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PROJECT
PHYSICS

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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

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

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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

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

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

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

^ - 'J

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

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

Teacher presentation:

Lab stations:

Small-group

Lab. Stations:

17th century Exp.

problem solving

Free Fall

Write up

Handbook: Survey

observa-

Text: 2.5-2.10

Handbook: E1-5

Chapter 2

tions

Teacher presentation:

Galileo & free fall

Quiz

Organize E1-1:

Naked-Eye

Astronomy

Handbook: Survey E1-1

Lab stations:

Vectors

Observe sky
Text: 3.1-3.4

Teacher

presentation:

Vectors

Observe sky

Text:

3.5-3.9

17
Lab stations:

force

mass
acceleration

Discussion:

Newton's laws

Ob«erve sky
Text: 3.10-3.11

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

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

Small-group

problem solving

session

Text: Reread 4.4

PSSC Film:

Frames of Reference

Small-group

discussion of film

Text: 4.7-4.8
Epilogue

Teacher presentation:

Satellites

Review Unit 1

Review

Individual student
evaluation

Test

Discuss test

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

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

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

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

200

100

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

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

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

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

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

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

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

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

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

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

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

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

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

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(!

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

(in intervals)

(mrr

on photo)

0.0
0.5

1.6

3.1

4.8

7.0

9.5

12.2

15.0

17.8

20.7

23.6

26.3

28.9

31.1

33.0

34.5

35.6

36.5

DEMONSTRATION NOTES

TABLE 2

Symmetric Intervals

A<y

v..

one interval

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

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

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.

>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

w„

one interval

(mm/Vso sec)

cm/sec

is Vso sec

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

.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.

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

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 —

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

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

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

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-

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

(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

(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

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

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

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

(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

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

Thermometer*

Thermometer

Ruler

Common calipers

Ruler

Magnifier

Ruler

Stopwatch

Graduated cylinder

Balance

Stopwatch

Balance or ruler

Voltmeter

Diameter

Temperature

Length

Diameter

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

■+■ —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

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)

t(sec)

f (sec)

t* (sec)

2.7

1.9

1.1

3.3

2.5

1.4

3.7

2.8

1.6

0.9

4.2

3.1

1.8

1.0

4.5

3.5

2.0

1.1

4.9

3.8

2.1

1.2

'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

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

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

20 cm

0.75 sec

1.26

1.54

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

hall#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,

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

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

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)

sec

sac

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

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

= V /2y ^ /v:

\ a

EXPERIMENT .VOTES

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' =

(1)

gives us

F cos - m

v' cos" 6
R cos d

which simplifies to

F -

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 —

-, 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

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^ =

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 =

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

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:

1 m

4.5 sec

4.45 m sec

9.85 m see

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

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

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

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

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

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

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

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

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

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

3,600 min

V«o sec

1.800 min

'/io sec

1,200 min

Vw sec

900 min

%»sec

600 mm

v.osec

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

^^^-/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

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

|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.

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.

mm.

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.

.+ r

— so / is not proportional to

blc ' ' ll.xed

EQUIPMENT NOTES

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

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\

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

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 = -

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

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

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

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.

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\ \(

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

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

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

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

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

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

then V - ——

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

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

«iop«' "—

tr*

^-g'^X-

IS'

>/ j CkX' -Ki.- O.,

•Jt-i

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

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

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

UMIT 1 / CONCKinS OF MOTION

6. la) First, we need to find the total time taken,
It

Ar, + It,
Ad, Ad,

V, v,

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

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_,^ =

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

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

0.75

0.57

0.85

0.63

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

^x-.

20 3D AO

tir->« (sec )

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

200 km - km
2.5 weeks - week

= 80 km\veek

Instantaneous speed at Q = — at Q

21. la)

40r

50-

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

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

23.

Data for graphs taken from photographs

Position

read on scale

1.5

9.0 cm

0.20 sec

45 cm sec

10.5

0.20

23.5

17.5

0.20

87.5

39.5

0.20

105

60.5

25.5

0.20

127

87,5

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_ =

36 m - m

6 sec

6 m/sec

\\j

(6 m/sec) (6 sed

36 m

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

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.

12.

Average

time

15 -I- 16 -^ 17 -h 18 -^ 19

— = 17 years

Average earning power

S8,000 -I- 512,000

S20,000

SIO.OOO

13.

lai a^^

V = 284 m sec
t = 5.0 sec

284 m/sec
5.0 sec

—

57 rri/sec"

SUGGESTED SOLLTIONS TO STUDY GUIDE PROBLEMS

(bid

= '/za/'
a = 57 m/sec^

(b)

f = 5.0 sec

100

= Vz X 57 nVsec^ x 15.0 sec)^

= 710 m

'•>.

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^

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

(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

40-

5 20

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

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 =

- at

- 20 m^sec"

= - 10 mysec^

= 1.0 sec

V =

m/sec - 10 m/sec"

X 1 sec

V =

m/sec

Ibid =

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 =

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

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

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

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 = - \',

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.

4 m/sec + nvsec

v„ =

\'„ = 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)

(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:

n - ^\

- 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

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 = -^ '

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.

-^^

^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'

2(20 m)

2 m sec*

= 20 sec'

= 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

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'

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

a) Three blocks east of the starting |)oiiit,

(b) (3 + 4 + 5 + 1 + 2) blocks = 15 blocks

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

20. la) a^,, = —

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

(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'

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

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"

.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

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 =

_ 1,000 kg m/sec^

" 50Tg

= 20 m/sec^

V = at

= (2 m/sec") (20 sec)

= 400 m/sec

= 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

m = -
a

_ 25 kg • nvsec^

5 nvsec*

= 5 kg

(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

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
=

a — —

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

4. when t

10 sec

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

\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

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.

