rec- 
ommended that \ou onl\' do demonstration D41. 
If onl\' E3-15 is done b\' the students, then D38, 
D41, and D42 are recommended. If onl\ E3-1S and 
E3-19 are carried out, D37, D39. D40. and D41 
should be considered. If no experiments are 

E3-15 

E3-16 E3-18 



planned, the demonstrations can suffice. In an\ 
case, D46 should be included. 

The chart below lists \arious pixjperties of waves 
that are demonstrated by the e.xperiments and 
demonstrations for Chapter 12 and the types of 
etjuipment that ma\' be used. 

Equipment Key: 

S — Slinky 

B— Bell \Va\e Machine 

T — Turntable Oscillator 

R — Ripple Tank 

P — Project Physics Equipment 



Wave Property 


E3-17 


E3-19 


D37 


D38 


D39 


D40 


D41 


D42 


D43 


D44 


D45 


046 


Pulse amplitude and length 
Pulse velocity 


S.R 
R 




S,B 
S,B,R 




















Traveling pulse as energy 

transport 
Absorption at a barrier 




P 




S,B,R 
B,R,P 


















Superposition 


S 








S,B 














T 



Reflection from a free end S 
Reflection from a fixed end S 

(phase inversion) 
Partial reflection at an S,R 

interface where v. = Vj 
Two-dimensional reflection R 

from barrier 



S,B 
S,B 

S,B,R 

R.P 



Transmission (impedance, 
or index match) 



Wave trains (reflection, 
refraction, etc.) 



B,R,P 



Refraction as a function of 

angle 
Refraction as a function of 

velocity 
Refraction through wave 

shaping elements 

(lenses) 



R,P 
R,P 
R,P 



Interference patterns 
Young's experiment 



R,P 
R 



Diffraction around obstacles R 
(frequency dependence) 

Single-, double-, and R 

multiple-slit diffraction 
patterns 



R,P 
R,P 



Standing waves R 

Longitudinal and transverse S 
waves 



S,B,R.P 
S,B,R,P 



DEMONSTRATION NOTES 



205 



D37 WAVE PROFACiATION 
Slinky 

Hie slinky is employed to demonstrate wa\e prtjp- 
agation because pulses pi-opagate slowly enough 
along it to be easily obsenetl. It can be used either 
by partnei-s or suspended hoiizontally by strings 
attached to a horizontal wire above it. If used by 
pailnecs, pull the slinky out to a length of about 
10 m on a smooth floor. Never let go of an end of 
the spring when it is stretched because the result- 
ing snarl is almost impossible to untangle. The 
snarling pitjblem is eliminated when the slinky is 
suspended pitjpeiiy. The strings suspending the 
slinky should be at least 1 m long, and spaced sev- 
eral centimetei-s apai1 along the entire length of 
the spring. One end of the spring may be tied to 
some support. Stretch the slink\' slightly by fasten- 
ing a light, 2-m string to the other end. 

Pulse amplitude and length can be demon- 
strated by sending different -sized pulses along the 
spring. The amplitude of a pulse may be defined 
as the maximum displacement of any point on the 
spring. Ask questions such as; In what dii-ection do 
the spring coils actually mo\e as the wave motion 
travels along the spring;^ Does the shape of the 
wave change as it travels along the spring? What 
determines the length of the pulse? Is there a re- 
lation between the length of the pulse and its am- 
plitude? Pulse velocity can be determined by meas- 
uring the time it takes the pulse to travel back and 
forth several times over a measui-ed distance. How 
is the velocity affected by changing the amplitude.' 
the pulse duration? the tension? What detemiines 
the pulse velocity? 

Transverse waves can be demonstrated bv grasp- 
ing one end of the spring and snapjiing it rapidiv 
at right angles to the length of the spring. Longi- 
tudinal waves are brought about by displacing the? 
end of the spring in a direction parallel to its 
length. A logical question to pose is: As the wave 
travels along the spring, in what dir-ection do the 
spring coils actually move? 

Bell IVave Machine 

txp. 1, (ietting Acquainted v\ith Waves." and 
txp. 4, "Wave Speed," in the book Similarities in 
Wave Behavior. Ihis book accompanies the Wave 
Machine, whtMher you i-ent or buv it. 

Kipple Tank 

Straight waves can be generated bv placing a 2-cm 
dowel or section of broomstick handle along one 
edge of the tank, rx)lling it backwaixl 1 or 2 cm, and 
then stopping, or bv (>mploying an electric rippler 
supplied bv most scientiMc apparatus houses. I's- 
ing a hand lor electronic stioboscope. detemiine 
the speed of the waves and verily the relatioti 
V = f\. Stn^ss the fact that this ivlationship is ap- 
plicable to waves both in the ripple tank and on a 
coil s|)iing islinkvi. Define the amplitude as the 
ma.ximum displacenKMit of anv jioint on the sur- 
face Make several measurt'ments of fn'(|uencv and 
w.ivclciigth when vou deteiinine the wave speed 



How is the frequency of your strobe related to the 
fi-equency of the waves? What can be said about 
the accuracy of your determination of the wave 
speed? 

D38 ENERGY TRANSPORT 
Slinky 

Energy transport is concerned with what happens 
to the amplitude of a traveling wave as the wave 
propagates. Inder the assumption that it takes 
"work" to defonn a medium, we associate the en- 
erg\' a wave possesses with its amplitude and ask 
such questions as: Does the shape of the wave 
change as it travels along the spring? Does the 
"size," or amplitude of the wave change? Whv does 
it change? What determines the amplitude of the 
pulse? The progressive loss in amplitude as the 
pulse travels along the spring is called damping. 
What happens to the energy lost? 

Bell Wave Vlachine 

See Kx|j. 2, Wave Damping," and Exp. 3, "Waves as 
Caniers of Enei^gv, in the book Similarities in Wave 
Behavior, which accompanies the Wave Machine. 
Absorption at a barrier can be demonstrated in 
Kxp. 10 where a dash-pot-and-piston arrangement 
is employed for a mechanical load. 

Ripple Tank 

Generate straight-fronted and circular-fronted waves. 
Does the amplitude of the straight-fronted wave 
change as the wave travels'' If so, how?" Whv^ Is any 
energv' lost? Energv absorption at a banner can be 
shown by allowing waves to strike the gauze fences 
and observing what happens to the amplitude. A 
discussion of reflection is not warranted at this 
time. 

Project Phisics Equipment 

The absoi-jjlion of audible sound, ultrasound, and 
micrxnvave Waves can be demonstrated with an 
assorlment of materials, such as pieces of metal, 
wood, glass, st\Tofoam, paraffin, masonite, etc. 
The fact that energ\' is transmitted can be asso- 
ciated with the effect these waves have on various 
receivers 

D39 SITPERPOSITIOX 

Slinky 

Ihe demonstration setup should be the same as 
for I)3T. 

Super-jjosition can be demonstrated b\ generat- 
ing two simultaneous pulses one from each end 
of the slinkv . 

Ask such questions as: What happens to the 
pulses as thev collide' When the pulses meet how 
does the resulting amplitude compar-e with the 
amplitude of each individual pulse when the 
pirlses arx* on the same side of the slinkv ' on op- 
posite sides? 

Bell Wave Machine 

See l.\p ."i ( li.sN-c i-ossinu of Waves" in .S/ni/V.inf/V'.s 
in Wave Mc/i.n ku 



I 



Z06 



UNIT 3 / THE TRIl'MPH OF MECH/WICS 



D40 REFLECTION 

Slinkj 

Reflection from a fixed end ipfiase inversion! is 
demonstrated b\' obsening the reflected pulse 
when one end of the slink\' is held rigidh' in place 
(infinite impedance:. The other case, reflection 
irom a free end izeix) impedance i can be obseived 
by ha\ing the end of the spring connected onl\' to 
a long thin thread. Observe these tvvo cases to see 
whether the displacement of the reflected pulse is 
on the same side or on the opposite side of the 
spring fix)m the incoming pulse. Partial reflection 
from an interface where \ , ~ \-, is then in\estigated 
by t\'ing together two coil springs on which \\a\es 
travel with different speeds. Send a pulse first in 
one direction and then in the other, asking what 
happens when the pulses reach the junction be- 
tween the two springs. B\ employing different 
springs, and thus different media, and observing 
the amplitude of the transmitted and reflected 
wa\es, one can qualitati\el\ demonstrate imped- 
ance, or inde.x-match, in ternis of the media veloc- 
ities. 

Bell Wa\e Machine 

Reflection from fi-ee or fixed ends is demonstrated 
in Exp. 6, while partial reflection at an interface is 
contained in Exp. 11 oi Similarities in Wave Behav- 
ior, which accompanies the \Va\e Machine. 

Ripple Tank 

The speed of water wa\ es depends on the depth 
of the water. Two different depths of water there- 
fore constitute two different media in which waves 
can be propagated. This situation can be brought 
about b\ a glass plate supported in the ripple tank. 
.Ask what will happen if straight wa\es generated 
in deep water, cit)ss the boundary between the two 
media i water depths' invohed. Here we are pri- 
marih- interested in the reflected ' wave. The 
transmitted wa\e and its refraction is presented in 
D42. Two-dimensional reflection fi-om an opaque 
barrier is alwa\s shown using paraffin blocks. The 
angle of incidence and angle of reflection can be 
measured. 



D42 REFRACTION 
Ripple Tank 

Refraction in the ripple tank can be observed b\' 
laying a sheet of glass in the center of the tank to 
make a shallow area. To make the refraction quite 
obvious, the frequency of the wa\e sfiould be low 
I less than 10 cxcles seci, and the water o\er the 
glass should be as shallow as possible. Use just 
enough water to co\er the glass. The waves refract 
at the edge of this ai"ea because they travel more 
slowK' whei'e the water is shallow. The wave that 
passes over the plate is the refracted wave: the 
acute angle between its fi-ont and the boundan of 
the new medium is the angle of refiaction, r. 

Trv- van ing the angle at which the pulse strikes 
the boundarv between deep and shallow water. 
Measure the angles of incidence and the corre- 
sponding angles of refraction over a wide range of 
values and determine how the angle of refraction 
varies as a function of the angle of incidence. De- 
termine the velocities of the wav es in the deep and 
shallow parts of the tank. What is the ratio of their 
velocities.'' of their wav elengths.' Compare with the 
data from angle measurements. Paraffin lenses can 
be cut or plastic ripple-tank lenses can be bought 
to demonstrate the focusing of waves. 

Project Physics Equipment 

Sound. Fill a lab gas balloon with carbon dio.vide. 
I Less dense gases do not work well. The resulting 
spherical lens wiU focus sound a few centimeters 
bev'ond the balloon. Explore the area near the bal- 
loon on the opposite side from the source. Trv- two 
or more frequencies. 

Ultrasound. .At the higher firequency of ultra- 
sound, the gas lens ma\ be too large. Experiment 
with various materials that are transparent to 
ultrasound and can be formed into a sphere, or 
hemisphere. Trv' the gas-filled balloon with this 
higher frequency'. 

A/jcrovvaves. From paraffin wax, cast a hemi- 
sphere or hemi-cvlinder of about 3 cm radius, per- 
haps in a small frozen juice can. It will act as a 
short focal-length lens. Observe the area behind 
the lens with the lens in position and while re- 
moved. 



Project Phv'sics Equipment 

The reflection of sound, ultrasound, and micro- 
waves can be demonstrated by reflecting them 
from an assortment of materials. From D3S you will 
have an idea of what materials provide optimal re- 
flection. 



D41 \\'A\T TRAINS 

Demonstrations of reflection and refraction are 
presented by employing a constant frequency 
source so as to distinguish between a pulse and a 
wave train. 

The Bell Wave Machine, Ripple Tank, or Project 
Phvsics equipment may be used 



D43 INTERFERENCE PATTERNS 
Ripple Tank 

The interference of wav es from two point sources 
is demonstrated best with the ripple tank, since 
the student can see the development of the nodal 
lines as waves progress from the point sources. It 
is enlightening to observe first a single circular 
pulse, then tvvo simultaneously generated pulses. 
Follow the path of the intersection of the two 
pulses. Ne.xt obsene two pulses that originated at 
different points and at slightly different times. The 
locus of the intersection points is seen to curb 
away from the sources. Next produce two or three 
successive pairs of simultaneous pulses and ob- 
serve their intersections. Finallv observe the inter- 



DE.MONSTRATIOV VCTTES 



207 



sections of continuous waves. By marking the po- 
sitions of the nodal lines as projected, it is possible 
to establish quantitatively the wavelength fr-om the 
douhle-slit equation. For detailed suggestions for 
demonstrations consult the PSSC Laboratory Guide, 
Experiments 11-8 through 11-13: or Lehrman and 
Swartz, laboratory' Experiments, \os. 36, 37, 39, 40, 
and 43; or Biinckerhoff and I aft, Modern Labora- 
tory Experiments in Physics, Nos. 31 and 35. 

Project Physics Equipment 

Sound. ConnecA the two loudspeaker's in series to 
the oscillator and mount them at the edge of the 
table about 25 cm apart. Observe the signal strength 
as the ear is moved along a horizontal line in fromt 
of the sourc:es. Move farther away from the sources 
and change the source separation to see what hap- 
pens. Note the effect of changing the frequency. 

This demonstration can be made quantitative by 
mounting a meter stick parallel to the line of the 
speakers so that the nodes can be located and their 
positions noted from the stick. Plot the positions 
I;<| of maxima and minima; also record D and d. 

l\vo-Source Interference iquantitativei. Again the 
lipple tank and film loops are best for shov\ing how 
the interference patterri is prxjduced by two waves. 
The double sources used in the other experiments 
(except lighti should be placed to minimize r-eflec- 
tions from hard surfaces, otherwise spurious nodcil 
points will be present. Set the source transducers 
at the edge of the tabletop and directed away from 
nearby walls. 

Mount the two sources so that the distance irom 
center to center can be measured, as well as the 
distance along the perpendicular bisector of the 
line connecting the sources. 

Ultrasound. Plug the second source into the am- 
plifier Ithe plugs "stack") and arrange the two 
sources about 5 cm apar1. Explor-e the field with 
the detector about 25 cm in front of the sources, 
and plot the maxima and minima positions (,vi. 
Also r-ecord D and d. 

Microwaves. A two-source extension horn is sirp- 
plied with the generator. Fit the two-sourx;e honi 
into the horn of the generator. It should fit snugl\ , 
but if necessary support it with a block of wood or- 
a rubber stopper. Explore the field about 25 cm in 
fr-ont of the two sources and plot the positions of 
maxima and minima l.vi. At least thr^ee maxima 
should be picked up on either- side of the central 
antinochv 

D44 DIFFRACTION 
Itipple Tank 

Demonstiate the behavior- of pulses and waves at 
opiMiings in bariier-s, around obstacles, and edges 
of barri('i-s iditlractioni. 

Project Physics Equipment 

Dilfraction around tthstaclvs anti etlf^es 

rlic obstacle must lie at least .i tew w.nclenmbs in 
size and \(M not too large 



Sound 

1. Stand on edge a piece of thick plywood or cel- 
otex about 25 cm high and at least equalK' long, 
with one of the \ertic^ll edges placed about 25 
cm in front of the source. Explore slowK the area 
about 75 cm beyond the obstacle, along the ,v, y, 
and z axes. Try other obstacles lear separation 
distances). Try other frequencies. 

2. Similarly, use a piece of wood placed about 25 
cm in front of the source, and with one \erticcil 
edge aligned with the center of the souix:e. Ex- 
plore the area in the "shadow zone" and im- 
mediately out of the shadow zone. How many 
"fringes" can be counted? 

Ultrasound 

1. Use an obstacle of about 3 cm width, placed 
about 10 cm in front of the source. The detector 
should be placed 5 to 10 cm beyond the obsta- 
cle. Prabe on the ,v, y, and z axes. 

2. Use a large screen to explore the edge diffraction 
pattern. 

Microwaves 

1. Use an obstacle about 4 or 5 cm wide, such as 
the narrow aluminum screen provided, placed 
about 10 or 12 cm in fr-ont of the source. ELxplore 
the field at 5 cm behind the screen and at greater 
distances. Observe the maximum in the center 
of the shadow. 

2. Mask one-half of the sourx;e with a large screen 
placed about 12 cm in front of the source. Ex- 
plor-e the intensity' of the field as the detector is 
moved parallel to the screen and about 5 cm 
behind it. \'ou might use the meter to record and 
plot intensity' as a function of ,v. \ou should be 
able to resolve at least two maxima. If the output 
is weak, use an amplifer to drive an ac i decibel i 
meter. Note that, at the first maximum, intensity- 
is greater than when ther« is no screen. 

045 STANDING U'A\XS 
Slinky 

The slinkv, pulled out and held rigid at one end, 
can show standing waves with one to sever-al 
nodes. 

Ripple Tank 

Place a straight banier- acrxiss the center of the tank 
parallel to straight-fixinted. advancing waves. Wlien 
the gener-ator speed and banier position arv piT)p- 
eriy related, standing waves ' will be foniied. How- 
does the length of a standing wave appear to com- 
pare with the length of a moving wave.' Can you 
measure the wavelength fi-om the standing-wave 
pattern' Change the depth of the water- and ask if 
a change in speed can be detected. 

Bell Wave Machine^ 

S«'e l.\p H, Inteifeivni-e and Standing; Wave*, in 
Siiuilnritirs in 1\.'j\y' Hrhn\ior 



ZOH 



I'MT 3 / THE TRll'MPH OF MECH/WICS 



Project Physics Equipment 

These demonstrations, which ai^e modified \er- 
sions of Melde s Experiment, " show that a wire or 
spring of gi\en length, mass-length ratio, and ten- 
sion can be made to oscillate at onl\' certain pre- 
dictable fitjquencies that depend upon mass, length, 
and tension. These frequencies ai-e related to each 
other b\ integei-s. This ma\ be used as an inti-o- 
duction to the concept of characteristic leigen^ fre- 
quencies, normal modes, and eigen\iilues. B\ anal- 
ogy , this prepares the student for the concept of 
quantum numbei-s. 

The demonstration ma\ also be used to show 
how currents and magnetic fields interact. This 
concept will be co\ ered in detail in Chapter 14 and 
should not be mentioned at this time. .A current- 
carr\ing wire placed in a stead\ magnetic field has 
a force acting on it pr-opoifional to the current. .An 
alternating current in the wii-e forces the wire to 
oscillate. Standing waxes can be set up in the wire 
or spring if the frequency is adjusted to one of the 
resonant frequencies of the wire or spring. 

Project Phxsics Equipment 

Transverse l\a\es. Standing trans\ei-se waxes in a 
wire under tension are produced in the arrange- 
ment shown schematicalh in the figure below. 
Clamp one end of a copper wii-e i#18 or #20' 2-5 
cm abo\e the tabletop and stretch the wii-e o\ er a 
pulle\- with a weight suspended on the other end. 



1 


^^«r S^v-f»'y 






L 




^ / 


\ 


^ 



A variable frequency alternating current of about 
3 A is required for this demonstration. This current 
is provided by the Project Physics TRAXSISIOR 
SWITCH, which pulses the current from a 6-\ 
power supph . The switch is dri\en b\' an audio 
oscillator set for square wa\e output. On the D.A- 
MON equipment, connect the oscillator to TRAN- 
SISTOR SWITCH INPLT and mo\e the slide switch 
to the TRANSISIOR SWTTCIH position. Ihe switched 
current output is available at the 0— *6\' ' termi- 
nals. 

.Altemati\el\', a power amplifier, rated at 20\\ or 
more, can be used. Add 30-60 cm of #30 nichrome 
wire in series with the output to piT)\ide a load of 
about 5 ohms. 

The magnet should provide as large a field as 
possible, and a surplus magnetron magnet senes 
\er\' well. Remember that the plane of oscillation 
is perpendicular to the field so the magnetic field 



must be horizontal if the standing waves are to be 
in the \ertical plane. 

Hold the length and tension constant, and vaiy 
the frequencx'. 1 he wire is fi.xed at both ends, and 
ma\' \ibrate in the fundamental mode and in man\' 
hamionic modes. Manx of these modes ai"e seen 
as the frequencN is s\N'ept through sexeral multiples 
of the fundamental frequenc\ . The amplitude is 
smaller at the higher frequencies. Haimonics are 
easy to obtain if you position the magnet so that 
it is at an antinode. The frequency /^, of the nth 
hamionic is gi\en b\': 



/„ = nf, 



111 



where f^ is the fundamental frequency. The fre- 
quency of the fundamental can be calculated from: 



/, 



i_ r 

2L\ (J 



(2) 



where L is the length of the wire in meters. T is the 
tension in ne\%lons, and ct is the mass per unit 
length in kg m. 

Vou can do the same experiment without an au- 
dio oscillator. Use a current source with fixed fre- 
quency' I the output of a 6-\' 3-.A filament trans- 
former! and adjust either the length of the wire or 
the tension on the wire for maximum-amplitude 
standing waves. The tension is adjusted simph' by 
changing the weights, and the length is easily 
changed b\ inserting a hardw ood w edge under the 
string between the fLxed edge and the pulle\ . 

A t\pical experiment \ields the following results: 
.A 2-m length of #24 nichrome wire has a mass of 
about 3.42 x 10"^ kg, so a is about 1.71 X 10"^ kg 
m. The wire is stretched between its supports, and 
L is measured as 64.3 cm. The tension is varied 
until the wire resonates at 60 Hz with a mass of 
1.020 kg attached to its free end. Substitute these 
values into Equation (2i: 



f, = 



T 



2L\ 



and \ou obtain a frequenc\- of /J = 59.5 sec. This 
is within 1% of the expected 60.00 sec \alue. 

Longitudinal Waves. Vou ma\' also pixjduce lon- 
gitudinal standing wa\ es in a stretched spring. The 
spring is mounted betiveen two fixed supports 
with electrical connections made frx)m each end to 
a power amplifier, a transfor-mer, or audio switch. 
A surplus magnetron magnet T-shape, not C- 
shapei has a c\iindrical ii"on slug about 3 cm long 
held magneticalh' to one pole. Insert the slug a 
short distance into the end of the coil. 

As the frequenc\' of the current is \aried. the 
spring responds at each of its resonant modes. 
Handwound coil springs are adequate for quali- 
tative demonstrations. For quantitati\e work a 
brass SHM spring iCenco #75490 or equi\ client i is 
desirable. The spring maxbe mounted horizontally 
nn the stage of an o\erhead projector 



DEMONSTRATION NOTES 



209 



The haimonic seines for a spring lor the series 
of frequencies at which standing waves appeari is 
given by Equation (li. The fundamental frequency 
f^ is given by the equation; 






(3) 



where k is the spring constant lA = AF/A/lfound 
experimentally by Hooke's law; and where m is the 
total mass of the spring being subjected to oscil- 
lation. It is possible, therefore, to compare a set of 
predicted values v\ith a set of experimental \alues. 
You will find that if you use a fixed frequency, 
you cannot "tune" the spring to lesonance by 
changing the tension or by changing the length of 
the spring by stretching. Once you have found the 
resonant condition, you can stretch the spring, and 
it continues oscillating in the same mode. V'ou can 
only tune the spiing by changing the mass of the 
portion of tht; spiing that is oscillating. lo do this, 
clamp the spring firmly at \arious points other 
than the end until resonance is found. 

046 TWO TURNTABLE OSCILLATORS 

Beats. Ivvo oscillators are set up so that the mo- 
tions of the two platforms are parallel. One oscil- 
lator has a pen attached to the platfomi; the other 
carries a chart recorder positioned so that the pa- 
per moves perpendicularly to the oscillations of 
the platfomi. The pen writes on the ma\ing pa[)er. 




n the first turntable only is switched on, the pen 
will draw a sine cui\e on the mo\ ing papcM". If the 
second lurntabh' only is switchtul on, the nuning 
paper will be drixen back and forth in simple har- 



monic motion under the stationarv' pen. and a sine 
curve will be dravNTi. If the two turntables are set 
to the same speed, the two cunes will ha\e the 
same wavelength. .\ow, if both turntables are 
switched on, the trace will be the result of the su- 
peqjosition of two sine cunes. Even if both oscil- 
lators have been set to nominally the same fre- 
quency, there will almost alwa\'s, in practice, be a 
detectable difference, which means that the re- 
sulting pattern will show beats. 

With the turntable set to a gi\en frequenc\' isav 
78 rpmi. small adjustments in frequency' can be 
achieved by loading down the platform to increase 
friction between the platform and its support. 
I Some phonograph motors can be adjusted o\er a 
small range by a V'ariac iPowerstati in the suppiv 
line.) Thus, one can change the beat frequencv bv 
adjusting the frequencv of one of the component 
oscillatoi-s.* 

Approach to Harmonic Synthesis. Two oscillators 
are set up as before, so that a pen attached to one 
writes on a chart recorder mounted on the other. 
If the frequency of one is a multiple of the other, 
then the resulting trace illustrates in a simple man- 
ner the elements of hannonic sxnthesis. One par- 
ticulaiy interesting trace that represents the addi- 
tion of the fii-st two temis in the Fourier s\nthesis 
of a square wave la sin0 + Vsa sin30i is shown in 
the figure below . The coarse" tuning was done by 
setting the two turntable speeds to 16 and 45 rpm 
See the section of this Resource Book entitled 
Equipment Notes for a description of the turntable 
oscillator. 




MncidentalK . the disrussion and anal\'sis of beats provides a 
good ()pporlunit> to point out to students the low precision of 
a result that is the diffon-nrp of two large numljers Measun' the 
wave numhei n-ciprocal wavelength of the two roinponeni 
osrillalions together with an estimate of the unrertaint> c:ai- 
<ulaH' the wave numlMT of the resultant Iwat bv taking the dif- 
ferenre of the two neartv equal component wave numl>ers The 
peirentage uncertaint> of this result will be vvrv lar|{e On the 
other hand the heat wave iuiml>er can l)e measunti <lin>ctlv 
with high pivnsion which denionstrales how sensitive the 
method of heaN is iii showing up small diffen-nces in fiT^uencv 



I MI :i / THK TRIlilVIPH OF MHf IH/MVICS 



Experiment Notes 



E3-1 COLLISIONS L\ OXE DLMEXSIOX. I 

Equipment. 
Method A 

EKnamics carts with a steel exploder spring 

for each pair 
10 X magnifier with scale 
Weights for changing masses of carts 
Either bell-timers with batteries and ticker 

tape for each cart 
or Polaroid camera and motor strobe with 

12-slotted disk 
or xenon strobe lamp 

Method B 

Air track and two or three gliders 

Blower for air track 

Polaixjid camera and tripod 

Either motor strobe with slotted disk 

or xenon strobe lamp 

White or metal straws or cardboaixJ pointei"s 
to be attached to gliders as markers for pho- 
tographic measurements 

10 X magnifier with scale 

General Discussion 

Since the word momentum is ne\er used and the 
concept of consen ation of momentum is ne\er as- 
sumed in the Handbook, the experiment may, if 
desired, be treated as a ' discoxen " lab. 

Experiments 3-3 and 3-4 Collisions in Two Di- 
mensions, ' may be done concurrentK if apparatus 
and student background permit. The instructions 
assume a knowledge of conservation of momen- 
tum in one dimension. 

Several different procedures are described in 
these experiments. They could be combined into 
one lab with as many different procedures being 
followed simultaneoush' as apparatus permits. 
After all the working groups ha\e finished, the\' can 
bring their findings to a common class discussion. 

Let students develop an intuiti\ e feeling for fric- 
tionless" collisions by "plaxing" with balloon pucks, 
disk magnets sliding on plastic iDxiite^ beads, balls, 
etc., before beginning the quantitati\e work with 
an air track, d\Tiamics carts, or film loops. 

Some of the procedures require strobe photog- 
raphy. Remember that the room need not be com- 
pleteK dark although a dark background is impor- 
tant. It is possible to ha\e t\vo groups working on 
photography in one part of the room, v\ithout mak- 
ing it impossible for the rest of the class to work." 

