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Full text of "Project Physics Handbook"

PROJECT 
PHYSICS 



M 



HANDBOOK 




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PROJECT 
PHYSICS 

m:] handbook 




Gerald Helton 

F. James Rutherford 

Fletcher G, Watsor^ 




Gerald flultun 

Department of Physics, Harvard University 

F. JameH Rutherford 

Depailriu.'nl (jf Science Education, New York University 

Fletf;her G. Watson 

Maivard Graduate School of Education 

Editorial Development: William N. Moore, Roland Cormier, Lorraine Smith-Phelan 

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

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

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

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



A(-kn()wl('(lgni«'nt.s appt-ar on pagv 1 
I'icliirr (ii'clits appear on page 1. 



Copyright © 1981, 1975, 1970 by Project Physics 

All High Is Reserved 

Printtul in the United States of America 

ISBN 0-D3-D5Smb-3 

01234-0r>!)-;>H7K54;J2 1 

Project Physics is a registered trademark 



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

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

I.I.RABI 
Nobel Laureate in Physics 



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

Preliminary results led to major grants from the U.S. Office of Education and the 
National Science Foundation. Invaluable additional financial support was also 
provided by the Ford Foundation, the Alfred P. Sloan Foundation, the Carnegie 
Corporation, and Harvard University. A large number of collaborators were 
brou^t together from all parts of the nation, and the group worked together in- 
tensively for over four years under the title Harvard Project Physics. The instruc- 
tors serving as field consultants and the students in the trial classes were also of 
vital importance to the success of Harvard Project Physics. As each successive ex- 
perimental version of the course was developed, it was tried out in schools 
throughout the United States and Canada. The instructors and students in those 
schools reported their criticisms and suggestions to the staflFin Cambridge. These 
reports became the basis for the subsequent revisions of the course materials. In 
the Preface to the Text you will find a list of the major aims of the course. 

Unhappily, it is not feasible to list in detail the contributions of each person 
who participated in some part of Harvard Project Physics . Previous editions of the 
Text have included a partial list of the contributors. We take particular pleasure in 
acknowledging the assistance of Dr. Andrew Ahlgren of the University of Min- 
nesota. Dr. Ahlgren was invaluable because of his skill as a physics instructor, his 
editorial talent, his versatility and enei^, and above all, his commitment to the 
goals of Harvard Project Physics. 

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

In the years ahead, the learning materials of the Project Physics Course will be 
revised as often as is necessary to remove remaining ambiguities, to clarify in- 
structions, and to continue to make the materials more interesting and relevant 
to students. 

Gerald Holton 

F. James Rutherford 

Fletcher G. Watson 



iU 



GOMTEMTS 



IntrfMlufdon 1 

Keeping Hecords 4 

Using Ihe Polaroid Land Camera 



Unit 1 /Concepts of Motion 



EXPERIMEIVTS 

1-1 Naked Eye Astronomy 



7 
13 



14 



1-2 Regularity and Time 
1-3 Variations in Data 13 
1-4 Measuring Uniform Motion 
1-5 A Seventeenth-Century 

Experiment 19 

Twentieth-Centuiy Version of Galileo's 

Experiment ZO 

Measuring the Acceleration of 

Gravity a, 21 

Newton's Second Law 24 

Mass and Weight 27 
1-10 Curves of Trajectories 28 
1-11 Prediction of Trajectories 30 
1-12 Centripetal Force 32 
1-13 Centripetal Force on a Turntable 



1-6 



1-7 



1-8 
1-9 



33 



AcmnTiES 

Checker Snapping 
Beaker and Hammer 
Pulls and Jerks 38 



38 



38 



Experiencing Newton's Second Law 35 
Make One of These Accelerometers 35 
Projectile Motion Demonstration 38 
Speed of a Stream of Water 38 
Photographing a Waterdrop Parabola 39 
Ballistic Cart Projectiles 39 
Motion in a Rotating Reference Frame 40 
Penny and Coat Hanger 41 
Measuring Unknown Frequencies 41 

FILM LOOP NOTES 

LI Acceleration Caused by Gravity. I 41 
L2 Acceleration Caused by Gra\ity. II 42 
L3 Vector Addition: Velocity of a Boat 42 
L4 A Matter of Relative Motion 44 
L5 Galilean Relativity: Ball Dropped from 

Mast of Ship 44 
L6 Galilean Relativity: Object Dropped 

from Aircraft 45 
L7 Galilean Relativity: Projectile Fired 

Vertically 46 
L8 Analysis of a Hurdle Race. I 47 
L9 Analysis of a Hurdle Race. II 48 



Unit 2 /Motion In the Heavens 



EXI'EKlME.VrS 

2-1 Naked-Eye Astronomy 80 

2-2 Size of the Earth 54 

2-3 The Distance to the Moon 57 

2-4 The Height of Piton, a Mountain on the 

Moon 57 
2-5 R('tn)gra(le Motion 60 

2-6 rhe Shapt! nt the Kailh s Ori)il 61 

2-7 Using Lenses to Make a 

Teh'scope 63 

2-H Ihe ()ri)il of Mars 67 

2-9 Inclination of Mars ()r1)it 70 

2-10 Ihe ()r»)it of Mei-curv 72 

2-11 Stepwise A{)pn)ximati()n to an 

Ort)it 75 
2-12 Model of the Or»)i« of Halley s 

Conicl 7!» 



ACTI\1TIES 

Making Angular Measurements 83 
Epicycles and Retrograde Motion 84 
Celestial Sphen* Model 86 
How Long is a Sidereal Da\ :* 87 
Scale Model of the Solar Sv'stem 88 
Build a Sundial 88 
Plot an Analemma 88 
Stonehenge 88 
Moon C'rater Names 89 
Literature 89 
Franie,s of Reference 89 
Demonstrating Satellite Ortiits 90 
Galileo 90 

Conic-Section Models 90 
Challenging Pnibleni: Finding Earth- Sun 
Distance from \'enus Photos 91 



Iv 



CONTENTS 



Measuring Irregular Areas 91 
Other Comet Orbits 91 
Drawing a Parabolic Orbit 91 
Forces on a Pendulum 92 
Trial of Copernicus 93 

FILM LOOP NOTES 

LlO Retrograde Motion: Geocentric 
Model 94 



Lll Retrograde Motion: Heliocentric 

Model 94 

L12 Jupiter Satellite Orbit 95 

L13 Program Orbit. I 97 

L14 Program Orbit. II 98 

LIS Centred Forces: Iterated Blows 98 

L16 Kepler's Laws 99 

L17 Unusual Orbits 100 



Unit S/The Triumph of Mechanlos 



EXPERIMENTS 

3-1 Collisions in One Dimension. I 102 
3-2 Collisions in One Dimension. II 104 
3-3 Collisions in Two Dimensions. I 110 
3-4 Collisions in Two Dimensions. 

II 113 

3-5 Conservation of Energy. I 118 
3-6 Conservation of Energy. II 121 
3-7 Measuring the Speed of a Bullet 122 
3-8 Energy Analv'sis of a Pendulum 

Swing 124 
3-9 Least Energy 124 
3-10 Temperature and 

Thermometers 126 
3-11 Calorimetry 128 
3-12 Ice Calorimetry 131 
3-13 Monte Carlo Experiment on Molecular 

Collisions 132 

3-14 Behavior of Gases 137 
3-15 Wave Properties 139 
3-16 Waves in a Ripple Tank 140 
3-17 Measuring Wavelength 141 
3-18 Sound 142 
3-19 Ultrasound 144 



ACmiTIES 

Is Mass Conserved? 147 

Exchange of Momentum Devices 147 

Student Horsepower 148 

Drinking Duck 148 

Mechanical Equivalent of Heat 149 

A Diver in a Bottle 149 

How to Weigh a Car With a Tire Pressure 

Gauge 150 

Perpetual Motion Machines? 150 

Standing Waves on a Drum and a 

Violin 152 
Reflection 152 



Moire Patterns 153 
Music and Speech Activities 154 
Measurement of the Speed of Sound 
Mechanical Wave Machines 155 



154 



FILM 

LI 8 
LI 9 
L20 

L21 
L22 
L23 

L24 
L25 

L26 

L27 

L28 
L29 
L30 
L31 

L32 
L33 
L34 

L35 

L36 
L37 
L38 
L39 
L40 
L41 
L42 
L43 



156 
157 



157 
158 



LOOP NOTES 

One-Dimensional Collisions. I 
One-Dimensional Collisions. II 
Inelastic One-Dimensional 
Collisions 157 
Two-Dimensional Collisions. I 
Two-Dimensional Collisions. II 
Inelastic Two-Dimensional 
Collisions 158 
Scattering of a Cluster of Objects 
Explosion of a Cluster of Objects 
Finding the Speed of a Rifle 
Bullet. I 160 
Finding the Speed of a Rifle 
Bullet. II 161 
Recoil 162 

Colliding Freight Cars 162 
Dynamics of a Billiard Ball 163 
A Method of Measuring Energy: Nails 
Driven into Wood 164 
Gravitational Potential Energy 
Kinetic Energy 165 
Conservation of Energy: Pole 
Vault 166 

Conservation of Energy: Aircraft 
Takeoff 167 
Reversibility of Time 168 
Superposition 168 
Standing Waves on a String 170 
Standing Waves in a Gas 170 
Vibrations of a Wire 171 
Vibrations of a Rubber Hose 172 
Vibrations of a Drum 173 
Vibrations of a Metal Plate 173 



158 
159 



164 



vi 



CONTENTS 



Dnlt 4 /LIshi and Eleotromagnetlsin 



KXIM:RIMKVrS 

4-1 Refraction of a I.igfit Beam 174 

4-2 Young's Kxperinient: The Wavelength 

of Ught 177 

4-3 Klectric Forres I 179 

4-4 Electric Forces. II: Coulomb's 

Law 181 
4-5 Forres on Currents 183 
4-6 Currents, Magnets, and Forres 187 
4-7 Electrxjn Beam Tube. I 190 

4-8 Electrxjn Beam Tubes. II 192 

4-9 Waves and Communication 



AcmnTiEs 

rhiii film Interference 200 
Handkerchief Diffraction Grating 
Photographing Diffraction Patterns 
Poisson's Spot 201 
Photographic Actr\ities 201 
Color 201 
Polarized IJglit 202 
Make an Ice Lens 203 



195 



200 
200 



203 



Detecting Electric Fields 
An lie Battery 204 
Voltaic Pile 204 
Measuring Magnetic Field Intensity 
More Perpetual Motion Machines 
Transistor Amplifier 206 
An Isolated North Magnetic Pole? 
Faraday Disk Dynamo 206 
Generator Jump Rope 207 
Simple Meters and Motors 207 
Simple Motor- Generator 
Demonstration 208 
Physics Collage 209 
Bicycle Generator 209 
I>apis Polaris, Magnes 209 
Microwave Transmission Systems 
Good Reading 210 



FILM LOOP NOTES 

L44 Standing Electromagnetic 



204 
205 



206 



210 



Waves 



211 



UhII 6 / Models of the Atom 



Exi>i:KiMi:vrs 

5-1 Klectrt)iysi.s 212 

5-2 The C!haige-I()-Mass Ratio for an 

EIectn)n 214 
5-3 The Measun'ment of Elementary 

Charge 217 
5-4 The Photoelectric Effect 219 
5-5 Spectmsc opy 222 



ACmVITIES 

Daltons Puzzle 226 
Electr^ilysis of Water 226 
Single-Electr-ode Plating 226 
Activitit's fr-om Scientific Aitirricnn 227 

Writing by or about Einstein 227 
Measuring (/Am for Ihi* Klectrxm 227 
Cathodtr Havs in a Clrxiokes I ube 227 



Ughting an Electric Lamp with a 

Match 227 
X Rays from a Crookes Tube 228 
Scientists on Stamps 228 
Measuring Ionization: A Quantum 

Effect 229 
Modeling Atoms with Magnets 230 
"Black Box' Atoms 231 
Standing Wa\es on a Band-Saw Blade 231 
Tunitahle Oscillator Patterns Re.sembling 

(le Bn)glie Wax-es 231 
Standing Waxes in a Wire Ring 232 

FILM LOOP NOTES 

1,45 Pixiduction of Sodium by 

Elect rol\-s is 233 

I.4f; Ihomson Model of the Atom 233 
1/47 Rutherionl Scattering 234 



COIVTENTS 



UhII C / The Muoleas 



EXPERIMENTS 

6-1 Random Events 236 

6-2 Range of oc and ;3 Particles 

6-3 Half-life. I 243 

6-4 Half-life. H 246 

6-5 Radioactive Tracers 248 



240 



6-6 Measuring the Energy of /3 
Radiation 2S0 

FILM LOOP XOTES 

L48 Collisions with an Object of Unknown 
Mass 2S3 



ACKNOWLEDGMENTS 

Page 52 Table 2-4 is reprinted from Solar and 
Planetary Longitudes for Years -2500 to +2500 pre- 
pared by William D. Stalman and Owen Gingerich 
(University of Wisconsin Press, 1963). 

Page 205 Smedlie, S. RavTnond, More Perpetual 



Motion Machines, Science Publications of Boston, 
1962. 

Page 206 I.F. Stacy, The Encyclopedia of Elec- 
tronics (Charles Susskind, Ed.), Reinhold Publishing 
Corp., New York, Fig. 1, p. 246. 



PICTURE CREDITS 



Unit 1, pp. 1, 15, 24, 25 HRW Photos by Russell 
Dian; pp. 10, 13, 34 (bottom), 35, 37 (cartoons) By 
permission of Johnny Hart and Field Enterprises, 
Inc.; p. 39 (left) Courtesy of Mr. Harold M. Waage, 
Palmer Physical Laboratory, Princeton University, 
(right and bottom) Courtesy of Educational De- 
velopment Center, Nev\ton, Mass. All photographs 
used with film loops courtesy of National Film Board 
of Canada. Photographs of laboratory equipment 
and of students using laboratory equipment were 
supplied with the cooperation of the Project Physics 
staff cind Damon Corporation. 

Unit 2, pp. 51, 65 (bottom), 66 (top), 67, 69, 75 
Mount Wilson and Palomar Observatories; pp. 53, 55, 
61, 67, 84, 87 (top) (cartoons) By permission of 
Johnny Hart and Field Enterprises, Inc.; p. 58 (top) 
Lick Observatory; p. 59 NASA; p. 61 sun film strip 
photograph courtesy of U.S. Naval Observatory; p. 65 
(top) Yerices Observatory; p. 73 Lowell Observatory; 
p. 85 (right) Photograph courtesy of Damon Corpo- 
ration, Educational Division. All photographs used 
with film loops courtesy of National Film Board of 
Canada. Photographs of laboratory equipment and 
of students using laboratory equipment were sup- 
plied with the cooperation of the Project Physics 
stciff and Damon Corporation. 

Unit 3, p. 110 (bottom) J. Ph. Charbonnier/Photo 
Researchers; p. 112 Wide World Photo; p. 129 HRW 
Photo by Russell Dian; pp. 141, 150, 151 (cartoons) By 
permission of Charles Gary Solin; p. 153 (top) 
"Physics and Music," Scient/flc /\mer/can, July 1948; 
p. 160 (cartoon) By permission of Johnny Hart and 



Field Enterprises, Inc.; p. 167 Cessna Aircraft. All 
Photographs used with Film Loops courtesy of Na- 
tional Film Board of Canada. Photographs of labora- 
tory equipment and of students using laboratory 
equipment were supplied with the cooperation of 
the Project Physics staff and Damon corporation. 

Unit 4, pp. 180 (bottom), 183 (bottom), 190, 210 
(bottom) Cartoons by Charles Gary Solin and repro- 
duced by his permission only; p. 197 HRW Photos by 
Russell Dian; p. 203 By permission of Johnny Hart 
and Field Enterprises, Inc.; p. 209 (bottom) "Physics " 
by Bob Lillich; p. 210 (top) Bumdy Library. All photo- 
graphs and notes with film loops courtesy of the 
National Film Board of Canada. Photographs of labo- 
ratory equipment and of students using laboratory 
equipment were supplied with the cooperation of 
the Project Physics staff and Damon Corporation. 

Unit 5, p. 217 Courtesy L.J. Lippie, Dow Chemical 
Company, Midland, Michigan; p. 220 HRW Photo by 
Russell Dian; p. 229 From the cover of The Science 
Teacher, Vol. 31, No. 8, December 1964. All photo- 
graphs used with film loops courtesy of National 
Film Board of Canada. Photographs of laboratory 
equipment and of students using laboratory equip- 
ment were supplied with the cooperation of the 
Project Physics staff and Damon Corporation. 

Unit 6. All Photographs used with film loops 
courtesy of National Film Board of Canada Photo- 
graphs of laboratory equipment and of students 
using laboratory equipment u'ere supplied with the 
cooperation of the Project Physics staff and Damon 
Corporation. 



INTRODUCTION 



This Handbook is your guide to observations, 
experiments, activities, and explorations, far 
and wide, in the n;alms of physics 

Prepare for challenging work, fun, and some 
surjjrises. One of the best ways to ieam physics 
is by doing physics, in the laboratory and out. 
Do not rely on reading alone. 

This Han(Ux)ok is different from laboratoiy 
manuals you may have worked with before. Far 
more projects are described here than you 
alone can possibly do, so you will need to pick 
and choose. 

Although only a few of the experiments and 
activities will be assigned, do any additional 
ones that interest you . Also, if an activity occurs 
to you that is not described here, discuss with 
your instructor the possibility of doing it. Some 
of the most interesting science you will experi- 



ence in this course will be the result of the 
activities you choose to pursue beyond the 
regular laboratory assignments. 

The many projects in this Handbook are 
divided into the following sections: 

The Experiments contain full instructions 
for the investigations you can do alone or with 
others in the laboratory. 

The Acrdvities contain many suggestions for 
construction projects, demonstrations, and 
other activities you can do by yourself in the 
laboratory or at home. 

The Film Loop Notes gi\-e instructions for 
the use of the variety of film loops that have 
been specially prepared for the course. 

Do as many of these projects as you can. 
Each one will give you a better grasp of the 
physical principles involved. 




INTRODUCTION 



E.XPER.1MEMT 



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On these two pages is shown an example of a student's graph shows at a glance how the extension of the 

lab notebook report. The table is used to record both rubber band changes as the force acting on it is 

observed quantities (mass, scale position) and calcu- increased. The notes in capital letters are comments, 
lated quantities (force, extension of rubber band). The 



UNIT 1 / INTRODUCTION 



IF THE Axes VA/EB-E NOT 
LABLED WOULD YOU 
PEME/^BER \A/MAT TME 
GPAPH WAS ALL ABOl/T? 




I 



)^ 



P 



EMPMASlZt 
FOIMT5 THAT 
REPRESENT 

DATA (eg. wifh 
cimles) 



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DOTTED LINES SHOW 
HYPOTHETICAL SAAOOTH 
GeAP>V4 



Ao ^iO e>o 70 



so 



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IMCtoDC POS&iBcC <?u«STlONS- ESPeciAU_Y ONe3 YOO CJ^C^'T A^HSW^B-fZ. 



INTRODUCTION 



Keeping Records 

Your records of observations made in the 
laboratory or at home can be kept in many 
ways. Regardless of the procedure followed, the 
key question for deciding what kind of record 
you need is: "Do I have a clear enough record 
so that I could pick up my lab notebook a few 
months from now and explain to myself or 
others what I did?" 

Here are some general rules to be followed in 
every laboratory exercise. Your records should 
be neatly uritten without being fussy. You 
should organize all numerical readings in 
tables, if possible, as in the sample lab write up 
on pages 6 and 7. You should always identify 
the units (centimeters, kilograms, seconds, etc.) 
for each set of data you record. Also, identify 
the equipment you are using, so that you can 
find it again later if you need to recheck your 
work. 

In general, it is better to record more rather 
than less data. Even details that may seem to 
have little bearing on the experiment you are 
doing — such as the temperature and whether 
it varied during the observations, and the time 
when the data were taken — may turn out to be 
information that has a bearing on your analysis 
of the results. 

You may have some reason to suspect that a 
particular datum is less reliable than other 
data. Perhaps you had to make the reading very 
hurriedly, or a line on a photograph was very 
faint. If so, make a note of that fact. Never erase 
a reading. When you think an entry in your 
notes is in error, draw a single line through it; 
do not scratch it out completely or erase it. You 
may find it was significant after all. 

There is no "wrong" result in an experiment, 
although results may be in considerable error. 
If your observations and measurements were 
carefully made, then your result will be reliable. 
Whatever happens in nature, including the 
laboratory, cannot be "wrong." It may have 
nothing to do with your investigation, or it may 
be mixed up with so many other events you did 
not expect, that your report is not useful. 
Therefore, you must think carefully about the 
interpretation of your results. 

Finally, the cardinal rule in a laboratory is to 
choose in favor of "getting your hands dirty" 
instead of "dry-labbing." In 380 B c, the Greek 
scientist, Archytas, summed it up this way: 

In subjects of which one has no knowledge, 
one must obtain knowledge either by learning 
from someone else, or by discovering it for 



oneself. That which is learnt, therefore, comes 
from another and by outside help; that which is 
discovered comes by one's own efforts and 
independently. To discover without seeking is 
difficult and rare, but if one seeks, it is frequent 
and easy; if, however, one does not know how to 
seek, discovery is impossible. 



Using the Polaroid Land Camera 

You will find the Polaroid Land camera a very 
useful device for recording many of your 
laboratory observations. Your textbook shows 
how the camera is used to study moving 
objects. In the experiments and activities 
described in this Handbook, many suggestions 
are made for photographing moving objects, 
both with an electronic stroboscope (a rapidly 
flashing xenon light! and uath a mechanical 
disk stroboscope (a slotted disk rotating in front 
of the camera lens). The setup of the rotating 
disk stroboscope with a Polaroid Land camera 
is shown below. 



Shotb&r 
button 



Cable r«ieQ6e Rim Selector 

socket I «• .c I 

Distance scale 



Lighten/ COf ken 







To ciexc ♦♦>€ t-amtm : 
r-cleaM or pr«Mdowvii on 



Below is a checklist of operations to help you 
use the modified Polaroid Land camera model 
210. For other models, your instructor will 
provide instructions. 

1. Make sure that there is film in the camera. 
If no white tab shows in the front of the door 
marked "4," you must put in new film. 

2. Fasten camera to tripod or disk strobe 
base. If you are using the disk strobe technique, 
fix the clip-on slit in front of the lens. 



UNIT 1 / INTRODLCTION 



3. Check film (speed) selector. Set to 
suggested position (75 for disk strobe or blinky; 
3000 for xenon strobe). 

4. If you are taking a "bulb" exposure, cover 
the electric eye. 

5. Check distance from lens to plane of object 
to be photographed. Adjust focus if necessary. 
Work at the distance that gives an image just 
one-tenth the size of the object, if possible This 
distance is about 120 cm. 

6. Ixjok through viewer to be sure that 
whatever pai1 of the event you are interested in 
will be recorded. (At a distance of 120 cm, the 
field of view is just under 100 cm long. I 

7. Make sure the shutter is cocked (by 
depressing the number 3 button). 

8. Run thmugh the experiment a couple of 
times without taking a photograph, to accus- 
tom yourself to the timing needed to photo- 
gra|)h the event. 

9. lake the picture; keep the cable release 
depressed only as long as necessary to record 
the event itself. Do not keep the shutter open 
longer than necessary. 

10. inill the white tab all the way out of the 
camera. Do not block the door (marked 4" on 
the camera). 



11. Pull the large yellow tab straight out, all the 
way out of the camera. Begin timing develop- 
ment. 

12. Wait 10 to 15 sec (for 3,000-speed black- 
and-white film). 

13. Ten to 15 sec after removing the film from 
the camera, strip the white print from the 
negative. 

14. Take measurements immediately. (The 
magnifier may be helpful.) 

15. After initial measurements have been 
taken, coat your picture with the preservative 
supplied with each pack of film Let the 
preservative drv' thoroughly. Label the picture 
on the back for identification and mount the 
picture in your (or a partner si lab report. 

16. The negative can be used, too Wash it 
carefully with a wet sponge, and coat with 
preservative. 

17. Recock the shutter so it will be set for the 
next use. 

18. Always be careful when moving around the 
camera that you do not inadvertently kick the 
tripod. 

19. Always keep the electric eye covered when 
the camera is not in use. Otherwise the 
batteries inside the camera will run down. 





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/ 
/ 
/ 

J 


.i • ^ 



Goneepis of Motion 

EXPERIMENTS 



Experiment 1-1 
NAKED-EYE ASTRONOMY 

This first experiment will familiarize you with 
the continually changing appearance of the 
sky. By watching the heavenly bodies closely 
day and night over a period of time, you will 
begin to understand what is going on in the slcy 
cind gain the experience you will need in Unit 2, 
"Motion in the Heavens." 

Do you know how the sun and the stars, the 
moon and the planets, appear to move through 
the sky? Do you know how to tell a planet from 
a star? Do you know when you can expect to 
see the moon during the day? Do you know 
how the sun and planets move in relation to 
the stars? 

The Babylonians and Egyptians knew the 
answers to these questions over 5,000 years ago. 
They found the answers by watching the 
ever-changing sky. Thus, astronomy began with 
simple observations of the sort you can make 
with your unaided eye. 

You know that the earth appears to be at rest 
while the sun, stars, moon, and planets are 
seen to move in various paths through the sky. 
The problem, as it was for the Babylonians, is to 



describe what these paths are and how they 
change from day to day, from week to week, 
and fham season to season. 

Some of these changes occur very slowly. In 
fact, that is why you may not have noticed 
them. You vvdll need to watch the motions in 
the sky carefully, comparing them to fixed 
points of reference that you establish. You will 
need to keep a record of your observations for 
at least four to six weeks. 

Choosing References 

To locate objects in the sky accurately, you first 
need some fixed lines or planes to which your 
measurements can be referred, just as a map 
maker uses lines of latitude and longitude to 
locate places on the earth. 

For example, you can establish a north - 
south line along the ground for your first refer- 
ence. Then, with a protractor held horizontally, 
you can measure the direction of an object in 
the sky around the horizon from this north - 
south line. The angle of an object around the 
horizon from a north- south line is called the 
object's azimuth. Azimuths are measured from 
the north point (0°) through east (90°) to south 



LMT I / CONCEPTS OF MOTION 



Alt.tude30' 




(180°) and west (270°) and around to north 
again (360° or 0°). See Fig 1-1. 

To measure the height of an object in the sky, 
you can measure the angle between the object 
and the horizon. When your horizon is 
obscured by trees or buildings, you can mea- 
sure from the zenith overhead 'altitude 90°) 
down to the object; its altitude is then 90° 
minus its zenith distance, for your second 



Fifl. 1-1 



abet 



xj November 20 d^^ 



*«&- 




Fig. 1-2 This chart of the stars will help you locate some 
of the bright stars and the constellations To use the 
map. face north and turn the chart until today's date is at 
the top. Then move the map up nearly over your head 
The stars will be in these positions at 8 P M. For each 
\\o\it earlier than 8 P.M., rotate the chart 15 degrees (one 



sector) clockwise For each hour/arer than 8 P M.. rotate 
the chart counterclockwise If you are observing the sky 
outdoors with the map, cover the glass of a flashlight 
with fairly transparent red paper to look at the map This 
will prevent your eyes from losing their adaptation to 
the dark when you look at the map 



EXPERIMENTS 



coordinate. The angle between the horizontal 
plane and the line to an object in the sky is 
called the altitude of the object. 

At night, you can use the North Star (Polaris) 
to establish the north- south line. Polaris is the 
one fairly bright star in the sky that moves least 
from hour to hour or with the seasons. It is 
almost due north of an obser\'er anywhere in 
the northern hemisphere. 

To locate Polaris, first find the "Big Dipper" 
which on a September evening is low in the sky 
and a little west of north. (See the star map, Fig. 
1-2.1 The two stars forming the end of the 
dipper opposite the handle are known as the 
"pointers," because they point to the North 
Star. A line passing through them passes very 
close to a bright star, the last star in the handle 
of the "Little Dipper." This bright star is the Pole 
Star, Polaris. 

Imagine a line from Polaris straight down to 
the horizon. The point where this line meets 
the horizon is nearly due north of you. See Fig. 
1-3. 




Fig. 1-3 

Now that you have established a north - 
south line, note its position with respect to 
fixed landmarks, so that you can use it day or 
night. 

You can establish the second reference, the 
plane of the horizon, and measure the altitude 
of objects in the sky from the horizon, with an 
astrolabe. An astrolabe is a simple instrument 
you can obtain easily or make yourself, very 
similar to those used by ancient viewers of the 
heavens. Use the astrolabe in your hand or on a 
flat table mounted on a tripod or on a 
permanent post. A simple hand astrolabe you 
can make is described in this Handbook, in the 
experiment dealing uith the size of the earth. 

Sight along the surface of the flat table to be 
sure it is horizontal, in line with the horizon in 
all directions. If there are obstructions on your 
horizon, a carpenter's level turned in all 
directions on the table will show when the 
table is level. 




Fig. 1-4 



Turn the base of the astrolabe on the table 
until the north- south line on the base points 
along your north- south line. You can also 
obtain the north- south line by sighting on 
Polaris through the astrolabe tube. Sight 
through the tube of the astrolabe at objects in 
the sky you wish to locate and obtain their 
altitude above the horizon in degrees from the 
protractor on the astrolabe. With some as- 
trolabes, you can also obtain the azimuth of the 
objects fixDm a scale around the base of the 
astrolabe. 

To follow the position of the sun with the 
astrolabe, slip a lai^ge piece of cardboard with a 
hole in the middle over the sky-pointing end of 
the tube. (CAUTION: NEVER look directly at the 
sun; it can cause permanent eye damage.) 
Standing beside the astrolabe, hold a small 
piece of white paper in the shadow of the large 
cardboard, several centimeters from the sight- 
ing end of the tube. Move the tube about until 
the bright image of the sun appears through 
the tube on the paper. (See Fig. 1-5.) Then read 




Fig. 1-5 



10 



I MT I / CONCEPTS OF MOTION 



the altitude of the sun from the astrolahe, and 
the sun's azimuth, if your instrument permits. 
Also record the date and time of your observa- 
tion. 

Observations 

Now that you know how to establish your 
references for locating objects in the sky, here 
are suggestions for observations you can make 
on the sun, the moon, the stars, and the 
|)lanets. Choose at lease one of these objects to 
obseive. Record the date and time of all your 
observations. Later, compare notes with 
classmates who observed other objects. 

A. Sun 

CAUTION: NEVER look directly at the sun; it 
can cause permanent eye damage. Do not 
depend on sunglasses or fogged photographic 
film for protection. It is safest to make sun 
observations on projected images. 

1. Observe the dii-ection in which the sun sets. 
Always make your observation from the same 
observing position. If you do not have an 
unobstructed view of the horizon, note where 
the sun disappears behind the buildings or 
trees in the evening. 

2. Observe the time the sun sets or disappears 
below your horizon. 

3. Try to make these observations about once a 
week. The first time, draw a simple sketch of 
the horizon and the position of the setting sun. 

4. Repeat the observations a week later. Note if 
the position or time of sunset has changed. 
Note if they change during a month. Try to 
continue these observations for at least two 
months. 

5. If you are up at sunrise, you can record the 
time and position of the sun's rising. (Check the 
weather forecast the night before to be reason- 
ably suri! that tin- sky will be clear.) 

6. Determine how the length of the day, from 
sunrise to sunset, changes during a week; 
during a month; or for the entire year. You 
might like to check your own observations of 
tlie times of sunrise and sunset against the 
times that an- often ivportrd in newspapers. 
Also, if the weather does not permit vou to 
observe the sun, the newspaper reports may 
help you to complete your records. 

7. During a single day, observe the suns 
azimuth at various times. Keep a HToitl of the 
a/.inuuh and the time of ()l)servation Deter- 
mine whether the a/.imutli changes at a 
constant rate during the day. or whether the 



B.C. 



By John H«rt 




■r p«rmliiLon o( Jotaa Sarc aad ft«ld tfii«rprl««f, lac. 

sun's apparent motion is more rapid at some 
times than at others. Find how fast the sun 
moves in degrees per hour. See if you can 
make a graph of the speed of the sun's change 
in azimuth. 

Similariy, find out how the sun's angular 
altitude changes during the day, and at what 
time its altitude is greatest. Compare a graph 
of the speed of the sun's change in altitude 
with a graph of its speed of change in azimuth. 
8. Over a period of several months, or exen an 
entire year, observe the altitude of the sun at 
noon or some other convenient hour. (Do not 
worTN' if you miss some observations.) Deter- 
mine the date on which the altitude of the 
sun is at a minimum. On what date would 
the sun's altitude be at a maximum? 

B. Moon 

1. Observe and record the altitude and 
azimuth of the moon and draw its shape on 
successive evenings at the same hour. Carry 
your observations through at least one cycle of 
phases, or shapes, of the moon, recording in 
your data the dates of any nights that you 
missed. 

For at least one week, make a daily sketch 
showing the appearance of the moon and 
another "overhead sketch of the relative posi- 
tions of the earth, moon, and sun. If the sun is 
below the horizon when \ou observe the moon, 
you will have to estimate the sun s position 
2. Ixuate the moon against the background of 
the stai-s. and plot its position and phase on a 
sky map supplied by your instructor. 



EXPERIMENTS 



11 



3. Find the full moon's maximum altitude. 
Find how this compares with the sun's 
maximum altitude on the same day. Determine 
how the moon's maximum altitude varies from 
month to month. 

4. There may be a total eclipse of the moon 
this year. Consult Table 1-1 on page 12, or the 
Celestial Calendar and Handbook , for the dates 
of lunar eclipses. Observe one if you possibly 
can. 




Fig. 1-6 This multiple-exposure picture of the moon was 
taken with a Polaroid Land camera. Each exposure was 
for 30 sec using 3,000-speed film. The time intervals 
between successive exposures were 15 min, 30 min, 30 
min, and 30 min. 



C. Stars 

1. On the first evening of star observation, 
locate some bright stars that will be easy to find 
on successive nights. Later you will identify 
some of those groups with constellations that 
are named on the star map (Fig. 1-2), which 
shows the constellations around the North 
Star, or on another star map furnished by your 
instructor. Record how much the stars have 
changed their positions compared to your 
horizon after an hour; after 2 hours. 
2. Take a time exposure photograph of several 
minutes of the night sky to show the motion of 
the stars. Try to work well away from bright 
street lights and on a moonless night. Include 
some of the horizon in the picture for refer- 
ence. Prop up your camera so it will not move 




Fig. 1-7 A time-exposure photograph of Ursa Major (The 
Big Dipper) taken with a Polaroid Land camera on an 
autumn evening. 



during the time exposures of an hour or more. 
Use a small camera lens opening (large 
/-number) to reduce fogging of your film by 
stray light. 

3. Viewing at the same time each night, find 
whether the positions of the star groups are 
constant in the sky from month to month. Find 
if any new constellations appear in the eastern 
sky after one month; after 3 or 6 months. 
Over the same periods, find out if some 
constellations are no longer visible. Determine 
in what direction and by how^ much the 
positions of the stars shift per week and per 
month. 

D. Planets 

The planets are located within a rather 
narrow band across the sky (called the zodiac) 
within which the sun and the moon also move. 
For detaUs on the location of planets, consult 
Table 1-1 on page 12, the Celestial Calendar and 
Handbook, or the magazine Sky and Telescope. 
Identify a planet and record its position in the 
sky relative to the stars at 2-week intervals for 
several months. 

Additional sky observations you may wish to 
make are described in Unit 2 of this Handbook. 



12 



LIVIT 1 / CONCEPTS OF MOTION 



Mercury 

Venus 

Mars 

Jupiter 

Saturn 



TABLE 1.1 A GUIDE FOR PLANET AND ECLIPSE OBSCTVATIONS.' 

1980 1981 1982 1983 

2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 , \0 . 1.2 2 4 6 . 8 10 12 



I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 

12 3 4 5 6 

S LpS L L L SLp 



1984 



1985 



1986 



1987 



2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12 



L = lunar eclipse 



S = total solar eclipse 



# = planetary notes 



Mercury 
Venus 
Mars 
Jupiter 


- 


' ' 1 ' ' 








— = 




1 1 1 1 1 1 1 


= 




1 1 1 1 , 


1 1 1 1 1 































Saturn 


1 1 1 


III,, 
7 


8.9 


S 


1 1 

10 


11 

L 


__ 

Ha 

L 


ley s Comet 

L 


SL 


12' 


S 


, 



(a) Lunar Edipses 

Lunar Central 

Data Typa TIma (EST) 



VisiMa 
from 



Data 



(b) Solar Edipsaa (Total) 



VIsiUafrom 



Duration 



1981 








1980 






July 17 


partial 


23:48 


Annerica (midnight) 


Feb. 16 


Central Africa, India 


4.4 min 


1982 








1981 






Jan. 9 


total 


14:56 


Asia 


July 31 


Siberia 


2.2 min 


July 6 


total 


02:30 


America (early 


1983 












morning) 


June 11 


Indonesia 


5.4 min 


Dec. 30 


total 


06:26 


America (sunrise) 


1984 






1983 








Nov. 22 


Indonesia, S. America 


2.1 min 


June 25 


partial 


02:25 


America (early 


1986 












morning) 


Oct 3 


near Greenland 


very brief 


1985 








1987 






May 4 


total 


14:57 


Asia 


March 29 


Central Africa 


0.3 min 


Oct. 28 


total 


12:43 


Asia 


1988 






1986 








March 18 


Philippines, Indonesia 


4.0 min 


April 24 


total 


07:44 


Pacific Ocean 


1990 






Oct 17 


total 


14:19 


Aaia 


July 22 


Finland, Arctic 


2.6 min 



Data 



(c) P lanetary Note* 
Events 



1980 1. February-March: Mars and Jupiter close, 1984 

passing March 1 in retrograde near oppo- 
sition. 

2. June 21: Mars passes Saturn. 

3. Nov. 1: Venus passes Jupiter and Saturn 
in early morning sky (spectacular). 

1961 4 February: Jupiter and Saturn close, near 1965 

opposition. 

1962 6. July 9: Mart passes Saturn in evening 

sky. 
6. Aug. 12: Mars passes Jupiter in evening 1966 
sky. 



7 June 18: Mars passes Saturn after oppo- 
sition in evening sky. 

8 Oct. 8: Venus passes Saturn in evening 
sky. 

9. Oct. 10: Mars passes Jupiter in evening 
sky. 

10. Jan 28: Venus passes Mars low in eve- 
ning sky. 

11. March- April: Mercury and Venus close 
in early dawn. 

12. Dec. 20: Mars passes Jupiter in evening 
sky — nothing spectacular. 



EXPERIMENTS 



13 



Experiment 1-2 
REGULARITY AXD TIME 

You will often encounter regularity in your 
study of science. Many natural events occur 
regularly, that is, o\er cind o\er again at equal 
time intervals. If you had no clock, how would 
you decide how regularly an event recurs? In 
fact, how can you decide how regular a clock 
is? 

Working with a partner, find several recur- 
ring events that you can time in the laboratorv'. 
You might use such events as a dripping faucet, 
a human pulse, or the beat of recorded music. 
(Do not use a clock or watch.) Select pairs of 
these events to compare. 

One lab partner marks each "tick" of Event A 
on one side of the strip chart recorder tape 
while the other partner marks each "tick" of 
Event B. After a long run has been taken, 
inspect the tape to see how the regularities 
compare. Record about 300 ticks of Event A. For 
each 50 ticks of that event, find on the tape the 
number of ticks of Event B; estimate to 1/10 of a, 
tick. Record \our results in a table something 
like this: 

EVENT A EVENT B 



1. What do you conclude about the regularity of 
Event B? If you think that the difference between A 
and B is larger than you would expect from 
measurement error, which of the two events is 
not regular? Explain. 

2. Which is more regular, Event B or Event C? In 
answering, what assumptions were you making 
about Event A? 

3. Now compare the regularity of one of your 
events to some device specifically designed to be 
regular, for example, an electric wall clock. What 
results do you get? How do you know the clock is 
regular? What standard could the clock be com- 
pared to? What about that standard? 



Experiment 1-3 
\'ARIATIOXS IN DATA 

If you count the number of chairs or people in 
an ordinary' sized room, you will probably get 
exactly the right answer. But if you measure the 
length of this page \Nith a ruler, your answer 
will have a small margin of uncertainty. That is, 
numbers read from measuring instruments do 
not give the e^act measurements. Every mea- 
surement is to some extent uncertain. 



First 50 ticks 
Second 50 ticks 
Third 50 ticks 
Fourth 50 ticks 



. ticks 

. ticks 
ticks 
ticks 



B.C. 



By John Hart 



Now repeat the procedure, compeiring \'Our 
Event A to at least one other periodic phenom- 
enon. Event C, and prepare a similar table. 



IN T>C NOWTM, KrtD ». 
TO TME NOftTii . 



^mr. 



F* 




<BE<»>tX« EVE C»« Tj«*T 

lULCVEBTMeBC TXe I 

4sON ^inul. BE KjSi>4^ I 

AM»-V11M<JTE '"CXV. I 




MOW DO>C5U Li»ce THAT'. 
TNE MI&eKABt-E TNIN*>'S 

BUST BO already! 




Fig. 1-8 



By perslsslon of John Hart and Field Cncerprlses. Inc. 

Moreover, if your lab partner edso measures 
the length of this page, the two answers uill 
probably be different. Does this mean that the 
length of the page has changed? Hardly! Then 
can \'ou possibly find the length of the page 
without any uncertainty in your measurement? 



14 



LMT 1 / CONCEPTS OF MOTION 



This lab exercise is intended to show you why 
the answer is "no." 

Various stations have been set up around the 
room, and at each one you are to make some 
measurement Record each measurement in a 
table lilte the one shown here. When you have 
completed the series, write your measurements 
on tlie board along with those of your 
classmates. Some interesting patterns should 
emerge if your measurements have not been 
influenced by anyone else. Therefore, do not 
talk about your results or how you got them 
until everyone has finished. 



STATION 


TO BE 
MEASURED 


MEASUREMENT 









1. How do you explain the variation m readings 
obtained by different students? 

2. Is there any way of identifying which value is 
the "true" one? 

3. Is the average value necessarily more correct 
than any of the actual measured values? If not, 
why is the average value often used in calcula- 
tions? 

4. How could you write the result of a series of 
differing measurements so as to indicate some- 
thing about the range of values? 



Experiment 1-4 
MEv\SlRL\C; 
UNIFORM MOTION 

If you roll a ball along a level floor or table, it 
eventually stops. Was the belli slowing down all 
the time, from the moment you gave it a push? 
Can you think of any things that have uniform 
motion in which their speed remains constant 
and unchanging? Could the diy-ice disk pic- 
tured in Sec. 1.3 of the text really be in uniform 
motion, even if the disk is called "frictionless '? 
Would the disk keep moving fore\'er? Does 
everything eventually come to a stop? 

In this experiment, you can check the 
answers to these questions for yourself You 
will observe very simple motion and make a 
photo record of it, or work with similar photos. 
You will measure the speed of an object <is 
precisely as you can, then record your data in 
tables and draw graphs from these data. From 
the graphs, you can decide whether the motion 
was uniform or not. 

Your decision may be harder to make than 
you would expect, since your experimental 
measurements can never be exact. TTiere are 
likely to be ups and downs in your final results. 
Your problem will be to decide whether the ups 
and downs are due partly to real changes in 
speed, to uncertainty in your measurements, or 
both. 

If the speed of your object appears to be 
constant, does this mean that you hax-e pro- 
duced an example of uniform motion? Do you 
think it is possible to do so? 



Doing the Experiment 

Various setups for the experiment can be made. 
It takes two people working together to photo- 
graph a disk sliding on a smooth surface 
covered with fine plastic beads, or a glider on 
an air track, or a steadily flashing light (called a 
blinkyl mounted on a small box pushed by a 
toy tractor. Your instructor will explain how to 
work with the setup you are using. Excellent 
photographs ran be made of any of them. 

If you do not use a camera at all. or if you 
work alone, then you may me.asure a transpar- 
enc>' or a mo\ie film projected on a lar^e piece 
of paper. (See. for e.xample. Film Loop 9 
Analysis of a Hurdle Race. II.") You may 
simply work from a prexiously prepared photo- 
graph such as Fig 1-11 If there is time, you 
might tr\' se\-eral of these methods. 



EXPERIMENTS 15 



Fig. 1-9 




Fig. 1-10 




16 



UNIT 1 / CONCEPTS OF MOTIOIV 




Fig 1-11 Stroboscopic photograph of a moving CO2 disk. 



One setup uses for the moving object a disk 
made of metal or plastic. A few plastic beads 
sprinkled on a smooth, dust-free table top (or a 
sheet of glass) provide a surface on which the 
disk slides with almost no friction. Make sure 
the surface is quite level, so that the disk will 
not start to move once it is at rest. 

Set up the Polaroid I^nd camera and the 
stroboscope equipment according to your in- 
structor's directions. See the Introduction for 
instructions on operating the Polaroid Land 
model 210, and for a diagram for mounting this 
camera uith a rotating disk stroboscope. A 
iTjler need not be included in your photograph 
as in Fig. 1-11. Instead, you can use a magnifier 
with a scale that is more accurate than a ruler 
for measuring the photograph. 

Either your instructor or a few trials will give 
you an idea of the camera settings and of the 
speed at which to launch the disk, so that the 
images of your disk are clear and well-spaced 
in the photograph. One student launches the 
disk while a second student operates the 
camera. A "dry run" or two without taking a 
pictuif will probably be needed for practice 
l)('f()n' you get a good picture. A good picture is 
one in which there are at least five sharp and 
clear images of your disk far enough apart for 
easy measuring on the photograph. 



Making Measurements 

Uliati'ver method you have used, your ne.xt 
step is to measuiv the spaces between succes- 
sive images of your moving object. For this, use 



a ruler with millimeter di\isions and estimate 
the distances to the nearest tenth of a millime- 
ter, as shown in Fig. 1-12. If you use a magnifier 
with a scale, rather than a ruler, you may be 
able to estimate these distances more precisely. 
List each measurement in a table like Table 1-2. 

Since the intervals of time between one 
image and the next are equcil, you can use that 
interval as a unit of time for analyzing the event. 
If the speed is constant, the distances of travel 
will be all the same, and the motion would be 
uniform. 

How would you recognize motion that is not 
uniform? 

Why is it unnecessaiy for you to know the 
time interval in seconds? 





TABLE 1-2 




Time 


Dista 


nc« Traveled in 


Interval 


Eech Time Interval 


1st 




0.48 cm 


2nd 




0.48 


3rd 




0.48 


4th 




0.48 


5th 




0.48 


6th 




0.48 



Table 1-2 has data that indicate uniform 
motion. Since the object traveled 0.48 cm 
during each time interval, the speed is 0.48 cm 
per unit time. 

It is more likely that your measurements go 
up and down as in Table 1-3. particularly if you 
measure with a ruler 



•4.fe&m 



Fig. 1-12 Estimating to one-tenth 
of a scale division. 



n 



TTTTJTTTT 



\cm 



EXPERIMENTS 



17 





TABLE 1-3 




Time 


Distance Traveled in 


Interval 


Each Time Interval 


1st 




0.48 cm 


2nd 




0.46 


3rd 




0.49 


4th 




0.50 


5th 




0.47 


6th 




0.48 



Is the speed constant in this case? Since the 
distances are not all the same, you might say 
"No, it is not constant." Perhaps you looked 
again at some of the more extreme data in 
Table 1-3, such as 0.46 cm and 0.50 cm, checked 
these measurements, and found them doubt- 
ful. Then you might say, "The ups and douTis 
are because it is difficult to measure to 0.01 cm 
with the ruler. The speed really is constant as 
nearly as I can tell." Which statement is right? 

Look carefully at the divisions or marks on 
your ruler. Can you read your ruler accurately 
to the nearest 0.01 cm? If you are like most 
people, you read it to the nearest mark of 0.01 
cm (the nearest whole millimeter! and estimate 
the next digit between the mari<s for the near- 
est tenth of a millimeter (0.01 cm), as illustrated 
in Fig. 1-12. 

In the same way, whenever you read the 
divisions of any measuring device, you should 
read accurately to the nearest division or marie 
and then estimate the next digit in the 
measurement. Then probably your measure- 
ment, including your estimate of a digit be- 
tween divisions, is not more than half a division 
in error. It is not likely, for example, that in Fig. 
1-12 you would read more than half a millime- 
ter away from where the edge being measured 
comes between the divisions. In this case, in 
which the divisions on the ruler are milli- 
meters, you are at most no more than 0.5 mm 
(0.05 cm) in error. 

Suppose you assume that the motion really is 
uniform and that the slight differences between 
distance measurements are due only to the 
uncertainty in reading the ruler. UTiat is then 
the best estimate of the constant distance the 
object traveled between flashes? 

Usually, to find the "best" value of distance 
you must find the average of the values. The 
average for Table 1-3 is 0.48 cm, but the 8 is an 
uncertain measurement. 

If the motion recorded in Table 1-3 really is 
uniform, the measurement of the distance 
traveled in each time interval is 0.48 cm plus or 
minus 0.05 cm, written as 0.48 ± 0.05 cm. The 



± 0.05 is called the uncertainty of your mea- 
surement. The uncertainty for a single mea- 
surement is commonly teiken to be half a scale 
division. With many measurements, this uncer- 
tainty may be less, but you can use it to be on 
the safe side. 

Now you can return to the key question: Is 
the speed constant or not? Because the num- 
bers go up and down you might suppose that 
the speed is constantly changing. Notice 
though that in Table 1-3 the changes of data 
above and below the average value of 0.48 cm 
are always smaller than the uncertainty, 0.05 
cm. Therefore, the ups and downs may all be 
due to the difficulty in reading the ruler to 
better than 0.05 cm, and the speed may, in fact, 
be constant. 

The conclusion from the data given here is 
that the speed is constant to within the 
uncertainty of measurement, which is 0.05 cm 
per unit time. If the speed goes up or down by 
less than this amount, you simply cannot 
reliably detect it with a ruler. 

Study your own data in the same way. 

Do they lead you to the same conclusion? If 
your data vary as in Table 1-3, can you think of 
anything in your setup that could have made 
the speed actually change? Even if you used a 
magnifier with a scale, do you still come to the 
same conclusion? 

Measuring More Precisely 

A more precise measuring instrument than a 
ruler or magnifier with a scale might show that 
the speed in this example was not constant. For 
example, if you used a measuring microscope 
whose divisions are 0.001 cm apart to measure 
the same picture again more precisely, you 
might arrive at the data in Table 1-4. Such 
precise measurement reduces the uncertainty 
from ±0.05 cm to ±0.0005 cm. 





TABLE 1-4 




Time 


Distance Traveled in 


Interval 


Each Time Interval 


1st 




0.4826 cm 


2nd 




0.4593 


3rd 




0.4911 


4th 




0.5032 


5th 




0.4684 


6th 




0.4779 



Is the speed constant when you measure to 
such high precision as this? 

The average of these numbers is 0.4804. All 
the numbers are presumably correct within 



18 



UNIT 1 / CONCEPTS OF MOTION 



half a division, which is 0.0005 cm. Thus, the 
best estimate of the true value is 0.4804 ± 0.0005 
cm. 

Draining a Graph 

If you have read Sec. 1.5 in the text, you have 
seen how speed data can be graphed. Your data 
provide an easy example to use in drawing a 
graph. 

Just as in the example on text page 19, mark 
off time intervals along the horizontal axis of 
the graph. Your units are probably not seconds; 
they are "blinks" if you used a stroboscope or 
simply "arbitrary time units" which mean here 
the equal time intervals between positions of 
the moving object. 

Next, mark off the total distances traveled 
along the vertical axis. The beginning of each 
scale is in the lower left-hand comer of the 
graph. 

Choose the spacing of your scale division so 
that your data will, if possible, spread across 
most of the graph paper. 



1. Does your graph show uniform motion? Ex- 
plain. 

2. If the motion in your experiment was not 
uniform, review Sec. 1.5 of the Text. Then from 
your graph find the average speed of your object 
over the whole trip. Is the average speed for the 
whole trip the same as the average of the speeds 
between successive measurements? 

3. Could you use the same methods you used in 
this experiment to measure the speed of a 
bicycle? a car? a person running? (Assume they 
are moving uniformly.) 

4. The divisions on the speedometer scale of 
many cars are 5 km/hr in size. You can estimate 
the reading to the nearest 1 km/hr. (a) What is the 
uncertainty in a speed measurement by such a 
speedometer? (b) Could you reliably measure 
speed changes as small as 2 km/hr? 1 km/hr? 0.5 
km/hr? 0.3 km/hr? 



Experiment 1-5 

A SEVENTEEIVTH-CEIVrrRY 

EXPERIMENT 

This exptMinuMit is similar to the one discussed 
by Galileo in the TVvo New Sciences. It \\i\\ give 
you fii-sthand experii'nre in working with tools 
similar to those of a sovrntpenth-crnturv scien- 
tist. You will mak«' quantitative nieasun*ments 
of the motion of a ball rolling dowii an incline, 
as described by Galileo. 



From these measurements, you should be 
able to decide for yourself whether Galileo's 
definition of acceleration was appropriate or 
not. You should then be able to tell whether 
Aristotle or Galileo was correct in his conclu- 
sion about the acceleration of objects of 
different sizes. 

Reasoning 

Behind the Experiment 

You have read in Sec. 2.6 of the Text how 
Galileo expressed his belief that the speed of 
free-falling objects increases in proportion to 
the time of fall, in other words, that they 
accelerate uniformly. But since free fadl was 
much too rapid to measure, Galileo assumed 
that the speed of a ball rolling down an incline 
increased in the same way as an object in free 
fall did, only more slowly. 

However, even a ball rolling down a low 
incline moved too fast to measure the speed 
cilong different parts of the descent accurately. 
So Galileo used the relationship d <^ t^ (or d/t^ 
- constant), an expression in which speed 
differences have been replaced by the total 
time t and total distance d rolled by the ball. 
Both these quantities can be measured. 

Be sure to study Text Sec. 2.7 in which the 
derivation of this relationship is described. If 
Galileo's original assumptions were true, this 
relationship would hold for both freely falling 
objects and rolling balls. Since total distance 
and total time are not difficult to measure, 
seventeenth-century scientists had a secondary 
hypothesis they could test by experiment: and 
so have you. Sec. 2.8 of the text discusses much 
of this material. 

Apparatus 

The apparatus that you will use is shown in Fig. 
1-13. It is similar to that described by Galileo. 

You will let a ball roll various distances down 
a channel about 2 m long and time the motion 
with a water clock. 

You will use a water clock to time this 
experiment because that was the best timing 
device available in Galileos time. The way your 
water clock works is very simple. Since the 
volume of water is proportional to the time of 
flow, you can measure daMime in milliliters of 
water Start and stop the flow with your fingers 
o\fr the upper end of the tube inside the 
funnel Whenewr you refill the clock, let a little 
water run through the tube to clear out the 
bubbles. 



EXPERIMENTS 



19 



Fig. 1-13 



openinq arv:A Cloiinq Phc 
tc>p or the. "tube wit-h 
'^our -finder* 




Stopping block 



paper cJip to 
adjusL -now to 
a. cor-)ve»-iicnt- 



tdpc ciown end 



0>ec>k efcr-aightneSS of 
cinannci fcy siqhtinq 
<>lonq ifc Sr)ci ddju-^ng 
Sopport "st-andSj 



Compare your water clock with a stopwatch 
when the clock is full and when it is nearly 
empty to determine how accurate it is. Does 
the clock's timing change? If so, by how much? 
Record this information in your notebook. 

It is almost impossible to release the ball with 
your fingers without giving it a slight push or 
pull. Therefore, restrain the ball with a ruler or 
pencil, and release it by quickly moving this 
barrier down the inclined plane. The end of the 
run is best marked by the sound of the ball 
hitting the stopping block. 

Brief Comment on Recording Data 

You should always keep neat, orderly records. 
Orderly work looks better and is more pleasing 
to you and everyone else. It may also save you 
fix)m extra work and confusion. If you have an 
oiiganized table of data, you can easily record 
and find your data. This will leave you free to 
think about your experiment or calculations 
rather than having to worry about which of two 
numbers on a scrap of paper is the one you 
want, or whether you made a certain mea- 
surement or not. A few minutes' preparation 
before you start work will often save you an 
hour or two of checking in books emd with 
friends. 

Operating Suggestions 

You should measure times of descent for 
several different distances, keeping the inclina- 
tion of the plcme constant and using the same 



bcdl. Repeat each descent about four times, and 
average your results. Best results are found for 
very small angles of inclination (the top of the 
channel raised less than 30 cm). At steeper 
inclinations, the ball tends to slide as well as to 
roll. 

From Data to Calculations 

Galileo's definition of uniform acceleration 
(text, page 531 was "equal increases in speed in 
equcd times " Galileo showed that if an object 
actually moved in this way, the totail distance of 
travel should be directly proportional to the 
square of the total time of fall, or d cc t^. 

If two quantities are proportional, a graph of 
one plotted against the other wUl be a straight 
line. Thus, making a graph is a good way to 
check whether two quantities ctre proportioned. 
Make a graph of d plotted against f^ in your 
notebook using your data. 



1. Does your graph support the hypothesis? 
Explain. 

2. How accurate is the water clock you have been 
using to time this experiment? If you have not 
already done so, check your water clock against a 
stopwatch for timing. In your judgment, how does 
the inaccuracy of your water clock affect your 
conclusion to Question 1 above? 



Going Further 

1. In Sec. 2.7 of the text, you learned that 
a = 2d/t^. Use this relation to calculate the 



20 



UNIT 1 / CONCEPTS OF MOTION 



actual acceleration of the ball in one of your 
runs. 

2. If you have time, find out whether Galileo or 
Aristotle was right about the acceleration of 
objects of various sizes. Measure d/t^ for several 
different sizes of balls, all rolling the same 
distance down a plane of the same inclination. 
Does the acceleration depend on the size of the 
ball? In what way does your answer refute or 
support Aristotle's ideas on falling bodies? 

3. Galileo claimed his results were accurate to 
1/10 of a pulse beat. Do you believe his results 
were that accurate? Did you do that well? How 
could you improve the design of the water 
clock to increase its accuracy? 

4. When Galileo first did the experiments with 
balls rolling down an incline he determined 
how far they went during equal time intervals. 
You can do this at least roughly by putting thin 
rubber bands or a similar small obstacle on the 
track, and listening for the bumps as the ball 
rolls down the track. Adjust the position of the 
rubber bands until the bumps all come at equal 
times. You might try to keep a regular rhythm 
by tapping on the table, or by listening to the 
drip of a faucet or other spigot (as in Method D 
of Experiment 1-7). Adjust the rubber bands 
until the bumps are at the same time intervals 
as the taps or drips. First try a few runs without 
the rubber bands, making chalk marks at the 
position of the ball, so that you have a good 
idea of where to put the rubber bands. 

When the rubber bands are properly spaced, 
measure the distance between them. Are these 
distances from the start in the ratio of 
1:3:5:7...? (There are several ways you can 
check. You could, for example, divide all the 
distances by the shortest one and see how 
close the ratios are to the odd integers. You 
could also divide each interval by the appro- 
priate odd integer and see how similar the 
ratios an».) 

This version of Galileo's experiments is 
described by Stillman Drake in his article 'The 
Role of Music in Galileo s Experiments," Scien- 
tific American. Vol. 232. No. 6 (June, 1975). pp. 
98- 104. 



Experiment l-(i 

tut: j>rri ETn-CEivriTR v \xrsioiv 
or <;alilf:o*s exfi:rimeivt 

Galileo's seventeenth-rentun «'.\periment had 
its limitations, as you n«ad in the text, Sec. 2.9. 
Ihe measurement of time with a water clock 



was imprecise and the extrapolation fitxim 
acceleration at a small angle of inclination to 
that at a verticiil angle (90°) was extreme. 

With more modern equipment, you can 
verily Galileo's conclusions; further, you can get 
an actual value for acceleration in finee fall (near 
the earth's surface). Remember that the idea 
behind the improved experiment is still 
Galileo's. More precise measurements do not 
always lead to more significant conclusions. 

Determine a g as carefully as you can. This is a 
fundamental measured value in modem sci- 
ence. It is used in mciny ways, from the 
determination of the shape of the earth and the 
location of oil fields deep in the earth's crust to 
the calculation of the orbits of earth satellites 
and spacecraft in space research programs. 

Apparatus and Procedure 

For an inclined plane use the air track. For 
timing the air track glider use a stopwatch 
instead of the water clock; otherwise, the 
procedure is the same as that used in Experi- 
ment 1-5. As you go to higher inclinations, you 
should stop the glider by hand before it is 
damaged by hitting the stopping block. 

Instead of a stopwatch, you may wish to use 
the Polaroid Land camera to make a strobe 
photo of the glider as it descends. A piece of 
white tape on the glider will show up well in 
the photograph; or you can attach a small light 
source to the glider. You can use a magnifier 
with a scale attached to it to measure the 
glider's motion as recorded on the photograph. 
Here the values of d will be millimeters on the 
photograph and ( will be measured in an 
arbitrary unit, the "blink" of the stroboscope, or 
the "slot" of the strobe disk. 



1. Plot your data as before on a graph of d versus 
t^. Compare your plotted lines with graphs of the 
preceding cruder seventeenth-century experi- 
ment, if they are available. Are there differences 
between them? Explain. 

2. Is dit^ constant for your air track glider? What is 
the significance of your answer? 

3. As a further challenge, if time permits, try to 
predict the value of a,, which the glider ap- 
proaches as the air track becomes vertical. What 
values do you get? The accepted value of a, is 9.8 
m/sec' near the earth's surface. 

4. What is the percentage error in your calculated 
value? That is, what percent of the accepted value 
is your error? 

percentage error 

accepted value calculated value ^ ^qq 
accepted value 



EXPERIMENTS 



21 



Therefore, if your value of a^ is 9.5m/sec^ your 
percentage error is 

9.8 m/sec^ - 9.5 m/sec ' „ mno/ 
9.8 nMsec^ "" ^°°/° 

(3/98) X 100% = 3% 
Notice that you cannot carry this 3% out to 
3.06% because you only know the 3 in the fraction 
3/98 to one digit. Therefore, you can only know 
one digit in the answer, 3%. A calculated value 
like this is said to have one significant digit. You 
cannot know the second digit in the answer until 
you know the digit following the 3. To be 
significant, this digit would require a third digit in 
the calculated values of 9.5 and 9.8. 
5. What are some of the sources of your error? 



Experiment 1-7 
MEASURING THE 
ACCELERATION OF GRAVITYg* 

Aristotle's idea that bodies falling to the earth 
are seeking out their natural places sounds 
strange today. After all, you know that gravity 
makes things fall. 

But just what is gravity? Newton tried to give 
operational meaning to the idea of gravity by 
seeking out the laws according to which it acts. 
Bodies near the earth fall toward it with a 
certain acceleration due to the gravitational 
"attraction" of the earth. How can the earth 
make a body at a distance fall toward it? How is 
the gravitational force transmitted? Has the 
acceleration due to gravity always remained the 
same? These and many other questions about 
gravity have yet to be answered satisfactorily. 

Whether you do one or several parts of this 
experiment, you will become more familiar 
with the effects of gravity by finding the 
acceleration of bodies in free fall yourself. You 
will learn more about gravity in later chapters. 

METHOD A: a^ bv Direct FaU* 

In this experiment, you will measure the 
acceleration of a falling object. Since the 
distance and therefore the speed of fall is too 
small for air resistance to become important, 
and since other sources of friction are very 
small, the acceleration of the falling weight is 
very nearly a^. 



'Adapted from R. F. Brinckerhoff and D. S. Taft, 
Modern Laboratory E^cperiments in Physics, by 
permission of Science Electronics, Inc., Nashua, New 
Hampshire. 



Doing the Experiment 

The falling object is an ordinary laboratory 
hooked weight of at least 200 g mass. (The drag 
on the paper tape has too great an effect on the 
fall of lighter weights.) The weight is suspended 
from about 1 m of paper tape. Reinforce the 
tape by doubling a strip of masking tape over 
one end and punch a hole in the reinforcement 
1 cm from the end. With careful handling, this 
tape can support at least 1 kg. 