7. (a) V = -

d
t = -

_ 25 m

10 m/sec
= 2.5 sec
lb) d = Vzat^

= Vz (-9.8 m/sec^)(2.5 sec)^
= - 30.6 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 =

(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

Displacement

Average speed
or constant
speed

Instantaneous
speed

Velocity

Acceleration

Acceleration of
gravity

Centripetal
acceleration

Frequency

Period

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„ =

m = 5.98 X 10"^ kg

SUGGESTED SOLUTIONS TO STUDTi GUIDE PROBLEMS

103

H = 1.495 X 10" m

2TTrt

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

= 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 =

, _ FR

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- =

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

—

0.1

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" = 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

= 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 = '-^^^
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.

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

Filmstrip

demonstration

E2-6, The Shape of

the Earth's Orbit

Prepare for debate

Handbook: E2-6

Help session on E2-6

Problem-solving

session

Finish orbit plot
Text: 6.6-6.8

Selected
Study
Guide
quest.

Small-group

problem solving

10
Quiz on Ch. 5 & 6

or
Other evaluation

Lab Stations:

Kepler

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

Small-group

problem solving

Handbook: E2-8
Finish Mars plot

Selected Study Guide
questions

Text: 8.1-8.4

Lab stations:

Newton

Handbook:
Survey Ch. 8
for options
days 20 &
21

Demonstrations

from day 16

Teacher presentation:

The

Newton Synthesis

Text: 8.5-8.7

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

Review

Other evaluation

23
Test

or
Other evaluation

Discuss test

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

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

Review

Review Unit 2

Evaluation

Test

Evaluation

or
Discuss test

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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*

Epicycle

Ratio

Deferent

Radius

Object

Sun/Planet

Sun

2,00

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,

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 ~~

— . 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 ^

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. "^

1 -t-

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

If we put the first of these expressions into Equa-
tion 111 and the; second into Kquation I2i, then we
get

3 . — 3„ — K

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"^

0°

re/K^ or euM o&set2_vATioN'5

-A.

.□"'^-x X

^ />^^^a. \

''/

□

,0/

□

□'

iio

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)

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^

1^0°

\70°

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.

JkO

.-_-

r —

•

» J

;^:

•

—

1 1
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A»«

o
1 ^

(

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1
•
1

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—

1 :

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i i

'.'

•

> A

'* <

•

rJL*'

fl ,

. •'

1 a "

im*

1 a '

> w 1

'• 1

' • «

r ,

' •.

■ a 1

> • .

' a <

1 a a

1 a <

,aH

,• <

-!:.

■ 9

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1

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t^'

k A

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1 4 .

* 4 i

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, A

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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'

J?iV

..--ir; .

i.^

D^ ■

—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.

\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.

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

Mar 21,

1931

118.6'

Feb 5,

1933

169.0°

Apr 20,

1933

151.4'-

Mar 8,

1935

204.4°

May 26,

1935

186.7°

Apr 12,

1937

245.7°

Sep 26,

1939

297.5°

Aug 4,

1941

016.5°

Nov 22,

1943

012.1°

Oct 11,

1943

080.1°

Jan 21,

1944

065.6°

Dec 9,

1945

123.r

Mar 19,

1946

107.6°

Feb 3,

1948

153.4°

Apr 4,

1948

138.3°

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

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.

Mar 21,

1931

118.6"

3.2N

150=

1.8N

Feb 5,

1933

169.0

4.0N

150

1.8N

Apr 20,

1933

151.4

2.5N

183

1.4N

Mar 8,

1935

204.4

3.0N

183

1.4N

May 26,

1935

186.7

0.6N

220

0.3N

Apr 12,

1937

245.7

0.7N

220

0.3N

Sep 16,

1939

297.5

4.6S

335

1.7S

Aug 4,

1941

016.5

4.1S

335

1.7S

Nov 22,

1941

012.1

0.6S

043

0.2S

Oct 10,

1943

080.1

0.5S

043

0.2S

Jan 21,

1944

065.6

2.7N

096

1.3N

Dec 9,

1945

123.2

2.9N

096

1.3N

Mar 19,

1946

107.6

3.0N

142

1.9N

Feb 3,

1948

153.4

4.4N

142

1.8N

Apr 4,

1948

138.3

3.1N

169

1.6N

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:

F y- —

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 =

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

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 ....

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 ....

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

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 ^ 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

(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.

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

but e = - , so c =

= 0.254a

«,-

.i - 0.254a

a + 0.254a
0.746a
1.254a
= 0.594

Therefore, -^

= 0.594

15. la) From Kepler's law, for bodies orbiting the
sun

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-

= 1

= 76'

= ^^W

= \^'57,700

= 17.9 AU

(b)fl^ =

Vza + Vzc but - ^ e

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.

(in '»H«dow) O

V 1 -f-rom shadow /

(jr, •9h»c*Ow)

Kid

I .Ml 2 .MOTION l\ Tllli HivUlJ.VS

ibi, ic)

Satellite

Europa

Ganymede

Calistro

/?.,

ff.v'

PlR" = k

(mm)*

days

hours

(X 10^ hours)

(X 10* hours')

(X 10^ mm')

Ihr^mm')"*

3.4

0.44

0.193

0.398

485

5.2

0.84

0.703

1.42

495

8.2

1.68

2.81

5.56

505

14.6

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

R^^

i2-!Tflr
T-

X —

■X-u-

13.9

X 10*1

128 X

24 X

36001

- sec^

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

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

•-»

*- •

•"-^

-••

»• ^'^

^ JL

" '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.

(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.

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

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

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'

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'

-)'

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

10"

X 15

X lO^^I

3.14'

X (8.65^

X 10*

6.67

5.98

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

— ; , 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

\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

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

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,

^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-

4tt-
.Gm.,/

which has the form T~
hamionic law.

= kR\ of Kepler's

— 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

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

Discuss day 9 lab

Problem solving:

Energy conservation

Teacher demonstration:

Conservation of energy

in inelastic collisions

Lab E3-10, E3-11, or

E3-12

Selected Study Guide
questions

Texii'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.

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

£ -

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.

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

+ 10

-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

192

±13.8

± 4

168

±12.9

± 6

128

±11.3

± 8

128

± 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

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 =

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

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

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

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 =

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

/^•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

S,B,R
B,R,P

Superposition

S,B

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

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.