To sa\'e time, ha\e the air track read\- and the 
camera in position at the beginning of the period. 
Set about 1 m from the track, the camera makes an 
image reduced in size about 10:1. Demonstrate the 
techniques of simultaneously opening the shutter 
and launching the glider and then closing the 
shutter after the interaction. 

"Note: Techniques of stroboscopic photography £ire described 
in I'nit 1 of this Resource Book 



Some limits must be set on the kinds of inter- 
actions to be photographed if the experiment is to 
be done in one class period. The photographs are 
less confusing to analyze when all interactions 
start with one glider stationary in the center of the 
track and the other launched toward it from the 
left. Tape the lights to the glider with one light 
higher than the other ibend up one lamp socket i 
so that their images can be distinguished on the 
photograph. 

As an alternative to the light sources, mount a 
white or metallized drinking straw on each glider 
and use a xenon strobe. Make one straw taller than 
the other. 

If the left-hand glider rebounds, the images will 
oxeriap and make measurement difficult. An inven- 
tive student could be encouraged to devise a way 
to distinguish the before pictures from the after" 
ones. Other-wise, photograph onl\ collisions in 
which the left-hand glider has a mass equal to or 
greater than the light-hand one so that the glider 
on the left does not rebound. 

Ways to distinguish the images of the rebound- 
ing glider include the following: With a piv ot fasten 
a small piece of colored transparent plastic or 
partlv exposed photographic negative on the glider, 
so that, when the gliders collide, the plastic will 
fall in front of the light isee belowi. The images 
formed after the collision by this glider will then 
be fainter. 




Alternatively, on the dxTiamics carts but not on 
the air-track gliders, the lamps themselves can be 
mounted vertically like the piece of plastic shown 
above, so that the entire lamp tips to a horizontal 
position upon collision. 

A third possibility is to have a student drop a 
filter in front of the camera lens at the instant of 
collision, thus dimming all subsequent images. 

/Although students must determine the mass of 
each cart with the light taped to it, the actual mass 
in kilograms is not needed. Onlv the relative mass, 
expressed as a multiple of the smelliest glider's 
mass, is important. 

The notes on photographv in Unit 1 suggest 
other ways of using the photographs. 

The collision experiment between carts is pos- 
sible as well, of course. 

If bell timer's and ticker tape are used to record 
the motion of the carts, notice that the tape at- 
tached to the left-hand cart passes under the right- 
hand cart to the timer. The timer lies far enough 



EXPERIMENT NOTES 



211 



to the right of the right-hand cart to allow room for 
the recoil motion of the right-hand cart. Con- 
versely, tape to the right-hand cart passes un- 
der the left-hand cart. The frequencies of the two 
timers may not he the same. Warn students to 
check this. 

When students tabulate their data, have them 
record speeds to the right as positive and to the 
left as negative, and i-emind them about the differ- 
ence between speed and velocity. 

Assemble data from all the photographs in a 
table to enable students to see the pattern that 
emerges. With glidei-s of equal mass, students may 
conclude from a simple elastic collision that speed 
is a conserved quantity. 

Use a linear explosion to show that the direc- 
tions in which the carts travel must be considered 
if momentum is to be conserved. 

The tabulated results make it clear to students 
that mv is the quantity that is conserved. Some 
students may notice that m\^ is also conserved for 
elastic collisions; point out that it is not conserved 
for inelastic ones lor explosions, since mv^ is a sca- 
lar, not a vector, quantity i. Defer further discussion 
of this point until after E3-5 is done. 

Stroboscopic photographs of one-dimensional 
collisions are discussed in the Activirv section of 
Unit 3 Handbook and iatei- in this section of the 
Resource Book. Film Loops IS and 19 also deal with 
one-dimensional collisions as does Transparency 
T19. 

Sample Results 




P' — momentum after the explosion 

P' = f^A< + m„v„' 

P' = 12.2 kgi (0.39 m/seci -I- (1.1 kg) i -0.75 m/seci 

P' = 8.6 kg m/sec - 8.2 kg m/sec 

P' = 

P = 0; there was no momentum befoi-e the 

explosion 
P = P'; momentum was conserved 



Ansu'ers to questions 

1. Speed is not conserved. 

2. Velocity is not conserAed. 

3. Momentum is conserved. The \ector simi of the 
momentimi is zeixj before and after the explo- 
sion. (See sample results above. I 



E3-2 COLLISIONS IN ONT DIMENSION. II 

MKI HOI) ,\. Iihu Loop 

See notes on Fihi} Uuip /.'). ILtndhook page 157 



XIETHOD B: Stroboscopic Photographs 

Equipment: 

Film Loops 19, 20. and 21 
Technicolor loop projector 
Graph paper, masking tape, ruler, strobo- 
scopic photographs lin the Handbook* 

STROBOSCOPIC PHOTOGRAPHS 
OF ONE-DEVIENSIONAL COLLISIONS: 
EVENTS 1-7 

/. M-\IEHL\LS PROVIDED AVD THEIR USES 
A set of stroboscopic photographs permits detailed 
quantitative study of se\en two-body collisions in 
one dimension. In the discussion that follows, 
these uall be called, simply. Events 1-7. The e\'ents 
are, primarily, illustrations of the principle of con- 
servation of momentum. 

The student may be assigned one or more of 
these events as take-home problems, or as study- 
period lor laboratorv-periodi tasks. It mav be ad- 
vantageous to assign a pair of students to a gi\en 
problem. Prints and student notes are pro\ided 
and may be kept by the student. 

Two o\erhead Transparencies, T19, show the 
stroboscopic photos of Events 1 and 2. W ith these 
the teacher can, in a few minutes, describe the 
problems qualitatively to the class before distrib- 
uting the assignment to students. 

The Transparencies of the two collision e\ents 
may also be used by the teacher to work out these 
examples in detail with the whole class, taking 
measurements directh' from the wall lor chalk- 
board I. 

The better students ma\' profit from working 
thix)ugh more than one e\ent. In fact, the series of 
e\ents was so chosen that there are certain rela- 
tions between events. In each pair, interesting dis- 
cussion questions can be raised. 

Events 1 and 2 form a pair because the\' are in- 
verse events involving the same balls. 

E\'ent 3 is a good exercise as long as it is not 
assigned alone but as a second e.xercise with Event 
1, 2, 4, or 5. This is due to the instructive character 
of error prxipagation in Event 3. A lai^e relati\-e er- 
ror arises when subtracting two neari\' equal quan- 
tities that are themselves known with small relative 
errors. 

Events 3 and 4 involve the same balls. Yet, in the 
former only one-half the kinetic energ\' is con- 
served, while the latter is almost perfectly elastic. 

Events 1 and 5 pixjceed similariy until the colli- 
sion takes place. The collision in 1 is elastic mot 
perfectlv elastic I. whereas in 5 it is perfectly in- 
elastic. 

A calculation of the kinetic rnertjv in the s\stem 
before and after collision is also of interest. It re- 
flects on the elasticity of the colliding balls Ibis 
energv is consenetl if the collision is pertectiv 
elastic. None of the examples is a perfectlv elastic 
collision, although kinetic enei-^v is *w.. c-onsiMAvd 
in Event 4 



212 



I'NIT 3 / THE TKIl'MPH OF VIFCIHAMCvS 



//. APP.ARATL'S USED TO OBTAIX' 

THE PHOTOGRAPHS 
The two colliding balls were, in each case, hung in 
bifilar suspension fix)m thin piano vN-ires as shown 
in Fig. 1. The\' were confined to mo\e on circular 
paths in the same vertical plane. The radii of these 
paths were the same for both balls i about 10 mi. 
Thus, the balls acted as pendulums of equal pe- 
riods labout 27T seci to excellent approximation. 
They were released simultaneousK b\' relays from 
chosen initial positions and therefore collided at 
the bottom of their swing i point B a quarter period 
lor about 1.6 seci later. 

A camera was placed in front of point B for a 
field of \iew as indicated. The i circular! path of the 
balls within this ftame w as illuminated b\' four s\ n- 
chronized General Radio Compam Stroboscopes 
I not shown I. The flash rate was alwa\s a simple 
integer submultiple of 60 Hz. This peimitted \eR' 
accurate calibration of this rate against the power 
compeinys time frequency b\' observing beats in a 
small neon bulb in the stroboscope circuits. 



BIFILAR STBEL WIRE 
^j SUSPENSION 




POINT OP COLLI SlON 



CAMEAAS PIELP CP VIEW 



C>3 KEFCecNCE BARS (im aporf) 



Fig. 1 Schematic diagram of the experiment (not to scale). 
The pendulums are very long. The height of the room in 
which the pictures were taken is 15 m. 

Two \ertical rods were placed in the field of \iew. 
The center points of the tops of these rods were, 
in all photos taken. 1 m apart las preciseK as pos- 
sible i. Therefore, the student can, by scaling, cal- 
culate actual distances fixjm the measurements on 
the photographic prints. 

Typical pictures can be found among the figures 
that follow. Clearly, the paths are not exacth' 
straight. Nor should the velocities found be exactly 
the same for a gi\en ball as it mo\es toward point 
B of collision, or away ftx)m B afterwards. But, if the 
amplitude of swing e.xceeds the portion within the 



camera fi'ame (about 0.5 m on each side of Bi by a 
sufficient factor, the \elocit>' variation will be small. 

The speed of the ball near the frame s edge is 
smaller than near the collision point B. iB\ the way, 
it is smaller by less than 1% if the full swing of the 
pendulum is 4 m: by 3% if it is 2 m: and so on." But 
that need not be of great concern because the stro- 
boscopic record allows one to check b\' how much 
the ball is slower near the frame s edge. 

If precision is affected b\' this \ariation fixjm the 
various exposures of ball position, then the best 
positions from which to calculate \elocities are 
those nearest the center. If this rule is followed, 
the eri-ors introduced b\ this aspect of the exper- 
imental setup are well within the estimated error 
of a reading of distance on the photograph by use 
of a good metric ruler with millimeter rulings. In 
none of the photographs is the "built-in error 
greater than 2% . 

///. DISCL'SSIO\' OF THE £\'EATS 
The se\en collision exents present similar analwi- 
cal problems. For any gi\'en event, the steps of this 
cinalysis follo\N-. 

A. A schematic diagram is pro\ided for each 
e\ent. It specifies qualitati\el\ what the conditions 
were before and after the collision. It also gi\es the 
masses of the colliding balls. 

B. The student can then proceed to a qu£ilitati\'e 
study of the stroboscopic photographs provided 
for the event and can actualh tell the order, in 
time, in which the stroboscope flashes occurred 
and number them. 

C. The student then must find values for the 
speeds of the balls before and after collision, by 
measurement. 

Here the student ma\' discover more than one 
time interval fhsm which measurement of displace- 
ment might be made. If so, "best choice must be 
made. 

Also, in some cases, conditions rule out an in- 
terval because the student can tell from a stud\' of 
the picture that the collision occurred during that 
interval. The displacement of a ball in that interval 
occurred at different speeds; the speed before, dur- 
ing, and after collision. 

A measurement of displacement is best made by 
measuring the distance between successive left lor 
right I edges of the ball in question. 

Each photograph shows two vertical rods. The 
centers of these rods are 1 m apart, with precision 
of 2 mm, a fr'action of 1%. Thus, measurements of 
displacement taken on the print can and should 
be converted to the actual values iin meters or cen- 
timeters i. This can be done by simple scaling. The 
student then finds the "real speeds of the balls 
before and after collision, which gives the exercise 
the flavor of the real experiment. 

We recommend that students be provided with 
a good-qualit\', but simple, see-through plastic 
ruler marked in millimeters; that they take care in 
positioning this ruler for evers measurement; and 



EXPERLMENT .VOTES 



213 



that they estimate to the tenth of the millimeter. 
They should be aware that this smallest significant 
digit may be in doubt. 

D. Thev can now proceed to the calculations re- 
quired by the problem. 

The teacher is provided with an overhead Trans- 
parency for Events 1 and 2. It is most strongly rec- 
ommended that the qualitative aspects of these 
events be discussed as prototypes before assigning 
events as problems for the student. Only the spe- 
cial aspects of Events 2 to 7 are discussed here. 

Event 1 

See also notes on Film Loop 18 i first example, page 
1561. 

Handbook, Fig. 3-4, page 105 shows, schemati- 
cally, the conditions befoi-e and after collision. In 
the stroboscopic photograph ^Handbook, Fig. 3-11, 
page 1071, we see two balls of different size. To 
reduce confusion, tuo dark stripes have been 
painted on the smaller ball iball B). The large ball 
(ball A) comes in from the right. The time and order 
of the flashes can therefore be analyzed. The photo 
at the bottom of this page corresponds to Hand- 
book, Fig. 3-11, page 107. 

Ball B clearly was at rest at center frame during 
flashes 1, 2, and 3. The collision must have oc- 
curred between flashes 3 and 4. Therefoj-e, the dis- 
placements experienced by either ball in interval 
"3 to 4" must be ruled out fiom the measurements 
useful for calculating speeds before or after colli- 
sion. Ball A was was consideiably slowed down by 
the collision. At the time of flash 7, ball B was al- 
ready out of frame. The camera shutter was closed 
before flash 8 occurred. 

The student has two choices i intervals "1 to 2" 
and "2 to 3"l in which to calculate the incoming 
speed of ball A and three choices i"4 to 5," "5 to 
6," "6 to 7") to measure its speed after collision. 

When using a luler marked in millimeter-s on an 
8 X 10 print of Fig. 3-4, the careful student will 
find that the displacement of ball A was about 0.5 



mm greater during time interval "1 to 2 ' than in 
interval "2 to 3." The difference amounts to about 
1%. 

The student should be encouraged to decide 
whether there is a "best choice" in finding the 
speed before collision in view of the experimental 
setup illustrated in Fig. 1. A student may. in this 
particular case, even decide with reasonable ar- 
guments that the average of both displacements is 
adequate. 

As a rule of thumb, of course, the interval closest 
to the instant of collision is the "best choice." By 
this reasoning, '4 to 5 ' is also best choice" of in- 
terval to find the speeds of both balls after collision. 

In any case, the student should convert each 
measurement of a displacement, taken from the 
print, to the actual displacement of the ball. This 
may be readily done by measuring the distance 
between the two vertical reference bars, which are 
1 m I ±2 mmi apart. 

Event 1 

Scale: 12.65 cm to 1 m on an 8 x 10 print 
Ball A: 0.532 kg 
Ball B: 0.350 kg 
Flash rate: 10 per sec 



Item 



Table 1 
Ball Time "Best Value" Direction 





A 


before 


3.37 m sec 


left 


velocity 


A 
B 


after 
before 


0.917 m sec 
m sec 


left 




B 


after 


3.72 m sec 


left 




A 


before 


1.79 kg m sec 


left 


momentum 


A 
6 


after 
before 


0.488 kg m sec 
kgm sec 


left 




B 


after 


1.30 kgm sec 


left 




A 


before 


3.02 joules 




kinetic energy 


A 
B 


after 
before 


0.223 joules 
joules 






B 


after 


2.42 joules 






(See Handbook, Fig. 3-11, page 107) Stroboscopic photograph of Event 110 flashes per second The numbers shown do 
not appear in the photograph given to the student. They correspond to the order in which the successive flashes occurred 
at 0.1 sec intervals. Each position of each ball can be associated with one of these numbers. The centers of the tops of the 
two rods appearing near the bottom of the picture are 1 m apart and serve as a scale reference (See also T19 and L 18. first 
example.) 



214 



I MIT 3 / THE TRIl'MPH OF MECHANICS 



Table 1 shows the results obtained b\' careful 
measurement from a photographic 8 x lO print. 
The Best \alue" is based on measurement of dis- 
placement with a good ruler, of 1 mm least count 
(estimating to the tenth of a millimeteri, during 
time intervals "2 to 3' before collision and 4 to 5' 
after collision. 

The momentum of the s\stem of two balls before 
collision is equal to the momentum of the s\ stem 
after collision to the number of significant digits 
avaUable: 1.79 kgm sec (directed to the left. On the 
other hand, the kinetic ener^- of the s\stem goes 
from 3.02 J before collision to 2.64 J after collision. 
It is only 87.4"b conserved owing to imperfect elas- 
ticit\'. 

Event 2 

See also notes on Film Loop 18 second example, 
page 156 1. 

Refer to Handbook, Fig. 3-5, page 105 and notice 
that the balls ha\e the same masses as in Event 1 
and that the collision is the inverse of E\ent 1. This 
was intentional. In Event 2, ball B is striking ball A 
at rest from the left. The incoming speed of ball B 



is. in fact, roughlv equal to that of ball A in Event 1. 

Moreover, note that in this case the incoming 
ball is reflected bv the collision. This is so because 
the ball it strikes has the greater mass. 

There are two stroboscopic photographs record- 
ing this event in Fig. 3-12. The first shows the event 
before collision, the second after collision. The pic- 
tures were taken bv allowing the event to control 
the camera shutter using electric relavs. A relav 
closed the shutter before collision and opened it 
just after collision. If some such division of this 
event had not been made, it would be difficult to 
diflferentiate exposures of ball B because this ball 
retraces its path. The transparency overlav of Fig. 
3-12 for overhead projectors provided to the teacher 
can be used to show this to the student. 

The measurement of displacement for ball B 
after collision is difficult in this case. See Fig. 3-12 
■'after.' It emerges from collision with small speed, 
which I as shown clearK' bv the picture i is decreas- 
ing visiblv. Interval 1 to 2 in Fig. 3-12 is clearlv 
best choice for both balls. Care is required to find 
the displacement of bcdl B in that interval. 



(See Handbook Fig. 3-12, page 107) Event 2. 10 flashes sec. Fig. 3-12 is also available as an 
overhead transparency, T19. (See also Film Loop 18, second example.) 




after 



EXPERIMENT .VOTES 



21i 



Our values of total momentum befor-e and after 
collision vver-e 1.14 and 1.16 kg rn/sec, respectively 
(dinu;ted to the right i. Ihe difficultv' in measuring 
the speed of hall B after collision and the large 
relative error arising from this difficulty do not 
greatly affect absolute errors. The reason is that the 
momentum of ball B aftei- collision is so small. Uis- 
coveiy by students of these aspects of measure- 
ment is in itself a worthwhile goal of our laboratory' 
instiTiction. 

Our values showed the total kinetic energy to be 
91% conseived. 

Event 3 

See also notes on Film Loop 19 (first example, page 
1571. 

As Handbook Fig. 3-6, page 105 and Fig. 3-13, 
page 108 illustrate, a massive ball A enters fmm the 
left, and ball B of considerably less mass comes 
from the right. Ihe speed of ball B compared to 
the speed of ball A before collision, however, is so 
large that the net momentum vector of the system 
actually points to the left. 

This has tw^o consequences that we feel to be of 
pedagogical value. One of these concerns the as- 
pects of measurement: The net momentum of the 
system is here a small difference of two large num- 
bers. The two "large numbei-s " are the momenta 
of the balls before or after collision. Most students 
can find these momenta with a precision of 1 or 
2%. However, in their small difference this becomes 



a large relative error. Matters are made worse be- 
cause of the rule that states that, in additions or 
subtractions, the entar in the result is the algebraic 
sum of the absolute errors of the two numbers in- 
voked. 

Our results were: 0.30 kgm/sec and 027 kgnvsec 
net momentum i directed toward the lefti before 
and after. This is a 10% difference. We have only 
two significant digits in the result, although each 
ball's momentum was itself known to three digits. 

The second consequence concerns a physical 
process. In this collision the impact of the steel 
balls was a xiolent one. Students can study this 
with their own data. Each ball s change in momen- 
tum, prt)duced by the collision, was of the order 
of 2.8 kgm/sec. (Change in momentum - ia\erage 
foree during collision) x (time of collisioni. With 
a conservative estimate of 0.01 sec for this time, the 
a\'erage force was 280 \, or the weight of 29 kg.] 

Obviously, the hard and elastic steel case of 
these case-har-dened balls was strongly deformed 
in this collision. This deformed the soft, and far 
less elastic, inner core. The result was a great deal 
of internal friction and heating. This e.xplains why 
the initial kinetic energx of the system is dissipated 
more here than in other events. We found it to be 
only 51% conserved. 

Event 4 imohes the same balls in a weak colli- 
sion, resulting in 98% conservation of kinetic en- 
ergy. 



(See Handbook Fig. 3-13, page 108.) Event 3. 10 flashes sec (See also Film Loop 19, first example.) 




before 




after 



216 



II\IT 3 / THE TRIl'MPH OF IV1ECH/\\IC:S 



Event 4 

See also notes on Film Loop 19 (second example, 
page 1571. 

Handbook Fig. 3-7, page 106 specifies this e\ent 
The massixe ball .A comes in tVom the left at high 
speed and o\ertakes the lighter ball B, which is 
going in the same direction. After collision, both 
balls are still going in the same direction, but ball 
B is tra\elling much faster than before, whereas 
ball A is slowed down relatixely little. 

Since each ball crosses the entire field of \iew 

ball A and ball B 



from left to right, a stroboscopic view of the whole 
exent leads to superposition of images created b\ 
flashing of the bulb at different times. iSee the fol- 
lowing sti-oboscopic photograph of ball A and ball 
B.I As a consequence, such a picture is difficult to 
analyze. 

We got araund this difficult\ b\ photographing 
Event 4 in its entiret\ rv\ice. Once iFig. 3-14, ball Ai 
ball B was painted black while ball A was painted 
white. The second time iFig. 3-14, ball Bi the colors 
were re\'ersed. 




Stroboscopic photograph of the entire Event 4. 10 flashes sec. Both balls cross the field of view in the same direction (fronn 
left to right) as time goes by. It is not easy to analyze this picture. Handbook Fig. 3-14 shows how this difficulty was resolved. 



Consider Fig. 3-14, ball A laboxe . It can be used 
to determine the speed of ball A before and after 
collision. However, position 3 is difficult to identifv . 
It could, at first inspection, have been befoie or 
after collision. A good student will be able to show, 
bv making measurements, that the flash which ex- 



posed position 3 occurred after collision. Distance 
"2 to 3" is slightly shorter than distance 1 to 2," 
while at the same time "3 to 4' is slightly longer 
than "4 to 5." If this is clear to the student, then 
the "best choice" for ball A is before collision "1 to 
2" and after collision "3 to 4." 



(See Handbook Fig. 3-14, page 108.) Event 4. 10 flashes sec. Whole event. Ball A (large mass) black. (See also Film Loop 
19, second example) 




ball A 



(See Handbook Fig. 3-15, page 109.) Event 5. 10 flashes sec (See also Film Loop 20, first example.) 




ball B 



EXPERI.MENT NOTES 



217 




(See Handbook Fig. 3-14, page 108.) Event 4. 10 flashes sec. Whole event. Ball B (small nnass) black. See also Fi/m Loop 
19, second example.) 



This is a fine point of the analysis that can be 
ignored. One can call position 3 doubtful. This 
rules out intervals "2 to 3" and "3 to 4." EqualK' 
precise results are obtained from Fig. 3-13 by 
choosing intervals "1 to 2" and "4 to 5." 

Analogous difficulties arise in analyzing Fig. 3-14, 
ball B. The internals before and after collision 
corresponding to safe choices are "3 to 4" and 
"6 to 7." 

We find, frx)m the safest choices, that the total 
momenta before and after collision are 8.35 and 
8.3« kgnVsec (to the right i, rt»spectr\ely. ! he total 
kinetic ener-gies before and after ar-e 15.9 and 15.5 
J, which correspond to 98% conservation of kinetic 
energy. 

This weak collision is much more elastic than 
another stronger collision between these balls, 
namely. Event 3. Event 4 forms an instructive com- 
panion to Event 3. The Handbook contains ques- 
tions about the suqjrising differences in appar-ent 
elasticity. 

Perfectly Inelastic Collisions 

In these examples, the colliding objects are steel 
balls covered by a thick layer of plasticene. Thev 
remain lodged together after collision. 

Event 5 

See also notes on Film Ixjop 20 (fir-st example, page 
157). 

Handbook Fig. 3-8, page 106 and Fig. 3-15. page 
109 show ball ,\, coming in from the right, striking 
ball B. which is at r-est. Our- measiir-ements vield 
momenta for- the system of balls of 2.(i(i kg ni sec 
befoi-e and 2.G7 kg msec after- collision idirt'cted to 
the lighti. The kinetic energv' after collision was 
61% of that befor-e collision. 

Event 6 

Set; also notes on I'ihu Unyp 20 (second example, 
|iage 1571. 

Handbook Fig. 3-9, [)age 10(i and Fig. 3-16, page 
109 tell the story. Ibis case involves the same balls 
as Event 5. Momentum of the svstem befoiv and 
after is 2.06 and 2 ()8 kgnvsec idirtjcted to the lefti 
by our nu'asurvmerits. 

Event 6 is an inslriictivc example, when consid- 
er-ed together- with Event 5. in which over 60% of 
the kinetic enei-gv was conseiAcd Here, oiilv 24"i. 



is conserved. The collision was more violent in 
Event 6. 

Event 7 

A very massive ball A is coming in from the left isee 
Handbook Fig. 3-10, page 107i at a speed that is 
small when compar-ecJ with the speed of ball B of 
small mass that enters from the tight. When the 
balls become lodged together, thev move off to the 
right, but the speed of ball A after collision is not 
ver\' gr-eatlv reduced from what it was before. This 
is analogous to the case of a head-on collision of 
a truck with a small car. 

Handbook Fig. 3-17, page 110 shows the strobos- 
copic r"ecords befor-e and after collision. 

Momenta before and after ar-e 6.58 and 6.71 kg m 
sec idir-ected to the righti, respectivelv. Kinetic en- 
ergv' after collision is about 27% of that before. 

E3-3 COLLISION'S IN TWO 
DLMEXSIOXS. I 

Equipment: 
Method A 

Ripple tank 

Plastic iDvlitei spheres 

Three or four balloon pucks without balloons. 

but with small labout 1-cmi white stxrofoam 

hemispheres glued to center of bottom of 

two of them, as markers 
Polaroid camera 
Mount for camera positioned venicallv above 

center of the ripple tank ifor example, on a 

step ladderi 
Either xenon strobe lamp 
or motor strobe v\ith 12-slotted disc and strong 

spotlight 
10 X magnifier with scale 

Method B 

Exactly the same equipment and setup as 
Method A except that the air pucks are re- 
placed bv three or four disc magnets, rwo of 
which have their centet-s marked by 1-cm 

white stv lofoam hemispheres. 

General DisruHNion 

Even if vour students do not do this experiment 
quantitalivelv, let them experiment with balloon 
[Jinks, putks on beads anil magnet pucks on 



ZIH 



IMI 3 rHli THIl'Mini OF IVIKCH/WICS 



beads to get an intuith'e feel for two-dimensional 
collisions. 

Two sizes of puck are supplied. The mass ratio 
is 2:1. They may be used as balloon pucks on an\ 
flat and fairK clean surface. iRemo\e the stopper 
v\ith the balloon when inflating.! You can impix)\e 
sanitation, and a\oid sali\a on the pucks. b\ plac- 
ing the puck tighth" up against the exhaust of a 
vacuum pump to inflate the balloon. 

The new rule is that momentum is a vector 
quantity. 

For strobe photographs, remo\e the balloons 
and slide the pucks on a low-friction surface made 
by sprinkling plastic iChlitei spheres on a ripple 
tank. Use a small piece of cla\ or tape to fi.\ a steel 
ball I such as that from the trajector\ apparatus' or 
a white sr\rofoam hemisphere at the center of the 
puck. lOther reflectors, such as a white-painted 
stopper, are also possible, but gi\e less satisfactory 
photographs.! 




The .xenon strobe must be at the side of the tank, 
not abo\e it . This is to reduce reflections tram the 
beads themsehes, which may ne\ertheless be a 
nuisance: don t use too mam beadsl 

To get the camera directh abo\e the working 
surface, extend one leg of the tripod more than the 
other. The tripod will then be unstable so the long 
leg must be held down, as shown in the sketch. 