Fig. 1-14 



When the suspended weight is allowed to 
fall, a vibrating tuning fork wall mark equal time 
intervals on the tape pulled down after the 
weight. 

The tuning fork must have a frequency 
between about 100 Hz (vibrations/second) and 
about 400 Hz. In order to mark the tape, the foric 
must have a tiny felt cone (cut from a marking 
pen tip) glued to the side of one of its prongs 
close to the end. Such a small mass affects the 
fork frequency by much less than 1 Hz. Saturate 
this felt tip with a drop or two of marking pen 
ink, set the fork in vibration, and hold the tip 
very gently against the tape. The falling tape is 
conveniently guided in its fall by two 
thumbtacks in the edge of the table. The easiest 
procedure is to have an assistant hold the 
weighted tape straight up until you have 
touched the vibrating tip against it and said 



22 



UIVIT 1 / CONCEPTS OF MOTION' 



"Go " After a few practice runs, you will become 
expert enough to mark many centimeters of 
tape with a wavy line as the tape is accelerated 
past the stationary vibrating fork. 

Instead of using the inked cone, you may 
press a comer of the vibrating tuning fork 
gently against a 2-cm square of carbon paper 
which the thumbtacks hold ink-surface- 
inwards over the falling tape. With some 
practice, this method can be made to yield a 
series of dots on the tape without seriously 
retarding its fall. 

Analyzing Your Tapes 

Label with an A one of the first wave crests (or 
dots) that is clearly formed near the beginning 
of the pattern. Count 10 intervals between wave 
crests (or dots), and mark the end of the tenth 
space with a B. Continue marking every tenth 
crest with a letter throughout the length of the 
record, which ought to be at least 40 waves 
long. 

At A, the tape already had a speed of v,,. From 
this point to B, the tape moved a distance d, in 
a time t. The distance d, is described by the 
equation for free fall: 

In covering the distance from A to C, the tape 
took a time exactly twice as long, 2f, and fell a 
distance dz described (by substituting 2f for f 
and simplifying) by the equation: 

d^ - 2v„f -t- ^^ 
2 

In the same way, the distances AD, AE, etc., are 
described by the equations: 

9a,r2 



dg = 3vof -t- 
d, = 4v„f -I- 



2 

16a „r- 



and so on. 

All of these distances are measured from A, 
the arbitrary starting point. To find the dis- 
tances fallen in each 10-rrest interval, you must 
subtract each equation from the one before it, 
getting: 

2 



AB = vj + 



BC = v„f + 



CD = v„f + 5£jiL 
2 

DE = v„f + ^^*^ 



From these equations, you can see that the 
weight falls farther during each later time 
interval. Moreover, when you subtract each of 
these distances, AB, BC, CD, . . . from the 
subsequent distance, you find that the increase 
in distance fallen is a constant. That is, each 
difference BC - AB = CD - BC = DE - CD = 
agf^. This quantity is the increase in the 
distance fallen in each successive 10-wave 
interval and thus is an acceleration. TTie 
formula describes a body falling with a con- 
stant acceleration. 

From your measurements of AB, AC, AD, etc., 
make a column of AB, BC, CD, DE, etc., and in 
the next column record the resulting values of 
a^f^. The values of a^t^ should all be equal 
(within the accuracy of your measurements). 
Why? Make all your measurements £is precisely 
as you can with the equipment you are using. 

Find the average of all your values of a^r^, the 
acceleration in centimeters/(10-vvave interval)*. 
You want to find the acceleration in cm/sec^. If 
you call the frequency of the tuning fork n per 
second, then the length of the time intervjil t is 
10/n sec. Replacing t of 10 waves by 10/n sec 
gives you the acceleration a^ in cm/sec^. 



1. What value do you get for aj What is the 
percentage error? (The ideal value of a^ is close to 
9.8 m/sec2.) 



METHOD B: a« from a Pendulum 

You can easily measure the acceleration due to 
gravity by timing the swinging of a pendulum. 
Of course, the pendulum is not falling straight 
down, but the time it takes for a round-trip 
swing still depends on a^. The time T it takes 
for a round-trip swing is 



r = 2Tr 



In this formula, / is the length of the pendulum. 
If you measure / with a ruler and T with a clock, 
you should he able to solve for a,. 

You may learn in a later physics course how- 
to derive the formula. Scientists often use 
formulas they hax-e not deri\-ed themselves, as 
long as they are confident of their \alidit>' 

Making the Measurements 

Tlie fonnula is derixed for a pendulum with all 
the mass concentrated in the weight at the 



EXPERIMENTS 



23 



bottom, caUed the bob. Therefore, the best 
pendulum to use is one whose bob is a metal 
sphere hung by a fine thread. In this case, you 
can be sure that almost all the mass is in the 
bob. The pendulum's length, /, is the distance 
from the point of suspension to the center of 
the bob. 

Your suspension thread can have any con- 
venient length. Measure / as accurately as 
possible in meters. 

Set the pendulum swinging with small 
sudngs. The formula does not work well for 
large swings, as you can test for yourself later. 

Time at least 20 complete round trips, 
preferably more. By timing many round trips 
instead of just one, you make the error in 
starting and stopping the clock a smaller 
fraction of the total time being measured. 
(When you divide by 20 to get the time for a 
single round trip, the error in the calculated 
value for one trip will be only 1/20 as large as if 
you had measured only one trip.) 

Divide the total time by the number of swings 
to find the time T of one swing. 

Repeat the measurement at least once as a 
check. 

Finally, substitute your measured quantities 
into the formula and solve for a^. 

If you measured / in meters, the accepted 
value of a„ is 9.80 m/sec^. 



1. What value did you get for aj 

2. What was your percentage error? You find 
percentage error by dividing your error by the 
accepted value and multiplying by 100: 

accepted value - your value ^ ,qq 
accepted value 

= YO^'"er'"o^ X 100 
accepted value 

With care, your value of a^ should agree within 

about 1%. 

3. Which of your measurements do you think was 
the least accurate? 



If you believe the answer to question 3 was 
your measurement of length and you think you 
might be ofi'by as much as 0.5 cm, change your 
value of/ by 0.5 cm and calculate once more the 
value of ag. Has ag changed enough to account 
for your error? Ilf ag went up and your value of 
ag was already too high, then you should have 
altered your measured / in the opposite 
direction. Try again!) 

If your possible error in measuring is not 
enough to explain your difference in ag, try 
changing your total time by a few tenths of a 



second; there may be a possible error in timing. 
Then you must recalculate T and therefore a^. 
If neither of these attempts works (nor both 
taken together in the appropriate direction), 
then you almost certainly have made an error 
in arithmetic or in reading your measuring 
instruments. It is most unlikely that ag in your 
school differs from 9.80 m/sec' by more than 
±0.01. 



METHOD C: Og v^ith Slow-Motion 
Photography (Fihn Loop) 

With a high-speed movie camera you could 
photograph an object falling along the edge of a 
vertical measuring stick. Then you could de- 
termine ag by projecting the film at standard 
speed and measuring the time for the object to 
fall specified distance intervals. 

A somewhat similar method is used in Film 
Loops 4 and 5. Detailed directions are given for 
their use in the Film Loop Notes on pages 59 
and 60. 



METHOD D: Og from Falling 
Water Drops 

You can measure the acceleration due to 
gravity ag simply with drops of water falling on 
a pie plate. 

Put the pie plate or a metal dish or tray on 
the floor. Set up a glass tube vvath a stopcock, 
valve, or spigot so that drops of water from the 
valve will fall at least 1 m to the plate. Support 
the plate on three or four pencils so that each 
drop sounds distinctly, like a drum beat. 

Adjust the valve carefully until one drop 
strikes the plate at the same instant the next 
drop from the valve begins to fall. You can do 
this most easily by watching the drops on the 
valve while listening for the drops hitting the 
plate. When you have exactly set the valve, the 
time it takes a drop to fall to the plate is equal to 
the time interval between one drop and the 
next. 

With the drip rate adjusted, find the time 
interval t between drops. For greater accuracy, 
you may want to count the number of drops 
that fall in 30 sec or 60 sec, or to time the 
number of seconds for 50 to 100 drops to fall. 

Your results are likely to be more accurate if 
you run a number of trials, adjusting drip rate 
each time, and average your counts of drops or 
seconds. The average of several trials should be 
closer to actual drip rate, drop count, and time 
intervals than one trial would be. 



24 



UNIT 1 / CONCEPTS OF MOTION 



Now you have all the data you need. You 
know the time t it takes a drop to fall a distance 
d from rest. From these data you can calculate 
a„ since you know that d = Vaflgf ^ for objects 
falling from rest. 



1. What value did you get for a J 

2. What is your percentage error? How does this 
compare with your percentage error by any other 
methods you have used? 

3. What do you think led to your error? Could it be 
leaking connections, allowing more water to 
escape sometimes? How does this affect your 
answer? 

Suppose the distance of fall was lessened by a 
puddle forming in the plate; how would this 
change your results? 

There is less water pressure in the tube after a 
period of dripping; would this increase or de- 
crease the rate of dripping? Do you get the same 
counts when you refill the tube after each trial? 

Would the starting and stopping of your count- 
ing against the watch or clock affect your answer? 
What else may have added to your error? 

4. Can you adapt this method of measuring the 
acceleration of gravity so that you can do it at 
home? Would it work in the kitchen sink? Would 
your results be more accurate if the water fell a 
greater distance, such as down a stairwell? 



With the table turning, the thread is burned 
and each ball, as it hits the carbon paper, will 
leave a mark on the paper under it. 

Measure the vertical distance between the 
balls and the angular distance between the 
marks. With these measurements and the 
speed of the turntable, determine the free-fall 
time. 



1. What value do you get for a,? 

2. What was your percentage error? 

3. What is the most probable source of error? 
Explain. 



METHOD F: Og uith Strobe 
Photography 

Photographing a falling light source with the 
Polaroid Land camera provides a record that 
can be graphed and analyzed to give an average 
value of a^. The 12-slot strobe disk gives a very 
accurate 60 slots per second. (A neon bulb can 
also be connected to the ac line outlet in such a 
way that it will flash a precise 60 times per 
second, as determined by the line frequency. 
Your instructor has a description of the approx- 
imate circuit for doing this. I 



METHOD E: a« with Falling Ball 
and Turntable 

You can measure a^ with a record-player 
turntable, a ring stand and clamp, carbon 
paper, two balls with holes in them, and thin 
thread. 

Ball X and ball Y are draped across the 
prongs of the clamp. Line up the balls along a 
radius of the turntable, and make the lower ball 
hang just above the paper, as shown in Fig. 
1-15. 



White Raper 
Turntable 



Fig 1-15 




1. What value do you get for a J 

2. What was your percentage error? 

3. What is the most probable source of error? 
Explain. 




Fig 1-16 



EXPERIMENTS 



25 



Experiment 1-8 
NEWTON'S SECOND LAW 

Newton's second law of motion is one of the 
most important and useful laws of physics. 
Re\iew text Sec. 3.7 on Xewton's second law to 
make sure you are familiar with it. 

Newton's second law is part of a much larger 
body of theory than can be studied with a 
simple set of laboratory experiments. Our 
experiment on the second law has two pur- 
poses. 

First, because the law is so important, it is 
useful to get a feeling for the behaxior of objects 
in terms of force (Fl, mass (ml, and acceleration 
(a). You will do this in the first part of the 
experiment. 

Second, the experiment permits you to 
consider the uncertainties of your mea- 
surements. This is the purpose of the latter part 
of the experiment. 

You will apply different forces to carts of 
different masses and measure the acceleration. 

Hou' the Apparatus Works 

You are about to find the mass of a loaded cart 
on which you then exert a measurable force. 
From Xewton's second law you can predict the 
resulting acceleration of the loaded cart. 



Arrange the apparatus as shown in Fig. 1-17. 
A spring scale is firmly taped to a dynamics 
cart. The cart, carrying a blinky, is puUed along 
by a cord attached to the hook of the spring 
scale. The scale therefore measures the force 
exerted on the cart. 




Fig. 1-17 

The cord runs over a pulley at the edge of the 
lab table, and from its end hangs a weight. The 
hanging weight can be changed so as to 
produce various tensions in the cord and thus 
various accelerating forces on the cart. 

Now You Are Ready to Go 

Measure the total mass of the cart, the blinky, 
the spring scale, and any other weights you 




Fig. 1-18 



26 



UNIT i / CONCEPTS OF MOTION 



may want to include with it to vary the mass m 
of the cart being accelerated. 

Release the cart and allow it to accelerate. 
Repeat the motion several times while watch- 
ing the spring-scale pointer. You may notice 
that the pointer has a range of positions. The 
niidfjoint of this range is a fairly good mea- 
surement of the average force F^,, producing 
the acceleration. Record Fav '" newlons OV). 

Faith in Newton's law is such that you can 
assume the acceleration is the same and is 
constant every time this particultir Fgv acts on 
the mass m. 

Substituting your known values of F and m, 
use Newton's law to predict what the average 
acceleration a^v vvas during the run. 

Then, from your record of the cart's motion, 
find a directly to see how accurate your 
prediction was. 

To measure the average acceleration a.^^, take 
a Polaroid photograph through a rotating disk 
stroboscope of a light source mounted on the 
cart. As alternatives, you might use a liquid 
surface accelerometer, described in detail on 
page 42, or a blinky. Analyze your results just as 
in the experiments on uniform and accelerated 
motion (1-4 and 1-5) to find a^^. 

This time, however, you must know the 
distance traveled in meters and the time 
interval in seconds, not just in blinks, flashes, 
or other arbitrary time units. 

You may wish to observe the following effects 
without actually making numerical mea- 
surements: (a) Keep the mass of the cart 
constant and observe how various forces affect 
the acceleration, (b) Keep the force constant 
and observe how various masses of the cart 
affect the acceleration. 



1. Does f .,, (as measured) equal ma,„ (as com- 
puted from measured values)? 

2. Do your other observations support Newton's 
second law? Explain. 



Experimental Errors 

It is unlikely that your values of F„^ and ma,,^ 
wen' equal. 

Does this mean that you ha\«' done a poor 
job of taking data? Not necessarily. As you think 
about it, you will see that there are at least two 
other possible reasons for the inequality. One 
niJiv bj' that you have not yet moasunnl 
everything neressarv in ortier to get an accu- 
rate value for each of your three quantities 



In particular, the force used in the calcula- 
tion ought to be the net, or resultant, force on 
the cart, not just the towing force that you 
measured. Friction force also acts on your cart, 
opposing the accelerating force. You can mea- 
sure it by reading (he spring scale as you tow 
the cart by hand at constant speed. Do it se\'eral 
times and take an average, F,. Since F, acts in a 
direction opposite to the towing force Fj, 

If Ff is too small to measure, then Fnpt = F^, and 
no correction for friction is needed. 

Another reason for the inequality of Fgv and 
maav rnay be that your value for each of these 
quantities is based on measurements and eveiy 
measurement is uncertain to some extent. 

You need to estimate the uncertainty of each 
of your measurements. 

Uncertainty' in Average Force F^v 

Your uncertainty in the measurement of Fav 
is the amount by which your reading of your 
spring scale varied above and below the average 
force, Fav- Thus, if your scale reading ranged 
from 1.0 to 1.4 N, the average is 12 N, and the 
range of uncertainty is 02 N. The value of F^^ 
would be reported as 12 ±02 N. Record your 
value of Fav and its uncertainty. 

Uncertainty in Mass m 

Your uncertainty' in m is roughly half the 
smallest scale reading of the balance with 
which you measured it. The mass consisted of 
a cart, a blinky, and a spring scale land possibly 
an additional mass). If the smallest scale 
reading is 0.1 kg, your record of the mass of 
each of these in kilograms might be as follows: 

mean = 0.90 ± 0.05 kg 

"^blinky = 0.30 ±0.05 kg 

"Ijcale = 0.10 ± 0.05 kg 

The total mass being accelerated is the sum of 
these masses. The uncertaintv' in the total mass 
is the sum of the three uncertainties. Thus, in 
this example, m = 1.30 ± 0.15 kg. Record your 
value of m and its uncertainty. 

Uncertaint}' in 

Average Acceleration a.,^ 

Finally, consider 3,,- You found this by 
measuring Ad/ At for each of the intervals 
between the points on your blinky photograph 

Suppose the points in Fig. 1-19 reprt'sent 
images of a light sourt^e photographed through 
a single slot, giviiig 5 images per second. 
Calculate Ad/ Af for sex'eral intervals. 



EXPERIMENTS 



27 



j^ Ad^-*|<— /vd^-^-« 6^3 — >1< L d^ 

Fig. 1-19 



If you assume the time between blinks to 
have been equal, the uncertainty in each value 
of Ad/ Af is due primarily to the fact that the 
photographic images are a bit fuzzy. Suppose 
that the uncertainty in locating the distance 
between the centers of the dots is 0.1 cm as 
shown in the first column of Table 1-5. 



TABLE 1-5 
Average Speeds Average Accelerations 



At/,/ M = 2.5 ± 0. 1 cm/sec 
Ac/j/Af = 3.4 ±0.1 cm/sec 
Id^l Af = 4.0 ± 0. 1 cm/sec 
Id J Af = 4.8 ± 0. 1 cm/sec 



A\^^/ Af = 0.9 ± 0.2 cm/sec2 
AvJ Af = 0.6 ± 0.2 cm/sec2 
AVj/ Af = 0.8 ± 0.2 cm/sec2 
Average = 0.8 ± 0.2 cm/sec^ 



When you take the differences between suc- 
cessive values of the speeds, Ad/ At, you get 
the accelerations, Av/ At, which are recorded 
in the second column. When a difference in 
two measurements is involved, you find the un- 
certainty of the differences (in this case, Av/ Af ) 
by adding the uncertainties of the two measure- 
ments. This results in an uncertainty in accel- 
eration of (±0.1) + (±0.1) or ± 0.2 cm/sec^ as 
recorded in the table. Determine and record 
your value of agv and its uncertainty. 

Comparing Your Results 

You now have the values of Fav< rn, and a^y, 
their uncertainties, and you considered the 
uncertainty of ma^y. When you have a value for 
the uncertainty of this product of two quan- 
tities, you will then compare the value of ma^y 
wath the value of F^^ and draw your final 
conclusions. For convenience, the "av" has 
been dropped from the symbols in the 
equations in the following discussion. When 
two quantities are multiplied, the percentage 
uncertainty in the product never exceeds the 
sum of the percentage uncertainties in each of 
the factors. In the example, m x a = 1.30 kg x 
0.8 cm/sec^ = 1.04 N. The uncertainty in a 
(0.8 ± 0.2 cm/sec^) is 25% (since 0.2 is 25% of 0.8). 
The uncertainty in m is 11%. Thus, the 
uncertainty in ma is 25% + 11% = 36%. The 
product can be written as ma — 1.04 N ± 36% 
which is, to two significant figures, 
ma = 1.04 ± 0.36 N 



(The error is so large here that it really is not 
appropriate to use the two decimal places; 
round off to 1.0 ± 0.4 N.) In the example, from 
direct measurement, F„^i = 1.2 ± 02 N. Are 
these two results equal within their uncertain- 
ties? 

Although 1.0 does not equal 1.2, the range of 
1.0 ± 0.4 overlaps the range of 1.2 ± 0.2. There- 
fore, the two numbers agree within the range 
of uncertainty of measurement. 

An example of the lack of agreement would 
be 1.0 ± 0.2 and 1.4 ± 0.1. These are presumably 
not the same quantity since there is no overlap 
of expected uncertainties. 

In a similar way, work out your own values of 
Fnet and ma^^. 



3. Do your own values agree within the range of 
uncertainty of your measurement? 

4. Is the relationship f „,., = ma^, consistent with 
your observations? 



Experiment 1-9 
MASS AND WTIIGHT 

You know from your own experience that an 
object that is pulled strongly toward the earth 
(for example, an automobile) is difficult to 
accelerate by pushing. In other words, objects 
with great weight also have great inertia. Is 
there some simple, exact relationship between 
the masses of objects and the gravitational 
forces acting on them? For example, if one 
object has twice the mass of another, does it 
also weigh twice as much? 

Measuring Mass 

The masses of two objects can be compared by 
obsendng the accelerations each experiences 
when acted on by the same force. Accelerating 
an object in one direction with a constant force 
for long enough to take measurements is often 
not practical in the laboratory. Fortunately 
there is an easier way. If you rig up a puck and 
springs between two rigid supports as shown 
in Fig. 1-20, you can attach objects to the puck 



■nnnnsTpmnnnnnmTri-o 



_9-STraTJ-5 ■*■*"« "3 6 6 SUT/^ 



Fig. 1-20 



28 



UNIT 1 / CONCEPTS OF MOTION 



and have the springs accelerate the object back 
and forth. The greater the inertial mass of the 
object, the less the magnitude of acceleration 
will be, and the longer it will take to oscillate 
back and forth. 

To "calibrate" your oscillator, first time the 
oscillations. The time required for five complete 
n)und trips is a convenient measure. Next tape 
pucks on top of the first one, and time the 
period for each new mass. (The units of mass 
an." not essential here; you will be interested 
only in the ratio of masses.) Then plot a graph 
of mass against the oscillation period, drawing 
a smooth curve through your experimental plot 
points. Do not leave the pucks stuck together. 

From your results, try to determine the 
relationship between inertial mass and the 
oscillation period. If possible, write an algebraic 
expression for the relationship. 

Weight 

To compare the gravitational forces on two 
objects, they can be hung on a spring scale. In 
this investigation, the units on the scale are not 
important because you are interested only in 
the ratio of the weights. 

Comparing Mass and Weight 

Use the puck and spring oscillator, and the 
calibration graph to find the masses of two 
objects (say, a dry cell and a stapler). Find the 
gravitational pulls on these two objects by 
hanging each from a spring scale. 



1. How does the ratio of the gravitational forces 
compare to the ratio of the masses? 

2. How would you conduct a similar experiment 
to compare the masses of two iron objects to the 
magnetic forces exerted on them by a large 
magnet? 



Comment 

You probabl\ will not be surprised to find that, 
to within your uik rrtainty of measuifment, the 
ratio of gravitational loixcs is the same as the 
ratit) of masses is this n-alK worth doing an 
e.vpcriment to find out, oris the answer obvious 
to begin with? Newton did not think it was 
obvious. He did a series of very precise 
experiments using many dilTeivnt substances 
to find out whether gravitational forte was 
alvvavs pnipjjrtional to inertial mass I'o the 
limits of his pnuision, Nev\1on found tin- 



proportionality,' to hold exactly. (Newlon's re- 
sults have been confirmed to a precision of 
±0.000000001% I 

Newton could offer no explanation from his 
physics as to why the attraction of the earth for 
an object should increase in exact proportion 
to the object's inertial mass. No other forces 
bear such a simple relation to inertia, and this 
remained a complete puzzle for two centuries 
until Einstein related inertia and gravitation 
theoretically. Even before Einstein, Ernst Mach 
made the ingenious suggestion tiiat inertia is 
not the property of an object by itself, but is the 
result of the gravitational forces exerted on an 
object by eveiything else in the universe. 



Experiment 1-10 

CUR\TS OF TRy^JECTORIES 

Picture a ski jumper. He leans forward at the 
top of the slide, grasps the railing on each side, 
and propels himself out onto the track. Streak- 
ing down the trestle, he crouches and gives a 
mighrv' leap at the takeoff lip, soaring up and 
out, over the snow-covered fields far below. The 
hill flashes into view and he lands on its steep 
incline, bobbing to absorb the impact. 

Like so many interesting events, this one 
involves a more complex set of forces and 
motions than you can conveniently deal with in 
the laboratory' at one time. Therefore, concen- 
trate on just one aspect: the fliglit through the 
air. What kind of a path, or trajectory', would a 
ski-jumping flight follow? 

At the moment of projection into the air a 
skier has a certain velocity (that is, a certain 




EXPERIMENTS 



29 



speed in a given direction i, and throughout the 
flight must experience the downward accelera- 
tion due to gra\ity. These are circumstances 
that can be duplicated in the laboratory. To be 
sure, the flight path of an actual ski jumper is 
probably affected by other factors, such as air 
velocity and friction; but you now know that it 
usually pays to begin experiments with a 
simplified approximation that allows you to 
study the effects of a few factors at a time. Thus, 
in this experiment you will launch a steel ball 
from a ramp into the air and try to determine 
the path it follows. 

How to Use 

the Equipment 

If you are assembling the equipment for this 
experiment for the first time, follow the man- 
ufacturer's instructions. 

The apparatus consists primarily of two 
ramps down which you can roll a steel ball. 
Adjust one of the ramps (perhaps with the help 
of a levell so that the ball leaves it horizontalK'. 

Tape a piece of squared graph paper to the 
plotting board with its left-hand edge behind 
the end of the launching ramp. 

To find a path that extends fully across the 
graph paper, release the ball from veirious 
points up the ramp until you find one from 
which the ball falls close to the bottom 
right-hand comer of the plotting board. Mark 
the point of release on the ramp and release the 
ball each time from this point. 

Attach a piece of carbon paper to the impact 
board, with the carbon side facing the ramp. 
Then tape a piece of thin onionskin paper over 
the carbon paper. 

Now when you put the impact board in its 
way, the ball hits it and lea\es a mark that you 
can see through the onionskin paper, and 
automatically records the point of impact 
between ball and board. (Make sure that the 
impact board does not mo\e when the ball hits 
it; steady the board with your hand if neces- 
sary.) Transfer the point to the plotting board 
by making a mark on it just next to the point on 
the impact board. 

Do not hold the ball in your fingers to release 
it; it is impossible to let go of the ball in the 
same way every time. Instead, restrain it with a 
ruler held at a mark on the ramp and release 
the ball by moving the ruler quickly away from 
it down the ramp. 

Try releasing the ball several times (always 
from the same point) for the same setting of the 




Fig. 1-21 



impact board. Do all the impact points exactly 
coincide? 

Repeat this for several positions of the impact 
board to record a number of points on the ball's 
path. Move the board equal distances every 
time and always release the ball from the same 
spot on the ramp. Continue until the ball does 
not hit the impact board any longer. 

Now remove the impact board, release the 
ball once more, and watch carefully to see that 
the ball moves along the points marked on the 
plotting board. 

The curve traced out by your plotted points 
represents the trajectory of the ball. By observ- 
ing the path the ball follows, you have com- 
pleted the first phase of the experiment. 

If you have time, you will find it worthwhile 
to go further and explore some of the proper- 
ties of your trajectory. 

Anal\'zing Your Data 

To help you analyze the trajectory, draw a 
horizontal line on the paper at the level of the 
end of the launching ramp. Then remove the 
paper from the plotting board and draw a 
smooth continuous cu^^'e through the points 
as shown in Fig. 1-22. 



30 



LMT 1 / CONCEPTS OF MOTION 




Fig 1-22 

You already know that a moving object on 
which no net force is acting will move at 
constant speed. There is no appreciable hori- 
zontal fort:e acting on the ball during its fall, so 
you can make an assumption that its horizontal 
motion is at a constant speed. Then, equally 
spaced vertical lines will indicate equal time 
intervals. 

Draw vertical lines through the points on 
your graph. Make the first line coincide with 
the end of the launching ramp. Because ofvour 
plotting procedure, these lines should be 
equally spac^ed. If the horizontal speed of the 
ball is unifoim, these vertical lines are drav\Ti 
thit)ugli positions of the ball separated by equal 
litnc inli'ivals. 

Now consider the vertical distances fallen in 
each timer interval. Measure down from your 
horizontal line the vertical fall to each of your 
plotted points Record your measurements in a 
column Alongside them rtu-oixl the corre- 
sponding hori/.ontal distances measured from 
the tliM vertical line. 



tion (see Sees. 2.5-2.8 in the Text and Experiment 
1-4). Use the data you have just collected to 
decide whether the vertical motion of the ball was 
uniformly accelerated motion. What do you con- 
clude? 

3. Do the horizontal and the vertical motions 
affect each other in any way? 

4. What equation describes the horizontal motion 
in terms of horizontal speed, v. the horizontal 
distance, Ax, and the time of travel, Ar? 

5. What equation describes the vertical motion in 
terms of the distance fallen vertically. A/, the 
vertical acceleration, a„, and the time of travel, 
Ar? 



Try These Yourself 

There are many other things you can do with 
this apparatus. Some of them are suggested by 
the following questions: 

1. What do you expect would happen if you 
repeated the experiment with a glass marble of 
the same size instead of a steel ball? 

2. What will happen if you next tr>' to repeat 
the experiment starting the ball from a different 
point on the ramp? 

3. What do you expect if you use a smaller or 
larger ball, starting always from the same 
reference point on the ramp? 

4. Plot the trajectorv' that results when you use 
a ramp that launches the ball at an angle to the 
horizontal. In what way is this curve similar to 
your first trajectory? 



Experiment l-ll 

PREDICTIOX OF TRAJECTTORIES 

You can predict the landing point of a ball 
launched horizontally from a tabletop at any 
speed. If you know the speed v of the ball as it 
leaves the table, the height of the table above 
the floor, Ay, and a^. you can then use the 
equation for projectile motion to predict where 
on the floor the ball will land. 

You know an equation for horizontal motion: 

A,x = V Af 

and you know an equation for free fall from 
rest: 

Av = ViaJAt)' 



1. What would a graph look like on which you 
plot horizontal distance against time? 

2. Earlier in your work with accelerated motion, 
you learned how to recognize uniform accelera- 



rhe time interval is difficult to measure. 
Besides, in talking about the shape of the path, 
all vou irallv need to know is how A>- n*lates to 
A,v. Since, as you found in the previous 



EXPERIMENTS 



31 



bal I must be 
caj^Ht. while. 
e>t\Ti in air 




Support- 
I Stand 



Fig. 1-23 



experiment, these two equations still apply 
when an object is moving horizontally and 
falling at the same time, you can combine them 
to get an equation relating Ay and Ax. without 
Af appearing at all. You can rewrite the 
equation for horizontal motion as: 



At = 



Ax 



Then you can substitute this expression for At 
into the equation for free fall, obtaining: 



Ay = 1/23^ 



Ax)' 



Thus, the derived equation should describe 
how Ay changes with Ax that is, it should give 
the shape of the trajectory. If you want to know 
how far out from the edge of the table the ball 
will land (Ax), you can calculate it from the 
height of the table ( Ayl, a^, and the balls speed 
V along the table. 



Doing the Experiment 

Find V by measuring with a stopwatch the time 
t that the ball takes to roll a distance d along the 
tabletop. (See Fig. 1-23.1 Be sure to have the ball 
caught as it comes off the end of the table. 
Repeat the measurement a few times, always 
releasing the ball from the same place on the 
ramp, and take the average value of v. 



Measure Ay and then use the equation for 
Ay to calculate Ax. Place a target, a paper cup, 
for example, on the floor at your predicted 
landing spot as shown in Fig. 1-24. How 
confident are you of your prediction? Since it is 
based on measurement, some uncertainty is 
involved. Mark an area around the spot to 
indicate your uncertainty. 

\ow release the ball once more. This time, let 
it roll off the table: if your measurements were 
accurate, it should land on the target as shown 
in Fig. 1-24. 

If the ball actuedly does fall within the range 
of values you have estimated for x. then you 
have supported the assumption on which your 
calculation was based, that vertical cind hori- 
zontal motion are not affected by each other. 



Measuring A x 



/ 



• ^^---Z-:^ ^ 



/ 



-^y 



~7 c^o 



Fig. 1-24 



32 



LMT I / CONCEPTS OF MOTION 






V^ 




Fig. 1-25 The path taken by a cannon ball according to a 
drawing by Ufano (1621). He shows that the same 
horizontal distance can be obtained by two different 
firing angles. Gunners had previously found this by 



experience. What angles give the maximum range? 
What is wrong with the way Ufano drew the trajec- 
tories? 



1. How could you determine the range of a ball 
launched horizontally by a slingshot? 

2. Assume you can throw a baseball 40 m on the 
earth's surface. How far could you throw the same 
ball on the surface of the moon, where the 
acceleration of gravity is one-sixth what it is on 
the surface of the earth? 

3. Will the assumptions made in the equations 
Ax vM and ly %a,(Ar)' hold for a ping- 
pong ball? If the table were 1,000 m above the 
floor, could you still use these equations? Why or 
why not? 



Experiment 1-12 
CEiXTRI FETAL FORCE 

Tin' inotioii of an earth satellite and ol a weight 
swijn^ ai-Mund your head on the end of a string 
are descrihed by the same laws of motion. Both 
aix' acrceleratiiig lowaril the center of their oriiit 
due to the action of ati unhalanced force. 



In the follouing experiment, you can dis- 
cover for yourself how this centripetal force 
depends on the mass of the satellite and on its 
speed and distance from the center. 

Hou- the Apparatus Works 

Your "satellite" is one or more rubber stoppers. 
When you hold the apparatus in both hands 
and swing the stopper around your head, you 
can measure the centripetal force on it with a 
spring scale at the base of the stick. The scale 
should ivatl in newlons; otherwise its readings 
should be conx-erted to newlons. 

You can change the length of the string to 
vai^' the radius fl of the circular orbit. Tie on 
more stoppers to varv the satellite mass m. 

The best way to set the frequencN- / is to 
swing the apparatus in time with some 
periodic sound from a metronome or an 
earphone attached to a blinky. Keep the rate 
constant by adjusting the swinging until you 
see the stopper cross the same point in the 
n)om e\erv tick. 



EXPERIMENTS 



33 




Fig. 1-26 



Hold the stick vertically and have as little 
motion at the top as possible, since this would 
change the radius. Because the stretch of the 
spring scale also alters the radius, it is helpful 
to have a maricer (knot or piece of tape) on the 
string. You can move the spring scale up or 
down slightly to keep the marker in the same 
place. 

Doing the Experiment 

The object of the experiment is to find out how 
the force F, read on the spring scale, varies with 
m, with /, and with R . 

You should only change one of these three 
quantities at a time so that you can investigate 
the effect of each quantity independently of the 
others. Either double or triple m, f, and R (or 
halve them, and so on, if you started with large 
values). 

Two or three different values should be 
enough in each case. Make a table and clearly 
record your numbers in it. 



Experiment 1-13 
CENTRIPETAL FORCE ON A 
TURNTABLE 

You may have had the experience of spinning 
around on an amusement park contraption 
knov\Ti as the Whirling Platter. The riders seat 
themselves at various places on a large, flat, 
polished wooden turntable about 12 m in 
diameter. The turntable gradually rotates faster 
and faster until everyone (except for the person 
at the center of the table) has slid off. The 
people at the edge are the first to go. Why do 
the people slide off? 

Unfortunately, you probably do not have a 
Whirling Platter in your classroom, but vou do 
have a Masonite disk that fits on a turntable. 
The object of this experiment is to predict the 
maximum radius at which an object can be 
placed on the rotating turntable without sliding 
off. 

If you do this under a vciriety of conditions, 
you will see for yourself how forces act in 
circular motion. 

Before you begin, be sure you have studied 
Sec. 4-6 in your text where you learned that the 
centripetal force needed to hold a rider in a 
circular path is given by F = mv^/R . 

Studying 
Centripetal Force 

For these experiments, it is more convenient to 
rewrite the formula F = mv^/R in terms of the 
frequency/. This is because/ can be measured 
more easily than v. You can rewrite the formula 
as follows: 



distance traveled number of revolutions 



in one revolution 
2nR x/ 



per second 



Substituting this expression for v in the formula 
for centripetal force gives: 



1. How do changes in m affect f when R and f are 
kept constant? Write a formula that states this 
relationship. 

2. How do changes in f affect F when m and /? are 
kept constant? Write a formula to express this 
relationship. 

3. What is the effect of R on f ? 

4. Now, put m, f, and R all together in a single 
formula for centripetal force, F. How does your 
formula compare with the expression derived in 
Sec. 4-6 of the Text? 



R 

_ 4iT^mRj^ 
R 

= ATT^mRp 



You can measure all the quantities in this 
equation. 



34 



LMT 1 / CONCEPTS OF MOTION 




Fig 1-27 



Friction on a Rotating Disk 

For objects on a rotating disk, the centripetal 
foix;e is provided by friction. On a frictionless 
disk, there could be no such centripetal force. 
As you can see from the equation just derived, 
the centripetal acceleration is proportional to fl 
and to p. Since the frequency of/ is the same 
for any object moving around on a turntable, 
then the c(!ntripetal acceleration is directly 
proportional to R, the distance from the center. 
The further an object is from the center of the 
turntable, therefore, the greater the centripetal 
forc:e must be to keep it in a circular path. 



You can measure the maximum force F^ax 
that friction can provide on the object, measure 
the mass of the object, and then calculate the 
maximum distance from the center flmax that 
the object can be without sliding off. Solving the 
centripetal force equation for R gives: 






ATT^mp 



Use a spring scale to measure the force needed 
to make some object (of mass m from 02 kg to 
1.0 kg) start to slide across the motionless disk. 
This will be a measure of the maximum friction 
force that the disk can exert on the object. 

Make a chalk marie on the turntable and time 
it (say, for 100 sec) or accept the marked value 
of revolutions per minute and calculate the 
frequency in hertz (Hz). 

Make your predictions of Rmax for turntable 
frequencies of 33 revolutions per minute (rpm), 
45 rpm, and 78 rpm. 

Then try the experiment! 



1. How great is the percentage difference be- 
tween prediction and experiment for each turn- 
table frequency? Is this reasonable agreement? 

2. What effect would decreasing the mass have 
on the predicted value of /?? Careful! Decreasing 
the mass has an effect on F also. Check your 
answer by doing an experiment. 

3. What is the smallest radius in which you can 
turn a car if you are moving 100 kmtir and the 
friction force between tires and road is one-third 
the weight of the car? (Remember that weight is 
equal to a^ x m.) 



by John Hart 



NO, MO, STliPiD- . THE 
OTHEff ENP. 




nl Ji4wi Had tmt ri«U lalatrcl***. !■>• . 



ACTIVITIES 



35 



ACTIVITIES 



CHECKER SNAPPING 



Stack several checkers. Put another checker on 
the table and snap it into the stack. On the basis 
of Newton's first law, can you explain what 
happened? 



BEAKER AND HAMMER 

Place a glass beaker half full of water on top of a 
pile of three wooden blocks. Three quick back- 
and-forth swipes (NOT FOUR!! of a hammer on 
the blocks leave the beaker sitting on the table. 



PULLS AND JERKS 



~r::^t~^ 



Fig. 1-28 



Hang a weight (such as 
a heavy wooden block) 
by a string that just 
barely supports it, and 
tie another identical 
string below the weight. 
A slow, steady pull on 
the string below the 
weight breaks the string 
above the weight. A 
quick jerk breaks it 
below the weight. Why? 



EXPERIENCING NEWTON'S 
SECOND LAW 

One way for you to get the feel of Newton's 
second law is actually to pull an object with a 
constant force. Load a cart with a mass of 
several kilograms. Attach one end of a long 
rubber band to the cart and, pulling on the 
other end, move at such a speed that the 



rubber band is maintained at a constant length, 
for example, 70 cm. Holding a meter stick above 
the band with its 0-cm end in your hand uill 
help you to keep the length constant. 

The acceleration vvill be very apparent to the 
person applying the force. Vary the mass on the 
cart and the number of rubber bands (in 
parallel) to investigate the relationship between 
F, m, and a. 



MAKE ONE OF THESE 
ACCELEROMETERS 

An accelerometer is a device that measures 
acceleration. Actually, anything that has mass 
could be used for an accelerometer. Because 
you have mass, you were acting as an acceler- 
ometer the last time you lurched forward in the 
seat of a car as the brakes were applied. 
With a knowledge of Newton's laws and certain 
information about yourself, anybody who mea- 
sured how far you leaned forward and how 
tense your muscles were would get a good idea 
of the magnitude and direction of the accelera- 
tion that you were undergoing; but it would be 
complicated. 

Here are four accelerometers of a much 
simpler kind. With a little practice, you can 
leam to read accelerations from them directly, 
without making any difficult calculations. 

A. The Liquid-Surface 
Accelerometer 

This dexice is a hollow, flat plastic container 
partly filled with a colored liquid. When it is not 
being accelerated, the liquid surface is horizon- 
tal, as shovvn by the dotted line in Fig. 1-29. But 
when it is accelerated toward the left (as 



B.C. 



by John Hart 




By permission o£ John Hart and Field Enterprises, Inc 



36 



UNIT 1 / CONCEPTS OF MOTION 



acccler-atJon 




Fig 1-29 



shown) with a uniform acceleration a, the 
surface becomes tilted. The level of the liquid 
rises a distance h above its normal position at 
one end of the accelerometer and falls the same 
distance at the other end. The greater the 
acceleration, the more steeply the surface of 
the liquid is slanted. This means that the slope 
of the surface is a measure of the magnitude of 
the acceleration a. 

The length of the accelerometer is 2/, as 
shown in Fig 1-29. So the slope of the surface 
may be found by: 

, vertical distance 

slope = 

horizontal distance 

2/ 

I 

Theory gives you a very simple relationship 
between this slope and the acceleration a; 

, h a 

slope = — = — 

/ a« 

Notice what this equation tells you. It says 
that if the instrument is accelerating in the 
direction shown with just a, (one common way 
to say this is that it has a "one-G acceleration" 
— the acceleration of gravity), then the slope of 
the surface is just 1; that is, /i = / and the 



whit^ f»per on 
bacJt of oeJI -, 

J V 



y^^^rie^ , I'l-ont of eel 

— — - ^-••■^SU — ' 



E. 



i-1 Ac^ 



Atc^ler^tcel liquid aurfAcc 
Fig 1-30 



surface makes a 45° angle with its normal, 
horizontal direction. If it is accelerating with 
Viag, then the slope will be Vz; that ish = V2I. In 
the same way, if /j = V4/, then a = Via,, and so 
on with any acceleration you care to measure. 
To measure h, stick a piece of centimeter 
tape on the front surface of the accelerometer 
as shown in Fig. 1-30. Then stick a piece of 
white paper or tape to the back of the 
instrument to make it easier to read the level of 
the liquid. Solving the equation abo\'e for a 
gives: 



a =ag X - 



Since a^ is very close to 9.8 m/sec^ at the 
earth's surface, if you place the scale 9^ scale 
units fiDm the center, you can read accelera- 
tions directly in meters per second-. For 
example, if you stick a centimeter tape just 9^ 
cm from the center of the liquid surface, 1 cm 
on the scale is equivalent to an acceleration of 1 
m/sec^. 

Calibration of the Accelerometer 

You do not have to trust blindly the theoiy 
just mentioned. You can test it for yourself. 
Does the accelerometer really measure acceler- 
ations directly in meters per second-? Strobo- 
scopic methods give you an independent 
check on the correctness of the prediction. 

Set the accelerometer on a dynamics cart 
and arrange strings, pulleys, and masses as you 
did in Experiment 1-9 to give the cart a uniform 
acceleration on a long tabletop. Put a block of 
wood at the end of the cart's path to stop it. 
Make sure that the accelerometer is fastened 
firmly enough so that it will not fly off the cart 
when it stops suddenly. Make the string as long 
as you can, so that you use the entire length of 
the table. 

Give the cart a wide range of accelerations by 
hanging different weights from the string. Use a 
stroboscope to record each motion. To mea- 
sure the accelerations from your stixjbe rec- 
ords, plot t^ against d, as you did in Elxperiment 
1-5. (What relationship did Galileo disco\-er 
between d/t'- and the acceleration?) Or use the 
method of analysis you used in Experiment 1-9. 

Compare your stroboscopic measurements 
with the readings on the accelerometer during 
each motion. It takes some cleverness to read 
the accelerometer accurately, particulariy near 
the end of a high-acceleration run One way is 
to Ua\f se\-eral students along the table observe 
the reading as the cart goes by: use the ax-erage 



ACTIVITIES 



37 



B.C 






of their reports. If you are using a xenon strobe, 
of course, the readings on the accelerometer 
will be visible in the photograph; this is 
probably the most accurate method. 

Plot the accelerometer readings against the 
stroboscopicaUy measured accelerations. This 
graph is called a calibration curve. If the two 
methods agree perfectly, the graph uill be a 
straight line through the origin at a 45° angle to 
each axis. If your curve has some other shape, 
you can use it to convert "accelerometer 
readings" to "accelerations" if you are udlling to 
assume that your strobe measurements are 
more accurate than the accelerometer. (If you 
are not willing, what can you do?) 

B. Automobile Accelerometer. I 

With a liquid-surface accelerometer mounted 
on the front-to-back line of a car, you can 
measure the magnitude of acceleration along 
its path. Here is a modification of the liquid- 
surface design that you can build for yourself. 
Bend a small glass tube (about 30 cm long) into 
a U-shape, as shown in Fig. 1-31. 



same reasoning as before. The tvvo vertical arms 
should be at least three-fourths as long as the 
horizontal arm (to avoid splashing out the 
liquid during a quick stop). Attach a scale to 
one of the vertical arms, as shown. Holding the 
long arm horizontally, pour colored water into 
the tube until the water level in the arm comes 
up to the zero mark. How can you be sure the 
long arm is horizontal? 

To mount your accelerometer in a car, fasten 
the tube with staples (carefully) to a piece of 
plywood or cardboard a little bigger than the 
U-tube. To reduce the hazard from broken glass 
while you do this, cover all but the scale (and 
the arm by it) with cloth or cardboard, but leave 
both ends open. It is essential that the acceler- 
ometer be horizontal if its readings cire to be 
accurate. When you are measuring acceleration 
in a car, be sure the road is level. Otherwise, 
you will be reading the tilt of the car as well as 
its acceleration. When a car accelerates, in any 
direction, it tends to tilt on the suspension. 
This will introduce error in the accelerometer 
readings. Can you think of a way to avoid this 
kind of error? 



0-4- 

oi 
o-z 



-0-2 f..' 

-o.5_|:; 

-O 4 ."^^ 

^ 



acxz^^r^ttcx^ 



Fig. 1-31 



j v .;^:^:-;;^!::;^ 



Calibration is easiest if you make the long 
horizontal section of the tube just 10 cm long; 
then each 5 mm on a vertical arm represents an 
acceleration of O.lg - (about) 1 m/sec^, by the 



C. Automobile Accelerometer. II 

An accelerometer that is more directly related 
to F = ma can be made from a 1-kg cart and a 
spring scale marked in newtons. The spring 
scale is attached between a wood frame and 
the cart as shown in Fig. 1-32. If the frame is 
kept level, the acceleration of the system can be 
read directly from the spring scale, since 1 N 




Fig. 1-32 



38 



UNIT 1 / CONCEPTS OF MO'I ION 



of foixre on the 1-kg mass indicates an accelera- 
tion of 1 m/sec^ (Instead of a cart, any 1-kg 
object can be used on a layer of low-friction 
plastic beads.) 

D. Damped-Pendulum 
Acceleroineter 

One advantage of liquid-surface accelerometers 
is that it is easy to put a scale on them and read 
accelerations directly frtjm the instrument. 
They have a drawback, though; they give only 
the component of acceleration that is parallel 
to their horizontal side. If you accelerate one at 
right angles to its axis, it does not register any 
acceleration at all. Also, if you do not know the 
direction of the acceleration, you have to use 
trial-and-error methods to find it with the 
accelerometers discussed up to this point. 

A damped-pendulum accelerometer, on the 
other hand, indicates the direction of any 
horizontal acceleration; it also gives the mag- 
nitude, although less directly than the previous 
instajnients do 

Hang a small metal pendulum bob by a short 
string fastened to the middle of the lid of a 1-L 
wide-mouthed jar as shown on the left-hand 
side of the sketch in Fig. 1-33. Fill the jar with 
water and screw the lid on tight. For any 
position of the pendulum, the angle that it 
makes with the vertical depends upon your 
position. What would you see, for example, if 
the bottle were accelerating straight toward 
you? Away from you? Along a table with you 
standing at the side? (Careful: This last ques- 
tion is trickier than it looks.) 





Fig 1 33 

To make a fascinating \ariation on the 
damped-pendulum accelerometer, simplv re- 
place Ihr prnduUiin bob with a <()ik aiul tuni 



the bottle upside down as shown on the 
right-hand side of the sketch. If you have 
punched a hole in the bottle lid to fasten the 
string, you can prevent leakage with the use of 
sealing wax, paraffin, or tape. 

This accelerometer will do just the opposite 
from what you would expect. The explanation 
of this odd behavior is a little be\'ond the scope 
of this course; it is thoroughly explained in 
The Physics Teacher, Vol. 2, No. 4 (April, 1964), 
p. 176. 



PROJECTILE MOTION 
DEMONSTRATION 

Here is a simple way to demonstrate projectile 
motion. Place one coin near the edge of a table. 
Place an identical coin on the table and snap it 
with your finger so that it flies off the table, just 
ticking the first coin enough that it falls almost 
straight down from the edge of the table. The 
fact that you hear only a single ring as both 
coins hit shows that both coins took the same 
time to fall to the floor from the table. 
Incidentally, do the coins have to be identical? 
Trv different ones. 




Fig. 1-34 



SPEED OF A STRE/iM 
OF WATER 

You can use the principles of projectile motion 
to calculate the speed of a stream of water 
issuing from a horizontal nozzle. Measure the 
vertical distance Ay from the nozzle to the 
ground, and the horizontal distance A,v from 
the nozzle to where the water hits the ground 
I'se the equation relating A.v and A\ that 
was deri\'ed in Experiment 1-11, soKing it forv: 



y = V2a, 






ACTIVITIES 



39 



and 



v2 = V3a,LM! 

y 



V = A;c 



2 Ay 



The quantities on the right can all be measured 
and used to compute v. 




Fig. 1-35 



PHOTOGRAPHING A 
WATERDROP PARABOLA 

Using an electronic strobe light, a doorbell 
timer, and water from a faucet, you can 
photograph a waterdrop parabola. The princi- 
ple of independence of vertical and horizontal 
motions will be clearly evident in your picture. 

Remove the wooden block from the timer. Fit 
an "eye dropper" barrel in one end of some 
tubing and fit the other end of the tubing onto a 
water faucet. (Instead of the timer you can use a 
doorbell uathout the bell.) Place the tube 
through which the water runs under the 
clapper so that the tube is given a steady series 
of sharp taps. This has the effect of breaking the 
stream of water into separate, equally spaced 
drops (see Fig. 1-36). 

To get more striking power, run the vibrator 
from a variable transformer (Variac) connected 
to the 110-volt ac, gradually increasing the 
Variac from zero just to the place where the 
striker vibrates against the tubing. Adjust the 
water flow through the tube and eye dropper 
nozzle. By \aevvang the drops uath the xenon 
strobe set at the same frequency as the timer, a 
parabola of motionless drops is seen. A spot- 
light and disk strobe can be used instead of the 




Fig. 1-36 



electronic strobe light, but it is more difficult to 
match the frequencies of vibrator and strobe. 
The best photos are made by lighting the 
parabola from the side (that is, putting the light 
source in the plane of the parabola). Figure 1-36 
was made in that way. With firont lighting, the 
shadow of the parabola can be projected onto 
graph paper for more precise measurement. 

Some heating of the doorbell coil results, so 
the striker should not be run continuously for 
long periods of time. 



BALLISTIC CART 
PROJECTILES 

Fire a projectile straight up from a cart or toy 
locomotive (as shouTi in Fig. 1-37) that is rolling 
across the floor with nearly uniform velocity. 
You can use a commercial device called a 
ballistic cart or make one yourself. A spring- 
loaded piston fires a steel ball when you pull a 
string attached to a trigger pin. Use the 
electronic strobe to photograph the path of the 
ball. 




Fig. 1-37 



40 



LMT 1 / CONCEI'TS Ol MOI ION 




Fig. 1-38 

Projectile trajectories of any object thrown 
into the air can be photographed using the 
electronic strobe and Polaroid Land camera. By 
fastening the camera (securely!) to a pair of 
carts, you can photograph the action from a 
moving frame of reference. 



MOTIOX IX A ROTATING 
RKFKKENCK FR^IME 

Here are three ways you can show how a 
moving object would appear in a rotating 
reference frame. 

METHOD I 

Attach a piece of paper to a phonograph 
turntable. Draw a line across the paper as the 
turntable is turning (see Fig. 1-39), using as a 
guide a meter stick supported on books at 
either side of the turntable. The line should be 
drawn at a constant speed. 

/ 




Fig 1-39 

METHOD U 

Place a Polai'oid I^and camera on the turn- 
table on the floor and let a Unifonn Motor 
Device (UMD) mn along the edge of ;i table. 



with a fliishlight bulb on a pencil taped to the 
UMD so that it sticks out over the edge of the 
table. (See Fig. 1-40.) 



■ light 50urc« 



r-counter-weighf 



uniform 
motion d»vic* 



turntoble- 



Fig 1-40 




METHOD HI 

How would an elliptical path appear if you 
were to view it fix)m a rotating reference 
system? You can find out by placing a Polaroid 
Land camera on a turntable on the floor, with 
the camera aimed upwards. (See Fig. 1-41.) For 
a pendulum, hang a flashlight bulb and an AA 
dry cell. Make the pendulum long enough so 
that the light is about 120 cm from the camera 
lens. 



"V-^yjiV^ 




i"ur»>rot)e 



■d^^- 




Fig 1-41 



With the lights out, give the pendulum a 
swing so that it swings in an elliptical path 
Hold the shutter open while the turntable 
makes one rex-olution. You can get an indica- 
tion of how fast the pendulum mox-es at 
different points in its swing by using a motor 
strob«> in fixrnt of the camera, or by hangiiig a 
blink\ 



ACTIVITIES 



41 



PENNY AND COAT HANGER 

Bend a coat hanger into the shape shown in 
Fig. 1-42. Bend the end of the hook sHghtly with 
a pair of pliers so that it points to where the 
fmger supports the hanger. File the end of the 
hook flat. Balance a penny on the hook. Move 
your finger back and forth so that the hanger 



•Turns O" -firiQCr here 




Fig. 1-42 



(and balanced penny) starts swinging like a 
pendulum. Some practice vAW enable you to 
swing the hanger in a vertical circle, or around 
your head, and still keep the penny on the 
hook. The centripetal force provided by the 
hanger keeps the penny from flying off on a 
straight-line path. Some people have done this 
demonstration successfully with a pile of as 
many as five pennies at once. 



MEASURING UNKNOWN 
FREQUENCIES 

Use a calibrated electronic stroboscope or a 
hand stroboscope and stopwatch to measure 
the frequencies of various motions. Look for 
such examples as an electric fan, a doorbell 
clapper, and a banjo string. 

On page 111 of the text you will find tables of 
frequencies of rotating objects. Notice the 
enormous range of frequencies listed, from the 
electron in the hydrogen atom to the rotation of 
the Milky Way galaxy. 



42 



UNIT 1 / CONCEPTS OF MOTION 



FILM LOOP NOTES 



Film lyoop 1 

ACCELERATION CAUSED BY 
GRAVTI'V. I 

A bowling ball in free fall was filmed in real time 
and in slow motion. Using the slow-motion se- 
quence, you can measure the acceleration of 
the ball as caused by gravity. This film was 
exposed at 3,900 frames/sec and is projected at 
about 18 frames/sec; therefore, the slow-motion 
factor is 3,900/18, or about 217. However, your 
projector may not run at exactly 18 frames/sec. 
To calibrate your projector, time the length of 
the entire film containing 3,331 frames. (Use the 
yellow circle as the zero frame.) 

To find the acceleration of the falling body 
using the definition 

, .. change in speed 

acceleration = ° *- 

time interval 

you need to know the instantaneous speed at 
two different times. You cannot directly mea- 
sure instantaneous speed from the film, but 
you can determine the average speed during 
small intervals. Suppose the speed increases 
steadily, as it does for freely falling bodies. 
During the fir-st half of any time interval, the 
instantaneous speed is less than the a\erage 
speed; during the second half of the interval, 
the speed is greater than average. Therefore, for 
uniformly accelerated motion, the average 
speed V;,^ for the interval is the same as the 
instantaneous speed at the midtime of the 
interval. 

If you find the instantaneous speed at the 
midtimes of each of two intervals, you can 
calculate the acceleration a from 



to move through each 0.5-m interval. Repeat 
these measurements at least once and then 
find the average times. Use the slow-motion 
factor to convert these times to real seconds; 
then, calculate the two values of v^^. Finally, 
calculate the acceleration a^. 

This film was made in Montreal, Canada, 
wliere the acceleration caused by gravity, 
rounded off to ±1%, is 9.8 m/sec^. Try to decide 
from the internal consistency of your data (the 
repeatability of your time measurements) how 
precisely you should write your result. 




Fig 1-43 



t, -t, 



where v , and v. are the average speeds during 
the two intervals, and where /, and /^ are the 
midtimes of these intervals. 

I\vo inteivals 0.5 m in length are shov\-n in 
the film I he ball falls 1 m before reaching the 
first marked interval, so it has some initial 
speed when it crosses the first line. Using a 
watch or clock with a sweep second hand, time 
the balls motion and n-cord the times at which 
the ball ctx)sses each of the four lines. You can 
make measurements using either the bottom 
edge of the ball or the top edge. With this 
information, you can detemiine the time (in 
apparent seconds! between the midtimes of the 
two intenals and the lime HMjuirfd for the ball 



Film Loop 2 

ACCELERATIOX CAUSED BV 
C;RA\1TV. II 

A bowling ball in free fall was filmed in slow 
motion. The film was exposed at 3,415 frames/ 
sec and it is projected at about 18 frames/sec. 
You can calibrate your projector by timing the 
length of the entire film. 3.753 frames. (Use the 
yellow circle as a reference mark.) 

If the ball starts from rest and steadily 
acquires a speed v after falling through a 
distance d. the change in speed A\ is \ - 0. or 
v. and the average s{)eed is: 

- + V , , 



The time required to fall this distance is: 



FILM LOOP NOTES 



43 






2d 



The acceleration a is given by: 



_ change of speed _ Av _ v 
time interval Af 2d/v 



2d 



Thus, if you know the instantaneous speed v of 
the falling body at a distance d below the 
starting point, you can find the acceleration. Of 
course, you cannot directly measure the in- 
stantaneous speed but only a\'erage speed over 
the interval. For a small interval, however, vou 
can make the approximation that the average 
speed is the instantaneous speed at the 
midpoint of the interval. (The average speed is 
the instantaneous speed at the midtzme, not 
the midpoint; but the error is small if you use a 
short enough interval.) 

In the film, small intervals of 20 cm are 
centered on positions 1 m, 2 m, 3 m, and 4 m 
below the starting point. Determine four aver- 
age speeds by timing the ball's motion across 
the 20-cm intervals. Repeat the measurements 
several times and axerage out errors of mea- 
surement. Convert your measured times into 
real times using the slow-motion factor. Com- 
pute the speeds, in meters per second, and 
compute the value of vV2d for each value of d. 

Make a table of calculated values of a, in 
order of increasing values of d. Is there any 
evidence for a systematic trend in the \alues? 
Would you expect any such trend? State the 
results by giving an average value of the 
acceleration and an estimate of the possible 
error. This error estimate is a matter of 
judgment based on the consistency of your 
four measured values of the acceleration. 



Film Loop 3 
\TCTOR ADDITIOX: 
\TLOCITY OF A BOAT 

A motorboat was photographed from a bridge 
in this film. The boat heads upstream, then 
downstream, then directly across stream, and 
at an angle across the stream. The operator of 
the boat tried to keep the throttle at a constant 
setting to maintain a steady speed relative to 
the water. Your task is to find out if he 
succeeded. 

First project the film on graph paper and 
mark the lines along which the boat's image 




Fig. 1-44 This photograph was tal<en from one bank of 
the stream. It shows the motorboat heading across the 
stream and the camera filming this loop fixed on the 
scaffolding on the bridge. 

moves. You may need to use the reference 
crosses on the maricers. Then measure speeds 
by timing the motion through some predeter- 
mined number of squares. Repeat each mea- 
surement several times, and use the average 
times to calculate speeds. Express all speeds in 
the same unit, such as "squares per second " (or 
"squares per centimeter " where centimeter 
refers to measured separations between marks 
on the moving paper of a dragstrip recorder). 
Why is there no need to convert the speeds to 
meters per second? Why is it a good idea to use 
a large distance between the timing marks on 
the graph paper? 

The head-to-taU method of adding vectors is 
illustrated in Fig. 1-45. Since velocity is a vector 
with both magnitude and direction, you can 
study vector addition by using velocity vectors. 
An easy way of keeping track of the velocity 
vectors is by using subscripts: 

v^ velocity of boat relative to earth 
Vb'v velocity of boat relative to water 
V\vE velocity of water relative to earth 
Then, , , 

^'be =Vbw +Vwe 




Fig. 1-45 The head-to-tail method of adding vectors. 



44 



UNIT I / CONCEPTS Ol MOITOV 



For each heading of the boat, a vector 
diagram can he drawn by drawing the velocities 
to scale. A suggested procedure is to record 
data (direction and speed) for each of the five 
scenes in the film, and then draw the vector 
diagram for each. 

Scene 1 

Two blocks of wood are dropped overboard. 
Time the blocks. Find the speed of the river, the 
magnitude of vJTk- 

Scene 2 

The boat heads upstream. Measure TJ^K' then 
find v^v using a vector diagram similar to Fig. 
1-46. 



BW 



WE. 



S: 



Fig 1-46 



^3E = ^BW ^ V 



Scene 3 

The boat heads downstream. Measure v^k- 
then find v^^ using a vector diagram. 

Scene 4 

The boat heads across stream and drifts 
downstr-eam. Measure the speed of the boat 
and the direction of its path to find v^y. Also 
measui-e the direction of vj^v the direction the 
boat points. One way to record data is to use a 



9o* 



heading a-' -5 _^ 



ISO 



Fig 1 47 




set of axes with the 0°- 180° axis passing 
through the markers anchored in the river. A 
diagram, such as Fig. 1-47, will help you record 
and analyze your measurements. i.N'ote that the 
numbers in the diagram are deliberately not 
correct.) Your vector diagram should be some- 
thing like Fig. 1-48. 




270* 



Fig. 1-48 

Scene 5 

The boat heads upstream at an angle, but 
moves directly across stream. Again find a value 
for v^v. 

Checking Your Work 

(a) How well do the four \alues of the 
magnitude of V^^,^, agree with each other? Can 
you suggest reasons for any discrepancies? 
(b) From Scene 4, you can calculate the heading 
of the boat. How well does this angle agree v\ith 
the observed boat heading? (c) In Scene 5, you 
determined a direction for v^^. Does this angle 
agree with the observed boat heading? 



Film Loop 4 

A .MATTER or RELATI\X 

MOTION 

Two carts of equal mass collide in this film. 
Three sequences labeled E\ent A, E\ent B, and 
Event C are shown. Stop the projector after 
each event and describe these ex-ents in words, 
as they appear to you. View the loop now, 
before reading further. 

Even though Ewnts ,\. B, and C are visibly 
difterent to the observer, in each the carls 
interact similarly. The laws of motion apply for 
each case. Thus, these e\'ents could be the 
same event observed fixim different reference 
frames. They are closely similar exents photo- 
graphed from difTen'iH frames of reference, as 
you see after the initial s«'(}uence of the film. 

The three extents are photographed by a 
camera on a cart, which is on a second ramp 
parallel to the one on which the colliding carts 
move, rhe camera is \'our frame of reference, 



FILM LOOP NOTES 



45 



your coordinate system. This frame of reference 
may or may not be in motion with respect to 
the ramp. As photographed, the three events 
appear to be quite different. Do such concepts 
as position and velocity have a meaning 
independently of a frame of reference, or do 
they take on a precise meaning only when a 
frame of reference is specified? Are these three 
events really similar events, viewed from dif- 
ferent frames of reference? 




You might think that the question of which 
cart is in motion is resolved by sequences at the 
end of the film in which an experimenter, 
Franklin Miller of Kenyon College, stands near 
the ramp to provide a reference object. Other 
visual clues may already have provided this 
information. The events may appear different 
when this reference object is present. But is 
this fi^ed frame of reference any more funda- 
mental than one of the moving frames of 
reference? Fixed relative to what? Is there a 
"completely" fixed frame of reference? 

If you have studied the concept of 
momentum, you can also consider each of 
these three events from the standpoint of 
momentum conservation. Does the total 
momentum depend on the frame of reference? 
Does it seem reasonable to assume that the 
carts would have the same mass in all the 
frames of reference used in the film? 