^^«r S^v-f»'y

^ /

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, =

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

before

3.37 m sec

left

velocity

A
B

after
before

0.917 m sec
m sec

left

after

3.72 m sec

left

before

1.79 kg m sec

left

momentum

A
6

after
before

0.488 kg m sec
kgm sec

left

after

1.30 kgm sec

left

before

3.02 joules

kinetic energy

A
B

after
before

0.223 joules
joules

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;

" — "^^

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

before

2.76 m sec

after

2.16

before

after

1.44

magnitude

before

1.01 kgm sec

after

0.774

momentum

before

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 l\ 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

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-

served ratio, — or fi, of hits to total trials was;
T

37 31^ 35^

50' 50 50
34 ± 3

The mean ratio is

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

32.5 cm
2X8

Shielding effect becomes important as A' is in-
c[-eased.

For,V =

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.

7
6
51-

3 -
X -
1 -

»-t»o l'^)

234

I'NIT 3 / THK TRIl'MPH OF MECH/WICS

I 0-8 -

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

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/'^

'If or> A

-"/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=

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 —

100

and \

cm/sec

2 95

= 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.

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

Event

it)

Nail Penetration

6.6

2.14

0.466

0.7 units

5.9

0.217

1.35

0.741

1.4 units

5.2

0.550

0.98

1.010

2.2 units

4.4

1.02

0.80

1.25

3.1 units

3.5

1.56

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

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:

29.5

42.5

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

k = — , n = 1, 2, 3,

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.

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

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:

1
vp

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'

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

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' =

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

ibi \\ = i\', + Avi = i\', ~ alv

50 kgni sec ION x 5 sec

5 kg 5 kg

= 20 m/sec

- p, + Ap

= 50 kg m sec -t- i5 sec x lO N

= 100 kgm sec

= m\ ,, therefore

_ El

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

1.2 X 10' -

= 4

X lo' \

0-20

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

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/ =

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

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^)

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 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

A,IKE)
F

40 J

4 N

10 m

6. Estimates may ha\e a wide range.

KE =

(kg)

(m/sec)

(joules)

(a)

0.15

900

67.5

(b)

lO"

300

9 X 10*

4.5 X 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

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

, 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

acceleration of earth = — =

m 6 X 10-"'kg
1.6 X 10'-^ msec'

d =

Vzaf
/id

t =

\ 1.6

X 10"^

\ 1.25
1.1 X

10'

10^^

■ sec

There are about 3 x lo sec in 1 yr so this
would be about 3 x lO^ vt.

lb) / =

\ 0.2 = 0.45 sec needed

\ a_ \ 9.8

for the stone to fall 1 m. During this time
the earth, with an acceleration of 1.6 x
10"^ msec", will mo\e 1.6 x 10 "'' m since

d - Vzar = Vz X 11.6 X io"'^i x 02
= 1.6 X 10"^ m
(cl The gravitational potential energy is as-
signed to the earth-rock s\stem because it
describes the PE of the rock relative to the
earth. PE alwavs has a frame of reference
"potential to something.'

15. The PE stored in the ball when it is initially
pulled aside is converted into KE during the
swing, which is then dissipated on impact in
KE of parts of the wall and w ork done in break-
ing structural bonds. lEnergv' is also converted
to other forms, such as heat and noise

16. lai If the boulder could somehow fall all the
wav to the center of the eaith, the greatest
decrease in PE of the isolated boulder-earth
system would be experienced.

(b) \o. The system possesses gravitational po-
tential enei^' with respect to the sun, for
example.

Convenient zero-levels would be

lai the lowest point in the pendulum swing

ibi the lowest point on the tracks

ici the midpoint of the oscillation

(d) The perihelion distance. One may consider
the mass of the sun to be concentrated at
its center. This would not be a suitable zero-
level because the gravitational force P ^ '^,

and approaches infinity when fl gets very
small.

Another possibilitv is to make the zero PE lev el
correspond \.o R^> ^ i infinity! . This is a natural
one if we

consider APE

= 1"

J Rr.

FdR

— constcmt

ffi dR

1 [r
— — constant —

R^ «o

= constant

- -)

a, Rl

where fl„ is designated as the R for which PE
= and must be ^. Of course, now all PE values
will be negative, but we reallv are only con-
cerned with changes in PE. Furthermore, this
would tell vou how much work vou must do to
free the planet from the sun.

2KE

R
2ma/i
R

17. KE = PE = = maji

According to the hint: ma^ ^

2h R

1 < — orh < -

R 2

18. The comet speeds up as it approaches the sun
and slows down as it recedes. That is, KE is
greatest when it is closest to the sun, smallest
when it is farthest awav. Conversely, PE is at a
minimum when the comet is closest and in-
creases as the separation increases. The total
amount of energv remains the same.

SUGGESTED SOLLTIONS TO STLD\ Gl IDE PROBLE.VIS

261

19. A primiti\'e lever system operates as follows:
When the piston r-eaches the top, pin 2 stiikes
thi(M; levei-s, two ot Which art; ciii-ectK' linked tf)
th(; \al\es B and C. thus opruiing theni and one
which is litiked thcouf^h a |)i\ot to A, thus clos-
ing A On the dounstioke, pin 2 releases \al\e
C;, which is clost^d h\ a spfing hut leaxes A and
B open. At the bottom of the stiT)ke. pin 1 strikes
the levei-s to A and B, opening A, closing B.

C:ams driven by a cranked whe(;l uould he
smocjther. They would haxt; to he shaped to
keep the \alves open for the appropriate du-
ration.

22. Various methods could be suggested, such as:

1. For lai and ibi the ideas of 21 could be
adapted to the bicycle or motorcycle traxeling
at constant speed on a le\el road.

2. lai, (bi, and ici could be supported so that a
pulley attached to the rotating pan ma\ lift a
weight at the end of a rope. Divide the work
done by the time i^equired.

3. A brake horsepower test consists of a brake
band pressed against a pulley on the motor
shaft so that the differ-ence in tension on the
two sides of the band and the speed at which
the pulley slips o\'er the band gh-es the power.