.•\ltemati\e procedure with ring magnets iwhich 
may be obtained fiDm Damon': Glue a piece of card 
over the top of the magnet. Put a piece of diA ice 
inside, or fill with dr\' snow from a CO, fire ex- 
tinguisher. Place the magnet on a flat surface i rip- 
ple tanki; it will float on the film of CO, gas. 



r~ f^i^ fm^u^r 




.Another possibilit\ here is an "explosion" be- 
tAveen magnets. Would two magnets mo\e apai1 in 
opposite directions along a straight line.' If the 
masses are equal will the speeds be equal? What 
happens with thi-ee or moi-e magnets held close 
together and released simultaneousK ? 

Students can draw their vector diagrams b\ 
pricking holes and drawing on the back of the pho- 
tograph, as shown below. 



K^' 



The conclusion that momentum is conserved in 

an interaction is not sufficient to enable us to pre- 
dict the final \elocities of the interacting bodies, 
e.xcept in a few special cases. Gi\en rriy m^. v^, Vg, 
the equation 

m\\ ^ mv^ = m\\' ^ ^nVf^' 

has two unknown quantities. \\ and \ „. 

Students should realize fi-om their knowledge of 
algebra that two equations are needed to find these 
quantities. Since a gi\en set of initial conditions 
always leads to the same pair of \alues of \\' and 
\b', some other requirement must be imposed on 
the system. Stress this point to prepare students 
to look for still another conserv ation law in the next 
e.xperiment. 

One special case is \er\' well illustrated b\ the 
magnet disc e.xperiments. In an i elastic' collision 
or interaction beUveen two equal-mass bodies, one 
of which is initialK at rest, the two \elocities after 
the collision ai-e perpendicular to each other. This 
is quite independent of the off-centeredness of 
the collision. 

Analysis of Data 

Three transparencies 1720 through T22i analyze a 
two-dimensional collision in detail. Furthermore, 
in this Resource Book, stroboscopic photographs 




EXPERI.MENT NtTTES 



219 



of two-dimensional events 8-14 are explained in 
the "Film Loop Notes" section. Use these refer- 
ences as samples of the results from E3-3. 

An additional analysis of i-esults follows: 
P^ = momentum of /A befoit? collision 
P„ = 

P^' = momentum of /A after collision 
F„' = momentum of B after collision 








M^v/- p; 






" — "^^ 


5 


iai/n,v\ ^ //j„r„ - m^v',, = m^v\' - 


^S^i' k 




= m^'V^A,' + ^Av' + ^A«'' + "^B '^ 
+ V' 'l 



+ V, 



(C)m^V'^ + "^b^'b + "^c^'c = '"a^'a' + "'b^'b' + 



p>. - p; ^ P5 



Answers to questions 

1. Student calculations. 

2. The sum of the mass x speed before the colli- 
sion is not equal to the sum of the mass x speed 
after the collision. 

3. The vector sum of the momentum before the 
collision is equal to the \'ector sum of the mo- 
mentum after the collision. 

4. One-dimensional collision conservation of mo- 
mentum is a special case of the two-dimensional 
problem. Two examples of three-dimensional 
conservation of momentum: 




Addendum to E3-3 

With a small modification, the Project Physics tra- 
jecton' apparatus ^EJ-S. Unit li can be used in a 
collision in a two-dimensional e.xperiment. 

The figure at the top of the ne.xl column shows 
how the tai-get ball is positioned at the end of the 
launching ramp. Ihe attachment can be swung 
about a vertical axis to vary the impact parameter 
lamount of olT-centeredness of the coUisioni. 

The plotting board is laid flat at the comer of a 
table. On it ai-e placed a sheet of caiijon paper icar- 
bon side upi, and o\'er the carbon paper a sheet of 
onionskin paper. 

VX'ith no tai^get ball in position, a ball is released 
from a point on the ramp and allowed to fall on 
the plotting board, which records its point of im- 
pact. Repeat a few times to get consistent results. 
A target ball is now placed on the support and the 
impact ball rvleased again fit)m the same point on 
the ramp. Both balls recoixl the positions at which 
the\ land on the impact lioaixl. 

Analysis of Data 

rhe balls all fall through the same \ertical distance 
from collision to the board, and aiv thei-efore in 
flight lor the same time Ihe horizontal distance 
that each ball trawls between collision and impact 
is thus [iniportional to its \elocitx Distances are 
ineasuii'd fn)m the collision point. 

Students can analxze for conservation of mo- 
mentum in the horizontal plane In drawing vector 
diagrams as shown below the photogra|)h. Ihe 
launch position mass ratio, and impact parameter 
can all be \aried to |>ix)\ide a wide range of situa- 
tions 

! he same data can l)e anal\/.ed for consel^ation 
ot kinetic energ\ in the horizontal plane 



zzo 



IMI :t IHIi TRIl'MPH (IF VIECHAMCS 











\ ^^^^^H 




-^d '' ° '^^'- ^3 



E3-4 COLLISIONS IN TWO 
DIMENSIONS. II 

Equipment; 

Film Loops: 22. 23. 24. 25. and 26 

Technicolor loop projector 

Graph paper, masking tape, ruler 

Stroboscopic photographs (in the Handbook^ 
Method A: Film Loops 

See notes on Film Loops 22-26 beginning on 
page 158. 
Method B: Stroboscopic photographs 



STROBOSCOPIC STILL PHOTOGRAPHS 
OF T\\'0-DLMENSIONAL COLLISIONS: 
E\TNTS 8-14 

1. \LAIERLALS PRO\ IDED .AND THEIR USES 

Measurements in E\ents 8-14 are essentialh' of 
the same nature as in the one-dimensional prob- 
lems. .Anahsis, on the other hand, requires con- 
struction of \ector diagrams and is for this reason 
a bit more complicated. 

The student ma\ be assigned one or more of 
these events as take-home problems, or as stud\- 
period lor laborator\ -period' tasks. It ma\ be ad- 
\antageous to assign a pair of students to a gi\en 
problem. 

Better students may profit from working through 
more than one example. E\ents 8 and 9 both in- 
\ol\e collisions in which one ball is initialK at rest. 
In E\ent 8. moreo\er, the balls have equal masses. 
E\ents 12 and 13 are peifectK inelastic collisions 
between balls of equal mass. The initial speeds are 
roughlv the same in these two examples, but in 
one the angle between incoming paths is acute, 
whereas in the other it is obtuse. Event 14 is the 
most ambitious problem. Here, a rapidh mo\ing 
ball scatters a cluster of six balls initialK- at rest. 
This problem should not be assigned unless the 
student has worked one of the simpler e\ents 
8-13'. 

The pixjblems are, primariK , illustrations of the 
principle of conservation of momentum. 

A calculation of the kinetic energy' in the s\ stem 
befoi-e and after collision is also of interest. It re- 
flects on the degree of \iolence of the collision and 
on the imperfect elasticit\ of the steel balls, iln 
E\ents 12 and 13, howe\er, the colliding balls are 
plasticene.! 

The stroboscopic photographs were made b\ the 
same procedures as those for E\'ents 1-7. 

n. DISCUSSION OF THE E\T\TS 

The vector diagram. 

\\ ith the print secureK taped to a comer of the 
large paper sheet the student uses the drafting 
triangle to draw a line parallel to the incident di- 
rection of motion of one of the balls. 

Choosing a convenient scale factor, the student 
measures off the momentum of this ball before col- 
lision. To the tip of this vector is now added the 
momentum of the next bail before collision, b\- the 
same procedure of drawing a parallel line and us- 
ing the same scale factor. Thus b\ adding vectors, 
the total momentum before collision can be drawn 
and its magnitude measured. 

The prxjcedure is repeated to find the total \ector 
momentum of the s\stem after the collision. 

It is recommended that the initial point of the 
procedure of adding \ectors tip to tip be the same 
point for both the before and after phases of 
the collision e\ent. It will then be relati\el\ eas\ to 
find a \alue for the angle between the two \ector 



EXPERIMENT NOTES 



221 



sums. This angle is one of the two measuies of 
error. The other is the percentage of diffei-ence in 
magnitudes. 

Kinetic energy. 

1 otal kin(!ti(; energy of the system hefore and 
after collision can l)e calculated from the gi\en 
masses and from the speeds determined by the 
measurements. 

In none of these events were the collisions per- 
fectly elastit;. K\'en though the hardened steel cases 
approach ideal elasticity, the soft steel inner- cores 
were fr-equently pennanently deformed by the col- 
lision. The loss in kinetic energy goes into the work 
of deformation and into heat. 

Clearly, the percentage of loss of kinetic energ\' 
should be larger for more violent collisions. If a 
student works through more than one exent les- 
pecially if the different events invoked the same 
balls, such as in Events 8 and 10, 9 and 11, or 12 
and 131 a comparison of the percentage of kinetic 
energv conserAed in the collision can lead to in- 
teresting discussions. 

Forget about rotation! 

In Events 8-14, each ball was suspended by a 
single wire. It is not likely that the balls were spin- 
ning appreciably before collision. However-, they 
may have been spinning considerably after- colli- 
sion. \o provision was made (by marks on the 
balls, or other- meansi to permit measurt'ment of 
these rates of spin. Only translational momentum 
and translational kinetic ener-^v is accessible to 
measur-ement by our- techni(|ues. 

In the perfectly inelastic collisions i Events 12 
and 131 care was taken to use photographs in 
which these spins ar-e essentially zer-o. See Hand- 
book Figure 9, page 20 and Figure 10, page 21. 
Hence, the possible complications pertaining to 
angular momentum land r-otatiorKil kinetic energvi 
do not enter in Events 12 and 13. 

Event 8 

See also TZU and notes on I'iliu Loop ZO i first ex- 
ample, page IJJTi. 

Fr-om the photograph of Event 8 iUnndbook Fig. 
3-27, page 1151 note that the balls have equal mass. 
Ball A comes in frtim the upper- left-hand comer- 
while ball B is initially at r-est. During the stixjbo- 
scope flashes nirmbered 1, 2, and 3. ball B is 
strx)ngly ex[)osed, [jhotographically, in its rx?st po- 
sition. The collision occurred in the inter-val be- 
twc^en flashes 3 and 4, Ibis inter-val is therefore 
inappr-()|)r'iate for- deterriiining ball spet^ls sincj' we 
arc iiitcr-eslcd in speeds belbi-e and after- collision. 

In a(-(-ordance with lh(> idea that internals near- 
est the collision points constitute best choices, 
it is cU^ar- that intenal "2 to 3 ' is 'best' beforv and 
that intei-val 4 to o" is "best" after (-ollision lor 
spe(Hl measiri-emenl . 

Table 2 sumniaii/.cs i-esuits lor a tvpiiai mc.is- 
uremctil 



The kinetic ener^gies of the two balls are scalars. 
rhey add numerically. By our measurements given 
in the table, the system possessed 1.39 J of trans- 
lational kinetic energy before collision and 123 J 
afterwards i88.o% conservation!. 

Event 8 

Scale: 23.1 cm to 1 m on an 8 x lo print 
Ball A, = Ball B„ = 0.367 kg 
Flashr-ate: 20 per sec 



Item 



Table 2 
Ball Time 



'Best Value' 



speed 


A 


before 


2.76 m sec 




A 


after 


2.16 




B 


before 







B 


after 


1.44 


magnitude 


A 


before 


1.01 kgm sec 


of 


A 


after 


0.774 


momentum 


B 


before 







B 


after 


0.527 



kinetic 
energy 



A before 

A after 

B before 

B after 



1.39 joules 

0.854 



0.379 



Because the masses of the balls are equal in this 
event, the photograph lends itself to an easy dem- 
onstr'ation of the prxjperties of the center of mass 
of the system. 

The results illustr-ate the following theorem: The 
center of mass of a system of particles, which is 
subject to zero net external force, travels uniformly 
along a straight line. 

It is not r-ecommended that the teacher present 
this theor-em and the constixrction to all students. 
Bather, we mention it hei-e because of its intrinsic 
inter-est and because it mav be apprxjpriate to ex- 
pose your best students to it. 

Event 9 

Our- discussion of this and subsequent events will 
emphasize onlv special points of interest. It will be 
less detailed than for Event 8 because procedures 
are identical. 

Event 9 isee Handbook Fig. 3-28. page 1 Hi' is sim- 
ilar to Event 8. .Again, one ball is initiallv at r-est. 
However-, the ball masses an> diftervnt in Event 9. 

Our r-esultant momenta before and after collision 
coincided in direction. Thev diffeivd in magnitirde 
bv 0.004 kg • m sec labout 0.5% i. 

Ihe kinetic energv of the svsterii fell from 0.727 
J before, to 481 J after collision ()nl\ Mi.2% of this 
energv is conser^ed Ibis should be compared to 
the 88"n conserAation in Event 8 

Notice that the balls A have the same mass in 
these two events Whv is the perventage loss of 
kinetic energv lai-ger in Event 9 ' Ihe answer is con- 
nect«'(i with the lar>jei- mass of ball B In Event 9, 
each ball expeiiences a greater change in momen- 
tum during collision than in Event 8 Hence the 
int«'iacli()n Ibix'es during collision wciv lar^^er on 



222 



ll\IT 3 THL TKI I'M I'll OF !V1l-X:HA\ii:S 



the average. The collision was more \iolent." The 
balls were pemianentK detoimed to a greater de- 
gree. 

Event 10 

See also notes on Film Loop 22 (second example, 
page 158). 

This e\ent is slightly more complicated than the 
two discussed so far because both balls are mo\ing 
before collision. Note. howe\er. that the balls again 
have identical masses as in E\ent 8. 

In our measurements, we found an angular sep- 
aration of 0.6° and 1.5% difference in magnitude 
between the total momenta before and after colli- 
sion. 

Total kinetic energ\' was 2.17 J before and 1.73 J 
cifter collision i79% conserved . 

Since E\ent 10 isee Handbook Fig. 3-29. page 117, 
like E\ent 8. uses balls of equal mass, it is again 
quite easy to locate the instantaneous images of 
balls A and B. It is clear that the path of the center 
of mass is a straight line and that its motion is 
uniform. 

One mereK measures the distance bet\veen two 
successi\e positions of the center of mass and con- 
verts to real distances b\ scaling. Recall that the 
reference rods at bottom frame ai^e 1.000 = 0.002 
m apart. I Successi\e positions are 0.05 sec apart in 
time. We find that the speed of the center of mass 
= 1.89 msec. 

The center of njass of a system of particles can 
be treated as if it were a particle whose mass equals 
the total mass of the system and whose momentum 
equals the total momentum of the system. This is 
a general result of theoretical mechanics. 

Since we know the speed of the center-of-mass 
particle," the magnitude of its momentum is 

10.367 -t- 0.3671 kg X ii.89i rasec 
= 0.734 kg X 1.89 msec 
= 1.39 kg • msec. 

Notice that this comes close to the xalues found 
for the magnitude of the total momentum of the 
s\stem b\- adding individual particle momenta \ec- 
torially. 

Furthermore, one could easily verifi' that the di- 
rection of the straight-line path taken b\ the center 
of mass is parallel to the total momentum \ector 
as found by the method of adding particle mo- 
menta. 

Event 11 

See also notes on Film Loop 21 ipage 157i. 

Event 11 isee Handbook Fig. 3-30, page 116' is 
analogous to Event 10 e.xcept that the masses of 
the two balls are not equal. We find a difference of 
1% between the magnitudes and an angle of less 
than 1' between the directions of the vector sums. 
Total kinetic energ\' before and after collision is 
3.10 and 2.65 J. respecti\el\' (85% conservation/. 

Events 12 and 13 IL23) 

See also notes on Film Loop 23 page 158i. 



In these events, the steel balls were covered bv 
a thick layer of plasticene. The\ remained lodged 
together after collision. The major difference be- 
tween these two events was the fact that the angle 
between the initial dii-ections of motion of the two 
balls was larger in Event 13 see Handbook Fig. 3- 
32, page 1171 than in Event 12 isee Handbook Fig. 
3-31, page 1171. 

■After collision, the particle svstem ' becomes a 
single, compound particle consisting of the coa- 
lesced balls. Its mass is the sum of the masses of 
the original balls. To find its speed and momen- 
tum, it is necessaiy to locate the center of mass. 
Even though the collision created noticeable de- 
formation, it is possible to locate this point easilv. 
Since the individual balls have identical masses, 
the compound paiiicle s center of mass lies at its 
geometrical center. This can be judged bv the na- 
ked eye and marked on the print bv a pencil dot. 











D^e dieted 
resu.\taot 



\ 



ot>sf ved 

r-GSu.l + a.nl' 



In Event 12. the total momenta before and after 
collision differed bv 0.5% and subtended an angle 
of 0.5° according to our data. See Fig. 1. Kinetic 
energv fell fham 3.10 J during collision 185% con- 
servation!. 

.As Fig. 2 shows, we found the dii'ections of the 
total momenta for Event 13. before and after colli- 
sion, to be the same. Their magnitudes differed by 
3%. Kinetic energv dixjpped from 2.81 J to 0.33%. 
Onlv 12% of this enei-gv' was conserved. 

\\h\ is the relative error significantlv larger in 
Event 13 than in Event 12? The principal reason 
cam be found bv an inspection of the vector dia- 
grams in Figs. 25 and 26. In Event 13. the total mo- 
mentum of the system before collision compared 
to Event 12, is a shod vector sum of tvvo vectors. 
On the other hand, the vectors added together to 
produce these sums are, in fact, of nearlv equal 
magnitudes in both events. Thev probably have 
nearly equal errors in them, too. W hen adding the 
vectors, the errors pile up. .And in the shorter 
vector sum of Event 13, the absolute error is about 
the same. So it is a larger relative enxjr. 



EXPERI.MENT NOTES 



223 



VVhy is there so much greater loss of kinetic en- 
ergy in Event 13? The answer is obvious from an 
inspection of the photogiaphs. The l)alis collide 
almost, but not (juite, heatl-on in Kvent i:J. Notice 
also, how much moi-e flattening takes place at the 
points of contact in Kvent 13 compared to Kvent 
12. The collision was more violent: more energ\' 
went into unrecoverable work of defomiation. 

Event 14 

See also notes on Film Ixjop 24 'page 158i. 

A stationary cluster of six balls is struck by a 
rapidly moving ball coming in fr-om the right isee 
Handbook Fig. 3-33, page 117). The initial position 
of the cluster is seen at the center of the figure. 

The total mass of the cluster is more than 4 kg. 
After the impact, ball speeds are relatively low. This 
explains why a flash rate of only 5 per sec was used 
and, also, why the speed of some of the balls after 
impact decreases noticeably as they mo\e toward 
the edge of the field of view. 

On the other hand, ball A is moving so fast befor-e 
impact that the strxjboscope, flashing only e\ery 0.2 
sec, did not captui-e within the frame more than 
one of its positions befor-e collision. For this reason, 
a second picture was taken. See Handbook Fig. 3- 
34, page 118. Ihe cluster of balls B to G was re- 
moved and ball A r-eleased from exactly the same 
initial point. Ihe flash rate was greater I20 per seci. 
The "impact speed" of ball A can be determined 
from measurements of displacement in the center 
of the photograph. 

Ihe analysis of Kvent 14 is analogous to the anal- 
ysis of the previous problems. It requires more 
measurements and comparison of the sum of 
seven momentum vectors lafter collisioni with the 
momentum of the in(X)ming ball. We r-ecommend 
that this problem be assigned only after- one of 
Kvents 8-13 has been worked out by the student, 
and then only if he or she is one of the better stu- 
dents in your- class. 

Notice that balls .A and C have identical radii i5 
cm). The same is traie of balls B and D I4.4 cmi. To 
distinguish between them, a mar-k in the shape of 
a cr'oss was taped to ball A, and a line-mar-k was 
taped to ball B. 

In determining speeds, the appropriate displace- 
ments must not involve the iriitial clirster-positions 
of the balls. "Best choice" is the distance iietween 
the fir-st and second "scattertHl positions. 

The reference laboiatorv distance of 1 m be- 
tween tip-centers of the reference r-ods may not 
appear- to be in th(' samt? scale as the two prints 
pr-ovided to the stirdent. Hence, the scaling factor's 
will not be (>(|ual. If this ditfert* nee in scaling factor-s 
is tak(Mi into account bv the students the calcu- 
lated actual distance" will rt* pix)iluce the labora- 
tory conditions 

After- deterniinlng the speeds and the magni- 
tudes of the individual momenta, the student must 
add the severi individual momentum vectors to- 
gether The vector- addition prxx-eeds most simpiv 



if the head-to-tail method is used. The order of the 
vectors being added is immaterial because vector 
addition is commutative." 

In our measurements and calculations, system 
momentum, befor-e and after, differed by 2.1% in 
magnitude and by 1.5° in angle. Kinetic ener^' fell 
from 5.22 J to 1.36% i26% conser^edl. This repre- 
sents a very violent set of collisions, especially 
when we remember that we used case-hardened 
steel balls. 

E3-0 CONSER\'ATIOX OF ENERGY . I 

Equipment. 
Method A 

One pair of dynamics carts with weak 10.025- 

cmi springs 
Four 1-kg masses 
Two light sources 
Polar-oid camera and film 
Motor str-obe unit 
Masking tape 

10 X magnifier- and scale and/or mm ruler 
Graph paper 

Method B 

Disc magnets 

Plastic iDvlitei beads 

Ripple tank 

Folar-oid camera and film 

Camer-a tripod, or other means of supporting 

camera above ripple tank 
Xenon strobe lor photo-flood lamps together 

with motor strobe uniti 
Steel or stvrofoam balls 
10 X magnifier and scale and or mm railer 
Graph paper 

Method C 

Air tr-ack and glider, blower 
Meter stick 
Sto|)watch 

lOptional; camera, light source, disc strx)be, 
graph paper' 

Students could analvze their data fttjm E3-3 for 
other conserved quantities, as a homewor-k assign- 
ment befor-e beginning this experiment. For the 
elastic collisions, thev will find that mv' is con- 
served, lit will not be conserved in the explosion, 
of cour"se. Other quantities like /n v and m'v may 
be conserved in special cases: for instance, m\ is 
conserved in an elastic collision between equal- 
mass objects.) 

Method A. I^rnaniirs (]arts 

Betor-t" the students stat1 photographing collisions, 
demonstrate a slow collision as follows. Remove 
one screw from the spring bumper of each cart 
and arrange the caris as shown in Fig. 1. .Add 1 oi- 
2 kg of extra mass to each carl. The combination 
of large mass and weak spring prxiduces an ex- 
tremclv slow interaction students can see clearly 
thai there is an instant v\ hen Ixitb carls are moving 



Z24 



I NIT 3 / THE TKIl'MPH OF MECH/WICS 



quite slowly. To take photographs, replace the 
screws so the bumpei-s are in their orij^inal shape. 
Two kinds of spring are supplied in the d\ namics 
cart kits; use the lighter one 1 0.025 cm: for this ex- 
periment. 



E 



3 



e=S)«=3 



Fig. 1 



Vou must limit the amount of compression of 
the bumpei-s b\ taping corks or rubber stoppers to 
the front of each cart, as shown in Fig. 1. Othei"v\ise 
too \ioient a collision ma\' bend the bumpers per- 
manentK' and make them unusable for subsequent 
experiments. Of course, a collision in which the 
bumper touches the stoppers will not be elastic 
and should not be analyzed. 

If the carts are mo\ing too slowh , successive 
strobe images will be too close together to meas- 
ure. 




- tn 



Fig. 2 Momentum of each cart, and total momentum, plot- 
ted against time. 




It is essential to be able to identify simultaneous 
images of the t\vo caits. The technique suggested 
to students in the Handbook is probabK' the sim- 
plest. Another is to close the camera shutter before 
either cai1 goes out of the field of \ iew, and \\ ork 
backwarxls from the last image of each cart. 

Transparency 23, slow collisions, is made from 
a photograph taken in this experiment. If your stu- 
dents don t succeed in getting a good photograph 
they could analyze data from the transparenc\ in 
the same wa\'. 

Method B. Magnets 

See £3-3 for an alternati\e method of floating" 
ring magnets. 

The following photograph iFig. 4i shows up the 
right angle' law \ety nicely. In an elastic collision 
between two equal-mass objects, the angle be- 
tween the rvvo \elocities after the collision is 90°. 
This is quite independent of the impact parameter 
(amount of off-centeredness i of the collision. 



Fig. 3 Kinetic energy of each cart, and total kinetic energy, 
plotted against time. 




Fig. 4 Photo of a two-dimensional "collision" between disc 
magnets. Strobe rate about 30 per sec. 

Students should be aware of this right angle law, 
but need not necessarily understand the proof, 
which involves both conservation of momentum 
and of kinetic energy. The proof follows: 

m^ = mg and v^ - 

The vectors v^, \\ . v^ can be represented by three 
sides of a triangle. 




Rg. 5 ^ 

When the masses are equal and the initial \eloc- 
ity of B is zero, the kinetic energy equation 



Vzm^v/ -I- ^/znn^v 
simplifies to 



= V2m^' 



-I- \' 



4- V2m„v„ 



EXPERI.ME.VT NOTES 



225 



This is the IMhagoi-ean equation for the thi-ee 
sides of a right triangle. If v^^ = v^"' -t- v,,'^, then 
the angle between i'/ and v,,' is 90°. 

In fact there usually is a steady loss of kinetic 
energy as the magnet moves across, \otice from 
the cuive plotted in Fig. (i that the rate of energy' 
loss is about constant. Althcnigh thei-e is a signifi- 
cant di-op in total kinetic energy during the inter- 
action, the KJi, after the interaction is about what 
it would have been if no interaction had taken 
place. 

This is not an easy experiment because the in- 
teraction is over so quickly, much more quickly 
than th(; collision between two carts with spring 
bumpers. In Fig. 6 la plot of data from the photo- 
graph reproduced in Fig. 4), thei-e is only one pair 
of values for velocities during the collision, but the 
kinetic energy minimum is nevertheless brief. The 
photograph was taken with the Stansi strobe rate 
at about 30/per sec. 



f 




Fig. 6 

Method C. Inclined Air Tracks 

The expression A lP£ I — - A iKii) describes "ideal' 
behavior rai-ely met in practice. Students will pi-ob- 
ably find that the air track is far from perfect. Thei-e 
may be some loss due to poorly fitting gliders. 
There may even be a gain in energyl The air pit^s- 
sure is slightly higher at the inlet end of the track, 
which tends to blow' the glider awa\ from that 
end when th»' track is horizontal, or push it uphill 
wIkmi th(^ inl(U is at the low end. It is moiv impor- 
tant that students analyze theii- results honest l\ 
and suggest reasons for an\' differeiu'e between 
A(/'/:i and - AlKKi than that they ' pmve that me- 
chanical energy is consei\ed. 

'/zmv^ = rna h 



x = \ 2,1 /i 



Note that v depends only on h, not on the slope 
of the track. In theory, therefore, v should be the 
same for all these trials. In pra ctice we found that 
V, is constant and is equal to Vza^/j to VNathin 10%. 

Disc strobe technique with light source on the 
glider, or xenon strobe technique with a white or 
metallized drinking straw mounted on the glider 
may be used. For this experiment it is essential to 
measure \'elocity in meters per second. Therefore 
the strobe rate and the scale factor i preferably 10:1 1 
must be known. 

In one trial iwith the air inlet at the low end of 
the trackl, we found a significant decrease in total 
energy. The kinetic enei-gy increase was less than 
the potential energy decrease, presumabK' due to 
the air pressure effect mentioned above. When the 
air inlet was at the top of the track we found the 
reverse: 

AlK£) > -A(P£) 

When the glider is at rest at the bottom of the 
track, all the energy' 's stored i potential energy i in 
the stretched rubber band. The stretching of the 
rubber band as the glider slows down and stops 
can be seen much more easily if the band is made 
quite slack. This is analogous to the slow collision 
between two dynamics cai1s with spring bumpers. 