If it be true that the impetus with which the 
ship moves remains indelibly impressed in the 
stone after it is let fall from the mast; and if it be 
fijrther true that this motion brings no impedi- 
ment or retardment to the motion directly 
downwards natural to the stone, then there 
ought to ensue an effect of a very wondrous 
nature. Suppose a ship stands still, and the time 
of the falling of a stone from the mast's round 
top to the deck is two beats of the pulse. Then 
afterwards have the ship under sail and let the 
same stone depart from the same place. Accord- 
ing to what has been premised, it shall take up 
the time of two pulses in its fall, in which time 
the ship m// have gone, say, twenty yards. The 
true motion of the stone will then be a 
transverse line (i.e., a curved line in the vertical 
plane), considerably longer than the ftrst 
straight and perpendicular line, the height of the 
mast, and yet nevertheless the stone will have 
passed it in the same time. Increase the ship's 
velocity as much as you will, the falling stone 
shall describe its transverse lines still longer 
and longer and yet shall pass them all in those 
selfsame two pulses. 

In the film a ball is dropped three times: 



7 




Film Loop 5 

GALILEAN RELATinTY: BALL 

DROPPED FROM MAST OF SHIP 

This film is a partial actualization of an 
experiment described by Sagredo in Galileo's 
Two New Sciences: 



Fig. 1-49 

Scene 1 

The ball is dropped from the mast. As in 
Galileo's discussion, the ball continues to move 
horizontally with the boat's velocity, and also it 
falls vertically relative to the mast. 



46 



Ll^lT 1 / CONCEPTS OF MOIION 



Scene 2 

Ihc ball is tipped off a stationary support as 
the boat goes by. It has no forward velocity, and 
it falls vertically relative to the water surface. 

Scene 3 

Ihe ball is picked up and held briefly before 
being rtjieased. 

Ihe ship and earth are frames of reference in 
constant relative motion. Each of the three 
events can be described as viewed in either 
frame of reference. The laws of motion apply for 
all six descriptions. The fact that the laws of 
motion work for both frames of reference, one 
moving at constant velocity with respect to the 
other, is what is meant by Galilean relativity. 
(The positions and velocities are relative to the 
frame of reference, but the laws of motion are 
not. A "relativity" principle also states what is 
not relative.) 

Scene 1 can be described from the boat frame 
as follows: "A ball, initially at rest, is released. It 
accelerates downward at 9.8 m/sec^ and strikes 
a point directly beneath the starting point." 
Scene 1 described differently from the earth 
frame is: "A ball is projected horizontally 
toward the left; its path is a jiarabola and it 
strikes a point below and to the left of the 
starting point.' 

To test your understanding of Galilean rela- 
tivity, you should describe the following: Scene 
2 from the boat frame; Scene 2 from the earth 
frame; Scene 3 from the boat frame; Scene 3 
from th(; earth frame. 



Film Loop 6 

C;ALILE/1N RELATI\^ TV: OiyECT 

»KOI*I>i:i> FROM AIRCRAFT 

A Cessna 150 air-craft 7 m long is mo\ing about 
30 m/sec at an altitude of about 60 m. The 
action is filmed from the ground as a flare is 
dn)pped fn)m the ainTaft. Scene 1 shows part 
of the flare's motion; Scene 2, shot fi-om a 
gn>ater distance, shows several flaivs di-oppiiig 
into a lake; Scene 3 shows the \erlical motion 
viewed head-on. Certain frames of the film are 
"fix)zen" to allow measurtMuents. The time 
interval between frt*eze frames is always the 
same. 

Seen fn)ni the earth's frame of reference, tin- 
motion is that of a projectile whose oiiginal 
velocity is the plane's velocity. If gravity is Ihe 
only force acting on the flan', its motion should 
he a parabola (('an \'ou check this 'i Relati\t' to 



the airplane, the motion is that of a body falling 
freely from rest. In the frame of reference of the 
plane, the motion is vertically downward. 




Fig 1-50 

The plane is flying approximately at uniform 
speed in a straight line, but its path is not 
necessarily a horizontal line. The flare starts 
with the plane's velocity, in both magnitude 
and in direction. Since it also falls freely under 
the action of gravity, you expect the flare's 
downward displacement below the plane to be 
d = Vzat'. The trouble is that \ou cannot be 
sure that the first freeze frame occurs at the 
very instant the flare is dropped. However, 
ther« is a way of getting around this difficulty. 
Suppose a time B has elapsed bet\veen the 
release of the flare and the first freeze frame. 
This time must be added to each of the freeze 
frame times (conveniently measured from the 
first freeze frame) and so you would have 

d = '/zai/ -t-Bi- 

To see if the flare follows an equation such as 
this, take the square root of each side: 

\/d~= (constantly -t-Bl 
U you plot Vdagainst f, you expect a straight 
line. Moreover, if B = 0, this straight line will 
also pass thix)ugh the 0-0 point 

Suggested Measurements 

(a) Verticiil Motion 

Project S(«MU' 1 on paper .At each freeze 
frame, when the motion on the screen is 
slopped brietlx , mark the positions of the flare 
and of the aircraft cockpit. Measure the dis- 
placement d of the flare below the plane. I'se 
any conx-enient units. The limes can be taken as 
inlegei-s, t = 0. 1. 2 designating successive 



FILM LOOP NOTES 



47 



freeze frames. Plot Vd~versus t. Is the graph a 
straight line? What would be the effect of air 
resistance, and how would this show up in 
your graph? Can you detect any signs of this? 
Does the graph pass through the 0-0 point? 
Analyze Scene 2 in the same way. 

(b) Horizontal Motion 

Use another piece of graph paper with time 
(in intervals) plotted horizontally and dis- 
placement (in squares) plotted vertically. Using 
measurements from your record of the flare's 
path, make a graph of the two motions in Scene 
2. What are the effects of air resistance on the 
horizontal motion? the vertical motion? Ex- 
plain your findings between the effect of air 
friction on the horizontal and vertical motions. 

(c) Acceleration Caused by Gravity 

The "constant" in your equation, d = 
(constant) (f -f- B)^ is Via; this is the slope of 
the straight-line graph obtained in part (a). The 
square of the slope gives Vza. Therefore, the 
acceleration is twice the square of the slope. In 
this way you can obtain the acceleration in 
squares per (interval)^. To convert your accel- 
eration into meters per second^, you can esti- 
mate the size of a "square" from the fact that 
the length of the plane is 7 m. The time interval 
in seconds between freeze frames can be found 
from the slow-motion factor. 



Film Loop 7 

GALILEAN RELATinXY: 

PROJECTILE FIRED \nERTICALLY 

A rocket tube is mounted on bearings that leave 
the tube free to turn in any direction. 

When the tube is hauled along the snow- 
covered surface of a frozen lake by a "ski-doo, ' 
the bearings allow the tube to remain pointing 
vertically upward in spite of some roughness of 
path. Equally spaced lamps along the path 
allow you to judge whether the ski-doo has 
constant velocity or whether it is accelerating. A 
preliminary run shows the entire scene; the 
setting is in the Laurentian Mountains in the 
Province of Quebec at dusk. 

Four scenes are photographed. In each case 
a rocket flare is fired vertically upward. With 
care you can trace a record of the trajectories. 

Scene 1 

The ski-doo is stationary relative to the earth. 
How does the flare move? 



Scene 2 

The ski-doo moves at uniform velocity rela- 
tive to the earth. Describe the motion of the 
flare to the earth; describe the motion of the 
flare relative to the ski-doo. 

Scenes 3 and 4 

The ski-doo's speed changes jifter the shot is 
fired. In each case describe the motion of the 
ski-doo and describe the flare's motion relative 
to the earth and relative to the ski-doo. In 
which cases are the motions a parabola? 




Fig. 1-51 

How do the events shown in this film 
illustrate the principle of Galilean relativity? In 
which frames of reference does the rocket flare 
behave the way you would expect it to behave 
in all four scenes, knouing that the force is 
constant, and assuming Newton's laws of 
motion? In which systems do Newton's laws 
fail to predict the correct motion in some of the 
scenes? 



Film Loop 8 

ANALYSIS OF A HURDLE RACE. I 

The initial scenes in this film show a regulation 
hurdle race, with 1-m high hurdles spaced 9 m 
apart. (Judging from the number of hurdles 
knocked over, the competitors were of some- 
thing less than Olympic caliber!) Next, a runner, 
Frank White, a 75-kg student at McGill Univer- 
sity, is shouTi in medium slow motion islow- 
motion factor 3) during a 50-m run. His time 
was 8.1 sec. Finally, the beginning of the run is 
shown in extreme slow motion islow-motion 
factor 80). "Analysis of a Hurdle Race. 11 " has 
two more extreme slow-motion sequences. 



48 



UNIT 1 / CONCEPTS OF MOTION 




Fig 1-52 

To study the* runner's motion, measure the 
average speed for each of the 1-m intervals in 
the slow-motion scene. A "drag-strip" chart 
recorder is particularly convenient for record- 
ing the data on a single viewing of the loop. 
VVliatc;v(!r method you use for measuring time, 
the small hut significant variations in speed \%ill 
be lost in experimental uncertainty unless you 
work very carefully. Repeat each measurement 
several times. 

The extreme slow-motion sequence shows 
the runner from m to 6 m. The seat of the 
runner's white shorts might serve as a refer- 
ence mark. (What are other reference points on 
the runner that could be used? Are all the 
reference points equally useful?) Measure the 
time to cover each of the distances: 0-1, 1-2, 
2-3, 3-4, 4-5, and 5-6 m. Repeat the 
measurements several times, view the film over 
again, and average your results for each inter- 
val. Your accuracy might be improved by 
forming a grand average that combines your 
average with others in the class. (Should you 
use a// the measurements in the class?) Calcu- 
late the average speed for each interval, and 
plot a graph of speed versus displacement 
Draw a smooth graph through the points. 
Discuss any interesting features of the graph. 

You might assume that the runner's legs 
push between the time when a foot is directlv 
beneath his hip and the time when that foot 
leaves the ground Is there any n»lationship 
between your graph of speed and the way the 
runner's fcH>t push on the track? 

The initial acceleration of the nmner can be 
estimated fn)m the time to move fn)m the 
starting point to the 1-m mark You can use a 
wat<h with a sw«'ep second hand C'alrulate the 
averagt' acceleration, in meters per second*, 
during this initial interval ilow does this 



forward acceleration compare with the mag- 
nitude of the acceleration of a falling body? 
How much force was required to give the 
runner this acceleration? What was the origin 
of this force? 

I'ilni Loop 9 

ANALYSIS OF A HURDLE RACE. II 

This film loop, which is a continuation of 
"Analvsis of a Hurdle Race. I, " shows two scenes 
of a hurdle race photographed at a slow- 
motion factor of 80. 

In Scene 1, the hurdler moves from 20 m to 26 
m, clearing a hurdle at 23 m. In Scene 2, the 
runner moves from 40 m to 50 m, clearing a 
hurdle at 41 m and sprinting to the finish line at 
50 m. Plot graphs of these motions, and discuss 
any interesting features. The seat of the run- 
ner's pants furnishes a convenient reference 
point for measurements. (See the film notes 
about "Analysis of a Hurdle Race. I " for further 
details.) 

No measurement is entirely precise; mea- 
surement error is always present, and it cannot 
be ignored. Thus, it may be difficult to tell if the 
small changes in the runner's speed are 
significant, or are only the result of measure- 
ment uncertainties. You are in the best tradi- 
tion of experimental science when you pay 
close attention to errors. 

It is often useful to display the experimental 
uncertainty graphically, along \%ith the mea- 
sured or computed values. 

For example, say that the dragstrip timer was 
used to make three different measurements of 
the time required for the first meter of the run: 
13.7 units, 12.9 units, and 13.5 units, which 
gives an average time of 13.28 units. (If you wish 
to convert these dragstrip units to seconds, it 
will he easier to wait until the graph has been 
plotted using \our units, and then add a 
seconds scale to the graph. I TTie lou-est and 
highest values are about 0.4 units on either side 
of the average, so you could report the time as 
13.3 ± 0.4 units. The uncertaint>' 0.4 is about 3*'b 
of 13.3: therefore, the percentage uncertainty in 
the time is 3*^. If you assume that the distance 
was exactly 1 m, so that all the uncertaintv is in 
the time, then the percentage uncertainty in 
the speed will be the same as for the time, 3%. 
The slow-motion speed is 100 cm/13.3 time 
imits. which equals 7 53 cm/unit Since 3*^ of 
7.53 is 0.23, the speed can be rt-ported as 
7.53 ± 0.23 cm/unit In graphing this sjwed 
value, plot a point at 7.53 and draw an error bar 



FILM LOOP NOTES 



49 



/O 



_ 

■: — -i 



^ 7 

6 



6 - 



\ 2 3 4 5 



Fig. 1-53 





1 1 


' 






' 










1.2 


- 






- 1 

< 


-N T 








- 


I.I 


- 




.; 


/ 


\ 


- 






- 


1.0 


— r 


^ 


K' 




« 


\ 


is 


sr 


- 


o.q 


1 


T 














' 


aft 


,. 1 J 














1 





5.1S 3.16 5.(7 5.1ft 3.19 5.20 3JII i.22 

Sfroboscopic Coincidence R?si-Hve 
Moons in Bromoform 
^OSC°" -^^-^^ MH3. 



extending 0.23 above and below the point, as 
shown in Fig. 1-54. Now, estimate the limit of 
error for a typical point on your graph and add 
error bars showing the range to each plotted 
point. 

Your graph for this experiment may well look 
like some graphs commonly obtained in scien- 
tific research. For example, in Fig. 1-54, a 
research team has plotted its experimental 
data; they published their results in spite of the 
considerable scattering of plotted points and 
even though some of the plotted points have 
errors as large as 5%. 

How would you represent the uncertainty in 
measuring distance, if there were significant 
errors here also? 



Fig. 1-54 




Motion in the Heavens 

EXPERIMENTS 



50 



Experiment 2<1 

1VAKEI>.EYE 

ASTKOxXOMY 

Weather permitting, you have been watching 
events in the day and night sky since this 
course started. Perhaps you have followed the 
sun's path, or viewed the moon, planets, or 
stars. 

From observations much like your own, 
scientists in the past developed a remarkable 
sequence of theories. The more aware you are 
of the motions in the sky and the more vou 
interijret them yourself, the more easily you 
can follow the development of these theories. If 
you do not have your own data, you can use the 
results ptT)\'id('d in the following sections. 

A. Oiu- l)a\ of 

Siui Olist'inatioiis 

A studjMit made the obsei-valions of the suns 
posit ion on September 23 as shown in fable 2-1 
If you plot altitude Iverticallyi against 
azimuth (horizontally) on a graph and mark the 
hours lor each point it will help sou to answer 
thr (|iicsti()iis that follow 



TABLE 2-1 



Eastern Standard 
Time (EST) 



Sun's 
Altitude 



Sun's 
Azimuth 



7:00 am. 








8:00 


08' 


097' 


9:00 


19 


107 


10:00 


29 


119 


11:00 


38 


133 


12:00 


45 


150 


1:00 PM 


49 


172 


2:00 


48 


197 


3:00 


42 


217 


4:00 


35 


232 


5:00 


25 


246 


6:00 


14 


257 


7:00 


03 


267 



1. What was the sun's greatest altitude during 
the day' 

2. What was the latitude of the observer? 

3. At what time (EST) was the sun highest? 

4. When during the day was the sun's direction 
(azimuth) changing the fastest? 

5. When dunng the day was the sun's altitude 
changing the fastest? 

6. At what time of day did the sun reach its 
greatest altitude? How do you explain the fact that 
it is not exactly at 12:00? 



EXPERIMENTS 



51 



B. A Year of Sun Observations 

A student made the follouing monthly observa- 
tions of the sun through a full year. 



Dates 



Sun's 

Noon 

Altitude 



TABLE 2-2 



Sunset 
Azimuth 



Time Between 

Noon and 

Sunset 



Jan 1 
Feb 1 
Mar 1 
Apr 1 
May 1 
Jun 1 
Jul 1 
Aug 1 
Sept 1 
Oct 1 
Nov 1 
Dec 1 



20° 

26 

35 

47 

58 

65 

66 

61 

52 

40 

31 

21 



238° 

245 

259 

276 

291 

300 

303 

295 

282 

267 

250 

239 



4''25'' 

4 50 

5 27 

6 15 

6 55 

7 30 
7 40 
7 13 
6 35 
5 50 
5 00 
4 30 



9. If the longitude was 71°W, what city was the 
observer near? 

10. Through what range (in degrees) did the 
sunset point change during the year? 

11. By how much did the observer's time of 
sunset change during the year? 

12. If the time from sunrise to noon was always 
the same as the time between noon and sunset, 
how long was the sun above the horizon on the 
shortest day? on the longest day? 

C. Moon Observations 

During October, a student in Las Vegas made 
the following observations of the moon at sun- 
set when the sun had an azimuth of about 255°. 



TABLE 2-3 



Date 



Angle from 
Sun to Moon 



Moon 
Altitude 



Moon 
Azimuth 



'h = hours, m = minutes. 



Use these data to make three plots (different 
colors or marks on the same sheet of graph 
paper) of the sun's noon altitude, direction at 
sunset, and time of sunset after noon. Place 
these data on the vertical axis and the dates on 
the horizontal axis. 



Oct 16 
18 
20 
22 
24 
26 
28 

?_ 



032° 

057 

081 

104 

126 

147 

169 



17° 
25 
28 
30 
25 
16 
05 



230° 

205 
180 
157 
130 
106 
083 



7. What was the sun's noon altitude at 
equinoxes (March 21 and September 23)? 

8. What was the observer's latitude? 



40° 



the 



13. Plot these positions of the moon on a chart 
similar to Fig. 2-1. 

14. From the data and your plot, estimate the 
dates of new moon, first quarter moon, and full 
moon. 

15. For each of the points you plotted, sketch the 
shape of the lighted area of the moon. 



_i ^ : . 


hr^ 


■i,:l,!i' 


' i ' 1 ■ ' ' • ■ 


— ■- ~ 


'. - J I.-- !- 

. i i ! , , , : : : : . : i : : . : . 


' ' 


i ' ' ■ ■ i ! ' 1 


1 i ! hi 1 


-■-^ 


: — 


, i ' : 1 1 ; 


1 , 1 1 1 . I . ! : : 1 . 1 i , ; i 1 1 : , : : .:,,[, 


t : 



;2o< 



150° 170° 190° 210° ??0° 2? 

AziT.uth 

Fig. 2-2 Phases of the moon: (1) 26 days, (2) 23 days, (3)17 days, (4) 5 days, (5) 3 days after new moon. 




52 



LMT 2 / MO'I ION IN I UK HEAVENS 



TABLE 2-4 PLANETARY LONGITUOES AT TEN-DAY INTERVALS 



im u*v 



ivri AM 



im MMi 



itn iw' 



l»l HA> 



Ml* ra «< 



V7 >14 114 



m ><} a< 



OCT 
OCT 

OCT 



tno ao m 
r« MO ai 

TXg HI «• 

mo 

11*0 



•arr 
««rT 






in* HMi a 

1*7* am 

IW* »»• II 

i*r* <Mi r 

l*>* MOT I 

1*7* IMT 11 

1*7* HAT r 

1*7* MAY II 

1*71 AM 10 

••7* A* B 

1*7* AM » 

1*1* Ail I* 

l«7i JUL a 

1*7* Ail a 

1*7* MIO • 

1(7* MM I* 

1*7* MIO a 

1*7* ■» • 



IT* 


1*7 


la 


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14 ia 


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Ml 


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1 la 


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la 


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1 147 


a* 


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IT* IM Ml IM 

la 174 til a* 

<M 177 a* n* 

n* 1*1 07 ti* 



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IIM ar a* ■> ■• 

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in 


4IM 


tn 


•a 


OI 


141 


IM 


174 


4IM 


a* 


M* 


■• 


IM 


1*7 


17* 


«a 


M* 


04 


MI 


1*1 


IM 


IT* 


4n* 


IM 


80 


t7« 


IM 


IM 


18 


48M 


a* 


ta 


IM 


IB 


•M 


18 


BM 


n* 


a> 


M* 


Ml 


IM 


177 


•a 


a* 


a* 


111 


1*4 


•W 


177 


OM 


a* 


m 


84 


IB 


IM 


18 



8* tT7 aa 



>n 114 84 a* 



a* IB 174 M* 



IB Pt IB 



•IM 
IIM 
• 17* 



1*77 AA 



1*77 ata M 



MM MA« I* 



•7 M* IM 



M* IB 18 



EXPBRIMEIVTS 



53 



B.C. 



By John Hart 



THINK lU ASK B.C 
WHAT THE MOON IS 




D. Locating the Planets 

Table 2-4, "Planetary Longitudes," lists the 
position of each major planet along the ecliptic. 
The positions are given, accurate to the nearest 
degree, for every 10-day interval. By interpola- 
tion, you can find a planet's position on any 
given day. 

The column headed "J.D." shows the corre- 
sponding Julian Day calendar date for each 
entry. This calendar is simply a consecutive 
numbering of days that have passed since an 
arbitrary "Julian Day 1" in 4713 BC: September 
22, 1983, for example, is the same as J.D. 
2,445,600. 

Julian dates are used by astronomers for 
convenience. For example, the number of days 
between March 8 and September 26 of this year 
is troublesome to figure out, but it is easy to 
find by simple subtraction if the Julian Days are 
used instead. 

Look up the sun's present longitude in the 
table. Locate the sun on your SC-1 Constella- 
tion Chart. The sun's path, the ecliptic, is the 
curved line marked off in 360 degrees of 
longitude. 

A planet that is just to the west of the sun's 
position (to the right on the chart) is "ahead of 
the sun," that is, it rises and sets just before the 
sun does. One that is 180° from the sun rises 
near sundown and is in the sky all night. 

When you have decided which planets may 
be visible, locate them along the ecliptic shown 



on your sky map SC-1. Unlike the sun, they are 
not exactly ecliptic, but they are never more 
than 8° Iham it. Once you have located the 
ecliptic on the Constellation Chart, you know 
where to look for a planet among the fixed 
stars. 

E. Graphing the Position of 
the Planets 

Here is a useful way to display the information 
in Table 2-4, "Planetary Longitudes." On ordi- 
nary graph paper, plot the suns longitude 
versus time. Use Julian Day numbers along the 
horizontal axis, beginning close to the present 
date. The plotted points should fall on a nearly 
straight line, sloping up toward the right until 
they reach 360° and then starting again at zero. 



Ifn 




















/ 












/ 






/ 






/ 






/ 




«» 


/ 




-T» 


/ 




s 


/ 


/ 


i- 


/ 










<T> 






c: 


/ 


/ 


o 




/ 












/ 
/ 

/ 













i 



ti'mt 



Fig. 2-3 



How long vvdll it be before the sun again has 
the same longitude it has today? Would the 
answer to that question be the same if it were 
asked three months from now? What is the 
sun's average angular speed (in degrees per 
day) over a whole year? When is its angular 
speed greatest? 

Plot Mercury's longitudes on the same graph 
(use a different color or shape for the points). 
According to your plot, how far (in longitude) 
does Mercury get from the sun? (This is 
Mercury's maximum elongation.) At what time 
interval does Mercury pass between the earth 
and the sun? 

Plot the positions of the other planets using a 
different color for each one. The data on the 
resulting chart are much like the data that 
puzzled the ancients. In fact, the table of 
longitudes is just an updated version of the 
tables made by Ptolemy, Copernicus, and 
Tycho. 



54 



UNIT 2 I MO I l()\ IN THE HEAVENS 



The graph contains a good deaJ of useful 
information For example, when will Mercury 
and Venus next be close enough to each other 
so that you can use bright Venus to help you 
find Mercury? Where are the planets, relative to 
the sun, when they go through their retrograde 
motions? 



Experiment 2-2 

SIZE OF THE EARTH 

You probably know that the earth has a 
diameter of about 12,800 km and a circumfer- 
ence of about 40,000 km. Suppose someone 
challenged you to prove W* How would you go 
about it? 

The first recorded calculation of the size of 
the earth was made a long time ago, in the third 
century BC , by Eratosthenes. He compared the 
lengths of shadows cast by the sun at two 
different jjoints in Egypt. The points were 
rather far apart, but were nearly on a north - 
south line on the earth's surface. The experi- 
ment you will do here uses a similar method. 
Instead of measuring the length of a shadow, 
you will measure the angle between the vertical 
and the sight line to a star. 

You will need a colleague at least 300 km 
away, due north or south of your position, to 
take simultaneous measurements. The two of 
you will need to agree in advance on the star, 
the date, and the time for your observations. 
See how close you can come to calculating the 
actual size of the earth. 

Assumptions and Theon' of 
the Experiment 

Ihe experiment is based on the following 

assumptions: 

1. The earth is a perfect sphere. 

3. A plumb line points toward the center of the 

earth. 

3. The distance from the earth to the stars and 

sun is very gi^at compared with, the «\ulh s 

diameter. 

The two observers must be located at points 
nearly north and south of each other. Suppose 
they are at points A and B. separated by a 
distance .s, as shown in lig 2-4 Ihe observer at 
A and the observer at H l)otb siglit on the same 
star at Ihe pii'arran^ed time, when the star is 
on or near their meridian, and measure the 
angle between the vertical of the plumb line 
and the sight line to the st.ii 




Fig 2-4 



Light rays ftx)m the star reaching locations A 
and B are parallel (this is implied by assump- 
tion 3). 

The difference between the angle ©^ at \ and 
the angle Q^ at B, is the angle <^ between the two 
radii, as shown in Fig. 2-5. 




Fig. 2-5 



In the triangle ABO 

«* = '^A - ^B> '!> 

If C is the circumference of the earth, and s is 
an arc of the meridian, you can make the 
proportion 

^ = A (21 

C 360° 

Combining equations ill and (2), you have 
C- 360° r 

where fl^ and ^b are measured in degrees. 

Doing the Experiment 

For best nvsults the two locations \ and B 
should be directl\- north and south of each 
other, and the observations should be made 
when the star is near its highest point in the 
sky. 



EXPEKIMEIVTS 



55 



You will need some kind of instrument to 
measure the angle 6. Such an instrument is 
called an astrolabe. If your instructor does not 
have an astrolabe, you can make one fairly 
easily from a protractor, a small sighting tube, 
and a weighted string assembled according to 
the design in Fig. 2-6. 



si raw 




-p-wood 

block. 



<- — protr aCtor 
-^f bread 
^■•-washer 



An estimate of the uncertainty in your 
measurement of d is important. Take several 
measurements on the same star and take the 
average value of d. Use the spread in values 
oi d to estimate the uncertainty of your obser- 
vations and of your result. 

Your value for the earth's circumference 
depends on the over-the-earth distance be- 
tween the two points of observation. You 
should get this distance from a map, using its 
scale. 



1. How does the uncertainty of the over-the-earth 
distances compare with the uncertainty in your 
value for 67 

2. What is your calculated value for the circum- 
ference of the earth and what is the uncertainty of 
your value? 

3. Astronomers have found that the average 
circumference of the earth is about 40,000 km. 
What is the percentage error of your result? 

4. Is this acceptable, in terms of the uncertainty of 
your measurement? 



Rg. 2-6 

Aim your astrolabe along the north -south 
line and measure the angle from the vertical to 
the star as it crosses the north -south line. 

If the astrolabe is not aimed along the 
north -south line or meridian, the star will be 
observed before or after it is highest in the sky. 
An error of a few minutes from the time of 
crossing the meridian will make little difference 
in the angle measured. 



B.C. 



By John Hart 




of John Hare and Field Enterprl 



The Size of the Earth; 
Simplified Version 

Perhaps, for lack of a distant colleague, you 
were unable to determine the size of the earth 
as described above. You may still do so if you 
measure the maximum altitude of one of the 
objects on the following list and then use the 
attached data as described below. 

In Santiago, Chile, Maritza Campusano Reyes 
made the following obserx'ations of the 
maximum altitude of stars and of the sun (all 
were observed north of her zenith): 

Antares (Alpha Scorpio) 83.0° 

Vega (Alpha Lyra) 17.5 

Deneb (Alpha Cygnus) 11.5 

Altair (Alpha Aquila) 47.5 

Fomalhaut (Alpha Pisces Austr.) 86.5 

Sun: October 1 59.4° 

15 64.8° 
November 1 70.7° 

15 74.8° 

Since Ms. Reyes made her observations when 
the objects were highest in the sky, the values 
depend only upon her latitude and not upon 
her longitude or the time at which the observa- 
tions were made. 

From a world atlas, find how far north you 
are from Santiago. Next, measure the maximum 
altitude of one or more of these objects at your 
location. Then calculate a value for the circum- 
ference of the earth. 



S6 



LMT 2 / MUTION 1\ I HE HEAVENS 




Fig. 2-7 Photographed by Kenneth R Policy, 6:00 P M 
CST on December 27, 1973, at Finley Air Force Station, 
North Dakota (Lat. 47.5''N, Long. 97.9°W). Exposure: 4 
sec at f5.6, 135 mm. 




Fig. 2-8 Photographori by David Farley. 6 00 P M CST on 
Decomlier 27, 1973, at Starkville. Mississippi (Lat. 
33 5 N, Long 88 7 W). Exposure 4 soc at f6.6. 105 mm 



EXPLKIMEM'S 



57 



Experiment 2>3 

THE DISTANCE TO THE MOON 

The moon is so near the earth that two widely 
separated observers see the moon in different 
positions against the background of fixed stars. 
(If you hold your thumb at arm's length and 
view it first with one eye and then the other, the 
apparent position of your thumb will also shift 
against the distant background! This shift in 
apparent position from the two ends of the 
baseline is an angle called paralla^c . 



1. If the object is moved farther away, will the 
parallax angle become larger or smaller? 

2. How will the parallax angle change if the 
baseline is made longer? shorter? 



On December 27. 1973, the moon and the 
planets \'enus and Jupiter were close together 
in the sky. Simultaneous photographs of the 
objects were taken bv two amateur astrono- 
mers in North Dakota and Mississippi iFigs. 
2-7 and 2-8). The photographers were 1,857 km 
apart. 

The moon was a thin crescent facing the sun; 
the remainder of the moon's disk was illumi- 
nated by earthshine. The "star points" of \'enus 
were caused by internal reflections in the 
camera. 

At first glance the two pictures appear 
identical, but notice the difference in the 
moon's position relative to the nearby star 
located between two short vertical lines. The 
apparent shift of the moon's position for these 
two observers can be found by tracing on thin 
paper or plastic the image of the moon, the 
centers of the images of Venus and Jupiter, and 
a star or two. Then match the tracing over the 
second picture. 



*? 

3. Why should you measure the displacement of 
several matching points on the two pictures of the 
moon? 

4. How do you know that the apparent position of 
the moon has changed and not the positions of 
Venus and Jupiter? 



Record the parallax angle in decimal degrees 
from the scale. Now you can use your own 
measurements of the parallax angle to obtain 
your distance to the moon in kilometers. How 
do your results compare with those of others? 



The linear diameter of the moon can also be 
found from measurements of its angular diam- 
eter on the photographs. In this geometrical 
relation, the moons diameter is now the 
baseline. 

TABLE 2-5 TABLE FOR CONVERTING 
SMALL ANGLES TO LENGTMS 



r 



Sin 9 



0.00 
0.05 
0.10 
0.15 
0.20 
0.25 
0.30 
0.35 
0.40 
0.45 
0.50 



0.00000 
.00087 
.00175 
.00262 
.00349 
.00436 
.00524 
.00611 
.00698 
.00785 
.00873 



AO.or 



0.000175 



5. What value do you compute for the diameter of 
the moon? 

6. How many significant figures should you 
report? 

7. If another object, such as the sun, a comet, or a 
star shows only a very small or no parallax, what 
may you conclude about its distance? 

8. Several approximations have been made in 
this analysis. How does each of the following 
affect your estimate of the distance to the moon? 

a. The baseline is not quite perpendicular to the 
direction of the moon. 

b. The over-land distance between observers is 
not the shortest distance between the observ- 
ers. 

c. The moon's lighted crescent was overexposed. 

d. The observer in Mississippi saw the moon 
several degrees higher in the sky than did the 
observer in North Dakota. 



Experiment 2-4 

THE HEIGHT OF PITOX, A 

MOUNTAIN ON THE MOON 

Closeup photographs of the moon's surface 
have been brought back to earth by the Apollo 
astronauts IFig. 2-9). Scientists are discovering a 
great deal about the moon from such photo- 
graphs, as well as from the landings made by 
astronauts in Apollo spacecraft. 

But long before the Space Age, indeed since 
Galileo s time, astronomers have been learning 
about the moon's surface. In this experiment, 
you will use a photograph iFig. 2-10) taken v\ith 
a large telescope in Ccilifomia to estimate the 



58 



UNIT 2 / MOTION IN THE HEAVENS 




Fig. 2-9 



height of a mountain on the moon. You will use 
a method similar to that used by Galileo, 
although you should be able to get a more 
accurate value than he could working with his 
small telescop>e (and without photographs!). 

The photograph of the moon in Fig. 2-9 was 
taken at the Lick Observatory very near the time 
of the third quarter. Tlie photograph does not 
show the moon as you see it in the sky at third 
quarter because an astronomical telescoj>e 
gives an inverted image, reversing top-and- 
bottom and left-and-right. iThus, north is at the 
bottom.) Figure 2-18 is a lOx enlai^ement of 
the area within the white rectangle in Fig. 2-9. 



Why Choose Piton? 

Piton, a mountain in the moon's northern 
hemisphere, is a slab-like pinnacle in a fairly 
flat area. When the photograph was made, with 
the moon near third-quarter phase, Piton was 
quite close to the line separating the lighted 
portion from the darkened portion of the 
moon. (This line is called the terminator, i 



Assumptions and Relations 

Figure 2-11 represents the third-quarter moon 
of radius r, with Piton P, its shadow of length /, 
at a distance d from the terminator. 




Fig. 2-10 



The ra\'s of light fmm the sun ran be 
considered to be parallel l>erau.se the motin is a 
great distance from the sun. Therefore, the 
angle at w+iich the sun's rev's strike Piton will 
not change if. in imagination, you rotate the 
moon on an axis that points toward the sun hi 



EXPERIMENTS 



59 



Fig. 2-12, the moon has been rotated to put 
Piton on the lower edge. In this position, it is 
easier to work out the geometiy of the shadow. 




JUIyt-I^MT- 



Rg. 2-12 



Figure 2-13 shows how the height of Piton 
can be found from similar triangles; h repre- 
sents the height of the mountain, / is the 
apparent length of its shadow, d is the distance 
of the mountain from the terminator, r is a 
radius of the moon (drawn from Piton at P to 
the center of the moon's outline at O). 



It can be proven geometrically land you can 
see from the drawingi that the small triangle 
BPA is similar to the large triangle PCO. The 
corresponding sides of similar triangles are 
proportional, so you can write 

h d ., , / X d 

- = - then, n = 

/ r r 

All of the quantities on the right can be 
measured from the photograph. 

The curvature of the moon's surface intro- 
duces some error into the calculations, but as 
long as the height and shadow are small 
compared to the size of the moon, the error is 
not great. 




Fig. 2-14 A ISO-km^ area of the moon's surface near the 
large crater, Goclenius. An unusual feature of this crater 
is the prominent rille that crosses the crater rim. 





Rg. 2-13 



Fig. 2-15 A 10-cm rock photographed on the lunar 
surface. 



60 



UNIT 2 / MO HON l,\ THE HEAVEN'S 



Measurements and Calculations 

Unless you are instructed otherwise, you 
should work on a tracing of the moon picture 
rather than in the book itself. Trace the outline 
of the moon and the location of Piton. If the 
photograph was made when the moon was 
exactly at third-quarter phase, then the moon 
was divided exactly in half by the terminator. 
The terminator appears ragged because high- 
lands cast shadows across the lighted side and 
peaks stick up out of the shadow side. Estimate 
the best overall straight line for the terminator 
and draw it on your tracing. Use a millimeter 
scale to measure the length of Piton's shadow 
and the distance from the terminator to Eton's 
peak. 

It probably will be easiest for you to do all the 
calculations in the scale of the photograph. 
Find the height of Piton in centimeters, and 
then change to the real scale of the moon. 



1. How high is Piton in centimeters on the 
photograph scale? 

2. The diameter of the moon is 3,476 km. What is 
the scale of the photograph? 

3. What value do you get for the actual height of 
Piton? 

4. Which of your measurements is the least 
certain? What is your estimate of the uncertainty 
of your height for Piton? 

5. Astronomers, using more complicated 
methods than you used, find Piton to be about 2.3 
km high (and about 22 km across at its base). 
Does your value differ from the accepted value by 
more than your experimental uncertainty? If so, 
can you suggest why? 



Experiment 2-S 
KETKOGK^IDE MOTION 

The fllmstrip you will use in this experiment 
presents photographs of the positions of Mars, 
fmm the lllrs of the Manard College Observa- 
torv, for thive oppositions of Mai's, in 1941, 
1943, and 1946. Ihe first scries of 12 fraiiuvs 
shows the positions of Mars before and after 
tin? opposition of October 10, 1941. The series 
begins with a photograph on August 3, 1941, 
and ends willi onr on Dim rinlx'r H, 1941 The 
second scric's shows positions of Mai-s beloiv 
and after the opposition of DcciMiiber 5. 1943 
This second seiies of sj'ven photographs l)egins 
on October 28, 1943, and ends on Febmaiy 19. 
1944 Tin* Ihiixl set of 11 picluifs, which shows 
Mai-s during nM.")-!**. an)und tin' opposition 
ol laiUMiA 14. 194(v begins with ()t tobcr Hi 



1945, and ends with Februaiy 23, 1946. Jupiter 
also shows in the second and third series. 

The photographs were taken by the routine 
Harvard Sky Patrol with a camera of 15-cm focal 
length and a field of 55'. During each expo- 
sure, the camera was driven by a clockwork to 
follow the daily western motion of the stars and 
hold their images fixed on the photographic 
plate. Mars was never in the center of the field 
and was sometimes almost at the edge because 
the photographs were not made especially to 
show Mars. The planet just happened to be in 
the star fields being photographed. 

The images of the stars and planets are not of 
equal brightness on all pictures because the sky 
was less clear on some nights and the expo- 
sures varied somewhat in duration. Also, the 
star images show distortions from limitations of 
the camera's lens. Despite these limitations, 
however, the pictures are adequate for the uses 
described below. 

Some of the frames show beautiful pictures 
of the Milky Way in Taurus (1943) and Gemini 
(1945). 

Using the Filmstrip 

1. The star fields for each series of frames have 
been carefully positioned so that the star 
positions are nearly identical, if the frames of 
each series are shoun in rapid succession, the 
stars will be seen as stationary' on the screen, 
while the motion of Mars among the stars is 
quite apparent. This would be like viewing a 
flip-book. Run the frames through quickly and 
notice the changing positions of Mars and 
Jupiter. 

2. Project the frames on a paper scn>en where 
the positions of various stars and of Mars can 
be marked. If the star pattern for each frame is 
adjusted to match that plotted from the first 
frame of that series, the positions of Mars can 
be marked accuratelv for the various dates. A 
continuous line through these points will be a 
track for .Mars. Kstimate the dates of the turning 
points. v\1ien Mars l)egins and ends its retro- 
grade motion. By using the scale (10°l shown 
on one frame, also find the angular size of the 
n'trograde loop (Compare your results with the 
average vahn^s in the lext. fiage 140, Unit 2. 

During 1943- 1944 and again in 1945- 1946. 
Jupiter came to opposition several months later 
than Mars did. .As a result. Jupiter appears in 
the frames and also shows its retrograde 
motion Jupiter's opjwsitions wvn* Januarv 11. 
1943; Fel>niar> 11. 1944 Manh 13. 1945; and 
.April 13. 1946 



EXPERIMENTS 



61 



Track Jupiter's position and find the dura- 
tion and size of its retrograde loop. Compare 
your results with the average values listed in 
the Text. This is the type of observational 
information that Ptolemy, Copernicus, and 
Kepler attempted to explain by their theories. 



Experiment 2-6 

THE SHAPE OF THE EARTH'S 

ORBIT 

Ptolemy and most of the Greeks thought that 
the sun revolved around the earth. But after the 
time of Copernicus, the idea gradually became 
accepted that the earth and other planets 
revolve around the sun. Although you probably 
believe the Copemican model, the evidence of 
your senses gives you no reason to prefer one 
model over the other. 

With your unaided eyes you see the sun 
going around the sky each day in what appears 
to be a circle. This apparent motion of the sun 
is easily accounted for by imagining that it is 
the ear^h that rotates once a day. But the sun 
also has a yearly motion with respect to the 
stars. Even if you argue that the daily motion of 
objects in the sky is due to the turning of the 
earth, it is stUl possible to think of the earth as 
being at the center of the universe, and to 
imagine the sun moving in a year-long orbit 
around the earth. Simple measurements show 
that the sun's angular size increases and 
decreases slightly during the year as if it were 
alternately changing its distance from the 
earth. An interpretation that fits these observa- 
tions is that the sun travels around the earth in 
a slightly off-center circle. 



During this laboratory exercise you will plot 
the sun's apparent orbit with as much accuracy 
as possible. 

Plotting the Orbit 

You know the sun's direction among the stars 
on each date that the sun is observed. From its 
observed diameter on that date, you can find its 
relative distance from the earth. So, date by 
date, you can plot the sun's direction and 
relative distance. When you connect your 
plotted points by a smooth curve, you will have 
drawn the sun's apparent orbit. 




Fig. 2-16 Frame 4 of the Sun Filmstrip. 

For observations you will use a series of sun 
photographs taken by the U.S. Naval Observa- 
tory at approximately one-month intervals and 
printed on a filmstrip. Frame 4, in which the 
images of the sun in January and in July are 
placed adjacent to each other, has been 
reproduced in Fig 2-16 so you can see how 
much the apparent size of the sun changes 
during the year. Note also how the apparent 
size of an object is related to its distance from 
you. 



By John H»rt 




By permission of John H«rc and Field Enterpri 



62 



UNII 2 / MOTION IN THE HEAVENS 




draw accurately a fan of lines radiating from the 
earth in these different directions. 



Fig. 2-17 As shown in the diagram, the angular size is 
inversely proportional to the distance — the farther 
away, the smaller the image. 



Procedure 

On a large sheet of graph paper (40 cm x 50 cm) 
make a dot at the center to represent the earth. 
It is particularly important that the graph paper 
be veiy large if you later plot the orbit of Mars in 
Experiments 2-8 and 2-9, which use the results 
of the present experiment. 

Take the 0° direction (toward a reference 
point among the stars) to be along the graph- 
paper lines toward the right. This will be the 
direction of the sun as seen fixim the earth 
on March 21 (Fig. 2-18). The dates of all the 
photographs and the directions to the sun, 
measured counterclockwise from this 0° direc- 
tion, are given in Table 2-6. Use a protractor to 



■O' 



.-l\r*.cttof 
of Vm*^/ 





TABLE 2-€ 






Direction 




Direction 




from Earth 




from Earth 


Data 


to Sun 


Data 


to Sun 


March 21 


000' 


Oct 4 


191' 


April 6 


015 


Nov. 3 


220 


May 6 


045 


Dec. 4 


250 


June 5 


074 


Jan. 4 


283 


Julys 


102 


Feb. 4 


315 


Aug. 5 


132 


March 7 


346 


Sept. 4 


162 







Measure carefully the diameter of the pro- 
jected image of each frame of the filmstrip. Vou 
can get a set of relative distances to the sun by 
choosing a constant and then di\iding it by the 
apparent diameters An orbit with a radius of 
about 10 cm will be a particularly convenient 
size for later use. If you measure the sun's diam- 
eter to be about 50 cm, a convenient constant to 
choose would be 500, since ^ = 10. A larger 
image 51.0 cm in diameter leads to a smaller 
earth- sun distance: 



500 



= 9.8 cm 



510 

Make a table of the relati\'e distances for each 
of the 13 dates. 

Along each of the direction lines you have 
drawn, measure the relative distance to the sun 
for that date. Through the points located in this 
way draw a smooth curve This is the apparent 
orbit of the sun relative to the earth i Since the 
distances are only relati\e, you cannot find the 
actual distance in kilometers from the earth to 
the sun from this plot.) 




EXPERIMENTS 



63 



1. Is the orbit a circle? If so, where is the center of 
the circle? If the orbit is not a circle, what shape is 
it? 

2. Locate the major axis of the orbit through the 
points where the sun passes closest to and 
farthest from the earth. What are the approximate 
dates of closest approach and greatest distance? 
What is the ratio of the largest distance to the 
smallest distance? 



earth's apparent direction from the sun on the 
13 dates, just by changing all the directions 180° 
and then making a new sun-centered plot. 
Another way is to rotate your plot until top and 
bottom cire reversed; this will change all of the 
directions by 180°. Relabel the 0° direction; 
since it is toward a reference point among the 
distant stars, it will still be towcird the right. You 
can now label the center as the sun, and the 
orbit as the earth's. 



A Heliocentric System 

Copernicus and his followers adopted the 
sun-centered model because they believed that 
the solar system could be described more 
simply that way. They had no new data that 
could not be accounted for by the old model. 

Therefore, you should be able to use the 
same data to turn things around and plot the 
earth's orbit around the sun. Clearly, the two 
plots will be similar. 

You already have a table of the relative 
distances between the sun and the earth. The 
dates of largest and smallest distances from the 
earth wiU not change, and your table of relative 
distances is still valid because it was not based 
on wtiich body was moving, only on the 
distance between them. Only the directions 
used in your plotting will change. 

To determine how the angles will change, 
remember that when the earth was at the 
center of the plot, the sun was in the direction 
0° I to the right) on March 21. 



3. This being so, what is the direction of the earth 
as seen from the sun on that date? (Figure 2-20 
will help explain how to change directions for 
a sun-centered diagram.) 




\ 



direction 



Experiment 2-7 

USING LEXSES TO .MAKE 

A TELESCOPE 

In this experiment, you will first examine some 
of the properties of single lenses. Then, you wiU 
combine these lenses to form a telescope, 
which you can use to observe the moon, the 
planets, and other heaxenh' las well as earth- 
bound) objects. 

The Simple Magnifier 

You know something about lenses already, for 
instance, that the best way to use a magnifier is 
to hold it immediately in front of the eye and 
then move the object you want to examine until 
its image appears in sharp focus. 

Examine some objects through several differ- 
ent lenses. Try lenses of various shapes and 
sizes. Separate the lenses that magnify from 
those that do not. What is the difference 
between lenses that magnify and those that do 
not? 



1. Arrange the lenses in order of their magnifying 
powers. Which lens has the highest magnifying 
power? 

2. What physical feature of a lens seems to 
determine its power or ability to magnify; is it 
diameter, thickness, shape, or the curvature of its 
surface? To vary the diameter, simply put pieces 
of paper or cardboard with various sizes of holes 
in them over the lens. 



Sketch side views of a high-power lens, of a 
low-power lens, and of the highest-power and 
lowest-power lenses you can imagine. 



If the sun is in the 0° direction from the earth, 
then from the sun the earth will appear to be in 
just the opposite direction, 180° away from 0°. 
You could make a new table of data giving the 



Real Images 

VV^th one of the lenses you have used, project 
an image of a ceiling light or an outdoor scene 
on a sheet of paper. Describe all the properties 



64 



UNIT 2 / MOTION IN THE HEAVENS 



of the image that you can observe. An image 
that can be projected is called a real image. 



3. Do all your lenses from real images? 

4. How does the size of the image depend on the 
lens? 

5. If you want to look 3t a real image without 
using the paper, where cJo you have to put your 
eye? 

6. The image (or an interesting part of it) may be 
quite small. How can you use a second lens to 
inspect it more closely? Try it. 

7. Try using other combinations of lenses. Which 
combination gives the greatest magnification? 



Making a Telescope 

With two lenses properly arranged, you can 
magnify distant objects. Figure 2-21 shows a 
simple assembly of two lenses to form a 
telescope. It consists of a large lens (called the 
objective) through which light enters and either 
of two interchangeable lenses for eyepieces. 

The following notes will help you assemble 
your telescope. 

1. If you lay the objective down on a flat clean 
surface, you will see that one surface is more 
curved than the other. The more curved surface 
should face the front of the telescope. 

2. Clean and dust off the lenses (using lens 
tissue or a clean handkerchief) before assembl- 
ing and try to keep fingerprints off of it during 
assembly. 

3. Wrap rubber bands around the slotted end 
of the main tube to give a convenient amount of 
friction with the draw tube, tight enough so as 
not to move once adjusted, but loose enough to 
adjust without sticking. Focus by sliding the 
draw tube with a rotating motion, not by 
moving the eyepiece in the tube. 

4. To use high power satisfactorily, a steady 
support (a tripod) is essential. 

5. Be sure that the lens lies flat in the 
high-power eyepiece. 




Use your telescope to observe objects inside 
and outside the lab. Low power gives about 
12 X magnification. High power gives about 30 x 
magnification, like Galileo's best telescope. 

Mounting the Telescope 

If no tripod mount is available, the teiescop>e 
can be held in your hands fui low-power 
observations. Grasp the telescop>e as far forward 
and as far back as possible and brace both arms 
firmly against a car roof, telephone pole, or 
other rigid support. 

With the higher power you must use a 
mounting. If a swivel-head camera tripod is 
available, the telescop>e can be held in a 
wooden saddle by rubber bands, and the 
saddle attached to the tripod head by the 
head's standard mounting screw. Because 
camera tripods are usually too short for 
comfortable viewing while you are standing, 
you should be seated in a reasonably comfort- 
able chair. 

Aiming and Focusing 

You may have trouble finding objects, esp>e- 
cially with the high-power e>'epiece. One tech- 
nique is to sight over the tube, aiming slightly 
below the object, and then to tilt the tube up 
slowly while looking through it and sweeping 
left and right. To do this well, you will need 
some practice. 

Focusing by pulling or pushing the sliding 
tube tends to mo\"e the whole telescop>e. To 
avoid this, rotate the sliding tube as if it were a 
screw. 

Eyeglasses will keep your eye farther from 
the eyepiece than the best distance. Far-sighted 
or near-sighted observers are generally able to 
view more satisfactorily by remo\ing their 
glasses and refocusing Obserxfrs with astig- 
matism ha\-e to decide whether or not the 
distorted image (without glasses) is more 
annoving than the reduced field of \iew (with 
glasses). 



i 




/>■"--' ' n./ff 



Fig. 2-21 






EXPERIMENTS 



65 



Many observers find that they can keep their 
eye in line with the telescope while aiming and 
focusing if the brow and cheek rest lightly 
against the forefinger and thumb. 

To minimize shaking the instrument when 
using a tripod mounting, remove your hands 
from the telescope while actually viewing. 

Limitations of Your Telescope 

You can get some idea of how much fine detail 
to expect when observing the planets by 
comparing the angular sizes of the planets uith 
the resolving power of the telescope. For a 
telescope with a 2.5-cm diameter object lens, to 
distinguish between two details, they must be 
at least 0.001° apart as seen fix)m the location of 
the telescope. The low-power Project Physics 
eyepiece may not quite show this much detail, 
but the high power will be more than sufficient. 
The angular sizes of the planets as viewed 
from the earth are: 



Venus 


0.003° 


(nninimum) 




0.016 


(maximum) 


Mars 


0.002 


(minimum) 




0.005 


(maximum) 


Jupiter 


0.012 


(average) 


Saturn 


0.005 


(average) 


Uranus 


0.001 


(average) 



Galileo's first telescope gave 3x magnifica- 
tion, and his "best" gave about 30 x magnifica- 
tion. (But he used a different kind of eyepiece 
that gave a much smaller field of view.) You 
should find it challenging to see whether you 
can observe all the phenomena he saw, which 
are mentioned in Sec. 7.7 of the Text. 

Observations You Can Make 

The objects suggested for observation have 
been chosen because they are ID fairly easy to 
find, (2) representative of what is to be seen in 
the sky, and (31 very interesting. You should 
observe ah objects with the low power first and 
then the high power. For additional informa- 
tion on current objects to observe, see the 
paperback Xew Handbook of the Heavens, or 
the last few pages of each monthly issue of the 
magazines Sky and Telescope, Natural History, 
or Science News . 

Venus 

No features will be visible on this planet, but 
you can observe its phases, as shown in Fig. 
2-22 (enlai^ed to equal sizes i and on page 196 




Fig. 2-22 Venus, photographed at Yerkes Observatory 

with the 205-cm telescope. 



of the Text. Ulien Venus is very bright you may 
need to reduce the amount of light coming 
through the telescope in order to see the shape 
of the image. A paper lens cap with a round 
hole in the center will reduce the amount of 
light land the resolution of detail !i. You might 
also try using sunglasses as a filter. 

SatuT'n 

This planet is so large that you can resolve 
the projection of the rings beyond the disk, but 
you probably cannot see the gap between the 
rings and the disk with your 30 x telescope (Fig. 
2-23). Compare your observations to the 
sketches on page 73 of the Text. 




Fig. 2-23 Saturn photographed with the 250-cm tele- 
scope at Mount Wilson. 



Jupiter 

Observe the four satellites that GalUeo dis- 
covered. Observe them several times, a few 
hours or a dav apart, to see changes in their 
positions. By keeping detailed data over several 
months time, you can determine the period for 
each of the moons, the radii of their orbits, and 
then the mass of Jupiter. (See the notes for the 
Film Loop, "Jupiter Satellite Orbit,' in this 
Handbook for directions on how to analyze 
your data.) 



66 



UNIT 2 / MOTIOIV IN THE HEAVEN'S 




Fig. 2-24 Jupiter photographed with the 500-cm tele- 
scope at Mount Palomar 



Jupiter is so large that some of the detail on 

its disk, like a broad, dark, equatorial cloud belt 
(Fig. 2-24), can be detected (especially if you 
know it should be there!). 



Moon 

Moon features stand out mostly because of 
shadows. The best observations are made 
around the first and third quarters. Make 
sketches of your observations, and compare 
them to Galileo's sketch on page 194 of your 
Text. Look carefully for walls, mountains in the 
centers of craters, bright peaks on the dark side 
beyond the terminator, and craters within 
other craters. 



The Pleiades 

The Pleiades, a beautiful little star cluster, is 
located on the right shoulder of the bull in the 
constellation Taurus. These stars are almost 
directly overhead in the evening sky in De- 
cember. The Pleiades were among the objects 
Galileo studied with his first telescope. He 
counted 36 stars, which the poet Tennyson 
described as "a swarm of fireflies tangled in a 
silver braid." 



The Hyades 

Ibis duster of stars is also in Taun.is, near 
the star Aldebaran, which foniis the bull's eye. 
The Hyades look like a v The high power may 
show that several stars are double. 



77ie Gretit Xehulti in Orion 

Look about halfway down the row of stars 
that fonii the sword of Orion. It is in the 
southeastern sky during December and 
January Use low power. f'g 2-25 



Algol 

I'his famous variable star is in the constella- 
tion Perseus, south of Cassiopeia. Algol is high 
in the eastern sky in December, and nearly 
overhead during January. Generally, it is a 
second-magnitude star, like the Pole Star. After 
remaining bright for more than ZVz days, Algol 
fades for 5 hours and becomes a fourth- 
magnitude star, like the faint stars of the Little 
Dipper. Then, the variable star brightens during 
the next 5 hours to its normal brightness. From 
one minimum to the next, the period is 2 days, 
20 hours, 49 minutes. 

Great Nebula in Andromeda 

Look high in the western sky in the early 
evening in December for this nebula, for by 
Januaiy it is low on the horizon. It will app>ear 
as a fuzzy patch of light, and is best viewed v\ith 
low power. Ihe light you see from this galaxv 
has been on its way for neariy 2 million years. 

The Milky Way 

This is pariicularly rich in Cassiopeia and 
Cygnus (if air pollution in your area allows it to 
be seen at all). 

Obserxir^ Sunspots 

CAUTION: Do not look at the sun through the 
telescope. The sunlight will injure your eyes. 
Figure 2-25 shows an arrangement of a tripod, 
the low-power telescope, and a sheet of paper 
for projecting sunspots. Cut a hole in a piece of 
cardboard so it fits snugly over the object end 
of the telescope. This acts as a shield so there is 
a shadow area where you can view the 
sunspots. First focus the telescopje. using the 
high-power eyepiece, on some distant object. 
Then, project the image of the sun on a piece of 




EXPERIMENTS 



67 



B.C. 



I TMINK THE ^V»N 
AND TUB CLWnI AftS. 
&OIH^ To CKASH . 



By John Hart 




By permission of John Hare and Field Enterprises, Inc. 



w^te paper about 60 cm behind the eyepiece. 
Focus the image by moving the draw tube 
slightly further out. When the image is in focus, 
you may see some small dark spots on the 
paper. To distinguish marks on the paper from 
sunspots, jiggle the paper back and forth. How 
can you tell that the spots are not on the 
lenses? 




Fig. 2-26 The sunspots of April 7, 1947. 




By focusing the image farther from the 
telescope, you can make the image larger and 
not so bright. It may be easier to get the best 
focus by moving the paper rather than the 
eyepiece tube. 



Experiment 2-8 

THE ORBIT OF MARS 

In this laboratory activity you will derive an 
orbit for Mars around the sun by the same 
method that Kepler used in discovering that 
planetary orbits are ellipticcd. Since the obser- 
vations are made from the earth, you will need 
the orbit of the earth that you developed in 
Experiment 2-6, "The Shape of the Earth's 
Orbit. " Make sure that the plot you use for this 
experiment represents the orbit of the earth 
around the sun, not the sun around the earth. 

If you did not do the earth-orbit experiment, 
you may use, for an approximate orbit, a circle 
of 10-cm radius drawn in the center of a large 
sheet of graph paper. Because the eccentricity 
of the earth's orbit is very small 10.0171, you can 
place the sun at the center of a circular orbit 
without introducing a significant error in this 
experiment. 

From the sun (at the center), draw a line to 
the right, parallel to the grid of the graph paper 
(Fig. 2-271. Label the line 0°. This line is directed 
toward a point on the celestial sphere called 
the vernal equinox and is the reference direc- 
tion from which angles in the plane of the 
earth's orbit (the ecliptic plane) are measured. 
The earth crosses this line on September 23. 
When the eeirth is on the other side of its orbit 
on March 21, the sun is between the earth and 
the vernal equinox. 




Inarch 






68 



LMT 2 / .VlOnO.V I\ I Ht IILAVE.VS 



Photographic Observarions of Mars 

You will use a booklet containing 16 enlai^ed 
sections of photographs of the sky showing 
Mars among the stars at various dates between 
1931 and 1950. All were made with a small 
camera used for the Harvard Observatory Sky 
Patrol. In some of the photographs, Mars was 
near the center of the field. In many other 
photographs Mars was near the edge of the 
field where the star images are distorted by the 
camera lens. Despite these distortions the 
photographs can be used to provide positions 
of Mars that are satisfactory for this study. 
Photograph Pisa double exposure, but it is still 
quite satisfactory. 

Changes in the positions of the stars relative 
to each other are extremely slow. Only a few 
stars near the sun have motions large enough 
to be detected after many years observations 
with the largest telescopes. Thus, you can 
consider the pattern of stars as fixed. 

Finding Mars' Location 

Mcirs is continually moving among the stars but 
is always near the ecliptic. From several 
hundred thousand photographs at the Harvard 
Observatory, 16 were selected, with the aid of a 
computer, to provide pairs of photographs 
separated by 687 days, the period of Mars' 
journey around the sun as determined by 
Copernicus. Thus, each pair of photographs 
shows Mars at one place in its orbit. 

During these 687 days, the earth makes 
nearly two full cycles of its orbit, but the 
interval is short of two full years by 43 days. 
Therefore, the position of the earth, from which 
you can observe Mars, will not be the same for 







the two observations of each pair. If you can 
determine the direction from the earth towards 
Mars for each of the pairs of observations, the 
two sight lines must cross at a point on the 
orbit of Mars. (See Fig. 2-28.1 

Coordinate System Used 

When you look into the sky you see no 
coordinate system. Coordinate systems are 
created for various purposes. The one used 
here centers on the ecliptic. Remember that the 
ecliptic is the imaginarv' line on the celestial 
sphere along wtiich the sun app>ears to mov-e. 

Along the ecliptic, longitudes are alwav-s 
measured eastward from the O' point ithe 
vernal equinox). This is toward the left on star 
maps. Latitudes are measured perpendicular to 
the ecliptic north or south to 90°. The small 
movement of Mars above and below the ecliptic 
is considered in Exfjeriment 2-9, "Inclination of 
Mars' Orbit " 

To find the coordinates of a star or of Mars, 
you must project the coordinate SN-stem upon 
the sky. To do this you ar^ provided with 
transparent overlays that show the coordinate 
system of the ecliptic for each frame. A to P. The 
positions of various stars are circled Adjust the 
overlay until it fits the star positions. Then you 
can read off the longitude and latitude of the 
position of Mars. Figure 2-29 shows how you 
can interpolate between marked coordinate 
lines. Because you are interested in only a small 
section of the sky on each photograph, you can 
draw each small section of the ecliptic as a 
straight line. For plotting, an accuracy of 0.5° is 
satisfactory. 




Fig. 2-28 Point 2 is the p>osition of the earth 687 days 
after leaving point 1 In 687 days. Mars has made exactly 
one revolution and so has returned to the same point on 
the orbit. The intersection of the sight lines from the 
earth determines that point on Mars' orbit. 



Fig. 2-29 Interpolation between coordinate lines In the 
sketch. Mars (Ml. is at a distance y from the 170 line 
Take a piece of paper or card at least 10 cm long Make a 
scale divided into 10 equal parts and lat>el alternate 
marks 0. 1.2. 3. 4. 5 This gives a scale in 5 steps 
Notice that the numbering goes from right to left on this 
scale Place the scale so that the edge passes through 
the position of Mars Now tilt the scale so that tfw and 
5 marks each fall on a grid line Read off the value of y 
from the scale In the sketch y 1 ^ ■ so tftat the 
longitude of M is 171.5'. 



EXPERIMENTS 



69 



TABLE 2-7. OBSERVED POSITIONS OF MARS 



Geocentric 
Long. Let. 



Mars-to- Mars-to- 

Earth Sun Heliocentric 

Distance Distance Long. Lat. 



A 


Mar. 21. 1931 


B 


Feb. 5, 1933 


C 


Apr. 20. 1933 


D 


Mar. 8. 1935 


E 


May 26, 1935 


F 


Apr. 12, 1937 


G 


Sept. 16. 1939 


H 


Aug. 4. 1941 


1 


Nov. 22. 1941 


J 


Oct. 11,1943 


K 


Jan. 21.1944 


L 


Dec. 9, 1945 


M 


Mar. 19.1946 


N 


Feb. 3. 1948 



Apr. 4, 
P Feb. 21, 



1948 
1950 



In a table similar to Table 2-7, record the 
longitude and latitude of Mars for each photo- 
graph. For a simple plot of Mars' orbit around 
the sun, you will use only the first column, the 
longitude of Mars. You will use the columns for 
latitude, Mars' distance from the sun, and the 
sun-centered coordinates if you also investigate 
the inclination, or tilt, of Mars' orbit. 

Finding Mars' Orbit 

When your table is completed for all eight pairs 
of observations, you are ready to locate points 
on the orbit of Mars. 

1. On the plot of the earth s orbit, locate the 
position of the earth for each date given in the 
16 photographs. You may do this by interpolat- 



ing between the dates given for the earth's orbit 
experiment. Since the earth moves through 
360° in about 365 days, you may use ±1° for 
each day ahead or behind the date given in the 
previous experiment. For example, frame A is 
dated March 21. The earth was at 166° on 
March 7; 14 days later on March 21, the earth 
will have moved 14° from 166° to 180°. Always 
work from the earth-position date nearest the 
date of the Mars photograph. 
2. Through each earth-position point, draw a 
"0° line" parallel to the line you drew from the 
sun toward the vernal equinox (the grid on the 
graph paper is helpful). Use a protractor and a 
sharp pencU to mark the angle between the 
0°-line and the direction to Mars on that date as 
seen frxjm the earth (longitude of Mars). The 
two lines drawn from the earth's positions for 
each pafr of dates will intersect at a point. This 
is a point on Mars' orbit. Figure 2-31 shows one 




(h>T I doy) 



Fig. 2-31 




Fig. 2-30 Photographs of Mars made with a 150-cm 
reflecting telescope (Mount Wilson and Palomar Obser- 
vatories) during closest approach to the earth in 1956. 