20. Power =

work done
time i"equir-ed

for-ce X distance
time

= F\

21. (al KE = Vimx'^ = Vz X (75 x lo'i x 16'

= 90 X 10" J

(bl power = forx:e x speed - 174 x lu'W

174 X 10" VV

force = = 11 X 10 \

16 m/sec

I water dragi

Ibis force would ha\c to do an amount of work
equal to the Kl. of the .ship to hiirig it to n-st.
I-I) - KK

KE _ 96 X 10" J
F 11 X 10''

= 88 X lo' m
KE 96 X lo"J

Ic) F = — - =

d 2 X 10'

48 X lO' \

Id) 1. inci^ease

2 incrvase

.\ (l('cr"<'ase
(" to ( liangc dir-cclion ol ship

23. (a) Power is work per unit time, P = work/sec,
measured in watts. The work done is then
uork = 140 W X 500 sec

= 7 X 10^ J
But woi-k is also equal to force x distance.
Therefore,

7 X 10" J

a = ;

70 kg X 9.8 m/sec "

= 102 m

ibi A human bod\' at rest produces about Hi)
kcal/hr. Therefore,

P =

(4,184 J kcaliiSOkcalhr
3,600 sec hr

= 93 \V

24. Both engines receive 100 J of heat.
Engine A: VV = Fd

= 5 N X 10 m
= 50 J output

efficiencx' =

100

= 50%

50 J

pu\^e^

10 sec

5 l\

tngine B: IV

2 \ X 20 m

40 J

efficiency

40
100

power

40'^.
40 J

5 sec

8l\

25. (al Neucomen s engine coirltl lie run at will
not dependent on tlu' pr-esence of wind In
addition it delivjMvd slightU grvater hor-se-
power,
ibi Watt s engine rvquiitul l»»ss fuel per- hor-se-
poxN-er-

26 PrTibabK the dut\ was more inipnilnnt hecatrse
ot fuel costs

ZtiZ

I'MT 3 THi; THII'MIMI Ol MivCH/WIC'.S

27. StudtMit ansvvei-s might inilucic the tollowing:

Desirable

Undesirable

(a) 1. Steam engine
power for mills
railroads
steamships
electric generators
profits and wealth
international trade

2. Gasoline engine

cheap, easy transportation

jobs

oil industry

(b) Nuclear power

another source of electricity

low cost (?)

reliable (?)

clean air

power for ships

jobs

need for wood, then coal
smoke and smog
crowded tenements
loss of workers'

independence
exploitation of natural

resources
waste ashes

smog

accidents

insurance

dependence on imported oil

potential massive hazards
disposal of radioactive

wastes
high cost of construction
delays in construction
centralized power sources

28. lai The maximum efficiency of an engine is de-
scribed b\' the change in temperature in
degrees Kehin di\ided b\ the initial tem-
perature. On 7"e,v/ page 295, the efficienc\' is
given as

efficiencx' =

T,i

= 1

For an engine operating on ocean water be-
tween 15 °C and 5 °C, the maximum effi-
ciency is

efficiency = 1

288°

3.5''

ibi If the engine operates at 3.5% efficiency, the
rate at w+iich water must be pumped thixjugh
the engine to yield 1 .\r\\' of mechanical en-
ergy' can be found as follows: One metric
ton of water cooled 1°C produces 4 MJ L\r\\'
seel of energx'. Therefore. 0.1 metric ton
cooled 10°C 115° - 5°i would produce 4 MW
of energy' and 0.025 metric ton lO.l 4 would
produce 1 MA\ . If the efficiencx- of the en-
gine is 3.5%, 10.025 0.0351 = 0.71 metric ton
sec of water would be required to produce
1 MVV of energy.

29. The coefficient of performance, described on

Test page 296 as T/lT, - T,i, applies to the air

conditioner and is iiluays gi"eater than 1.

lai With an outside temperature T, of 40°C

l313°Ki and an inside temperature '/"_, of 21°C

(294°Ki, the coefficient of peiformance is

T, 294

(bi If the outside temperatuit; 7', is raised 10°C
to 50"C i323°Ki and the inside temperatui-e
r, is lowered 10°C: to iTC i284°Ki the coef-
ficient of performance is

284

= 7.3

323 - 284

The coefficient of performance is deci-eased
by 8.2.

ic) If the outside temperature 7 , is raised b\'
another 10°C: to 60°C i333°Ki, the coeflicient
of performance becomes

284

333

d84

= 5.8

313

294

= 15

30. No. The distances are in a ratio of appro.xi-
mately 1 : 2 ; 3; for a given weight, the latios of
costs are not 1 : 2 : 3; for a gi\en distance the
costs are not pixjpoilional to the weights.

31. Clausius would argue in the sequence: c, a, e,
b, d, f.

32. .An ideal engine has no heat loss, is therefoi-e
reversible, and has a maximum efficiency of
100% .

33. The efficiency of a completely reversible ideal
engine must be 100%. For this to occur, the
temperature of the cold side must be absolute
zero iO°K).

34. lai 1 kcal = 4,184 J. To raise the temperature

of water '/2°C requires 2.5 kcal or 10.46 x
10"* J. Each descent of the 1-kg weight cor-
responds to 1 X 9.8 X 1 J. Hence, more
than 1,000 such descents would be re-
quired.

ibi 1. use less water

2. use a larger mass

3. use a longer distance

4. use a liquid of lower specific heat ifor ex-
ample, mercuiyi.

I Note: Method 4 has not been discussed in this
te.xt but some students may be aware of this
possibilir\M

35. Consider 1 kg of water going over the falls. In

a drop of 50 m. 1 x 9.8 x 50 = 490 J will be

490

generated, which is equivalent to kcal ot

4.180

Vk kcal available to warm the 1 kg of water.
Thus, the temperature would be raised b\
about ' /«°C. The answer would be the same re-
gardless of the amount of water since, although
more water would |)ro\ide more joules, there
would be a corresponding increase in the num-
ber of Calories required to heat the larger mass
of w atei-

SLGGESTtU SOLITIO.NS TO STLDV OLIDL PROBLEMS

263

:m. rhn effiricncv of a powfi- |)lant is gi\f;n hv

r. - 7 ,

cHiciciicv =

'/■,

vvher-e 7' is in clogit'es KeKin lor llie nuclear
plant.