The simplest wa\' to measure the energy lost at 
each rebound would be to measure the height of 
successive i^bounds. 

£, = ma^h^ 

:AE = ma^^h 

These film loops could be used: 

L18 One-dimensional collisions. I 

L19 One-dimensional collisions. II 

L21 Two-dimensional collisions. I 

L22 Two-dimensional collisions II 

L31 A method of measuring ener^': nails driven 

into wood 
L32 Gra\itational potential enei^ 
L33 Kinetic energ\ 

L34 Consei-vation of energ\ : pole \ault 
L35 ConseiAation of energy; aircraft takeoff 

Discussion 

The different obsei-\ations made by the \arious stu- 
dent gixjups should be summarized in a class dis- 
cussion after the experiments have been done. 
Some of the points to bring out are: 

1. I'nlike momentimi, kinetic energy is not con- 
sened in all the situations investigated It is not 
consei-ved in elastic collisions or e.xplosions, for 
examphv 

2. Although the kinetic enei-g\ after an interaction 
ifor example, slow collision' ma\ return to its 
initial value, therv is a temporal disappearance 
of kinetic enei^' during the interaction, that is, 
while the springs aiv touching or while the mag- 
nets aiv close to each other Ibis is the time to 



ZZ(> 



IIIVIT 3 / THli THlliMPH OF MECH/WICS 



mention potential, or stored, energy and some 
of its forms, elastic and magnetic. 
3. The air-track glider pushed up the track comes 
to rest momentarih' near the top, increases 
speed coming down again, comes to rest mo- 
mentarily at the rubber band, mo\es back up 
again, and so repeats the cycle. The original ki- 
netic energ\' is comerted to gra\itational poten- 
tial energy, back to kinetic energ\', to elastic po- 
tential, back to kinetic energv', and so on. 

Ansivers to questions 

METHOD A 

1. Ves. 

2. The momentum was almost completely con- 
served. 

3. Yes. 

4. It would be displaced downward and have a 
smaller energ\' dip. 

5. Yes. Student answei-s will \ar\'. 
METHOD B 

1. Energ\' is almost conserved. 

2. 10%. 

3. Then energy is not conserved. 

4. No. Some energy is stored in the magnetic fields. 
METHOD C 

1.-2 Note abo\e discussion. 



E3-6 COXSER\'ATION OF ENERGY. II 

METHOD A: Fam Loops 

See notes on Film Loops 19-36 beginning on Hand- 
book page 157. 

METHOD B: Stroboscopic Photographs 

See previous notes on the photographs. 
Equipment: 

An\' of Film Loops 1 9-35 

Film loop projector 

Stopwatch, or strip recorder i"dragstrip"l 

Stroboscopic photographs lin the Handbook) 



E3-7 MEASLTUxXG THE SPEED 
OF A BLT.LET 

Equipment: 
Method A 

Air track and glider 

Gun and projectile 

Juice can plus cotton wadding 

Stopwatch 

Meter stick 
Method B 

Can plus wadding, or soft block 
If the bullet bounces back a bit, the cart will gain 
more momentum. Frtjm the i-elation 

..\/ -r m 



we have 




1.57 X 10"' 

V = X 9 X 10 - 

3.5 X 10"' 


J'" 

Vl.78 


= 0.404 X lO' \ 5.6 




= 4.04 X 2.36 




= 9.55 




= 9.6 m/sec 





Answers to questions 

METHOD A: 

1. Student answer. 

2. If the bullet bounces back a bit, the cai1 will gain 

(A/ + miv' 

more momentum. Using v = would 

m 

lead to too high a \ alue of \ . 

3. (ai Use time-of-flight technique with two beams 

of light and X\\o photocells as described in 

notes on the use of oscilloscope, Resource 

Book. 
lb) Fire horizontally and measure the horizontal 

distance traveled as it falls to ground. 
Id Fire verticalK' upwcuds: measure maximum 

height attained. 
idi Time-of-flight measurement described in 

Handbook Unit 1, page 30. 

METHOD B: 

1. Student answer. 

2. A large fraction of the bullet's kinetic enei^ has 
been converted into work tearing apart the wood 
and generating heat. 

3. Same as 3 above. 

Using Method B, the Ballistic Pendulum, a light can 
be attached to a pendulum and photographed. 
From the photograph, the distance d may be meas- 
ured easily and v computed, given the masses of 
bullet and block. 



Ea-8 ENERGY ANALYSIS OF A 
PENDLXLTVI S\\1NG 

Equipment: 

Pendulum i about 1 m longi 

Polaroid Land camera 

AC blinky, or light source and motor strobe 
This is a relativelv straightfonvard investigation 
that relates the kinetic and potential energies of a 
pendulum bob, and suggests that the total energy 
is constant. The potential enei^ of the raised bob 
is calculated from: 

PE = ma^lh 

Since the mass is constant during the experiment, 
the relative kinetic and potential energies can be 
derived by neglecting the mass. 

The largest uncertainty is likely to be in the 
measurement of the velocity of the bob at the bot- 
tom of the swing. 



LXPERI.MENT NOTES 



227 



E3-9 LEAST ENERGY 

Eqiii|)in(;nt: 

1 in of headed cliain 
Graph papec, luler 
Weights and string 

The principle of least energy, illustrated by this 
laboratory experiment, is a very useful principle. 
As Feynnian states in his Lectures, Vol. 2, page 2, 
" The average kinetic eneig\' less the average poten- 
tial energv' is as little as possible for the path of an 
object going from one point to another." Although 
sometimes tenned the principle of least action, the 
moi-e general idea of least energy applies to soap 
bubbles as well as hanging chains. 

The hanging chain that has a uniform density, 
similar- to a cable or telephone wire, takes the 
shape of a catenary. It is not a parabola. The article 
"Suspension Bridges" by Thomas B. Gi-eenslade, Jr., 
in the Rhvsics Teacher for- January 1974 discusses 
the differ-ence. When a dead weiglit like a roadway 
is supported at regular horizontal intervals along 
a suspended cable, the curAe through the points 
of suspension do fall along a par-abola. The two 
curves have similar' but not identical shapes. 

A catenary curve has the least potential ener-g\'. 

The paper by Greenslade suggests a variety of 
interesting and relatively simple activities of poten- 
tial interest to students. Building real as well as 
mathematical models may prT)\'e stimulating. 



E3-10 TEMPERATLTRE AND 
THERMOMETERS 

Equipment: 

As many as possible of the following: 

Uncalibrated merx:ury in glass thermometer 

Gas-pr-essur-e thermometer 

Gas-\'olume thermometer 

Thennistor- plus ampliHeivpower supply plus 

meter 
rhermocouple 
Lar-ge baths of boiling water, ice water, and 

four or fi\e water baths at intermediate tem- 

peratui-es 
Millimeter scales inot plastici to attach to un- 

calibr-ated thermometers 
As many differ-ent gases (CX),, \_,, O^, \_.0, etc.i 

as possible 

Temperature 

Students irsualK' ha\'e not thought aboirt what tem- 
peiatiire means, ar"e not awarv that an\' pri)blem 
of defining lempcratirre e.xists, and ma\ e\en be 
unwilling to admit that it does. Indeed man\' te.xts 
dismiss the iiroblcm with a statement like, "The 
temperature of a l){)d\- is the scale reading on a 
suitable thermometer," 

This e.xperinu'nt uill have succeeded if the stu- 
dents have been hrt)ught, [jossibly for the fii-st time, 
to think about the natun* of tlie concept of trni- 
prrminr The object of the expriiinent is NO! to 



have students cadibrate other thermometers against 
the mercury-in-glass thermometer but to realize 
that on first consideration the other devices ar« in 
themselves just as valid for use as thermometers 
land for use w+ien constructing temperature scales i. 
It is only much later, w+ien we have some theor-et- 
ical basis for our ideas of heat and temperature, 
that we will have any r-eason other than conven- 
ience for choosing one device over another 

What follows is a more than usuallv lengthv out- 
line of the sort of prelab and postlab discussion 
that we believe would make the students aware of 
the prxiblem. The title we have given to the e.xper- 
iment r-eflects the order in which the subject has 
to be developed: from the crude subjective sensa- 
tion of hotness and coldness, through the inven- 
tion of some objective device sensitive to changes 
in hotness, to the establishment of a temperature 
scale. 

The major point to bring out is this: Tempera- 
ture, like all other ideas that have been of great 
value to science, is an invented concept. .Acceler- 
ation is another example, which could have been 
defined as AvAs rather than Av A/, and indeed 
Galileo considered this. But \v \l turns out to be 
a much more useful definition. 

One point may summarize the whole problem. 
We frequently say and teach that Charles (or Gay- 
Lussac) found by experiment that the volume of a 
sample of gas is proporlional to its absolute tem- 
perature. He probably used a mercurv expansion 
thermometer. Suppose he had used a diffei^ent sort 
of thermometer lone whose expansion was very 
nonlinear with respect to the expansion of mer- 
curyi. What kind of law would he have discov- 
ered? 

Galileo used a thermoscope consisting of a 
glass flask with a long neck dipping into water. The 
water level in the tube rxjse or fell as the air in the 
bulb was cooled or heated. \'ou might want to add 
this to the v arietx' of thermometers used in the ex- 
periment. 

Any convenient, easily measured propertv' of any 
substance that changes with hotness (the original, 
subjective sensation tmm which we start i could be 
used to construct a thermometer, to define tem- 
perature quantitativeh . Notice that we cannot at 
this stage of the argument sav that the expansion 
of mercury ifor instance i is a good svstem to 
choose "because mercurv expands linearlv with 
temperatur-e. Onlv the rever'se is possible: We 
coirld irse the merx'uiA expansion system to define 
temperatirre Birt the choice is not obvious and a 
wide V arietx of s\ stems is available. 

,An optical pvr-ometer measures the temper-aturv 
of hot bodies bv comparing the light the\ emit with 
a standarxl ihot wirvi. 

We could not put a gas bulb between t\vo collid- 
ing carls lor insert it under a patient s tonguei and 
hope to measure a temperature rise We cannot 
irse a mer-cirr-x ther-mometer- below the freezing 
point or above the hoilrrig pomt of ineieuiA 



VZH 



l'\IT 3 / THK TRIl'MPH OF MECHANICS 



The original fixed points on the Fahrenheit scale 
were: 0°, the temperature of a mixture of equal 
parts of ice and salt, and 100", human bod\ tem- 
perature. Toda\ . the Fahi-enheit scale is defined bv: 
32°F, the temperature at which ice melts, and 
212°F. the temperatui-e at which water boils. 

There is no reason at this stage, either- theoretical 
or experimental, to suppose that the two themiom- 
eters should necessarilv agree. There is a small but 
significant difterence between a mereurv expan- 
sion themiometer and an alcohol expansion ther- 
mometer, rhere is a \er\' striking difference be- 
tween the thermistor thermometer and almost an\' 
other thermometer. 

Use as wide a \'ariet\' of dexices as possible: un- 
callbrated mereuiy. uncalibrated alcohol, gas \ol- 
ume, gas pressure, themiistor. thermocouple, etc. 
Assign a pair of devices to each group of students. 

Plotting a graph helps students to distinguish 
betv\een systematic differences and random ones. 

Ha\e one ice bath and one boiling water bath 
and at least fi\e other numbered baths covering 
the intermediate range. Note tha. except for the 
ice and boiling water, the baths need not be main- 
tained at constant temperature. Each group must, 
of coui-se, take the readings of its rvvo devices at 
the same time. 



Thermometers 

A. Uncalibrated mercury-in-glass. For e.vample, 
Cenco cat. no. 77320i. The volume of a confined 
sample of mercurv' is indicated bv a thin thi-ead of 
the sample that is free to rise in an empfv tube. .A 
centimeter scale should be fixed along the ther- 
mometer. A 30-cm wooden ruler iwith the brass 
strip removed! is satisfactorv when attached with 
rubber bands as in the figure below. The extra 
length provides a convenient handle. It is impor- 
tant that the scale be close enough to the mereun' 
thread to facilitate reliable readings. 

B. Uncalibrated alcohol-in-glass. iFor example, 
Macalaster cat. no. 2666i. The same principle is il- 
lustrated as in .A. A 15-cm steel scale is long enough 
to ser\ e here. Tape does not hold up well in boiling 
water. Rubber bands mav be used, but care should 
be taken to bind them tightlv enough to the ther- 
mometer and scale to prevent them from slipping 
out of alignment. .A small stick or glass i"od can be 
bound together with the thermometer and scale to 
help position the scale close to the liquid thread. 
With some maneuvering, the scale can be made to 
show through the glass directly behind the liquid 
thread. 

C. Thermistor. The electrical conductivitv of a 
semiconductor decreases rapidlv with increase in 
temperature. The ' lOOK" thermistor supplied by 
Damon has a resistance of about 400Kn at 0°C. 
lOOKfi at 25°C, and 4Ki). at lOO'C. Use a volt-ohm- 
milliammeter to measure resistance directlv, or 
connect a 6-V drv cell and a milliammeter in series 
with the thermistor and use current as a measure 



Pc»ver Supo^ij 



Amplir ler 



MeTtr 




tKcrmifttGr^^ ' 



of temperature. The maximum current will be 
about 1.5 mA at 100°C. 

It would also be possible to use the Damon am- 
plifier along with the thermistor as is indicated in 
the above schematic diagram. Caution: Turn the 
ac-dc switch to dc, wire the microammeter acixjss 
the temiinals marked meter output, and use the 
1.5-\' battery used in the pen-light souix:e. Then the 
OFF SET control of the amplifier can be adjusted 
so that the microammeter will read zero for any 
temperature sampling. 

The thermistor and leads to it must be insulated 
to pr-event conduction through the water. Slide a 
length of spaghetti or shrink tubing over it, or, if 
you use the thermistor embedded in lead, apply a 
coating of nail polish. 

D. Thermocouple. The amplifier unit can also be 
used to detect the differ-ence in voltage at the two 
junctions of a paii' of wires made of different met- 
als. .An irxan-constantan couple develops about 5 
m\ for a temperature difference of lOO'C. With the 
amplifier GAIN set to 100 the maximum output 
voltage will be about 0.5 \ . 



Power Supjoiv^ 



^p.iztcr 



+ 0-0 



<? 



gain n 
offset I 



constdita 




Crvft+Snt^n 



ijnk.no/vn ice water 
TCM-iperatore. 

Vou could simplifv the circuit by connecting the 
iron wire directlv to the amplifier ter-minal. This 
contact then becomes the r-eference junction and 
if it is less than r-oom temperature, the voltage will 
be negative 



tL\PERIME\T .NOTES 



229 



E. Gas Volume. Use a piece of capillary tubing 
closed at one end with a plug of silicone ixibbei- 
cement. A mercuiy index tiaps a fixed amount of 
air (see Pai1 B of £3-14, Ihe Beha\ioi of Gases). Use 
as many different gases as possible. 

F. Gas Pressure. "John s law' apparatus la toilet 
itjseivoir float plus absolute pressui-e gauge, for ex- 
ample, Welch Scientific, cat. no. 1602) can be used 
here. Use as many diffei-ent gases as possible. 



E3-11 CALORLMETRY 

Equipment: 

Foam plastic iStyrofoam) cups 

Thermometer 0-100° C 

Hot and cold water 

Ice (cracked or small cubes) 

Balance 

Metal samples 

This experiment introduces students to the idea 
that heat flow can be measured by observing the 
change in temperature of some standard sub- 
stance, for example, liquid water. Using this method, 
students learn that measured amounts of heat pro- 
duce differnt temperature changes in different 
substances. Also, students measure the latent heat 
of ice and the specific heat of a metal. 

The foam plastic IStyrofoam i cups are extremely 
useful for heat experiments and are inexpensive. 
Keep a good supply on hand, and encourage stu- 
dents to use them to improxise experiments other 
than the ones specifically described hert'. 

These cups are such good insulators that exper- 
iments can generally be done v\ith the cup uncov- 
ered. For example, when water originally 15°C be- 
low room temperature is placed in an uncxnered 
cup, it increases in temperature by less than 0.2 
°C/min. Since the marks on theimometer scales are 
usually a degree or more apart, the error intro- 
duced by heat leakage in the brief experiments sug- 
gested her-e is not much more than the un(-ei1ainr\' 
in the temperature measurement. The cups are 
also very light (between 2 and 3 gl and therefore 
absorb very little heat. It is not necessaiy at this 
stage to require that students connect their calcu- 
lations for this heat loss although the cori-ection 
could be made later as a separate experiment 

The prt!liminary experiment establishes the ap- 
pr-oximate rate at uhich heat leaks to or- fr-om the 
calorinn't('rs. and also prt'par"{\s students for a dis- 
cussion of lat(Mit heat ]i\\vr in th«^ period This pi-e- 
liminary work can be starled in the beginning of 
the period and carried along with \hv other- exper- 
iments, or- can be done the da\ belbrv It will gi\e 
stud(Mits some feeling tbi- the insulating character- 
istics of th(?ir cir|)s and also impress them uith the 
fact that a water-and-ic(* mixtur-e rxMiiains at exactly 
0° C until all the ice is melted. Claution students 
that the\' should stir Ihe water- gentl\ with the thet - 
rnonietcr heloi-e taking (M(-Ii i-eading 



Background 

1 he caloririieter was first used quantitatively to 
measure heat by Joseph Black 1 1728-1799). He 
made the three assumptions about the nature of 
heat outlined in the Handbook. 

Underlying the third assumption is the idea that 
temperature is a quantit\' that can be measured. 
The adoption of a temperature scale and basic im- 
pro\'ements in thermometer design in the early 
eighteenth centurx' by Fahrenheit and others made 
Black's work possible. 

Ar-ound the beginning of the nineteenth centuiy, 
the first assumption was shown to be inadequate 
to explain the r-elationship between work and heat. 
However, the caloric theory', as it was called, was 
extremely plausible: it served \ery well in devel- 
oping early ideas about heat, and to this day is stilJ 
implicit in many intuiti\e ideas used in calorime- 
iry. 

Mixing Hot and Cold Liquids 

Students know frtim experience that when the two 
quantities of water are put together, the tempera- 
ture of the resulting mixture will be between the 
two starting tempeiatur-es. Ask if they can predict 
the exact temperature of the mLxtur-e. This will be 
easy if the cups of hot and cold water ha\e iden- 
tical masses, but not so eas\' if their masses differ. 
Ulien unequal quantities of hot and cold water 
are mixed, the relationship 

holds. 

The calorie is the CGS unit of heat. It is a con- 
venient unit for the calorimeter experiments de- 
scribed her'e, but is too small for most practical 
applications. The MKS unit, the kilocalorie ikcal or 
Call, is the heat that enter-s or leaves 1 kg of water 
when the temperatur-e changes b\ 1° C. This unit 
is the same as the Clalorie used by dietitians. 

Answers to questions 

1.-17. Student answers. 

Measuring Heat Capacity 

The next rvasonable qirestion to ask is whether the 
constant of prT)por1ionalit\ c in the heat equation 
is the same for other materials as it is for water. 
Provide students with small samples of \arious 
metals and ask them to predict the equilibrium 
temperature when a hot metal sample is mixed 
with cold water in a calorimeter. 

The thrvad holding the metal samples should be 
tied to a \N-ooden stick so that hands do not get too 
close to the steam. 

Specific Heat Capacities of Metals 



aluminum 


0.22 cal g C 


brass 


0.08 


copper 


0.093 


iron 


0.11 


lead 


0031 


mercury 


0033 



'z:w 



HIVIT 3 / THE TRIlilVIPII OF MECHANICS 



Measuring Latent Heat 

The ice-and-water mixture may still be at 0° C. Even 
if the ice has all melted, it is pi-obabK' cooler than 
the cup originalK containing onl\ ice water. Since 
heat is leaking into both cups, what is happening 
to the heat that entei-s the ice-and-water mLxture? 
Because this added heat does not produce a 
change in temperature of the water while the ice 
is melting, this heat is called latent, or hidden, heat. 

After the ice melts, the resulting ice water also 
absorbs heat as it warms up to its final tempera- 
ture. The necessity' of consideiing this additional 
amount of heat in the equation seems to be diffi- 
cult for man\ students. Perhaps a comparison to 
compounding interest in a bank account, getting 
interest on interest, would be helpful. 

If students use ordinan ice cubes, their \alues 
will generalK' be less than the accepted \ alue of 80 
cal g. This is probably due to the presence of air 
bubbles and impurities. Excellent results can be 
obtained using the plastic cups if you make ice 
cubes from distilled water. Start with the cup about 
half full of \\ ater at a few degrees abo\e ix)om tem- 
perature and record its mass. .Add an ice cube that 
has been placed in a container of cold water for a 
few minutes so its temperature will be 0" C. rather 
than about — 10° C as it comes from the freezer. 
Dr\ off the ice with a paper towel to a\oid trans- 
ferring excess water to the calorimeter. 

Rate of Cooling 

Newton's law of cooling predicts that the rate of 
cooling is appro.vimateh' proportional to the dif- 
ference in temperature of a sample and its sur- 
roundings. This is an empirical result due to the 
combination of se\eral ph\'sical processes. Stu- 
dents should be able to report qualitati\ el\ that the 
greater the temperature difference between the 
sample and its surroundings the quicker its fall or 
risei of temperature. But don t expect a formal 
statement of Newton's law. 

Students who did not do the e.xperiment on la- 
tent heat will find that the ice water with ice in it 
warmed up less than the ice water. Use this ob- 
servation to introduce latent heat, or let those stu- 
dents who did that experiment report their results 
to the rest of the class. 

Because the conditions of each experiment 
iquantitA' of water, amount of stirring, etc.i are dif- 
ferent, students can onl\ get an estimate of the 
error due to loss of heat from the calorimeter. In 
one trial, 100 g of water 25° C abo\ e room temper- 
ature lost about 2° C lAH = 200 cali per minute. 

Questions for Discussion 

1. Students could be urged to look for other fac- 
tors that can intixjduce errors in calorimeter ex- 
periments, and to suggest possible remedies. Here 
are a few examples: 

lai The thermometer must take up some of the 
heat in the calorimeter. If we knew the specific heat 
of mercur\' and glass and the mass of each that is 
immersed in the water, we could make allowance 



for this factor. This would be \er\' difficult to do 
because the relative amounts of glass and mercury 
are unknown. Perhaps it would be possible to 
measure the heat capacit\' of a thermometer di- 
rectly b\' experiment, but in an\' case the enx)r re- 
sulting fixjm neglecting this is quite small. 

ibi Heat is lost while samples are being trans- 
ferred: the hot water and metal samples will lose 
some heat while being transferred: some ice will 
melt after it has been dried but before it is put in 
the calorimeter. These will be difficult to estimate, 
but they make relati\ely small contributions to the 
total error. 

ici The major source of uncertaint\' is probablv 
the thermometer reading. Note that the larger the 
change in temperature during the experiment, the 
smaller will be the fractional error in \t due to 
uncertaint\' in reading the thermometer. 

Another source of uncertaintv' in the thermom- 
eter readings is the wa\ in which the thermome- 
ters are calibrated. Some are calibrated to read cor- 
rectly when the entire themiometer is immersed, 
while others should ha\e onl\ the bulb in the liq- 
uid. In the latter case, it is possible to correct for 
the length of the exposed thread of mercur\'. This 
correction will, howe\er, be very small. 

2. .After students ha\e measured the latent heat 
of melting ice, the\' should consider wa\'s to meas- 
ure the latent heat of steam. Section 10.6 of the 
Te\t emphasizes the importance of Watt s im- 
proxement of the steam engine, which was based 
on his realization that the condensation of steam 
releases a large amount of heat. In fact, the latent 
heat of \ aporization of water is about se\ en times 
the latent heat of fusion of ice. The latent heat is 
measured by bubbling steam through cold water 
in a calorimeter cup: the procedure is the same as 
for ice. Howexer, \ou ma\' not wish to expose stu- 
dents to the possibility of burns from the li\e 
steam. 

The following notes on Experiment 3-12 suggest 
a relati\ eh- safe wa\ to estimate the heat of \ apor- 
ization of water. 



E3-12 ICE CALORLMETRV 

Equipment: 

Three St> rofoam cups 

Small light bulb, wiring, electrical source 

Ehe I such as India inki 

In this experiment, the student becomes ac- 
quainted with the concept of heat of fusion, or 
melting. Onl\ the ice-water s\stem is explored, but 
students should expect that similar heat ex- 
changes occur when other materials ha\ e a change 
of state. 

Like Joseph Black, they may also conclude that 
a sizable quantity' of heat is required to change 
water into steam. Black noted that a pan of water 
could be heated to boiling, but that continued 
heating was required to boil oft" the water Ihe 



EXPERIMENT NOTES 



231 



water did not all instantly turn into steam when 
its temperatui-e reached the boiling [joint. 

Students might speculate on the histoiy of the 
eailh and possible technohjgical consequences if 
water had no lu;ats of fusion and of vapoiization. 
Some itjugh estimate of this heat of vapoiization 
foi- water can be made when a pan of water is put 
on a constant heat souixre. A time sequence of ther- 
mometei- leadings as the known quantity of water 
heats up will define the rate of heat injjut. The 
amount of time n;(juiit;d to boil off the water then 
indicates the relative quantity of heat needed to 
vafjorize each unit of water. Again, students should 
be encouraged to assume that (Jther mateiials ha\e 
similar heats of vaporization. 



Three experiments of 50 trials each were made 
with iV I number of target marbles) = 8. The meas- 
ured length of the line of 8 marbles is 11.1 cm. 

Thus, d = — — =1.4 cm. The target field should 

8 
be at least 25 cm from the launching board oth- 
ervxise the bombarding mai-bles will not be mo\ing 
in parallel and too many hits will result. The ob- 

LJ 

served ratio, — or fi, of hits to total trials was; 
T 

37 31^ 35^ 

50' 50 50 
34 ± 3 



The mean ratio is 



50 



E3-13 MOiNTE CARLO EXPERLMENT OX 
MOLECLTLAR COLLISION'S 

Equipment: 
Game I 

12 marbles 

Board studded with nails 
Board marketl off with coordinates 
Some sticky wax 
Game II 

Large piece of graph paper on a drawing board 

Both the formula for mean free path and that for 
viscosity are approximate. A rigorous deri\'ation 
may be found in an intemiediate text on kinetic 
theoiy. 

You cannot give a precise definition of random- 
ness nor a clear-cut exposition of this technicjuc 

You might get a random set of numbei-s In- open- 
ing a metropolitan telephone diit^ctory and taking 
the last two digits from the call numbei-s as they 
occur. Another way is to take a linear expression, 
say ll,v -I- 7. Plug in a number for ,v, say 54. Com- 
pute the value of the expression, namely 11 x 54 
+ 7 = 601. Take the last two digits of this \alue 
for the next number of the random series and pro- 
ceed as before. Thus, 11 x 01 -I- 7 = 18, 11 x 18 
+ 7 = 205, etc. The series is 54, 1, 18, 5, 62, 89, 86, 
53 . . . etc. The table of random numbers gi\en is 
from Slatisticiil rnhlcs. by lishei- and Nates. Ihe 
fii-st digit of the number taken from the table is the 
abscissa. 

in (ianu* I, both target molecules and hombaiti- 
ing mohuiile have the same Unite diameter-; but in 
(Jam(! II only the target molecules have a finite di- 
ameter-. The bombar-ding molecule is a test parlicle 
with no diameter-. 



Then d = 



HP _ HD_ _ 

2A'T ~ TZ\' ~ 
= 1.38 ± 0.13 cm 



34 



31 



50 



32.5 cm 
2X8 



Shielding effect becomes important as A' is in- 
c[-eased. 



For,V = 



88 



H 
12, — = 

T 150 



gi\ing d = 



88 
150 



32.5 cm 
2 X 12 



= 0.80 cm. 