Left: August 10; right: Sept. 11. Note the shrinking of 
the polar cap. 



70 



LIMIT 2 / MOTION l\ IHE HEAVENS 



point on Mars' orbit obtained from the data of 
the first pair of photographs. By drawing the 
intersecting lines from the eight pairs of 
positions, you establish eight points on Mars' 
orbit. 

3. You will notice that there are no points in 
one section of the orbit. You can fill in the 
missing part because the orbit is symmetrical 
about its major axis. Use a compass and, by trial 
and error, find a circle that best fits the plotted 
points. 

Now that you have plotted the orbit, you have 
achieved what you set out to do: You have used 
Kepler's method to determine the path of Mars 
around the sun. 

Kepler's Law from Your Plot 

If you have time to go on, it is worthwhile to 
see how well your plot agrees with Kepler's 
generalization about planetary orbits. 



1. Does your plot agree with Kepler's conclusion 
that the orbit is an ellipse? 

2. What is the average sun-to-Mars distance in 
astronomical units (AU)? 

3. As seen from the sun, what is the direction 
(longitude) of Mars' nearest and farthest posi- 
tions? 

4. During what month is the earth closest to the 
orbit of Mars? What would be the minimum 
separation between the earth and Mars? 

5. What is the eccentricity of the orbit of Mars? 

6. Does your plot of Mars' orbit agree with 
Kepler's law of areas, which states that a line 
drawn from the sun to the planet sweeps out 
areas proportional to the time intervals? From 
your orbit, you see that Mars was at point B' on 
February 5, 1933, and at point C on April 20, 1933, 
as shown in Fig. 2-32. There are eight such pairs 
of dates in your data. The time intervals are 
different for each pair. 

Connect these pairs of positions with a line 
to the sun, Fig. 2-32. Find the areas of squares 
on the graph paper (count a square when more 
than half of it lies within the arc^al. Di\ide the 
area (in squares) by the number of days in the 
interval to find an "area per day" value. Are 
these values nearly the same? 



7. How much (by what percentage) do they 
vary? 

8. What is the uncertainty in your area mea- 
surements? 

9. Is the uncertainty the same for large areas as 
for small? 

10. Do your results bear out Kepler's law of 
areas? 



n^l*^ 



oRBiT 




fA*rr«'s c*6'^ 



APKiL 33 \°.i3 



Fig. 2-32 In this example, the time interval is 74 days. 

Experiment 2-9 

INCLINATION OF MARS' ORBIT 

When you plotted the orbit of .Mars in t.xperi- 
ment 2-8, you ignored the slight mo\-ement of 
the planet above and below the ecliptic This 
movement of Mars north and south of the 
ecliptic shows that the plane of its orbit is 
slightly inclined to the plane of the earth's 
orbit. Now you may use the table of values for 
Mars' latitude i which you made in Experiment 
2-81 to determine the inclination of .Mars' orbit 
First make a three-dimensional model of twx> 
orbits to see what is meant by the inclination of 
orbits. You can do this quickly with two small 
pieces of cardboard (or index cards). On each 
card draw a circle or ellipse, but hax-e one larger 
than the other Mark clearly the position of the 
focus I sun I on each card Make a straight cut to 
the sun. on one card from the left, on the other 
from the right. Slip the cards together until the 
sun-points coincide. Tilt the two cards (orbit 
planes) at various angles (Fig. 2-331. 




>\#V^ > 



y- 




Rg. 2-33 



EXPERIMENTS 



71 



Theoiy 

From each of the photographs in the set of 16 
that you used in Experiment 2-8, you can find 
the observed latitude (angle from the ecliptic) of 
Mars at a particular point in its orbital plane. 
Each of these angles is measured on a photo- 
graph taken from the earth. As you can see 
from Fig. 2-33, however, it is the sun, not the 
earth, that is at the center of the orbit. The 
inclination of Mars' orbit must, therefore, be an 
angle measured at the sun. It is this angle (the 
heliocentric latitude) that you wish to find. 

Figure 2-34 shows that Mcirs can be repre- 
sented by the head of a pin whose point is 
stuck into the ecliptic plane. Mars is seen fix)m 
the earth to be north or south of the ecliptic, 
but you want the north - south angle of Mars as 
seen from the sun. The following example 
shows how you can derive the angles as if you 
were seeing them irom the sun. 



V 



KMS 



tc:u9T\c v>u».\ie \ 




AlAftS 



3.3" 



3.3' 



Fig. 2-34 



1.71 AU 



d -- lA 



In plate A (March 21, 1933), Mars was about 
3.2° north of the ecliptic as seen from the earth. 
But the earth was considerably closer to Mars 
on this date than the sun was. The angular 
elevation of Mars above the ecliptic plane as 
seen from the sun will therefore be considera- 
bly less than 3.2°. 

For very small angles, the apparent angular 
sizes are inversely proportional to the dis- 
tances. For example, if the sun were twice as far 
from Mars as the earth, the angle at the sun 
would be one-half the angle at the earth. 

Measurement on the plot of Mars' orbit 
(Elxperiment 2-8) gives the earth- Mars dis- 
tance as 9.7 cm (0.97 AU) and the sun- Mars 



distance as 17.1 cm (1.71 AU) on the date of the 
photograph. The heliocentric latitude of Mars 
is, therefore. 



9.7 



X 3.2°N = 1.8°N 



17.1 

You can check this value by finding the 
heliocentric latitude of this same point in Mars' 
orbit on photograph B (February 5, 1933). The 
earth was in a different place on this date so the 
geocentric latitude and the earth- Mars dis- 
tance wiU both be different, but the heliocentric 
latitude should be the same to within your 
experimental uncertainty (Fig. 2-35). 




Fig. 2-35 On February 5, the heliocentric longitude (X^) of 
point B on Mars' orbit is 150°; the geocentric longitude 
(\g) measured from the earth's position is 169°. 



Making the Measurements 

Turn to the table of data you made for 
Experiment 2-8, on which you recorded the 
geocentric latitudes A.g of Mars. On your Mars' 
orbit plot from Experiment 2-8, measure the 
corresponding earth- Mars and sun- Mars 
distances and note them in the same table. 

From these two sets of values, calculate the 
heliocentric latitudes as explained above. The 
values of heliocentric latitude calculated from 
the two plates in each pair (A and B, C and D, 
etc.) should agree within the limits of your 
experimental procedure. 

On the plot of Mars' orbit, measure the 
heliocentric longitude \h for each of the eight 
Mars positions. Heliocentric longitude is mea- 
sured from the sun, counterclockwise from the 
0° direction (direction toward vernal equinox). 

Complete the table given in Experiment 2-8 
by entering the earth- Mars and sun- Mars 
distances, the geocentric and heliocentric 
latitudes, and the geocentric and heliocentric 
longitudes for all 16 plates. 

Draw a graph that shows how the heliocen- 
tric latitude of Mars changes with its heliocen- 
tric longitude (see Fig. 2-36). 



72 



UNIT 2 / MOTION IN THE HEAVENS 




Lovciruoe 



5 

Fig. 2-36 Change of Mars' heliocentric latitude with 
heliocentric longitude. Label the ecliptic, latitude, as- 
cending node, descending node, and inclination of the 
orbit in this drawing. 



From this graph, you can find two of the 
elements that describe the orbit of Mars with 
respect to the ecliptic. The point at which Mars 
crosses the ecliptic from south to north is 
called the ascending node. (The descending 
node, on the other side of the orbit, is the point 
at which Mars crosses the ecliptic ftxim north 
to south.) 

The angle between the plane of the earth's 
orbit and the plane of Mars' orbit is the 
inclination of Mars' orbit /. When Mars reaches 
its maximum latitude above the ecliptic, which 
occurs at 90° beyond the ascending node, the 
planet's maximum latitude equals the inclina- 
tion of the orbit /. 

Elements of an Orbit 

Two angles, the longitude of the ascending 
node, 11, and the inclination i, locate the plane 
of Mars' ori)it with rt\spect to the plane of the 
ecliptic. One mon* angle is needed to orient the 
ortiit of Mars in its ort)ital plane. This is the 
argument of perihelion a>, shown in Fig. 2-37, 
which is the angle in the orbit plane between 
the ascending node and perihelion point. On 



your plot of Mars' orbit, measure the angle 
firom the ascending node il to the direction of 
the perihelion to obtain the argument of the 
perihelion a». 

If you have worked along this far, you have 
determined five of the six elements that define 
any orbit: 
a: semimajor cixis, or average distance (which 

determines the period i 
e : eccentricity ( shapie of orbit as given by c/a in 

Fig. 2-37) 
i: inclination (tilt of orbital plane) 
O: longitude of ascending node (wtiere orbital 

plane crosses ecliptic) 
a»: argument of perihelion (which orients the 

orbit in its plane) 

These five elements (shown in Fig. 2-37) fix 
the orbital plane of any planet or comet in 
space, tell the size and shape of the orbit, and 
also give its orientation within the orbital 
plane. To compute a complete timetable, or 
ephemeris , for the body, you need only to know 
T, a zero date when the body was at a particular 
place in the orbit. Generally, T is gix'en as the 
date of a perihelion passage. Photograph G was 
made on September 16, 1933. From this you can 
estimate a date of perihelion passage for Mars. 

Experiment 2-10 

THE ORBIT OF MERCITIY 

Mercury, the innermost planet, is never very far 
from the sun in the sky. It can be seen only 
close to the horizon, just before sunrise or just 
after sunset, and \iewing is made difficult by 
the glare of the sun. 

Except for Pluto, which differs in sev-eral 
respects from the other planets. Mercuiy has 
the most eccentric planetary orbit in the solar 



Pkin« of orbi* 



Plort«c^ ecliptic 



Fig. 2-37 The five elements of an orbit. You can 
familiarize yourself with these elements of an orbit by 
adding them to the three-dimensional model of two 
orbits, assuming that the earth's orbit is in the plane of 
the ecliptic. 




■V yf ^ - to vttrr<alequi'«OX 



EXPERIMENTS 



73 



system (e = 0.206). The large eccentricity of 
Mercury's orbit has been of particular impor- 
tance, since it has led to one of the tests for 
Einstein's general theory of relativity. For a 
planet with an orbit inside the earth's, there is a 
simpler way to plot the orbit than by the pair 
observations you used for Mars. In this experi- 
ment, you will use this simpler method to get 
the approximate shape of Mercury's orbit. 

Mercur\''s Elongations 

Assume a heliocentric model for the solcir 
system. Mercury's orbit can be found from 
Mercury's maximum angles of elongation east 
and west from the sun as seen from the earth 
on various known dates. 

The angle (Fig. 2-38) between the sun and 
Mercury, as seen from the earth, is called the 
elongation. Note that when the elongation 
reaches its maximum value, the sight lines from 
the earth are tangent to Mercury's orbit. 

Since the orbits of Mercury and the earth are 
both elliptical, the greatest value of the elonga- 
tion varies from revolution to revolution. The 
28° elongation given for Mercury in the Text 
refers to the maximum value. Table 2-8 gives 
the angles of a number of these greatest 
elongations. 



Fig. 2-38 The greatest western elongation of Mercury, 
May 25, 1964. The elongation had a value of 25' West. 



TABLE 2-8. SOME DATES AND ANGLES 
OF GREATEST ELONGATION FOR MERCURY* 

Date 



January 4, 1963 
February 14 
April 26 
June 13 
August 24 
October 6 
December 18 
January 27, 1964 
April 8 
May 25 



Elongation 


19° 


E 


26 


W 


20 


E 


23 


W 


27 


E 


18 


W 


20 


E 


25 


W 


19 


E 


25 


W 



*from the American Ephemeris and Nautical 
Almanac 



tftRTM > OKBif. 



X 



VSV^^^^I^OP^ 



/ 

/ 



/ 



25 







Fig. 2-39 Mercury, first quarter phase, taken June 7, 1934, at the Lowell Observatory, Flagstaff, Arizona. 



74 



UNIT a / MOTION IN THE HEAVENS 



Plotting the Orbit 

Vou can work from the plot of the earth's orbit 
that you established in Experiment 2-6. Make 
sure that the plot you use for this experiment 
represents the orbit of the earth around the 
sun, not of the sun around the earth. 

If you did not do the earth s orbit experi- 
ment, you may use, for an approximate earth 
orbit, a circle of 10-cm radius drawn in the 
center of a sheet of graph paper. Because the 
eccentricity of the earth's orbit is very small 
(0.017), you can place the sun at the center of 
the orbit without introducing a significant error 
in the experiment. 

Draw a reference line horizontally from the 
center of the circle to the right. Label the line 0°. 
This line points toward the vernal equinox and 
is the reference from which the earth's position 
in its orbit on different dates can be estab- 
lished. The point where the 0° line from the sun 
crosses the earth's orbit is the earth's position 
in its orbit on September 23. 

The earth takes about 365 days to move once 
around its orbit (360°). Use the rate of approxi- 
mately 1° per day, or 30° per month, to establish 
the position of the earth on each of the dates 
given in Table 2-8. Remember that the earth 
moves around this orbit in a counterclockwise 
direction, as viewed from the north celestial 
pole. Draw radial lines from the sun to each of 
the earth positions you have located. 

From these positions for the earth, draw 
sight lines for the elongation angles. Be sure to 
note, fixjm Fig. 2-38, that for an eastern 
elongation, Mercury is to the left of the sun as 
seen from the earth. F"or a western elongation. 
Mercury is to the right of the sun. 

You know that on a date of greatest elonga- 
tion Mercuiy is somewhere along the sight line, 
but you do not know exactly where on the line 
to place the planet. You also know that the 
sight line is tangent to the orbit. A reasonable 
assumption is to put Mercury at the point 
along the sight \mv closest to the sun. 

You can now find the orl)it of Mercury' b\' 
drawing a smooth curve through, or close to, 
these points. Remember that the orbit must 
touch each sight line without crossing any of 
them. 

Finding R,,, 

The average distance of a planet in an elliptical 
orbit is equal to one-half the long diameter of 
the ellipse, the semimajor a.xis. 

To find the size of the semimajor axis a of 
Mercury s ori)it (Fig. 2-40), relatix-e to the earth s 




Fig 2-40 

semimajor axis, you must first find the aphelion 
and perihelion points of the orbit. You can use 
a drawing compass to find these points on the 
orbit farthest from and closest to the sun. 

Measure the greatest diameter of the orbit 
along the line perihelion- sun- aphelion. 
Since 10 cm corresponds to 1 AU (the 
semimajor axis of the earth's orbit), you can 
now obtain the semimajor axis of Mercury's 
orbit in astronomical units. 

Calculating Orbital EccentriciU' 

Eccentricitv' is defined as e =c/a (Fig. 2-411. 
Since c, the distance from the center of 
Mercury's ellipse to the sun, is small on your 
plot, you lose accuracy if you liy to measure it 
directly. 



- -^ 




Fig 2-41 

From Fig 2-41, you can see that c is the 
difference between Mercury's f>erihelion dis- 
tance Ap and the semimajor axis a That is: 

c =a - R^ 



Therefore, 



c 

e = - 

a 



a-n. 



e = 1 - ^ 

.1 



EXPERIMENTS 



75 



You can measure flp and a with reasonable 
accuracy fix)m your plotted orbit. Compute e, 
and compare your value with the accepted 
value, e == 0.206. 

Kepler's Second Law 

You can test Kepler's equal-area law on your 
Mercury orbit in the same way as that de- 
scribed in Experiment 2-8, "The Orbit of Mars." 
By counting squares, you can find the area 
swept out by the radial line fiDm the sun to 
Mercury between successive dates of observa- 
tion, such as January 4 to February 14, and June 
13 to August 24. Divide the area by the number 
of days in the interval to get the "area per day." 
This should be constant, if Kepler's law holds 
for your plot. Is it constant? 



Experiment 2-11 

STEPfllSE APPROXIMATION 

TO AN ORBIT^ 

You have seen in the Text how Newton 
analyzed the motions of bodies in orbit, using 
the concept of a centrally directed force. On the 
basis of the discussion in the Text Sec. 8.4, you 
are now ready to apply Newton's method to 
develop an approximate orbit of a satellite or a 
comet around the sun. You can also, from your 
orbit, check Kepler's law of areas and other 
relationships discussed in the Text. 

Imagine a ball rolling over a smooth level 
surface such as a piece of plate glass. 



1. What would you predict for the path of the 
ball, based on your knowledge of Newton's laws 
of motion? 

2. Suppose you were to strike the ball from one 
side. Would the path direction change? 

3. Would the speed change? Suppose you gave 
the ball a series of "sideways" blows of equal 
force as it moves along. What do you predict 
its path might be? 



Reread Sec. 8.4 of the Text if you have 
difficulties answering these questions. 



"This experiment is based on a similar one de- 
veloped by Dr. Leo Lavatelli, University of Illinois, 
American Journal of Physics, Vol. 33, p. 605. 




Fig. 2-42 Photograph of the comet Cunningham made at 
Mount Wilson and Palomar Observatories December 21, 
1940. Why do the stars leave trails and the comet does 
not? 

Your Assumptions 

A planet or satellite in orbit has a continuous 
force acting on it. As the body moves, the 
magnitude and direction of this force change. 
To predict exactly the orbit under the applica- 
tion of this continually changing force requires 
advanced mathematics. However, you can get a 
reasonable approximation of the orbit by break- 
ing the continuous attraction into many small 
steps, in which the force acts as a sharp "blow" 
toward the sun once every 60 days. (See Fig. 
2-43.) 




Fig. 2-43 A body, such as a comet, moving in the vicinity 
of the sun will be deflected from its straight-line path by 
a gravitational force. The force acts continuously but 
Newton has shown that we can think about the orbit as 
though it were produced by a series of sharp blows. 



76 



UNIT 2 I MOTION IN THE HEAVENS 



Fig. 2-44 



The application of repeated steps is known 
as iteration; it is a powerful technique for 
solving problems. Modem high-speed digital 
computers use repeated steps to solve complex 
problems, such as the best path (or paths) for a 
space probe to follow between earth and 
another planet. \t^ 

You can now proceed to plot an approximate 
comet orbit if you make these additional 
assumptions: 

1. The force on the comet is an attraction 
toward the sun. 

2. The force of the blow varies inversely with 
the square of the comet's distance from the 
sun. 

3. The blows occur regularly at equal time •JjV' 
intervcds — in this case, 60 days. The mag- 
nitude of each brief blow is assumed to equal 
the total effect of the continuous attraction of 
the sun throughout a 60-day intervjil. 



Rg. 2-45 



i: 






Effect of the Central Force 

From Newton's second law you know that the 
gravitational force will cause the comet to 
accelerate toward the sun. If a force F acts for a 
time interval Af on a body of mass m, you 
know that 

F = ma = m 

Af 



and, therefore, 



Ar=^ 



Ar 



This equation relates the change in the 
body's velocity to its mass, the force, and the 
time for which it acts. The mass m is constant; 
so is Af (assumption 3 abovel. The change in 
velocity is therefore proportional to the force, 
Av^oc p. But remember that the force is not 
constant in magnitude: it varies in\ersely with 
the square of the distance from comet to sun. 



<r 



Fig 2-46 



<lr 



Fig. 2-47 







V 



4. Is the force of a blow given to the comet when 
it is near the sun grerater or smaller than one 
given when the comet is far from the sun? 

5. Which blow causes the greatest velocity 
change? 



In Fig. 2-44. the vector v^ represents the 
comet's velocity at tin- point .A During tin* finit 
60 days, the comet moves from A to B (Fig. 2-451 
At B a blow causes a velocity change AvplFig 
2-461. The new velocity after the blow is v7= v^ 
+ Av7, and is found by coinplrting the \'ector 
triangle (Fig. 2-47). 



A 



Fig. 2-48 

The comet therefore leax^es point B with 
xelocity v^ and continues to mo\f with this 
velocity for another 60-day interval Because the 
time intervals l>etw?>en blows are always the 
same (60 days), the displacement along the 
path is proporiional to the velocity, v* You 
therefore use a length proportional to the 



EXPERIMENTS 



77 



comet's velocity to represent its displacement 
during each time interval (Fig. 2-48). 

Each new velocity is found, as above, by 
adding to the previous velocity the Av^ven by 
the blow. In this way, step by step, the comet's 
orbit is built up. 

Scale of the Plot 

The shape of the orbit depends on the initial 
position and velocity, and on the acting force. 
Assume that the comet is first spotted at a 
distance of 4 AU from the sun. Also assume that 
the comet's velocity at this point is v*= 2 AU/yr 
at right angles to the sun- comet distance R. 
The following scale factors will reduce the 
orbit to a scale that fits conveniently on a 40 x 
50 cm piece of graph paper. 

1. Let 1 AU be scaled to 6.3 cm so that 4 AU 
becomes about 25 cm. 

2. Since the comet is hit every 60 days, it is 
convenient to express the velocity in astronom- 
ical units per 60 days. Adopt a scale factor in 
which a velocity vector of 1 AU/60 days is 
represented by an arrow 6.3 cm long. 

The comet's initial velocity of 2 AU/yr can be 
given as 2/365 AU per day, or 2/365 x 60 = 0.33 
AU/60 days. This scales to an arrow 2.14 cm 
long. This is the displacement A to B of the 
comet in the first 60 days. 

Computing Av 

On the scale, and with the 60-day iteration 
interval that has been chosen, the force field of 
the sun is such that the Av^given by a blow 
when the comet is 1 AU fhjm the sun is 1 AU/60 
days. 



To avoid computing Av for each value of fl, 
you can plot Av against fl on a graph. Then for 
any value of fl, you can immediately find the 
veilue of Av. 

Table 2-9 gives values of fl in astronomiciil 
units and in centimeters to fit the scale of your 
orbit plot. The table also gives for each value of 
fl the corresponding value of Av in AU/60 days 
and in centimeters to fit the scale of your orbit 
plot. 

TABLE 2-9. SCALES FOR R AND Iv 



Distance from 

the Sun, R 

AU cm 


Change 
in Speed, Iv 
AU/60 
days cm 


0.75 


4.75 


1.76 11.3 


0.8 


5.08 


1.57 9.97 


0.9 


5.72 


1.23 7.80 


1.0 
1.2 
1.5 


6.35 
7.62 
9.52 


1.00 6.35 
0.69 4.42 
0.44 2.82 


2.0 


12.7 


0.25 1.57 


2.5 
3.0 


15.9 
19.1 


0.16 1.02 
0.11 0.71 


3.5 


22.2 


0.08 0.51 


4.0 
4.5 


25.4 
28.6 


0.06 0.41 
0.05 0.38 



Graph these values on a separate sheet of 
paper at least 25 cm long, as illustrated in Fig. 
2-49, and carefully connect the points with a 
smooth curve. 

You can use this curve as a simple graphical 
computer. Cut off the bottom margin of the 
graph paper, or fold it under along the fl axis. 
Lay this edge on the orbit plot and measure the 
distance from the sun to a blow point (such as 



Rg. 2-49 




78 



UNIT 2 / MOTION I\ THE HEAVENS 




t-^ 



B in Fig. 2-50). With dividers or a drawing 
compass, measure the value of Av correspond- 
ing to this R and plot this distance along the 
radius line toward the sun (see Fig. 2-50). 

Making the Plot 

1. Mark the position of the sun S hal^vay up 
the large graph paper (held horizontallyl and 30 
cm from the right edge. 

2. Locate a point 25 cm (4 AU) to the right 
from the sun S. This is point A where you first 
find the comet (Fig. 2-51). 



Fig. 2-S2 



^^t^ 



Ro 



Fig. 2-51 

3. To represent the comets initial \elocity, 
draw vector AB perpendicular to SA (Fig. 2-52). 
B is the comet's position at the end of the first 
60-dav interval. At B a blow is struck that causes 
a change in velocity A\ ,. 

4. Use your Av graph to measure the dis- 
tance of B from the sun at S, and to find Av, for 
this distance (Fig. 2-50). 

5. The fon-e. and therefon* the change in 
velocity, is always directed towarti the sun. 



i A 



From B lay off Av, toward S. Call the end of this 
short line M (Fig 2-52). 

6. Draw the line BC', which is a continuation 
of AB and has the same length as AB. That is 
where the comet would have gone in the next 
60 days if there had been no blow at B 

7. The new velocity after the blow is the 
vector sum of the old wlocity i represented by 
BC') and Av (represented b\ BM' To find the 
new velocity v, draw the line C'C parallel to BM 
and of equal length (Fig. 2-53). The line BC 
represents the new \-elocit>' x-ector v,, the 
velocity with which the comet leax'es point B. 

8. .-Vgain the comet moves with uniform 
\-elocity for 60 da^-s, arriving at point C. Its 
displacement in that time is Ad, = v, x 60 
da\'s. ami Ix'cause of the scale factor chosen, 
the displacement is n'presented by the line BC 
(Fig. 2-54). 



EXPERIMENTS 



79 



«lr^' 



Fig. 2-53 





Fig. 2-54 



9. Repeat steps 1 through 8 to establish point 
D, and so forth, for at least 14 or 15 steps (25 
steps gives the complete orbit). 
10. Connect points A, B, C . . . with a smooth 
curve. Your plot is finished. 

Prepare for Discussion 



6. From your plot, find the perihelion distance. 

7. Find the center of the orbit and calculate the 
eccentricity of the orbit. 

8. What is the period of revolution of your 
comet? (Refer to Text, Sec. 7.3.) 

9. How does the comet's speed change with its 
distance from the sun? 

It is interesting to go on to see how well the 
orbit obtained by iteration obeys Kepler's laws. 

10. Is Kepler's law of ellipses confirmed? (Can 
you think of a way to test your curve to see how 
nearly it is an ellipse?) 

11. Is Kepler's law of equal areas confirmed? 



To answer question 11, remember that the 
time interval between blovv^ is 60 days, so the 
comet is at positions B, C, D . . . , etc., after equal 
time intervals. Draw a line from the sun to each 
of these points (include A), and you have a set 
of triangles. 

Find the area of each triangle. The area A of a 
triangle is given hyA = Vzafa, where a and b are 
altitude and base, respectively. You can also 
count squares to find the areas. 



More Things to Do 

1. The graphical technique you have prac- 
ticed can be used for many problems. You can 
use it to find out what happens if different 
Initial speeds and/or directions are used. You 
may wish to use the 1/fl^ graph, or you may 
construct a new graph. To do this, use a 
different law (for example, force proportional to 
1/fl^, to 1/fl, or to fl ) to produce different paths; 
actual gravitational forces are not represented 
by such force laws. 

2. If you use the same force graph but reverse 
the direction of the force to make it a repulsion, 
you can examine how bodies move under such 
a force. Do you know of the existence of any 
such repulsive force? 



Experiment 3-12 

MODEL OF THE ORBIT OF 

HALLEyS COMET 

Halley's comet is referred to several times in 
your Text. If you construct a model of it, you 
will find that its orbit has a number of 
interesting features. 

Since the orbit of the earth around the sun 
lies in one plane and the orbit of Halley's comet 
lies in another plane intersecting it, you will 
need two large pieces of stiff cardboard for 
planes on which to plot these orbits. 

The Earth's Orbit 

Draw the ecirth's orbit first. In the center of one 
piece of cardboard, draw a circle with a radius 
of 5 cm (1 AU) for the orbit of the earth. On the 
same piece of cardboard, also draw approxi- 
mate (circular) orbits for Mercury (radius 0.4 
AU) and Venus (radius 0.7 AU). For this plot, you 
can consider that all of the planet orbits lie in 
the same plane. Draw a line from the sun at the 
center and mark this line as 0° longitude. 

The table on page 4 of this Handbook lists the 
apparent position of the sun in the sky on 12 
dates. By adding 180° to each of the tabled 
values, you can get the positions of the earth in 
its orbit as seen from the sun. Mark these 
positions on your drawing of the ecirth's orbit. 
(If you wish to mark more than those 12 
positions, you can do so by using the technique 
described on page 4.) 

The Comet's Orbit 

Figure 2-55 shows the positions of Halley's 
comet near the sun in its orbit, which is very 



80 



LMT 2 / MOTION IN THE HEAVENS 




/ 



T 



/,t^ OP ;F 




Rg. 2-55 



nearly a parabola. You will construct your own 
ori)it of Halley's comet by tracing Fig. 2-55 and 
mounting tbe tracing on stiff cardboard. 

Combining the Two Orbits 

Now you have the two orl)its, the comets anil 
the earth's, in their planes, each of which 
contains the sun. You need only to fit the two 
together in accordance with the elements of 
ort)its that you may have used in the experi- 
ment on the Inclination of Mars' Orliit ' 

The line along which the comet's ortiital 
plane cuts the ecliptic plane is called the line of 
nodes. Since you have the major axis drawn, 
you can locate the ascending node, in the 



orbital plane, by measuring a;, the angle from 
perihelion in a direction opposite to the 
comet's motion (see Fig. 2-551. 

To fit the two orbits together, cut a narrow 
slit in the ecliptic plane i earth s oriiili along the 
line of the ascvnciin^ node in as far as the sun. 
The longitude of the comet s ascending mule II 
was at 57° as shown in Fig. 2-56. Tlien slit the 
comet's orbital plane on the side of the 
descendinfi node in as far as the sun (see Fig 
2-551 Slip one plane into the other along the 
cuts until the sun-points on the twx) planes 
come together 

To establish the model in three dimensions 

you must now fit the twt) planes together at the 



EXPERIMENTS 



81 




Fig. 2-56 



^ / /iSCtAJOINfr 



ii^qo ^ 




Rg. 2-57 t[lov£> 



correct angle. Remember that the inclination i, 
162°, is measured upward (northward) from the 
ecliptic in the direction of fl + 90° (see Fig. 
2-57). When you fit the two planes together, you 
will find that the comet's orbit is on the 



underside of the cardboard. The simplest way 
to transfer the orbit to the top of the cardboard 
is to prick through with a pin at enough points 
so that you can draw a smooth curve through 
them. Also, you can construct a small tab to 
support the orbital plane at the correct angle of 
18° (180° - 162°) as shown in Fig. 2-57. 

Halley's comet moves in the opposite sense 
to the earth and other planets. Whereas the 
earth and planets move counterclockwise 
when viewed from above (north of) the ecliptic, 
Halley's comet moves clockwise. 




Fig. 2-58 



If you have persevered this far, and your 
model is a fairly accurate one, it should be easy 
to explain the comet's motion through the sky 
shown in Fig. 2-59. The dotted line in the figure 
is the ecliptic. 

With your model of the comet orbit you can 
now answer some very puzzling questions 
about the behavior of Halley's comet in 1910. 



I 




I I i 


' *^ 








1 • • 




T 


I 




' 


• 


• 




• * • 


• 


• 


• 


• 


•yS^Hl 


^^ 






• 

• 
• 




• 






• 






J^ 




--4 
• 


"^ 


• Nov 1 


Dec 1 


7- 


-^^ 


^ 


^^ 


•^C^ *""" • ^^ 






• 

• 








• • 


• 
• 

• • 




• 


• 


• 




J^ 












^ 


• .• 


• 










• *■ 


y^ 




• • 

• 


• 








. * 


• 


• 


. . • 




. 


• • • 


^ 




« 










* 




• 








, 


^ 


• 






• 


• 




• 










• 




• • < 












• 


• 






• 






• ,'f 


" 














« « 










, 


/ . 


• 






• 


• 






1 ' 




• ,• 


• 




1 


1 1 



Fig. 2-59 IVlotion of Halley's comet in 1909-1910. 



H2 



LM I 2 / VIO'I ION 1% THE HEAVENS 



1. Why did the comet appear to move westward 
for many months? 

2. How could the comet hold nearly a stationary 
place in the sky during the month of April 1910? 

3. After remaining nearly stationary for a month, 
why did the comet move nearly halfway across 
the sky during the month of May 1910? 

4. What was the position of the comet in space 
relative to the earth on May 19th? 

5. If the comet's tail was many millions of miles 
long on May 19th, is it likely that the earth passed 
through part of the tail? 

6. Were people worried about the effect a com- 
et's tail might have on life on the earth? (See 
newspapers and magazines of 1910!) 

7. Did anything unusual happen? How dense is 
the material in a comet's tail? Would you expect 
anything to have happened? 



The elements of Halley's comet are, ap- 
proximately: 

a (semi-major axis) 17.9 AU 

e (eccentricity) 0.967 

I (inclination of orbit plane) 162° 
fl (longitude of ascending 

node) 057° 

io (angle to perihelion) 112° 

Most recent perihelion date April 20, 1910 



From these data, you can calculate that the 
period is 76 years, and the perihelion distance 
is 0.59 AL'. Halley's comet is again expected at 
perihelion on Februaiy 9, 1986. 



1986 Comet Return 

The three-dimensional model for the orbits of 
the earth and Halley's comet in 1909- 1910 can 
be used to establish the positions of the earth 
and the comet in 1986. The comet will come to 
perihelion again on February 9, 1986. If you 
locate the earih in its orf)it on that date and 
change the dates of the comet's position before 
and after perihelion, you will find that at 
perihelion the comet is almost directly behind 
the sun, far away and difficult to observe. 

Since the tail will trail outward behind the 
comet, you can also determine how the tail will 
be viewed from the earth at various dates: 
sideways, foreshortened, or almost head-on. 
Consider also the effects of moonlight on the 
visibility of the tail; a full moon will occur on 
January 25 and Februaiy 22 in 1986 When do 
you expect the comet and its tail to be most 
readily observed? 



ACTIVITIES 



83 



ACTIVITIES 



MAKING ANGULAR 
MEASUREMENTS 

For astronomical observations, and often for 
other purposes, you need to estimate the angle 
between two objects. You have several instant 
measuring devices handy once you calibrate 
them. Held out at arm's length in front of you, 
they include: 

1. your thumb; 

2. your fist, not including thumb knuckle; 

3. two of your knuckles; and 

4. the span of your hand from thumb-tip to tip 
of little finger when your hand is opened wide. 

For a first approximation, your fist is about 8° 
and thumb-tip to little finger is between 15° and 
20°. 

However, since the lengths of people's arms 
<md the sizes of their hands vary, you can 
calibrate yours using the follovvdng method. 

To find the angular size of your thumb, fist, 
and hand spem at your arm's length, you can 
make use of one simple relationship. An object 
viewed from a distance that is 57.4 times its 
diameter covers an angle of 1°. For example, a 
1-cm circle viewed from 57.4 cm away has an 
angular size of 1°. 

Set a 10-cm scale on the blackboard chalk 
tray. Stand with your eye at a distance of 5.74 m 
from the scale. From there, observe how many 
centimeters of the scale are covered by your 
thumb, etc. Make sure that you have your arm 
straight out in front of your nose. Each 10 cm 
covered corresponds to 1°. Find some conven- 
ient measuring dimensions on your hand. 

A Mechanical Aid 

You can use an index card and a meter stick 
to make a simple instrument for measuring 
angles. Remember that when an object with 
a given diameter is placed at a distance fiDm 
your eye equal to 57.4 times its diameter, it 
forms an angle of 1°. This means that a 1-cm 
object placed at a distance of 57.4 cm from 
your eye would form an angle of 1°. If a 1-cm 
diameter object placed at a distance of 57.4 cm 
from your eye covers an angle of 1°, at this same 
distance, a 2-cm diameter object would cover 
2°, and a 5-cm object 5°. 

Now you can make a simple device that you 
can use to estimate angles of a few degrees. 

Cut a series of stepwise slots (as shown in 
Fig. 2-60) in the index card. Mount the card 



~2.C.rr) 




Fig. 2-60 

vertically at the 57-cm mark on a meter stick. 
Cut flaps in the bottom of the card, fold them to 
fit along the stick, and tape the card to the stick 
(Fig. 2-61). Hold the zero end of the stick against 
your upper lip and observe. (Keep a stiff upper 
lip!) 




Fig. 2-61 

Things to Observe 

1. what is the visual angle between the point- 
ers of the Big Dipper (see Fig. 2-62)? 



J{ 






Rg. 2-62 



2. What is the angular length of Orion's belt? 

3. How many degrees away from the sun is the 
moon? Observe on several nights at sunset. 

4. What is the angular diameter of the moon? 
Does it change between the time the moon 



84 



LMT 2 I MOTION IN THE HEAVENS 



B.C. 



Bjr John Hart 




By p<rali«lon o( John Hart and Field Entrrprlici, I 



rises and the time when it is highest in the sky 
on a given night? To most people, the moon 
seems larger when near the horizon. Is it? See 
"The Moon Illusion," Scientific American , July, 
1962, p. 120. 



EPICYCLES AND RETROGRADE 
MOTION 

The hand-operated epicycle machine allows 
you to explore the motion produced by two 
circular motions. You can vary both the ratio of 
the turning rates and the ratio of the radii to 
find the forms of the different curves that may 
be traced out. 

The epicycle machine has three possible gear 
ratios: 2 to 1 (producing two loops per revolu- 
tion), 1 to 1 (one loop per revolutionl, and 1 to 2 
(one loop per two revolutions). To change the 
ratio, simply slip the drive band to another set 
of pulleys. The belt .should be twisted in a figun* 
8 so the deferent arm (the long arm) and the 
epicycle arm (the short arm) rotate in the same 
direction. 



AniS 



3g^ 



u 



n 



Fig. 2-63 






Tape a light source (penlight cell, holder, and 
bulb) securely to one end of the short, epicycle 
arm and counterweight the other end of the 
arm with another (unlit) light source iFig. 2-63) 
If you use a fairiy high rate of rotation in a 
darkened room, you and other observers 
should be able to see the light source mox-e in 
an epic%'cle. 

The form of the curve traced depends not 
only on the gear ratio but also on the relative 
lengths of the arms. As the light is moved closer 
to the center of the epicycle arm, the epicycle 
loop decreases in size until it becomes a cusp 
(Fig. 2-64). UTien the light is \-er>' close to the 
center of the epicycle arm, as it would be for 




Rg 2-«4 



ACTIVITIES 



85 



the motion of the moon around the earth, the 
curve will be a slightly distorted circle, as 
shown in Fig. 2-65. 




Fig. 2-65 



To relate this machine to the Ptolemaic 
model, in which planets move in epicycles 
iiround the earth as a center, you should really 
stand at the center of the deferent arm (earth) 
and view the lamp against a distant fixed 
background. The size of the machine, however, 
does not allow you to do this, so you must view 
the motion from the side. (Or, you can glue a 
spherical glass Christmas-tree ornament at the 
center of the machine; the reflection you see in 
the bulb is just what you would see if you were 
at the center.) The lamp then goes into 
retrograde motion each time an observer in 
front of the machine sees a loop. The retro- 
grade motion is most pronounced when the 
light source is far ftxjm the center of the 
epicycle axis. 



Photographing Epicycles 

The motion of the light source can be photo- 
graphed by mounting the epicycle machine on 
a turntable and holding the center pulley 
stationary with a clamp iFig. 2-661. Alternatively, 
the machine c£in be held in a burette clamp on 
a ringstand and turned by hand. 






Fig. 2-67 




Fig. 2-68 




Fig. 2-69 



Fig. 2-66 An epicycle demonstrator connected to a 
turntable. 




Fig. 2-70 



84> 



L'MT 2 / MO'I IO\ IN THE HEAVENS 



Running the turntable for more than one 
revolution may show that the traces do not 
exactly overlap. (This prcjbably occurs because 
the drive band is not of uniform thickness, 
particularly at its joint, or because the pulley 
diameters are not in exact proportion.) As the 
joining seam in the band runs over either 
pulley, the ratio of speeds changes 
momentarily and a slight displacement of the 
axes takes place. By letting the turntable rotate 
for some time, the pattern wall eventually begin 
to overlap. 

A time photograph of this motion can reveal 
very interesting geometric patterns. You might 
enjoy taking such pictures as an after-class 
activity. Figures 2-67 to 2-70 show four exam- 
ples of the many different patterns that can be 
produced. 



'zOjCiiot 




Fig. 2-71 



CELESTIAL SPHERE MODEL* 

You can make a model of the celestial sphere 
with a round-bottom flask. With it, you can see 
how the appearance of the sky changes as you 
go northward or southward and how the stars 
appear to rise and set. 

To make this model, you will need, in 
addition to the round-bottom flask, a one-hole 
rubber stopper to fit its neck, a piece of glass 
tubing, paint, a fine brush (or grease pencil), a 
star map or a table of star positions, and 
considerable patience. 

On the bottom of the flask, locate the point 
opposite the center of the neck. Mark this point 
and label it N for north celestial pole (Fig. 2-711. 
With a string or tape, determine the circumfer- 
ence of the flask, the greatest distance around 
it. This wall be 360° in your model. TTien, 
starting at the north celestial pole, mark points 
that are one-quarter of the circumference, or 
90°, from the North Pole point. These points lie 
around the flask on a line that is the celestial 
equator. You can mark the equator with a 
grease pencil (china marking pencill, or with 
paint. 

To locate the stars accurately on your "globe 
of the sky," you will need a coonlinate system. If 
you do not wish to have the coordinate system 
marked pennanently on your model, put on 
the lines with a grease pencil. 



'Adaptod from Ymi and Srirncr. by Paul F. Brand- 

vvi-in. rl ;il . llaiTourl Brace. Jo\an(i\icti 



Mark a point 23.5° from the North Pole labout 
one-quarter of 90°). This will be the pole of the 
ecliptic, marked E.P. in Fig. 2-71. The ecliptic 
(path of the sunt will be a great circle 90° from 
the ecliptic pole Tlie point where the ecliptic 
crosses the equator from south to north is 
called the vernal equinos. the position of the 
sun on March 21. All positions in the sk\' are 
located eastward fiDm this point, and north or 
south from the equator. 

To set up the north- south scale, measure 
off eight circles, 10° apart, that run east and 
west in the northern hemisphere parallel to the 
equator. These lines are like latitude on the 
earth but are called declination in the sky. 
Repeat the construction of these lines of 
declination for the southern hemisphere. 

A star's east- west position, called its right 
ascension, is recorded in hours eastward from 
the vernal tK^uinox. To set up the east- w^est 
scale, mark intervals of l/24th of the total 
circumference starting at the x'enial {*quinox 
Tliese marks are 15° apart i rather than 10°) 
since the sky turns through 15° each hour. 

From a table of star positions or a star map, 
you can locate a star's coordinates, then mark 
the star on \our globe ,-MI east - WTst positions 
are rxpnvssed eastward or to the nght of the 
XTnial equinox as you face \our globe 

lo finish the model, put the glass tube into 
the stopper so that it almost reaches across the 
flask and points to your North Pole point. Then 
put enough ink\ water in the flask so that wlien 
you hohl the neck straight down, the water just 
comes up to the line of the equator. For safety. 



ACTIVITIES 



87 



B.C. 



By John Hart 



Piciceo A LOO&c 
THE COMSTEUATiOMS 




wrap wire around the neck of the flask and over 
the stopper so it will not fall out (Fig. 2-72). 




eauo^'ci^ 



CoHK 



Fig. 2-72 



Now, as you tip the flask, you have a model of 
the sky as you would see it from different 
latitudes in the Northern Hemisphere. If you 
were at the earth's North Pole, the north 
celestial pole would be directly overhead and 
you would see only the stars in the northern 
half of the sky. If you were at latitude 45° N, the 
north celestial pole would be halfway between 
the horizon and the point directly overhead. 
You can simulate the appearance of the sky at 
45° N by tipping the axis of your globe to 45° 
and rotating it. If you hold your globe with the 
tixis horizontal, you would be able to see how 
the sky would appear if you were at the 
equator. 



HOW LONG IS A SIDEREAL DAY? 

A sidereal day is the time interval in which a 
star travels completely around the sky. To 
measure a sidereal day you need an electric 
clock and a screw eye. 

Choose a neighboring roof or fence towards 
the west. Then fix a screw eye as an eyepiece in 
some rigid support such as a post or a tree so 
that a bright star, when viewed through the 
screw eye, wall be a little above the roof (Fig. 
2-73). 




Fig. 2-73 

Record the time when the star viewed 
through the screw eye just disappears behind 
the roof, then record the time again on the next 
night. How long did it take to go around? What 
is the uncertainty in your measurement? If you 
doubt your result, you can record times for 
several nights in a row and average the time 
intervals; this should give you a very accurate 
measure of a sidereal day. (If your result is not 
exactly 24 hours, calculate how many days 
would be needed for the difference to add up to 
24 hours.) 



»H 



UNIT 2 / MOTION IN THE HEAVENS 



SCALE MODEL 

OF rHE SOLAK SYSTEM 

Most drawings of the solar system £ire badly out 
of scale, because it is impossible to show both 
the sizes of the sun and planets and their 
relative distances on an ordinary-sized piece of 
paper. Constructing a simple scale model will 
help you develop a better picture of the real 
dimensions of the solar system. 

Let a tennis ball about 7 cm in diameter 
represent the sun. The distance of the earth 
from the sun is 107 times the sun's diameter or, 
for this model, about 7.5 m. (You can confirm 
this easily. In the sky the sun has a diameter of 
0.5°, about half the width of your thumb when 
held upright at arm's length in front of your 
nose. Check this, if you wish, by comparing 
your thumb to the angular diameter of the 
moon, which is nearly equal to that of the sun; 
both have diameters of 0.5°. Now hold your 
thumb in the same upright position and walk 
away from the tennis ball until its diameter is 
about half the wndth of your thumb. You will be 
between 7 and 8 m from the ball!) Since the 
diameter of the sun is about 1,400,000 km, in 
the model 1 cm represents about 200,000 km. 
From this scale, the proper scaled distances 
and sizes of all the other planets can be 
derived. 

The moon has an average distance of 384,000 
km from the earth and has a diameter of 3,476 
km. Wliere is it on the scale model? How large 
is it? Completion of the column for the 
scale-model distances in Table 2-101 will yield 
some surprising results. 



TABLE 2-10. 



Obl«ct 



A SCALE MODEL OF 
THE SOLAR SYSTEM 



Solar 
Distance 



Diameter 



Sample 
Object 



AU Model km Model 

(cm) (approx.) (cm) 

Sun 1,400,000 7 tennis ball 

Mercury 0.39 4,600 

Venus 0.72 12,000 

Earth 1.00 750 13,000 pinhead 

Mars 152 6 600 

Jupiter 5.20 140,000 

Saturn 9.45 120,000 

Uranus 19.2 48,000 

Neptune 30.0 45,000 

Pluto 39.5 6,000 
Nearest 
star 2 7 x 10» 

The average distance betwriMi tlu* «Mrlh and 
sun is callrd tin* astronomicjil unit (Ain. This 



unit is used to describe distance within the 
solar system. 



BUILD A SUNDLIL 

If you are interested in building a sundial, there 
are numerous articles in the 'Amateur Scien- 
tist" section of Scientific American that you will 
find helpful. See particularly the article in the 
issue of August 1959. Also see the issues of 
September 1953, October 1954, October 1959, or 
March 1964. The book Sundials by Mayall and 
Mayall ICharles T. Branford Co., pjublishers, 
Boston) gives theory and building instructions 
for a wide variety of sundials. Encyclop>edias 
also have helpful articles. 



PLOT AN ANALEM.\1A 

Have you seen an analemma? Elxamine a globe 
of the earth, and you will usually find a 
graduated scale in the shapje of a figure 8. with 
dates on it. This figure is called an analemma. It 
is used to summarize the changing positions of 
the sun during the year. 

You can plot your own analemma. Place a 
small mirror on a horizontal surface so that the 
reflection of the sun at noon falls on a 
south-facing wall. Make observations each day 
at e-xactly the same time, such as noon and 
mark the position of the reflection on a sheet of 
paper fastened to the wall. If you remo\-e the 
paper each day, you must be sure to replace it 
in exactly the same position. Record the date 
beside the point The north- south motion is 
most e\ident during Septemlxr- October and 
March -April You can find more atxiut the 
east -west migration of the marks in as- 
tronomy texts and encyclopedias under the 
subject "equation of time " 



STONEHENGE 

Stonehenge (s«h» Text p 130) has been a mvster>' 
for centuries Some scientists haw thought that 
it was a pagan temple, others that it was a 
monument to slaughteretl chieftains I-egends 
invoked the powvr of Merlin to explain how the 
stones were brought to their present location. 
Recent studies indicate that Stonehenge may 
haw been an astronomical olwervator^' and 
eclipse computer. 



ACTIVITIES 



89 



Read "Stonehenge Physics," in the April, 1966 
issue of Phvsics Today; Stonehenge Decoded, 
by G.S. Hawkins and J.B. White (Dell, 1966); or 
see Scientific American, June, 1953. Make a 
report and/or a model of Stonehenge for your 
clftss. 



MOON CRATER NAMES 

Prepare a report about how some of the moon 
craters were named. See Isaac Asimov's Bio- 
graphical Encyclopedia of Science and Technol- 
ogy for material about some of the scientists 
whose names were used for craters. 



LITERATURE 

The astronomical models that you read about 
in Chapters 5 and 6, Unit 2, of the Text strongly 
influenced the Elizabethan \iew of the world 
and the uni\erse. In spite of the ideas of Galileo 
and Copernicus, vvriters, philosophers, and 
theologians continued to use Aristotelian and 
Ptolemaic ideas in their worics. In fact, there are 
many references to the crystal-sphere model of 
the universe in the writings of Shakespeare, 
Donne, and Milton. The references often are 
subtle because the ideas were commonly 
accepted by the people for whom the works 
were written. 

For a quick overview of this idea, with 
reference to many authors of the period, read 
the paperbacks The Elizabethan World Picture, 
by E.M.W. Tillyard (Vintage Press) or Basil 
VVilley, Seventeenth Century Background 
(Doubledayi. 

An interesting specific example of the prevail- 
ing view, as expressed in literature, is found in 
Christopher Marlowe's Doctor Faustus, when 
Faustus sells his soul in return for the secrets of 
the universe. Speaking to the devil, Faustus 
says: 



FRAiMES OF REFERENCE 

1. Two students, A and B, take hold of opposite 
ends of a meter stick or a piece of string one or 
two meters long. If A rotates about on one fixed 
spot so that A is always facing B while B walks 
around A in a circle, A vvdll see B against a 
background of walls and furniture. How does A 
appear to B? Ask B to describe how A appears 
against the background of walls and furniture. 
How do the reports compare? In what direction 
did A see B move, toward the left or right? In 
which direction did B see A move, toward the 
left or right? 

2. A second demonstration invokes a camera, 
tripod, blinky, and turntable. Mount the camera 
on the tripod (using a motor-strobe bracket if 
the camera has no tripod connection) and put 
the blinky on a turntable. Aim the camera 
straight down (Fig. 2-74). 



\ 









Fig. 2-74 



* 



Take a time exposure with the camera at rest 
and the blinky mo\ing one revolution in a circle 
(Fig. 2-75). If you do not use the turntable, move 



"...Come, Mephistophilis, let us 

dispute again 
And argue of divine astrology. 
Tell me, are there many heavens 

above the moon? 
Are all celestial bodies but one 

globe 
As is the substance of this centric 

earth? . . ." 



Fig. 2-75 




90 



UNIT 2 / MOTION' IN THE HEAVENS 



the blinky by hand around a circle drawn 
faintly on the background. Then take a second 
print with the blinky at rest and the camera, on 
time exposure, moved steadily by hand about 
the axis of the tripod. Try to move the camera at 
the same rotational speed as the blinky moved 
in the first photo. 

Can you tell, just by looking at the photos, 
whether the camera or the blinky was moving? 



DEMONSTRATING SATELLITE 
ORBITS 

A piece of thin plastic or a rubber sheet can be 
stretched tight and clamped in an embroidery 
hoop about 55 cm in diameter. Place the hoop 
on some books and put a heavy ball, for 
example, a 5-cm-diameter steel ball bearing, in 
the middle of the plastic IFig. 2-76). The plastic 
will sag so that there is a greater force toward 
the center on the ball when it is closer to the 
center than when it is farther away. 










r/»;^c''//VA'y //•■r^/ 



Fig. 2-76 

You can use a smaller hoop on the stage of an 
overhead projector. Use small ball bearings, 
marbles, or heads as "satellites." Then you will 
have a shadow prf)jection of the large central 
mass, with the small satellites racing around it. 
Be careful not to drop the hall through the 
glass. 

If you take strolw photos of the motion, you 
can check whether Kepler's three laws are 
satisfied; you can see where satellites travel 
fastest in their ortiits, and how the orbits 
themselves turn in space. To take the picture, 
set up the hoop on llir floor with black pap«>r 
under it. 

You can use eitlier the electninic stn)b«' liglit 
or the slotted disk stroboscope to take the 
pictures. In either case, place the camera 
directly over the hoop and the light source at 
the side, slightly enough above the [ilane of the 
hoop so that the floor under the h(»op i.s not 




Rg. 2-77 



well lighted (Fig. 2-77). A ball bearing or marble 
will make the best pictures. 

Here are some questions to think about: 

1. Does your model gi\e a true representation 
of the gravitational force around the earth? In 
what ways does the model fail? 

2. Is it more difficult to put a satellite into a 
perfectly circular orbit than into an elliptical 
one? What conditions must be satisfied for a 
circular orbit? 

3. Are Kepler's three laws really v-erified? 
Should they be? 

For additional detail and ideas see "Satellite 
Orbit Simulator," Scientific American. October. 
1958. 



GALILEO 

Read Bertolt Brecht's play, Galileo and pre- 
sent a part of it for the class There is some 
controx-ersy about whether the play truly 
reflects what historians believe were Galileo's 
feelings. For comparison, \x>u could read The 
Crime of Galileo, by Giorgio de Santillana: 
Galileo and the Scientific Revolution. Itv I>aura 
Fermi: The Galileo Quadricentennial Supple- 
ment in Sky and Telescope. Februar>- 1964: or 
articles in the April, 1966 issue of The Physics 
Teacher, "Galileo: Antagonist." and Galileo 
Galilei: An Outline of His Life " 



conic: SEcmoNs models 

Obtain from a mathematics teacher a demon- 
stration cone thas has l>een cut along se\-eral 
difTenMit planes so that when it is taken apart 
the planes fonn the four conic sections. 



ACTIVITIES 



91 



If such a cone is not available, tape a cone of 
paper to the front of a small light source, such 
as a flashlight bulb. Shine the light on the wall 
and tilt the cone at different angles with respect 
to the wall. You can make cill the conic sections 
shown in Sec. 7.3 of the Text. 

If you have a wall lamp with a circular shade, 
the shadows cast on the wall above and below 
the lamp are usually hyperbolas. You can check 
this by tracing the curve on a Icirge piece of 
paper, and seeing whether the points satisfy 
the definition of a hyperbola. 




CHALLENGING PROBLEM: 
FINDING EARTH - SUN 
DISTANCE FROM VENUS PHOTOS 

Here's a teaser: Assume that Venus has the 
same diameter as the earth. Also assume that 
the scale of the pictures on page 196 of the Text 
is 1.5 seconds of arc per millimeter. 

Determine the distance from the earth to the 
sun in kilometers. 



MEASTOEVG 
IRREGIXAR AREAS 

Are you tired of counting squares to measure 
the area of irregular figures? A device called a 
planimeter can save you much drudgery. There 
cire several styles, ranging from a simple pocket 
knife to a complex arrangement of worm gears 
and pivoted arms. See the "Amateur Scientist" 
section of Scientific American, August, 1958 and 
February, 1959. 



Fig. 2-78 M. Babinet prevenu par sa portiere de la visite 
de la comete. A lithograph by the French artist Honore 
Daumier (1808-1879), Museum of Fine Arts, Boston. 



From these data you can calculate that the 
perihelion distance flp is 0.33 AU and the 
aphelion distance Rg is 4.11 AU. 

The comet of 1680 is discussed extensively in 
Newton's Principia, where approximate orbital 
elements are given. The best parabolic orbital 
elements known are: 

T = Dec. 18, 1680 / = 60.16° 

oj = 350.7° Rp = 0.00626 AU 

n = 272.2° 

Note that this comet passed veiy close to the 

sun. At perihelion, it must have been exposed 

to intense destructive forces like the comet of 

1965. 

Comet Candy had the following parabolic 
orbital elements: 

T = Feb. 8, 1961 / ^ 150.9° 

w = 136.3° Rp = 1.06 AU 

n = 176.6° 



OTHER COMET ORBITS 

If you enjoyed making a model of the orbit of 
Hcdley's comet, you may want to make models 
of some other comet orbits. Data are given 
below for several other comets of interest. 

Encke's comet is interesting because it has 
the shortest period known for a comet, only 3.3 
years. In many ways, it is representative of aill 
short-period comets that have orbits of low 
inclination and pass near the orbit of Jupiter, 
where they are often strongly deviated. The full 
ellipse can be drawn at the scale of 10 cm for 1 
AU. The orbital elements for Encke's comet are: 
n = 335° 



a = 222 AU 
e = 0.85 
I = 15° 



(t) = 185° 



DRAWING A PARABOLIC ORBIT 

The parabola is an unusual conic section 
whose eccentricity is exactly 1. Geometrically, it 
has the interesting property that aR points on a 
parabola are equidistant both from the focus 
and from a line perpendicular to the major axis 
and twice the perihelion distance from the 
focus. This construction line is called the 
directrix. The geometrical property permits a 
quick development of points on a parabola, as 
Fig. 2-79 indicates. 

Along the major axis, locate a point that is 
twice the distance to the perihelion. At that 
point draw a line perpendicular to the major 
axis. Then with a drawing compass swing an 
arc of any length from the focus. Without 



92 



UNIT 2 / MOTION IN THE HEAVENS 




Fig. 2-79 Parabola for orbit with perihelion distance q 
0.20 AU. 



changing the size of the arc, locate on the 
directrix a point such that an arc drawn from 
there will intersect the first arc as far as 
possible from the directrix; the line from the 
directrix to that intersection will be parallel to 
the major axis. By changing the size of the arc, 
you can establish a series of points on the 
parabola. Then draw a smooth curve through 
the points. 

The number of days for a body mox-ing 
around the sun in a pcirabolic orbit to move 
from a given solar distance to perihelion is 
given in the accompanying table (Table 2-11). 
With it, and the date of perihelion, you can 
establish the dates at which a comet was at any 
point on its parabolic orbit. 









TABLE 2-11 








Solar 
















Distance 




Pwihalion Dista 


no* q 


AU) 




X AU 


00 


0.2 


0.4 


0.6 


0.8 


1.0 


1.2 


2.0 


77.8 


88.1 


97.1 


105.8 


108.0 


109.6 


107.8 


1.8 


66.1 


78.2 


84.3 


98.0 


932 


930 


886 


1.6 


56.1 


64.8 


72.0 


767 


782 


76.0 


694 


1.4 


45.4 


56.0 


60.3 


638 


634 


590 


468 


1.2 


382 


43.9 


48.9 


50.7 


495 


50.0 


0.0 


1.0 


294 


363 


380 


362 


319 


0.0 




0.8 


196 


234 


27.8 


24.5 


0.0 






0.7 


16.7 


21.3 


22.5 


14.5 








0.6 


18.8 


17.5 


17.2 


0.0 








0.5 


9.7 


15.5 


11.3 










0.4 


6.9 


9.8 


0.0 










0.3 


4.5 


6.1 












0.2 


2.6 


0.0 












0.1 


0.9 















FORCES ON A PENDULUM 

If a pendulum is drawn aside and released with 
a small sideways push, it will move in an almost 
elliptical path. This looks vaguely like the 
motion of a planet about the sun, but there are 
some differences. 

To investigate the shape of the pendulum 
orbit and see whether the motion follovN-s the 
law of areas, you can make a strobe photo with 
the setup shown in Fig. 2-80. Use either an 
electronic strobe flashing from the side, or use 
a small light and AA battery cell on the 
pendulum and a motor strobe disk in front of 
the lens. If you put tap>e over one slot of a 
12-slot disk to make it half as wide as the rest, it 
will make every twelfth dot fainter, gi\ing a 
handy time marker, as shown in Fig. 2-81 Vou 
can also set the camera on its back on the floor 
with the motor strobe above it, and suspend 
the pendulum overhead. 



Rg. 2-80 





Fig 2-81 

Are the motions and the forces similar for the 
pendulum and the planets' The center of force 
for planets is located at one focus of the ellipse 
VMiere is the center of fon^e for the pendulum' 
Measure your photos to determine w+iether the 
pendulum bob follows the law of areas for 
motion under a central force. 



ACTIVITIES 



93 




Fig. 2-82 



In the case of the planets, the force varies 
inversely with the square of the distance 
between the sun and the planet. From your 
photograph, you can find how the restoring 
force on the pendulum changes with distance 
R from the rest point (Fig. 2-82). Find Av 
between strobe flashes for two sections of the 
orbit, one near and one far from the rest point. 
How do the accelerations as indicated by the 
Av's compare with the distances R? Does the 
restoring force depend on distance in the same 
way as it does for a planet? If you have a copy 
of Newton's Principia available, read Prop- 
osition X. 



TRIAL OF COPERXICUS 

Hold a mock trial for Copernicus. Have two 
groups of students represent the prosecution 
and the defense. If possible, ha\e English, social 
studies, and language instructors serve as the 
jury for your trial. 



94 



LNIT 2 / MOTION IN THE HEAVENS 



FILM LOOP NOTES 



Film Loop 10 
RETROGRADE MOTION: 
GEOCEISTTRIC MODEL 

The film illustrates the motion of a planet such 
as Mars, as seen from the earth. It was made 
using a large "epicycle machine" as a model of 
the Ptolemaic system (Fig. 2-83). 

First, from above, you see the characteristic 
retrograde motion during the "loop" when the 
planet is closest to the earth. Then the studio 
lights go up and you see that the motion is due 
to the combination of two circular motions. 
One arm of the model rotates at the end of the 
other. 

The earth, at the center of the model, is then 
replaced by a camera that points in a fixed 
direction in space. The camera views the 
motion of the planet relative to the fixed stars 
(the rotation of the earth on its ^lxis is being 
ignored). This is the same as if you were looking 
at the stars and planets from the earth toward 
one constellation of the zodiac, such as Sagit- 
tarius. 

The plcmet, represented by a white globe, is 
seen along the plane of motion. The direct 
motion of the planet, relative to the fixed stars, 
is eastward, toward the left (as it would be if 
you were facing south). A planet's retrograde 
motion does not always occur at the same 
place in the sky, so some retrograde motions 
are not visible in the chosen direction of 
observation. To simulate obser\'ations of 
planets better, an additional three retrograde 
loops were photographed using smaller bulbs 
and slower speeds. 



Note the changes in apparent brightness and 
angular size of the globe as it sweeps close to 
the camera. Actual planets app>ear only 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 
focused only on positions in the sky. Film Loop 
11 shows a similar model based on a heliocen- 
tric theoiy. 



Film Loop 11 
RETROGRADE MOTION: 
HELIOCENTRIC MODEL 

This film is based on a lai^e heliocentric 
mechanical model. Globes represent the earth 
and a planet mo\ing in concentric circles 
around the sun (represented by a yellow globe). 
The earth (represented by a light blue globe) 
passes inside a slower mo\ing outer planet 
such as Mars (represented by an orange globe). 

Then the earth is replaced by a camera 
ha\ing a 25° field of view. TTie camera points in 
a fixed direction in space, indicated by an 
arrow, thus ignoring the daily rotation of the 
earth and concentrating on the motion of the 
earth relative to the sun. 

The view from the moving earth is shown for 
more than one year. First the sun is seen in 
direct motion, then Mars comes to opp>osition 
and undergoes a retrograde motion loop, and 
finally you see the sun again in direct motion 

Scenes are viewed from above and along the 
plane of motion. Retrograde motion occurs 
whenever Mars is in opposition, that is 




Fig. 2-«3 



FILM LOOP NOTES 



95 



v\'hene\'er Mars is opp>osite the sun as \iev\ed 
from the earth. Not ah these oppositions take 
place when Mars is in the sector the camera 
sees. The time between oppositions a\'erages 
about 2.1 years. The film shows that the earth 
mo\«s about 2.1 times around its orbit between 
opjxjsitions of Mau^. 

You can calculate this value. The earth makes 
one c^'cle around the sun per year and Mars 
makes one cycle around the sun every 1,88 
years. So the frequencies of orbital motion are: 

/«artt = 1 cvcAt and/^^an = 1 cyc/1.88yr 
= 0-532 cyc/yr 
The frequency' of the earth relati\-e to Mars is 

JtarOi ~ Jmnn- 

ftat^ -fmm = 1-00 cyc/yT - 0,532 cyc/yr 
= 0.468 c\cAt 
That is, the earth catches up with and passes 
Mars once every 

= 2.14 years. 