BOO - ;}0()

efficiency =

3()(J

= 0..'>0
loi the fossil liici [jiant,

efficiency =

750 - 300
300

= 0.60
(ai To produce 1 MVV of electrical fjower at
thiese efficiencies, the heat tiumpetl into the
environment is the total pouei |)i'otluced
minus th(» useful output. Ihe total |)o\ver is
the output (li\icl(Hl hv tlu; efficiency. Then

OUt|)Ut

heat loss - - outjjut

efficiency

- output I - 1 I

\ efficiency /

For- the nucl(!ai plant, the heat loss is

1 M\\ (-^ - 1 J - 1 .\I\\

For the fossil fuel plant the heat loss is

1 M\\ ( 1 I - ()(S7 M\\

\0.6 /

(h) The rate of flow of cooling water to ci-eale

an e.xit flow at 303 "K is given hv

rate itons se(N = i7', - 303i x

heat loss (MWI

rate

IHOO

3031

4.2 MJ/sec (per ton per °C)

f-or the nuclear plant,

/ 1 \l.l sec \
\4:i MJ sec/
= 70 tons/sec

Foi- the fossil fuel plant.

/0.{J7 .MJ sec\

rate = (7.10 - 303 1

\ 4.2 MJ/sec /

= 71.5 tons/sec

37. Digging iv(|uires 400 Cal hr oi 200 Clal for 0.5
hr. With 20",. efficiency. 1. 000 tlal must he su|)-
plied. Ilamhurger supplies 4,000 Cal kg so 0.25
kg will he i-etiuii-ed

3K. One-half kilogram of animal fat \ iclds 300 ( .il
^ 4 3 X lo' Cal; 22.5 kg of anintal fat wtnild
\icld 21.5 X lo' C:al. If food intake is cut h\
1,000 (!al da\ 21 5 da\s will he n«(|uiii'd lo lose
22 5 kg

39. (a) Fewer joules of enei-gv' are needed to move

smaller- masses. .Note that the r-atio of weights
(3 : 2i is ahout equal to the ratio of Calorie
requirements.

ihi Fewer- Calories art? needed to maintain hody
temperatur-e in a warm climate.

(CI Therf! is a higher- per-centage of children in
the Indian population whose masses would
he significantly less than the adult 495 kg
and who consecjuenth' would ha\e lower
Calorie needs for mechanical wor-k. (Jn the
other hand. the\' would have gi-eater intake
needs for- gixjv\1h: it is not c-lear how these
would compar-e.i

40. Our consumer needs of food, fuel, electricity',
gasoline, etc., can all he expressed in joules or
Calories. With our dollar-s we aif thus hu\ ing
energv'. However, the cost per joule lor Caloriei
is not the same for all the various forms of en-
ergv' and it would not he practical to adopt the
dollar- as an energv unit.

41 Ihe upward momentum of the rocket equals
the dowTiwartl momentum of the e.xliaust gases.
The chemical potential energv- of the fuel is
converted into kinetic energv' of the rocket,
heat, light, and sound ener;gv . plus the gravi-
tational pot(>ntial energv that the rocket ac-
(|uir-es.

42. Ihe stat(!ment is hasicallv triie I he cluMnical
ener-gv' eventually hecomes heat ener^- since
at the end of the trip the car is hrought to rest,
hence having no final KF, i How ever-, if the stop-
ping place is at a higher elevation than the
starting place, gravitational PK will he pr-ovided
fr-om the original chemical ener-gv of fuel.' .\nv
energv radiated as light or sound will eventu-
allv warm some ahsor-fjer.

43. \Mien a hot and a cold hodv aiT* placed in (-on-
tact. the entrxjpv change is

HIT. - TJ

A.S

T/l\

(a) By firing its rvtr-or-ockets the space capsirU*
can Induce its speed and will then he pulled
closer lo the central hod\ ahout which it is
or-liitiiig. For stahle or-l)its. centripetal forte
equals gravitational forve:

mv'

= C,

Alter lh»' sp»'C(i is it'duct'tl frx)m that niH'ded
in the higher oriirt the capsule will In* ac-
cvlei-ated as it spir-als inwarxl When it n-aches
the desirvd lower altitude its speed and di-
rfclion must again he adjusted hv >uilahle
liiirm ol Its iDckels

Zii-l

I'lVIT 3 / Till-: TKII'MIMI Ol MLCH/WICS

lb) In the lower orbit, the required speed will
be greater tlian in the higher oi4)it, since \'H
= Gm, = constant . So KK in tlie lower- oH)it
is greater than KL in the higher.

Id PE will be less when it is closer to the cen-
tral bod\'.

idi Less: A total E = AFK - AKt = -2 AKE
+ AKE = -AKE

lei The difference was dissipated in i-ocket fir-
ings.

45. lai i. Total energ\' of system is decreased.

ii. The heat absorlied b\ the s\stem is less
than heat gi\en otf In the s\'steni.
iii. Moi-e work is done b\ the s\stem tlian
on the sx'stem.

ibi i. .All three are negati\e; the man does
work and gi\es otf hod\ heat,
ii. .All three ai^ negati\e: the batter\ does
work turning o\er the motor, some heat is
gi\en otf, and the internal chemical poten-
tial enei"g\' is decreased,
iii. Ah is negati\'e since heat and light are
gi\en off.

AlV is positixe lin this case the work is
electrical'.

\E is slightK positixe pos. Al\ > neg.
AHi.

i\ . Al\ is positi\e: Af/ is negati\e, but now
an equilibrium e.xists, so A£ = 0.
\ . .All thi-ee are negati\e; work is done mo\-
ing the pails: heat and light are gi\en oft.
\i. lai When first starting: Al\ positi\e: \H
negati\e, but greater than Al\, so At
is negative,
ibi .After stabilizing: Al\ positive: AH
negative, but now — Al\ = Al\ , so
A£ - 0.