H 43 

For iV = 20, — was — 
7' 50 

43 32^ cm 

gixing d — — X 

50 2 X 20 



= 0.70 cm. 




Sample ReNiilts (Game II 

laigct marbles wrn' set up, using random rurm- 
b(Ms, on a 6-bv-6 grid (5-cm spacingi Bombar-ding 
marbles wei-e r-eleased throirgh the pinbail ma- 
chine ar-ra\ of nails illustrated in tin* Hniidhook 



Wandering horse in a snow-covered field 



Z3'Z 



iJ\iT 3 / i'hl: rKiiiMPii or mkcH/WICS 



Sample Results (Game II) 
50 




lo a> so ' 4o 

Random array of square target molecules. 



Answers to questions 

1.-2. Student answeni. 

sum of path lengths 



3.L = 



4.L - 



number of paths 
A 

\'d 

A _ I2,500i 

XL ~ 



1,354 
50 



27.1 units 



= 2.31 units 



I27II40I 

The approximation by means of the Clausius re- 
lationship is half as much as it should be. Per- 
haps what is uTong is that a dimensionless par- 
ticle is used rather thcin another square molecule. 

A A 

Therefore, if L = — is changed to L = -, 

Xd ^ A'(2di 

the calculation in 4 will turn out to be 1 unit. 
Since the incident molecule must ha\e a dimen- 
sion, the result is a reduction in the mean free 
path. If the molecular diameter-s are equal, the 
mean free path is hal\ed. 



E3-14 BEHAMOR OF GASES 

Equipment: 

I. Bo\le s law apparatus: Either conventional 
J-tube, 

or simple s\Tinge t\pe 

iMacalaster #30220; Linco #6250: Damon 
#99129 

Set of weights for use with s\ringe, hooked 
or flat 

Other gases such as CO,, \,, O,, i\,0 if pos- 
sible 

II. Capillaiy tubes 
Mercury 

Silicone rubber sealant 
Millimeter scales imetal or wood, not plas- 

tici 
Beaker of water 
Bunsen burner or hot plate 
rhemiometer-s 0-100" C 



EXPERIMENT VOTES 



233 



For an extended account of Boyle's work see 
Harvard Case Histories in E^cperimental Science, 
edited by J. B. Conant. Foi an edited, annotated 
version see Great E^cperiments in Physics, h\' Monis 
Shamos. 

Detailed instiiictions for use of the conventional 
Boyle's law apparatus ai-e found in most physics 
and chemistry laboratory manuals. 

The simple plastic: syiinges are preferable be- 
cause they are inexfjensive and easy to use: stu- 
dents can obtain data frtjm them v\ith a minimum 
of preliminary disc;ussion. No mercurv' is used: nor 
need one explain the relationship between height 
of mercuiy column and pressure. Here pressure is 
simply force (weight on piston i divided by the area 
of the piston. You will probably need to remind 
students that weight is measui-ed in newtons and 
is equal to mass times a^. The numbers WTilten on 
"weights" 1100 g, oOO g, etc.) are actually masses . 

The data obtained with this equipment are not 
very precise, but since the primary purpose of the 
experiment is to show how the data are analyzed 
and interprtited, the syringes art; adequate. 

Grtjase the piston with glycerine or vaseline to 
r-educe friction. Make sure that no water gets into 
the CA'linder. 

Another way to imprtne the data is to record two 
readings for each forx:e apfilied to the piston: one 
when the plunger is lifted slightK' and released, 
and the other when the plunger is depressed 
slightly and released. The mean of the two readings 
is the value used. 



I. Volume and Pressure 

To get a l\^ \eisus \' plot that is comincingly not 
a straight line, one must work o\er a rather wide 
prt!ssur-e range using weights of up to a few kilo- 
grams. But to get a good \alue for the inter-cept of 
the P^^ ver-sus lA' plot, data at relati\el\' low P^^ val- 
ues are needed ibelow 1 kg of added weighti Ir^' 
to have some students work at both high and low 
pr-essur-es. 

A common difficulty students encounter- in 
working with gases is the tendency to confuse ab- 
solute pressure? with gauge pr-essure. The prxjce- 
dure described is designed to show that Boyle's 
law as it is usually written, 

PV = constant 

is only true if P is the absolute pressure. Students 
vary the foixre exerled on an air sample by placing 
weights on the piston. Ihcn the\ are to conxert 
these forx-es to pr-essiri-e b\' measuring the diameter 
of the piston and computing its area. Ihis step 
would not be necessary if the purpose of the ex- 
periment weiv simply to demonstiat«; a linear re- 
lationship between 1 \ and /' HowcNcr, when stu- 
dents use \ alues for /' obtained fi-om their- data, not 
taking into account atmospheiic |)ressur-e, the\' 
lind that their- graph of 1 \ against /' does not pass 
Ihrough the origin Instead it should pass thix)ugh 



the P-axis at about - 10 \/cm. Be sure students 
understand the significance of this intercept value. 
Point out that if a constant term iF^ = 10 N/cmi 
is added to each value of the calculated pressure 
P^, the graph would pass thixjugh the origin. This 
means that 



iP^ + p ) = k- 
V 



(P^ + PjV = constant 




Explain that P^ is the additional, constant pres- 
sure exerled on the piston b\' the atmosphere, and 
that the total pressure of the gas in the syringe is 
iP^^ -I- Pj. This is called the absolute pressure. The 
quantity P„ is usually called gauge pressure be- 
cause it is the pressur-e that is most commonK- 
measured by a pressure gauge. 

The similar beha\ior of different gases is an im- 
por-tant point because it suggests that the same 
model ikinetic theoryi could be used to explain the 
behavior of all gases. 

Have different students work with different gases 
if at all possible. 

Answers to questions 

V^arious Way's of Plotting the Results. 



8 

7 
6 
51- 



cc 



3 - 
X - 
1 - 






»-t»o l'^) 



234 



I'NIT 3 / THK TRIl'MPH OF MECH/WICS 



I 0-8 - 



8 

V 



Ob - 



2 - 



o-Z 









4-00 



«oo 



goo 



— 1_— 
looo 



i,xoc 



~p (_fl>1>£:>> n/^s^- &frA^-'S_) 



II. \oluine and Temperature 

In place of this experiment, \ou can substitute a 
quick and effective demonstration. A small flask or 
test tube is fitted with a one-hole stopper. .A long 
capillar^' tube with a mercur\ pellet near one end 
is pushed through the stopper. The flask must be 
perfectK' dr\- and the mercun pellet near the lower 
end of the tube at room temperature. The e.xpan- 
sion of air is made \i\idl\ apparent as the flask is 
heated in a water bath. 

Of course, if you want to make quantitati\e meas- 
urements of change in \olume with temperature 
you must relate the \olume of the flask to the \ ol- 
ume iper unit lengthi of the capilla^^• tube. 



The plastic sxringes are supported b\' a ring- 
stand and immersed in a beaker of cold water. Stu- 
dents record temperatures and \olumes as the\' 
slowi\- heat the water. Friction between the piston 
and syringe is again the major source of uncer- 
taintA' in measuring the volume. The change in 
temperature must be \er\- gradual and the water 
continuousK stirred so that the air in the s\Tinge 
will be at about the same temperature as the water. 
This apparatus will do little more than show that 
air expands as it is heated. Because of the large 
amount of friction in the s\ringe, no pretense 
should be made that students can determine the 
value of absolute zero by extrapolating their V-T 
cur\e to find the intercept on the T axis. 



J I 1 1 1 1 1 1 i 



ltx> <^oo ^00 goo '000 



"p 



EXPERIMENT VOTES 233 



Detailed dii-ections are given below for- prepaiing 
the constant-pr-essui-e gas theniiometers referred 
to in the llnnclhonk. 

Answers to questions 

4. Even if a graph of V against 7' is a straight line, 
this shows only that air expands with increasing 
temperature in somewhat the same way that 
mercury in the glass themiometer does. If the 
optional experiment "Temperature and Ther- 
mometers" iE3-10\ is not done, there should be 
a brief discussion of temperature scales as out- 
lined earlier. 

5. The behavior of a solid is usually linear over only 
a limited temperature range, which, for some 
solids, is too small to be of any use in thennom- 
eters. V\'ithin th(> limits of oui experimental error 
any liquid will gi\e a straight line if the temper- 
ature is not near its boiling or freezing point. 

The graphs for gases are more nearly straight 
than for li(|uids, if the temperature is not near 
the point where the gas liquifies. 

I'he failure* of a material to expand or contract 
in a periectly linear manner is largely a conse- 
quence of the forces of attraction acting between 
its molecules. Since these forces are least in 
gases and greatest in solids, gases pm\ide the 
most nearly perfect themiometers, but not the 
most convenient in size. Liquids are more con- 
venient, and for most purposes, sufficiently "lin- 
ear." 
6.-7. The lowei- limit is reached when the gas is so 
cold that the molecules are motionless and in 
contact with each other like maHiles in a box. 
While we cannot reach this state of affairs, our 
straight-line graph of 7" vei-sus I'V will identifv 
this temperatuit; when it is extrapolated to very 
small volume. 

8. When weights are added to the piston and the 
pressure increased, the temperature goes up be- 
cause work has been done on the gas. However, 
since the sample in our case is small and is not 
in an insulated container, it quickly returns to 
room temperature. 

9. The relationship between volume and temper- 
ature will continue to l)e a linear one within the 
accuracn' of our experiment as long as the tem- 
perature is well above that at which the gas 
liquifies. 



I^quipment Vote: Assembling a ('onstant- 
l*ressur»; (ias 'I'heriiioiiieler 

About 15 cm of ( aijillai^ tube makes a theiinom- 
eter of convenient size Ihc dimensions of the tube 
are not critical, but it is veiA' important that the 
bore be drv. It can be diied bv heating, bv rinsing 
with alcohol and waving lapidlv , oi- better still, bv 
connecting it to a vacuum jnimp foi- a few mo- 
ments. 

Ihe dI^ ( aijiliaiA tube i.s dipped into a containei 
of mei-cuiA , and the end sealed with the lingertip 



as the tube is v\ithdrawn, so that a pellet of mer- 
cury r-emains in the lower end of the tube. 

The tube is held at an angle and the end tapped 
gently on a hard surface until the mercury pellet 
slides to about the center of the tube. 

One end of the tube is sealed with a dab of sili- 
cone 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 boiling water for a few 
moments. 

A scale now must be positioned along the com- 
pleted tube. The scale will be directlv over the bore 
if a stick is placed as a spacer next to the tube and 
bound together with r-ubber bands. A long stick 
makes a convenient handle. The zero of the scale 
should be aligned carefully with the end of the gas 
column, that is, the end of the silicone steel. 






I ■ , VI \.^,V.1-^^-»^ 



^llt > I llt > 



' Stt «w 



■)f ica.CA. 



In use, the thermometer should be completely 
immei-sed in whatever one wishes to measure the 
temperature of, and the end tapped against the 
side of the container gentlv to allow the mereury 
to slide to its final resting place. 

To fill the thermometer with some gas other than 
air, connect a capillary tube to the gas suppK by 
a shor1 length of rubber tubing. Open the gas valve 
slightly to flush out the tube, and fill it with gas. 
Detach the rubber tube. Pick up a [lellet of mereury 
as befor-e. Keep your finger over the far end of the 
tube while you r-eplace the rubber tube igas valve 
shut I. Lav the thermometer down flat. Work the 
mercurA' pellet to the center- of the tube and. open- 
ing the gas valve slightlv and ver^' cautiouslv. re- 
lease your finger for an instant. 

Remove from the gas supply, seal off the end that 
was connected to the gas suppiv with silicone arid 
attach the scale. 



E3-lo WAVE PROPERTIES 
Special \ote: 

Espcrimcnt 3-15 begins a series of laboraton- ac- 
tivities dealing with waves. These experiments aiv 
not onlv inteivsting in themselves, but thev build 
towarxl a discirssion of the naturae of light as inter- 
teiiMice phenomena lead to a wave theory for light 
(iradirallv the student pr-ogr-esses from obvious 
pirlses and waves in springs to waves in a liquid, 
then to audible soirnd waves, and on to inaudible 
irltrasonic waves. .At e>ach step, incrvased reliance 
is made upon instr-umental detection Ihe com- 
mon pi-opj'i1i»'s of wavelength standing waves, and 
interfei-eru-es shoirld be strvssed In I'nits 4 and .■> 
other discussit)ns and «'\peiimeiits will li-e.il the 
similar' properties of light 



Z36 



UIVIT 3 THE TRH'MPH OF .MECHAMCS 



Equipment: 

Waves in springs 

"Slinkx'" spring 

8-10 m of rope (clothesline) or a different 
spring that can be stretched to this length 

Waves in a ripple tank 

Ripple tank setup, complete with light source, 
beaches, and \ariable-speed \va\e generator 
with straight wa\e and two point soui-ces 

Paraffin blocks for wa\e barriei-s 

Rubber tubing i about 50 cmi for use as a wave 
"mirror 

dowel (20 cmi or broomstick handle i30-50 cm 
longi to generate straight pulses 

Sheet of glass with one edge 30-50 cm long to 
fit in tank, with washers as comer supports 
to adjust height (for refraction i 

Hand-dri\en stroboscope lor motor strobo- 
scope to be dri\en b\' handi 

Meter stick 

Clock or watch with second hand 

Large beaker or jai- for filling and emptying rip- 
ple tank 

Students should experiment with longitudinal 
pulses long enough to appreciate the difference 
between these and the trans\'erse pulses with 
which the rest of the experiment deals. 

Answers to questions 

1. The amplitude changes because of friction: The 
amplitude of the pulse is a function of its energ\' 
and, as the energ\' is dissipated by friction, the 
amplitude of the pulse decreases. 

2. That depends on whether the far end is fi^e or 
fixed. Presumably the other person is holding it 
down, so that the reflected pulse will be upside 
down. 

3. The speed of a wa\'e along a slinky, rope, or sim- 
ilar de\ice, is proportional to 

V[tension (D/mass per unit length i |Jii]. 

Therefore, increasing the tension should in- 
crease the speed of the wa\ e along the slink\'. 

4. The c onc lusions ought to be consistent with 
v X \/T'\x, but of course one would not expect 
quantitatixe results here. 

5.-7. The pulses pass through one another v\ithout 
being altered. During their collision, the location 
of a point on the spring at an\' instant is simpK' 
the \ ector sum of the two separate locations that 
the point would ha\ e occupied if the pulses had 
passed o\er it separateh'. 

8. They are imersely proportional. More precisely, 
their pixjduct, fX, is a constant whose \alue is 
the \elocir\' of the wa\es. 

E3-16 \\'A\TES IN A RIPPLE TANK 
Ansuers to questions 

1. Their directions make equal angles with the 
perpendicular to the barrier. 

2. If the barrier is conca\e, the reflected straight 



waves become curved and converge on a small 

area or a point. 
3. A pulse started at the focus should look like a 

mo\ ie of the reflected pulse run backwai-d. 

It might be intei-esting for some of your students 
to note that the intensity' of the diffraction pattern 
has a graph of roughly the following shape: 




4. The wave speed is less oxer the shallow area. 

5. The wa\'e direction is turned away toward a 
perpendicular to the boundaiy. 

6. The angle with the perpendicular and the 
speed both decrease as the wave crosses the 
boundary into the shallower area. More pre- 
ciselv 



sin 9, 
sin 9, 



where 9 is the angle the wa\'e direction makes 
with the perpendicular to the boundaiy. 

7. Since fX is a constant, increasing one quantity 
decreases the other. 

8. See figures on re,vf pages 364 and 373. 

9. As the wa\elength increases, the pattern 
emerging from the slits opens out like an un- 
folding fan, and the nodal lines make larger an- 
gles with the direction perpendicular to the 
barrier. 

10. The fewer wavelengths along the width of the 
barrier the less the distortion of the wax e train 
"downstream" from it. 

11. The smaller the opening the greater the angle 
of spread. 

12. The longer the waxelength the greater the angle 
of spread. 



E3-17 MEASURING \\'A\TELENGTH 

Three methods of measuring waxelength are de- 
scribed. The firet method lai utilizes a stroboscope 
to freeze" a pattern of waxes. Measurement of the 
distance between crests xxill proxide a xalue for the 
xvavelength. The second method ibi employs 
standing xxaxes. 

The third method ici utilizes the nodal and an- 
tinodal lines formed xxhen two sources haxe the 
same frequency of xibration. ,AJong antinodal lines, 
the distance to the two sources differ b\ a whole 
number of waxelengths. Students should start xxith 
a central antinodal line, such as point A in the 
Handbook illustration, and then measure the dis- 
tances to the txx'o sourc'es at increasing angles off 
center. 

The wax'elengths obtained will xaiy xxith the set- 
ups. 



EXPERIMENT ,\OTES 



237 



E3-18 SOLTVD 

Equipment: 

Amplifier power supply 
Oscillator plug-in unit 

2 small loudspeakers 

Funnel lor thistle tube) and 50 cm of rubber 
tubing to fit it ifor ear trumpet i 

V'aiious sheets of Styrofoam, metal, glass, par- 
affin, masonite, wood, celotex, etc., for ab- 
sorption tests 

Spherical balloon and CO, source 

3 ringstands v\ith adjustable clamps 
Meter stick 

Again, there is a lot of material here. One could 
easily spend several lab periods on these experi- 
ments. 



Ansivers to questions 

1. The sound is absorbed. Heavy cloth, such as 
velvet or tenycloth, does \'erv well. 

2. Student answer. 

3. The best patterns are probably obtained when 
the board is perpendicular to the line straight 
out from the speaker. The narrow board i two- 
edge diffraction I should be svmmetrical]\' placed, 
while the wide board i single-edge diffraction) 
should have its edge on that center line. The 
diffraction patterns are those familiar from op- 
tics. They should roughly have the following 
forms: 



8. The positions of the maxima are separated by 
a distance 





ihtens/'^ 



y 



'If or> A 



-"/i ''/> 



i 




4. Sound waves will be heaixl at the edge of the 
shadowed art»a. 

5. Neither the spacing of the minima nor the 
wavelength should depend on the loudness. 

6. The waxelcngth iiui-eases when the frequenc\ 
is decivased 

Note that these experiments parallel those done 
with a ripple tank in K.VIb' imuch mor-e (jualita- 
ti\ely. of course). 
7 I he closer \.hv two sources the more wi(iel\ 
spacc'd ail' the nodes 



-(^)-fe) 



where v is the speed of sound. Thus, changing 
d and changing /have completely equivalent 
effects. 
9. The wavelength changes inversely with the fre- 
quency. 

10. No. Speed is independent of intensity. 

11. Yes 



E3-19 LT.TRASOL^TJ 

Equipment: 

Amplifier power supply 

Oscillator plug-in unit 

3 ultrasonic transducers 

Oscilloscope or 

Amplifier power supply, microameter. and 

diode 1100,000 ohm resistor, optional' 
Sheets of test materials as listed under Sound 

above 
Meter stick 

With invisible, inaudible ultrasound, this exper- 
iment parallels the activities of the previous e.xper- 
iments. Students explore the transmission, reflec- 
tion, and diffraction of these waves. They then 
create standing waves and mav estimate the wave- 
length as with audible sound waves. Finallv, the 
students investigate the interference patterns they 
can establish. 

Primaiy emphasis is upon the qualitative char- 
acteristics of the waves rather than upon quanti- 
tative results, although those may be of interest. A 
vcilue for the speed of ultrasound permits compar- 
isons with the speed of audible sound. 

Answers to questions 

1. Energv that is neither transmitted nor reflected 
is absorbed. 

2. Ultrasound will be diffracted like audible sound, 
but the pattern of diffraction will be smaller be- 
cause the wavelength is shorter. 

3. The spacing of nodes for standing waves is not 
related to the intensitv' of the sound. 

4. In the two-source itwo-sliti setup, the spacing of 
the nulls will decrease as the separation of the 
sources increases 

5. I'se of the equation assumes among other 
things, that this is a wave phenomenon, that the 
frequency is stable, that the intensitv' is not sig- 
nificant, etc. 

6 rhe speed of ultrasound is indep)endent of the 
intensitv. 

7. The speed of ultrasound is the same as that of 
audible sound labout 333 m see'. Ihe actual val- 
ues mav vaiy somewhat with humidit> .Also the 
fivquenrv of the <»s(illat(>r mav he unceil.iiii bv 
as much as 10' 



238 



UNIT 3 / THE TRIl'MPM Ol MliCKAMCS 



Film Loop Notes 



SPECIAL NOTE ON THE 
USIT 3 FILM LOOPS 

In a number of cases, a single collision e\ent is 
described by a film loop, a transparencx', and a 
stixjboscopic photograph. Each stixDlioscopic photo 
ma\ be found in the Acti\it>' section of Handbook 
Unit 3 and a discussion of its use is included in 
this Resource Book in the section entitled Experi- 
ment \'otes. 

One way that a teacher might take advantage of 
this duplication is by solving a momentum con- 
servation pix)blem on a transparency while the stu- 
dents complete the same problem at their seats. 
FinalK , b\' projecting the film loop on the chalk- 
board, the actual collision can be demonstrated or 
resoKed. Other imaginative teaching strategies can 
be devised through the use of these three media 
to teach conservation of momentum and energ\ . 

/. I\'TRODi'CTlO\' 

Six different two-body collisions, occurring along 
one dimension, were filmed with a high-speed 
motion-picture camera. Each loop, L18-L20. con- 
tains tvvo of these collision events. LIS. in addition, 
shows establishing scenes, filmed at nomial cam- 
era speeds, which instruct the viewer about the 
experimental setup used to produce these colli- 
sions. L19 and L20 contain high-speed sequences 
only. The projector shows the high-speed footage 
in slow motion. 

Student notes for each loop ai-e also provided in 
the Handbook. One or more events can be assigned 
as study-period lor laboratorv-periodi problems. 
Students should work in pairs. 

The apparatus used in these filmed e.xperiments 
is described brieflv in the Handbook and in more 
detail in the Experiment \'otes of this Resource 
Book. 

The films should be projected upon a white 
sheet of paper taped to the wall. The 45-bv-60 cm 
desk pads often found in offices are a good source 
of such sheets. Even better are 43-bv-55 cm quad- 
rille sheets, lightly ruled, 4 squares to the centi- 
meter, available through drafting supply stores. 
The projected image should fill the paper sheet. 
Students also need a good plastic see-through 
ruler, marked in millimetei"s, and a stopwatch or 
other timing device precise to the tenth of a sec- 
ond. A pair of drafting triangles will be helpful for 
drawing accurately parallel lines. Precision up to 
three significant digits is possible if measurements 
are carefully made. 

The experiments involve marking pairs of vertical 
lines down on the paper to serve as "timing bars." 
Their positions land the distance between the 
lines I are to be chosen after initial viewings of the 
film and then accurately measured. Ihe passage of 
the balls on their horizontal paths can now be 



timed. Care must be taken not to move the projec- 
tor during the experiment. 

Students calculate speeds of the balls before and 
after collision in relative units; for example, in 
terms of the apparent speed acrxass the paper. Sec- 
tion II, which discusses the six events in detail, also 
describes fui-ther a tvpical series of such measure- 
ments and calculations. Ihe masses of the balls are 
given. 

Each collision is governed by the principle of 
conservation of momentum.' But these loops may 
be used to teach students far more than mere ver- 
ification of this principle, especially if a pair of 
events is assigned as problems, if only one event 
is assigned, this event should not be the fii'st ex- 
ample in L19. In many cases the assignment of one 
event may be enough. But better students mav well 
find time to study with profit a pair of events. 

The events occurring in each loop do, in fact, 
constitute matched pail's that mav be used to teach 
interesting additional lessons. In LIS the two col- 
lisions are mutually inverse. In one collision, one 
ball is reflected, whereas this is not the case in the 
other, a matter which often surprises the novice. 
In each loop, the same balls collide strongly and 
weakly, respectiv elv. Calculation of kinetic energ}', 
before and after collision, can lead to interesting 
discussion on the more imperfectlv elastic nature 
of the balls involved. Moreover, study of the two 
events in L19 contains a useful lesson about error 
propagation. These aspects ai-e discussed further 
in Section II, which follows. 

//. THE SIX EVE\'TS. A\'D HOW TO TAKE DATA 
The collision events present similar analytical 
problems. Measurements consist of timing each 
ball's motion past "timing bars" whose position 
and separation are chosen bv the students. In our 
discussion, we give details for the first example in 
LIS only. Onlv the special aspects of the other five 
events are discussed. 

The simplest problem of the six is the fii-st ex- 
ample of L20. There we deal with a perfectly in- 
elastic collision of two balls with one ball initially 
at rest. After collision, the balls move together as 
one "compound" particle. Onlv tAvo timings are re- 
quired. 

A schematic diagram is provided for each event 
both in the Experiment Xotes of the Resource Book 
and in the Handbook. It specifies the masses of the 
balls and qualitative conditions before and after 
collision. 



"Momentum is a \ector quantiH'. Since the collisions in this 
series of film loops are one-dimensional, the vectors in any one 
problem are all parallel or antiparallel. This simplifies the cal- 
culations. lx>ops 21, 22. 23. and 24 as well as one series of stro- 
boscopic still photographsi in\ol\e two-dimensional collisions 



FILM LOOP \OTES 



239 



L18 OXE-DCVIEXSIOXAL COLLISIONS. I 

I'irst csmnpltt. Sr-c llic IaciiI I |)li()t()^iii|)li on page 
214. Hall H is inilialK at rest. AlKir ( (jliisicjii, it iii(j\(;s 
oil to tin; Ictt at rcnif^lily the same s|3('(;cl as tlio 
initial specKl of hall A. liall A coiik^s in tioni the 
rif^ht and (mikm-^iss Iroin collision with iinehanf^ed 
dii^ection ol nioti(jn 

ThitU! timings an; r(!(|uiied. luo ol these i initial 
velocity of A, (inal \'elo(Mty ol Hi are simple since 
the speeds are r«!lati\(!ly large. The following de- 
scrihes a more detailed typical jjracedui-e than will 
\w found in the student notes. 

1 Align the projector with the paper sheet on 
the wall. This alignment must not he disturbed 
until measui-ements ait; completed . Kun the loop 
at least once for oricMitation. 

2. To find the initial velocity of A, two fine \er- 
tical lines must he drawn on the sheet. Hei-e the 
lailings of gra|)h pa|)ei- sIkmMs are useful. 1 he two 
lines must he; placed to the right of the collision 
point as clost; to that point as possible. 

rh(! s(!paration h»!tw(!(!n these parallel lines 
should 1)(! as large as possible so that the distance 
between them can be m(!asur(!d with reasonable 
precision. Since a rul(!r marked in millim(!ters is 
used, it is possible; to estimate the tenth of a mil- 
limeter, although this digit is a doubtful figure. 
Hence, a distance of at least 10 cm is pi-eferable, 
for then it can hv measured with |>r(U'ision of thi-ee 
significant digits. 

The separation bcMween these lines, on thi; otlnM- 
hand, should Ix; as small as possible because; the 
hall, as it mo\es toward the collision point, gains 
slightly in spe(;d. If the image from the projector 
fills the pap(;i- she(;t as completeK as possible , this 
source of (;rr'or' c()ntril)iit(;s onl\' a small fraction of 
1%. The; stiid(;nts could measure; the separation 
several limes, estimating tin; near(;st tenth of a 
millim(;t(;r-, and use an average. 

Similarly, lO-cm "timing bar's" can be used foi- 
hall H after- collision. But ball ,\ mo\es slowly after- 
collision. 