0.468 

Note the increase Ln apparent size and 
brightness of the globe representing Mars when 
it is nearest the earth Mewed with the naked 
eye, Mars shows a lai^e variation In brightness 
(ratio erf cibout 50:1 1 but always apf>ears to be 
only a pwint of light. Through a telescop>e, the 
angular size also \'aries as predicted b\' the 
model. 

The heliocentric model is in some ways 
simpler than the geocentric model of Ptolemy, 
and gi^-es the general features observed for the 
planets: angular p>osition, retrograde motion, 
and \ariation in brightness. Howe\-er detailed 
numerical agreement between theon.- and oh>- 
serva^tion cannot be obtained using circular 
orbits. 



Fflm Loop 12 

JtTITER SATELLITE ORBIT 

This time-Lapse stud\' of the orbit of Jupiter's 
satellite, lo, was filmed at the Lowell Observat- 
ory in Flagstaflf, Arizona, using a 60-cm refractor 
t^escope. 

Exposures were made at 1-min intervals 
during 5e\«n nights. ,An almost complete orbit 
of lo is reconstructed using all these exposures. 

The film first shows a segment of the orbit as 
photographed at the telescop>e a clock shows 
the p>a&sage of time Due to smaD errors in 
guidiing the telescopje arid atmospheric turbu- 
lence, the highly inagni6ed images of Jupiter 
and its satellites dance abouL To remove this 




Rg. 2-84 Business end of :.' e ■.'.■:■' 'efractor at Lowell 
Observatory. 



unsteadiness, each irruige lover 2100) was 
optically centered in the frame. The stabilized 
images were joined to giv-e a continuous record 
of the motion of lo. Some variation in bright- 
ness was caused by haze or cloudiness. 

The four Galilean satellites are listed in Table 
2-12. On Feb. 3, 1967, the>' had the configuration 
shown in Fig. 2-85. The satellites mo\e nearly in 
a plane viewed almost edge-on; thus, the\' seem 




Rg. 2-«5 



96 



UNIT 2 / MOTION IN THE HEAVENS 







TABLE 2-12. 


SATELLITES OF JUPfTB* 










Radius of 












Orbit 


Eccantridty 


Diamatar 




Nam* 


Period 


(kilo mat era) 


of Orbit 


(kilomatars) 


1 


lo 


1''18''28- 


419,200 


0.0000 


3,200 


II 


Europa 


3" 13" 14"" 


667,200 


0.0003 


2,880 


III 


Ganymede 


7" 3" 43'" 


1,065,600 


0.0015 


4,960 


IV 


Callisto 


16" IS" 32'" 


1,873,600 


0.0075 


4.480 



to move back and forth along a line. The field of 
view is large enough to include the entire orbits 
of I and II, but III and IV are outside the camera 
field when they are farthest from Jupiter. 

The position of lo in the last frame of the Jan. 
29 segment matches the position in the first 
frame of the Feb. 7 segment. However, since 
these frames were photographed nine days 
apart, the other three satellites had moved, so 
you can see them pop in and out while the 
image of lo is continuous. Lines identify lo in 
each section. Fix your attention on the steady 
motion of lo and ignore the comings and 
goings of the other satellites. 

Interesting Features of the Film 

1. At the start, lo appears almost stationaiy at 
the right, at its greatest elongation: another 
satellite is moving toward the left and overtakes 
it. 

2. As lo moves toward the left (Fig. 2-86), it 
passes in front of Jupiter, a transit. Another 
satellite, Ganymede, has a transit at about the 
same time. Another satellite moves toward the 
right and disappears behind Jupiter, an occula- 
tion. It is a very active scene! If you look closely 
during the transit, you may see the shadow of 




Ganymede and perhaps that of lo, on the left 
part of Jupiter's surface. 

3. Near the end of the film, lo (mo\ing toward 
the right) disappears; an occulation begins. 
Look for lo's reappearance; it emerges from an 
eclipse and appears to the right of Jupiter. Note 
that lo is out of sight part of the time because it 
is behind Jupiter as viewed from the earth, and 
part of the time because it is in Jupiter's 
shadow. It cannot be seen as it moves from D to 
E in Fig. 2-87. 



^Ht*l>tiu> Of 




Fig 2-86 Still photograph from Film Loop 12 showing 
tho positions of three satellites of Jupiter at the start of 
the transit and occultation sequence. Satellite IV is out 
of the picture, far to the right of Jupiter. 



FiQ. 2-87 

4. Jupiter is seen as a flattened circle because 
its rapitl ix)tation period O'' S."}"") has caused it 
to flatten at the poles and bulge at the equator. 
The effect is quite noticeable: The equatorial 
diameter us 142,720 km and the polar diameter 
is 133.440 km 

Measurements 

1. Period of orbit. Time the motion between 
transit and orrulation ifrom B to D in Fig 2-871. 
half a re\oliitit)n. Ui find the jM'riod The film is 
projected at about 18 frames/sec. so that the 
speed-up factor is 18 x 60, or 1.080. How ran 
you ralihrate your projector more arcurateU :» 



FILM LOOP NOTES 



97 



(There are 3.969 frames in the loop.) How does 
your result for the period compare with the 
value given in Table 2-12? 

2. Radius of orbit. Project on paper and mark 
the two extreme positions of the satellite, 
farthest to the right (at A) and farthest to the left 
(at C). To find the radius in kilometers, use 
Jupiter's equatorial diameter for a scale. 

3. Mass of Jupiter. You can use your values for 
the orbit radius and period to calculate the 
mass of Jupiter relative to that of the sun (a 
similcir calculation based on the satellite Cal- 
listo is given in the Text). How does your 
experimental result compare with the accepted 
value, which is mjm^ — 1/1,048? 



Film Loop 13 
PROGRAM ORBIT I 

A student (Fig. 2-88, right) is plotting the orbit of 
a planet, using a stepwise approximation. His 
instructor (left) is preparing the computer 
program for the same problem. The computer 
and the student follow a similar procedure. 




Fig. 2-88 



Then the program instructs the computer to 
calculate the force on the planet from the sun 
from the inverse-square law of gravitation. 
Newton's laws of motion are used to calculate 
how far and in what direction the planet moves 
after each blow. 

The computer's calculations can be dis- 
played in several ways. A table of X and Y values 
can be typed or printed. An X- Y plotter can 
draw a graph from the values, similar to the 
hand-constructed graph made by the student. 
The computer results can also be shown on a 
cathode-ray tube (CRT), similar to that in a 
television set, in the form of a visual trace. In 
this film, the X-Y plotter was the mode of 
display used. 

The dialogue between the computer and the 
operator for Trial 1 is as follows. The numerical 
values are entered at the computer typewriter 
by the operator after the computer types the 
message requesting them. 

Computer: GIVE ME INITIAL POSITION IN 

AU . . . 
Operator: X = 4 

Y = 

Computer: GIVE ME INITIAL VELOCITY IN 

AU/YR . . . 
Operator: XVEL = 

YVEL = 2 

Computer: GIVE ME CALCULATION STEP IN 

DAYS . . . 
Operator: 60 

Computer: GIVE ME NUMBER OF STEPS FOR 

EACH POINT PLOTTED . . . 
Operator: 1 

Computer: GIVE ME DISPLAY MODE . . . 
Operator: X-Y PLOTTER 



The computer 'language" used was Fortran. 
The FORTRAN program (on a stack of punched 
cards) consists of the "rules of the game": the 
laws of motion and of gravitation. These rules 
describe precisely how the calculation is to be 
done. The program is translated and stored in 
the computer's memory. 

The calculation begins with the choice of 
initial position and velocity of the planet. The 
initial position values of X and Y are selected 
and also the initial components of velocity 
XVEL and YVEL (XVEL is the name of a single 
variable, not a product of four variables). 



You can see that the orbit displayed on the 
X-Y plotter, like the student's graph, does not 
close. This is surprising, as you know that the 
orbits of planets are closed. Both orbits fail to 
close exactly. Perhaps too much error is 
introduced by using such large steps in the 
step-by-step approximation. The blows may be 
too infrequent near perihelion, where the force 
is largest, to be a good approximation to a 
continuously acting force. In Film Loop 14, 
"Program Orbit II," the calculations ai"e based 
upon smaller steps, and you can see if this 
explanation is reasonable. 



98 



UNIT 2 / MOTION IN THE HEAVENS 



Film lAHtp 14 
PROGRAM ORBIT II 

In this continuation of the film "Program Orbit 
I," a computer is again used to plot a planetary 
orbit with a force inversely proportional to the 
square of the distance. The computer program 
adopts Newton's laws of motion. At equal 
intervals, blows act on the body. The orbit 
calculated in the previous film probably failed 
to close because the blows were spaced too far 
apart. You could calculate the orbit using many 
more blows, but to do this by hand would 
require much more time and effort. In the 
computer calculation, you need only specify a 
smaller time interval between the calculated 
points. The laws of motion are the same as 
before, so the same program is used. 

A portion of the "dialogue" between the 
computer and the operator for Trial 2 is as 
follows: 

Computer: GIVE ME CALCULATION STEP IN 

DAYS . . . 
Operator: 3 
Computer: GIVE ME NUMBER OF STEPS FOR 

EACH POINT PLOTTED . . 
Operator 7 

Computer: GIVE ME DISPLAY MODE . . . 
Operator: X- Y PLOTTER 

Points are now calculated every three days (20 
times as many calculations as for Trial 1 on the 
"Program Orbit I" film), but to avoid a graph 
with too many points, only one out of seven of 
the calculated points is plotted. 

The computer output in this film can also be 
displayed on the face of a cathode-ray tube 
(CRT). The CRT display has the ad\'antage of 
speed and flexibility and is used in the other 
loops in this series, Film Loops 15, 16, and 17 
On the other hand, the permanent record 
produced by the X- Y plotter is sometimes veiy 
convenient. 

Orbit Program 

The computer program for orbits is written in 
FORTRAN II and includes "ACCEPT" (data) state- 
ments used on an IBM 1620 input txpewriter 
(Fig. 2-89). 

With slight modification, the program 
worketi on a CDC 3100 and CDC: 3200, as shown 
in Film I^xips 13 and 14, "Pn)grani Orbit I and 
"Program Orbit II," With additional slight 
modifications (in statement 16 and the three 
sucrmling statiMurnt.s in Fig 2-89) it ran I>e 



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flx»l Il>t r>- - TO Ml (ML> t/i OCLI* 

iniM/tll I'.I'.K 
ir >«iL • >VIL • 0.1 • tACCCL • JClT* 

TVtl • T»ll. • O.i • TACOL • eCLT* 

WJ TO 1« 
OfLI* V • *(!.( LIMIT llX Tints OILT* I 
!• »»IL • »Hl • •ACClL • Ot.i.1* 

i»lt • tin. ' TiCCtl. • oetT* 
DfLI» » • I»ILCX.|T1 Ilut^ OtLT* T 
I« « • « • I»IL • ttti* 

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IM>t« • IM<t> • I 

I'lKOtl - KfaLLSl It. II. II 



used for other force laws. A similar program is 
presented and explained in FORTK\.\for Physics 
by Alfred M Bork ( Addison- V\"esley, 19671 

Note that it is necessary' to ha\¥ a subroutine 
MARK. In this case, it is used to plot the points 
on an X-Y plotter, but MARK could be 
replaced by a PRINT statement to print the X 
and Y coordinates. 



Film Loop tS 
CENTRAL FORCEkS: 
ITERATED BLOH'S 

In Chapter 8 and in Experiment 2-11 and Film 
Loop 13 on the stepwise approximation of 
orbits. Kepler's law of areas applies to objects 
acted on b\ a central force The force in each 
case was attrarti\f and was either constant or 
varied smootWy according to some pattern. But 
suppose the central force is repulsiw; that is. 
directed awav from the center' Suppose it is 
sometimes attractiw and sometimes repulsix-e? 
Wliat if the amount of force applied each time 
varies unsvstrmatically' Under these circum- 
stances, would the law of areas still hold? You 
can use this film to find out. 

The film was made by photographing the 
face of a cathode-ray tube (CRT) that displayetl 



FILM LOOP NOTES 



99 



the output of a computer. It is important to 
realize the role of the computer program in this 
film. It controlled the change in direction and 
change in speed of the "object" as a result of a 
"blow." This is how the computer program uses 
Newton's laws of motion to predict the result of 
applying a brief impulsive force, or blow. The 
program remained the same for all parts of the 
loop, just as Newton's laws remain the same 
during all experiments in a laboratory. How- 
ever, at one place in the program, the operator 
had to specify how he wanted the force to vary. 



Random Blows 

Figure 2-90 shows part of the motion of the 
body as blows are repeatedly applied at equal 
time intervals. No one decided in advance how 
great each blow was to be. The computer was 
programmed to select a number at random to 
represent the magnitude of the blow. The 
directions toward or away from the center were 
also selected at random, although a slight 
preference for attractive blows was built in so 
that the pattern would be likely to stay on the 
face of the CRT. The dots appear at equal time 
intervals. The intensity and direction of each 
blow is represented by the length of line at the 
point of the blow. 

Study the photograph. How many blows 
were attractive? How many were repulsive? 
Were any blows so small as to be negligible? 




Rg. 2-90 



You can see if the law of areas applies to this 
random motion. Project the film on a piece of 
paper, mark the center, and mark the points 
where the blows were applied. Now measure 
the areas of the triangles. Does the moving body 
sweep over equal areas in equal time intervals? 

Force Proportional to Distance 

If a weight on a string is pulled back and 
released with a sideways shove, it moves in an 
elliptical orbit with the force center (lowest 
point) at the center of the ellipse. A similar path 
is traced on the CRT in this segment of the film. 
Notice how the force varies at different dis- 
tances firom the center. A smooth orbit is 
approximated by the computer by having the 
blows come at shorter time intervals. In frame 
2(a), four blov^ are used for a full orbit; in 2(b) 
there are nine blows, and in 2(c), 20 blows, 
which give a good approximation of the ellipse 
that is observed with this force. Geometrically, 
how does this orbit differ from planetaiy orbits? 
How is it different physically? 

Inverse-Square Force 

A similar program is used wath two pl<inets 
simultaneously, but udth a force on each 
varying inversely as the square of the distance 
from a force center. Unlike the real situation, 
the program assumes that the planets do not 
exert forces on one another. For the resulting 
ellipses, the force center is at one focus 
(Kepler's first law), not at the center of the 
ellipse as in the previous case. 

In this film, the computer has done 
thousands of times faster what you could do if 
you had enormous patience and time. With the 
computer you can change conditions easily, 
and thus investigate many different cases and 
display the results. Once told what to do, the 
computer makes fewer calculation errors than 
a person. 



Film Loop 16 
KEPLER'S LAWS 

A computer program similar to that used in the 
film "Central Forces: Iterated Blows ' causes the 
computer to display the motion of two planets. 
Blows directed toward a center (the sun), act on 
each planet in equal time intervals. The force 
exerted by the planets on one another is 
ignored in the program; each is attracted only 
by the sun, by a force that varies inversely as the 
square of the distance from the sun. 



lOO 



L.MT 2 / MOTION IN THE HEAVENS 



Initial positions and initial velocities for the 
planets were selected. The positions of the 
planets are shown as dots on the face of the 
cathode-ray tube at regular intervals. (Many 
more points were calculated between those 
displayed.) 

You can check Kepler's three laws by project- 
ing on paper and marking successive positions 
of the planets. The law of areas can be verified 
by drawang triangles and measuring areas. Find 
the areas swept out in at least three places: 
near perihelion, near aphelion, and at a point 
approximately midway between perihelion and 
aphelion. 

Kepler's third law holds that in any given 
planetary system the squares of the periods of 
the planets are proportional to the cubes of 
their average distances from the object around 
which they are orbiting. In symbols, 

where T is the period and flav is the average 
distance. Thus, in any one system, the value of 
TVflav^ ought to be the same for all planets. 

You can use this film to check Kepler's law of 
periods by measuring T for each of the two 
orbits shown, and then computing T^/R^y^ for 
each. To measure the periods of revolution, use 
a clock or watch with a sweep second hand. 
Another way is to count the number of plotted 
points in each orbit. To find flav for each orbit, 
measure the perihelion and aphelion distances 
(flp and fla) and take their average (Fig 2-91). 




Fig. 2-91 The mean distance R^, of a planet P orbiting 
about the sun is (/?,. + R,)l2 

How close is the iigrccmeiit luMween \our 
two values of TVfl.v'? Which is the greater 
souH'e of em)r, the measurement of T or of fl „ :* 



To check Kepler's first law, see if the orbit is 
an ellipse with the sun at a focus. You can use 
string and thumbtacks to draw an ellipse. 
Locate the empty focus, svmmetrical with 
respect to the sun's position. Place tacks in a 
board at these two points. Make a loop of string 
as shown in Fig. 2-92. 




Rg. 2-92 



Put your pencil in the string loop and draw 
the ellipse, keeping the string taut. Does the 
ellipse match the observed orbit of the planet? 
What other methods can be used to find 
whether a curve is a good approximation of an 
ellipse? 

You might ask whether checking Kepler's 
laws for these orbits is just busy-work, since the 
computer already "knew " Kepler's laws and 
used them in calculating the orbits. But the 
computer was not given instructions for 
Kepler's laws. V\'hat you are checking is 
whether Newton's laws lead to motions that fit 
Kepler's descriptive laws. The computer 
"knew" (through the program gi\"en it) only 
Newlon's laws of motion and the in\-erse- 
square law of gra\itation. This compulation is 
exactly w+»at Nev\1on did, but without the aid of 
a computer to do the routine work. 



Film Loop 17 
UNl^SrAL ORBITS 

In this film, a modification of the computer 
program described in "Central Forres: Iterated 
Blows' is used. There are two sequences: The 
first shows the eflr€»ct of a disturbing force on an 
ortiit prtMluretl by a central inwrse-square 
force: the second shows an orbit produced by 
an in\'erse-cul>e force. 

The word prrturbation refers to a small 
variation in the motion of a celestial bod\' 



FILM LOOP NOTES 



101 




Fig. 2-93 



Fig. 2-94 



caused by the gravitational attraction of 
another body. For example, the planet Neptune 
was discovered because of the perturbation it 
caused in the orbit of Uranus. The main force 
on Uranus is the gravitational pull of the sun, 
and the force exerted on it by Neptune causes 
a perturbation that changes the orbit of Uranus 
very slightly. By working backward, astrono- 
mers were able to predict the position and 
mass of the unknown planet from its small 
effect on the orbit of Uranus. This spectacular 
"astronomy of the invisible" was rightly re- 
garded as a triumph for the Newtonian law of 
universal gravitation. 

Typically, a planet's entire orbit rotates 
slowly, because of the smedl pulls of other 
planets and the retarding force of friction due 
to dust in space. This effect is called advance of 
perihelion . Mercury's perihelion advances 
about 500 seconds of eirc (0.14°) per century. 
Most of this was explained by perturbations 
due to the other planets. However, about 43 
seconds per century remained unexplained. 
When Einstein reexamined the nature of space 
and time in developing the theory of relativity, 
he developed a new gravitational theory that 
modified Newton's theory in crucial ways. 
Relativity theory is important for bodies moving 
at high speeds or near massive bodies. Mer- 
cury's orbit is closest to the sun and therefore 
most affected by Einstein's extension of the law 
of gravitation. Relativity was successful in 



explaining the extra 43 seconds per century of 
advance of Mercury's perihelion. 

The first sequence shows the advance of 
perihelion due to a small force proportional to 
the distance fl, added to the usual inverse- 
square force. The "dialogue " between operator 
and computer sttirts as follows: 

PRECESSION PROGRAM WILL USE 

ACCEL = G/(R'R} + P'R 
GIVE ME PERTURBATION P 

P = 0.66666 
GIVE ME INITIAL POSITION IN AU 

X = 2 

Y = 
GIVE ME INITIAL VELOCITY IN AUATl 

XVEL = 

YVEL = 3 

The symbol * means multiplication in the 
FORTRA.\ language used in the program. Thus, 
G/tfl'fl) is the inverse-square force, and P'R is 
the perturbing force, proportional to R . 

In the second part of the film, the force is an 
inverse-cube force. The orbit resulting from the 
inverse-cube attractive force, as from most force 
laws, is not closed. The planet spirals into the 
sun in a "catastrophic " orbit. As the planet 
approaches the sun, it speeds up, so points are 
separated by a large firaction of a revolution. 
Different initial positions and velocities would 
lead to quite different orbits. 




The Triumph of lUleehaniGS 



EXPERIMENTS 



102 



Experiment 3-1 

COLLISIOXS 

IN OXE UIMEIVSIOX. I 

In this experiment, you will investigate the 
motion of two objects interacting in one 
dimension. The interactions (explosions and 
collisions in the cases treated here) are called 
one-diiiuMisional because the objects move 
along a single straight line. Your purpose is to 
look for quantities or combinations of quan- 
tities that remain unchanged before and after 
the interaction, that is, quantities that are 
conserved. 

Your experimental explosions and collisions 
may seem not only lame but also artificial and 
unlike the ones you see around you in eveiyday 
life. But this is tyjiical of many scientific 
experiments, which simplifv the situation so as 
to make it easier to makj* meaningful mea- 
sun'MU-nls and to tlisctner patterns in the 
obseivetl behavior. The underlying laws an* the 
same for all phenomena, whether or not they 
an» in a laboratory. 

Two clifTenMit ways of observing interactions 
an» described hen* land two others in Kx|M'ri- 



ment 3-21. You will probably use only one of 
them. In each method, the friction between the 
interacting objects and their surroundings is 
kept as small as possible, so that the objects are 
a nearly isolated system Whichev-er method 
you do follow, you should handle your results 
in the way described in the final section. 
"Analysis of Data." 

METHOD A. Dxnamics Carts 

"Explosions are easily studied using the 
low-friction dvTiamics carts. Squee«' the loop 
of spring steel flat and slip a loop of thn»ad over 
it to hold il compn'ssed Wi\ the compressetl 
loop between two carts on the floor or on a 
smooth table (Fig 3-11 \Mien you release the 
spring by burning the thread, the carts fly apart 
with wlocities that \ou can measure from a 
strobe (ihotograph or b\ ain of the techniques 
you learned in eariier exjK'riments 

U)ad the carts with a variety of weights to 
create simple ratios of masses, for example. 2 to 
1 or 3 to 2. Take data for as great a x-ariety of 
mass ratios as time permits Because friction 
will gradualK slow the carts down, vou should 



EXPERIMENTS 



103 



COKK 



^ 



.£2 



'l_^ 



CS^ 



CID 







Fig. 3-1 



make measurements on the speeds im- 
mediately after the explosion is over (that is, 
when the spring stops pushing). 

Since you are interested only in comparing 
the speeds of the two carts, you can express 
those speeds in any units you wish, without 
worrying about the exact scale of the photo- 
graph and the exact strobe rate. For example, 
you can use distance units measured directly 
from the photograph I in millimeters) and use 
time units equal to the time interval between 
strobe images. If you follow that procedure, the 
speeds recorded in your notes will be in 
millimeters per interval. 

Remember that you can get data &x)m the 
negative of a Polaroid picture as well as from 
the positive print. 

METHOD B. Air Track 

The air track allows you to observe collisions 
between objects, "gliders," that move with 
almost no friction. You can take stroboscopic 
photographs of the gliders either with the 
xenon strobe or by using a rotating slotted disk 
in front of the camera. 

The air track has three gliders: two small 
ones with the same mass, and a larger one with 
just twice the mass of a small one. A small and a 
large glider can be coupled together to make 
one glider so that you can have collisions 
between gliders whose masses are in the ratio 
of 1:1, 2:1, and 3:1. (If you add light sources to 
the gliders, their masses will no longer be in the 
same simple ratios. You can find the masses 
from the measured weights of the glider and 
light source.) 

You can arrange to have the gliders bounce 
apart after they collide (elastic collision) or stick 
together (inelastic collision). Good technique is 
important if you are to get consistent results. 
Before tciking any pictures, try both elastic and 
inelastic collisions with a variety of mass ratios. 
Then, when you have chosen one type to 



analyze, rehearse each step of your procedure 
with your partners before you go ahead. 

You can use a good photograph to find the 
speeds of both carts, before and after they 
collide. Since you are interested only in com- 
paring the speeds before and after each colli- 
sion, you can express speeds in any unit you 
wish, without worrying about the exact scale of 
the photograph or the exact strobe rate. For 
example, if you use distance units measured 
directly ftxjm the photograph (in millimeters) 
and time units equal to the time interval 
between strobe images, the speeds recorded in 
your notes will be in millimeters per interval. 

Remember that you can get data from the 
negative of your Polaroid picture as well as 
ftx)m your positive print. 



^ 



-^ 



"^ 




Fig. 3-2 

Analysis of Data 

Assemble all your data in a table having 
column headings for the mass of each object, 
m^^ and m^, the speeds before the interaction, 
v^ and Vg (for explosions, v.^ = v% = 0), and the 
speeds after the collision, v^' and Vg'. 

Examine your table carefully. Search for 
quantities or combinations of quantities that 
remain unchanged before and after the interac- 
tion. 



1. is speed a conserved quantity? That is, does 
the quantity (v^ + v„) equal the quantity (v^' -r 

2. Consider the direction as well as the speed. 
Define velocity to the right as positive and velocity 
to the left as negative. Is velocity a conserved 
quantity? 

3. If neither speed nor velocity is conserved, try a 
quantity that combines the mass and velocity of 
each cart. Compare (m^^v^ + n^,^v^^) with {m^v^' 
+m„Vn') for each interaction. In the same way 
compare mlv, mv, m^v, or any other lilcely 
combinations you can think of, before and after 
interaction. What conclusions do you reach? 



104 



UNIT 3 / THE TRILMPii OF MECHANICS 



Experiment 3-2 

f:oLM Slows 

IN ONE DIMENSION. II 

METHOD A. Film Loops 

Film Loops 18, 19, and 20 show one- 
dimensional collisions that you cannot easily 
perform in your own laboratory. They were 
filmed with a very high-speed camera, produc- 
ing the effect of slow motion when projected at 
the standarxl rate. You can make measurements 
directly from the pictures projected onto graph 
paper. Since you are interested only in compar- 
ing speeds before and after a collision, you can 
express speeds in any unit you wish, that is, 
you can make measurements in any convenient 
distance and time units. 

Notes for these film loops are located on 
pages 156 and 157. If you use these loops, read 
the notes carefully before taking your data. 

METHOD B. Stroboscopic Photo- 
graphs 

Strxjboscopic photographs showing seven dif- 
ferent examples of one-dimensional collisions 
appear on the following pages.* They are useful 
here for studying momentum and again later 
for studying kinetic energy. 

For each event, you should find the speeds of 
the balls before and after collision. From the 
values for the mass and speed of each ball, 
calculate the total momentum before and after 
collision. Use the same; values to calculate the 
total kiin'tic cncrgv before and after collision. 

You should read Section 1 before analyzing 
any of the events, in order to find out what 
measurements to make and how the collisions 
were prt)duced. After you have made your 
measurements, turn to Section II for questions 
to answer about each event. 

I. The Measurements Vou Will Make 

To make the necessary' measurements, you will 
need a metric ruler marked in millimeters, 
preferably of transparent plastic with sharp 
scale markings Before starting your work, 
consult Fig. 3-2 for suggestions on imprx)\ing 
your measuring technique. 

Figur^e 3-3 shows schematically that the 
colliding balls were hung from very long wires. 
The balls were released from rest, and their 
double-wii'j' (bifilar) suspensions guided them 
to a s(juaii'ly head-on collision Sti-oboscopes 
illuminaU'tl the 90 cm x 120 cm rx'ctangle that 

•Rnpnnhirod by (icnnisiiiun of National Film Boarti 
(if ('aiKida 



is edge c^ imagm 




was the field of \iew of the camera. The 
stroboscop>es are not shown in Fig. 3-3. 

Notice the two rods whose tops reach into 
the field of view. These rods were 1 m (±2 mm) 
apart, measured ftxim top center of one rx>d to 
top center of the other. The tops of these rods 
are visible in the photographs on which you 
will make your measurements. This enables 
you to convert your measurements to actual 
distances if you wish. However, it is easier to 
use the lengths in millimeters measured di- 
rectly on the photograph if you are merely 
going to compare momenta 

The balls speed up as the\' mo\e into the 
field of view. Likewise, as they leave the field of 
view, thev slow down. Therefore successive 



^ 



^ 




^^ 



Fig. 3-3 Setup for photographing or>»-dim«nsion«l 
collisions 



EXPERIMENTS 



105 



displacements on the stroboscopic photo- 
graph, each of which took exactly the same 
time, will not necessarily be equal in length. 
Check this with your ruler. 

As you measure a photograph, number the 
position of each ball at successive flashes of the 
stroboscope. Note the interval during which 
the collision occurred. Identify the clearest 
time interval for finding the velocity of each ball 
(a) before the collision and (b) after the 
collision. Then mark this information necir each 
side of the interval. 

II. Questions To Be Answered About 
Each Event 

After you have recorded the masses (or relative 
masses) given for each baU and have recorded 
the necessciry measurements of velocities, ans- 
wer the following questions. 

1. What is the total momentum of the system 
of two balls before the collision? Keep in mind 
here that velocity and, therefore, momentum 
are vector quantities. 

2. What is the total momentum of the system 
of two balls after the collision? 

3. Was momentum conserved within the limits 
of precision of your measurements? 

Event 1 

The photographs of this Event 1 and aU the 
following events appear as Figs. 3-11 to 3-17. 
This event is also shown as the first example in 
Film Loop 18, "One-Dimensional Collisions. I." 

Figure 3-4 shows that ball B was initially at 
rest. After the collision, both balls moved off to 
the left. (The balls are made of hardened steel.) 

B-VENT 1 



before 



&Vg-NT 2. 



be-fc 



^ o 

i)50 yam^ 632 grams 



a-ft^r 
Fig. 3-5 



O 



2>50 ^amt> 52>2 jranris 



collision reversed the direction of motion of 
ball B and sent ball A off to the right. (The balls 
are of hardened steel.) 

As you can tell by inspection, bail B moved 
slowly after collision, and thus you may have 
difficulty getting a precise value for its speed. 
This means that your value for this speed is the 
least reliable of your four measurements of 
speed. Nevertheless, this fact has only a small 
influence on the reliability of your value for the 
total momentum after collision. Can you ex- 
plain why this should be so? 

Why was the direction of motion of ball B 
reversed by the collision? 

If you have already studied Event 1, you wall 
notice that the same balls were used in Events 1 
and 2. Check your velocity data, and you will 
find that the initial speeds were nearly equal. 
Thus, Event 2 was truly the reverse of Event 1 . 
Why, then, was the direction of motion of ball A 
in Event 1 not reversed although the direction 
of ball B in Event 2 was reversed? 

Event 3 

This event is shown as the first example in 
Film Loop 19, "One-Dimensional Collisions. II." 
Event 3 is not recommended until you have 
studied one of the other events. Event 3 is 
especially recommended as a companion to 
Event 4. 

Figure 3-6 shows that a massive ball A 
entered from the left. A less massive ball B came 



Fig. 3-4 



O 



e-^/pKiT 2. 



Event 2 

This event, the reverse of Event 1, is showTJ as 
the second example in Film Loop 18. 

Figure 3-5 shows that ball B came in fixjm the 
left and that ball A was initially at rest. The 



be-Pc 



aft-c-»' 



I 80 icilogK-afn 632 gratis 



Fig. 3-6 



106 



UNIT 3 / THE TRIUMPH OF MECHANICS 



in from the right. The directions of motion of 
both balls were reversed by the collision. (The 
balls are made of hardened steel.) 

When you compare the momenta before and 
after the collision, you will probably find that 
they differed by more than any other event so 
far in this series. Explain why this is so. 

Event 4 

This event is also shown as the second 
example in Film Loop 19. 



beforiE f j — *- Cj — »- 



Rg. 3-7 



O o- 



E /eMT 5 



b«rfc3r-« 



o O 



Fig. 3-8 



-o 



Event 6 

This event is shown as the second example 
in Film Loop 20. 

Figure 3-9 shows that balls .A and B mo\'ed in 
from the right and left, resp>ecti\'ely, before 
collision. The balls are made of a soft material 
(plasticene). They remained stuck together after 
the collision and moved oft" together to the left. 
This is another perfectly inelastic collision, like 
that in Event 5. 



Figure 3-7 shows that two balls came in ftxjm 
the left, that ball A was far more massive than 
ball B, and that ball A was moving faster than 
ball B before collision. The collision occurred 
when A caught up with B, increasing Bs speed 
at some expense to its own speed. (The balls 
are made of hardened steel.) 

Each ball moved across the camera's field 
from left to right on the same line. In order to 
be able to tell successive positions apart on a 
stroboscopic photograph, the picture was 
taken twice. The first photograph shows only 
the progress of the large ball A because ball B 
had been given a thin coat of black paint (of 
negligible mass). Ball A was painted black when 
the second picture was taken It will help you 
to analyze the collision if you number white- 
ball positions at successive stroboscope flashes 
in each picture. 

Ewnt 5 

This event is also shov\n as the first example 
in Film Loop 20, "Inelastic One-Uimensional 
Collisions." You should find it interesting to 
analyze this event or Ewnt 6 or Ewnt 7, but it is 
not necessary to do mon* than one 

Figuiv 3-H shows that ball A came in from the 
right, striking ball B which was initially at nvst 
The balls are made of a soft material (plas- 
ticene). Tliey n'mained stuck together after the 
collision and moved off to the left as one. A 
collision of Ihis type is called /wrferf/y inelastic 



eVEMT (S 



toc^ore. 



A-^t^r- 



Fig. 3-9 



c^-O 



445 jr»rr^ 



(h&l S^ams 



A-B 



-cO 



This event was photographed in two parts. 
The first print shows the conditions before 
collision, the second print, after collision Had 
the j)ictiire been taken with the camera shutter 
open throughout the motion, it would be 
difficult to take measurements because the 
combined balls (A -t- Bi after collision retraced 
the path that ball B followed Iw'fore collision 
You can numl>er the positions of each liall 
l)efore collision at successiw flashes of the 
strobosco|>e lin the fir-st photol; you can do 
likewise for the combined balls (A + BI after the 
collision in the second photo. 

Event 7 

Figuiv 3-10 shows that balls A and B moxipd 
in from op|H)site dinvtion.s iK'fon* collision 
I'he balls an> made of a soft material (plas- 
ticenei Th('v remained stuck together after 
collision and mo\t*d off together to the right 
Ibis is another jK'rfeclly inelastic collision. 



EXPERIMENTS 



107 



B-VeNiT 7 



\o<.-forc 
Hg. 3-10 



4.79 kilogram--. (^GOgf^ms 



A+ B 



o> 



This event was photographed in two parts. 
The first print shows the conditions before 
collision, the second print, after collision. Had 



the picture been made with the camera shutter 
open throughout the motion, it would be 
difficult to take measurements because the 
combined balls (A + B) return along the same 
path as incoming ball B. You can number the 
positions of each ball before collision at succes- 
sive flashes of the stroboscope (in the first 
photograph); you can do likewise for the 
combined balls (A + B) after collision in the 
second photograph. 

Photographs of the Events 

The photographs of the events are shown in 
Figs. 3-11 through 3-17. 




Rg. 3-12 Event 2 
10 flashes/sec. 




Bnni-K :o 




EXPERIMENTS 109 




Fig3-15 Event 5, 10 flashes/sec. 




• 


i# •« «# 


c 



Fig3-16 Event 6, 10 flashes/sec. 



no 



UNIT 3 / THE THIUMPH OF MECHANICS 



• •• 




Fig. 3-17 Event 7, 10 flashes/sec. 



Expcrinicnt 3-3 
C:f)MJSIf)XS IX TUO 
niMEXSIOXS. I 

Collisions rarely occur in only one dimension, 
that is, along a straight line. In billiards, 
baskj-thall, and tennis, the ball usually re- 
bounds at an angle to its original direction. 
Oixlinaiy e.\plosions (which can be thought of 
as collisions in which initial \elocities art? all 
zero) send pieces Hying off in all directions 

Ibis e-xperinient deals with collisions that 
occur in two dimensions, that is, in a single 
plane, instead of along a single straight line It 
assumes that you know what momentum is 
and understand what is meant l)y conserva- 
tion of momentum in one tlimension In this 
experiment, you will discover a general fonn of 
the nil,, for one dimension that applies also to 
the conservation of momentum in cases when> 
the parts of the system move in two km lhn>ei 
dimensions 

Iwo methods of getting data on two- 
dimensional cdllisions are described in 
Melliods A and H (and two others in Kxperi 
ment 3-41, but you will pix)bably want to M\o\k 
only one method \Vhiche\fr method vou use 
handle your results in the wa> ih-s( hImhI in 
"Analvsis of Data." 




EXPERIMENTS 



111 



METHOD A. Colliding Pucks 

On a carefully leveled glass tray covered with a 
sprinkling of Dylite spheres, you can make 
pucks coast with almost uniform speed in any 
direction. Set one puck motionless in the 
center of the table and push a second similar 
one toward it, a little off-center. You can make 
excellent pictures of the resulting two- 
dimensional glancing collision with a camera 
mounted directly above the surface. 

To reduce reflection from the glass tray, the 
photograph should be taken using the xenon 
stroboscope uath the light on one side and 
almost level with the glass tray. To make each 
puck's location clearly visible in the photo- 
graph, attach a steel ball or a small white 
Styrofoam hemisphere to its center. 

You can get a great variety of masses by 
stacking pucks one on top of the other and 
fastening them together with tape (avoid having 
the collisions cushioned by the tape). 

Two people are needed to do the experi- 
ment. One experimenter, after some prelimi- 
nary practice shots, launches the projectUe 
puck while the other experimenter operates 
the camera. The resulting picture should 
consist of a series of white dots in a rough "Y" 
pattern. 

Using your picture, measure and record all 
the speeds before and after collision. Record 
the masses in each case too. Since you are 
interested only in comparing speeds, you can 
use any convenient speed units. You can 
simplify your work if you record speeds in 
millimeters per dot instead of trying to work 
them out in centimeters per second. Because 
friction does slow the pucks down, find speeds 
as close to the impact as you can. You can also 
use the "puck " instead of the kilogram as your 
unit of mass. 

METHOD B. Colliding Disk Magnets 

Disk magnets wall also slide freely on Dylite 
spheres as described in Method A. 

The difference here is that the magnets need 
never touch during the "collision." Since the 
interaction forces are not really instantaneous 
as they are for the pucks, the magnets follow 
curving paths during the interaction. Con- 
sequently the "before" velocity should be 
determined as early as possible and the "after" 
velocities should be measured as late as 
possible. 

Following the procedure described above for 
pucks, photograph one of these "collisions." 



Again, small Styrofoam hemispheres or steel 
balls attached to the magnets should show up 
in the strobe picture as a series of white dots. 
Be sure the paths you photograph are long 
enough so that the dots near the ends are along 
straight lines rather than curves (see Fig. 3-18). 




Fig. 3-18 



Using your photograph, measure and record 
the speeds and record the masses. You can 
simplify your work if you record speeds in 
millimeters per dot instead of working them out 
in centimeters per second. You can use the disk 
instead of the kilogram as your unit of mass. 

Analysis of Data 

whichever procedure you used, you should 
analyze your results in the same way. 



1. Multiply the mass of each object by its 
before-the-collision speed, and add the products. 

2. Do the same thing for each of the objects in the 
system after the collision, and add the after-the- 
collision products together. Does the sum before 
the collision equal the sum after the collision? 



Imagine the collision you observed was an 
explosion of a cluster of objects at rest; the total 
quantity mass-times-speed before the explo- 
sion will be zero. But surely, the mass-times- 
speed of each of the flying fragments after the 
explosion is more than zero! "Mass-times- 
speed" is obviously not conserved in an explo- 
sion. You probably found it was not conserved 
in the experiments with pucks and magnets, 
either. You may already have suspected that 
you ought to be taking into account the 
directions of motion. 



112 



U.MT 3 / THE TRIUMPH OF MECHANICS 



To see what is conserved, proceed as follows. 

Use your measurements to construct a 
drawing like Fig. 3-19, in which you show the 
directions of motion of all the objects both 
before and after the collision. 




Fig. 3-19 ^ 

Have all the direction lines meet at a single 
point in your diagram. The actual paths in your 
photographs will not do so, because the pucks 
and magnets are lai^e objects instead of points, 
but you can still draw the directions of motion 
as lines thnjugh the single point P. 

On this diagram draw a vector arrow whose 
magnitude (length) is proportional to the mass 
times the speed of the projectile before the 
collision. (You can use any convenient scale.) In 
Fig. 3-20, this vector is marked ni^v^. 



d6FOf!e 



Fig. 3-20 




C0LLI5/ON 



Below your first diagram draw a second one 
in which you once more draw the directions of 
motion of all the objects exactly as before. On 
this second diagram, construct the vectors for 
mass-times-speed for each of the objects 
leaving Pafter the collision. For the collisions of 
pucks and magnets, your diagram will resem- 
ble Fig. 3-21. Now construct the "after-the- 
collision" vector sum. 







Collision 



Fig. 3-21 

rhe ItMigth of each of your arrows is giv-en by 
the pnxhict of mass aiul speed. Since each 
arrow is drawn in the liirrction of the speed, 
the arrows n^present the pnuluct of mass and 



velocity mv which is called momentum. The 
vector sums "before" and "after" collision, 
therefore, represent the total momentum of the 
system of objects before and after the collision. 
If the "before" and "after" arrows are equal, 
then the total momentum of the system of 
interacting objects is conserved. 



3. How does this vector sum compare with the 
vector sum on your before-the-collision figure? 
Are they equal within the uncertainty? 

4. Is the principle of conservation of momentum 
for one dimension different from that for two, or 
merely a special case of it? How can the principle 
of conservation of momentum be extended to 
three dimensions? SIcetch at least one example. 

5. Write an equation that would express tfie 
principle of conservation of momentum for colli- 
sions of (a) three objects in two dimensions, (b) 
two objects in three dimensions, (c) three objects 
in three dimensions. 




Fig. 3-22 A 1.350 kg steel ball s^ung by a crane against 
the walls of a condemned building What happens to the 
momentum of the l>all7 



EXPERIMENTS 



113 



Experiment 3-4 
COLLISIONS IN TWO 
DIMENSIONS. II 

METHOD A. Film Loops 

Several Film Loops (21, 22, 23, 24, and 251 show 
tw'o-dimensional collisions that you cannot 
conveniently reproduce in the laboratory. 
Notes on these films appear on pages 77-79. 
Project one of the loops on the chalkboard or 
on a sheet of graph paper. Trace the paths of 
the moving objects and record their masses 
and measure their speeds. Then go on to the 
analysis described in notes for FUm Loop 21. 

METHOD B. Stroboscopic 
Photographs 

Stroboscopic photographs* of seven different 
two-dimensional collisions in a plane are used 
in this experiment. The photographs (Figs. 3-27 
to 3-34) are shown on the pages immediately 
following the description of these events. They 
were photographed during the making of Film 
Loops 21-25. 

I. Material Needed 

1. A transparent plastic ruler, marked in mil- 
limeters. 

2. A large sheet of paper for making vector 
diagrams. Graph paper is convenient. 

3. A protractor and two large drawing triangles 
are useful for transferring directional vectors 
fixim the photographs to the vector dicigrams. 

II. How the Collisions Were Produced 

Balls were hung on 10-m wires, as shown 
schematically in Fig. 3-23. They were released 
so as to collide directly above the camera, 
which was facing upward. Electronic strobe 
lights (shown in Fig. 3-26) illuminated the 
rectangle shown in each fi^ame. 

Two white bars are visible at the bottom of 
each photograph. These are rods that had their 
tips 1 m (±2 mml apart in the actual situation. 
The rods make it possible for you to convert 
your measurements to the actual distance. It is 
not necessary to do so, if you choose instead to 
use actual on-the-photograph distances in 
millimeters (as you may have done in your 
study of one-dimensional collisions). 

Since the balls are pendulum bobs, they 
move faster near the center of the photographs 

'Reproduced by permission of National Film Board 
of Canada. 




Fig. 3-23 Setup for photographing two-dimensional 
collisions. 



than near the edge. Your measurements, there- 
fore, should be made near the center. 

III. A Sample Procedure 

The purpose of your study is to see to what 
extent momentum seems to be conserved in 
two-dimensional collisions. For this purpose 
you need to construct vector diagrams. 

Consider an example: In Fig. 3-24, a 450-g and 
a 500-g ball are moving toward each other. Ball 
A has a momentum of 1.8kg-m/sec, in the 
direction of the ball's motion. Using the scale 
shown, draw a vector 1.8 units long, parallel to 
the direction of motion of A. Similarly, for ball B 
draw a momentum vector 2.4 units long, 
parallel to the direction of motion of B. 

The system of two balls has a total momen- 
tum before the collision equal to the vector sum 
of the two momentum vectors for A and B. 

The total momentum after the collision is 
also found the same way, by adding the 
momentum vector for A after the collision to 
that for B after the collision (see Fig. 3-25). 

This same procedure is used for any event 
you analyze. Determine the momentum (mag- 
nitude and direction) for each object in the 
system before the collision, graphically add 
them, and then do the same thing for each 
object after the collision. 



114 



UNIT 3 / THE THILMPH OF MECHAMtS 



\ A 



4-50 ^ 



Scale 
1 Itg. m/sc-o 
I i 



500 




Fig. 3-24 Two balls moving in a plane. Their individual 
momenta, which are vectors, are added together 
vectorially in the diagram on the lower right The vector 
sum is the total momentum of the system of two balls. 
(Your own vector drawings should be at least twice this 
size.) 



\ / 

> \ 



/ 



Scale 
lux?. rn^CO 

V^ 1 



fe 



/ 4i 



For each event that you analyze, consider 
whether momentum is conserved. 

Events 8, 9, 10, and 11 

Event 8 is shown as the first example in Film 
Loop 22, "Two-Dimensional Collisions. II." 

Event 10 is shown as the second example in 
Film Ixjop 22. 

Event 11 is also shown in Film Loop 21, 
"Two-Dimensional Collisions. I." 

These are all elastic collisions. Events 8 and 
10 are simplest to analyze hecause each shows 
a collision of equal masses. In Events 8 and 9, 
one hall is initially at ivst. 

A small sketch next to each photograph 
indicates the direction of motion of each ball. 
The mass of each ball and the strobe rate are 
also given. 



b«-fore 




af "tcr 



Fig. 3-25 The two balls collide and move away Their 
individual momenta after collision are added vectorially. 
The resultant vector is the total momentum of the 
system after collision. 



Events 12 and 13 

KviMit 12 is shown as the first example in Film 
Ix)()p 23, 'Inelastic Two-Dimensional C'olli- 
sions. ' 

Event 13 is shown as the second example in 
Film Loop 23. 



Since Events 12 and 13 are similar, there is no 
need to do lK)th. 

Ewnts 12 and 13 show inelastic collisions 
l)etween two plasticene balls that stick together 
and move ofT as one compound object after the 



EXPEKI.ME.VTS 



115 



collision. In Event 13 the masses are equal; in 12 
they are unequal. 

CAUTION: You may find that the two objects 
rotate slightly about a common center cifter the 
collision. For each image after the collision, you 
should make marks halfway between the cen- 
ters of the two objects. Then determine the 
velocitv' of this center of mass, and multiply it 
by the combined mass to get the total momen- 
tum after the collision. 

Event 14 

Do not try' to analN'ze Event 14 unless you 
have done at least one of the simpler Events 
8 through 13. 

Event 14 is shown on Film Loop 24 "Scatter- 
ing of a Cluster of Objects." 

Figure 3-26 shows the setup used in photo- 
graphing the scattering of a cluster of balls. 
The photographer and camera are on the floor; 
and four electronic stroboscope lights are on 
tripods in the lower center of the picture. 

Use the same graphical methods as you used 
for Events 8-13 to see if the conservation of 
momentum holds for more than two objects. 
Event 14 is much more complex because you 
must add seven vectors, rather than two, to get 
the total momentum after the collision. 

In Event 14, one ball comes in cmd strikes a 
cluster of six balls of various masses, which 
were initially at rest. Two photographs are 
included: print 1 shows only the motion of ball 
A before the event: print 2 shows the positions 
of cill seven balls just before the collision and 
the motion of each of the seven balls after the 
collision. 




Fig. 3-26 Catching the seven scattered balls to avoid 
tangling the wires from which they hang. The photo- 
grapher and the camera are on the floor. The four 
stroboscopes are on tripwads in the lower center of the 
picture. 



You can analyze this event in two different 
ways. One way is to determine the initial 
momentum of ball A fixjm measurements taken 
on print 1 and then compare it to the total final 
momentum of the system of seven balls from 
measurements taken on print 2. The second 
method is to determine the total final momen- 
tum of the system of seven balls on print 2, 
predict the momentum of ball A, and then take 
measurements of print 1 to see whether baU A 
had the predicted momentum. Choose one 
method. 

The tops of prints 1 and 2 lie in identical 
positions. To relate measurements on one print 




AQ3C73 



■^^ 



e>Q2>G7 2 



Rg. 3-27 Event 8, 20 flashes/sec. 



116 



UNIT 3 / THE TRIDMI'H Ol MECHANICS 



to measurements on the other, measure a bail's 
distance relative to the top of one picture with a 
ruler; the ball would lie in precisely the same 
position in the other picture if the two pictures 
could be superimposed. 

There are two other matters you must 
consider. First, the time scales are different on 
the two prints. Print 1 was taken at a rate of 5 
flashes/sec, and print 2 was taken at a rate of 20 



flashes/sec. Second, the distance scale may not 
be exactly the same for both prints. Remember 
that the distance from the center of the tip of 
one of the white bars to the center of the tip of 
the other is 1 m i ±2 mm) in real space. Check 
this scale carefully on both prints to determine 
the conversion factor. 

The stroboscopic photographs for Events 
8- 14 appear in Figs. 3-27 to 3-34 






367< 



Fig. 3-28 Event 9, 20 flashes/sec. 






Fig. 3-29 Event 10, 20 flashes/sec. 



A 



^ ^P 



Fig. 3-30 Event 11, 20 flashes/sec. 




EXPERIMENTS 117 




Fig. 3-31 Event 12, 20 flashes/sec. 







5003 



X3 



Fig. 3-32 Event 13, 10 flashes/sec. 




Fig. 3-33 Event 14, print 1, 20 flashes/sec 



AO16OO2 



118 UNIT 3 / THE THILMPIf OF MECHANICS 




Rfl. 3-34 Event 14, print 2, 20 flashes/sec. 



Experiment 3-5 
CONSERVATION OF ENERGY. I 

In the previous experiments on conservation of 
momentum, you recorded the results of a 
numher of collisions involving carts and gliders 
having different initial velocities. You found 
that within the limits of experimental uncer- 
tainty, momentum was conserved in each case. 
You can now use the results of these collisions 
to leam about another extremely useful con- 
servation law, the conservation of energy. 

Do you have any reason to believe that the 
product of m and v is the only conserved 
quantity? In the data obtained fhjm your 
photographs, look for other combinations of 
quantities that might be conserved. Find values 
for tn/v, m V, and mv^ for each cart before and 



after collision, to see if the sum of these 
quantities for both carts is conser\'ed. Comp>are 
the results of the elastic collisions v\ith the 
inelastic ones. Consider the "explosion," too. 

Is there a quantit>' that is conserved for one 
type of collision but not for the other? 

There are several alternative methods to 
explore further the answer to this questioniyou 
will probably wish to do just one. Check your 
results against those of classmates who use 
other methods. 

METHOD A. Dynamics Carts 

To take a closer look at the detaUs of an elastic 
collision, photograph two d\Tiamics carts as 
you may ha\-e done in a pre\ious exjjeriment. 
Set the carts up as shown in Fig. 3-35. 




liQhtvxjncesot s((Qb+ly different hcIah+$ 

^ ^ 



^^ 




Run off \f^ c£r\ier of table 
( I35cm minimum width} 



Or^ of 12 disc slots faped 
Qlrnost holf closed 



Fig 3-3S 



EXPERIMENTS 



119 



The mass of each cart is 1 kg. Extra mass is 
added to make the total masses 2 kg and 4 kg. 
Tap>e a light source on each cart. So that you 
can distinguish between the images formed by 
the two lights, make sure that one of the bulbs 
is slightly higher than the other. 

Place the 2-kg cart at the center of the table 
and push the other cart toward it from the left. 
If you use the 12-slot disk on the stroboscope, 
you should get several images during the time 
that the spring bumpers are touching. You will 
need to know which image of the right-hand 
cart was made at the same instant as a given 
image of the left-hand cart. Matching images 
will be easier if one of the 12 slots on the 
stroboscope disk is half-covered with tape (Fig. 
3-36). Images formed when that slot is in front 
of the lens will be fainter than the others. 



r-^ 




'//'' 



V 



Rg. 3-36 



Now compute values for the scalar quantity 
mv- for each cart for each time interval, and add 
them to your table. On another sheet of graph 
paper, plot the values of mv- for each cart for 
each time interval. Connect each set of values 
with a smooth curve. 

Now draw a third curve that shows the sum 
of the two vjilues of mv^ for each time interval. 



3. Compare the final value of mv^ for the system 
with the initial value. Is m\/^ a conserved quantity? 

4. How would the appearance of your graph 
change if you multiplied each quantity by Vi? (The 
quantity Vimv^ is called the kinetic energy of the 
object of mass m and speed v.) 



Compute values for the scalar quantitv' V2mv- 
for each cart for each time inter\'al. On a sheet 
of graph paper, plot the kinetic energy of each 
cart as a function of time, using the same 
coordinate axes for both. 

Now draw a third curve that shows the sum 
of the two values of Vimv- for each time interval. 



5. Does the total amount of kinetic energy vary 
during the collision? If you found a change in the 
total kinetic energy, how do you explain it? 



Compute the values for the momentum mv 
for each cart for each time intenal while the 
springs were touching, plus at least three 
intervals before and after the springs touched. 
List the values in a table, making sure that \ou 
pair off the vcdues for the two carts correctly. 
Remember that the lighter cart was initially at 
rest while the hea\ier one moved toward it. 
This means that the first few \'alues of mv for 
the lighter cart will be zero. 

On a sheet of graph paper, plot the momen- 
tum of each cart as a function of time, using the 
same coordinate axes for both. Connect each 
set of values with a smooth curve. 

Now draw a third curve that shows the sum 
of the two values ofmv ithe total momentum of 
the svstem) for each time interval. 



1. Compare the final value of mv for the system 
with the initial value. Was momentum conserved 
in the collision? 

2. What happened to the momentum of the 
system while the springs were touching; was 
momentum conserved during the collision? 



METHOD B. Magnets 

Spread some Dylite spheres (tiny plastic beads) 
on a glass tray or other hard, flat surface. A disk 
magnet will slide freely on this low-friction 
surface. Level the surface carefully. 

Put one disk magnet at the center and push a 
second one toward it, slightly off" center. You 
want the magnets to repel each other without 
actually touching. Try varying the speed and 
direction of the pushed magnet until you find 
conditions that make both magnets move off 
after the collision with about equal speeds. 

To record the interaction, set up a camera 
directly above the glass tray i using the motor- 
strobe mount if your camera does not attach 
directly to the tripod) and a xenon stroboscojae 
to one side as in Fig. 3-37. Mount a steel ball or 
a Styrofoam hemisphere on the center of each 
disk VNith a small piece of clay. The ball will give 
a sharp reflection of the strobe light. 

Take strobe photographs of several interac- 
tions. There must be several images before and 
after the interaction, but you can vary the initial 
speed and direction of the mo\ing magnet to 



120 



I MT 3 / THE TRIUMPH OF MECHANICS 




Fig. 3-37 

get a variety of interactions. Using your photo- 
graph, calculate the "hefore" and "after" sjjeeds 
of each disk. Since you are interested only in 
comparing speeds, you can use any convenient 
units of measurement for speed. 



1. Is mi/2 a conserved quantity? Is V2mv^ a 
conserved quantity? 



If you find that there has been a decrease in 
the total kinetic energy of the system of 
interacting magnets, consider the follov\ing: 
The surface is not perfectly frictionless and a 
single magnet disk pushed across it \%ill slow 
down a bit. Make a plot of Vzfnv- against lime 
for a moving disk to estimate the rate at which 
kinetic energ\' is lost in Ibis way. 



2. How much of the loss in '/jmw' that you 
observed in the interaction can be due to friction? 

3. What happens to your results if you consider 
kinetic energy to be a vector quantity? 



Wlicn the two disks ar»> close together (but 
not touching) there is quite a strong force 
between them pushing them apart. If you put 
the two disks down on the surface close 
together and n«lease them, they will fly apart: 
Ibe kinetic en«Tg\ of the system has incnMseil 

If \()u have lime to go on, nou should In lo 
find out what happens lo the total ({uanlity 
Viniv* of the disks while they an* close together 



during the interaction. To do this you will use a 
fairiy high strobe rate, and push the projectile 
magnet at fairly high speed, without letting the 
two magnets actually touch, of course. Close 
the camera shutter before the disks are out of 
the field of view so that you can match images 
by counting backward from the last images. 

Now, working backward from the last inter- 
val, measure v and calculate VT/nv- for each 
disk. Make a graph in which you plot Vimv' for 
each disk against time. Draw smooth curves 
through the two plots. 

Now draw a third curve that shows the sum 
of the two Vzmv- values for each time interval. 



4. Is the quantity ^/imv^ conserved during the 
interaction, that is, while the repelling magnets 
approach very closely? 
Try to explain your observations. 



METHOD C. Inclined Air Tracks 

Suppose you give the glider a push at the 
bottom of an inclined air track. As it moves up 
the slope it slows down, stops momentarily, 
and then begins to come back dovx-n the track. 

Clearly the bigger the push you give the 
glider (the greater its initial velocity v,), the 
higher up the track it v\ill climb before 
stopping. From expierience you know that there 
is some connection between v, and d. the 
distance the glider moves along the track 

According to physics texts, when a stone is 
thrown upwartl, the kinetic ener^ that it has 
initially (V2n7i',-) is transformed into gravita- 
tional potential energv' (ma,/jl as the stone 
moves up. In this experiment, you will test to 
see whether the same relationship applies to 
the bj'havior of the glider on the inrlinetl air 
track In paHicular, your task is to find the 
initial kinetic cnei^' and the increase in 
potential energy of the air track glider and to 
compare them. 

rhe puqjose of the first set of measurements 
is to find the initial kinetic energv •j/rj*,*. You 
cannot measutv \ , dinTtlv , but vou van find it 
from vour calculation of the. i\rr,ige vvlocity »',, 
as follows. In the case of unifonii acceleration 
v,v = Vi(v, + v,l. Since final velocity' v, = at the 
top of the track. v„ = V^»'i or v, = 2v,,. 
Remember that v., = Ad/Al, so v, = 2( Ad/ All: 
A</ and At an» easy to measure with vnur 
ap|>aratus 

To measure Ad and Af. three f>eople are 
needed: one gives the glider the initial push. 



EXPERI.ME.V IS 



121 



another marks the highest point on the track 
reached by the glider, and the third uses a 
stopwatch to time the motion from push to 
rest. 

Raise one end of the track a few centimeters 
above the tabletop. The launcher should prac- 
tice pushing to produce a push that will send 
the glider nearly to the raised end of the track. 

Record the distance traveled and time taken 
for several trials, and weigh the glider. Deter- 
mine and record the initial kinetic energj-. 

To calculate the increase in gra\itational 
potential energx, you must measure the vertical 
height h through which the glider moves for 
each push. Vou will probabK' find that you 
need to measure from the tabletop to the track 
at the initial hi and final /i j points of the glider's 
motion (see Fig. 3-38), since h - hf - h^. 
Calculate the potential energy increase, the 
quantity maji, for each of your trials. 




Fig. 3-38 

For each trial, compcire the kinetic energv' 
loss v\ith the potential energs' increase. Be sure 
that you use consistent units: m in kilograms, v 
in meters/second, a^ in meters/second-, h in 
meters. 



1. Are the kinetic energy loss and the potential 
energy increase equal within your experimental 
uncertainty? 

2. Explain the significance of your result. 



Here are more things to do if you have time to 
go on: 

(a) See if your answer to Question 1 continues 
to be true as you make the track steeper and 
steeper. 

(b) Ulien the glider rebounds from the rubber 
band at the bottom of the track, it is momentar- 
ily stationaiy; its kinetic energy is zero. The 
same is true of its gravitational potential energy, 
if you use the bottom of the track eis the zero 
level. Yet the glider will rebound from the 
rubber b<md (regain its kinetic energy! and go 
quite a way up the track (gaining gravitational 
potential energy! before it stops. See if you can 
explain what happens at the rebound in terms 
of the conversation of mechanical energy. 



(c) The glider does not get quite so far up the 
track on the second rebound as it did on the 
first. There is evidently a loss of energy. See if 
you can measure how much energy is lost each 
time. 



Experiment 3-6 
CONSERVATION OF ENERGY. II 

METHOD A. Film Loops 

You may have used one or more of Film Loops 
18-25 in your study of momentum. You will 
find it helpful to view these slow-motion films 
of one- and two-dimensional collisions again, 
but this time in the context of the study of 
energy. The data you collected previously will 
be sufficient for you to calculate the kinetic 
energv' of each ball before and after the 
collision. Remember that kinetic energy V2mv- 
is not a vector quantity, and therefore you need 
only use the magnitude of the velocities in your 
calculations. 

On the basis of your analysis you may wish to 
try to answer such questions as: Is kinetic 
energy conserved in such interactions? If not, 
what happened to it? Is the loss in kinetic 
energv' related to such factors as relative speed, 
angle of impact, or relative masses of the 
colliding balls? Is there a difference in the 
kinetic energy lost in elastic and inelastic 
collisions? 

The FUm Loops were made in a highly 
controlled laboratoiy situation. ,After you have 
developed the technique of measurement and 
analysis from Film Loops, you may want to turn 
to one or more loops dealing with things 
outside the laboratorv' setting. FUm Loops 
26-33 involve freight cars, billiard balls, pole 
vaulters, and the like. Suggestions for using 
these loops can be found on pages 81-87. 



METHOD B. Stroboscopic 
Photographs of Collisions 

V\'hen studying momentum, you may have 
taken measurements on the one-dimensional 
and two-dimensional collisions shown in 
stroboscopic photographs on pages 107-110 
and 115-118. If so, you can now easily reex- 
amine your data and compute the kinetic 
energy Vzmv- for each ball before and after the 
interaction. Remember that kinetic energy is a 
scalar quantity, and so you will use the mag- 
nitude of the velocity but not the direction in 



laa 



UNIT 3 / THE TRIUMPH Ol MECHANICS 



making your computations. You would do well 
to study one or more of the simjiler events (for 
example, Events 1, 2, 3, 8, 9, or 10) before at- 
tempting the more complex ones involving 
inelastic collisions or several balls. Also, you 
may wish to review the discussions given 
earlier for each event. 

If you find there is a loss of kinetic energy 
beyond what you would expect from mea- 
surement error, try to explain your results. 
Some questions you might try to answer are: 
How does kinetic enei^ change as a function 
of the distance from impact? Is it the same 
before and after impact? Mow is energy conser- 
vation influenced by the relative speed at the 
time of collision? How is energy conservation 
influenced by the angle of impact? Is there a 
difference between elastic and inelastic interac- 
tions in the fraction of energy conserved? 



of the projectile before impact. There are at 
least two ways to find v'. 



METHOD A. Air Track 

The most dirtnt way to find v ' is to mount the 
target on the air track and to time its motion 
after the im|)act. (See Fig. 3-39.) Mount a small 
can, lightly packed with cotton, on an air-track 
glider. Make sure that the glider will still ride 
freely with this extra load. Fire a "bullet" la 
pellet from a tov gun that has been checked for 
safety by your instructor) horizontalK', parallel 
to the length of the air track. If M is lai^e 
enough, compared torn, the gliders speed will 
be low enough so that you can use a stoprwatch 
to time it over a 1-m distance. Rep>eat the 
measurement a few times until you get consis- 
tent results. 