46. lai sun s energ\

— ♦ evaporation of water -
rainfall -^ plants — » fuel
oil, gas)

coal,

electric generator-
water heat

stove heat

ibt sun's energ\'
aiel-*

plants

{gravitational
KE

♦ heat of friction

(el sun's eneigv —* water evaporation and con-
densation — > wind's mei:hanical energv of
the pump -^ KE land possibly PEi of water

lai Since H\ ~ H, — h, and ener-gv is conserved,
then H!, - H, ^ h also.

ibi The total entr-opv change AS. defined as AS
= Ah v. is composed of the entr-opv change
on the hot side AS, and of that on the cold
side AS, I AS = AS, + AS,'. But

h; - h, h, - h.

and AS, ■

48,

AS,

Since H[
have

h and //;

7,
H, = h.

AS =

h_ _ h_
T, 'l\

Id Because T, on the cool side is less than 7',
on the hot side, hl\ is greater than h T, and
AS must be positive.

The entropv change of the melting ice will be
AS = AH /' = 3.4 MJ 273 'K = 12.5 kJ ^K. Ihe
entropv change of the warming water- will be
equal but negative: — 12.5 kJ °K. Since no heat
is lost, the entropy of the ice-water universe
does not change.

CHAPTER 11

2. Conducting political polls, W ratings, gambling
casino games.

3. Several concepts accepted in piesent-dav ki-
netic theorA art? contained in the quotation bv
Lucr-etius, such as continual random motion of
atoms that niiiv involve elastic collisions. On the
other hand, according to Lucr-etius, some col-
lisions were not elastic: all atoms wer-e identical
and difter-ent substances wei-e for-med bv their
becoming entangled: and sunlight was atomic
in natur-e.

4. ,A distribution is a mathematical description
stating that the items have a certain character-
istic in common. Statistical distributions aie

more likelv to agr-ee with e.xperimental insults
when the number of items is large.

lai Ther-e is a most pr-obable lor average i speed

for the molecules,
ibi The higher the tempeiatur-e the larger the

average speed.
Ici \o molecules can have negative speeds, so

there is a definite cut-off at zero, birt no

such sharp limit on the cur\'es at high

speeds.

No. In the transmission of sound the forwaixl
and back motion is superimposed on the nor-
mal molecular motion.

CMaiisius gave the [larticies ol tlic simple theory

SLGGESILD SULl TIOVS TO S Tl D\ GLIDE PKOBLE.XLS

•mi

appi-eciable size. This meant that frequent col-
lisions would occur among them. He could
then ex|jlain the slowness of diffusion.

8. (a) Ihf; volume of oil etjuais the acea in contact

with the water times the thickness of the
layer: V = A x h

V 10 "^m" _y

h = - = — -. — r = 10 m
A lO' m''

(1 cm^ = 10 " m^i

(b) The thickness of the layer is 10 " m since

the layer is assumed to be one molecule

thick.

9. la) Total volume equals the volume of one mol-

ecule times the number of molecules: V =

V X N

V 10 "m'

N = - ^ —-r. — J = 10-'

(b) Density =

10 - m"

mass

so thei'e would be

volume V
0.001 as many molecules in any volume of gas
as compannl with the same volume of liquid.
One cubic centimeter of a gas uill contain lO"*
molecules. This is in agi-eement with the large
numbei"s mentioned in the text.

10. A lift pump will not operate at all on the moon
since the moon has no atmosphere.

11. If the ail- had constant density, 1,000 x 10.5 m
(about 10 km I would balance the 10.5-m col-
umn of water. Actually, the densit\' decreases
with altitude and the atmosphere has no sharp
cut-olT point. We have evidence of its existence
at more than 100 km above the earth's surface.

12. (al P, = 100 \/m' X 2 X Vj

= 66 N/m-
(bl r, = l\{P^P^nD,/D^)

= 100 °C X 2 X '/z
= 100 °C

F weight

13. /' = - = ;

A ai-ea of base

1 aim is about 10 Wcm^ (10 Pal

For a 529-N person standing on:

1. Two shoes, each about 2."> cm long and 8 cm

wide

P =

529

1
= 1.3 N'/cm" = about -atm

2 X 25 X 8

2. Two skis, each about 200 cm long antl 8 cm
wide

529 , 1

P = = 0.17 N/cm = about — atm

2 X 200 X 8 60

3. l\vo skates, each about 30 cm long and 0.3
cm wide

529

14. Starting with Boyle's law as P = kB and sub-

xi kM

stituting D = — , gives P = — or P\' = kSl =

constant for a given mass of gas M.
P = kD it + 273°)

15. The ideal gas law is P = kDT, where

P »: D for constant 7"
P ^ T for constant D
D -x 1/7" for constant P

The ideal gas law does not apply to ver\' dense
gases or to any gas at a phase transition to a
liquid or solid.

16. 1. If the temperature does not change, U +
273°) = constant orP ^ D, which is Boyle's law.
2. If the pressure is kept constant,

P = kD it + 273°i = If -(- 273 ( and

kXl

If + 273°

Comparing the volumes at two different tem-
peratures,

V^ - V, = [f^ -I- 273 - If, -I- 2731)

— If,
P -

f.i

or change in volume is proportional to
change in temperature, which is Ga\-Lus-
sac's law.

17. The ideal gas law P - kDT becomes P ^ T ior

a constant mass of air confined in the volume

of the tire. iThis \olume is assumed not to

IP
change appreciably.) A specific ratio of -—

(such as the 12-kPa drop per 10° C citedi would
have to be associated with a particular amount
of air trapped in the volume available. If the
ideal gas law is applied to this situation, it
would predict a total pressure of 34 \ cm* i24
N cm^ on the gaugei for a temperature of 4°C.
(Remember that absolute zero con^sponds to
- 273° C.) This is a fairlv high pressure for most
passenger car use.

Next consider the two statements made and
convert the data given to a fonn in which it can
easily be checked for consistence'.