When the speed of a hall is small, the amplitude 
of its swing las a pendulum, s(;e Kv(;nt 1, page 214i 
is small compared to the field of \iev\'. Hence, it 
loses considerabK in speed while still in the field 
of vitivv. To r-educe this sour-ce of err'or', the timing 
bars" must be pla(-ed as near- as possible to the 
(collision and must be s(>|)ar-at(;d b\ as little as pos- 
sible. Ibis raises another- souixe of (;r-rx)i-: now the 
separation cannot be measuied with the same pi-e- 
cision. \\v (hose timing bars" S(>par-ated In ^^ cm. 
and when this distaru-e is measui-ed In oui- ruler 
we ma\ ha\j; onl\ two significant iligits 

;j. Ihrce xcloc-ities must be deterruiru'tl, oru; loi 
ball 15 and two ibef()i-(> and after-i for- hall ,\. One 
\alue for- each of the thi-«>e times of passage acr-oss 
the corifsponding |)air of timing bar-s is needed 
V\'ith a sto|nvatch, obtain thr-ee\alues for- each time 
inter-val and calculate the axciage. Ihis will (-e(|uin' 
r-epealcd pr-ojeetion ol the film loop 

Hepclilioii of the mcismcincrits ol distance w ilh 



our ruler, in i2i above, may conveil the tenths of 
the millimeter- fiom doubtful to significant. .-\ like 
conversion of digits from doubtful to significant 
may it;sult from rt;peated time measui-ements. 

4. Calculate velocities, momenta, and kinetic 
energies fiom the data. The table that follows lists 
these for a typical itju of the e.vpeiiment fJo not 
assume that the student will get the same numer- 
ical values as appear in the table. These values de- 
pend on how largt; the image is on the paper and 
on the frame rate delivered bv a piojector. Ilie lat- 
ter is not guaranteed equal bv the manufactui^rs 
in all models On the other hand, the conclusions 
reached i including those about enoi-si from the ta- 
ble are roughly those the student should i-eath. 

5. In the table, we have intentionallv omitted 
mean deviations to keep dis(-ussion simple We 
have also, for simplicitv, failed to take advantage of 
the gain in significant digits obtained through av- 
eraging i-epeated measui-ements. .\ good student 
can get better pi-ecision than is demonstrated here. 

Loop 18 

I'irst K.vamph;: On(;-Dimensional Collisions 

Ball ,\: Ti'Sl grams 

Ball B: ^.lO grams 

A line under- a digit means doubtful. 



Item 



Ball Time Average Values 



Direc- 
tion 



velocity 



A before 0.813 cm sec left 

A after 0.252 cm sec left 

B before cm sec — 

B after 0.885 cm sec left 



momentum 



A before 433. g cm sec left 

A after 134. gem sec left 

B before g cm sec — 

B after 310. gem sec left 



kinetic energy 



A before 

A after 

8 before 

B after 



176. ergs 

16.9 ergs 

ergs 

137. ergs 



not 



The small velocitv of ball .A after collision iTdures 
the number of significant digits to two and affects 
the coir-esponding momentum in the same wav. 
lotal momentum alter- (-ollision is 4 4 -"^ H)' gem 
sec, and belbi-e collision it is 4 H v lO" gem sec to 
the same number- of signilicant digits Ihe differ- 
ence is 2.3";. of the av erage 

The average values depend on the size of the 
pr-ojected image Students will not get these values 
noi- will differi'nt grx)ups get e(|ual values 

Ihe kinetic enerf^v calculation henvever does 
not suffer- frx)m a rvduction fi-om thrx'e to two sig- 
nilicant digits. Ihe kinetic ein-rgv of the system is 
17(i ergs befoix; and 1.14 ergs after collision Kinetic 
energv is 87.5";. ronservetl. The collision is not per- 
lectK elastic. The balls wen* case-harxlened steel 
which means that tln'v have an inner coiv of soft 
steel rh(> collision is suflicienllv str-ong to penna- 
nenllv delor-m this v^n^' slightlv with subse(|uent 



Z40 



l'\IT A 'mi; TKII'MI'll OF \lliCH.\Mf:S 



loss of mechanical ener^\ into internal ener|^\ , 
much of it heat. 

When a stntient studies both examples in this 
loop, the peix^entiiges of enei^ consen ation should 
be compared and will lead to an inteiestin^ dis- 
cussion. See below. 

Second c,\umplc. .Again, one ball comes in and 
strikes anothei" at rest. iSee K\ent 2 photo on page 
215 of this Resource Book.' .Again, three sets of tim- 
ing measurements are required. Ikit this time it is 
the moi-e massi\e ball that is initialK at i-est. Ihis 
i-exei^ses the direction of motion of the incoming 
ball iifter collision. 

This event is, initialh , the e.xact reverse of the 
previous e.vample. The initial speed of ball 15 is the 
same as it was for ball A in the first example, to 
three significant digits. 

After collision, ball B moves to the left ver\ 
slovvlv. The "timing bai"s" need to be brought even 
more closelv together than in the first example. 
The number of significant digits again dixjps fi-om 
three to two. 

But hei-e there is no loss in significant digits in 
the calculation of total momentum. After collision 
our measurements gave: 

P„ = 23. gem sec to the left 
P^ =317. g cm sec to the right 



IP = 294. gem sec to the right 

(Numerical values given here are not representative 
of those in student data.) 

The fact that there was no loss in significant dig- 
its does not, however, necessarily insure greater 
precision in the results. Our value for total mo- 
mentum before collision is 285 gem sec to the 
right. The difference is 3.1%. 

This difference is larger than it was for the first 
example, but note that it is laiger only in relative, 
not in absolute, value . The absolute difference is, 
in fact. 10 g cm sec in the fii-st, 9 g cm sec in the 
second example. Since the measuiements were 
equivalent in these two examples and equal in 
number, the fact that absolute diffeiences are 
nearlv equal should not surpiise us. 

Our data also revealed that only 82% of the ki- 
netic energv' sunives the collision. V\'hv was this 
percentage higher t87.5% i when the same balls col- 
lided in the opposite manner.'' 

In our pair of examples, this (|uestion is an- 
swered in a surprising way. We should expect 
greater energv loss if two balls collide with each 
other harder.' The average collision force of in- 
teraction is proportional to momentum charige ex- 
perienced bv either ball. In the first example, these 
changes shown in the table ar-e 3.0 x 10" and 3.1 
X 10" gem sec r-especti\ely; in the second example 
isee data abovei, 3.08 x 10" and 3.17 x 10". Thus, 
the second collision was slightly greater, and, 
ther"efor"e, a higher- per-centage of energv' was lost 



However, this conclusion must be discarded. 
For, if differ-ence of momentum, which is in prin- 
ciple zero, was 3% on basis of the data, this indi- 
cates an experimental error- of at least that size. In 
calculating kinetic energv . we s(|uai-e the measur-ed 
speeds and this doubles the relative erroisl Ihe 
apparent difference of 5% in energv loss is ex- 
plained b\' the err-ors of our obser\ations and may 
be spurious. 



LI 9 Ox\E-DL\IEXSIOXAL COLLISION'S. II 

The two examples of this loop involve the same 
pair- of balls in difVerent collision. (Jne ball is more 
than thr-ee times as massive as the other-. 

In the first example, the two balls c'ome at eac:h 
other- fiom opposite directions at considerable r-el- 
ativ e V elocity. In the second example, both balls ar-e 
moving across the field of view in the same direc- 
tion, one ball catching up with the other, but the 
relative veiocitv' is small. 

In both cases, four velocities must be found from 
the film footage, and four pair-s of timing bar-s " ar-e 
i-ecjuiied. 

The fii'st example should not be assigned alone, 
together with the second, it fomis a highlv instrarc- 
tive problem. 

First example. See Event 3 on p. 21(5. Both balls 
reverse their direction of motion upon c-ollision. 
Ball A is moving slow Iv befor-e and after-, and a sep- 
aration of less than 10 cm ma\ be needed betweern 
the timing biirs. Because ball .A is massive, how- 
ever, the magnitude of its momentum i before as 
well as after! is c-omparable to the corr-esponding 
values of momentum cjf ball B. 

Since the momenta of the two balls are oppo- 
sitely directed iboth before and after the collisioni, 
the net momentum of the sxstem can be calculated 
onlv as a small dilTerence between large number's . 
Hence, a large relative enor mav be expected. 

When we performed these measurements, we 
found a differ-ence of 8.7% between the net result 
for the momentum before coUision and for the net 
momentum after-. The actual numerical momen- 
tum values I in gem sed we are about to cite from 
our data will not be the same as thcjse found by 
your students. Nevertheless, we quote them here: 
133 net momentum befor-e, 122 after-, both direc:ted 
tcj the left . But the individual momenta werx; much 
larger-. Before collision: ball A. 448 to right: ball B, 
581 to left. .After- collision: ball A. 454 to left: ball B, 
332 to right. 

The large relative err-or- is complelelv accounted 
for b\ the circumstances of this collision. Ihe large 
individual momenta are far mor-e preciseh known. 
Not so the difference! Had it been the sum , the 
r-elative error would be small. 

I he second point of inter-est herx? rvlates to the 
strength of the collision and is of parlicular- value 
when this example is studied with the other event 
in this Film Loop. 1 he student will find that only 
about 43"o of the total initial kinetic- energv sui-\ives 



FILM LOOP NOILS 



241 



the collision. Fn the other e\'ent, this energv is 97% 
(.•unserved; that is, the collision is almost perfectly 
elastic. Why? Because the soft cor-e of these case- 
harxlened steel balls is sorely tested in a r-eal col- 
lision as occui-s in the first example, but it is hardly 
touched in a weak collision. A good deal of any 
deformation of the inner cor-e is permanent with 
resulting loss of the "work of deformation ' from 
the initial sujjply of kinetic energy. The loss goes 
into internal energv' imostly heati. 

Another way to explain the idea of strength ' in 
a collision is the following: The average force of 
interaction, multiplied by the time of contact in 
collision, equals the change in momentum of one 
of the balls. This is the "impulse theorem," a de- 
rived fonn of Newton's second law. Hence, change 
in momentum is a measurx; of the deforming force 
that occurs during contact in the collision. Note 
from the above data that ball As momentum 
changes from 448 to the right to 454 to the left, that 
is, by 902 g cm/sec iby our figures, not the stu- 
dent's!. In the second example the change is onl\' 
about 400 g cm/sec. 

Second example. See Event 4 on p. 217. All veloc- 
ities and momenta have the same directions (to the 
right) in this event. Four sets of measurements 
must be made to find the four speeds. All the 
speeds are large enough to allow 10 cm separation 
between "timing bars' Hence, there is chance for 
three-digit pi-ecision. 

Net momentum (before or afteri is here calcu- 
lated by addition of individual magnitudes (and 
not by subtraction, as was the case in the pre\ious 
event I. Therefore, the relative difference between 
momenta before and after will be small. Our value 
for it is 0.48% . 

Kinetic energ\' is 97.2% consened. I'he collision 
is almost perfectly elastic. This should be com- 
pared with the 43% consenation found when the 
same balls collided in the pre\ious event. 



L20 INELASTIC OXE-DEVIEXSIONAL 
COLLISIONS 

In these e\ents, the colliding objects are steel balls 
covered by a thick layer of plasticene. They remain 
lodged together after collision. Both collisions in- 
volve the same balls. The second collision is far 
stronger than the first. 

The first example is the simplest of all sL\ prob- 
lems in this set of thr-ee film loops. It requires only 
two sets of timing measurements. The second ex- 
ample requires three. 

Precision is excellent in both e\ents. 

First e.xample. See Event 5 on p. 217. Ball A is 
initially at rest. Our \alue for the relati\e difference 
between net momentum before collision and after 
is 0.4%. Kinetic energ\' is 59.4% conserved. Three- 
digit precision thi-oughout. 

Second example. See E\ent 6 on p. 218. Here the 
balls come in from opposite directions. They move 
off to the left after collision. Three-digit precision 
throughout To three significant digits, we found 
identical values for net momentum before and 
after. But kinetic energv' is only 32.7% conserved 
because the collision is more violent here than in 
the first example. 

NOTE: 

Teacher notes for Film Loops 21, 22, 23, and 24 are 
included in the section of this Resource Book en- 
titled Esperiment S'otes. The discussion deals with 
the use of the identical stroboscopic photographs 
that are reproduced both in this book and in the 
Handbook. 



L21 T^VO-DLMENSIONAL COLLISIONS. 
L22 T\VO-DLMENSIONAL COLLISIONS. 
L23 LVELASTIC T\\'0-DLMENSIO.\AL 

COLLISIONS 
L24 SCATTERING OF A CLl'STER 

OF OBJECTS 



L25 EXPLOSION OF A CLUSTER OF OBJECTS 




Pd - *^o< 



Note No *dju»tmen-t^ M»nt ir«:lc W 't^ 
".low motoo jrtd tfi« loalc cf projec^iot^ 




242 



r\IT 3 / THE TRIl'MPH OF IVIKCH/XMCS 



Sample Data 

Identification 



Mass (kg) 

Projected distance traveled (cm) 


0.357 
1.55 


0.539 
3.05 


0.366 
2.90 


1.78 
3.00 


1.02 
1.25 


Direction of travel 


342= 


56 


86= 


179= 


284 


Time to travel designated distance 
(sec) 


15.0 
15.3 


17.4 
17.0 


11.3 
11.4 


31.7 
31.8 


10.9 
11.8 



Notes 

Given 

This will depend upon the size of 

projection. 

Up is taken as 0°. Direction is 

measured clockwise. 

This is measured time disregarding 

slow-motion factor. 



Sample Calculations 

Identification 



Notes 



Average time traveled for 
designated distances 
Speed (cm sec) 
Velocity (cm sec, degrees) 
Momentum 



15.2 


17.3 


11.4 


31.8 


11.3 


0.102 


0.176 


0.254 


0.0945 


0.111 


0.102, 342= 


0.176, 056= 


0.256, 080= 


0.0945, 179= 


0.111, 284= 


3.60, 342= 


9.49, 056= 


9.42, 086= 


16.8, 179= 


11.3, 284 



Ad 
St 



From a comparison of e.xpected and observ ed 
values for the momentum of ball E, it is clear that 
momentum is not conserved within the system of 
the five balls. There is too much predicted mo- 
mentum. This may lead the student to realize that 
something is earning a\\'a\' momentum fTX)m the 
svstem in a dii^ction opposite to ball E. Looking at 
the film again, the student ma\' note that the waste 
matter from the powder charge ma\' be the un- 
measured recipient of the required momentum. 

An ad\anced student ma\' wish to stud\' the 
chemical potential energ\ stored within the pow- 
der charge. Howe\er, as this can onl\ be done in 
arbitrary units, it ma\ ha\e little significant mean- 
ing. 

An interested student might want to repeat this 
experiment in order to improve the technique of 
measuring. One idea might be to time the balls 
from the explosion to when the\ are last seen. Fur- 
thermore, greater care might be taken to be moi'e 
accurate in recording the direction of ball E in the 
smoke. 

An error of 2° and at least 1 sec was in\ ol\ ed in 
the above measurements of balls A, B, C, and D. .An 
error of 3° and 2 sec was inv oh ed in measuring the 
motion of ball E. 

L26 AND L27 FEVDLNG THE SPEED OF A 
RIFLE BIXLET: METHOD I 
AND METHOD H 

/. IXTRODLCTIOX 

These films are not meant to provide a precise lab- 
oratoiy e.xercise. While it is possible to obtain rep- 
resentative values for the muzzle velocities, the 
main intention is to bring a few conserv ation prin- 
ciples into the context of a real experimental prob- 
lem, the ballistic pendulum. 

The bullets speed is calculated from the logs 
speed after impact by use of conservation of mo- 
mentum. In Method I, the latter speed is deter- 
mined directly. In Method II, on the other hand, 
the log s speed after impact must also be calcu- 



lated. The student can measure onlv the log's full 
height of rise during its swing awav from impact. 
To relate the two quantities, we invoke conserva- 
tion of the sum of the log s kinetic energv' and its 
gravitational potential energv during the swing. 

Finallv, one can lin each filmi compare the ki- 
netic energv' of the bullet before impact with that 
of the log I with the bullet embedded in iti after 
impact, and ask the student: How was energv' con- 
served here? 

//. GEXER^AL DESCRIPTIOX 

In Method I, there is a slow-motion sequence that 
permits timing the motion of the log just after the 
bullet strikes. The circular path of the log has a 
verv- large radius and the film sequence is a close- 
up of the log for a field of view of about 30 cm x 
40 cm. Hence, the motion of the log can be consid- 
ered uniform along a horizontal straight line. The 
student must convert distance as well as time 
measurements taken from the projected image of 
the slow-motion scene to actual distance and 
real time. The information necessarv for this con- 
version appears in the film. Now the student can 
calculate the bullet's speed by invoking momen- 
tum conservation, as follows 

mu = (A/ + ml V, (1) 

m — mass of the bullet 

M — mass of the log 

V = speed of the log plus bullet after impact 

u — speed of the bullet 

The values of .\/ and m are given in each film. 

In Method II, the measurements are simpler; 
thev involve distances onlv . No measurements of 
time are required. The film contains a slow-motion 
sequence showing the log close up, as it goes 
through its full pendulum swing after impact. The 
student can measure the vertical height of rise, h, 
of the log. That is to say, h is the distance from the 
log s lowest initial position before impact to the 
highest point of its swing, which is readilv identi- 



FIL.M LOOP NOTES 



243 



flable because it re\'erses its dii-ection of motion at 
that point. 

Since the iof^s kinetic ener^\' just after imfiact 
(at the stall of its swingi must equal the gra\'ita- 
tional pcjtential energv' fequin;d to raise it vertically 
through a distance h lat the end of its svvingi, then 

Therefore, v^ = 2a ^/i 

V = speed of log iplus bullet) 

just after impact 
h = height of vertical rise of log 

during suing 
a^ = acceleration of gra\it\' 

Having calculated v from Equation i2i, u can be 
calculated bv use of Equation lli. 

Method I 

In the close-up slow-motion sequence that is in- 
tended for taking data, the student must make two 
horizontal mai-ks on the paper sheet taped to the 
wall while the log is at rest in the image. These 
marks must span the vertical dimension of the ft)d 
that is given in another portion of the film to cor- 
respond to 15 cm "actual distance." By later meas- 
uring the distance between these marks, conver- 
sion of distances measured on the paper to actual 
laboratory distances becomes possible by scaling. 

A strip of white adhesive is taped to the log. 
Either of its vertical edges can be used as a refer- 
ence line for timing the log's horizontal motion 
after impact. l\vo verlical lines are drawn on the 
paper as timing bar-s. The distance berween them 
is measur-ed, and converted by scaling. The motion 
is timed by stopwatch. 

"Film time" can be converted to "real time" if the 
slow-motion factor is known. This factor is the ratio 

frames/second taken by the camera 

frames/second delivered by the projector 

The quantity in the numerator is given in the film. 

The (iuantit\ in the denominator must be meas- 
ured, since the manufactui-er of the prx)jector does 
not guarantee this rate to within less than 10%. 
The total number of frames is printed on the car- 
tridge into which the film is looped. There is a 
single black frame with a lar^ge white cirrle in the 
"black strt'tch ' between the end and start of the 
film loop that is visil)le on the scr-een as a brief 
flash I bus, the student can time the length of the 
loop. 

When we took the indicatcHi measurvments of 
this film loop. \\v found that oui- loop of ;}.849 
frames ran 207.3 sec: a projection rate of 1857 
frames/ sec. The slow -motion factor- was thei'efoi'e 



2,850 



Furthermore, we found the speed of log plus bul- 
let after impact was v — 106 era sec, and the speed 
of bullet was u = 6.43 x 10^ cm/sec -^ 2 or 322 
X 10^ cm/sec. 

The kinetic energv' of the bullet before impact 
was 2,890 J. The kinetic energv' of the log iplus bul- 
leti after impact, on the other hand, was onlv 2-50 
J I Most of the initial kinetic ener^' suppiv is dis- 
sipated in tearing wood and producing heat in this 
inelastic collision. 

Method II 

In the close-up. slow-motion sequence that is in- 
tended for measurement, the student again marks 
off the vertical dimension H of the log while it is at 
rest to serve as a scaling reference. The actual value 
of H is given in the film as 9.0 cm. 

There are two hor-izontal strips of adhesive taped 
to the log. Any horizontal edge of these strips can 
now serve to mark off the initial position of the log 
and the final position at full swing to determine h. 

Our measurements ion the filmi \ielded h - 5.33 
cm, V = 102. cm/sec, and u = 5.8 x lO"* cm sec. 

The kinetic energv' of the bullet before impact 
was 1,200 J. The kinetic ener^' of the log with the 
bullet embedded in it was only 42 J. 

L28 RECOIL 

1 he film is valuable fomi both qualitative and 
quantitative aspects bv illustrating the real life 
recoil of an actual cannon and a laboratory' cannon 
suspended from strings. Relative measurements of 
momentum can be made to test the conservation 
laws. 

With the high-speed camera, a delav is observed 
between fuse ignition and the emergence of the 
bullet fham the cannon barrel. During this delav 
the bullet travels thr-ough the barrel from its initial 
position to the end of the muzzle. 

To travel a distance of 20 cm on our paper, the 
projectile requir-ed 2.95 sec. The bullet's mass is 
3.50 g and its relative momentum 3.5 i20 2.95i g cm 
sec. Momentum conservation in one dimension 
predicts 

Thus, the velocitv of the cannon v should be given 
by 



\; 


= 




ere — 




1 


m, 




100 


and \ 


= 


20 

cm/sec 

2 95 



18 



= 153.4 



Times measur-ed on the (ilni .ir-e converted to ab- 
solute time In dividing bv tlic .slow -motion lac-tor 



The predicted velocitv of the cannon is 0.06 cm sec 
or 75 sec for 5 cm with an error in timing of 10% 
for the bullet and a subsequent errxir of 10% in the 
prediction. Our data gav-e an experimental value of 
67 5 sec for the cannon to move 5 cm in the op- 
|)osit»> iliivction which is within tin* mar-gin of er- 



Z44 



I'NIT 3 / THK TRII MPH OF MECHANICS 



ror. One might expect a lower \ alue since the pow- 
der charge has some momentum . .Also, the pi-ojec- 
tor manufacturer guarantees no less than 10% er- 
ixjr of uniformit\ in pix)jection rate. 

The kinetic energ\' of the bullet is Vzm^\-'. or. 
v\ith our data, '2i3.5ii20 2.95r = 80.5 gem" sec', 
while the cannon has a kinetic energ\' of '2 i350i 
(5 65.71" or 9.6 g cm" sec". The kinetic energ\ of the 
bullet is not equal to that of the cannon: nor would 
we e.xpect it to be, for this is not an elastic collision 
but an e.xplosion with energ\' lost both to the pow- 
der charge and thix)ugh frictional losses within the 
barrel. Kinetic energ\' is not conserved. 

L29 COLLIDIXG FREIGHT CARS 

rhe test of coupling strength was made b\ the Up- 
lands Railwa\' Laboratory for Canadian Pacific Rail- 
road. The test engineers i-eport for the trial shown 
in the film s slow-motion sequence gi\es the peak 
coupling force as 4,784,850 \; hammer cars \eloc- 
it\' after impact is 1.3 m sec. 

An alternati\ e method of finding \', , in\ ohing less 
accuracy but easier to understand, is to measui-e 
the time for the hammer car to come to rest after 
the collision. Then the initial \elocit\' is just t\\ice 
the a\erage \elocit\'. 

Measurements from the film lin arbitrarv' units 1 
gave 286 units for total momentum before collision, 
280 units after collision. Kinetic energ\ of the sys- 
tem decreased fixjm 390 units to 167 units. 

L30 D^'NAMICS OF A BILLIARD BALL 

The film has \alue e\en if used onl\ qualitati\el\- 
to illustrate conservation of momentum in a real- 
life situation. Measurements can be made and in- 
terpreted at two le\ els of difficult\'. 

1. Students should ha\e no difficulty with 
straightfonvard conseiAation of linear momentum, 
as outlined in the Handbook. For best results the 
\elocities after impact should be measured over 
short distances to a\ oid complications due to fric- 
tion. The cue ball s fonvard linear \elocit\- is neg- 
ligible just after collision, but this ball does gain 
forward speed as its rotational speed decreases 
due to friction. In a t\pical measurement, balls 
were timed as their leading edges mo\ed forvvard 
a distance equal to one radius. The measured 
speeds 1 hence also the measured momenta agreed 
within 1%. 

2. The following anal\'sis is gi\en primarilx' for 
teacher background. The balls rotate as well as 
translate, so we must consider both translational 
and rotational momentum. The force of friction 
between the ball and the table surface aftects the 
motion of a ball whenever there is slipping ithat is, 
a relative motion between the ball's lower surface 
and the table 1. A basic assumption is that the coef- 
ficient of sliding friction i|xi is the same for each 
ball and is independent of the speed of slipping. 
We use Newton's second law for translation if — 
mat and for rotation 17 = lai where t = torque, / 
= moment of inertia i = -:,mr for a sphere rotating 



about an axis through its center 1, and a is the an- 
gular acceleration. When a ball is rolling without 
slipping, its linear \ elocitx v and its angular \ elocity 
o) ai"e i-elated by the equation \ — no. 

At the moment of impact, the only foree on each 
ball is that due to the other ball, acting along the 
line of centei"s. Because these forces ha\e no le\er 
amis, the\' can cause no changes in angular \ eloc- 
ity' at the moment of impact . The cue ball, which 
was rolling, must continue to spin with the same 
o), and the target ball, which had no initial co, must 
start to slide with no rotation. 1 hese conditions do 
not persist, howe\er, because a frictional torque 
acts on each ball while its lower surface is slipping 
on the table. Time of spinning of cue ball. .A fric- 
tional force y^ma^ acting toward the right on the 
bottom surface of the cue ball does two things: It 
causes the ball's rotational \elocit\' to decrease , 
and it causes the ball s translational velocit\ to in- 
crease in the for^vard dir-ection. .After a time f, the 
\elocit\' of the ball s lower surface equals the for- 
ward \elocit\' of the ball: 

noj, + atJ = + at. 



where 



then 



y^mar 



iima^ 
- f. 



.After this time has elapsed, the cue ball continues 
to roll without slipping. 

Time for target ball to slide. While all this is going 
on, the target ball stalls to slide with \elocir\' \',. 
Friction acting to the left causes the ball s trans- 
lational speed to decrease , and it causes the ball s 
rotational speed to inci-ease from zero up to some 
\alue. The ball starts to roll without slipping after 
a time f, when the linear \elocit\ of the ball s sur- 
face I due to rotation I becomes equal to the ball's 
foiAvard translational \elociU': 



no -I- af,i = v, ■+■ at.. 



ii.ma r 
r[0 + ^f. 



iima^ 



— ^', 



where t, = 



2\', 



Now we can explain the strange beha\ior of the 
balls. From the law of conservation of linear mo- 
mentum, V = V,, hence t, and f, are equal. The 
changeoxer to rolling without slipping occurs at 
the same time for both balls: the cue ball seems to 
'know' what the target ball is doing. 
Conservation of angular momentum. Change in an- 
gulaj' momentum equals itor-quei x itimei. While 
the cue ball slides, it loses angular momentum 
ifjunapriifji. While the target ball slides, it gains an- 
gular momentum i(xma^rMf,). We have seen that f, 
= f,: hence, ther-e is no net change in angular mo- 



FIL.M LOOP VOTES 



245 



mentum. Since llie monientuin ot mi'itia is tlie 
samr; for ea('h hall, this nmans that w = w, -^ w^, 
wheit! CD is llie initial an^ulai' \el(j(it\ ol the cue 
hall, and to, and i>}, an; the arif^ulai- \elocilies meas- 
ured ahei hoth halls an; rollinf^ \%ith(jut slip|)inf^ 
A tyi)ical nieasuff merit from the (ilm conlinns this 
tf) within ahnut HV. 

l.orfjiricnl nj friclion. rh«? lime for slipping was 
found to he 

2v. 
t 

from which p. can he found if /, and \., are meas- 
ured in real-time and real-distance units, i The 
slow-motion factoi- is 1H7 and each hall has a di- 
amet(;r .■j.24 cm.i 

Perhaps it is easier to work with distances than 
with times. The distance foi' tht; target hall to slide 
is 



Vit' 



- V. (r^) 



V, - 



m / \7 ma J 



which simjjlifies to d - 



\2v' 



49 ma. 