Experiment 3-7 

MEASl KING THE SPEED OF 

A BULLET 

In this experiment you will use the principle of 
the conservation of momentum to find the 
speed of a bullet. Sections 92 and 9.3 in the 
Text discuss collisions and define momentum 

You will use the general equation of the 
principle of conservation of momentum for 
two-body collisions: m^^ + ^b^^ - ^^a^a "^ 
m,^„. 

The experiment consists of firing a projectile 
into a can packed with cotton or into a hea\y 
block that is free to move horizontally. Since all 
velocities before and af^er the collision are in 
the same direction, you may neglect the v-ector 
nature of the (equation above and work only 
with speeds. To avoid subscripts, call the mass 
of the target M and the much smaller mass of 
the projectile /7i. Before impact, the tai^et is at 
rest, so you have only the speed v of the 
pn)jectile to consider After impact, both the 
target and the added pi-ojectile move with a 
common sjieed \ ' Thus, the general equation 
becomes 

mv = (Af + mlv' 
or 

{M +m)v' 

lloth masses aiv easy to measure Iherefore. if 
the comparatiwiy slow speed v' can 1m' found 
after impact, you can compute the high s|>eed \ 




Fig. 3-39 



1. What IS your value for the bullet's speed? 

2. Suppose the collision between bullet and can 
was not completely inelastic, so that the bullet 
bounced back a little after impact. Would this 
increase or decrease your value for the speed of 
the bullet? 

3. Can you think of an independent way to 
measure the speed of the bullet? If you can. make 
the independent measurement. Then see if you 
can account for any differences between ttie two 
results. 



MFTHOn B. R;illis!ic Ptnduluiii 

rliis was lh«' onginal methixi ot deti>niiining 
the sjM'ed of bullets inwnled in 1742 and still 
used in stime oixinance lal>oratnn»»s .A mo\ablr 
block is susptMuled as a tn»ely swinging pen- 
dulum whose motion nnvals the bullet s s[>eed. 

Ohtaiiiiii^ tlu* S|M*e(l Kqiiatioii 

Ihe collision is inelastic, so kinetic enepfjv w 
n«it conserved in the inifKicI But during the 
neari\- frirtionle5is swing of the |x>ndulum af^er 
the impact mechanical enerRN' is conser%T»d 



EXPERIMENTS 



123 



that is, the increase in gravitational potential 
energy of the pendulum at the end of its 
upward swing is equal to its kinetic energy 
immediately after impact. Written as an equa- 
tion, this becomes 

(M +m)aan = 

* 2 

where h is the increase in height of the 
pendulum bob. 

Solving this equation for v' gives: 



in the 



v' = V2ag/i 

Substituting this expression for v' 
momentum equation above leads to 

;V/ + m .. /r — r 



Now you have an equation for the speed v of 
the bullet in terms of quantities that are known 
or can be measured. 

A Useftil Approximation 

The change h in vertical height is difficult to 
measure accurately, but the horizontal dis- 
placement d may be 10 cm or more and can be 
found easily. Can h be replaced by an equiva- 
lent expression involving d? The relation bet- 
ween h and d can be found by using a little 
plane geometry. 

In Fig. 3-40, the center of the circle, O, 
represents the point from which the pendulum 
is hung. The length of the cords is /. 




In the triangle OBC, 

SO l^^d^ +l^-2lh +h' 

and 2//i = d^ + h^ 

For small swings, h is smcill compared with / 
and d, so you may neglect h^ in comparison 
with d', and write the close approximation 

2lh - d' 
or h = d'-/2l 

Putting this value of h into your last equation 
forv above and simplifying gives: 



_(M +m)d^ AT" 
m V T 



If the mass of the projectUe is small com- 
pared with that of the pendulum, this equation 
can be simplified to another good approxima- 
tion. How? 

Finding the Projectile's Speed 

Now you are ready to begin the experiment. 
The kind of pendulum you use will depend on 
the nature and speed of the projectile. If you 
use pellets from a toy gun, a cylindrical 
cardboard carton stuffed lightly with cotton 
and suspended by threads from a laboratory 
stand will do. If you use a good bow and arrow, 
stuff straw into a fairly stiff corrugated box and 
hang it irom the ceiling. To prevent the target 
pendulum irom twisting, hang it by parallel 
cords connecting four points on the pendulum 
to four points direcdy above them, as in Fig. 
3-41. 



^-[:fM 



D 




Hg. 3-40 



Rg. 3-41 



124 



UNIT 3 / THE TRILMPH OF MECHANICS 



To measure d, a light rod la pencil or a soda 
straw) is placed in a tube clamped to a stand. 
The rod extends out of the tube on the side 
toward the pendulum. As the pendulum 
swings, it shoves the rod back into the tube so 
that the rod's final position marks the end of 
the swing of the pendulum. Of course the 
pendulum must not hit the tube and there 
must be sufficient friction between rod and 
tube so that the rod stops when the pendulum 
stops. The original rest position of the pen- 
dulum is readily found so that the displace- 
ment d can be measured. 

Repeat the experiment a few times to get an 
idea of how precise your value for d is. Then 
substitute your data in the equation for v, the 
bullet's speed. 



1. What is your value for the bullet's speed? 

2. From your results, compare the kinetic energy 
of the bullet before impact with that of the 
pendulum after impact. Why is there such a large 
difference in kinetic energy? 

3. Can you describe an independent method for 
finding v^? If you have time, try it, and explain any 
difference betwen the two values of v. 



Experiment 3-8 
ENERGY ANALYSIS OF A 
PENDULUM SWING 

According to the law of conservation of energy, 
the loss in gravitational potential energy of a 
simple pendulum as it swings from the top of 
its swing to the bottom is completely trans- 
ferred into kinetic energy at the bottom of the 
swing. You can check this with the following 
photographic method. A 1-m simple pendulum 
(measured from the support to the center of the 
bob) with a 0.5-kg bob works w«ll. Release the 
pendulum from a position where it is 10 cm 
higher than at the bottom of its swing. 

To simplify the calculations, set up the 
camera for 10:1 scale reduction. Two different 
strobe arrangements have proved successful: 
(1) tape an ac blinky to the bob, or (2) attach an 
AA cell and bulb to the bob and use a 
motor-strobe disk in front of the camera lens. In 
either case, you may need to use a two-string 
suspension to prj'vent the pendulum lM)b from 
spinning while swinging. Make a time exposurt* 
for one swing of the pendulum. 

You can either measure directly fnnn your 
print (which should look something like the 
one in Fig. 3-42) or make pinholes at the center 




no 3-*2 



of each image on the photograph and project 
the hole images onto a larger sheet of paper. 
Calculate the instantaneous speed v at the 
bottom of the swing by di\iding the distance 
traveled between the images nearest the bot- 
tom of the swing by the interval between the 
images. The kinetic ener^ at the bottom of the 
swing, Vimv-, should equal the change in 
potential ene^gv' from the top of the swing to 
the bottom. If A/i is the difference in \-ertical 
height between the bottom of the swing and 
the top, then 

V = V2a, A/i 

If you plot both the kinetic and potential 
energ\' on the same graph (using the bottom- 
most point as a zero lex^l for gra\itational 
potential enepgv), and then plot the sum of K£ 
-t- PE, you can check whether total enei^ is 
conserved during the entire swing. 



Experiment 3-9 
LEAST ENERGY 

Concepts such as momentum, kinetic energy. 
potential eneix>', and the conservation laws 
often tuni out to be unexpjectedly useful in 
helping you to understand what at first glance 
seem to l>e unrelated phenomena ThLs exper- 
iment offers just one such case in point: How- 
can you explain the observation that if a chain 
is allowed to hang freely from its two ends, it 
always assumes the same shajx'^ Hang a 1-m 
lei\gth of l>eaded chain, the ty|X' u.sed on light 
sockets. fn)m two (M)ints as shown in Fig 3-43 
What shajH' does the chain assume' .At first 
glance it seems to be a parabola 

('heck whether it is a paratxila b\' finding the 
equation for the parabola that wi>uld go 
through the vertex and the twii fixed points 



EXPERIMENTS 



125 



would be an excellent computer problem.) 
Draw vertical parallel lines about 2 cm apart 
on the paper behind the chain (or use graph 
paper). In each vertical section, make a mark 
beside the chain in the middle of that section 
(see Fig. 3-44). 



\ 



Fig. 3-43 

Determine other points on the parabola by 
using the equation. Plot them and see whether 
they match the shape of the chain. 

One way to plot the parabola is cis follows. 
The vertex in Fig. 3-43 is at (0,0) and the two 
fixed points are at (-8, 14.5) and (8, 14.5). All 
pcirabolas symmetric to the y axis have the 
formula y = /gc^, where fc is a constant. For this 
example, you must have 14.5 = ^(8)^, or 14.5 — 
64/c. Therefore, k = 0227, and the equation for 
the parabola going through the given vertex 
and two points is y = 0.227}i^. Now substitute 
values forx, producing a table of}i andy vcdues 
for the parabola. When you plot these values on 
the graph paper behind the chain, do the chciin 
and the plotted points coincide? 

A more interesting question is why the chain 
assumes the particular shape it does, which is 
called a catenary curve. Recall that the gravita- 
tional potential energy of a body mass m is 
defined as ma^h, where ag is the acceleration 
due to gravity, and h is the height of the body 
above the reference level chosen. Remember 
that only a difference in energy level is mean- 
ingful; a different reference level only adds a 
constant to each value associated with the 
original reference level. In theory, you could 
measure the mass of one bead on the chain, 
measure the height of each bead above the 
reference level, and total the potential energies 
for all the beads to get the total potential energy 
for the whole chain. 

In practice, that would be quite tedious, so 
you will use an approximation that will still 
allow you to get a reasonably good result. (This 




Fig. 3-44 

The total potential energy for that section of 
the chain will be approximately Mag/jav- where 
/lav is the average height marked, and M is the 
toted mass in that section of chain. Notice that 
near the ends of the chain there are more beads 
in one horizontal interval than there are near 
the center of the chain. To simplify the solution 
further, assume that M is always an integral 
number of beads that you can count. 

In summary, for each interval multiply the 
number of beads by the average height for that 
interval. Total all these products. This total is a 
good approximation of the gravitational poten- 
tial energy of the chain. 

After doing this for the finely hanging chain, 
pull the chain uath thumbtacks into different 
shapes such as those shown in Fig. 3-45. 



Fig. 3-45 




lae 



UNIT 3 / THE TRIUMPH OF MECHANICS 



Calculate the total p>otential energy for each 
shape. Does the catenary curve (the freely 
formed shape) or one of these others have the 
minimum total potential enei^? 

If you would like to explore other in- 
stances of the minimization principles, try 
the following: 

1. When various shapes of wire ai^ dipjjed into 
a soap solution, the resulting film always forms 
so that the total surface area of the film is a 
minimum. For this minimum surface, the total 
potential energy due to surface tension is a 
minimum. In many cases the resulting surface 
is not at all what you would exp>ect. An 
excellent source of suggested exp>eriments with 
soap bubbles, and recipes for good solutions, is 
the paperback Soap Bubbles and the Forces that 
Mould Them, by Charles V. Boys, (Dover, 1959). 
Also, see "The Strange World of Surface Film," 
The Physics Teacher (Sept., 1966). 

2. Rivers meander in such a way that the work 
done by the river is a minimum. For an 
explanation of this, see "A Meandering River," 
in the June, 1966 issue of Scientific American . 

3. Suppose that points A and B are placed in a 
vertical plane as shown in Fig. 3-46. You want to 
build a track between the two points so that a 
ball will roll from A to B in the least possible 
time. Should the track be straight or in the 
shape of a circle, parabola, cycloid, catenary, or 
some other shape? An interesting property of a 
cycloid is that no matter where on a cycloidal 
track you release a bail, it will take the same 
amount of time to reach the bottom of the 
track. You may want to build a cycloidal track 
in order to check this. Do not make the track so 
steep that the ball slips instead of rolling. 




Fig. 3-46 

A more complete treatment of this principle 
of h'iist action is given in the Fevnman Lectures 
on rhvsics, \'ol. 2 

Expi*rinient 3-10 

TEiviPi:Ry\TiTRi: Axn 

THE KMOM ETE KS 

You can usually tell just by louih which of two 
similar bodies is the hotter. But if you want to 
toll «vxartly how hot something is or to com- 



municate such information to somebody else, 
you have to find some way of assigning a 
number to "hotness. ' This number is called 
temperature, and the instrument used to 
measure this number is called a thermometer 

Standard units for measuring intervals of 
time and distance, the day and the meter, are 
both familiar. But try to imagine yourself living 
in an era before the invention of thermometers 
and temp)erature scales, that is, before the time 
of Galileo. How would you describe, and if 
p>ossible give a number to, the "degree of 
hotness" of an object? 

Any property (such as length, volume, den- 
sity, pressure, or electrical resistancei that 
changes with hotness and that can be mea- 
sured could be used as an indication of 
temjjerature; any device that measures this 
prop)erty could be used as a thermometer 

In this experiment you will be using ther- 
mometers based on prop)erties of liquid exptan- 
sion, gas expansion, and electrical resistance. 
(Other common kinds of thermometers are 
based on electrical voltages, color, or gas 
pressure.) Each of these devices has its own 
particular merits that make it suitable, from a 
practical point of view, for some applications, 
and difficult or imp>ossible to use in others. 

Of course, temp)erature estimates gnvn by 
two different types of thermometers must agree 
over the range that they are to be used in 
common. In this exp)eriment you will make 
your own thermometers, put temperature 
scales on them, and then compare them to see 
how well they agree with each other. 

Defining a Temperature Scale 

How do you make a thermometer? First, you 
decide what property (length, volume, etc. I of 
what substance (mercury, air, etc. I to use in 
your thermometer. Then you must decide on 
two fixed points in order to arri\« at the size of 
a degn»e. A fixed point Ls liased on a ph\'sical 
phenomenon that always occurs at the same 
degn*e of hotness Iwo con\Tnient fixed (X)inls 
to use are the melting (X)int of ice and the 
boiling point of water. On the Celsius i centi- 
grade I scale they are assigned the values O'C and 
lOO^'C at oixlinarv atmospheric pressure 

Wlien you are making a themiomeler of any 
sort, you haw to put a scale (»n it against which 
you can read the ln)tness-srnsiliw quantity. 
Often a piece of centimeter-marked tape or a 
short piece of ruler will do Submit \x)ur 
thermometer to two fixed points of hotness (for 
example a luUh of Ixiiling water and a bath of 



EXPERIMENTS 



127 




Rg. 3-47 Any quantity that varies with hotness can be 
used to establish a temperature scale (even the time it 
takes for an antacid tablet to dissolve in water!). Two 



"fixed points" (such as the freezing and boiling points of 
water) are needed to define the size of a degree. 



ice water) and mark the positions on the 
indicator. 

The length of the column can now be used to 
define a temperature scale by assuming that 
equal temperature changes cause equal 
changes along the scale between the two 
fixed-point positions. Suppose you marked the 
length of a column of liquid at the freezing 
point and again at the boiling point of water. 
You can now dhade the total increase in length 
into equcil parts and call each of these peirts 
"one degree" change in temperature. 

On the Celsius scale, the degree is 1/100 of 
the temperature range between the boiling and 
freezing points of water. 

To identify temperatures betw^een the fixed 
points on a thermometer scale, mark ofif the 
actual distance between the two fixed points 
on the vertical axis of a graph and equal 
intervals for degrees of temperature on the 







t^ 


^lOO 


- 


-jjX' 


o> 




^^ 


c 




^^^ 


•■n A 


v* 


^ 


o 


^y^ 




« 


^^ 




i^" 


r '«^ 






^ 




s 




. — . — , — , — , — , — , — . — .— >. 



lOO 



Fig. 3-48 



•t-cmperature 



horizontal axis, as in Fig. 3-48. Then plot the 
fixed points, X, on the graph and draw a 
straight line between them. 

The temp>erature on this scale, correspond- 
ing to any intermediate position /, can be read 
from the graph. 

Other properties and other substances can 
be used (the volume of different gases, the 
electrical resistance of different metals, and so 
on), and the temperature scale defined in the 
same way. All such thermometers will have to 
agree at the two fixed points, but do they agree 
at intermediate temperatures? 

If different physical properties do not change 
in the same way with hotness, then the 
temperature values you read from thermome- 
ters using these properties will not agree. Do 
similar temperature scales defined by different 
physical properties agree anywhere besides at 
the fixed points? That is a question that you 
can answer from this lab experience. 

Comparing Thermometers 

You uill make or be given two "thermometers" 
to compare. Take readings of the appropriate 
quantities, such as length of liquid column, 
volume of gas, electrical resistance, or thermo- 
couple voltage, when the devices are placed in 
an ice bath and again when they are placed in a 
boiling-water bath. Record these values. Define 
these two temperatures as 0° and 100° and 
draw the straight-line graphs that define inter- 
mediate temjjeratures as described above. 



128 



LNIT 3 / THE TRIUMPH OF MECHANICS 



Now put your two thermometers in a series 
of baths of water at intermediate temperatures, 
and again measure and record the length, 
volume, resistance, etc., for each bath. Put both 
devices in the bath at the same time in case the 
bath is cooling down. Use your graphs to read 
the temperatures of the water baths as indi- 
cated by the two devices. 

Do the temperatures measured by the two 
devices agree? 

If the two devices give the same readings at 
intermediate temperatures, then you can appar- 
ently use either as a thermometer. But if they 
do not agree, you must choose only one of 
them as a standard thermometer. Give what- 
ever reasons you can for choosing one rather 
than the other before reading the following 
discussion. If possible, compare your results 
with those of classmates using the same or 
different kinds of thermometers. 

There wall, of course, be some uncertainty in 
your measurements, and you must decide 
whether the differences you observe between 
the two thermometers might be due only to 
this uncertainty. 

The relationship between the readings from 
two different thermometers can be displayed 
on another graph, where one axis is the reading 
on one thermometer and the other axis is the 
reading on the other thermometer. Each bath 
will give a plot on this graph. If the points fall 
along a straight line, then the two thermometer 
properties must change in the same way. If, 
however, a fairly regular smooth curve can be 
drawn through the points, then the two 
thermometer properties probably change with 
hotness in different ways. (Figure 3-49 show^ 
possible results for two thermometers.) 



> 100 \- 

I ^ 



(«;/.' 



«> 



U'lUr 



ice 

Uitti 



Fig. 3-49 






U*i,B 



• ba-ih 4- 






100 



Discussion 

If you were to compare many gas thermome- 
ters at constant volume as well as pressure, and 
use different gases and different initial volumes 
and pressures, you would find that they all 
behave quantitatively in veiy much the same 
way with resp>ect to changes in hotness. If a 
given hotness change causes a 10% increase in 
the pressure of gas A, then the same change 
will also cause a 10% increase in gas B's 
pressure. Or, if the volume of one gas sample 
decreases by 20% when transferred to a particu- 
lar cold bath, then a 20% decrease in xtjlume 
will also be observed in a sample of any other 
gas. This means that the temperatures read 
from different gas thermometers all agree. 

This sort of close similarity of behavior 
between different substances is not found as 
consistently in the expansion of liquidft or 
solids, or in their other prop>erties, and so these 
thermometers do not agree, as you may have 
just discovered. 

This suggests two things. First, there is quite 
a strong case for using the change in pressure 
(or volume) of a gas to define the temjierature 
change. Second, the fact that, in such experi- 
ments, all gases do behave quantitatively in the 
same way suggests that there may be some 
underlying simplicity' in the behavior of gases 
not found in liquids and solids, and that if one 
wants to learn more about the way matter 
changes with temp)erature, one would do well 
to stari with gases. 



Experiment 3-11 
CALORIMETRY 

Speedometers measure speed, voltmeters mea- 
sure voltage, and accelerometers measure ac- 
celeration. In this experiment, you will use a 
device called a calorimeter As the name 
suggests, it measures a quantity connected 
with heat 

Unfortunately, heat enerf^v' cannot be mea- 
sured as directly as some of the other quan- 
tities mentioned abov«. In fact, to measure the 
heat ener^^V' absort>ed or given off b\- a sub- 
stance you must measure the change in 
leni|>eratun' of a second substance chosen as a 
standard I'he heat exchange takes place inside 
a calorimeter, a container in which measured 
quantities of materials can be mixed together 
without an appreciable amount of heat being 
gained fhjm or lost to the outside. 



EXPERIMENTS 



129 




Fig. 3-50 

A Preliminari' Experiment 

The first experiment will give you an idea of 
how good a calorimeter's insulating ability 
really is. 

Fill a calorimeter cup (a Styrofoam coffee cup 
does nicely) about half full of ice water. Put the 
same amount of ice water uith one or two ice 
cubes floating in it in a second cup. Into a third 
cup, pour the same amount of water that has 
been heated to nearly boiling. Measure the 
temperature of the water in each cup, and 
record the temperature and the time of obser- 
vation. (See Fig. 3-50.) 

Repeat the observations at about 5-min 
intervals throughout the period. Between ob- 
servations, prepare a sheet of graph paper wdth 
coordinate axes so that you can plot tempera- 
ture as a function of time. 

Mixing Hot and Cold Liquids 

(You can do this experiment while continuing 
to take readings of the temperature of the water 
in your three cups.) You will make several 
assumptions about the nature of heat. Then 
you wiU use these assumptions to predict what 
will happen when you mix two samples that 
are initially at different temperatures. If your 
prediction is correct, you can feel some confi- 
dence in your assumptions; at least, you can 
continue to use the assumptions until they 
lead to a prediction that turns out to be wrong. 
First, assume that, in your Ccilorimeter, heat 
behaves like a fluid that is conserved. Assume 
that it can flow from one substance to another, 
but that the toted quantity of heat H present in 
the calorimeter in any given experiment is 
constant. Then the heat lost by a warm object 
should just equal the heat gained by a cold 
object. In symbols, 

- AH, = AH2 



Next, assume that if two objects at different 
temperatures are brought together, heat will 
flow from the warmer to the cooler object until 
they reach the same temperature. 

Finally, assume that the amount of heat fluid 
Ah that enters or leaves an object is propor- 
tional to the change in temperature AT and to 
the mass of the object, m. In symbols, 

AH = cm AT 

where c is a constant of proportionality that 
depends on the units, and is different for 
different substances. 

The units in which heat is measured have 
been defined so that they are convenient for 
calorimeter experiments. The Calorie (cal) is 
defined as the quantity of heat necessary to 
change the temperature of 1 kg of water by one 
Celsius degree. (This definition has to be 
refined somewhat for very precise woric, but it 
is adequate for your purpose.) In the expres- 
sion 

AH = cm AT 

when m is measured in kilograms of water and 
T in Celsius degrees, H wiU be the number of 
Calories. Because the Ccdorie was defined this 
way, the proportionality constant c has the 
value 1 Cal/kg C° when water is the only 
substance in the calorimeter. In metric units, 1 
Cal = 4.19 kJ. Therefore, you could also 
measure directly in joules: 1 J of energy heats 1 
g of water by 1/4.19, or 0.240 Celsius degree. 



Checking the Assumptions 

Measure and record the mass of two empty 
plastic cups. Then put about one-half cup of 
cold water in one and about the same amount 
of hot water in the other, and record the mass 
and temperature of each. (Subtract the mass of 
the empty cup.) Now mix the two together in 
one of the cups, stir gently uath a thermometer, 
and record the final temperature of the mix- 
ture. 

Multiply the change in temperature of the 
cold water by its mass. Do the same for the hot 
water. 



1. What is the product (mass x temperature 
change) for the cold water? 

2. What is this product for the hot water? 

3. Are your assumptions confirmed, or is the 
difference between the two products greater than 
can be accounted for by uncertainties in your 
measurement? 



130 



UNIT 3 / THE TRIUMPH OF MECHANICS 



Predicting 

from the Ajisumptions 

Try another mixture using different quantities 
of water, for example one-quarter cup of hot 
water and one-half cup of cold. Before you mix 
the two, try to predict the final temp>erature. 



4. What do you predict the temperature of the 
mixture will be? 

5. What final temperature do you observe? 

6. Estimate the uncertainty of your thermometer 
readings and your mass measurements. Is this 
uncertainty enough to account for the difference 
between your predicted and observed values? 

7. Do your results support the assumptions? 



Melting 

The cups you filled with hot and cold water at 
the beginning of the period should show a 
measurable change in temperature by this 
time. If you are to hold to your assumption of 
conservation of heat fluid, then it must be that 
some heat has gone fixjm the hot water into the 
room and from the room to the cold water. 



Measuring Heat Capaciti' 

(While you are doing this experiment, continue 
to take readings of the temperature of the water 
in your three test cups.) Measure the mass of a 
small metal sample. Put just enough cold water 
in a calorimeter to cover the sample. Tie a 
thread to the sample and susp>end it in a beaker 
of boiling water. Measure the temp>erature of 
the boiling water. 

Record the mass and temp>erature of the 
water in the calorimeter. 

V\'hen the sample has been immersed in the 
boiling water long enough to be heated uni- 
formly 12 or 3 min), lift it out and hold it just 
above the surface for a few seconds to let the 
water drip off, then transfer it quickly to the 
calorimeter cup. Stir gently with a thermometer 
and record the temperature when it reaches a 
steady value. 



10. Is the product of mass and temperature 
change the same for the metal sample and for the 
water? 

11. If not, must you modify the assumptions 
about heat that you made earlier in the experi- 
ment? 



8. How much has the temperature of the cold 
water changed? 

9. How much has the temperature of the water 
that had ice in it changed? 



The heat that must have gone from the room 
to the water-ice mixture evidently did not 
change the temperature of the water as long as 
the ice was present. But some of the ice melted, 
so apparently the heat that leaked in melted 
the ice. Kvidently, heal was needed to cause a 
"change of state" (in this case, to change ice to 
water) tjven if there was no change in tempera- 
ture. The additional heat required to melt 1 g of 
ice is called Intent heat ofmeltins^.Lutent means 
hidden or dormant. Ihe units are C^alories per 
gram; ihen* is no temperature unit here 
because the temperature does not change. 

Next, you will do an experiment mixing 
materials other than liquid water in the 
calorimeter to see if your assumptions about 
heal as a fluid can still be used l\vo such 
(vxp«'nm<Mils are descrilM-d brlow. 'Measuring 
Meat (lapacily ' antl MtM.siiring l„it««nl Meal If 
you have lime for only one of ihem, choo.se 
either one. Finally, do "Rale of Cooling" to 
complete your pn*liminarv rxperimrnt 



In the expression AH = cm AT, the constant 
of proportionality c (called the specific heat 
capac/f>') may l>e different for diflferent mate- 
rials. For water Ihe constant has the value 1 
cal/kgC° or 0.240 J/gC. Vou can find a \alue of r 
for the metal by using the assumption that heal 
gained by the water equals the heat lost by the 
sample Writing subscripts for water H^ and 
metal sample //», AW» = — AW, 
Then Cy,my, Af« = -r/ri.Af, 



and 



c. 



^ -c,^, Af, 
m,A/, 



12. What IS your calculated value for the specific 
heat capacity ci for the metal sample you used? 



If your assumptions about heat being a fluid 
are valid, you now ought lo be able to pretlict 
the final lem|)«'rature of am mixture of water 
and your malenal 

Ity lo \vrif\ the usefulness of your value 
Predict the final temperature of a mixture of 
water and a heated piece of your material 



EXPERIMENTS 



131 



using different masses and different initial 
temperatures. 



13. Does your result support the fluid model of 
heat? 



Measuring Latent Heat 

Use your calorimeter to find the latent heat of 
melting of ice. Start with about one-half cup of 
water that is a little above room temperature, 
and record its mass and temperature. Take a 
small piece of ice from a mixture of ice and 
water that has been standing for some time; 
this will assure that the ice is at 0°C and will not 
have to be warmed up before it can melt. Place 
the small piece of ice on paper toweling for a 
moment to dry' off water on its surface, and 
then transfer it quickly to the calorimeter. 

Stir gently with a thermometer until the ice is 
melted and the mixture reaches an equilibrium 
temperature. Record this temperature and the 
mass of the water plus melted ice. 



water, the latent heat is found to be 80 cal/g, or 
335 J/g. How does your result compaire with the 
accepted value? 

Rate of Cooling 

If you have been measuring the temperature of 
the water in your three test cups, you should 
have enough data by now to plot three curves 
of temperature against time. Mark the tempera- 
ture of the air in the room on your graph too. 



16. How does the rate at which the hot water 
cools depend on its temperature? 

17. How does the rate at which the cold water 
heats up depend on its temperature? 



Weigh the amount of water in the cups. From 
the rates of temperature change (degrees/ 
minute) and the masses of water, calculate the 
rates at which heat leaves or enters the cups at 
various temperatures. Use this information to 
estimate the error in your earlier results for 
latent or specific heat. 



14. What was the mass of the ice that you added? 
The heat given up by the warm water is 

The heat gained by the water formed by the 
melted ice is 

Hi = c«/niAti 

The specific heat capacity c^ is the same in 
both cases; that is, the specific heat of water. 

The heat given up by the warm water first 

melts the ice, and then heats the water formed 

by the melted ice. Using the s>Tnbol AWl for the 

heat energy required to melt the ice, 

- AH„ = AHi -I- AHj 

So the heat energy needed to melt the ice is 

AHl = - AH« - AHi 

The latent heat of melting is the heat energy 
needed per gram of ice, so 

latent heat of melting = — - 



15. What is your value for the latent heat of 
melting of ice? 



When this experiment is done with ice made 
&x)m distilled water with no inclusions of liquid 



Experiment 3-12 
ICE CALORIMETRY 

A simple apparatus made up of thermally 
insulating Styrofoam cups can be used for 
doing some ice calorimetiy experiments. Al- 
though the apparatus is simple, careful use will 
give you excellent results. To determine the 
heat transferred in processes in which heat 
energy is given off, you will be measuring either 
the volume of water or the mass of water from a 
melted sample of ice. 

You will need either three cups the same 
size, or two large and one slightly smaller cup. 
Also have some extra cups ready. One large cup 
serves as the collector, A (Fig. 3-511, the second 
cup as the ice container, I, and the smaller cup 
(or one of the same size cut back to fit inside 
the ice container as shown) as the cover, K. 



Fig. 3-51 




Rg. 3-52 



132 



UNIT 3 / THE TRIUMPH OF MECHANICS 



Cut a hole about 0.5 cm in diameter in the 
bottom of cup I so that melted water can drain 
out into cup A. To keep the hole from becoming 
clogged by ice, place a bit of window screening 
in the bottom of I. 

In each experiment, ice is placed in cup 1. 
This ice should \ie carefully prepared, free of 
bubbles, and dry, if you plan to use the known 
value of the heat of fusion of ice. However, you 
can use ordinary crushed ice, and, before doing 
any of the (!xp(!riments, deteiniine experimen- 
tally the effective heat of melting of this 
nonideal ice. (Why should these two values 
differ?) 

In some experiments that require some time 
to complete (such as Experiment b),you should 
set up two identical sets of apparatus (same 
quantity of ice, etc.), except that one does not 
contain a source of heat. One will serve as a fair 
measure of the background effect. Measure the 
amount of water collected in this control 
apparatus during the same time, and subtract 
this amount from the total amount of water 
collected in the experimental apparatus, 
thereby correcting for the amount of ice melted 
just by the heat leaking in from the room. An 
efficient method for measuring the amount of 
water is to place the arrangement on the pan of 
a balance and lift cups I and K at regular 
intervals (about 10 min) while you weight A 
with its contents of melted ice water. 

(a) Heat of melting of ice 

Fill a cup about one-half to one-third full 
with crushed ice. (Crushed ice has a larger 
amount of surface area, and so will melt more 
quickly, thereby minimizing errors due to heat 
from the rt)om.) Bring a small measured 
amount of water (about 20 mU to a boil in a 
beaker or large test tube and |)our it over the 
ice in the cup. Stir briefly with a poor heat 
conductor, such as a glass rod, until equilib- 
rium has been reached. Pour the ice- water 
mixture through cup I. Collect and measure the 
final amount of water (/n,) in A If m„ is the 
original mass of hot water at 100°C with which 
you started, then ni, - m„ is the mass of ice that 
was melted. The heat enepgy absorbed by the 
melting ice is the latent heat of melting for ice, 
/.|, limes the mass of meltetl ice: L^inif — m„). 
This will be e(|ual to the heat eiiergN lost by the 
boiling water cooling fn»m 100''(." to 0''C. There- 
fore, 

L|(/n, - m„) = m„ AT 



and 



L, = 



m. 



lOOC 



m, - m„ 



Note: This derivation is correct only if there is 
still some ice in the cup afterwards. If you start 
with too little ice, the water will come out at a 
higher temperature. 

For crushed ice that has been standing for 
some time, the value of L, will vary l>et\veen 70 
and 75 Cal/g. 

(b) Heat exchange and transfer by 
conduction and radiation 

For several possible experiments you will 
need the following additional apparatus. Make 
a small hole in the bottom of cup K and thread 
two wires, soldered to a lightbulb, through the 
hole. A flashlight bulb that operates with an 
electric current between 300 and 600 milliam- 
peres (mA) is preferable; but e\'en a GE #1130 
6-volt automobile headlight bulb (which draws 
2.4 A) has been used with success. (See Fig. 
3-52.) In each experiment, you are to observe 
how the difl^erent apparatus affects heat trans- 
fer into or out of the system. 

1. Place the bulb in the ice and turn it on for 5 
min. Measure the ice melted. 

2. Repeat 1, but place the bulb above the ice for 
5 min. 

3. and 4. Repeat 1 and 2, but cover the inside of 
cup K with aluminum foil 

5. and 6. Rep>eat 3 and 4, but. in addition, co\-er 
the inside of cup I with aluminum foil. 
7. Prepare "heat-absorbing" ice by freezing 
water to which you hax-e added a small amount 
of dye, such as India ink Repeat any or all of 
experiments 1 through 6 using this specially 
prepared ice. 

Some questions to guide your observations: 
I3oes any heat escape w+jen the bulb is 
immersed in the ice^ What arrangement keeps 
in as much heal as possible? 



ExpiTinient 3-13 

MOiVTE <./\KLO i:\Pi:RI.\IKVr 

ON .\10I.FC:i'I^\K <:OI.LISIONS 

.\ model tor a gas consisting ot a larg«' niiml>er 
of wi'N small particles in rapid random motion 
has many advantages One of these is thai it 
makes it possible to estimate the pn)j>erties of a 
gas as a \\ hole from the l>eha\ior of a cx)mf>ara- 
tivT>ly small random sample of its molecules In 
this ex|>eriment. you will not use actual gas 
particles, but instead employ analogs of 
molecular collisions The technique is named 
the Montr Carlo mrthod after that famous 



EXPERIMENTS 



133 



gambling casino in Monaco. The experiment 
consists of two games, both of which involve 
the concept of randomness. You will probably 
have time to play only one game. 

Game I: Collision Probability for a 
Gas of Marbles 

In this part of the experiment, you wall tiy to 
find the diameter of marbles by rolling a 
"bombarding marble" into an curay of "tai^et 
marbles" placed at random positions on a level 
sheet of graph paper. The computation of the 
marble diameter vvHl be based on the propor- 
tion of hits and misses. In order to assure 
randomness in the motion of the bombarding 
marble, release each marble fix)m the top of an 
inclined board studded with nails spaced 
about 2.5 cm apart: a sort of pinball machine 
(Fig. 3-53). To get a fairly even, yet random, 
distribution of the bombarding marble's mo- 
tion, move its release position over one space 
for each release in the series. The launching 
board should be about 0.5 m from the target 
board; from this distance, the bombarding 
marbles will move in nearly parallel paths 
through the target board. 




Rg. 3-53 



First you need to place the tJirget marbles at 
random. Then draw a network of crossed-grid 
lines spaced at least two marble diameters 
apart on your graph paper. (If you are using 
marbles whose diameters are 1 cm, these grid 
lines should be spaced 3—4 cm apart.) Number 
the grid lines as shown in Fig. 3-54. 



9 

8 

7 
b 

t ' 
3 

I 





!4- 



'-^-4— ^- ■: — - 



J-^^. 



-f--- 1 — r~'7 



X — 

Fig. 3-54 Eight consecutive two-digit numbers in a table 
of random numbers were used to place the marbles. 



One way of placing the marbles at random Is 
to turn to the table of random numbers at the 
end of this experiment (Table 3-1). Each 
student should start at a different place in the 
table and then select the next eight numbers. 
Use the first two digits of these numbers to 
locate positions on the grid. The first digit of 
each number gives the^ coordinate, the second 
gives the y coordinate, or vice versa. Place the 
tai^et marbles in these positions. Books may be 
placed around the sides of the graph paper to 
serve as containing walls. 

With your array of marbles in place, make 
about 50 trials with the bombcirding marble. 
From your record of hits and misses compute 
fl, the ratio between the number of runs in 
which there £ire one or more hits to the total 
number of runs. Remember that you are 
counting "runs with hits," not hits. Therefore, 
several hits in a single run are still counted as 
"one." 

Inferring the size of the marbles 

How does the ratio fl lead to the diameter of 
the tau^et object? The theory applies just as 
well to determining the size of molecules as it 
does to marbles, although there would be 10^° 
or so molecules instead of eight "marble 
molecules." 

If there were no tai^get marbles, the bombard- 
ing marble would get a clear view of the full 
width, D, of the target eirea. There could be no 
hit. If, however, there were target marbles, the 
100% clear view would be cut down. If there 



134 



UNIT 3 / THE TRILMPII OF MECHANICS 



were N target marbles, each with diameter d, 
then the clear path over the width D would be 
reduced by JV x d. 

It is assumed that no tai^et marble is hiding 
behind another. (This corresp)onds to the 
assumption that the sizes of molecules are 
extremely small compared to the distances 
between them.) 

The blocking effect on the bombarding 
marble is, however, greater than just Nd. The 
bombarding marble will miss a target marble 
only if its center passes more than a distance of 
one radius on either side of it. (See Fig. 3-55.) 
This means that a target marble has a blocking 
effect equal to twice its diameter (its own 
diameter plus the diameter of a bombarding 
marble), so the total blocking effect of N 
marbles is ZNd. Therefore, the expected ratio fl 
of hits to totiil trials is 2Nd/D (total blocked 
width to total width). Thus, 
ZNd 
D ' 



fl = 



which can be rearranged to give an expression 
ford: 



d = 



RD 
IN 



r 

2d 









2d 



Fig. 3-55 A projectile will clear a target only if it passes 
outside a center-to-center distance d on either side of it 
Therefore, thinking ot the projectiles as points, the 
effective blocking width of the target is 2d. 



measurement of the marbles. (For example, line 
up your eight marbles against a book Measure 
the total length of all of them together and 
divide by eight to find the diameter d of one 
marble.) 



1. What value to you calculate for the marble 
diameter? 

2. How well does your experiment prediction 
agree with direct measurement? 



Game II: Mean Free Path Between 
Collision Squares 

In this part of the exp)eriment you play with 
blacked-in squares as target molecules in place 
of marble molecules in a pinball game. On a 
sheet of graph f)ap>er 50 units on a side 12,500 
squares), you will locate, by the Monte Cario 
method, between 40 and 100 molecules. ELach 
student should choose a different number of 
molecules. 

You will find a table of random numbers 
(fix>m to 50) at the end of this exp>eriment 
(Table 3-1). Begin anywhere you wish in the 
table, but then proceed in a regular sequence. 
Let each pair of numbers bv thr .v and v 
coordinates of a point on your graph, ilf one of 
the pair is greater than 49, you cannot use it 
Ignore it and take the next pair.) Then shade in 
the squares for which these points are the 
lower left-hand comers (Fig. 3-561 You now 
have a random array of square target 
"molecules." 




(03, 02) 



Rg. 3-56 



To check the acruracy of the Montr C'arlo 
method, rompan» the value ford obtained from 
the foiiiuila abou* with that obtained by direct 



Rules of the fiamr 

rbe w.iy a bombarding particle passes 
Ihmugli this array, it is Imund to collide with 
some of the target jvarticles There are fi\« rules 



EXPERIMENTS 



135 



for this game of collision. All of them are 
illustrated in Fig. 3-57. 

(a) The particle can travel only along lines of 
the graph paper, up or down, left or right. A 



L.i 



rri 






4-^ 




r-f-l--f 4 




Rg. 3-57 

particle starts at some point (chosen at ran- 
dom) on the left-hand edge of the graph paper. 
The particle initially moves horizontally fix)m 
the starting point until it coUides with a 
blackened square or another edge of the graph 
paper. 

(b) If the particle strikes the upper left-hand 
comer of a target square, it is diverted upward 
through a right angle. If it should strike a lower 
left-hand comer it is diverted downward, again 
through an angle of 90°. 

(c) When the path of the particle meets an 
edge or boundary of the graph paper, the 
particle is not reflected directly back. (Such a 
reversal of path would make the particle retrace 
its pre\ious path.) Rather it moves tvvo spaces 
to its right along the boundary edge before 
reversing its direction. 

(d) There is an exception to rule (c). Whenever 
the particle strikes the edge so near a comer 
that there is no room for it to move two spaces 
to the right vdthout meeting another edge of 
the graph paper, it moves two spaces to the left 
cdong the boundeuy. 

(e) Occasionally two target molecules may 
occupy adjacent squares and the particle may 
hit touching corners of the two target 
molecules at the same time. The rule is that this 
counts as two hits and the particle goes straight 
through without changing its direction. 



Finding the mean free path 

With these collision rules in mind, trace the 
path of the particle as it bounces about cimong 
the random array of target squares. Count the 
number of collisions with targets. Follow the 
path of the particle until you get 50 hits with 
tfirget squares (collisions with the edge do not 
count). Next, record the 50 lengths of the paths 
of the peirticle between collisions. Distances to 
and from a boundary should be included, but 
not distances a/ong a boundary (the two spaces 
introduced to avoid backtracking). These 50 
lengths are the fi^e paths of the particle. Total 
them and divide by 50 to obtain the mean ft-ee 
path, L for your random two-dimensional cirray 
of square molecules. 

In this game, your molecule analogs were 
pure points, that is, dimensionless. In his 
investigations, Clausius modified this model by 
giving the particles a definite size. Clausius 
showed that the average distcince L a molecule 
travels between collisions, the so-called mean 
ft'ee path, is given by 



L = 



Na 



where V is the volume of the gas, N is the 
number of molecules in that volume, and a is 
the cross-sectional area of an individual 
molecule. In this two-dimensional game, the 
particle was moving over an area A, instead of 
through a volume V, and was obstructed by 
targets of width d, instead of cross-sectional 
area a. A two-dimensional version of Clausius 's 
equation is 



where N is the number of blackened square 
"molecules." 



3. What value of L do you get from the data for 
your runs? 

4. Using the two-dimensional version of 
Clausius's equation, what value do you estimate 
for d (the width of a square)? 

5. How does your calculated value of d compare 
with the actual value? How do you explain the 
difference? 



136 



UNIT 3 / THE TRIUMPH OF MECHANICS 



TABLE 3-1 

TABLE OF 1000 RANDOM TWO-DIGIT NUMBERS 
(FROM to 50) 



03 47 


44 22 


30 30 


22 00 


00 49 


22 17 


38 30 


23 21 


2011 


24 33 


16 22 


36 10 


44 39 


46 40 


24 02 


19 36 


38 21 


45 33 


1423 


01 31 


33 21 


03 29 


08 02 


20 31 


37 07 


03 28 


47 24 


11 29 


49 06 


10 39 


34 29 


34 02 


43 28 


03 43 


43 40 


26 08 


28 06 


50 14 


21 44 


47 21 


32 44 


11 05 


05 05 


05 50 


23 29 


26 00 


09 05 


27 31 


08 43 


04 14 


18 18 


04 02 


48 39 


48 22 


38 18 


15 39 


48 34 


50 28 


37 21 


1509 


23 42 


31 08 


19 30 


06 00 


20 18 


30 24 


1533 


10 07 


14 29 


05 24 


35 12 


11 12 


11 04 


01 10 


25 39 


48 50 


24 44 


03 47 


34 04 


44 07 


12 13 


42 10 


40 48 


45 44 


42 35 


41 26 


41 10 


23 05 


06 36 


08 43 


37 35 


12 41 


02 02 


19 11 


06 07 


42 31 


23 47 


47 25 


10 43 


1238 


16 08 


18 39 


03 31 


49 26 


07 12 


1731 


17 31 


35 07 


44 38 


40 35 


31 16 


10 47 


38 45 


28 40 


33 34 


24 16 


42 38 


1909 


41 47 


50 41 


32 43 


45 37 


30 38 


22 01 


30 14 


02 17 


45 18 


29 06 


13 27 


46 24 


27 42 


03 09 


08 32 


24 02 


05 49 


18 05 


22 00 


23 02 


44 43 


43 20 


00 39 


05 03 


49 37 


23 22 


33 42 


26 29 


00 20 


12 03 


1005 


02 39 


11 27 


39 32 


13 30 


36 45 


09 03 


46 40 


22 07 


03 03 


05 39 


03 46 


35 24 


22 49 


1733 


35 01 


01 32 


1809 


47 03 


39 41 


36 23 


19 41 


16 20 


38 36 


29 48 


07 27 


48 14 


34 13 


07 48 


39 12 


20 18 


19 42 


38 23 


33 26 


15 29 


20 02 


21 45 


04 31 


48 13 


23 32 


37 30 


09 24 


45 11 


27 07 


39 43 


13 05 


47 45 


47 45 


00 06 


41 18 


05 02 


03 09 


18 00 


14 21 


49 17 


30 37 


25 15 


04 49 


24 19 


40 23 


24 17 


17 16 


20 46 


06 18 


45 07 


06 28 


49 44 


10 08 


43 00 


38 26 


34 41 


11 16 


05 26 


50 25 


38 47 


39 38 


42 45 


10 08 


16 06 


43 18 


34 48 


27 03 


21 19 


13 42 


16 04 


00 18 


16 46 


13 13 


1629 


44 10 


29 18 


22 45 


41 23 


03 10 


35 30 


24 36 


38 09 


25 21 


08 40 


20 46 


39 14 


37 31 


34 50 


20 14 


21 46 


38 46 


12 27 


20 44 


46 06 


01 41 


30 49 


18 48 


39 43 


13 04 


24 15 


08 22 


13 29 


04 05 


42 29 


50 47 


01 50 


01 48 


18 14 


04 43 


27 46 


23 07 


19 28 


07 10 


23 19 


41 45 


25 27 


19 10 


09 47 


34 45 


08 45 


25 21 


49 21 


18 46 


16 40 


35 14 


41 28 


41 15 


44 17 


04 33 


15 22 


12 45 


39 07 


34 27 


14 47 


35 33 


42 29 


47 47 


40 33 


42 45 


07 08 


38 15 


08 25 


22 06 


07 26 


32 44 


03 42 


42 34 


33 27 


10 45 


18 40 


11 48 


48 03 


07 16 


32 25 


20 25 


44 22 


39 28 


06 09 


04 26 


14 35 


36 03 


15 22 


02 07 


46 48 


45 12 


47 11 


30 19 


33 32 


34 25 


45 17 


13 26 


03 37 


33 35 


08 13 


15 26 


09 18 


34 25 


42 38 


40 01 


43 31 


30 33 


39 11 


49 41 


27 44 


11 39 


06 19 


47 23 


1506 


22 08 


50 44 


50 11 


18 16 


00 41 


07 47 


34 25 


28 10 


50 03 


22 35 


49 36 


44 21 


25 12 


19 44 


31 51 


49 18 


40 36 


00 27 


22 12 


31 04 


32 17 


08 23 


38 32 


01 47 


43 53 


44 04 


10 27 


16 00 


1633 


39 00 


01 50 


07 28 


35 02 


38 00 


46 47 


33 29 


28 41 


09 23 


47 48 


37 32 


07 02 


07 48 


07 41 


22 13 


37 27 


27 12 


34 21 


07 04 


49 34 


05 03 


36 07 


10 15 


21 48 


14 44 


39 39 


1509 


23 23 


37 31 


00 25 


17 37 


13 41 


13 39 


40 14 


19 48 


34 18 


08 18 


08 06 


44 26 


12 45 


32 24 


24 30 


29 13 


34 39 


27 44 


11 20 


37 40 


36 46 


35 22 


09 09 


07 45 


29 12 


48 35 


05 38 


43 11 


45 18 


28 14 


04 37 


48 38 


43 12 


14 08 


04 04 


18 17 


1033 


04 32 


27 37 


33 42 


3441 


07 41 


49 14 


31 38 


08 31 


38 30 


42 10 


08 09 


17 32 


46 15 


15 43 


15 31 


46 45 


42 34 


46 31 


29 03 


08 32 


11 06 


20 21 


24 16 


13 17 


29 34 


42 31 


1600 


02 48 


10 34 


32 14 


25 39 


29 31 


18 37 


28 50 


07 26 


06 24 


20 15 


60 11 


21 31 


20 49 


07 35 


41 16 


16 17 


43 36 


20 26 


39 38 


00 49 


14 10 


29 01 


49 28 


21 30 


40 15 


01 07 


16 04 


1909 


36 12 



EXPERIMENTS 



137 



Experiment 3-14 
BEHA\10R OF GASES 

Air is elastic or springy. You can feel this when 
you put your finger over the outlet of a bicycle 
pump and push down on the pump plunger. 
You can tell that there is some connection 
between the volume of the air in the pump and 
the force you exert in pumping, but the exact 
relationship is not obvious. About 1660, Robert 
Boyle performed an experiment that disclosed 
a veiy simple relationship betueen gas pres- 
sure and volume, but not until two centuries 
later was the kinetic theory of gases developed, 
which satisfactorily accounted for Boyle's law. 
The purpose of these experiments is not 
simply to show that Boyle's law and Gay 
Lussac's law I which relates temperature and 
volume) are "true. ' The purpose is also to show 
some techniques for analyzing data that can 
lead to such laws. 

I. Volume and Pressure 

Boyle used a long glass tube in the form of a J to 
investigate the "spring of the air." The short 
arm of the J was sealed, and air was trapped in 
it by pouring mercury into the top of the long 
arm. 

A simpler method requires only a small 
plastic syringe, calibrated in milliliters, and 
mounted so that you can push down the piston 
by piling weights on it (see Fig. 3-58). The 
volume of the air in the syringe can be read 
directly fix)m the calibrations on the side. The 
pressure on the air due to the weights on the 
piston is equal to the force exerted by the 
weights divided by the area of the face of the 
piston: 




Rg. 3-58 



Because "weights " are usually marked with 
the value of their mass, you will have to 
compute the force firom the relation Fgrav = 
magrav (It vvUl help if you answer this question 
before going on: Uliat is the weight, in newtons, 
of a 0.1-kg mass?) 

To find the area of the piston, remove it fiDm 
the syringe. Measure the diameter (2fl) of the 
piston face, and compute its area ftx)m the 
familiar formula A — ttR'. 

You will want to both decrease and increase 
the volume of the air, so insert the piston about 
halfway down the barrel of the syringe. The 
piston may tend to stick slightly. Give it a twist 
to fi^e it and help it come to its equilibrium 
position. Then record this position. 

Add weights to the top of the piston and each 
time record the equilibrium position, after you 
have given the piston a twist to help overcome 
friction. 

Record your data in a table with columns for 
volume, weight, and pressure. Then remove the 
weights one by one to see if the volumes are the 
same with the piston coming up as they were 
going down. 

If your apparatus can be turned over so that 
the weights pull out on the plunger, obtain 
more readings this way, adding weights to 
increase the volume. Record these as negative 
forces. (Stop adding weights before the piston is 
pulled aU the way out of the barrel!) Again 
remove the weights and record the values on 
returning. 



Interpreting Your Results 

You now have a set of numbers somewhat 
like the ones Boyle reported for his experiment. 
One way to look for a relationship between the 
pressure P„ and the volume V is to plot the 
data on graph paper. Plot volume V (vertical 
axis) as a function of pressure P„ (horizontal 
axis I. Then draw a smooth curve that gives an 
overall "best fit ' Because errors of measure- 
ment affect each plotted point, your smooth 
curve need not go through all the points. 

Since V decreases as P„ increases, you can 
tell before you plot it that your curve represents 
an 'inverse " relationship. As a first guess at the 
mathematical description of this curve, tiy the 
simplest possibility, that 1/V is proportional to 
P«. That is, lA' oc p^. if i/v is proportional to 
P^, then a plot of 1/V against P^. will lie on a 
straight line. 

Add another column to your data table for 
values of lA^ and plot this against P^.. 



138 



UNIT 3 / THE TRIUMPH OF MBCHANICS 



1. Does this curve pass through the origin? 

2. If not, at what point does your curve cross the 
horizontal axis? (In other words, what is the value 
of P^ for which MV would be zero?) What is the 
physical significance of the value of P^7 



In Boyle's time, it was not understood that air 
is really a mixture of several gases. Do you 
believe you would find the same relationship 
between volume and pressure if you tried a 
variety of pure gases instead of air? If there are 
other gases available in your laboratory, flush 
out and refill your apparatus with one of them 
and try the experiment again. 



Draw a straight line as nearly as possible 
through the points on your V-T graph and 
extend it to the left until it shows the approxi- 
mate temperature at which the volume would 
be zero. Of course, you have no reason to 
assume that gases have this simple linear 
relationship all the way down to zero volume. 
(In fact, air would change to a liquid long before 
it reached the temperature indicated on your 
graph for zero volume.) However, some gases 
do show this linear behavior over a wide 
temperature range, and for these gases the 
straight line always crosses the T axis at the 
same point. Since the volume of a sample of gas 
cannot be less than zero, this point represents 
the lowest possible temperature of the gases, 
the absolute zero of temp>erature. 



3. Does the curve you plot have the same shape 
as the previous one? 



7. What value does your graph give for absolute 
zero? 



II. Volume and Temperature 

Boyle suspected that the temperature of his air 
sample had some influence on its volume, but 
he did not do a quantitative experiment to find 
the relationship between volume and tempera- 
ture. It was not until about 1880, when there 
were better ways of measuring temperature, 
that this relationship was established. 

You could use several kinds of equipment to 
investigate the way in which volume changes 
with temperature. Such a piece of equipment is 
a glass bulb with a J tube of mercury or the 
syringe described above. Make sure the gas 
inside is dry and at atmospheric pressure. 
Immerse the bulb or syringe in a beaker of cold 
water and record the volume of gas and 
temperatuiv of the water (as measured on a 
suitable thermometerl periodically as you 
slowly heat the water. 

Inter|)reting Yoiu- Results 



4. With either of the methods mentioned here, 
the pressure of the gas remains constant. If the 
curve is a straight line, does this "prove" that the 
volume of a gas at constant pressure is propor- 
tional to its temperature? 

5. Remember that the thermometer you used 
probably depended on the expansion of a liquid 
such as mercury or alcohol. Would your graph 
have been a straight line if a different type of 
thermometer had been used? 

6. If you could continue to cool the air, would 
there be a lower limit to the volume it would 
occupy? 



III. Questions for Discussion 

Both the pressure and the temp>erature of a gas 
sample affect its volume. In these experiments, 
you were asked to consider each of these 
factors separately. 



8. Were you justified in assuming that the tem- 
perature remained constant in the first experi- 
ment as you varied the pressure? How could you 
check this? How would your results be affected if, 
in fact, the temperature went up each time you 
added weight to the plunger? 

9. In the second experiment, the gas was at 
atmospheric pressure. Would you expect to find 
the same relationship between volume and tem- 
perature if you repeated the experiment with a 
different pressure acting on the sample? 



Gases such as hydrogen. ox>'gen. nitrogen 
and rarljon dio.xide are wrv different in their 
rhrmiral lM'ha\ior Vet thr\ all show the same 
simple relationships l>et\veen wlumo, pres- 
sure, and tem|H'rature that you founH in these 
exjM'riments. cnvr a fairly wide range of pres- 
sun\s and t«MiijM>ratiiPPS This suggests that 
jM»rliaps thtMi" IS a simpir ph\siral iiKMlrl that 
will explain the l)eha\ior of all gas<'s within 
these limits of trm}>eraturp and pressure. 
Chapter 11 of the Text desrrilx*s just such a 
simple model and its im|Kir1anre in the 
dp\flopment of physics. 



EXPERIMENTS 



139 



Experiment 3'15 
WAVE PROPERTIES 

In this laboratory exercise you will become 
familiar with a variety of wave properties in 
one- and two-dimensional situations.* Using 
ropes, springs, Slinkies, or a ripple tank, you 
can find out what determines the speed of 
waves, what happens when they collide, and 
how waves reflect and go around comers. 

Waves in a Spring 

Many waves move too fast or are too small to 
watch easily, but in a long "soft" metal spring 
you can make big waves that move slowly. With 
a partner to help you, pull the spring out on a 
smooth floor to a length of about 6-9 m. Now, 
with your free hand, grasp the stretched spring 
about 50 cm from the end. Pull the spring 
together toward the end and then release it, 
being careful not to let go of the fixed end with 
your other hand! Notice the single wave, called 
a pulse, that travels along the spring. In such a 
longitudinal pulse, the spring coils move back 
and forth along the same direction as the 
wave travels. The wave carries energv', and thus 
could be used to carry a message from one end 
of the spring to the other. 

You can see a longitudinal wave more easily 
if you tie pieces of string to several of the loops 
of the spring and watch thefr motion when the 
spring is pulsed. 

A transverse wave is easier to see. To make 
one, practice moving your hand very quickly 
back and forth at right angles to the stretched 
spring, until you can produce a pulse that 
travels down only one side of the spring. This 
pulse is called transverse because the indi- 
vidual coils of wire move at right angles to 
(transverse to) the length of the spring. 

Perform experiments to answer the following 
questions about transverse pulses. 



Next observe what happens when waves go 
from one material into another, an effect called 
refraction. To one end of your spring attach a 
length of rope or rubber tubing (or a different 
kind of spring) and have your partner hold this 
end. 



4. The far end of your first spring is now free to 
move back and forth at the joint. What happens to 
a pulse (size, shape, speed, direction) when it 
reaches the boundary between the two media? 



Have your partner detach the extra spring 
and once more grasp the far end of your 
original spring. Then you both send a pulse on 
the same side, at the same instant, so that the 
two pulses meet in the center. The interaction 
of the two pulses is Ccdled interference . 



5. What happens (size, shape, speed, direction) 
when two pulses reach the center of the spring? (It 
will be easier to see what happens in the 
interaction if one pulse is larger than the other.) 

6. What happens when two pulses on opposite 
sides of the spring meet? 

As the two pulses pass on opposite sides of the 
spring, can you observe a point on the spring that 
does not move at all? 

7. From these observations, what can you say 
about the displacement caused by the addition of 
two pulses at the same point? 

By vibrating your hand steadily back and 
forth, you can produce a train of pulses, a 
periodic wave. The distance between any two 
neighboring crests on such a periodic wave is 
the wave/engf/j . The rate at which you vibrate 
your hand will determine the ^equency of the 
periodic wave. Use a long spring and produce 
short bursts of periodic waves so you can 
observe them without interference by reflec- 
tions from the far end. 



1. Does the size of the pulse change as it travels 
along the spring? If so, in what way? 

2. Does the pulse reflected from the far end 
return to you on the same side of the spring as the 
original pulse, or on the opposite side? 

3. Does a change in the tension of the spring have 
any effect on the speed of the pulses? When you 
stretch the spring farther, in effect you are 
changing the nature of the medium through 
which the pulses move. 



♦Adapted from R.F. BrinckerhofT and D.S. Taft, Mod- 
ern Laboratory Experiments in Physics, by per- 
mission of Science Electronics, Nashua, N Ji. 



8. How does the wavelength seem to depend on 
the frequency? 

You have now observed the reflection, refrac- 
tion, and interference of single waves, or pulses, 
traveling through different materials. These 
waves, however, moved only along one dimen- 
sion. So that you can maike a more realistic 
comparison with other forms of traveling 
energy, in the next experiment you will 
examine these same wave properties spread 
out over a two-dimensional surface. 



140 



UNIT 3 / THE TRIUMPH OF MECHANICS 



Experimeni 3-16 

WAV ES IN A RIPPLE TANK 

In the laboratoiy, one or more ripple tanks will 
have been set up. To the one you and your 
partner are going to use, add water (if neces- 
saiy) to a depth of 6-8 mm. Check to see that 
the tank is level so that the water has equal 
depth at all four comers. Place a large sheet of 
white paper on the table below the ripple tank, 
and then switch on the overhead light source. 
Disturtjances on the water surface are pro- 
jected onto the paper as light and dark 
patterns, thus allowing you to "see" the shape 
of the disturbances in the horizontal plane. 

To see what a single pulse looks like in a 
ripple tank, gently touch the water with your 
fingertip, or, better, let a drop of water fall into it 
from a medicine dropper held only a few 
millimeters above the surface. 

For certain purposes, it is easier to study 
pulses in water if their crests are straight. To 
generate single straight pulses, place a dowel, 
or a section of a broom handle, along one edge 
of the tank and roll it backward a fraction of a 
centimeter. By rolling the dowel backward and 
forward with a uniform frequency, a periodic 
wave, a continuous train of pulses, can be 
formed. 

Use straight pulses in the ripple tank to 
observe refiection, refraction, and difi^raction, 
and circular pulses from point sources to 
observe interference. 

Reflection 

Generate a straight pulse and notice the 
direction of its motion. Now place a barrier in 
the water so that it intersects that path. 
Generate new pulses and ()bser\'e what hap- 
pens to the pulses when they strike the barrier. 
Try different angles between the barrier and the 
incoming pulse. 



1. What is the relationship between the direction 
of the incoming pulse and the reflected one? 

2. Replace the straight barrier with a curved one. 
What is the shape of the reflected pulse? 

3. Find the point where the reflected pulses run 
together. What happens to the pulse after it 
converges at this point? At this point, called the 
focus, start a pulse with your finger or a drop of 
water. What is the shape of the pulse after 
reflection from the curved barrier? 

Refraction 

\j\y a sheet of glass in the center of the tank, 
supported by coins if necessary, to make an 



area of very shallow water. Tiy varying the angle 
at which the pulse strikes the boundary 
between the deep and shallow water. 



4. What happens to the wave speed at the 
boundary? 

5. What happens to the wave direction at the 
boundary? 

6. How is change in direction related to change in 
speed? 



Interference 

Arrange two point sources side by side a few 
centimeters apari. When tapped gently, they 
should produce two pulses. You will see the 
action of interference better if you vibrate the 
two point sources continuously w\\h a motor 
and study the resulting pattern of waves. 



7. How does changing the wave frequency affect 
the original waves? 

Find regions in the interference pattern where 
the waves from the two sources cancel and leave 
the water undisturbed. Find the regions where the 
two waves add up to create a doubly great 
disturbance. 

8. Make a sketch of the interference pattern 
indicating these regions. 

9. How does the pattern change as you change 
the wavelength? 



Diflfraction 

With two-dimensional waves you can observe a 
new phenomenon: the behavior of a wax^e when 
it passes around an obstacle or through an 
opening. The spreading of the wax^e into the 
"shadow" area us called diffraction . Generate a 
steady train of wav-es by using the motor dri\en 
straight-pulse source. Place a small l>arrier in 
the path of the wa\-es so that it intercepts j>ar1. 
but not all, of the wax^e front Observe what 
happens as the waves pass the edge of the 
barrier Now \ar>- the wawlength of the incom- 
ing wave train by changing the speed of the 
motor on the sourx^. 



10. How does the interaction with the obstacle 
vary with the wavelength? 

Place two long barriers in the tank, leaving a 
small opening between them 

11. How does the angle by which the wave 
spreads out beyond the opening depend on the 
size of the opening? 

12. In what way does the spread of the diffraction 
pattern depend on the length of the waves? 



EXPERIMENTS 



141 



Experiment 3-17 
MEASURING WAVELENGTH 

There are three ways you can conveniently 
measure the wavelength of the waves generated 
in your ripple tank. You should try them all, if 
possible, and cross-check the results. If there 
are differences, indicate which method you 
believe is most accurate and explciin why. 

METHOD A: Direct 

Set up a steady train of pulses using either a 
single-point source or a straight-line source. 
Observe the moving waves with a stroboscope, 
and then adjust the vibrator motor to the 
lowest frequency that will "freeze" the wave 
pattern. Place a meter stick across the ripple 
tank and measure the distance between the 
crests of a counted number of waves. 