Absolute
Gauge Actual temp

pressure Temp. pressure (related
(N/cm'l (°C) {Niem') to "C) Prr

299

0087

- 1

272

0.085

288

0.090

244

0082

2 X 30 X 0.3

- 29 » .\ ciir = about 3 atm

Note that for the lir-st set of data the internal
agrvement is otT bv ai»()ut 4 '.^ Init the second
set agr-ee exact Iv

•ZMi

I Ml a rni-; thiimph of vilcha.mcs

The statements as phrased in the quotation
suggest two additional comments:

1. The two sets of data cannot be for the scime
amount of air since in one case 16 \/cm^ cor-
responds to 26 °C and in the other to 15 °C.

2. The fii-st case describes a 27°C temperature
drop and a pressure drop of 3 Wcm" in agree-
ment with the rule stated at the outset. The
second case describes a 44° drop in tempera-
ture (15° to -29° I and only a 4 \ cm" drop in
pressure.

The purpose of this question is to encourage
students to read such statements criticalK' and
to recognize correlations between eveivdav
concerns Isuch as adequate tire pressure! and
physics formulas.

18. A working model copies the real thing on a re-
duced scale: a theoretical model idealizes the
actual situation to facilitate the mathematical
analysis.

19. Speeds relatix e to laboratory
particle before collision = v
particle after collision = \''
piston at time of collision = u

Speeds relatixe to the piston:

speed of the particle before collision
- V — u

(particle and piston headed in same direc-
tion!

speed of the particle after collision
= v' + u

(particle and piston now headed in oppo-
site directions)

These speeds before and after collision are
equal to each other, according to the Galilean
relati\ity principle, so

+ u = V
v' = V

u
2u

^<x

Thus, the speed (relative to the laboratory! after
collision v' is smaller than the original speed
r by twice the speed of the piston u.

20. The temperature would not change since there
is no work done.

21. Pressure, mass, \'olume, temperature, viscosity,
rate of diffusion, color, and odor all might be
suggested.

22. The relations among pr-essure, volume, and
temperature: the rate of diffusion. In addition,
the sizes and speeds of the particles can be
estimated and temperature could be gi\en a
mechanical interpretation.

23. Much of the material in the can is in the gas-
eous phase under pressur-e. The kinetic theory
relates an increase in temperature to an in-
crease in speed of the parlicles and hence an
increase in pressure. The can may burst!

24. Work is done on the gas by the piston. By the
first law of thermodvnamics, the internal en-
ergy' of the gas must incr-ease and so the tem-
perature rises. Fr-om the kinetic theory, a mol-
ecule striking the mo\ing piston has its speed
incr-eased. This results in a higher average ki-
netic energ\' for all the particles, which corre-
sponds to a higher temperature. The air mol-
ecules eventually cool to the temperatur-e of
their sur-r-oundings by means of successive col-
lisions with the particles of the walls around
them. The extra energv' is not destroyed but is
e\ entually shared by so many par-ticles that no
lasting temperature rise is detected.

25. la) Perfectly insulating would mean that no

heat could escape, so the gas would have to
stay hot. Energy transferred to the container
walls by colliding molecules would be e\ en-
tually transferred back to other molecules.
lb! The chemical energ\ r-eleased in the burn-
ing of the fuel in a gas sto\ e lor the electrical
energ\' converted into heat in an electric
sto\ei fir-st speeds up the metal particles of
the kettle, which then pass on e.xtra energ\'
to the water, raising its temperature until
boiling occurs. At the boiling temperature,
the liquid molecules have sufficient energy
to escape as a gas.

26. Lucretius and Newton seem to be in basic
agreement. However, even though both speak
of explaining evervthing in terms of particles in
motion, N'evv-ton's idea is primariK' determin-
istic ithe motions of the particles should be
determinable irom lawsi, while Lucr-etius idea
seems to be mor-e statistical lonly the chance
conjunction of large numbers of particles, a
conjunction which is slightly prx)bablei.

27. The three statements of the law are: Hi Heat
will not flow by itself frxim a cold body to a hot
one. 12! It is impossible to fully convert a given
amount of heat into work. (3! The entropx' of an
isolated system tends to incr-ease lit cannot de-
crease!. To show that statements Hi and i2! are
equivalent requires two steps: Step A. Any vio-
lation of 12) is also a violation of H i. Step B. Any
violation of ID is also a violation of i2).

Step A. Imagine an engine that can fully con-
vert heat into work (statement '2^]. Then imag-

Sl GGESTED SOLLTIO.NS TO STL'DY GLIDE PROBLEMS

267

ine a reversible engine that acts as a refrigera-
tor, and put tlie two (uigines together The net
effect of the two engines is to transfer- heat froni
a cold l)ody to a hot body without doing an\
work, riiis vicjiates statement H).

Stcfj li. Imagine an engine that \iolates state-
ment Hi; that is, it causes heat to flow from a
cold to a hot body with no other- changes taking
place. Then take a heat engine that works be-
tween the same two temperatur-es as the first
engine. Run both engines at once. Because the
resulting engine picks up the heat that is
dumped by the r-ever-sible engine and returns
it to the hot r-eservoir, the net effect is to converl
a given amount of heat entirely into work. This
violates statement i2).

28. It would not work. Energ\' would ha\e to be
expended to deer-ease the temperatur-e of the
water. The only time you can get work done at
the expense of heat energy is when you ha\'e
a reservoir- of heat at a higher temperature than
the surroundings.

29. When the ball is released, the initial potential
energy is

PE = mgh

= iO.l kgl (10 m/sec^) (1 ml

= 1 J
The PE is converled to KE; therefore, just before
the first bounce, KE = 1 J. During the fir-st
bounce, some KE is converted to heat. The re-
mainder brings the ball to a lesser height, after
which the ball continues to bounce. When the
ball finally stops bouncing, its KE = 0, PE = 0,
and all its energ\' has been turned into heat: Q
= 1 J (2.4 X 10 ' Call.