Thus, (JL can he found In measuring v, and (/. C^al- 
culated \alu(!s of p. aif ahout 0.33. This agit^es with 
a direct measurement of ahoul 0.3 for the coeffi- 
cient of sliding friction not shown in the film'. 

L31 A METHOD OF ME/\SntIXG 

ENXRGV: XAILS DRHEX INTO 
WOOD 

As the nail penetrates deepK into the wood, tlie 
foice of friction inci-eases somewhat, so the pene- 
tration is less than would he e.xpected on the hasis 
of the first few hlows. I herefore, the giapli is 
cuiM'd downward, as shown in the lUmdbmjk. I'or 
man\ purposes this effect can he ignoivd and the 
energv of the ohjecl striking tin* nail assumed to 
he directK proportional to the depth of penetra- 
tion. 

L32 (;Ry\\'ITATIO\/VL POTENTLVL 
EXER(;\ 

In testing and defining gi'a\ilational potential en- 
erg\ . we use the graphs from lAM that r-elate kinetic 
energv' to nail penetration Since then' is no loss of 
eneig\ as tin* hodies fall, we can sav that the po- 
tential energ\ of the ohject is the same as the en- 
erg\' just l)«'l()n' impact and then measure the nail 
perKMration to lind that eneigv 

nail peiictralion — » energs — * potential energ\ 

I urthei-. ' 2 m\'^ - lunji. wIhmv ni is the mass, \ its 
\clocit\' just l)efoie impact, h tin* initial lieight 
aho\«' the /,ei"o position i the nail lop \ and a^ the 
acceleration of gra\it\ 

In the liisl sei|uencj' we measur-e the nail posi- 



tion hefore and after collision to find the penetra- 
tion and plot this versus weight ma^ As in L3i. \ou 
mav expect the graph t(j hend dowiiwaril slightly. 
iSee Fig l.i 




Fig. 1 Weight (newtons) 

There is a direct relation hetween weight and 
penetration, tfieit?for-e hetween weiglit and energ\ 
and hetween weight and giaxitational potential en- 
ergy. I lie height aho\e zeix) position has b<'en held 
constant at 2 m. 

In the second setjuence the mass is the same 
while the initial height, h. is varied frxim 1 ni to 4 
m aho\'e the zero position. I o graph this data, one 
might measure values of h irom the sci-een or use 
the given data as we have done. 



o 

4 



f> cm. 



s 3 
^ I 



I ? 3 4 

he.gM of \c.n (m) 
Fig. 2 Height of a fall (m) 

The Film Utop does not gi\e a starting measuiv- 
ment in the second sequence to estahlish the ini- 
tial position of the nail so we have usetl the difter- 
ences hetween the data to extrapolate an initial 
[)osition 

|-ix)m Fig Z. we can conclude that there is a di- 
rect relation hetween peru'tration and distance ol 
fall which implies a direct relation Iwlween gr-:iv- 
itational |)olential energv and distante of tall We 
can conclude that gravitational potential enei-gv is 
directiv prxiponional to the pi-oduct otweiglil and 
initial height 

PE y "J.i^/i or /•/. = K mnjy 

where K is a constanl whose value depends on our 
chosen units 



ZMi 



I'MT 3 THi: THIl'\ll'M Ol MliCII/WICS 



The slow-motion factors i33 for the first se- 
quence and 10 for the second' are ghen. Ihe moi-e 
ambitious student ma\ wish to use these to deter- 
mine r-elati\e \elocities and tiie relations of \ and 
h or ener-ffx and \elocit\ . 



L33 KINETIC ENERGY 

L31 suggests that nail penetration is a con\enient 
method for studying the enei-g\ of an object that 
hits the nail. In L33. we examine the enei-g\' in its 
kinetic form to find a relation between kinetic en- 
ergy- and \elocit\'. 

B\ following the Handbook, the students should 
ha\e no tix)uble in timing the passage of the objects 
across the reference lines: the\ must be consistent 
in using the same edges for all objects. 

Measurements fixjm the film gi\e the following 
data: 





Time 


1 






1 


Event 


it) 


t 


Nail Penetration 


H 













6.6 




1 


2.14 


0.466 


0.7 units 


5.9 


0.217 


2 


1.35 


0.741 


1.4 units 


5.2 


0.550 


3 


0.98 


1.010 


2.2 units 


4.4 


1.02 


4 


0.80 


1.25 


3.1 units 


3.5 


1.56 


5 


0.71 


1.41 


4.3 units 


2.3 


1.98 



Since \elocit\- is distance time and the distance 
is constant, the \elocit\- in each case is in\erseK 
proportional to the time. Tliat is. v ^ 1 t. The stu- 
dent should plot two graphs: kinetic energ> KE 
mail penetration' \ei-sus \-^ [or H P"l. and another 
of KE \ersus \ 1 /'. 

The two plots for our data are gi\en below . We 
can conclude that there is a direct relationship be- 
t\\een KE and v'. Howexer, the plot of KE \ei-sus \ 
is a parabola. iFig. 4.i 

The interested student ma\ wish to deteniiine 
the relati\e mass of the last unmarked object b\ 
measuring its penetration and \elocit> and then 
establishing ratio beUveen these results and those 
obtained in the first part. Our data would suggest 
that: 



f = 1.2 and \ 



0.7 



- S 
5 2 



S" r /.c /.£ 1.4 lA '-S ^O 



.o 

5 ^ 






-r 3 





■■* i S 1.0 ,.2 /.V /,i 



Fig. 4 



Inteipolating from Fig. 3. the penetration should 
be 1.6 units if the objects were alike in mass. The 
filmed sequence suggests that the nail penetration 
is 5 units so we ma\ conclude that object 6 has a 
mass three times that used in the first sequence. 



L34 COXSER\'ATIOX OF ENERGY: 
POLE \'ALLT 

The film can be used qualitati\ el\ . The intermittent 
freeze-frame sequences are long enough so that 
the teacher can mention the dilferent fonns into 
which the total energ\ has been transfoimed while 
the\ are actualh happening. Quantitati\e meas- 
urements are good to about ICc. It is best to con- 
centrate on comparing energ\ at position 1 with 
that at position 3 and to leaxe the more difficult 
check at position 2 for students who enjo\ this 
t\pe of measurement. E\en if no measurements are 
made at all. \ou can tell the class that the enei-gv 
is dixided appi-o.vimateK as follows: 



1. initial kinetic energ\ 

2a. kinetic energ\ 

2b. graxitational potential enei-g\ 

2c. elastic potential energ\ 



1300 joules 

450 joules 

650 joules 

300 joules 

1400 joules 



Fig. 3 



3. final graxitational potential en- 

erg\' 1800 joules 

less muscular woi-k -400 joules 

1400 joules 

In position 2, each of the three fonns of enei^' is 
a significant portion of the total. 

The intemiittent freeze-frame technique is used 
because even the slow-motion action is too fast for 
accurate timing of a 1-m displacement. B\ this new- 
technique, a student can measure the speed to 
within a few percent b\' counting. sa\-. 20 frames 
1 1 frame out of 20 is only a 5% errort. 

Some fine points can be raised for class discus- 
sion: 111 Ihe gra\itational potential energ\- of the 
pole itself is onl\ about 10 J in position 3. and can 
be neglected. i2i To be precise, we should calculate 
the woi-k done in bending the pole 



FIL.M LOOP \OTLS 



24' 



L35 COXSER\'ATION' OF EXERGV: 
AIRCRAFT TAKEOFF 

Air resistance docs depend on speed and therefore 
does d<;crease somewhat as the planes speed de- 
creases. Make clear to the student that we are us- 
ing an apprtjxiniation when we ignore this effect. 
The appmximation is justified b\' the large mass 
linertiai of the filane. For instance, a typical meas- 
urement shows that at the upper le\el the speed 
was only two-thirds the vaue at gmund le\el. This 
means that the foree of air resistance at the upper 
level is only two-thirds the \'alue at ground level. 
If, as intended, the engine supplies a constant for- 
ward force, a small net foi-ward force builds up 
when the plane re'aches the upper le\el. But the 
acceleration caused by this unbalanced force is 
small because of the large mass of the plane: a = 
F/m. It is this sluggishness of response to changes 
in air i-esistance that allows us to make the simple 
energy analysis outlined for student measure- 
ments. 

If this seems unreasonable, reflect on the fact 
that the plane will, if it flies long enough at the 
upper le\el, regain its original gixjund speed when 
it again i-eaches teniiinal \'elocit>': but there is no 
sign of such an increase during the time the plane 
remains \isible in the film. The planes inertia is 
simply too large to allow a rapid change of speed 
due to such a small unbalanced foree. 

During preparation of the film, five trials were 
carefully analyzed. The trial selected for reproduc- 
tion ga\e results appixj.ximateK as follows 'height 
in meters, energies in units of lo' joulesi: 



KE 



PE 






43 





43 


24 


29.5 


13 


42.5 


46 


20 


24.5 


44.5 
f,v 43.3 



As the table shows, the total enei^' remained con- 
stant to within about 4%. 

L36 RE\TRSIBILIT\' OF TIME 

The film conctuitiates on fwo t\'pes of sports that 
depend u|)on a substanti\e lack of friction: figure 
skating and pool. Although the expert in each sport 
may easily detect the direction of time through 
minor losses of energy or by a "feeling" for timing 
and position, the a\erage teacher and student may 
be momcntaiily at a loss as to the diivction of time. 
It is the loss of (Micr-gx and timing that often 
pnnides clues to the direction of time. An example 
of the loss of energ\ is clear in the next to last 
sequence wheix* a billiaixl ball is stmck and then 
slows down lo a stop, i he iv\ei-se motion probablx 
is a e-e\(M-sal of the film, liming is in\<)l\ed in the 
analysis of the motion of thive balls, one cue ball 
and rvvo othei-s. Iix)m experience, we may feel 
that it is unlikely for two balls to hit the cue so 
pix'cisely, and we an* apt to belirxe that it was the 
cur l)all that (lid the hilliiii; 



In the more complicated event in which the cue 
splits the set, it appears that time can only flow in 
one direction. The balls lose their kinetic enei^ 
and come to a halt; it is statistically unlikely that 
the\' will then come together. The chances of the 
resulting random energ\' in the air. the table, and 
the internal structure of the billiard balls e\'er re- 
turning to its original form is statisticalK' remote. 
Vet over short periods of time it seems as if the 
re\'ersibilit\' of .\ew1onian laus does hold. It seems 
that conservation of momentum and energ\' are 
"invariant" to within experimental error. When 
these short periods are added together the result- 
ing continual losses of energv' suggest that time is 
not reversible. 

L37 SLTERPOSITIOX 

The amplitudes of component waxes are intention- 
ally \aried somewhat irregularis' while setting up" 
a superposition. This is to remind the student that 
a human operator is recilly causing these changes 
on the face of an oscilloscope. Be sure that the 
student understands that these are not anima- 
tions. 

Among the less familiar aspects of superposition 
is the fact that when rwo sinusoidal wa\es ha\ing 
the same wavelength but different phases and am- 
plitudes are combined, the resultant is again a sim- 
ple sinusoidal wave of the same wa\elength but 
uith an intemiediate phase. The phenomenon of 
beats can also be seen to result from the super- 
position of two waves having slightly different 
wavelengths. 

L38 STA.XDLXG \\'A\XS ON A STRING 

The film loops on standing waves iL30, LAO. and 
IA1\ are designed to emphasize the underKing fea- 
tures common to all standing waxes. The source 
is at the left i tuning fork, loudspeaker, or dipole 
antennai; a reflector is at the right iwooden rod, 
piston, or aluminum miiTori. .At the end of L4I, the 
three txpes of standing waves are compared in one 
composite picture in which the wa\'elengths are 
the same and the distances berween nodes igrven 
b\' ' ,\i are the same. The wa\e speeds and the 
frequencies differ by as much as a factor of 10*^. 

L39 STANT)L\G \VA\'ES IN A GAS 

In s\mbols. where L is the length of the tube, 

L = (n -t- VzKVzX) 
2L 



k = 



Since 



In + Vz) 



F = - then F = — in + Mil 
\ 2L 



in -I- V2I 2L 



= — = constant 



24N 



I'lVIT 3 / THE TRIUMPH Ol MHCH/WICS 



For this pipe,//(n + V2I is about 151 \ib sec, finom 
which \' can be found to be 348 ni sec if L is gi\en 
as 1.15 m. 



L40 \lBRATIONS OF A WIRE 

The wire was actualK a standard brass welding rod 
2.4 mm in diameter. A short horizontal right-angle 
bend near the clamped end of the rod was essen- 
tial to allow that point to sene as a node without 
undue restraint of vertical vibrations at neighbor- 
ing points. 

A surplus radar magnet was placed near an an- 
tinode of the wires \ibration. The magnetic force 
is perpendicular both to the current and to the 
magnetic field. 

,-\n audio frequency source of high current and 
low \oltage was needed. An audio oscillator fed a 
20-V\' hi-fi amplifier whose output was matched to 
the wire b\' a suiplus power transformer used in 
a re\erse or stepdown connection. The high- 
voltage" I plate I winding was connected to the am- 
plifiers 16-ohm output, and the filament winding 
was connected to the \ibrating wire. Audio cur- 
rents of se\eral amperes passed through the wire. 

For the wire, the obsened frequencies of the first 
four modes were 8, 24, 48, 78 \ib sec. 

An example will make clear the camera tech- 
niques used. For the time exposure' or blurred 
shot, the camera speed was 3 ftames sec and the 
shutter was set at a full opening of 200° lout of 
360° I. Each frame thus was exposed for '200 360 1 
IV3) = 1 5.4 sec. During this time the wii^ had a 
chance to make enough \ibrations to gi\e the de- 
sired blurred effect, simulating what the e\'e sees. 
To obtain the "slow-motion" sequence for, sav, the 
48 \ib sec mode, the camera speed was set at 45 
ft'ames sec, and the shutter closed to 20°. Each ex- 
posure was therefore i20 360) il45i = 1/810 sec, 
which was short enough to freeze the wire's mo- 
tion. The strobe rates was 48 — 45 = 3/sec as pho- 
tographed. This becomes about 1/sec \\hen pro- 
jected in the classroom at 18 frames sec. 

The circular wire was actualK' clamped at fwo 
points very close together, which served as binding 
posts for the current. For the cii"cular wire, the ob- 
served frequencies of the first four modes were 10 
24, 55, and 101 \ib sec. 

In discussing the Bohr atom frtim the point of 
\iew of de Broglie waves lUnit 5i, a familiar argu- 
ment is that in the nth energ\' state there are n 
wavelengths in a complete circle of radius r. Then, 
since \ = h/mv, we ha\e nth/mv — Zirr, whence 
m\T = nhZiT. This is Bohr's quantum condition for 
angulcir momentum. But the analog\ is not as pow- 
erful as it seems. The Heisenbeig uncertaints' prin- 
ciple prexents us from knowing simultaneousK 
both the angular momentum and the direction of 
the normal to the plane of an oi+)it. Therefore, the 
planetar\' model of an electrons plane orbit is not 
a valid one, although it is useful in many cases as 
a first step. Ihe film of a \ibrating cirtrular wire can 



certainly be used to show the student how a sim- 
ple mechanical system with circular s\inmetiA has 
a discrete beha\ior. By analog\', this makes plau- 
sible the ai-gument that a simple atom might be- 
have in a similar discrete fashion. 



L41 MBRATIOXS OF A Rl^BER HOSE, 
L42 MBRATION'S OF A DRL'M, AND 

L43 MBRATIOXS OF A METAL PLATE 

/. IXTRODUCTIOX 

This is a set of three qualitative demonstration 
films. \o work notes for the students are provided. 
The subjects supplement the study of waves. The\' 
may be shown in class by the teacher, or viewed 
by students individually, after the concept of the 
standing wave has been covered in Chapter 12. 

F/7m Loops make use of the concept of standing 
waves and extend it. L41 should be shown fii-st. .All 
the loops demonstrate the following ideas: 

1. The vibrations of bodies can be explained in 
terms of standing waves. 

Suppose you have something lan elastic bodyi ca- 
pable of a certain tvpe of vibration. Then you will 
find that: 

2. The body can vibrate in more than one mode of 
this tvpe of vibration. Each mode corresponds 
to a fixed, but different, frequency of vibration. 

Moreover, the films, especially L41, are so con- 
structed that they suggest the following fact: 

3. In principle, the number of possible modes of 
this tvpe of vibration is infinite. 

In each film, we drive the body at continuouslv 
increasing frequency iwith a motor in L41, with a 
loudspeaker in L42 and L43i. When the driving fre- 
quencv passes thi-ough one of the fixed ft^quencies 
of vibration of which the bodv is capable, some- 
thing happens. This event is shown in detail and 
illustrates the concept of 

4. resonance. 

//. GEXERAL DESCRIPTION 

L41 \lbrations of a Rubber Hose 

Unit 3 presents the concept "standing wave" in 
connection with one-dimensional transverse waves, 
such as are found in stretched strings. 

Whenever two identical transvei"se traveling sine 
waves pass over the string in opposite directions, 
the supeiposed wave pattern appeai-s to be stand- 
ing. To put it in another wav, the string is vibrat- 
ing. The vibration occurs in loops. A loop is exactly 
one-half wavelength long. If the string vibrates in 
more than one loop, neighboring loops vibr-ate in 
opposite phases. Loops are sepaiated bv points on 
the string that do not move at all, called nodes. 
Successive nodes are separated by one-half the 
wavelength of the moving wave. 



FILM LOOP .VOTES 



249 



The lubher hose is driven by a variable speed dc 
motor connected through an eccentric linkage to 
point A at the bottom of the hose. 1 he motoi 
shakes point A in a sideways oscillatory manner-, 
but the ain|)litude of this motion is so small that 
point A can he trt;ated as a node when considering 
the waves in the hose. Motor speed is contr-olled 
by a X'ariac. 

The film opiuis with a scene in which the hose 
is strt!tched to prt)duce tension. The value 7' of this 
tension, together with the mass jjl, per- unit length 
of the hose, determine the wa\'e speed v: 

V = T/\i. 

It follows that the wavelength \ is r-estricted by 
the length of the hose and by the fact that end 
points A and B must be nodes. The wavelength \ 
can only take on the discr-ete values 

ZL 2L 2L 

K = IL, L, — , — , — , . . . , or 
3 4 5 

2L 

k = — , n = 1, 2, 3, 

n 

If/ is the corresponding frequency, and for sine 
waves \/ = V, 

f = — ,n = 1.2,3 

■'" 2L 

The overtones ar-e all integer multiples of the fun- 
damental frequency/ - v/2L. In our- hose,/ = 2 
cycles/sec. 

The above numerical details need not and jirob- 
ably should not be presented by you in class. It 
would be better to keep the discussion on a qual- 
itative level. 

The main sequence of the film r-ecords what hap- 
pens after the motor is turned on and as its speed 
is continuously incr-eased fr-om zerx). As the speed 
appr-oaches 2 rps I not given in the filmi, the am- 
plitude steadily r-ises IrcsonanceK Motor speed 
continues to incr-ease, a transition to the second 
harmonic takes place, and so on. 

The film shows the fir-st 15 tiansxer-se modes of 
the rubber hose. 

L42 Vibrations of a Drum 

The \il)ialiiig bod\ is now a circular rubber mem- 



brane under tension, i The wave speed in this case 
is the square root of the ratio of surface tension to 
mass per unit area.i 

Here we are dealing with two-dimensional waves 
that pass radially inward or outward as well as 
"angularly' artjund the circle. The standing waves 
are now not the simple sinusoidal loops we saw in 
L41. LAZ was made primariK to show what two- 
dimensional standing waves might look like qual- 
itatively. 

The "drum" is also capable of tr-ansverse vibra- 
tion in an infinity of modes. The characteristic fre- 
quencies ar-e now not integer multiples of each 
other-. In the model dr-um we used, they were 50, 
152, 258, and 373 cyclessec for the first four snth- 
metric modes shown, and 100 and 205 c\des sec 
for the two antisxTnmetric modes. 

The drum head was photographed, in some of 
the sequences, with a variable-speed, motion-pic- 
ture camer-a. For each of the sL\ modes, the speed 
of the camer-a was slowK' \'aried from just below to 
just above the characteristic frequenc\' of the mode 
in question, while keeping the drum in steady res- 
onant \ibration. The effect is stroboscopic. The vi- 
bration appear-s slowed down, revealing the shape 
of the membrane for each mode. 

Forlunately for the viewer of this film, the film is 
silent. The loudspeaker driving the drum was run- 
ning at 30 W ruot-mean-square power. 

L43 Vibrations oi a Metal Plate 

A squar-e aluminum plate is clamped tight at its 
center-. A loudspeaker drives the plate from below 
at incr-easing fr-equenc>-. 

This system resonates for manv differ-ent fre- 
quencies. The amplitudes of the two-dimensional, 
standing-wave patteriis in the plate are too small 
to make them visible. 

Sand is sprinkled on the plate. It vibrates fu- 
riously. When the output frequencv' of the loud- 
speaker reaches resonance for one of the plate's 
modes, the sand collects along lines, curves in a 
symmetric pattern, and is quiescent there. These 
lines and curves are the geometrical loci of the 
nodes of the two-dimensional standing waves in 
the plate inorf.i/ lincs\. The patterns of nodal lines 
ar-e c:alled the Chladt}! figures. 



Equipment Notes 

TURNTABLE OSCILIvVIOH 

A veilical r-od is attachetl to a rotating phonograph r-otates. the platform moves with simple har- 

turntai)l('. ibis r-od extends up thn)ugh a long slot mcinic motion iSH.Mi. This combination of tuin- 

cut into a i-ectangular platfoim. The platfoi-in is table and platform is rvfenvtl to as a turntable os- 

(U>nsti-ain(>d to move in a dii-i'ilion parallel to the cillator- iFig. 1 1 Various phenomena illustiated with 

slot and in a horizontal plaiuv ,\s tin* tuintable lh«' tirrntable oscillator aiv desiiibed below 



2.10 



UNIT 3 / THE TKIl'VIPII Oi ,\li:(:H/\\ICS 




Fig. 1 



Sine Curves 

In Fig. 1, the turntable oscillator is shown in op- 
eration v\ith a pen attached to the platform. A sine 
cune, drav\Ti by the pen, can be seen on the strip 
chart recorder on the left. Figure 2 is a repixsduc- 
tion of the trace, which displays the SHM as a func- 
tion of time. 



Figure 4 is a sketch of six such hand-drawn 
traces. Students were here asked to produce traces 
of lai a sine cune, ibi a square wa\e, ici a sawtooth, 
idi an exponential relaxation curve, lei i-ectified half 
waves, and ifi rectified full wa\es. It can be seen 
that quite respectable-looking traces can be pro- 
duced in this way, and it seems likeK that efforts 
to produce the traces will impro\e understanding 
of the kinematics of such motion in physical sys- 
tems. Ask students to give actual examples of each 
motion they are graphing. 



u 


_ 


n 

1 





Fig. 4 

Harmonic SvTithesis 

The superposition of two wa\es can be demon- 
strated qualitati\ely and quantitatively with two 
turntable oscillators arranged so that the recip- 
rocating motions are parallel to each other isee 
Fig. 51. 



Fig. 2 

In order to emphasize oscillations and periodic 
functions in a simple way, students should be en- 
couraged to produce a few hand-drawn traces on 
moving strip charts. The strip chart recorder has 
a plate with two slots mounted above the moving 
paper isee Fig. 3i. The student, using a sharp pencil 
or felt-tipped pen, should make periodic move- 
ments back and forth. 





Fig. 3 



Fig. 5 

,A pen mounted on one oscillator leaves a trace 
on a strip chart recorder mounted on the second 
oscillator. The paper of the recorder mov es per- 
pendicularly to the direction of oscillation. If the 
frequency' of one oscillator is multiple of the other, 
the resulting trace illustrates the element of har- 
monic synthesis: that is, the production of complex 
periodic functions by the addition of rvvo or more 



EQUIPMEiVr \OTES 



251 



frequencies. TTie traces pi-oduced in this way are 
shov\ii in I-ig. H tlntjugh I-ig. 9. 

The ain|)litiJtl(! of the oscillation is incceased by 
mo\'ing the \ei1ical peg towaid the rini of the turn- 
table. Adjust th(! amplitude stj that (;ithei- trace 
alone gives only half-papei- width. 



Fig. 6 




Fig. 7 




Fig. 8 




Fig. 9 

I he fn'quetKy of oscillation depends upon the 
rotational spetul of the turntables. Ihe turntable 
spc'ed sj'lector- pr-ovidcs the ' Coai-se tuning at Hi 
;i;j, 45, and 78 ir\' inin line tuning of one tunitable 
can bv atu^omplished b\' using a \ ariac or power- 
stal \-ollage conti-ol. Alwa\s rrdiicr the s|)eed In 
lowering the xollage. X'oltagcs in excess of 120 \ 



may damage the phonograph motor. (CALTION: 
Do not use the transistorized speed controls now 
available for drills, and other ac-dc motors, as they 
may be damaged when used with phono motor.) 
The speed can also be adjusted i slowed i b\' me- 
chanically loading the motor, i.e., b\' adding weights 
to the platform. Weights should be placed in pairs. 
symmetrically on the two sides of the platform. The 
phase relationship can be altered by adjusting the 
positions of the two recoixlers before switching on, 
and then turning them both on simultaneously. 

ACTrVlTIES 

Some student activities associated with the cou- 
pled oscillators follow: 

1. Pi-oduce superposed traces of two sine curves 
of different frequencv ratios, amplitudes, and phases. 

2. Attempt to analyze superposed traces in order 
to identify the components. 

3. Compare the sum of two sine curves with the 
original curves: The orxlinates of the original curves 
should be first added arithmetically, point for 
point, and the resulting theoretical curve can 
then be compar-ed with the one obtained b\ actual 
mechanical addition of two sinusoidal motions. 

4. Apply the skills learned from the above to the 
analysis of oscilloscope traces of simple sound 
waveforms fix)m musical instruments, tuning forks, 
combined output of X\\o audio oscillators, etc. 

Beats 

The outputs of two coupled oscillators produce 
noticeable beats if the frequencies f^ and f, are 
nearly the same. Beats are commonK' demon- 
strated by simultaneously sounding two tuning 
forks of slightly different frequencies. Beats can 
easily be produced on a piano or oi^an b\' plaving 
two adjacent notes (a black key and a white keyi 
simultaneously. 

The superposition of two waxes to produce beats 
can be demonstrated quantitati\el\ by coupling 
the out|Juts of the two turntable oscillators as de- 
scribed in the previous section. Set each oscillator 
for equal amplitude, and the coarse tuning con- 
ti-ols for the same fitujuencx . Ihe amplitude oieach 
oscillatoi- should not be greater than one-half the 
paper width. 

Figui-t^ 10 shows the trace with, first, onl\ oscil- 
lator #1 running at fj and, then, onl\' oscillator 
#2 running at /]. The beats are prxjduced on the 
strip chart i-ecorder when both oscillators are op- 
erated simultaneously. 







Fig 10 



ZS'Z 



I'MT 3 / THK THIIIIVIPH OF MKCILWICS 



When two functions have the form v, = sin a 
and V, - sin b and these functions are added, the 
result is: 

v = \', + v, = sin a - sin fa 

- 2 COS (^j sin (^) 1 

If a = Zirf^t and fa = 2Tif,t and/, > C Equation 
111 becomes 



V = [2 cos 2tt 



;4^),...,f^) 



t 2 



Vou now see that y is a periodic function v\ith 
cin a\erage frequency,^, gi\'en by 



L = 



(3' 



StmultaneousK', the amplitude of v \ aries in time 
with the lower firequencv': 

Jamp ^ 



the amplitude frequenc\i I4i 



One complete c\ cle at this frequency is marked 
amplitude frequency' in Fig. 10. 