METHOD B: Standing Waves 

Place a straight barrier across the center of 
the tcink parallel to the advancing waves. When 
the distance of the barrier fix)m the generator is 
properly adjusted, the superposition of the 
advancing waves and the waves reflected firom 
the barrier wall produce standing waves. In 
other words, the reflected waves are, at some 
points, reinforcing the originail waves, vvtiUe at 
other points there is cilways cancellation. The 
points of continued canceUation are called 
nodes . The distance between nodes is one-half 
wavelength. 

METHOD C: Interference Pattern 

Set up the ripple tank with two point 
sources. The two sources should strike the 
water at the same instant so that the two waves 
will be exactly in phase and of the same 
frequency as they leave the sources. Adjust the 
distance between the two sources and the 
frequency of vibration until a distinct pattern is 
obtained, such as in Fig. 3-59. 

As you study the pattern of ripples, you will 
notice lines along which the waves cancel 




Fig. 3-59 An interference pattern in water. Two point 
sources vibrating in phase generate waves in a ripple 
tank. A and C are points of maximum disturbance (in 
opposite directions) and B is a point of minimum 
disturbance. 



almost completely so that the amplitude of the 
disturbance is almost zero. These lines are 
called nodal lines, or nodes. You have already 
seen nodes in your earlier experiment with 
standing waves in the ripple tank. 

At every point along a node the waves 
arriving from the two sources are half a 
wavelength out of step, or "out of phase." This 
means that for a point (such as B in Fig. 3-59) to 
be on a line of nodes it must be V2 or 1V2 or 2V2 
. . . wavelengths farther from one source than 
from the other. 

Between the lines of nodes are regions of 
maximum disturbance. Points A and C in Fig. 
3-59 are on lines down the center of such 
regions, called anfinoda/ lines . Reinforcement of 
waves from the two sources is at a maximum 
along these lines. 

For reinforcement to occur at a point, the 
two waves must eirrive in step or "in phase." 
This means that any point on a line of 
antinodes is a whole number of wavelengths 0, 
1, 2, . . . farther from one source than from the 
other. The relationship between crests, 
troughs, nodes, and antinodes in this situation 
is summarized schematically in Fig. 3-60. 




142 



UNIT 3 / THE TRIUMPH OF MECHANICS 




Fig. 3-60 Analysis of interference pattern similar to that 
of Fig. 3-59 set up by two in-phase periodic sources 
(Here Si and Sj are separated by four wavelengths ) The 
letters A and N designate antinodal and nodal lines The 
dark circles indicate where crest is meeting crest, the 
blank circles where trough is meeting trough, and the 
half-dark circles where crest is meeting trough. 



distance L from the sources, then d.L, and X are 
related by the equations 

d L 

or 

L 

where fc is the distance between neighboring 
antinodes lor neighboring nodes). 

You now have a method for computing the 
wavelength X from the distances that you can 
measure precisely. Measure x» d, and L in your 
ripple tank and compute X. 

Experiment 3-18 
SOUND 

In previous exp>eriments, you observed how 
waves of relatively low frequency behave in 
different media. In this expjeriment, you will try 
to determine to what extent audible sound 
exhibits similar projaerties. 

At the laboratory station where you work, 
there should be the following: an oscillator, a 
power supply, two small loudsp>eakers. and a 
group of materials to be tested. A loudsp>eaker 
is the source of audible sound waves, and your 
ear is the detector. First connect one of the 
loudspeakers to the output of the oscillator and 
adjust the oscillator to a frequency of about 
4000 Hz. Adjust the loudness so that the signal 
is just audible 1 m away from the speaker. The 




Most physics textbooks drvelop the 
mathematical ai^ument of the n>lati()nship of 
wavelength to the geometry of the interference 
pattern. (Si^e, for example, p. 119 in Unit 3 of the 
Text.) If the di.stanre between the sources is d 
atul the detector is at a comparatively greater 



Fig. 3-61 Sound from the speaker can be detected by 
using a funnel and rubber hose, the end of wtiich is 
placed to the ear The oscillator's banana plug )scks 
must be inserted into the 8 V. ♦ 8 V and ground holes 
of the power supply Insert the speaker's plugs into the 
sine wave ground receptacles of the oscillator Select 
the audio range by means of the top knob of the 
oscillator and then turn on the power tupply. 



EXPERIMENTS 



143 



gain-control setting should be low enough to 
produce a clear, pure tone. Since reflections 
from the floor, tabletop, and hard-surfaced 
walls may interefere with your observations, set 
the sources at the edge of a table. Put soft 
material over any close hard surface that could 
cause reflective interference. 

You may find that you can localize sounds 
better if you make an "ear trumpet" or stetho- 
scope from a small funnel or thistle tube and a 
short length of rubber tubing (Fig. 3-61). Cover 
the ear not in use to minimize confusion when 
you are hunting for nodes and maxima. 

Transmission and Reflection 

Place samples of various materials at your 
station between the speaker and the receiver to 
see how they transmit the sound wave. In a 
table, record your qualitative judgments as 
best, good, poor, and so on. 

Test the same materials for their abUity to 
reflect sound and record your results. Be sure 
that the sound is really being reflected and is 
not coming to your detector by some other 
path. You can check how the intensity varies at 
the detector when you move the reflector and 
rotate it about a vertical axis (see Fig. 3-62). If 
suitable materials are available to you, also test 
the reflection from curved surfaces. 






Hg. 3-62 



1. On the basis of your findings, what generaliza- 
tions can you make relating transmission and 
reflection to the properties of the test materials? 



Refraction 

You have probably observed the refraction or 
"bending" of a wave front in a ripple tank as the 



wave slowed down in passing from water of 
one depth to shaUower water. 

You may observe the refraction of sound 
waves using a "lens" made of gas. Inflate a 
spherical balloon with carbon dioxide gas to a 
diameter of about 10-15 cm. Explore the area 
near the bedloon on the side away from the 
source. Locate a point where the sound seems 
loudest, and then remove the balloon. 



2. Do you notice any difference in loudness when 
the balloon is in place? Explain. 



Diffraction 

In front of a speaker set up as before; place a 
thick piece of hard material about 25 cm long, 
mounted vertically about 25 cm directly in front 
of the speaker. Slowly probe the area about 75 
cm beyond the obstacle. 



3. Do you hear changes in loudness? Is there 
sound in the "shadow" area? Are there regions of 
silence where you would expect to hear sound? 
Does there seem to be any pattern to the areas of 
minimum sound? 



For another way to test for dififraction, use a 
large piece of board placed about 25 cm in front 
of the speaker with one edge aligned with the 
center of the source. Now explore the area 
inside the shadow zone and just outside it. (See 
Fig. 3-63.) 

Describe the pattern of sound interference 
that you detect. 



4. Is the pattern analogous to the pattern you 
observed in the ripple tank? 



Wavelength 

METHOD A: Standing Wave 

Set your loudspeaker about 0.5 m above and 
facing toward a hard tabletop or floor, or about 
that distance from a hard, smooth plaster wall 
or other good sound reflector. Your ear is most 
sensitive to the changes in intensity of faint 
sounds, so be sure to keep the volume low. 

Explore the space between the source and 
reflector, listening for changes in loudness. 
Record the positions of minimum loudness, or 
at least find the approximate distjmce between 



144 



UNIT 3 / THE TRIUMPH OF MECHANICS 



two consecutive minima. These minima are 
located one-half wavelength apart. 



5. Does the spacing of the minima depend on the 
intensity of the wave? 

Measure the wavelength of sound at several 
different frequencies. 

6. How does the wavelength change when the 
frequency is changed? 

METHOD B: Interference 

Connect the two loudspeakers to the output 
of the oscillator and mount them at the edge of 
the table about 25 cm apart. Set the frequency 
at about 4,000 Hz to produce a high-pitched 
tone. Keep the gain setting low during the 
entire experiment to make sure the oscillator is 
producing a pure tone, and to reduce re- 
flections that would interfere with the experi- 
ment. 



9. Does the wavelength change with frequency? 
If so, does it change directly or inversely? 



Calculating the Speed of Sound 

The relationship between speed v, wave- 
length X, and frequency / is v = Kf. The 
oscillator dial gives a rough indication of the 
frequency (and your instructor can advise you 
on how to use an oscilloscop>e to make precise 
frequency settings). Using your best estimate of 
X, calculate the speed of sound. If you ha\'e 
time, extend your data to answer the following 
questions. 



10. Does the speed of the sound waves depend 
on the intensity of the waves' 

11. Does the speed depend on the frequency? 




Pig. 3-63 

Move your ear or "stethoscope" along a line 
parallel to, and about 50 cm from, the line 
joining the sources. Can you detect distinct 
maxima and minima? Move farther away from 
the sources; do you find any change in the 
pattern spacing' 



7. What effect does a change in the source 
separation have on the spacing of the nodes? 

8. What happens to the spacing of the nodes if 
you change the frequency of the sound? To make 
this experiment quantitative, work out for yourself 
a procedure similar to that used with the ripple 
tank. 



Measure the separation d of the source 
centers and the distance ,x l)etween nodes and 
use these data to calciilatr the wawlength X. 



Expc>riment 3-19 
IXIRrlSOl'XD 

The equipment needed for this exp>eriment is 
an oscillator, power suppK , and three ultra- 
sonic transducers (cr^-stals that transform elec- 
trical impulses into sound waves, or\ice versal, 
and several materials to be tested. The signal 
from the detecting transducer can be displayed 
with either an oscilloscope (as in Fig. 3-641 or an 



r 




hi 


Ml 


^^mSi 




f ^Hf 


H^^^^H 


lita^^^ 


^HH 



Rg. 3-64 Complete ultrasound equipment Plug the -8 
V. 8 V ground jacks from the amplifier and oscillator 
into the power supply Plug the coaxial cable attached to 
the transducer to the sine wave output of the oscillator 
Plug the coaxial cable attached to a second transducer 
into the input terminals of the ampliher Be sure that the 
shield of the coaxial cable is attached to ground Turn 
the oscillator range switch to the 5 kHi-50 position Turn 
the horizontal frequency range switch of the oscillo- 
scope to at least 10 kHz Turn on the oscillator and 
power supply Tune the oscillator for maximum recep- 
tion, about 40 kHz 



EXPERIMENTS 



145 




Fig. 3-65 Above, ultrasound transmitter and receiver. 
The signal strength is displayed on a microammeter 
connected to the receiver amplifier. Below, a diode 
connected between the amplifier and the meter, to 
rectify ttie output current. The amplifier selector switch 
should be turned to ac. Jhe gain control on the amplifier 
should be adjusted so that the meter will deflect about 
full-scale for the loudest signal expected during the 
experiment. The offset control should be adjusted until 
the meter reads zero when there is no signal. 



amplifier and meter (Fig. 3-65). One or two of 
the transducers, driven by the oscillator, are 
sources of the ultrasound, while the third 
transducer is a detector. Before you proceed, 
have the instructor check your setup and help 
you get a pattern on the oscilloscope screen or 
a reading on the meter. 

The energy output of the transducer is 
highest at about 40,000 Hz, and the oscillator 
must be carefully "tuned" to that frequency. 
Place the detector a few centimeters directly in 
front of the source and set the oscillator range 
to the 5-50 kHz position. Tune the oscillator 
carefully around 40,000 Hz for maximum de- 
flection of the meter or the scope track. If the 
signal output is too weak to detect beyond 25 
cm, plug the detector transducer into an 
amplifier and connect the output of the 
amplifier to the oscilloscope or meter input. 

Transmission and Reflection 

Test the various samples at your station to see 
how they transmit the ultrasound. Record 
your judgments as best, good, poor, etc. Hold 
the sample of the matericil being tested close to 
the detector. 

Test the same matericils for their ability to 
reflect ultrasound. Be sure that the ultrasound 
is really being reflected and is not coming to 
your detector by some other path. You can 
check this by seeing how the intensity varies at 
the detector when you move the reflector. 

Make a table of vour observations. 



1. What happens to the energy of ultrasonic 
waves in a material that neither reflects nor 
transmits well? 



Diffraction 

To observe diffraction around an obstacle, put a 
3-cm wide piece of hard material about 8 or 10 
cm in front of the source (see Fig. 3-66). Explore 
the region 5-10 cm behind the obstacle. 




Fig. 3-66 Detecting diffraction of ultrasound around a 
barrier. 



2. Do you find any signal in the "shadow" area? 
Do you find minima in the regions where you 
would expect a signal to be? Does there seem to 
be any pattern relating the places of minimum 
and maximum signals? 

Put a larger sheet of absorbing material 10 cm 
in front of the source so that the edge obstructs 
about one-half of the source. 

Again probe the "shadow" area and the area 
near the edge to see if a pattern of maxima and 
minima seems to appear. 

Measuring Wavelength 

METHOD A: Standing Wave 

Investigate the standing waves set up bet- 
ween a source and a reflector, such as a hard 
tabletop or metal plate. Place the source about 
10 to 15 cm from the reflector with the detector. 

Find the approximate distance between two 
consecutive maxima or two consecutive 
minima. This distance is one-half the 
wavelength. 



3. Does the spacing of nodes depend on the 
intensity of the waves? 



146 



UNIT 3 / THE TRILMPIl OF MECHANICS 



METHOD B: Interference 

For sources, connect two transducers to the 
output of the oscillator and set them ahout 5 
cm apart. Set the oscillator switch to the 5-50 
kHz position. For a detector, connect a third 
transducer to an oscilloscope or amplifier and 
meter as described in Method A of the experi- 
ment. Then tune the oscillator for maximum 
signal from the detector when it is held near 
one of the sources (about 40,000 Hz). Move the 
detector along a line parallel to and about 25 
cm in front of a line connecting the sources. Do 
you find distinct maxima and minima? Move 



closer to the sources. Do you find any change 
in the pattern spacing? 



4. What effect does a change in the separation of 
the sources have on the spacing of the nulls? 



To make this experiment quantitath'e, work 
out a procedure for yourself similar to that 
used with the ripple tank. Measure the appropn 
riate distances and then calculate the 
wavelength using the relationship 




Fig. 3-67 Setup for determination of wavelength by the 
interference method. 



derived earlier for interference patterns in a 
ripple tank. 



5. In using that equation, what assumptions are 
you making? 



The Speed of Litrasound 

The relationship between sp)eed v, wavelength 
k, and frequency/ is v = Kf. Using your best 
estimate of X, calculate the sjjeed of sound. 



6. Does the speed of the ultrasound waves 
depend on the intensity of the wave? 

7. How does the speed of sound in the inaudible 
range compare with the speed of audible sound? 



ACTIVITIES 



147 



ACTIVITIES 



IS MASS CONSERVED? 

You have read about some of the difficulties in 
establishing the law of conservation of mass. 
You can do several different experiments to 
check this law. 

Antacid Tablet 

You will need the following equipment: antacid 
tablets; 2-L flask, or plastic jug (such as is used 
for bleach, distilled water, or duplicating fluid); 
stopper for flask or jug; warm water; balance 
(sensitivity better than 0.1 g); spring scale 
(sensitivity better than 0.5 g). 

Balance a tablet and 2-L flask containing 
200-300 mL of water on a sensitive balance. 
Drop the tablet in the flask. When the tablet 
disappears and no more bubbles appear, 
readjust the balance. Record any change in 
mass. If there is a change, what caused it? 

Repeat the procedure above, but include the 
rubber stopper in the initial balancing. Im- 
mediately after dropping in the tablet, place the 
stopper tightly in the mouth of the flask. (The 
pressure in a 2-L flask goes up by no more than 
20%, so it is not necessary to tape or wire the 
stopper to the flask. Do not use smaller flasks in 
wtiich proportionately higher pressure would 
be built up.) Is there a change in mass? Remove 
the stopper after all reaction has ceased. What 
happens? Discuss the difference between the 
two procedures. 

Brightly Colored Precipitate 

You will need: 20 g lead nitrate; 11 g potassium 
iodide; Erlenmeyer flask, 1000 mL with stopper; 
test tube, 25 x 150 mm; balance. 

Place 400 mL of water in the Erlenmeyer 
flask, add the lead nitrate, and stir until 
dissolved. Place the potassium iodide in the 
test tube, add 30 mL of water, and shcike until 
dissolved. Place the test tube, open and up- 
ward, carefuUy inside the flask and seal the 
flask with the stopper. Place the flask on the 
balance and bring the balance to equilibrium. 
Tip the flask to mix the solutions. Replace the 
flask on the balance. Does the total mass 
remain conserved? What does change in this 
experiment? 

Magnesium Flashbulb 

On the most sensitive balance you have avail- 
able, mejisure the mciss of an unflashed mag- 



nesium flashbulb. Repeat the measurement 
several times to make an estimate of the 
precision of the measurement. 

Flash the bulb by connecting it to a battery. 
Be careful to touch the bulb as little as possible, 
so as not to wear away any material or leave any 
fingerprints. Measure the mass of the bulb 
several times, as before. You can get a feeling for 
how small a mass change your balance could 
have detected by seeing how large a piece of 
tissue paper you have to put on the balance to 
produce a detectable difference. 



EXCHANGE OF MOMENTUM 
DEVICES 

The four situations described below are more 
complex tests for conversation of momentum, 
to give you a deeper understanding of the 
generality of the conservation law and of the 
importance of your frame of reference. 
(a) Fasten a section of HO-gauge model rail- 
road track to two rings stands cis shown in Fig. 
3-68. Set one truck of wheels, removed fiDm a 
car, on the track and from it suspend an object 
with mass roughly equal to that of the truck. 
Hold the truck, puU the object to one side, 
parallel to the track, and release both at the 
same instant. What happens? 




Fig. 3-68 



Predict what you expect to see happen if you 
released the truck an instant after releasing the 
object. Try it. Also, try increasing the sus- 
pended mass. 



148 



UNIT 3 / THE TRIUMPH OF MECHANICS 



(b) An air track supported on ring stands can 
also be used. An object of 20 g mass was 
suspended by a SO-cm string from one of the 
small air-track gliders. (One student trial con- 
tinued for 166 swings.) 

(c) Fasten two dynamics carts together with 
four hacksaw blades as shown in Fig. 3-69. Push 
the top one to the right, the bottom to the left, 
and release them. Try giving the bottom cart a 
push across the room at the same instant you 
release them. 




Fig. 3-69 



account for this increase? {Hint: Set the wedge 
on a piece of cardboard supported on plastic 
beads and try the exp>eriment.) 

STUDENT HORSEPOU'ER 

When you wtilk up a flight of stairs, the work 
you do goes into frictional heating and increas- 
ing gravitational potential energy. The 
AlP£)|„v. in joules, is the product of your 
weight in newtons and the height of the stairs 
in meters. 

Your useful power output is the average rate 
at which you did the lifting work, that is, the 
total change in IP£)b»v- di\ided by the time it 
took to do the work. 

Walk or run up a flight of stairs and have 
someone time how long it takes. Determine the 
total vertical height that you lifted yourself by 
measuring one step and multiplying by the 
number of steps. 

Calculate your useful work output and your 
power, in both units of watts and in horse- 
power. (Take 1 horsepower to be equal to 746 
watts.) 



What would happen wlien you released the 
two if there were 10 or 20 ball bearings or small 
wooden balls hung as pendula from the top 
cart? 

(d) Push two large rubber stoppers onto a short 
piece of glass tubing or wood (Fig. 3-70). Let the 
"dumbbell" roll down a wooden wedge so that 
the stoppers do not touch the table until the 
dumbbell is almost to the bottom. When the 
dumbbell touches the table, it suddenly in- 
creases its linear momentum as it moves along 
the table. Principles of rotational momentum 
and enei^ are involved here that are not 
covered in the Text, but even without extending 
the Text, you can deal with the "mysterious' 
increase in linear momentum when the stop- 
pers touch the table. 




^ M 



DRINKING DUCK 

A toy ctilled a Drinking Duck demonstrates ver\' 
well the conN-ersion of heat ener^- into ener^gv' 
of gross motion by the processes of evaporation 
and condensation. The duck will continue to 
bob up and down as long as there is enough 
water in the cup to wet the beak (see Fig. 3-71). 




Fig 3 70 



Rg. 3-71 



Using wtiatyou have learned about conserva- 
tion of tnoiiKMitum. what do vou tliink could 



Rather than dampen vour spirit of ad\-enture, 
we will not tell you how it works. If vou cannot 
figure out a (lossible mechanism for yourself. 



ACTIVITIES 



149 



George Gamow's book, The Biography of 
Physics, has a very good explanation. Gamow 
also calculates how far the duck could raise 
water in order to feed itself. An interesting 
extension is to replace the water with rubbing 
alcohol. What do you think will happen? 



MECHANICAL EQUIVALENT 
OF HEAT 

By dropping a quantity of lead shot ftxim a 
measured height and measuring the resulting 
change in temperature of the lead, you can get 
a value for the ratio of work units to heat units, 
the mechanical equivalent of heat. 

You will need the following equipment: 
cardboard tube; lead shot (1-2 kg); stoppers; a 
thermometer. 

Close one end of the tube with a stopper, and 
put in 1- 2 kg of lead shot that has been cooled 
about 5°C below room temperature. Close the 
other end of the tube with a stopper in which a 
hole has been drilled and a thermometer 
inserted. Carefully roll the shot to this end of 
the tube and record its temperature. Quickly 
invert the tube, remove the thermometer, and 
plug the hole in the stopper. Now invert the 
tube so the lead falls the fuU length of the tube 
and repeat this quickly 100 times. Reinsert the 
thermometer and measure the temperature. 
Measure the average distance the shot falls, 
vv+iich is the length of the tube minus the 
thickness of the layer of shot in the tube. 

If the average distance the shot falls is h and 
the tube is inverted N times, the work you did 
raising the shot of mass m is 

AW = N X mag x h 

The heat AH needed to raise the temperature 
of the shot by an amount AT is 
AH = cm AT 

where c is the specific heat capacity of lead, 3.1 

X 10-* cal/gC° (or 0.13 J/gC°). 
The mechanical equivalent of heat is 

Aw/ AH. The accepted experimental vjilue is 
4.184 Nm/Cal (or Nm/J). 



rainbow. He and his astronomer friend Gas- 
sendi were a bulwark agfiinst Aristotelian 
physics. Descartes belonged to the generation 
between Gcilileo and Newton. 

On the lighter side, Descartes is known for a 
toy called the Cartesian diver that was veiy 
popular in the eighteenth centuiy when very 
elaborate ones were made. To make one, first 
you will need a column of water. You may find 
a large cylindrical graduate in the laboratory, 
the taller the better. If not, you can improvise 
one out of a large jug or any other tall glass 
container. Fill the container almost to the top 
with water. Close the contciiner in a way that 
permits you to change the pressure in it. For 
example, take a short piece of glass tubing with 
fire-polished ends, lubricate the glass tubing 
jmd the hole in the stopper wath water, and 
carefully insert the glass tubing. Fit the rubber 
stopper into the top of the container as shown 
in Fig. 3-72. 



rubber tufct 




'1*^ rubbtr stopper 
I — glass tube. 



pili beetle. 



ciiv«r rifli>t-sid« up 



A DIVER IN A BOTTLE 

Descartes is a name well known in physics. 
When you graphed motion in Text Sec. 1.5, you 
used Cartesian coordinates, w^ch Descartes 
introduced. Using Snell's law of refraction, 
Descartes traced 1,000 rays through a sphere 
and came up with an explanation of the 




Fig. 3-72 

Next construct the diver. You may limit 
yourself to pure essentials, namely a small pill 
bottle or vial, which may be weighted with wire. 



ISO 



UNIT 3 / THE THIliMPH OF MECHANICS 



Partially fill it with water so it just barely floats 
upside down at the top of the water column. If 
you are so inclined, you can decorate the bottle 
so it looks like a real underwater swimmer (or 
creature, if you prefer). The essential things are 
that you have a diver that just floats and that 
the volume of water in the diver can be 
changed. 

Now to make the diver perform, blow 
momentarily on the rubber tube. According to 
Boyle's law, the increased pressure (transmit- 
ted by the water) decreases the volume of 
trapped air and water enters the diver. The 
buoyant force decreases, according to Ar- 
chimedes' principle, and the diver begins to 
sink. (Archimedes' principle simply says that 
the buoyant force on an object is equal to the 
weight of the liquid displaced. This is the 
reason objects can float.) 

If the original pressure is restored, the diver 
rises again. However, if you are lucky, you will 
find that as you cautiously make it sink deeper 
and deeper down into the column of water, it is 
more and more reluctant to return to the 
surface as the additional surface pressure is 
released. Indeed, you may find a depth at 
which the diver remains almost stationary. 
However, this apparent equilibrium, at which 
its weight just equals the buoyant force, is 
unstable. A bit above this depth, the diver will 
freely rise to the surface, and a bit below this 
depth it will sink to the bottom of the water 
column from which it can be brought to the 
surface only by vigorous sucking on the tube. 

If you are mathematically inclined, you can 
compute what this depth would be in terms of 
the atmospheric pressure at the surface, the 
volume of the trapped air, and the weight of the 
diver. If not, you can juggle with the volume of 
the trapped air so that the point of unstable 
equilibrium comes about halfway down the 
water column. 



The diver raises interesting questions. Sup- 
pose you have a well-behaved div-er that 'floats ' 
at room temperature just halfway down the 
water column. Where will it "float' if the 
atmospheric pressure drops? Where will it 
"float " if the water is cooled or heated? If the 
ideal gas law is not enough to answer this 
question, you may have to do a bit of reading 
about the vapor pressure of water. 



HOW TO VITIIGH A CAR WITH A 
TIRE PRESSITRE GAUGE 

Reduce the pressure in all four auto tires so 
that the pressure is the same in each and 
somewhat below recommended tire pressure. 

Drive the car onto four sheets of graph paper 
placed so that you can outline the area of the 
tire in contact with each piece of pap>er. The car 
should be on a reasonably flat surface igarage 
floor or smooth drivewayl. Then sprav water on 
the graph paper. After the car is moved off the 
pap>er, you can measure the dry area The 
flattened part of the tire is in equilibrium 
between the vertical force of the ground 
upward and the downward force of air pres- 
sure within. 

Measure the air pressure in the tires, and the 
area of the flattened areas If you use centime- 
ter graph pap>er, you can determine the area in 
square centimeters by counting squares. 

Pressure P (in pascals, Pa) is defined as F/A, 
where F is the downward force lin newlons) 
acting perpendicularly on the flattened area A 
(in square metersi. Since the tire pressure 
gauge indicates the pressure abow the normal 
atmospheric pressure of 101 kPa. \x)U must add 
this value to the gauge reading Compute the 
four forces as pressure times area. Their sum 
gix'es the weight of the car. 




PERI»ETrAL-.\IO HON 
.\1AC:IIINES? 

You must ha\-e heard of "perjjetual- motion" 
machines which, once started, will continue 
nmning and doing useful work forp\"er These 
proposed de\ires are inconsLstenI with the 
laws of themiodMiamirs ilt is tempting to say 
that thrv unlatr the laws of thpmi()d\-namics, 
hut this implies that laws are rules bv which 
Nature fnu5f run. instead of descriptions scien- 
tists haw thought up i It is now beliex-ed that It 
is in principle impossible to build such a 
machine. 



ACTIVITIES 



151 



But the dream dies hard! New proposals are 
made edmost daily. Thus, S. Raymond Smedile, 
in Perpetual Motion and Modem Research for 
Cheap Power (Science Publications of Boston, 
1962), maintains that this attitude of "it can't be 
done" negatively influences the sejirch for new 
sources of cheap power. His book gives 16 
examples of proposed machines, of wWch two 
are shown here. 

Number 5 (Fig. 3-73) represents a wheel 
composed of 12 chambers marked A. Each 
chamber contains a lead ball B, which is free to 
roll. As the wheel turns, each ball rolls to the 
lowest level possible in its chamber. As the balls 
roll out to the right edge of the wheel, they 
create a preponderance of turning effects on 
the right side as against those balls that roll 
toward the hub on the left side. Thus, it is 



claimed that the wheel is driven clockwise 
perpetuiilly. If you think this machine wHl not 
work, explain why not. 

Number 7 (Fig. 3-74) represents a water- 
driven wheel marked A. D represents the 
buckets on the perimeter of the waterwheel for 
receiving water draining from the tank marked 
F. The waterwheel is connected to pump B by a 
belt and wheel. As the overshot wheel is 
operated by water dropping in it, it operates 
the pump that sucks water into C from which it 
enters into tank F. This operation is supposed 
to go on perpetually. If you think otherwise, 
explain why. 

If such machines could operate, would the 
conservation law^ necessarily be wrong? 

Is the absence of perpetual-motion machines 
due to "theoretical" or "practical" deficiencies? 





Fig. 3-73 Number 5. 



Fig. 3-74 Number 7. 





The cartoons on pages 150-151 (and others of the 
same style that are scattered through the Handbook) 
were drawn in response to some ideas in the Project 
Physics course by a cartoonist who was unfamiliar with 
physics. On being informed that the drawing on the 



left did not represent conservation because the candle 
was not a closed system, he offered the solution 
above. (Whether a system is "closed" depends, of 
course, upon what you are trying to conserve.) 



152 



UNIT 3 / THE TRIUMPH OF MECHANICS 



STANDING WAVES ON A DRUM 
AND A \^OLIN 

You can demonstrate many different patterns 
of standing waves on a rubber membrane using 
a method very similar to that used in Film Loop 
42, "Vibrations of a Drum." If you have not yet 
seen this loop, view it, if possible, before setting 
up the demonstration in your lab (see Fig. 3-75). 




Fig. 3-75 




Rg. 3-76 



Figure 3-76 shows the apparatus in action, 
producing one pattern of standing waNfs. The 
drumhead in the figure is an ordinary 17.5-cm 
embroidery hoop with the end of a lai^e 
balloon stretched over it. If you make your 
drumhead in this way, use as lai^e and as 



strong a balloon as possible, and cut its neck off 
with scissors. A flat piece of sheet rubber 
(dental dam) gives better results, since e\'en 
tension over the entire drumhead is much 
easier to maintain if the rubber is not curved to 
begin with. Try other sizes and shapes of 
hoops, as well as other drumhead materials, 
such as a plastic wrap. 

A 10-cm, 45-ohm speaker, lying under the 
drum and facing upward toward it, drn-es the 
vibrations. Connect the speaker to the output of 
an oscillator. If necessary, ampUK' the osciUator 
output. 

Turn on the oscillator and sprinkle salt or 
sand on the drumhead. If the frequency is near 
one of the resonant frequencies of the surface, 
standing waves will be produced. The salt will 
collect along the nodes and be thrown off from 
the antinodes, thus outlining the pattern of the 
vibration. Vary the frequency' until you get a 
clear pattern, then photograph or sketch the 
pattern and move on to the next frequency 
w+iere you get a pattern. 

When the speaker is centered, the \ibration 
pattern is symmetrical around the center of the 
surfare. In ordrr to get antis\inmetric nodes of 
vibration, move the speaker toward the edge of 
the drumhead. Elxperiment with the spacing 
between the speaker and the drumhead until 
you find the position that gives the clearest 
pattern; this position may be different for 
different frequencies. 

If your patterns are distorted, the tension of 
the drumhead is probably not uniform. If you 
have used a balloon, you may not be able to 
remedy the distortion, since the curvature of 
the balloon makes the edges tighter than the 
center. By pulling gently on the rubber, how- 
ever, vou mav at Ira.st hv ablf to make the 
tension eN-en all around the edge. 

A similar procedure, used 150 years ago and 
still used in anaK'zing the performance of 
violins, is shown in Fig. 3-77, reprinted from 
Scientific American. "PhN-sics and Music." 



REFLECmON 

Two-dimensional water surface waves exhibit a 
fascinating variet>' of reflection phenomena If 
you have never watched closely as water waves 
are reflerted from a fixed barrier, you should do 
so ,'\ny still pool or waler-fill€»d wash basin or 
tub will do Watch the circular wavT« radiate 
outwartl, refleiM from rocks or walls, run 
through each other, and finally die out. Dip 
your fingeriip into and out of the water quickly. 



ACTIVITIES 



153 




»l 



Rg. 3-77 Chladni plates indicate the vibration of the 
body of a violin. These patterns were produced by 
covering a violin-shaped brass plate with sand and 
drawing a violin bow across its edge. When the bow 
caused the plate to vibrate, the sand concentrated along 
quiet nodes between the vibrating areas. Bowing the 
plate at various points, indicated by the round white 
marker, produces different frequencies of vibration and 




different patterns. Low tones produce a pattern of a few 
large areas; high tones a pattern of many small areas. 
Violin bodies have a few such natural modes of 
vibration that tend to strengthen certain tones sounded 
by the strings. Poor violin bodies accentuate squeaky 
top notes. This sand-and-plate method of analysis was 
devised over 150 years ago by the German acoustical 
physicist Ernst Chladni. 



or let a drop of water fall from your finger into 
the water. Now watch the circular wave ap- 
proach and then bounce off of a straight wall or 
board. The long side of a bathtub is a good 
straight barrier. (See the illustrations on p. 376, 
Unit 3.) 

MOIRE PATTERNS 

You will probably notice a disturbing visual 
effect from the patterns in Figs. 3-78 and 3-79. 



Some types of art depend on similar effects, 
many of which are caused by moire patterns. 

If you make a photographic negative of the 
pattern in Fig. 3-78 or Fig. 3-79 and place it on 
top of the same figure, you can use it to study 
the interference pattern produced by two point 
sources. 

There are an increasing number of scientific 
applications of moire patterns. Because of the 
great visual changes caused by very small shifts 
in two regular overlapping patterns, they can 





Rg. 3-78 



Rg. 3-79 



184 



UNIT 3 / THE TRIUMPH OF MECHANICS 



be used to make measurements to an accuracy 
of +0.0000001%. Some specific examples of the 
useof moire patterns are visualization of two- or 
multiple-source interference patterns, the 
measurement of small angular shifts, mea- 
surements of diffusion rates of solids into 
liquids, and representations of electric, mag- 
netic, and gravitational fields. Some of the 
patterns created still cannot be expressed 
mathematically. 

Scientific American (May, 1963) has an excel- 
lent article, "Moire Patterns" by Gerald Oster 
and Yasunori Nishijima. The Science of Moire 
Patterns, a book by G. Oster, is available from 
tdmund Scientific Co., Barrington, N. J. Ed- 
mund also has various inexpensive sets of 
different patterns, which save much drawing 
time, and that are much more precise than 
hand-drawn patterns. 



MUSIC AND SPEECH 
ACTIVITIES 

(a) Frequency ranges: Set up a microphone 
and oscilloscope so you can display the 
pressure variations in sound waves. Play differ- 
ent instruments and see how "high C" differs 
on them. 

(b) Some beautiful oscilloscope patterns result 
wlien you display the sound of computer 
music records, which use sound synthesizers 
instead of conventional instruments. 

(c) For interesting background, see the follow- 
ing articles in Scienfj/jcAmer/'can; "Physics and 
Music," July, 1948; "The Physics of Violins," 
November, 1962; "The Physics of Wood-winds, ' 
October, 1960; and "Computer Music," De- 
cember , 1959. 

(d) The Bell Telephone Company has an in- 
teresting educational item, which may be 
available through your local Bell Telephone 
office. A 33V3 LP record, "The Science of 
Sounds," has 10 bands demonstrating different 
ideas about sound. For instance, racitig cars 
demonstrate the Doppler shift, and a soprano, 
a piano, and a factory whistle all sound alike 
when overtones are filtered out electronically 
The record is also available on the Folkways 
label FX6136. 

(e) "Test records" are available for stereo hi-fi 
equipment. Many of these records let you 
check the "frequency response" of a sN^item by 
giving a series of steady tones at various 
frequencies. TYy playing one of these records 
and checking for nodes Be sure to place the 
speakers far enougli ;ipai1 (How f.ir- ap.in do ihc 



speakers have to be so that you get two nodes 
between them? Compute this for/ = 100; 
500; 1,000; and 5,000 Hz.) Listen for nodes by- 
moving your head back and forth. 



MEASUREMENT OF THE SPEED 
OF SOUND 

For this experiment you need to work outside 
in the vicinity of a large flat v\'all that produces a 
good echo You also need some source of loud 
pulses of sound at regular intervals, about one 
per second or less. A friend beating on a drum 
or something with a higher pitch will do. The 
important thing is that the time between one 
pulse and the next does not vary, so a 
metronome would help. The sound source 
should be fairly far away from the wall, about 
200 m in front of it. 

Stand somewhere between the reflecting wall 
and the source of pulses (see Fig. 3-801. You will 
hear both the direct sound and the sound 
reflected from the wall. The direct sound will 
reach you first because the reflected sound 
must travel the additional distance from you to 
the wall and back cigain. As you approach the 
wall, this additional distance decreases, as does 
the time interval between the direct sound and 
the echo. Movement away from the wall 
increases the interval. 




f— vtiH 



Hg. 3-80 

If the distance from the source to the wall is 
great enough, the added time taken by the echo 
to reach you ran amount to more than the time 
between drum beats You will l>e able to find a 
position at wliich you hear the rcho of one 
pulse at the same time vou hear the direct 
sound of the next pulse Then vou know that 
the .sound took a time equal to the interval 
between pul.ses to travel from vou to the wall 
and bark to vou. 

Measure your distanc<' fnmi the wall Find 
the time interval lM»tween pulses by measuring 
the time for a lai^e number of pulses I'se these 
twii values to calculate the sj>eed of sound 



ACTIVITIES 



155 







9uls^ 






Rg. 3-81 



■iime. 



(If you cannot get far enough away from the 
wall to get this synchronization, shorten the 
interval between pulses. If this is impossible, 
you may be able to find a place where you hear 
the echoes exactly halfway between the pulses 
as shown in Fig. 3-81. You will hear a pulse, 
then an echo, then the next pulse. Adjust your 
position so that these three sounds seem 
equally spaced in time. At this point you know 
that the time taken for the return trip from you 
to the wall and back is equal to /la/f the time 
interval between pulses.) 



MECHANICAL WAVE MACHINES 

Several types of mechanical wave machines are 
described below. They help a great deal in 
understanding the various properties of waves. 

Slinky 

The spring called a Slinky behaves much better 
when it is freed of friction from the floor or 
table. Hang a Slinky horizontally from strings at 
least 1 m long tied to rings on a wire stretched 



Stretched CJite. ^ -- Curtain J?injs 



Ij V fv 



, J I r , r 



; ! 



! I 



I I 



(f^mlumm^mmMmw^ 



s/;nk 



V 



from two solid supports. Tie strings to the 
Slinky at every fifth spiral for proper support. 
(See Fig. 3-82.) 

Fasten one end of the Slinl^ securely and 
then stretch it out to about 5- 10 m. By holding 
onto a 3-m piece of string tied to the end of the 
Slinky, you can illustrate the "open-ended" 
reflection of waves. 

See Experiment 3-15 for more details on 
demonstrating the various properties of waves. 

Rubber Tubing and 
Welding Rod 

clamp both ends of a 1-m piece of rubber 
tubing to a table so it is under slight tension. 
Punch holes through the tubing every 2.5 cm 
with a hammer and nail. (Put a block of wood 
under the tubing to protect the table.) 

Obtain enough 30-cm lengths of welding rod 
for all the holes you punched in the tubing. 
Unclamp the tubing, and insert one rod in each 
of the holes. Hang the rubber tubing vertically, 
as shown in Fig. 3-83, and give its lower end a 
twast to demonstrate transverse waves. Perfor- 
mance and visibility are improved by adding 
weights to the ends of the rods or to the lower 
end of the tubing. (See Fig. 3-83.) 




Rg. 3-82 



Fig. 3-83 



A Better Wave Machine 

An inexpensive paperback, Similarities in Wave 
Behavior, by John N. Shive of Bell Telephone 
Laboratories, has instructions for building a 
better torsional-wave machine than that de- 
scribed above. 



156 



UNIT 3 / THE TRIUMPH OF MECHANICS 



FILM LOOP NOTES 



FUm Loop 18 

ONE-DIMENSIOXAL COLLISIOxXS. 
I 

Two different head-on collisions of a pair of 
steel balls are shown. The balls hang from long, 
thin wires that confine each ball's motion to the 
same circular arc. The radius is large compared 
with the part of the arc, so the curvature is 
hardly noticeable. Since the collisions take 
place along a straight line, they can be called 
one-dimensional. 




You will find it useful to mark the filmed 
crosses on the pwper on which you are 
projecting, since this will allow you to correct 
for projector movement and film jitter. Vou 
might want to give some thought to measuring 
distances. You may use a ruler with marks in 
millimeters, so you can estimate to one-tenth of 
a millimeter Is it wise to tiy to use the zero end 
of the ruler, or should you use p>ositions in the 
middle? Should you use the thicker or the 
thinner marks on the ruler? Should you rely on 
one measurement, or should you make a 
number of measurements and av-erage them? 

Estimate the uncertainty in distance and 
time measurements, and the uncertainty- in 
\elocity. What can you learn finom this about the 
uncertaintv in momentum? 



>T 



In the First example, ball B, weighing 350 g, is 
initially at rest. In the second example, ball A, 
with a mass of 532 g, is the one at rest. 

With this film, you can make detailed mea- 
surements of the total momentum and enei^ 
of the balls before and after collision. Momen- 
tum is a vector, but in'this one-dimensional 
case you need only worrv about its sign. Since 
momentum is the product of mass and velocity, 
its sign is detennined by the sign of the velocity. 

You know the masses of the balls. Velocities 
can be measured by finding the distance 
traveled in a knov\ii |K?riod of time. 

After viewing the film, you can decide on 
what strategN' to use for distance and time 
measuit'iTients. One possibility' wouUl be to 
time the motion through a given distance with 
a stopwatch, perhaps making two lines on the 
pap<>r You need the velocity just Iwfore and 
after the collision Since the balls an- hanging 
fnim win«s, their velocity is not constant On 
the other hand, using a small an- incn-ases the 
chances of distance- time uncertainties /\s 
with most measuring situations, a number of 
conflicting factors must be considered 



J, 



>Ji 



in 



Rg. 3-84 

Wlien you compute the total momentum 
befon* and after collusion (the sum of the 
momentum of each balli. remember that you 
must consider the direction of the momentum 

.\rv the differences l>etwven the momentum 
bt^fore and after collLsion significant, or are the\' 
within the experimental error already estl- 
matetP 

Saw the data you coll«t so that later you CAn 
make similar calculations on total kinetic 
eneixy for both balls just before and just 
collusion 



FILM LOOP NOTES 



15T 



Film Loop 19 

ONE-DIMENSION COLLISIONS. 
II 

Two different head-on collisions of a pair of 
steel balls aire shoun, with the same setup as 
that used in Film Loop 18, "One-Dimensional 
Collisions. I. " 

In the first example, ball A with a mass of 1.8 
kg collides head on VNith ball B, with a mass of 
532 g. In the second example, ball A catches up 
with ball B. The instructions for Film Loop 18 
may be followed for completing this investiga- 
tion also. 

Film Loop 20 

EVELASTIC ON^-DLMENSIONAL 

COLLISIONS 

In this film, two steel balls covered with 
plasticene hang from long supports. Two 
collisions are shown. The two balls stick 
together after colliding, so the collision is 
inelastic. In the first example, ball A, weighing 
443 g, is at rest when ball B, with a mass of 662 
g, hits it. In the second example, the same two 
balls move toward each other. Two other films, 
"One-Dimensional Collisions. I' and "One- 
Dimensional Collisions. 11" show collisions in 
which the two balls bounce off^ of each other. 
What different results might you expect from 
measurements of an inelastic one-dimensional 
collision? 

The instructions for Film Loop 18 may be 
followed for completing this investigation. 

Are the diff^erences between momentum 
before and after collision significant, or are the\' 
within the experimental error already esti- 
mated? 

Save your data so that later you can make 
similcir calculations on total kinetic energy for 
both balls just before and just after the 
collision. Is whatever diff^erence you may have 
obtained explainable by experimental error? Is 
there a noticeable difference between elastic 
and inelastic collisions as far as the conserva- 
tion of kinetic energy is concerned? 

Film Loop 21 

TWO-DIMENSIONAL COLLISIONS. 
I 

Two hard steel bsdls, hanging irom long, thin 
wires, collide. Unlike the collisions in Film 
Loops 18 and 20, the balls do not move along 
the same straight line before or after the 



collisions. Although the balls do not all strictly 
move in one plane because each motion is an 
arc of a circle, to a good approximation 
everything occurs in one plane. Therefore, the 
collisions are two-dimensional. Two collisions 
are filmed in slow motion, with ball A having a 
mass of 539 g, and ball B having a mass of 361 g. 
Two more cases are shown in Film Loop 22. 

Using this film, you can find both the 
momentum and the kinetic energy' of each ball 
before and after the collision, and thus study 
total momentum and total kinetic energy 
conservation in this situation. Thus, you should 
save your momentum data for later use when 
studying energy. 

Both direction and magnitude of momentum 
should be taken into account, since the balls do 
not move on the same line. To find momentum 
you need velocities. Distance measurements 
accurate to a fraction of a millimeter, and time 
measurements to about one-tenth of a second 
are suggested, so choose measuring instru- 
ments accordingly. 

You can project directly onto a large piece of 
paper. An initial problem is to determine lines 
on which the balls move. If you make many 
marks at the centers of the balls, running the 




Fig. 3-85 



158 



LIMIT 3 / THE TRIUMPH OF MECHANICS 



film several times, you may find that these do 
not form a straight line. This is due both to the 
inaccuracies in your measurements and to the 
inherent difficulties of high-speed photo- 
graphy. Cameras photographing at a rate of 
2,000 to 3,000 frames a second "jitter," because 
the film moves so rapidly through the camera 
that accurate frame registration is not possible. 
Decide which line is the "best" approximation 
to determine direction for velocities of the balls 
before and after collision. 

To find the magnitude of the velocity, the 
sf)eed, measure the time it takes the ball to 
move across two lines marked on the paper. 
For the sake of accuracy, take a number of 
different measurements to determine which 
values to use for the speeds and how much 
error is present. 

Compare the sum of the momentum before 
collision for both balls with the total momen- 
tum after collision. If you do not know how to 
add vector diagrams, you should consult your 
teacher or the Programmed Instruction 
Booklet, "Vectors II." The momentum of each 
object is represented by an arrow whose 
direction is that of the motion and whose 
length is proportional to the magnitude of the 
momentum. Then, if the head of one arrow is 
placed on the tail of the other, moving the line 
parallel to itself, the vector sum is represented 
by the arrow that joins the "free" tail to the 
"free" head. 

What can you say about momentum conser- 
vation? Remember to consider measurement 
errors. 

Film Loop 22 
TWO-DIi\IFNSIOIVAL 
COLLISIONS. II 

Two hard steel balls, hanging from long, thin 
wires, collide. Unlike the collisions in Film 
Loops 18 and 20, the balls do not move along 
the same straight line before or after the 
collisions. Although all the balls do not strictly 
move in one plane, as each motion is an arc of a 
circle, eveiything occurs in one plane There- 
fore, the collisions are two-dimensional I\vo 
collisions are filmed in slow motion, with Ixith 
balls having a mass of 367 g. IVvo other cases 
are shown in Film Loop 21 

I'sing this film, you ran find both the kinetic 
energv and thr inomenliiin of each ball Iwfore 
and afirr the collision, and thus study total 
moinenlutn and total enrrg\ conserNation in 
this situation. Follow the instructions gi\-en for 
Film l,oop 21 in completing this in\rstigation 



Fflm Loop 23 ' 

INELASTIC TH'O-DLMENSIONAL 
COLLISIONS J 

Two hard steel balls, hanging from long, thin 
wires, collide. Unlike the collisions in Film 
Loop>s 18 and 20, the balls do not move along 
the same straight line before or after the 
collision. Although all the balls do not strictly 
move in one plane, as each motion is an arc of a 
circle, to a good approximation the motion 
occurs in one plane. Therefore, the collisions 
are two-dimensional. Two collisions are filmed 
in slow motion. E^ch ball has a mass of 500 g. 
The plasticene balls stick together after colli- 
sion, and move as a single mass. 

Using this film, you can find both the kinetic 
energy and the momentum of each ball before 
and after the collision, and thus study total 
momentum and total energy conservation in 
this situation. Follow the instructions given for 
Film Loop 21 in completing this investigation. 



Film Loop 24 

SCATTERLNG OF A CLUSTER 

OF OmECTS 

This film and also Film Loop 25 each contain 
one of the Events 8- 13 of the series. Strobo- 
scopic Still Photographs of Two-Dimensional 
one of the Events 8- 13 of the series, "Strobo- 
scopic Still Photographs of Two-Dimensional 
Collisions, ■ or one of the examples in Film 
Loops 22 and 23 All these examples invoK* 
two-body collisions, whereas the film described 
here involves seven objects and Film Loop 25, 
five objects. 

In this film, se\-en balls are suspended from 
long, thin wires. The camera sees only a small 
portion of their motion, so the balls all mo\^ 
approximately along straight lines The slow- 
motion camera is above the balls Six balls are 
initially at rest. A hardened steel ball strikes the 
cluster of resting balls The diagram in Fig 3-86 
shows the mass of each of the l^a\ls 

Part of the film is photographed in slow- 
motion at 2.000 frames j)er second Bv project- 
ing this section of the film on paper sewral 
times and making measurements of dLstancc^s 
and times, you can determine the dixvctions 
antl magnitutlcs of the wiocities of each of the 
balls Distance and time measurements are 
lUMHled Discussions of how to make such 
measurements are contained in the Film Loop 
Notes for one-dimensional and two- 



FILM LOOP NOTES 



159 




Rg. 3-86 




Rg. 3-87 

dimensional collisions. (See FUm Loops 18 and 
21.) 

Compare the total momentum of the system 
both before and after the collision. Remember 
that momentum has both direction and mag- 
nitude. You can add momenta after collision by 
representing the momentum of each ball by an 
arrow, and "adding" arrows geometrically. 
WTiat can you say about the accuracy of your 
calculations and measurements? Is momen- 
tum conserved? You might also wish to con- 
sider energy conservation. 

Film Loop 25 

EXPLOSIOX OF A CLUSTER 

OF OBJECTS 

Five balls are suspended independently fix)m 
long, thin wires. The balls are initially at rest, 



with a small cylinder containing gunpowder in 
the center of the group of balls. The masses and 
initial positions of the balls are shown in Fig. 
3-88. The charge is exploded and each of the 
balls moves off in an independent direction. In 
the slow-motion sequence, the camera is 
mounted directly above the resting balls. The 
camera sees only a small part of the motion, so 
that the paths of the balls are almost straight 
lines. 




^- 



Rg. 3-88 



In your first viewing, you might try to predict 
where the "missing" balls will emerge. Several 
of the balls are hidden at first by the smoke 
from the charge of powder. All the balls except 
one are visible for some time. What information 
could you use that would help you make a 
quick decision about where this last ball will 
appear? What physical quantity is important? 
How can you use this quantity to make a quick 
estimate? When you see the ball emerge from 
the cloud, you can determine whether or not 
your prediction was correct. The animated 
elliptical ring toward the end of the film 
identifies this final ball. 

You can also make detailed measurements, 
simUar to the momentum conservation mea- 
surements you may have made using other 
Project Physics Film Loops. During the slow- 
motion sequence, find the magnitude and 
direction of the velocity of each of the balls after 
the explosion by projecting the film on paper, 
and measuring distances and times. The notes 
on previous films in this series. Film Loops 18 
and 21, will provide you with information about 
how to make such measurements if you need 
assistance. 

Determine the tot£d momentum of all the 
balls after the explosion. What was the momen- 



160 



UNIT 3 / THE TRIUMPH OF MECHANICS 



B.C. 



by John Hart 




ty ^raliilon of John Hart and rtald Intarprliai, Inc. 



turn before the explosion? You may find these bullet imbeds itself in the log. The two bodies 



results slightly puzzling. Can you account for 
any discrepancy that you find? Watch the film 
again and pay close attention to what happens 
during the explosion. 



move together after this violent collision. The 
height of the log is 15.0 cm. You can use this 
information to convert distances to centime- 
ters. The setup is illustrated in Fig. 3-89. 



KEVETIC ENERGY 
CALCULATIOIVS 

You may have used one or more of Film Loops 
18 through 25 in your study of momentum. You 
will find it helpful to view these slow-motion 
films of one- and two-dimensional collisions 
again, but this time in the context of the study 
of energy. The data you collected previously 
will be sufficient for you to calculate the kinetic 
energy of each ball before and after the 
collision. Remember that kinetic energy Vzmv^ 
is not a vector quantity; therefore, you need 
only use the magnitude of the velocities in your 
calculations. 

On the basis of your analysis you may wish to 
tiy to answer such questions as: Is kinetic 
energy consumed in such interactions? If not, 
what happened to it? Is the loss in kinetic 
energy rt^lated to such factors as relative speed, 
angle of impact, or relative masses of the 
colliding balls? Is then^ a difference in the 
kinetic energy lost in elastic and inelastic 
collisions? 

Film I.4M>p 26 

FINIIINC; THE SPEED OF A 

RIFLE BULLET. I 

In this film, a rifle bullet of 13.9 g is fired into an 
8.44-kg log Ilie log is initially at rest, and the 




r^---»1 




Fig. 3-89 Schematic dicgram of ballistic pendulum (not 

to scale) 

You can make measurements in this film 
using the extreme slow-motion sequence The 
high-speed camera used to film this sequence 
operated at an awrage rate of 2A50 frames per 



FILVI LOOP NOTES 



161 



second; if your projector runs at 18 frames per 
second, the slow-motion factor is 158. Although 
there was some variation in the speed of this 
camera, the average frame rate of 2,850 is quite 
accurate. For velocity measurements in cen- 
timeters per second (a convenient unit to use in 
considering a rifle bullet) convert the apparent 
time of the film to seconds. Find the exact 
duration with a timer or a stopwatch by timing 
the interval from the yellow circle at the 
beginning to the one at the end of the film. 
There are 3,490 frames in the film, so you can 
determine the precise speed of the projector. 

Project the film onto a piece of white paper 
or graph paper to make your measurements of 
distance and time. View the film before making 
decisions about which measuring instruments 
to use. As suggested above, you can convert 
your distance and time measurements to 
centimeters and seconds. 

After measuring the speed of the log after 
impact, ciilculate the bullet speed at the 
moment when it entered the log. What physical 
laws do you need for the calculation? Calculate 
the kinetic energy given to the bullet, and edso 
calculate the kinetic energy of the log after the 
bullet enters it. Compare these two energies 
and discuss any differences that you might 
find. Is kinetic energy conserved? 

A final sequence in the film allows you to find 
a lower limit for the bullet's speed. Three 
successive frames are shown, so the time 
between each is 1/2,850 sec. The frames are 
each printed many times, so each is held on the 
screen. How does this lower limit compiire with 
your measured velocity? 



Film Loop 27 

FINDING THE SPEED OF A 

RIFLE BULLET. II 

The problem proposed by this film is to 
determine the speed of the bullet just before it 
hits a log. The wooden log with a mass of 4.05 kg 
is initially at rest. A bullet fired ftxim a rifle 
enters the log I Fig. 3-90). The mass of the bullet 
is 7.12 g. The bullet is imbedded in the thick log 
and the two move together after the impact. 
The extreme slow-motion sequence is intended 
for tciking measurements. 

The log is suspended from thin wires so that 
it behaves like a pendulum that is free to swing. 
As the bullet strikes the log, the log starts to 
rise. When the log reaches its highest point, it 
momentarily stops, and then begins to swing 




Fig. 3-90 



back down. This point of zero velocity is visible 
in the slow- motion sequence in the film. 

The bullet plus the log after impact form a 
closed system, so you would expect the total 
amount of mechanical energy of such a system 
to be conserved. The total mechaniccil energy is 
the sum of kinetic energy plus potential energy. 
If you conveniently take the potential energy as 
zero at the moment of impact for the lowest 
position of the log, then the energy at that time 
is all kinetic energy. As the log begins to move, 
the potential ener^ is proportional to the 
vertical distance above its lowest point, and it 
increases while the kinetic enei^, depending 
upon the speed, decreases. The kinetic energy 
becomes zero at the point where the log 
reverses its direction because the log's speed is 
zero at that point. All the mechanical energy at 
the reversal point is potential energy. Because 
energy is conserved, the initial kinetic energy at 
the lowest point should equal the potentiiil 
energy at the top of the swing. On the basis of 
this result, write cm equation that relates the 
initial log speed to the final height of rise. You 
might check this result with your teacher or 
with other students in the class. 

If you measure the vertical height of the rise 
of the log, you can calculate the log's initial 
speed, using the equation just derived. What is 
the initial speed that you find for the log? If you 
wish to convert distance measurements to 
centimeters, it is useful to know that the 
vertical dimension of the log is 9.0 cm. 

Find the speed of the rifle bullet at the 
moment it hits the log, using conservation of 
momentum. 

Calculate the kinetic energy of the rifle bullet 
before it strikes and the kinetic enei^gy of the log 



182 



UNIT 3 / THE TRIUMPH OF MECHANICS 



plus bullet after impact. Compare the two 
kinetic enei^ies, and discuss any difference. 



Film Loop 28 
RECOIL 

Conservation laws can be used to determine 
recoil velocity of a gun, given the exjjerimental 
information that this film provides. 

The preliminary scene shows the recoil of a 
cannon firing at the fort on St. Helene Island, 
near Montreal, Canada (Fig. 3-91). The small 
brass laboratoiy "cannon" in the rest of the film 
is suspended by long wires. It has a mass of 350 
g. The projectile has a mass of 3-5 g. When the 
firing is photographed in slow motion, you can 
see a time lapse between the time the fuse is 
lighted and the time when the bullet emei^es 
from the cannon. Why is this delay observed? 
The camera used here exposes 8,000 frames per 
second. 

Project the film on paper. It is convenient to 
use a horizontal distance scale in centimeters. 
Find the bullet's velocity by timing the bullet 
over a lar^ge fraction of its motion. (Only relative 
values are needed, so it is not necessary to 
convert this velocity into centimeters per 
second.) 

Use momentum conservation to predict the 
gun's recoil velocity. The system (gun plus 
bullet) is one dimensional; all motion is along 
one straight line. The momentum before the 



gun is fired is zero in the coordinate system in 
which the gun is at rest, so the momentum of 
the cannon after firing should be equal and 
opp>osite to the momentum of the bullet 

Test your prediction of the recoil velocity by 
running the film again and timing the gun to 
find its recoil velocity experimentally. What 
margin of error might you exp)ect? Do the 
predicted and observed values agree? Give 
reasons for any difference you observe. Is 
kinetic enef^ conserved? Elxplain your answer. 



Film Loop 29 

CX)LLIDCVG FREIGHT CARS 

This film shows a test of freight-car coupling. 
The collisions, in some cases, w«re \iolent 
enough to break the coupUngs. The "hammer 
car," coasting down a ramp, reaches a speed of 
about 10 km/hr. The momentary force between 
the cars is about 4,400,000 \. The photograph 
(Fig. 3-921 shows coupling pins that were 
sheared off by the force of the collision. The 
slow-motion collision allows you to measure 
speeds before and after impact, and thus to test 
conservation of momentum. The collisions are 
partially elastic, as the cars separate to some 
extent after collision. The masses of the cars 
are: hammer car: m , = 95,000 kg: tai^get car: m, 
= 120,000 kg. To find velocities, measure the 
film time for the car to mo\'e through a giwn 




Fig 3-91 



FILM LOOP NOTES 



163 



Rg. 3-92 Broken coupling pins from colliding freight 



distance. (You may need to run the film several 
times.) Use any convenient units for velocities. 

Simple timing will give v, andv2. The film was 
made on a cold winter day jmd ftiction was 
appreciable for the hammer car after collision. 
One way to allow for friction is to make a 
velocity time graph, iissume a uniform negative 
acceleration, and extrapolate to the instant 
after impact. 

An example might help. Suppose the ham- 
mer car coasts 3 squares on graph paper in 5 
sec eifter collision, and it also coasts 6 squares 
in 12 sec after collision. The average velocity 
during the first 5 sec was Vj = (3 squares)/(5 
sec) = 0.60 squares/sec. The average velocity 
during any short interval approximately equals 
the instantaneous velocity at the midtime of 
that interval, so the car's velocity was about v, 
= 0.60 squares/sec at t = 2-5 sec. For the 
interval 0-12 sec, the velocity was v, = 0.50 
squares/sec at f = 6.0 sec. Now plot a graph like 
that shown in Fig. 3-93. This graph shows, by 
extrapolation, that v, = 0.67 squares/sec at t = 
0, just after the collision. 



.70 


- 


— ^ 






\ 




^ 




% 




•V 




\ 




v 


.60 


"*\ 


.50 


X 



3 

sr 



( 2 3 -4 5 <i. 7 
t (seconds) 

Fig. 3-93 Extrapolation backwards in time allow for 
friction in estimating the value of v^ immediately after 
the collision. 



Compare the total momentum of the system 
before collision with the toted momentum after 
collision. Calculate the kinetic energy of the 
freight cars before and after collision. What 
fraction of the hammer car's original kinetic 
enei^ has been "lost"? Can you account for 
this loss? 



Film Loop 30 

DYNAMICS OF A BILLIARD BALL 

The event pictured in this film is one you have 
probably seen many times, the striking of a ball, 
in this case a billiard ball, by a second ball. (See 
Fig. 3-94.) Here, the camera is used to "slow 
down" time so that you can see details in this 
event that you probably have never observed. 
The ability of the camera to alter space and 
time is important in both science and art. The 
slow-motion scenes were shot at 3,000 fremies 
per second. 




Fig. 3-94 Billiard balls near impact. The two cameras 
took side views of the collision, which are not shown in 
this Film Loop. 

The "world" of your physics course often has 
some simplifications in it. Thus, in your 
textbook, much of the discussion of mechanics 
of bodies probably assumes that the objects are 
point objects, with no size. Clearly these 
massive billiard balls have size, as do all the 
things you encounter. 

For a point particle, you can speak in a 
simple, meaningful way of its position, its 
velocity, and so on. But the particles photo- 
graphed here are billiard balls and not points. 
What information might be needed to describe 
their positions and velocities? Lxjoking at the 
film may suggest possibilities. What motions 
can you see besides simply the linear forward 
motion? Watch each ball carefully, just before 



164 



LNIT 3 / THE TRIUMPH OF MECHANICS 



and just after the collision. Watch not only the 
overall motion of the ball, but also "internal" 
motions. Can any of these motions by appro- 
priately described by the word "spin"? Can you 
distinguish the cases where the ball is rolling 
along the table, so that there is no slippage 
between the ball and the table, from the 
situations where the ball is skidding along the 
tahlt! without rolling? Does the first ball move 
immediately after the collision? You can see 
that even this simple phenomenon is more 
complex than you might have expected. 

Can you write a careful verbal description of 
the event? How might you go about giving a 
more careful mathematical description? 

Using the slow-motion sequence, you can 
make a momentum analysis, at least partially, 
of this collision. Measure the velocity of the cue 
ball before impact and the velocity of both balls 
after impact. Remember that there is friction 
between the ball and the table, so velocity is not 
constant. Since the balls have the same mass, 
conservation of momentum predicts that 

velocity of cue ball just before collision = 
sum of velocities of the balls just after 
collision 
How closely do the results of your measure- 
ments agree with this principle? What reasons, 
considering the complexity of the phenome- 
non, might you suggest to account for any 
disagreement? VVliat motions are you neglect- 
ing in your analysis? 



Film LcMip 31 

a methoii of measihring 
i:xi:k(;y: nails driixn into 
uoou 

Some physical quantities, such as distance, can 
be measured directly in simple ways. Other 
concepts can be connected with the world of 
experience only through a long series of 
measurements and calculations. In certain 
situations, simple and reliable methotls of 
deteniiining rncri^v an* possible Hcnv you are 
concenied with the energv' of a nuning object 

This film allows you to check the validity of 
one way of measuring mechanical energv' If a 
moving object strikes a nail, the objeit will lose 
all of its enei>(\'. This ener-gN has some effect, in 
that the nail is dri\Tn into the wood The 
enei-g\ of the objecl becomes wm-k done on the 
nail, driving it into the block ot wood 

The first scenes in tlie film show a construc- 
tion site A pile drixTr strikes a pile owr and 



ov-er again, "planting" it in the ground. The 
laboratory' situation duplicates this situation 
under more controlled circumstances. Each of 
the blows is the same as any other because the 
massive object is always raised to the same 
height above the nail. The nail is hit 10 times. 
Because the conditions are kept the same, you 
exp)ect the energv' of the impact to be the same 
for each blow. Therefore, the work from each 
blow is the same. L'se the film to determine if 
the distance the nail is driven into the wood is 
proportional to the energv' or work. How can 
you find the energy exjjended if you know the 
depth of penetration of the nail? 