The entr-opy change equals the amount of
heat divided by the temperature, so at 300°K,

^S = 0/1

= (1 Jl / (30010
= 3.3 X lO^J/^K

It does not matter how much heat goes to
the ball assuming exciA thing is nearly at 300°K
I he ('nliopx' change of the uni\erse is the sum
of all the entrx)py changes:

-^un = '^ha., + ^,a.,io + ••

And since T,,„„ = T,,,,,,. = 300°K,

30. Eventuallx', all tempeiatuif s will become e(|ual-
ized abo\e the absolute zcmi) Energ\ cannot be
destrxjyed.

31. //"thert* werv such a being, he could get ar-ound
the second law of iheiinodN iiamics

Maxwell's assumptions:

1 a finite being who knows the paths and nc
locities ot ,ill Ihc particles

2. the ability to open and close a mass-less

slide.
Both assumptions are impossible to realize in
any laboratory.

32. The paradox lies in what is meant by long
enough. The chances of return to a former
state are very slight on account of the large
number of molecules. The second law, on the
other hand, takes an o\'erall point of view and
says it never- happens.

33. (ai Newton used a mathematical time. His laws

are equations of the first degree ilineari in
time. Reverse the sign of the time and the
process runs backwards. Exeiy purely me-
chanical process is rexersible.
(bi When heat and temperature are introduced,
the law of the increase of entrop\' with time
gives time a definite direction and processes
no longer are re\'ersible.

34. The melting of ice is an irrev'ersible process be-
cause the ordered arrangement of molecules in
the ice crystals is lost, and therefore entiiopy
increases.

35. It would seem as though time were running
backwards.

36. The assumptions involved can be formulated
as follows:

1. The long-range histor>' of the world is c\'clic
lancient belief i.

2. The univer-se has a definite quantity- of en-
erg\'.

3. There is a finite number of molecules.

4. Ther-e is a finite number of possible curange-
ments of the molecules.

5. Time is infinite.

6. All possible combinations of molecules have
at least some prxjbability.

7. The kinetic theory- full\- and adequately de-
scribes nature.

Students ma\' differ widel\' in what they believe
to be "true " in a discussion such as this. It
should be a good intellectual experience for
them to examine why they accept or reject
statements such as these.

37. In discussing this problem as a phvsicist, one
must take an anthr-opomorphic or materialistic
view of religion. Population increase prvdudes
a cyclical process for individuals The number
of individuals, although numbennl in terms of
billions, is infinitesimal comparvd with billions
of billions of molecules. \ou pn)bablv contain
within V our bodv a few molecules of Julius Cae-
sar!

It is dangerous to extrapolate ideas that arise
in one context tt) a differvnt area or to give them
universal validitv . but even vague analogies
hold an undeniable fascination.

;<f< \evv1onian mechanics could not explain the
.i|)p.U(Mit in i'\ CI sit)ililv of iii.K-roM-opic pn>c-

•MiH

I'MT 3 THi; TKIUMPH OF MECHANICS

esses, as well as the processes in\ol\ed within
the molecules and the atoms that compose
them.

39. Adxantages:

The\' correlate a number of disparate facts.
The\' suggest new experiments.
The\' lead to new results.

Their inadequacies suggest the form that newer
and more perfect models must take.

Disad\antages:

The\' are in\ariabl\ idealizations, and onl\ ap-
pro.ximations to the real world.

The\ tend to channel thinking, create a para-
digm, which is difficult to relinquish, e\en
when it becomes obvious that the model is
inadequate.

40. The pressure on the liquid at an\ depth is pro-
portional to the weight of the water abo\e. Be-
cause the pressure is exerted in all directions

(sideways on the tank as well as vertically i, the
lower section of the tank must be stronger to
offset the greater out\vard pressure.

41. lai The pressure per unit area at the bottom of
the cube is P lat topi ■*■ DgL.

ibi The force on the top of the cube is the pres-
sure per unit area multiplied b\ area, or PL'.
directed downward.

ici The force on the bottom of the cube is P
+ DgL' X L', directed upwaixl.

id I The net force on the cube is the difference
between the upward force on the bottom
and the downward force on the top, which
is

[IP -^ DgLi X L'] - PL- = DgL"

lei The net force iDgL^i equals the weight of the
fluid displaced which has a weight of Dg
per unit \olume for the total volume L^ of
the cube.

CHAPTER 12

2. If you put \our hand at one end of a long metal
rod and someone taps the other end shaipK ,
you will feel the vibration almost immediately.

3. If \ou swing \ our arms as vou walk the follow-
ing parts will be in phase: right arm and left
leg, left arm and right leg; '2 c\cle out of phase:
right and left arms, right and left legs. There do
not seem to be am generally agreed upon parts
that are V4 c\cle out of phase.

4. Use the principle of superposition. The shape
at the end of the first and third intervals is the
same, a truncated triangle. At the central time
a triangle of twice the height would result.

The wave shapes at the end of the first and
third intervals are reflections of each other.
Waves cancel at the central time.

,N ,'•- /\
A /, \
X ^

_2_ J. 2,-
3 s 2'-

V ., >,

1' -^ / 3

N /
' V

/r T

\ /

V
Exacth' the same shape, but with displacement
below the line of the undisturbed rope.

No. Kinetic energv does not obev the super-
position principle for two reasons. First, the
mass of the medium in motion ma\ change as
tvvo waves superpose. Second, although the ve-
locities superpose and v, -*- v, is the velocitv of
the superposed wave, the squeires of the speeds
do not superpose. That is,

V,- -t- \',- * (V, -I- v,f

F'beats

F. - F,

SLGGESTED SOLLTIONS TO STtT)V Gl'IDE PROBLE.MS

369

9. They would be surfaces in three dimensions
formed by the intersections of a series of hem-
ispheres analogous to the semicircles in the fig-
ure on page 366. Nodal surfaces would rxjquire
th«; inter-section of a crest hemisphei-e v\ith a
trough hemisphere. (This assumes that the
speakers are emitting the same uniform tone.)

10. The intensity would gradually decrease, until
it r-eached zer« at \',. Then it would gradually
incr-ease again and r-each a maximum at point
A,. Then the cycle could begin all over again.

11. The perpendicular bisector of the line joining
the sources is always