1 l.i 




Fig. 11 

\\ ith two oscillators operating at slightlx' differ- 
ent frequencies, one gains on the other. .Assume 
that the\ start at the same time but 180" out of 
phase Fig. 11 1; their outputs add to zero a null . 
.As #1 oscillator overtakes #2 they come into 
phase and their outputs add as in b . As #1 con- 
tinues to gain on #2, they again go out of phase 
[the null at ic]. The two are again in phase at d 
but now in the opposite phase: and again out of 
phase at ei; and so on. Since the positions at a 
and e are identical, one complete amplitude c\cle 
has elapsed. The beat frequenc\' c\'cle is between 
consecuti\e nulls or maxima and occurs tv\ice dur- 
ing each amplitude c\de marked amplitude fre- 
quencA' in Fig. 10. 

Spatial Frequencj' and M a\e .Number 

From the trace on a strip chart recorder one can 
measure the spatial frequency," i.e., the number 
of oscillations per centimeter on the chart. The 
svmbol V ' I Greek letter nu'i is used for spatial 
frequencN': the units are cm~V (The spatial fre- 
quency is quite analogous to the familar time fre- 
quencv / i.e., the number of oscillations per sec- 
ond, which is measured in sec'Vi Just as time 
frequency' is the reciprocal of period i/ = 1 Ti, so 
spatial fipequenc\' is the reciprocal of wavelength. 
V — 1 K. Incidentalh , the term wave number, used 



b\- spectroscopists. is also the reciprocal of wave- 
length. 

If the tape mo\es through the recorder at a uni- 
form rate, v ^ f. If v is constant then 

v 

Since \'ou already know that y is a periodic func- 
tion with a time frequency/ Equation 5' shows 
that y must also be a periodic function with the 
spatial frequency v^. Equations 3 and 4i can now 
be written in terms of spatial frequency: 



V, - V, 



and 



I the a\erage frequency) 



V, — V, 



161 



17) 



I the amplitude frequence" 

From Equations 6^ and i", di\ision gives the 
number of oscillations in e\erv two beats: 






i8i 



These three relationships can be \erified with rec- 
ords similar to the one shown in Fig. 12. 




Fig. 12 

Lissajous Figures 

Lissajous figures can be produced b\- coupling t\vo 
oscillators with the output of one perpendicular to 
the output of the other see Fig. 13 . Demonstrate 
Lissajous figures first with two turntable oscilla- 
tors and then with an audio oscillator and an os- 
cilloscope. Set the oscilloscope sweep control to 




Rg. 13 



EQUIPMENT NOTES 



253 



"line" position and adjust tli<; sweep width to 
ahout oiif!-iialt tin; scit;en diainetec. (Connect the 
audio oscillator- to the vertical in[)ut, and adjust 
tin; amplitude of th<; signal until the; sif^nal is ahout 
one-half scit;en diani(!ter. Stational IJssajous fig- 
ures will a()|)(;ar on the oscilloscope scttuMi when 
the; oscillator lrr;(ju(!ncy is exacth' a multiple, or 
suhniultifjie, of the line frequency. If the phase 
changes slightly, the shape of the figure is altered 
(see Fig. 14). 




Fig. 14 Same frequency; phase difference changing from 
zero (straight line) to 90" (circle). 



If the frequencies of the oscillatoi-s are propor- 
tional to whole numhei-s, the trace closes u|)on it- 
self and may he repeated again and again. Figure 
15 is a i-eproduction of actual traces pixjduced hv 
a pair of tumtahle oscillators. 



Some of the traces show that a given frequency 
ratio can produce a varietA' of figures if the relative 
amplitude and starting phase are changed. 




Fig. 16 Frequency 2 to 1 with zero phase difference. 




Fig. 17 Frequency 2 to 1 with 90 phase difference 




Fig. 15 Left picture shows almost equal frequencies, 
drifting in phase. Right picture shows frequency 3 to 1, drift- 
ing in phase. 




Fig. 18 Frequency 3 to 1 with 90 phase difference. 




Fig. 19 Frequency 3 to 1 with zero phase difference 



254 



l'\ir 3 / lill-; TKIl'MPH OF IVIBCIIAMCS 



THERMISTOR 



A typical thermistor has a resistance of about 100 
K at 25°C, and its temperature coefficient is ap- 
proximately — 5%/'C [i.e, resistance at 26°C is 0.95 
X 100 K; at 27°C it is 0.95- x 100 K, and at I25 + 
t\°C it is 0.95' X 100 K]. A plot of resistance R versus 
temperature on semilog paper is a straight line: 



lO 



1 
vp 



lo 



to- 



too 



20 Ao 

TtMPtRATUR.E 

Sample Calibration Curi'es for 
Thermistor— Amplifier Combination 

The graphs are intended as examples only the 
slope of the line isensitix-it^i will \ar\ with the gain 
setting and the intercept will depend on the actual 
thermistor used, the gain, and dc offset settings. 

Xote that o\er small temperature ranges the re- 
sponse is linear. 




-6^ 




DC OFFSET ■ H> ^c If 



UiFftyerrc vo.m 

.nioJ.I 99S-0O4- 



, aoo - 



H 1' 



?0 




oo35o ,o\('c^a.\e 
ta5.o4-a4 5S")c' 

I 'i.OQm'J 

= 3oo "^ 'j/c ° 



^P M PE_R.A-rO «.£. 




•^e+er on Sou A^cale, 



_4q*«-4 



To measure small changes at other tempera- 
tures, adjust the gain and dc offset as necessar\' to 
get a near-zero reading on a sensiti\e scale. 

To measure temperatures o\er a larger range it 
is simplest to measure the thermistor's resistance 
directly in ohms with a volt-ohm-milliammeter (see 
also Experiment 3-10: Thermometers and Tem- 
perature' i. The resistance \ersus temperature ("O 
plot is not linear o\er a wider range isee above i. 



EQUIPMENT NOTES 



255 



Sufjfjested Solutions to Study Guide Problems 



CHj\inEl\ 9 



2. He called tliis stateiiKMit an uncoiilestahle ax- 
iom" because in all cases whei-e lie had made 
careful measurements he cDuld find no de\ia- 
lion from it. Recall Xiju-ton s ' Kuh; ol F^easoning 
HI" quoted in Chapter «. 11 mass \\(;i-e not con- 
seiA(!d in all reactions it would ha\e been a 
remarkal)l(! coincidence il La\'oisiei- had hap- 
pcuied to test onh reactions where it was. 

3. (ai Yes; an inccease of 2 x lO' parts in K x lO"' 

(less than a rTiillionth of a millionth of a mil- 
lionthli falls far- below the limits of accuracy 
of any exp(Miment. 
ihi The solar- system would \u'. sufficiently large 
if the stat(!rnent given about the soirrce of 
meteoric dust is acceptj;d. (Assume the so- 
lar' system to tixtend out to l.H x lo" km to 
include comets. i 

4. I lu! |)ur pos(! ol this questifin is to foict! stu- 
dents to (!xamine thcnr- own thinking with r-e- 
gard to the durability of sci(!ntilic statemcMits. 
(If (;xperim(!nts are limited to ordinary chemi- 
cal reactions, the; ansutM- would be; rio, based 
on consid(Mation of mass-(MK;rg\' transforma- 
tion. Most stirthuits will not be; av\'are of this, 
howexer', before studying (;hapt(M' 20. i 

.'5. i\o; a diflerence in \\(!ight ipull of gia\it\ i is not 
the same as a dilfer-ence in mass liruMtiai 



(b) The momenta of the discs are: 

A. 40 kgm/sec west 

B. 150 kg in/sec north 

C. 20 kg-m/sec east 

D. 20 kg-m/sec east 

(c) The momenta east and west cancel. After 
the collision, the total mass is 2.'> kg Since 
ATjjV, = m,v',, 

150 kgm'sec norlh 

V, = 

25 kg 

= B m sec norlh 

(di Because momentum is a vector quantity . 
momenta in one dir-ection can cancel those 
in another dirxntion, as for discs A = C -(- 
I) in jjart ibi. 

10 lai {;i\en: m^ = 10" kg = 1.5 x 10" 

v\ = 2.0 msec rn^v^ = \\' = x„' 

m^v^ + m„v„ = m^\\' + m„v„' 

m^\\ -t- m„0 = ifn^ -(- ^n^^f\\' 



ibi v' = 



(C) V, = 



m, -t- m„ 
10' X 2 



2 X 10' 



10' -(- 1.5 X 10" 2 5 X 10' 
= 0.8 msec 



tt. Place the snake within a contairier- (Hiirippcd 
with a device for- igniting the |)ill by reriujte con- 
tr-ol (for- example, an electric spaik). Seal the 
container and deteniiine its mass before and 
alt(!i- ignition. 

7. (ai Mass of li(|irid remaining = sum of initial 
masses minus mass of precipitate - (19.4 
+ 100 -I- 3:M + 1001 - 32.3 = 220.2 g. 
(bl The solids going into reaction weighed 
19.4 g + 33.1 g = 52.5 g total 
The yellow jirecipitate weighed 32.3 g dry 
and the white pi-etipitatc w(>ighed 20.2 g for- 
a lota! of 52.5 g of solid. NO mass has been 
added bv or- lost fr'om the water- 



«. (at 
(bl 



(el 
9 lai 



The total mass is (iO g on the earth and on 
the moon. 

Mass is an attribute of material. Weight is a 
description of the graxitatiorial attraction of 
a large body (earth or- mooiH on a smaller 
mass. Mass does not (hange, but weight de- 
pends on location 
The statement reporls nothing about ma.s.ses. 

l-'or- an isolated s\stem Iher-e will Ix- no 
change in the total moinciiinrn 



1 1. ,\(-corcling to Wrhslcrs I'hird Intvrwilioiuil Die- 
tionur}'. momentum is 

( 1 1 a pr-opertN of a moving bods that determines 
the length of time recjuired to bring it to rest 
wIkmi under- the actit)n of a constant for-ce 
or- moment 
(2i moment 

(31 (a I the foix'e of motion acquiretl b\ a mo\- 
ing bod\' as a result of the continuance 
of its motion 
(hi something held to rvsemble such foix-e 
of motion of a moxing bod\-. 

None of these statements resemble the lech- 
nital definition of mass a veltuitN but delini- 
tion ' li abo\e is tied to Sec. 9.4 discussion: /-A/ 
- A/nv. 

12. lo ha\t? e(|ual impact tlu* momenta of the can- 
non ball and the light par1i(-|e w-ould ha\e to 
!)(• e(|ual 

mass of light panicle x ij x lo"i 
= 10 X KM) kg rn sec 

10 X 10^ 

mass of light p.ir tr(-le = r 

:< ^ 10 

= 3.3 X 10 ' ku 



2S« 



niviT a / Till: liiii'ivii'ii or iviEt:ii \\it:s 



13. la I The carts exert forces on each other. 

ibi Ves, to the sxstem as a whole assuming 
there is no friction between carts and track. 

(ci Take a stroboscopic picture of the collision. 
Note that there is no well-defined instant of 
collision and that the carts do not ha\e uni- 
fomi speeds before or after the collision. So 
momentum comparisons will ha\e to be 
made for corresponding time intervals or 
when the carts are so far apai1 that the re- 
pulsixe force is \er\- small. 

14. The closed s\stem must include the ball and 
the earth; as the ball rises the earth mo\es awa\ 
in the opposite direction, as the ball falls the 
earth falls toward the ball. Of course the 
mass of the earth is so much greater than the 
mass of the ball that the earth s speed in both 
cases is too small to detect. 

15. Ves. Corollar\' III states: The quantit\ of motion 
which is obtained b\ taking the sum of the mo- 
tions directed towarxis the same parts, and the 
difference of those that are directed to contrary' 
parts, suffers no change from the action of bod- 
ies among themsehes.' Definition II states: 
"The quantitN of motion is the measure of the 
same, arising fhjm the \elocit\ and quantit\ of 
matter conjointK . 



16. lai Since Ap - F\t, 

Ap 80 kgm sec 
AF ~ 20 N 
- 4 sec 



At 



ibi \\ = i\', + Avi = i\', ~ alv 

50 kgni sec ION x 5 sec 





5 kg 5 kg 




= 20 m/sec 


Pi 


- p, + Ap 




= 50 kg m sec -t- i5 sec x lO N 




= 100 kgm sec 


Pi 


= m\ ,, therefore 


1- 


_ El 




m 




100 kgm sec 




5kg 




— 20 m/sec 



let Some problems are easier to solve with the 
momenta formula, but the\' are not more 
basic. 

lai Both ocean liners and planes have large 
momenta: undergoing a significant change 
in direction implies a \elocit\ and momen- 
tum i change. Such a change requires a force 
F operating for a time Af. Water and air do 
not ha\e a fiim enough consistency to allow 
for a sizeable F hence Af must be large. 

h' The argument is similar to that abo\e for 



planes, unless large rocket blasts are also 
reported. 

18. Initial momentum = 60 kg x 20 m/sec 
= 1.2 X lo' kg'msec 
FAf = 



-- A 


momentum 




A 


momentum 






Af 






1.2 X 10' - 







3 




= 4 


X lo' \ 




Av 


0-20 




Af 


3 





acceleration rate - -^ - 



— - 6.6 m/sec 
d = v,f -I- Vz ar = 30.3 m 
Her momentum is imparted to the earth. 

19. Estimate the scale of the photograph, assuming 
the golf clubs to be 1 m long. The sketch rep- 
resents a trace from the photograph on page 
27 of Unit 1. Three successive relative positions 
of club and ball are shown; the first before con- 
tact, the second and third after contact. 




(a) The distance between ball positions 2 and 
3 is about the same as the length of the club, 
hence the speed of the ball after impact is 



1 m 



0.01 sec 



100 m sec 



ibi momentum = 0.046 kg x lOO msec 
= 4.6 kgm sec 

(c) In the interval between the first and second 
strobe flash lO.Ol seci the club must move 
to the ball and the ball away from the club. 
Thus, duration of impact must be less than 
about '/» of 0.01 sec or 0.003 sec. 



idi F = 



Amomentum 



It 



4.6 
0.003 



1.5 X 10 \ lat 
least) 



20. Yes; when the statement is made that mA\' 
Am\. 



21. 



^\ *" AB "^ 'AC 

P =z F -^ F 



SUGGESTED SOLITIONS TO STl D\ Gl IDL PROBLEMS 



257 



Since the interaction time Af is the same for all, 
the ;:hanges in momentum uili he as follows; 

^Py = /-A ^t = '•\« ^' + l-'u ^' 

Ap„ = F„ M = F„, A/ + F,„ A/ 
Ap, = F, At = F,^ Af + F, „ Af 

Total momentum = A/;^ ^ A/;,, -^ A/;, 
change = (/.■ -). /r + c- + p 

" 'ah ^ ' a\ ' \< ' t A 

+ F,^ + f.H' A/ 
= since F^„ - -F„^etc. 

22. lai F Ar = Imv = miv^, - v) 



A/ = 

F 

ih) Exhaust momentum would be equal to mo- 
mentum change of capsule: mu,, - » ' 
(cl Mass of fuel x v.. = mlv„ - v) 



mw.. - VI 



mass of fuel = 



23. lai liv the consei-vation of momentum (jrinci- 
ple, Alfrj^\"\i -t- AifT7|,r„i = 0, assuming m^ 
and m^^ to he constant. 

Then, m^ A\\ = -m„A\'„ 

-Ai'„ m, 

(hi Let A he the pellet, B the; howling hall then 

, m,. 
m„ > m. and — > 1 

'"ii 
Av'^ = A\'|, hence A\\ > A\„ 

m — rti 

(c) v.' = 



If m„ > niy we can neglect the contrilnition of 

m^ to both the numerator and denominator 

-m. 

aho\e, so that v^ becomes \\ = -\\ 

m„ 

That is, the speed is the same hut with ix^vei-se 

dii-ection. 



24. m, v, + m^\\ = n1^\\' + niS\' 
(a) V', = v\ = 
(bl Total momentum is since both bodies are 

at it^st. 
(c) Total momentum is In the law of the con- 

seivation of momentum. 
(di Mat^niludrs of /n, r, ' anti inS\' arc e(|ual 

since their- sum is e(|ual to 



lei = m^\\ ' + /n^\\' 



10 kg 1 



/(I, 1(10(1 kt; 1(10 



V, = X 1000 m/sec 

^ 100 

= - 10 m/sec 

25. lai A momentum = F A/ i35-28i x lo*" x 1.5 

X 10' = 10.5 X 10" kg msec. 
(b) The amount of fuel that was expended. 

26. In cases lai, ibi, and idi Af is lengthened, 
thereby decreasing F. In case ici, Af is short- 
ened making F large. 

27. la) \o. Momentum is conseiAed The center of 

gra\it\ of the s\stem i-emains fixed. As the 
ball swings foi-ward the cart moves back- 
ward and \ice versa when the ball swings 
back. 

ihi The cart would continue to move forward 
with the slight oscillation described in lai. 

ICI The cait uould continue to mo\e backward 
with the slight oscillation described in lai. 

28. lai The car on the left was traveling faster. 

ibi The speed of one car lor the masses of the 
cars I, the distance they slid after- the colli- 
sion, and the retarding frictional foix'e of the 
gix)und. 

ici The frictional force between the grxjund and 
the car's is constant. 

29. Assume the per-son is fixed to the earth. While 
the bullet is in flight, per-son. target, and earlh 
collecti\el\ ha\e the same backwarxl momen- 
tum as the bullet fol^valx^. When the bullet hits 
the target, the speeds of all the components 
become zeit). Momentum is conserved in lai, 
ibi, and ici. In all thr-ee cases the total momen- 
tum = 0. 

30. lai 0.8 X mass of ball 1 toward the cushion 1 
(hi 0.8 X mass of ball laway from cushioni 
Id A momentum = 2 x 0.8 x m = 1.6 m 

Id) No, for the s\stem that contains onh the 
ball. 
Yes, for the system includes the hall and the 
earlh I hen the compensating A momentum is 
supplied In the cushion-tahle-<'a(lh svstem 

31. lai As he leaps into the air the asteroid will re- 

coil in the opposite direction. 
lb) The asteroid will always move in the op- 
posite diivction to his motion, so it will spin 

32. lai The total kinetic energv' of an isolated s\s- 

tem involving onlv elastic collisions is con- 
stant 
ibi The total kinetic energv- does not change 

10 (KE), = V,£ m.v^ -t- '1 my, 

= Vj X 5 kgi4 nvseci" 

+ 'i X 10 kg(3 mseci* 

= 85 kg-m sec" 

= 85 J 
'd' \ft«*r elastic ri'l)ound the lol.ii kinetit en- 
fitrv rs not changed 



ZAM 



UNIT 3 / nil-; TRIl'MPH OF IVH^CII AMCS 



33. 



Object 



(kg) 



(m sec) 



mv 
(kgm sec) 





Vzmv' 


(kg 


m'sec^) 


63 




214 




0.056 




1.79 ' 


10* 


4.0 X 


10 * 


12.5 X 


10* 



30.0 

50.0 

1.5 

49.6 

0.4 

5.0 

\ole:.Ans\\ei-s to be supplied b\ students ai-e in bold r\pe. 



baseball 


0.14 


hockey puck 


0.17 


super ball 


0.050 


light car 


1460 


mosquito 


5 X 10' 


football player 


100 



4.2 

8.55 
0.075 

7.25 X 10* 

2.0 ■ 10 ^ 

500 



34. lai The total mass is4g-'-6g + 8g=18g. 
The total niomentuni is 

i4 X 20i gem sec noilh + i6 x 31 gem 
sec east + 18 x lOi gcm^sec south 
which totals 18 gem sec east. 
The total kinetic energ\- is 
K£ = '2 X 412O1- -t- '2 X 6131- 
+ V2 X 8il0r 
= 1,227 J 
lb' Kinetic 

Mass Momentum Energy 



36. 



1. open system 
elastic collisions 

2. open system 
inelastic collisions 

3. closed system 
elastic collisions 

4. closed system 
inelastic collisions 



18 g unknown unknown 

18 g unknown unknown 

18 g 18 gem sec 1.227 J 

east 

18 g 18 gem sec 1,227 J 

east 



35 Given; m, = 3 m^ \\ = -\„ 

v^' = v„'= -2v„ 

To show conserv ation of momentum: 

3m' -\„i -+- m \ „ = '3m 0> -^ m^ ~-^'b 

- 2m \- = - 2m \ „ 



lo show consen ation ot kinetic energ\ : 
'/2i3mi v^' + Vzm v^" = '/2i3mii0i" -t- '^2 mi2 vg 



41 '/2m \',5 I = ' 2m4\, 
2;n v' = 2m v' ' 



m^ - m 
\\ - \ 



m„ = 3/77 
V '„ = 



Conservation ot momentum: 

777\ = m\\' + 3771l„' 

\ = \\' + 3\„' 
V, - v - 3\„' 

Since the collision is head-on, the \ector aspect 
of this equation will i-educe to onl\ the possi- 
bility of different directions for the \'s that show 
up algebraicallx as a ± sign. 

Consen ation of kinetic enei^': 

Vzmv' - '/2m\\'' -s- '/23m\ „'" 
i' = v,'- + 3v„" 

Substituting for \\' from the result aboxe 
V' = v~ — 6\Tb' + 9\h'' -•- 3\'„'' 

6^'V'b' = 12v,'- 
\ 
*'b' = - assuming \V / 

v V 

\ , = \' — 3i R — \— 3-= -- 

2 2 

rhus the small ball rebounds with one-half its 
original speed and the lai-ger ball mo\es in the 
original direction with one-half the original 
speed of the small ball. 



CHAPTER 10 



2. \o work is done since the dii-ection of the force 
is perpendicular to the direction of motion. 
I Students may argue that he does get tired. 1 



Force 



direction of motion 



3. lai The speeds in the new reference frame are 
IV, — ui and i\', — uK 
lb I \o. In the new reference frame, they are 
KE,' = '/2Ani\-, - ur and KE,' = Vzmiv, - 



Id \o. 

AKE' = KE,' - KE,' 

= '/2ml V, — ui" — V2 7ni\', —ID' 

= Vzrriv,' - 2m', + u" - \'," -t- 2m', - u" 

= ['/amv," - '/2/Tiv,^ -I- lmu\\ - 777m- J 

This is not the same as before vxhen 

AKE = l2771\',- - V2 7T7X-,- 

.Note the additional tenns; 777m , - mu\\ 

Id) \o. W = F' X d 

Since F' = F. and d' = d - ut. 



SLGGESTED SOLITIOVS TO STl DV GLIDE PROBLEMS 



259 



W ^ Fd - Fut 

= Fd - ufFt> 

= Fd - uimAvi 

= Fd - umiv., - v,i 
This is not the same as before when u = Fd 
(e) Yes. W - Fd - umW^ - v,l 

= AKE - umiv^ - v,l 
= [Vzmv./ - y^mv^^\ + \um\\ — umvj 
which is the same as AKE'. 
(fl (July liiii. rlic r(;iati(}nship Fd - Ai'/vmv'^i 
igi One would tiiini< tlial sonH;thing an object 
has" vvouici be a pio|j(Mt\ of the object it- 
self, and not be dependent on anything out- 
side of the object, as, for example, the i-ef- 
erence frame from which it is ineasui-ed. 

4. KE = V2m\r 

= VkO.l X 10"^')(2 X 10*) 
= 1.8 X 10 '^ J 
IJ 



N = 



1.8 X 10 '^ J per electron 
= 5.5 X lo'^ electrons 

5. The final KE is 80 J. f he initial KE is 

IKE), = Vimx^ 

= Vz X 5 kgl4 m/sec)^ 
= 40J 

rheit;foi-e, 40 J must also ha\e been added by 
the 40-N force. 



A(K£I 


= 


Fd 


d 


= 


A,IKE) 
F 




= 


40 J 

4 N 




= 


10 m 



6. Estimates may ha\e a wide range. 



KE = 





m 




V 




2 




(kg) 




(m/sec) 


v" 


(joules) 


(a) 


0.15 




30 


900 


67.5 


(b) 


lO" 




300 


9 X 10* 


4.5 X 10' 


(0 


75 




10 


10' 


3.75 X 10^ 


(d) 


6 X 


10^* 


3 X 10" 


9 V 10° 


2.7 X 10" 



ibi v\or-k done ^ Fd ^ 400 x 900 
= 36 X 10* J 
= AKE = '/2200v'^ = 10 V 
v^ = 36 X lo'; \' = 60 m-sec 
The r-esult is the same as in lai. 

8. (al AKE = KE,„,, - KE.,,,,,,, 
= - Vzmv,^ 
= - '/z(0.002)l300l- 
= -Vz X 2 X 10"^ X 13 X 10^)^ 
= -90J 
(hi Work done by tree = -AKE of bullet = 

90 J 
ici V\' = F X d 

W 



F = 



90 J 
0.05 m 



= 18 X 10- 



9. W = F X d = AKE of ball 

= Vzm\'^ = Vz X 4.6 

X 10"' X lOO' 
= 2.3 X 10- J 

The ball does the same amount of woi-k on the 
club in the opposite direction. 

10. mass of penny = 3g = 3xio~^kg 

thickness = 1.5 mm = 1.5 x I0~^m 
The top penn\ is 49 thicknesses or 74 x 10~^ 
m abo\e the bottom one. 
lai PE of top penny = ma^ = i3 x lo 'i 

X 9.8 

X 17.4 X 10 -I 
= 2.2 X 10 ^Joules 
ibi a\'erage height of pile - 

25 X 1.5 X 10"^ = 37 X 10'^ m 
total PE — no. of pennies x mass of each 
X a^ X a\erage height 
= 50 X 13 X 10" ^1 X 9.8 X 

(37 X 10 "'l 
= 5.4 X 10 -J 



11. lai d 



1 J 
5 \ 



= 0.2 m 



(bi Enei^' required = work done lifting the 

plane to cniising altitude 

- weight of plane x 
height 

= 7 X lo' \ X 10* m 

= 7 X 10" J 



F 400 , 

7. (a) a = — = - 2 m/sec 

m 200 

2 

, 2d 2 X 900 

t^ = — = = 900 

H 2 

/ = ;U) sec 

\ .jf - 2 X 30 = 60 m sc( 



12. 1. Prtnide a standatti mass raised to the pixijwr 
InMght rhe ptx)per height can be delenuined 
for a 1-kg mass as follows: 



PE = ma^ = 1 J 
d = 



1 J 



= 10.2 V 10 



1 kg X 9.8 
- 10.2 cm 

Note Ibis height (h'pcnds on value of a .it tin- 
|).ii'tii'uiar' iuc.ition 



inn 



UNIT 3 / THH TKIl'MPH OF IVlEf:H/\\ICS 



2. Pro\ide a standard mass tra\eling with the 
proper speed. For 1 kg. the speed can be de- 
termined as follows; 

KE = Vzmv- = 1 J 

2 X 1 J 



1kg 
1 .4 m sec 



\ote: This method has the disadvantage that 
the energv' cannot be stoi-ed. 

3. Provide a compressed spring on which 1 
J of work has been done. 

\ote: This method requires a knowledge of the 
beha\ior of springs, including the fact that they 
can lose their elasticir\' thixjugh aging. 

13. lai The stored energ\' can be measured as a 

change in potential energ\', PE - mgh, 
where h is the stretch A,x of the rubber band, 
ibi A iPEi = -AKEi Id -As the weights bob 
up and down, the total energ\- of the s\stem 
is distributed in changing fractions between 
kinetic energ\', graxitational potential en- 
ergy, and elastic (stretch) energy. 

14. lai Force exerted on earth by stone 

= force exerted on stone by earth 

= 1 kg X 9.8 m-sec" 

= 9.8 X 

mass of earth = 6 x l(f^ kg 

accelera