The simplest way to display the measux^e- 
ments made with this film may be to plot the 
depth of nail p>enetration versus the number of 
blows. Do the experimental points that you 
obtain lie approximately along a straight line? If 
the line is a good approximation, then the 
energy is about proportional to the depth of 
penetration of the nail. Thus, depth of penetra- 
tion can be used in the analysis of other films to 
measure the energy of the striking object 

If the graph is not a straight line, you can still 
use these results to calibrate your enei^- 
measuring device. By use of f>enetration versus 
the number of blows, an observed penetration 
(in centimeters, as measured on the screen), 
can be converted into a number of blows, and 
therefore an amount proportional to the work 
done on the nail, or the energv' transferred to 
the nail. Thus, in Fig. 3-95, a jjenetration of 3 cm 
signifies 5.6 units of enepRk' 




Rg 3-95 

Film I^<M»p '.12 
GRyW'ITAliOXAL POXrVTIAL 

f:xi:kc;y 

Introduiton phvsics courses usualK do not 
give a complete definition of potential ener]^ 
because of the mathematics invnlwd Only 



FILM LOOP NOTES 



165 



particular kinds of potential energy, such as 
gravitational potential energy, are considered. 

The expression for the gravifaf /on a/ poten- 
tial energy of an object near the earth is the 
product of the weight of the object and its 
height. The height is measured from a location 
chosen arbitrarily as the zero level for potential 
energy. It is almost impossible to "test" a 
formula v\ithout other physics concepts. Here, 
a method of measuring energy is needed. The 
previous Film Loop 31, "A Method of Measuring 
Energy," demonstrated that the depth of pene- 
tration of a nail into wood, due to a blow, is a 
good measure of the energy at the moment of 
impact of the object. 

Although you are concerned with potential 
energy, you will calculate it by first finding 
kinetic energy. Where there is no loss of energy' 
through heat, the sum of the kinetic energy and 
potential energy is constant. If you measure 
potential energy from the point at which the 
weight strikes the nail, at the moment of 
striking all the energy' will be kinetic energy. On 
the other hand, at the moment an object is 
released, the kinetic energy is zero, and all the 
energy is potential energy. These two must, b\ 
conservation of energy, be equal. 

Since totcil energy is conserved, you can 
determine the initial potential energy that the 
object had from the depth of penetration of the 
nail by using the results of the measurement 
connecting energy and nail penetration. 

Two types of measurements are possible 
with this film. The numbered scenes are all 
photographed fix)m the same position. In the 
first scenes (Fig. 3-961, you can determine how 
gravitational potential energy depends upon 
weight. Objects of different masses fall from the 
same distance. Project the film on paper and 
measure the positions of the nailheads before 




and after the impact of the falling objects. Make 
a graph relating the penetration depth and the 
weight nia^. Use the results of the previous 
experiment to convert this relation into a 
relation between gravitational potential energy 
and weight. VVTiat can you learn from this 
graph? What factors are you holding constant? 
What conclusions can you reach fixjm your 
data? 

Later scenes (Fig. 3-97) provide information 
for studying the relationship between gravita- 
tional potential energy and position. Bodies of 
equal mass are raised to different heights cmd 
allowed to fall. Study the relationship between 
the distance of fall and the gravitational poten- 
tial energy. What graphs might be useful? What 
conclusion can you reach fi:x)m your measure- 
ments? 




Fig. 3-96 



Fig. 3-97 

Can you relate the results of these measure- 
ments to statements in the Text concerning 
gravitational potential energy? 



Film Loop 33 
KEVETIC ENERGY 

In this film, you can test how kinetic energy KE 
depends on speed v. You will measure both KE 
and V, keeping the mass m constant. 

The penetration of a nail driven into wood is 
a good mecisure of the work done on the nail, 
and thus is a measure of the energy lost by 
whatever object strikes the nail. The speed of 
the moving object can be measured in several 
ways. 

The preliminary scenes show that the object 
falls on the nail. Only the speed just before the 
object strikes the nail is important. The scenes 
intended for measurement were photographed 



166 



LNIT 3 / THE TRIUMPH OF MECHANICS 



with the camera on its side, so the body 
appears to move horizontally toward the nail. 
The speeds can be measured by timing the 
motion of the leading edge of the object as it 
moves from one reference mark to the other. 
The clock in the film (Fig. 3-98) is a disk that 
rotates at 3,000 revolutions per minute. Project 
the film on paper and mark the positions of the 
clock pointer when the body crosses each 
reference mark. The time is proportional to the 
angle through which the pointer turns. The 
speeds are proportional to the reciprocals of 
the times since the distance is the same in each 
case. Since you are testing only the form of the 
kinetic energy dependence on speed, any 
convenient unit can be used. Measure the 
speed for each of the five trials. 




Fig. 3-98 

The kinetic energy of the moving object is 
transformed into the work required to drive the 
nail into the wood. In Film Loop 31, you related 
the work to the distance of penetration. Mea- 
sure the nail penetration for each trial, and use 
your results fn)m the prtnious film. 

How does KE depend on v? The conservation 
law derived from Newton's laws indicates that 
KE is pniportional to v^, the square of the 
speed, not proportional to v. Test this by 
making two graphs. In one graph, plot KE 
verticalK and plot v- horizontally. For compari- 
son, plot KE versus v. What can you conclude? 
Do you have any assurance that a similar 
rtrlation will hold if the speeds or masses are 
ver\' (lilTenMit fi-om those found here'' How 
might \(>u go about determining this.' 



Film Loop 34 

CONSERVATION OF ENERGY: 
POLE VAULT 

This quantitative film ran help you study 
conservation of enei^. A pole vaulter (mass 6« 



kg, height 180 cm) is shown (first at normal 
speed and then in slow motion! clearing a bar 
at 3.45 m. You can measure the total energy of 
the system at two moments in time: (II just 
before the jumper starts to rise and (21 part of 
the way up, when the pole is bent. The total 
energy of the system is constant, although it is 
divided differently at different times. Since it 
takes work to bend the pole, the pole has elastic 
potential energy when bent. This elastic energy 
comes from some of the kinetic energy the 
vaulter has as he runs horizontally before 
inserting the pole in the socket. Later, the 
elastic potential ener^ of the bent pole is 
transformed into some of the jump>ers gravita- 
tional potential energy when he is at the top of 
the jump (Fig. 3-99). 






UHls iUH^H 



Fig. 3-99 

Position 1 

The energy is entirely kinetic enerig^, yimv*. 
To help you measure the runner's speed, 
successive frames are held as the runner moves 
past two markei^ 1 m apart. ELach "freeze 
frame" represents a time interval of 1/Z50 sec. 
the camera speed Find the runner's aver^age 
speed over this meter, and then find the kinetic 
ener^'. If m is in kilograms and v is m meters 
per second. £ will be in joules. 

Position 2 

The jumper's center of gravitv' is about 1 02 m 
alKJve the soles of his feet Three types of energv 
are involwd at the intermediate positions I'se 
the stop-frame sequence to obtain the speed of 
the jumper Tlie seat of his pants can be used 
as a n'ferrnce Calculate the kinetic energV' and 
gravitational potential enerx>' as already de- 
scrilied. 



FILM LOOP NOTES 



167 



The work done in deforming the pole is 
stored as elastic potential energy. In the final 
scene, a chain windlass bends the pole to a 
shape similar to that which it assumed during 
the jump in Position 2. When the chain is 
shortened, work is done on the pole: work = 
(average force) x (displacement!. During the 
cranking sequence, the force varied. The aver- 
age force can be approximated by adding the 
initial and final values (found from the scale) 
and dividing by two. Convert this force to 
newtons. The displacement can be estimated 
from the number of times the crank handle is 
pulled. A close-up show^ how far the chain 
moves during a single stroke. Calculate the 
work done to crank the pole into its distorted 
shape. 

You now can add and find the total energy. 
How does this compare with the original 
kinetic energy? 

Position 3 

Gravitational potential energy is the work 
done to raise the jumper's center of gravity. 
From the given data, estimate the verticjil rise of 
the center of gravity as the jumper moves from 
Position 1 to Position 3. (His center of gravity 
clears the bar by about 0.3 m.) Multiply this 
height by the jumpers weight to get potential 
energy. If vveight is in newtons and height is in 
meters, the potential energy will be in joules. A 
small additioncil source of energy is in the 
jumper's muscles; judge for yourself how far he 
lifts his body by using his arm muscles as he 
nears the highest point. This is a small 
correction, so a relatively crude estimate will 
sufiBce. Perhaps he pulls with a force equal to 
his own weight through a vertical distance of 
0.7 m. 

How does the initial kinetic energy, plus the 
muscular energy expended in the pull-up, 
compare with the final gravitational potential 
energy? (An agreement to within about 10% is 
about as good as you can expect firom a 
measurement of this type.) 



Film Loop 35 

CXJNSERVATIOX OF ENERGY; 
AIRCRAFT TAKEOFF 

The pilot of a Cessna 150 (Fig. 3-100) holds the 
plane at constant speed in level flight, just 
above the surface of the runway. Then, keeping 
the throttle fixed, the pilot pulls back on the 
stick, and the plcme begins to rise. With the 




Fig. 3-100 



same throttle setting, the plane levels off at 
about 100 m. At this altitude, the ciircraft's 
speed is less than at ground level. You can use 
this film to make a crude test of energy 
conservation. The plane's initial speed was 
constant, indicating that the net force on it was 
zero. In terms of an approximation, £iir resis- 
tance remained the same after lift-off. How 
good is this approximation? What would you 
expect ciir resistance to depend on? When the 
plane rose, its gravitational potential energy 
increased, at the expense of the initial kinetic 
energy of the plane. At the upper level, the 
plane's kinetic energy is less, but its potential 
energy is greater. According to the principle of 
conservation of energy, the total energy IKE + 
PE) remained constant, assuming that air 
resistance and any other similar factors are 
neglected. But are these factors negligible? Here 
are the data concerning the film and the 
airplane: 

length of plane: 75 m 

mass of plane: 550 kg 

weight of plane: 

550 kg X 9.8 m/sec^ = 5400 N 

camera speed: 45 fi"ames/sec 

Project the film on paper. Mark the length of 
the plane to calibrate distances. 

Stop-frcime photography allows you to mea- 
sure the speed of 45 frames per second. In 
printing the measurement section of the film, 
only eveiy third frame was used. Each of these 
frames was repeated ("stopped ") a number of 
times, enough to iillow time to mark a position 
on the screen. The effect is one of "holding" 
time, and then jumping one-fifteenth of a 
second. 

Measure the speeds in all three situations, 
and also the heights above the ground. You 
have the data needed for calculating kinetic 
energy (Vimv^) and gravitational potential 



168 



L.\IT 3 / THE TRILMPH OF MECHANICS 



energy (ma„/i) at each of the three levels. 
Calculate the total energy at each of the three 
levels. 

Can you make any comments concerning air 
resistance? Make a table showing (for each 
level) KE, PE, and £ totals. Do your results 
substantiate the law of conservation of energy 
within experimental error? 



Film Loop 36 
REVERSIBILITY OF TIME 

It may sound strange to speak of "reversing 
time." In the world of common experience, we 
have no control over time direction, in contrast 
to the many aspects of the world that we can 
modify. Yet physicists are much concerned 
vAXh the reversibility of time; perhaps no other 
issue so clearly illustrates the imaginative and 
speculative nature of modem physics. 

The camera provides a way of manipulating 
time. If you project movie film backwards, the 
events pictured happen in reverse time order. 
This film has sequences in both directions, 
some shown in their "natural" time order and 
some in reverse order. 

The film concentrates on the motion of 
objects. Consider each scene from the stand- 
point of time direction: Is the scene being 
shown as it was taken, or is it being reversed 
and shown backward? Many sequences tire 
paired, the same film being used in both time 
senses. Is it always clear which one is forward 
in time and which is backward? With wiiat 
types of events is it difficult to tell the "natural" 
direction? 

The Newtonian laws of motion do not 
depend on time direction. Any filmed motion of 
particles following strict Newtonian laws 
should look completely "natural" whether seen 
forward or backward. Since Newtonian laws are 
"invariant" under time reversal (changing the 
direction of time), you could not tell by 
examining a motion obeying these laws 
whether the sequence is forward or backward. 
Any motion that could occur forward in time 
can also occur, under suitable conditions, with 
the events in the opposite order. 

With more complicated phv-sical systems, 
with an extremely large number of particles, 
the situation changes. If ink were dn)pp«nl into 
water, you would haw no ditfirult>' in deter- 
mining wtiich sequence was photographed 
forward in time and which backwartl So 
certain physical j)henomena at least appear to 
be imn'ersible, taking place in only one lime 



direction. Are these processes fundamentally 
irreversible, or is this only a limitation on 
human powers? This is not an easy question to 
answer. It could still be considered, in spite of a 
50-year histoiy, a frontier problem. 

Reversibility of time has been used in many 
ways in twentieth-century physics. For exam- 
ple, an interesting way of viewing the two kinds 
of charge in the unix-erse, positive and negati\'e, 
is to think of some particles as "mo\ing" 
backward in time. TT»us, if the electron is 
viewed as mo\ing forward in time, the positron 
can be considered as exactly the same particle 
moving backward in time. This backward 
motion is equivalent to the forward-moving 
particle having the opposite chaise! This was 
one of the keys to the dev-elopment of the 
space- time view of quantum electrodynamics 
which R. P. Feynman described in his Nobel 
Prize lecture. 

For a general introduction to time reversibil- 
ity, see the Martin Gardner article, "Can Time 
Go Backward?" originally published in Scien- 
tific American (January, 1967). 



Film Loop 37 
SUPERPOSITION 

Using this film, you will study an important 
physical idea, superposition. The film was 
made by photographing patterns displayed on 
the face of the cathode-ray tube iCRTi of an 
oscilloscope, similar to a television set. Vou may 
have such an oscilloscope in your laboratory. 
Still photographs of some of these patterns 
appearing on the CRT screen are shown in Figs. 
3-101 and 3-102, The two patterns at the top of 
the screen are called sinusoidal The\' ar^ not 
just any wavy lines, but lines generated in a 
pi^ecise fashion. If you are familiar with the sine 
and cosine functions, you will recognize them 
here. TTie sine function is the special case 
where the origin of the coortlinate system is 
located where the function is zero and starting 
to rise. No origin is shown, so it is arbitrary as to 
w+iether one calls these sine curves, cosine 
curves, or some other sinusoidal curve V\hat 
physical situations might lead to curves of this 
type'' lYou might want to consult books about 
simple harmonic oscillators i Hert* the curves 
are pnuluced by electronic cirxniits that gener^ 
ale an electrical voltage changing in time so as 
to cause the curve to be displayed on the 
cathode-ray tube The oscilloscope operator 
can adjust the magniludi«s and phasi«s of the 
twi) top functions. 



FILM LOOP NOTES 



169 




Rg. 3-101 



Fig. 3-104 




Rg. 3-102 



Fig. 3-105 




Rg. 3-103 



The bottom curve is obtained by a point-by- 
point adding of the top curves. Imagine a 
horizontal axis going through each of the two 



top curves, and positive and negative distances 
measured vertically from this axis. The bottom 
curve is at each point the algebraic sum of the 
two points above it on the top curves, as 
measured ftxjm their respective axes. This 
point-by-point algebraic addition, when 
applied to actual waves, is called superpos/fion . 
Two cautions are necessary. First, you are 
not seeing waves, but models of waves. A wave 
is a disturbance that propagates in time, but, at 
least in some of the cases shown, there is no 
propagation. A model always has some limita- 
tions. Second, you should not think that all 
waves are sinusoidal. The form of whatever is 
propagating can be any shape. Sinusoidal 
waves constitute only one important class of 
waves. Another common wave is the pulse, 
such as a sound wave produced by a sharp 
blow on a table. The pulse is not a sinusoidal 
wave. 



170 



UNIT 3 / THE TRILMPH OF MECHANICS 



Several examples of superposition are shown 
in the film. If, as approximated in Fig. 3-101, two 
sinusoidal curves of equal period and 
amplitude are in phase, both having zeroes at 
the same places, the result is a double-sized 
function of the same shape. On the other hand, 
if the curves are combined out of phase, so that 
one has a positive displacement while the other 
one has an equal negative displacement, the 
result is zero at each point (Fig. 3-102). If 
functions of different periods are combined 
(Figs. 3-103 to 3-105), the result of the superpo- 
sition is not sinusoidal, but more complex in 
shape. You are asked to interpret, both verbally 
and quantitatively, the superpositions shown 
in the film. 



A high-speed snapshot of the string at any 
time would show its instantaneous shape. 
Sections of the string move, except at the 
nodes. The eye sees a blurred or "time- 
exposure" superjjosition of string shapes be- 
cause of the frequenc>' of the string. In the film, 
this blurred eflfect is reproduced by photo- 
graphing at a slow rate; each frame is exfxised 
for about 1/15 sec. 

Some of the vibration modes are photo- 
graphed by a stroboscopic method. If the string 
vibrates at 72 vib/sec and frames are exf>osed in 
the camera at the rate of 70 times p)er second, 
the string seems to go through its complete 
cycle of vibration at a slower frequency when 
projected at a normal speed. In this way, a 
slow-motion effect is shown. 



Film Loop 38 

STANIJUVC; WAVES ON A STRING 

Tension determines the speed of a wave 
traveling down a string. When a wave reaches a 
fixed end of a string, it is reflected back again. 
The reflected wave and the original wave are 
superimposed or added together. If the tension 
(and therefore the speed) is just right, the 
resulting wave will be a standing wave. Certain 
nodes will always stand still on the string. 
Other points on the string will continue to 
move in accordance with superposition. V\'hen 
the ttMision in a vibrating string is adjusted, 
standing waves appear when the tension has 
one of a set of "right" values. 

In the film, one end of a string is attached to a 
tuning fork with a frequency of 72 vibrations 
per second. The other end is attached to a 
cylintler. The tension of the string is adjusted 
by sliding the cvlindrr hack and forth 

Several standing wav<' |)atterns i\rv shown 
For example, in the third mode the string 
vibrates in three segments with two nodes 
(points of no motion) between the nodes at 
each end. The nodes are half a wavelength 
apart. Between the nod(>s are points of 
maxinuirn p()ssil)le vibration called antinodes 

Vou tune the strings of a violin or guitar b\ 
changing the tension on a string of fixed length, 
higher tension com'sponding to higher pitch. 
Dilleit'nt notes an* pfoduced In pl.uing a finger 
on the sliinfi to shoilen ihv vibrating pari In 
this film, the litHjutMicv of vibration of a string is 
fixed, b»H"ause the string is always drivvn at 72 
vil)/sec. When the fivquj'ncy ivmains constant, 
the wavelength changes as the tension is 
adjusted because veloc it\ depends on tension 



Film Loop 39 

STAXDLXG ll'A\XS L\ A GAS 

Standing waves are set up in air in a large glass 
tube (Fig. 3-1061 The tube is closed at one end 
by an adjustable piston. A loudsp>eaker at the 
other end supplies the sound wave. The 
speaker is driven by a variable-finequenc\' oscil- 
lator and amplifier. About 20 watts of audio- 
power are used, telling ever>'one in a large 
building that filming is in progress! The wavt^s 
are reflected frxjm the piston. 




Fig. 3-106 



A standing wave is foniied when the fn*- 
quenc\' of the oscillator is adjusted to one of 
several discrete values Most frequencies do not 
givv standing waws l^esonance is indicated in 
each mode of vibration by ntxies and an- 
tinodes rhere is always a nmle at the fixed end 
iwhere air molecules cannot mow) and an 
aniinode at tlie speaker (wfiere air is set into 
motion) Between the fixetl end and the speaker 
then* niav l>e additintial nodes and antinodes 



FILM LOOP NOTES 



171 



The patterns can be observed in several ways, 
two of which are used in the film. One method 
of making visible the presence of a stationary 
acoustic wave in the gas in the tube is to place 
cork dust along the tube. At resonance, the dust 
is blown into a cloud by the movement of air at 
the antinodes; the dust remains stationary at 
the nodes where the air is not moving. In the 
first part of the film, the dust shows standing 
wave patterns for various frequencies (Table 
3-2). 

Frequency Number of half- 

(vib/sec) wavelengths 

230 1.5 

370 2.5 

530 3.5 

670 4.5 

1900 12.5 

The pattern for/= 530 is shown in Fig. 3-107. 
From node to node is VaX, and the length of 
the pipe is 3 A. + VSzX (the extra V2X is from the 
speaker antinode to the first node). There are, 
generally, (n -I- V2) half-wavelengths in the fixed 
length, so A. a i/(n -I- V2). Since/ oc i/X, / oc 
(n -I- V2). Divide each frequency in the table by 
(n -I- V2) to find wiiether the result is reasonably 
constant. 




Fig. 3-107 

In all modes, the dust remains motionless 
near the stationcuy piston that is a node. 

In the second part of the film, nodes and 
antinodes are made visible by a dififerent 
method. A wire is placed in the tube near the 
top. This wire is heated electrically to a dull 
red. When a stemding wave is set up, the wire is 
cooled at the antinodes, because the air carries 
heat away from the ware when it is in vigorous 
motion. So the wire is cooled at the antinodes 
and glows less. The bright regions correspond 
to nodes where there are no air currents. The 
oscillator frequency is adjusted to give several 
standing wave patterns with successively smal- 
ler wavelengths. How many nodes and an- 
tinodes are there in each case? Can you find 
the frequency used in each case? 



Film Loop 40 
VIBRATIONS OF A WIRE 

This film shows standing wave patterns in thin 
but stiff wires. The wave speed is determined 
by the wire's cross section and by the elastic 
constants of the metal. There is no external 
tension. Two shapes of wire, straight and 
circular, are used. 

The wire passes between the poles of a 
strong magnet. When a switch is closed, a 
steady electric current from a battery is set up 
in one direction through the wire. The interac- 
tion of this current and the magnetic field leads 
to a downward force on the wire. When the 
direction of the current is reversed, the force on 
the wire is upward. Repeated rapid reversal of 
the current direction can make the wire vibrate 
up and down. 

The battery is replaced by a source of 
variable-ft^quency alternating current whose 
frequency can be changed. Wlien the frequency 
is adjusted to match one of the natural 
frequencies of the wire, a standing wave buUds 
up. Several modes are shown, each excited by a 
different frequency. 

The first scenes show a straight brass wire, 
2.4 mm in diameter (Fig. 3-108). The "boundary 
conditions" for motion require that, in any 
mode, the fixed end of the wire is a node and 
the free end is an antinode. (A horizontal 
plastic rod is used to support the wire at 
another node.) The wire is photographed in 
two ways: (1) in a blurred "time exposure," as 
the eye sees it; and (2) in "slow motion," 
simulated through stroboscopic photography. 




Fig. 3-108 



Study the location of the nodes and an- 
tinodes in one of the higher modes of vibration. 
They are not equally spaced along the wire, as 
for vibrating string (see FUm Loop 38). This is 
because the wire is stiff whereas the string is 
perfectly flexible. 



172 



UNIT 3 / THE TRIUMPH OF MECHANICS 



In the second sequence, the wire is hent into 
a horizontal loop, supported at one point iFig. 
3-109). The boundary conditions require a node 
at this point; there can be additional nodes, 
equally spaced around the ring. Several modes 
are shown, both in "time exposure" and in 
"slow motion." To some extent, the vibrating 
circular wire is a helpful model for the wave 
behavior of an electron oHiit in an atom such as 
hydrogen; the discrete modes correspond to 
discrete energy states for the atom. 



film. An eccentric arm attached to the motor 
shakes the bottom end of the hose. Thus, this 
end moves slightly, but this motion is so small 
that the bottom end also is a node. (See Fi^. 
3-110.) 




Fig. 3-109 



Film Loop 41 

VIBRATIONS OF A RUBBER HOSE 

You can generate standing waves in many 
physical systems. When a wave is set up in a 
medium, it is usually reflected at the bound- 
aries. Characteristic patterns will be formed, 
depending on the shape of the medium, the 
frequency of the wave, and the material. At 
certain points or lines in these patterns, there is 
no vibration because all the partial waves 
passing through these points just manage to 
cancel each other through superposition (as 
you saw in the ripple tank). 

Standing wave patterns only occur for certain 
frequencies. Tlie physical process selects a 
spectrum of frequencies from all the possible 
ones. Often there are an infinite number of 
such discnUe frequencies. Sometimes there are 
simple mathematical relations between the 
selected frequencies, but for other bodies the 
relations are more complex. Several films in this 
series show vibrating systems with such pat- 
terns. 

This film uses a rubber hose, clamped at the- 
top. Such a stationary point is called a node 
The bottom of the stretched hose is attached to 
a motor whose speed is increased during the 




Fig 3-110 



The motor begins at a frequency below that 
for the first standing wave pattern. As the motor 
is graduaUy speeded up, the amplitude of the 
\ibrations increases until a well-defined steady 
loop is formed between the nodes. This ioon 
has its maximum motion at the cen'er. The 
pattern is half a wa\-elength long. Increasing the 
speed of the motor leads to other harmonics, 
each one being a standing wave pattern with 
both nodes and antinodes, points of maximum 
vibration. These resonances can be seen in the 
film to occur only at certain sharp frequencies. 
For other motor frequencies, no such simple 
pattern is seen. Vou can count as many as 11 
loops with the highest frequency- case shown. 

It would be interesting to ha\« a sound track 
for this film. The sound of the motor is by no 
means constant during the process of increas- 
ing the frequency The stationary resonance 
pattern corresponds to points v>tiere the motor 
is running much more quietly, because the 
motor does not need to fight against the hose. 
This sound distinction is particularly notice- 
able for the higher harmonics. 

If you play a xiolin, cello, or other stringed 
instnmient, vtiu might ask how the harmonics 
observed in this film are related to musical 
properties of \ilH-ating strings What can be 
done with a violin string to change the 
frequency of vibration' What musical relation 
exists between two notes if one of them is twice 
the frequency of the other? 



FILM LOOP NOTES 



173 



What would happen if you kept increasing 
the frequency of the motor? Would you expect 
to get arbitrarily high resonances, or would 
something "give"? 



Film Loop 42 
\^BRATIO^S OF A DRU\I 



The standing wave patterns in this film <ire 
formed in a stretched, circular rubber mem- 
brane driven by a loudspeaker (see Fig. 3-111). 
The loudspeaker is fed large cmiounts of power, 
about 30 watts — more power than you would 
want to use with a television set or phono- 
graph. The frequency of the sound can be 
changed electronically. The lines drawn on the 
membrane make it easier for you to see the 
patterns. 




Fig. 3-111 



The rim of the drum cannot move, so in all 
cases it must be a nodal circle — a circle that 
does not move as the waves bounce back and 
forth on the drum. By operating the camera at a 
frequency only slightly different firom the 
resonant frequency, a stroboscopic effect en- 
ables you to see the rapid vibrations in slow 
motion. 

In the first part of the film, the loudspeaker is 
directly under the membrane and the vibratory 
patterns are symmetrical. In the fundamental 
harmonic, the membrane rises and falls as a 
whole. At a higher frequency, a second circular 
node shows up between the center and the 
rim. 

In the second part of the film, the speaker is 
placed to one side so that a different set of 
modes is generated in the membrane. You can 
see an asymmetrical mode where there is a 
node along the diameter, with a hill on one side 
and a valley on the other. 

Various symmetric and cisymmetric vibration 
modes are shown. Describe each mode, iden- 
tifying the nodail lines and circles. 

In contrast to the one-dimensional hose in 
FUm Loop 41, there is no simple relation of the 
resonant frequencies for this two-dimensional 
system. The frequencies are not integral multi- 
ples of any basic frequency. The relation 
between values in the frequency spectrum is 
more complex than that for the hose. 



Film Loop 43 

VIBRATIONS OF A METAL PLATE 

The physical system in this film is a square 
metal plate (see Fig. 3-112). The various 
vibrational modes are produced by a 
loudspeaker, as with the vibrating membrane in 
Film Loop 42. The metal plate is clamped at the 
center, so that point is always a node for each 
of the standing wave patterns. Because this is a 
stiff metal plate, the vibrations are too slight in 
amplitude to be seen directly. The trick used to 
make the patterns visible is to sprinkle sand 
along the plates. This sand moves away from 
the parts of the plates that are in rapid motion, 
and tends to fall filong the nodal lines, which 
£ire not moving. The beautiful patterns of sand 
are known cis Chladni figures. These patterns 
have often been much admired by artists. 
These and similar patterns are also formed 
when a metcil plate is caused to vibrate by 
means of a violin bow, as seen at the end of this 



174 



UNIT 3 / THE TRIUMPH OF MECHANICS 




Fig. 3-112 



film and in the Activity, "Standing Waves on a 
Drum and a Violin." 

Not all frequencies will lead to stable pat- 
terns. As in the case of the drum, these 
harmonic frequencies for the metal plate obey 



complex mathematical relations, rather than 
the simple arithmetic progression seen in a 
one-dimensional string; but, again, they are 
discrete e\'ents. Only at certain well-defined 
frequencies are these pattenns produced. 




1 




.^' 


f 









Light and Eleetromagnetisiii 

EXPERIMENTS 



Experiment 4-1 

REFRACTTION OF A LIGHT BEAM 

You can easily demonstrate the behavior of a 
light beam as it passes ftxjm one transparent 
materia] to einother. All you need is a semicircu- 
lar plastic dish, a lens, a small light source, and 
a cardboard tube. The light source from the 
Millikan apparatus (Unit 5) and the telescope 
tube with objective lens (Units 1 and 2) will 
serve nicely. 

Making a Beam Projector 

To begin with, slide the Millikan apparatus light 
source over the end of the telescope tube (Fig. 
4-1). When you have adjusted the bulb- lens 
distance to produce a peircdlel beam of light, the 
beam will form a spot of constant size on a 
sheet of paper moved toward and away from it 
by as much as 50 cm. 

Make a thin flat light beam by sticking two 
pieces of black tape, about 1 mm apart, over the 
lens end of the tube, creating a slit. Rotate the 
bulb filament until it is parallel to the slit. 

When this beam projector is pointed slightly 
downward at a flat surface, a thin path of light 
falls across the surface. By directing the beam 



1 r^rn apo<*' i"^ 




or 

(iaht Source. telescope tube. 



' ) 



q^l 



Fig. 4-1 



parallel 
beam 



into a plastic dish filled vAxh water, you can 
observe the path of the beam emerging into the 
air. The beam direction can be measured 
precisely by placing protractors inside and 
outside the dish, or by placing the dish on a 
sheet of polar graph paper (Fig. 4-2). 



Fig. 4-2 




175 



176 



UNIT 4 / LIGHT AND ELECTROMAGNETISM 



Behavior of a Light Beam at the 
Boundary l)etv^'een Two Media 

Direct the beam at the center of the flat side of 
the dish, keeping the slit vertical. Tilt the 
projector until you can see the path of light 
both before it reaches the dish and £ifter it 
leaves the other side. 

To describe the behavior of the beam, you 
need a convenient way of referring to the angle 
the beam makes with the boundary. In physics, 
the system of measuring angles relative to a 
surface assigns a value of 0° to the perpendicu- 
lar, or strciight-in, direction. The angle at which 
a beam strikes a suri"ace is called angle of 
incidence id^); it is the number of degrees away 
from the straight-in direction. Similariy, the 
angle at which a refracted beam leaves the 
boundary is called the angle of refraction idf). It 
is measured as the deviation from the straight- 
out direction (Fig. 4-3). 



Refraction Angle and Change 
in Speed 

Change the angle of incidence in S* ste(>s from 
0° to 85°, recording the angle of the refracted 
beam for each step. As the angles in air get 
larger, the beam in the water begins to spread, 
so it becomes more difficult to measure its 
direction precisely. You can avoid this difficulty 
by directing the beam into the round side of the 
dish instead of into the flat side. This will give 
the same result since, as you have seen, the 
light path is reversible. 



3. On the basis of your table of values, does the 
angle in air seem to increase in proportion to the 
angle in water? 

4. Make a plot of the angle in air against the 
angle in water. How would you descrit>e the 
relation between the angles? 




Fig. 4-3 

Note the direction of the refracted beam for a 
particular angle of incidence. Then direct the 
beam perpendicularly into the rounded side of 
the dish where the refracted beam came out 
(Fig. 4-4). At what angle does the beam now 
come out on the flat side? Does reversing the 
path like this have the same kind of effect for all 
angles? 



1. Can you state a general rule about the 
passage of light beams through the medium? 

2. What happens to the light beam when it 
reaches the edge of the container along a radius? 



Change the angle of incidence and observe 
how the angles of the reflected and refracted 
beams change (It may be easiest to leave the 
projector supported in one place and to rotate 
the sheet of paper on which the dish rests. I You 
will see that the angle of the reflected beam is 
always equal to the angle of the incident beam, 
but the angle of the refracted beam is not 
related to the angle of incidence in so simple a 
fashion. 



According to both the simple w^ve and 
simple particle models of light, it is not the ratio 
of angles in two media that will be constant, but 
the ratio of the sines of the angles. Add two 
columns to your data table and, referring to a 
table of the sine function, record the sines of 
the angles you observed. Then plot the sine of 
the angle in water against the sine of the angle 
in air. 



5. Do your results support the prediction made 
from the models? 

6. Write an equation that describes the relation- 
ship between the angles. 



According to the wa\ e model, the ratio of the 
sines of the angles in two media is the same as 
the ratio of the light speeds in the two media. 



7. According to the wave model, what do your 
results indicate is the speed of light in water? 



Color Differences 

You haw prol)abl\' observed in this ejiperiment 
that different cxilors of light are not rrfracted by 
the same amount (This effect is called disper- 
sion.) This is most noticeable when you direct 
the beam into the round side of the dish, at an 
angle such that the rvfracted Ix'am leaving the 
flat side lies very close to the flat side. The 



EXPERIMENTS 



177 



different colors of light making up the white 
beam separate quite distinctly. 



8. What color of light is refracted most? 

9. Using the relation between sines and speeds, 
estimate the difference in the speeds of different 
colors of light in water. 



Other Phenomena 

In the course of your observations you probably 
have observed that for some angles of inci- 
dence, no refiracted beam appears on the other 
side of the boundary. This phenomenon is 
called total internal reflection . 



10. When does total internal reflection occur? 

By immersing blocks of glass or plastic in the 
water, you can observe what heppens to the 
beam in passing between these media and 
water. (Liquids other than water can be used, 
but be sure you do not use one that will 
dissolve the plastic dish!) If you lower a smaller 
transparent container upside-down into the 
water so as to trap air in it, you ctin observe 
what happens at another water- air boundary 
(Fig. 4-5). A round container so placed will 
show wiiat effect an air bubble in water has on 
light. 



Fig. 




Fig. 4-5 



Experiment 4-2 

YOUNG'S EXPERIMENT: 

THE WAVELENGTH OF LIGHT 

You have seen how ripples on a water surface 
are diffracted, spreading out after having 
passed through an opening. You have also seen 
wave interference w+ien ripples, spreading out 
from two sources, reinforce each other at some 
places and cancel out at others. 

Sound and ultrasound waves behave like 
water waves. These diffraction iind interference 
effects cire characteristic of all wave motions. If 
light has a wave nature, must it not also show 
diffraction and interference effects? 

You may shake your head when you think 
about this. If light is diffracted, this must mean 
that light spreads around comers. But you 
learned in Unit 4 that "light travels in straight 
lines." How can light both spread around 
comers and move in straight lines? 

Simple Tests of Lig^t Waves 

Have you ever noticed light spreading out after 
passing through an opening or around an 
obstacle? Try this simple test: Look at a narrow 
light source several meters away from you. (A 
straight-filament lamp is best, but a single 
fluorescent tube far away will do.) Hold two 
fingers in front of one eye and parallel to the 
light source. Look at the light through the gap 
between your fingers (Fig. 4-6). Slowly squeeze 
your fingers together to decrease the width of 
the gap. What do you see? What happens to the 
light as you reduce the gap between your 
fingers to a very nannw slit? 



.ight source 




11. Before trying this last suggestion, make a 
sketch of what you believe will happen. If your 
prediction is wrong, explain what happened. 



^W 



Fig. 4-6 

Evidently light can spread out in passing 
through a very narrow opening between your 
fingers. For the effect to be noticeable, the 
opening must be small in comparison to the 
wavelength. The opening must be much 
smJiller than those used in the ripple tank, in 
the case of light, or with sound waves. This 
suggests that light is a wave, but that it has a 



178 



LNIT 4 / LIGHT AND ELECTROMAGNETISM 



much shorter wavelength than the ripples on 
water, or sound or ultrasound in the air. 

Do light waves show interference? Your 
immediate answer might be "no." Have you ever 
seen dark areas formed by the cancellation of 
light waves from two sources? 

As with diffraction, to see interference you 
must arrange for the light sources to be small 
and close to each other. A dark photographic 
negative with two clear lines or slits running 
across it works very well. Hold up this film in 
front of one eye with the slits parjillel to a 
narrow light source. What evidence do you see 
of interference in the light coming from the two 
slits? 



magnifying eyepiece and scale unit in the end 
toward your eye and look through it at the light. 
(See Fig. 4-8.) What you see is a magnified view 
of the interference pattern in the plane of the 
scale. Try changing the distance between the 
eyepiece and the double slits. 



strolght /~\^ 
lOmp 

Fig 4-8 







tciescop* tubes 



Two-Slit Interference Pattern 

To examine this interference pattern of light in 
more detail, fasten the film with the double slit 
on the end of a cardboard tube, such as the 
telescope tube without the lens. Make sure that 
the end of the tube is "light-tight," except for 
the two slits. (It helps to cover most of the film 
with black tape.) Stick a piece of translucent 
"frosted" tape over the end of a narrower tube 
that fits snugly inside the first one. Insert this 
end into the wider tube, as shown in Fig. 4-7. 



In an earlier exf>eriment, you calculated the 
wavelength of sound from the relationship 

The relationship was deriv-ed for water waN-es 
from two in-phase sources, but the mathemat- 
ics is the same for any kind of wave. iThe use of 
two closely spaced slits gives a reasonably good 
approximation of in-phase sources.) See Fig. 
4-9. 



Stroight 
filament 
lomp 



n\ 



■ dojble 
Slit 



■ fros-^ed'tope 



^"X-© 




telescope tubes 



Fig 4-7 



Set up your double tube at least 1^ m away 
from the narrow light source with the slits 
parallel to the light source. With your eve about 
30 cm away from the open end of the tube, 
focus your eye on the tape "screen." Then» on 
the screen is the interference pattern formed by 
light from the two slits. 



1. Describe how the pattern changes as you 
move the screen farther away from the slits. 

2. Try putting different colored filters in front of 
the double slits. What are the differences between 
the pattern formed in blue light and the patterns 
formed in red or yellow light? 



Measurement of \Va\elength 

Ri'inovr th<» translur«»nt tap«» schmmi fnmi the 
inside end of the narrow tube Insert a 



Fig. 4-9 

I'se the formula to find the wa\-elength of the 
light transmitted by the different colored filters. 
To do so, measure ,<, the distance between 
neighboring dark fringes, with the measuring 
magnifier (Fig 4-101. (Remember that the small- 
est divisions on the scale are 1 mm i Vou can 
also use the magnifier to measure d. the 
distance between the two slits Place the film 
against the scale and then hold the film up to 




Fig 4-10 



EXPERIMENTS 



179 



the light. In the drawing, / is the distance from 
the slits to the plane of the pattern you measure. 
The speed of light in air is approximately 
3 X 10* m/sec. Use your measured values of 
wavelength to calculate the approximate light 
frequencies for each of the colors you used. 



3. Could you use the method of "standing 
waves" (Experiment 3-18, "Sound") to measure 
the wavelength of light? Why? 

4. Is there a contradiction between the statement, 
"Light consists of waves" and the statement, 
"Light travels in straight lines"? 

5. Can you think of a common experience in 
which the wave nature of light is noticeable? 



Suggestions for Some More 
Experiments 

1. Examine light diffracted by a circular hole 
instead of by a ncurow slit. The light source 
should now be a small point, such as a distant 
flashlight bulb. Look also for the interference 
effect with light that passes through two small 
circular sources (pinholes in a card) instead of 
the two narrow slits. (Thomas Young used 
circular openings rather than slits in his 
original experiment in 1802.) 

2. Look for the diffraction of light by cm 
obstacle. For example, use straight wires of 
various diameters, parallel to a narrow light 
source. Or use circular objects such as tiny 
spheres, the head of a pin, etc., and a point 
source of light. You can use either method of 
observation: the translucent tape screen or the 
magnifier. You may have to hold the magnifier 
fciirly close to the diffracting obstacle. 

3. Try some of the experiments listed under 
"Two-Slit Interference Pattern," using four or 
six slits instead of two. Note any differences in 
results. 

Instructions on how to photograph some of 
these effects are in the activities that follow. 



Experiment 4-3 
ELECTTRIC FORCES. I 

If you walk across a carpet on a dry day and 
then touch a metal doorknob, a spark may 
jump across between your fingers and the 
knob. Your hair may crackle as you comb it (see 
Fig. 4-11). You have probably noticed other 
examples of the electrical effect of rubbing two 
objects together. Does your hair ever stand on 
end after you puU a sweater over your head? 




Fig. 4-11 

(This effect is particularly strong if the sweater 
is made of a synthetic fiber.) 

Small pieces of paper are attracted to a 
plastic comb or ruler that has been rubbed on a 
piece of cloth. Try it. The attractive force is 
often large enough to lift scraps of paper off 
the table, showing that the attractive force is 
stronger than the gravitational force between 
the paper and the entire earth! 

The force between the rubbed plastic and the 
paper is an electriceil force, one of the four basic 
forces of nature. 

In this experiment, you will make some 
observations of the nature of the electrical 
force. If you do the next experiment, "Electric 
Forces. 11," you v^ make quantitative mea- 
surements of the force. 

Forces between Electrified Objects 

Stick a 20-cm length of transpjirent tape to the 
tabletop. Press the tape down well with your 
finger leaving about 2.5 cm loose as a "handle." 
Carefully remove the tape ftx)m the table by 
pulling on this loose end, preventing the tape 
from curling up around your fingers. 

To test whether or not the tape became 
electrically charged when you stripped it from 
the table, see if the nonsticky side wall pick up a 
scrap of paper. Even better, will the paper jump 
up from the table to the tape? Is the tape 
charged? Is the paper charged? 

So far, you have considered only the effect of 
a charged object (the tape) on an uncharged 
object (the scrap of paper). What effect does a 
charged object have on another charged ob- 
ject? Here is one way to test it. 

Charge a piece of tape by sticking it to the 
table and peeling it off as you did before. 
Suspend the tape from a horizontal wooden 
rod, or over the edge of the table. (Do not let the 
lower end curl around the table legs.) 



ISO 



LNIT 4 / LIGHT AND ELECTROMAGNETISM 



Now charge a second strip of tape in the 
same way and bring it close to the first one, 
Have the two nonsticky sides facing (see Fig. 

4-12). 




Fig. 4-12 

Do the two tapes affect each other? Is the 
force between the tapes attractive or repulsive? 

Hang the second tape about 10 cm away from 
the first one. Proceed as before and electrify a 
third piece of tape. Observe the reaction 
between this tape and your first two tapes. 
Record all your observations. Leave only the 
first tape hanging from its support; you will 
need it again shortly. Discard the other two 
tapes. 

Stick a new piece of tape (A) on the table and 
stick a second tape (B) over it. Press them down 
well. Peel the stuck-together tapes from the 
table. To remove the net charge the pair will 
have picked up, run the nonsticky side of the 
pair over a water pipe or your lips. Check the 
pair with the original test strip to be sure the 
pair is electrically neutral. Now carefully pull 
the two tapes apart. 



1. As you separated the tapes did you notice any 
interaction between them (other than that due to 
the adhesive)? 

2. Hold one of these tapes in each hand and bring 
them slowly towards each other (nonsticky sides 
facing). What do you observe? 

3. Bring first one, then the other of the tapes near 
the original test strip. What happens? 



Mount A and B on the rod or table edge to 
serve as test strips. If you have rods of plastic, 
glass, or rubbor available, or a plastic comb, 
ruler, etc., rhargi* each one in turn by rubbing 
on cloth or fur. Bring the rod or comb close to A 
and then B. 



Although you cannot pro\-e it from the 
results of a limited number of exp>eriments, 
there seem to be only two classes of electrified 
objects. No one has ev'er produced an elec- 
trified object that either attracts or repels both 
A and B (wtiere A and B are themselves 
electrified objects i The two classes are called 
positive ( + ) and negative i-i. Write a general 
statement summarizing how all members of 
the same class behave with each other (attract, 
repel, or remain unaffected by) and with all 
members of the other class. 

A Puzzle 

Your system of two classes of electrified objects 
was based on observations of the way chained 
objects interact. How can you account for the 
fact that a chained object 'like a rubbed combi 
will attract an unchained object (like a scrap of 
paper)? 



4. Is the force between a charged body (either ♦ 
or -) and an uncharged body always attractive, 
always repulsive, or is it sometimes one, some- 
times the other? 

5. Can you explain how a force arises between 
charged and uncharged bodies and why it is 
always the way it is? The clue here is the fact that 
the negative charges can move about slightly, 
even in materials called nonconductors, like 
plastic and paper. 




EXPERIMEIVTS 



181 




Fig. 4-13 



Experiment 4-4 
ELECTTRIC FORCES. II: 
COULOMB'S LAW 

You have seen that electrically charged objects 
exert forces on each other, but so far your 
observations have been qualitative; you have 
observed but not measured. In this experiment, 
you will find out how the amount of electrical 
force between two charged bodies depends on 
the amounts of electrical charges and on the 
sepeiration of the bodies. In addition, you will 
experience some of the difficulties in using 
sensitive equipment. 

The electric forces between charges that you 
can conveniently produce in a laboratory are 
smaU. To measure them at all requires a 
sensitive balance, good technique, and a day 
that is not too humid (otherwise the chai^ges 
leak off too rapidly). 

Constructing the Balance 

(If your balance is already assembled, you need 
not read this section; go on to "Using the 
Balance.") A satisfactory balance is shown in 
Fig. 4-13. 

Coat a small Styrofoam ball with a conduct- 
ing paint and fix it to the end of a plastic sliver 
or toothpick by sticking the pointed end of the 
sliver into the ball. Since it is veiy important 
that the plastic be clean and dry (to reduce 
leakage of chcirge along the surface), handle the 
plastic slivers as little as possible, and then only 
with clean, dry fingers. Push the sliver into one 
end of a soda straw leaving at least 2.5 cm of 
plastic exposed, as shown in Fig. 4-14. 

Next, fill the plcistic support for the balance 
with glycerin, oil, or some other liquid. Cut a 
shallow notch in the top of the straw about 2 
cm from the axle on the side away from the 
sphere (see Fig. 4-14). 

Locate the balfmce point of the straw, ball, 
and sliver unit. Push a pin through the straw at 



this point to form an axle. Push a second pin 
through the straw directly in front of the axle 
and perpendicular to it. (As the straw rocks 
back and forth, this pin moves through the fluid 
in the support tube. Friction wdth the fluid 
reduces the swings of the balance.) Place the 
straw on the support, the pin hanging down 
into the liquid. Now, adjust the balance by 
sliding the plastic slrver slightly in or out of the 
straw, until the straw rests horizontally. If 
necessary, stick small bits of tape to the straw 
to make it balemce. Make sure the bedance can 
swing freely while meiking this adjustment. 

Finally, cut five or six small, equal lengths of 
thin, bare wire (such as #30 copper). Each 
should be about 2 cm long, and they must all be 
as close to the same length as you can make 
them. Bend them into small hooks (Fig. 4-14) 
that can be hung over the notch in the straw or 
hung from each other. These are your 
"weights." 



rf^ ttiirt Kook u>ct«nT 




Rg. 4-14 

Mount another coated ball on a pointed, 
pliistic sliver and fix it in a clamp on a ring 
stand, as shown in Fig. 4-13. 

Using the Balance 

Charge both balls by wiping them with a 
rubbed plastic comb or ruler. Then bring the 
ring-stand ball down toward the beilance ball. 



182 



UNIT 4 / LIGHT AND ELECTROMAGNETIS VI 



1. What evidence have you that there is a force 
between the two balls? 

2. Is the force due to the charges? 

3. Can you compare the size of the electrical 
force between the two balls with the size of 
gravitational force between them? 



Repeat until you hav'e used all the hooks; do 
not reduce the air space between the balls to 
less than 0-5 cm. Then quickly retrace your 
steps by removing one hook (or more) at a time 
and raising the ring-stand ball each time to 
restore balance. 



Your balance is now ready, but in order to do 
the experiment, you need to solve two techni- 
cal problems. During the experiment you will 
adjust the position of the ring-stand ball so that 
the force between the charged balls is balanced 
by the wire weights. The straw will then be 
horizontal. First, therefore, you must check 
quickly to be sure that the straw is balanced 
horizontally each time. Second, measure the 
distance between the centers of the two balls. 
You cannot put a ruler near the charged balls, 
or its presence will affect your results; however, 
if the ruler is not close to the balls, it is very 
difficult to make the measurement accurately. 

Here is a way to make the measurement. 
With the balance in its horizontal position, you 
can record its balanced position with a mark on 
a folded card placed near the end of the straw 
(at least 5 cm away from the charges). (See Fig. 
4-13.) 

How can you avoid the parallax problem? Try 
to devise a method for measuring the distance 
between the centers of the balls. Ask your 
instructor if you cannot think of one. 

You are now ready to make measurements to 
see how the force between the two balls 
depends on their separation and on their 
charges. 



Doing the Experiment 

From now on, work as quickly as possible but 
move carefully to avoid disturbing the balance 
or creating air currents. It is not necessary to 
wait for the straw to stop moving before you 
record its position. When it is swinging slightly, 
but equally, to either side of the balanced 
position, you can consider it balanced. 

Charge both balls, touch them together 
briefly, and move the ring-stand ball until the 
straw is returned to thr balanced position The 
weight of one hook now balances the electric 
force between the charged spheres at this 
separation. Record the distance between the 
balls. 

Witliout nu'harging the balls, atld a second 
hook and n'adjiist the .system until balance is 
again restored. Recortl this new sepanition 



4. The separations recorded on the "return trip" 
may not agree with your previous measurements 
with this same number of hooks. If they do not, 
can you suggest a reason wtiy? 

5. Why must you not recharge the balls between 
one reading and the next? 



Interpreting Your Results 

Make a graph of your measurements of force F 
against separation d between centers. Cleariy F 
and d are inversely related; that is, F increases 
as d decreases. You can go further to find the 
relationship between F and d. For example, it 
might be F oc i/d. F a 1/c/*, or F a: l/d^ etc. 



6. How would you lest which of these relation- 
ships best represents your results? 

7. What relationship do you find between f and 
d7 



Further Investigation 

In this experiment, you can determine how the 
force F varies with the charges on the spheres 
when d is kept constant. 

Chaise both balls and then touch them 
together briefly. Since they are neariy identical, 
it is assumed that when touched, they v\ill 
sharp the total charge almost equally 

Hang four hooks on the balance and movT* 
the ring-stand ball until the straw is in the 
balanced position. Note this position. 

Touch the upper l)all with your finger to 
discharge the ball If the two balls are again 
brought into contact, the charge left on the 
balance ball will be shared equally between the 
two balls. 



8. What IS the charge on each ball now las a 
fraction of the original charge)? 



Return the ring-stand ball to its previous 
f)osition how many houks must you removx* to 
restore the balance? 



EXPERIMENTS 



183 



9. Can you state this result as a mathematical 
relationship between quantity of charge and 
magnitude of force? 

10. Consider why you had to follow two precau- 
tions in doing the experiment: 

(a) Why can a ruler placed too close to the charge 
affect results? 

(b) Why should you get the spheres no closer 
than about 0.5 cm? 

11. How might you modify this experiment to see 
if Newton's third law applies to these electric 
forces? 



Experiment 4*5 
FORCES ON CURRENTS 

If you did Elxj>eriment 4-4, you used a simple 
but sensitive balance to investigate how the 
electric force between two charged bodies 
depends on the distance between them and on 
the fimount of charge. In this and the next 
experiment you will examine a related effect: 
the force between moving charges, that is, 
between electric currents. You wUl investigate 
the efifect of the magnitudes and the directions 



of the currents. Before starting the experiment, 
you should have read the description of 
Oersted's and Ampere's work (Text Sections 
14.11 and 14.12). 

The apparatus for these experiments (like 
that in Fig. 4-15) is similar in principle to the 




Fig. 4-15 




184 



UNIT 4 / LIGHT AND ELECTROMAG.NETISM 



balance apparatus you used to measure elec- 
tric forces. The current balance measures the 
force on a horizontal rod suspended so that it 
is free to move sideways at right angles to its 
length. You can study the forces exerted by a 
magnetic field on a current by bringing a 
magnet up to this rod while there is a current 
in it. A force on the current-carrying rx)d causes 
it to swing away finom its original position. 

You can also pass a current through a fixed 
wire parallel to the pivoted rod. Any force 
exerted on the rod by the current in the fixed 
wire will again cause the pivoted rod to move. 
You can measure these forces simply by 
measuring the counterforce needed to return 
the rod to its original position. 

Adjusting the Current Balance 

This instrument is more complicated than 
most of those you have worked with so far 
Therefore, it is worthwhile spending a little 
time getting to know how the instrument 
operates before you start taking readings. 

1. You have three or four light metal rods bent 

into I I or '-vj-' shapes. These are the movable 

"loops." Set up the balance with the longest 
loop clipped to the pivoted horizontal bar. 
Adjust the loop so that the horizontal part of 
the loop hangs level with the bundle of wires 
(the fixed coil) on the pegboard frame. Adjust 
the balance on the frame so that the loop and 
coil are parallel as you look down at them. They 
should be at least 5 cm apart. Make sure 
the loop swings freely. 

2. Adjust the "counterweight" cylinder to bal- 
ance the system so that the long pointer arm is 
approximately horizontal. Mount the 
S -shaped plate (zero-mark indicator) in a 
clamp and position the plate so that the zerx) 
line is opposite the horizontal pointer (Fig. 
4-16). (If you are using the equipment for the 
first time, draw the zero-index line yourself.) 




^/W 




zero line 



Ftg. 4-16 Set ttie zero mark level with the pointer when 
there is current in the balance loop and no current in the 
fixed coil 



3. Now set the balance for maximum sensitiv- 
ity. To do this, move the sensiti\ity clip up the 
vertical rod (Fig. 4-171 until the loop slowly 
swings back and forth. These oscillations may 
take as much as 4 or 5 sec p>er swing. If the clip 
is raised too far, the balance may become 
unstable and flop to either side without 
"righting" itself. 




Fig 4-17 

4. Make sure that the pivots (knife-edge con- 
tacts) are clean and shiny luse fine-grade 
abrasive paper i, and remain clean throughout 
the experiment; otherwise, they will not let the 
current pass reliably. Now connect a 6 \'/5 A 
max power supply that can supply up to 5 A 
through an ammeter to one of the flat horizon- 
tal plates on which the pivots rest Connect the 
other plate to the second terminal of the power 
supply I Fig. 4-18). To limit the current and keep 
it frxjm tripping the circuit breaker, it may be 
necessary to put one or two l-fl resistors in the 
circuit. I If your power supply does not ha\e 
variable control, it should be connected to the 
plate thrxjugh a rheostat.) 




Fig 4 18 



EXPERIMENTS 



185 



6. Set the variable control for minimum cur- 
rent, and turn on the power supply. If the 
ammeter deflects the wrong way, interchange 
the leads to it. Slowly increase the current to 
about 4.5 A. 

6. Now bring a smaU magnet close to the 
pivoted conductor. 



1. How must the magnet be placed to produce 
the greatest effect on the rod? What determines 
the direction in which the rod swings? 



You will make quantitative measurements of 
the forces between magnet and current in the 
next experiment, "Currents, Magnets, and 
Forces." The rest of this experiment is con- 
cerned with the interaction between two cur- 
rents. 

7. Connect a similar circuit consisting of a 
power supply, ammeter, and rheostat (if there 
is no variable control on the power supply) to 
the fixed coil on the vertical pegboard (the 
bundle of ten wires, not the single wire). The 
two circuits (fixed-coU and movable-hook) must 
be independent. Your setup should now look 
like the one shown in Fig. 4-19. Only one meter 
is actually required, because you can move it 
from one circuit to the other as needed. It is, 
however, more convenient to work with two 
meters. 



'^^1 







"^ ffHEcr>rrT 



Rg. 4-19 Current balance connections using rheostats 
when variable power supply is not available. 



8. Turn on the currents in both circuits and 
check to see which way the pointer rod on the 
btilance swings. It should move up. If it does 
not, see if you can miike the pointer swing up 
by changing something in your setup. 



2. Do currents flowing in the same direction 
attract or repel each other? What about currents 
flowing in opposite directions? 



9. Prepare some "weights" fix)m the thin wire 
given to you. You will need a set that contains 
wire lengths of 1 cm, 2 cm, 5 cm, and 10 cm. 
You may want more than one of each; you can 
make more as needed during the experiment. 
Bend the wires into small S-shaped hooks so 
that they can hang from the notch on the 
pointer or from each other. This notch is the 
same distance from the axis of the balance as 
the bottom of the loop; therefore, when there is 
a force on the horizonteil section of the loop, 
the total weight F hung at the notch will equal 
the magnetic force acting horizontally on the 
loop. (See Fig. 4-20). 



hooks 



i 



=--=&<> 



f <- 



C 1 001 



Rg. 4-20 Side view of a balanced loop. The distance 
from the pivot to the wire hook is the same as the 
distance to the horizontal section of the loop, so the 
weight of the additional wire hooks is equal in mag- 
nitude to the horizontal magnetic force on the loop. 



These preliminary adjustments tire common 
to all the investigations. But from here on, there 
are separate instructions for three different 
experiments. Different members of the class 
wiU investigate how the force depends upon: 

(a) the current in the wires, 

(b) the distance between the wires, or 

(c) the length of one of the wires. 

When you have finished your experiment, 
read the section "For Class Discussion." 

(a) How Force Depends on Current 
in the Wires 

By keeping a constant separation between the 
loop and the coil, you can investigate the effects 
of varying the currents. Set the balance on the 
frame so that, as you look down at them, the 
loop and the coil are parcdlel £ind about 1 cm 
apart. 

Set the current in the balance loop to about 3 
A. Do not change this current throughout the 
experiment. With this current in the balance 
loop and no current in the fixed coil, set the 
zero mark in line with the pointer rod. 



186 



UNIT 4 / LIGHT AM) ELECTROMAG.N'ETISM 



Starting with a relatively small current in the 
fixed coil (about 1 A), determine how many 
centimeters of wire you must hang on the 
pointer notch until the pointer rod returns to 
the zero mark. 

Record the current /, in the fixed coil and the 
length of wire added to the pointer arm. The 
weight of wire is the balancing force F. 

Increase I, step by step, checking the current 
in the balance loop as you do so, until you 
reach a current of about 5 A in the fixed coil. 



3. What is the relationship between the current 
in the fixed coil and the force on the balance loop? 
One way to discover this is to plot force F against 
current /,. Another way is to find what happens to 
the balancing force when you double, then triple, 
the current /,. 

4. Suppose you had held /, constant and mea- 
sured F as you varied the current in the balance 
loop /,,. What relationship do you think you would 
have found between F and /,,? Check your answer 
experimentally (for example, by doubling /,,) if 
you have time. 

5. Can you write a symbolic expression for how 
F depends on both /, and /,,? Check your answer 
experimentally (by doubling both /, and /,,) if you 
have time. 

6. How do you convert the force, as measured in 
centimeters of wire hung on the pointer arm, into 
the conventional unit for force (newtons)? 



(b) How Force Varies uith the 
Distance between Wires 

To measure the distance between the two 
wires, you have to look down. F»ut a scale on the 
wooden shelf below the loop. Because there is a 
gap between the wires and the scale, the 
number you read on the scale changes as you 
move your head hack and forth. This effect is 
called parallax, and it must be reduced if you 
are to get good measurements. If you look 
down into a mirror set on the shelf, you can tell 
when you are looking straight down because 
the wire and its image will line up. Tr^' it I Fig. 
4-21). 



WTfOP jJtJf 




< 



V 



.vi..i.. ^ 



iim^t 



na^t « 



Fig. 4-21 Only when your eye is perpendicularly above 
the moving wire will it line up with its reflection in the 
mirror. 



Stick a length of centimeter tap>e along the 
side of the mirror so that you can sight down 
and read off the distance between one edge of 
the fixed wire and the corresponding edge of 
the balance loop. Set the zero mark with a 
current /(, of about 4-5 A in the balance loop and 
no current /, in the fixed coil. Then adjust the 
distance to about 0.5 cm. 

Begin the experiment by adjusting the cur- 
rent passing through the fixed coil to 4^ A. 
Hang weights on the notch in the pointer arm 
until the pointer is again at the zero position. 
Record the weight and distance carefully. 

Repeat your measurements for four or fh-e 
greater separations. Between each set of mea- 
surements make sure the loop and coil are still 
parallel; check the zero position, and see that 
the currents are still 4.5 A. 



7. What is the relationship between the force F 
on the balance loop and the distance d between 
the loop and the fixed coil? One way to discover 
this is to find some function of d (such as 1 d'. 1 d. 
d^, etc.) that gives a straight line when plotted 
against F. Another way is to find what happens to 
the balancing force F when you double, then 
triple, the distance d. 

8. If the force on the balance loop is F, what is 
the force on the fixed coil? 

9. Can you convert the force, as measured in 
centimeters of wire hung on the pointer arm, into 
force in newtons? 



(c) Hou Force \'aries with the 
Length of One of the Ulres 

By keeping constant currents /, and It, and a 
constant separation d, you can im-pstigate the 
effects on the length of the wires. In the current 
balance setup, it is the bottom, horizontal 
section of the loop that interacts most strxingly 
with the coil. Ixjops with sewral different 
lengths of horizontal segment are piDMded 

To measure the distances l>etwtM'n the two 
wires, you have to look down at them Put a 
scale on the wooden shelf l>elow the loop 
Because there is a gap between the wires and 
the scale, what you read on the scale changes 
as you mow your head I>ack and forth This 
effect is called paralla,\ parallax must be 
reduced if you are to get giMxi measurements. If 
you look down into a mirror set on the shelf, 
you can tell when you are looking straight 
down l>ecause the wire and its image will line 
up Tr> it I Fig 4-21) 

Stick a length of centimeter tape along the 
side of the mirror Then you can sight dowii 



EXPERIMENTS 



187 



and read off the distance between one edge of 
the fixed wire and the corresponding edge of 
the balance loop. Adjust the distance to about 
0.5 cm. With a current I^ of about 4.5 A in the 
balance loop and no current /f in the fixed coil, 
set the pointer at the zero mark. 

Begin the experiment by passing 4.5 A 
thrxDugh both the bcilance loop and the fixed 
coil. Hang weights on the notch in the pointer 
arm until the pointer is again at the zero 
position. 

Recorxl the vcdue of the currents, the distance 
between the two wir^s, and the weights added. 

Turn off the currents, and carefully remove 
the balance loop by sliding it out of the holding 
clips (Fig. 4-22). Measure the length / of the 
horizontcil segment of the loop. 



For Class EHscussion 

Be prepared to report the results of your 
particular investigation to the rest of the class. 
As a class, you wiU be able to combine the 
individual experiments into a single statement 
about how the force varies with current, with 
distance, and with length. In each part of this 
experiment, one factor was varied while the 
other two were kept constant. In combining the 
three separate findings into a single expression 
for force, you are assuming that the effects of 
the three factors are independent . For example, 
you are assuming that doubling one current 
will always double the force, regardless of what 
constant values d and / have. 



Fig. 4-22 




13. What reasons can you give for assuming such 
a simple independence of effects? What could you 
do experimentally to support the assumption? 

14. To make this statement into an equation, 
what other facts do you need; that is, what must 
you know to be able to predict the force (in 
newtons) existing between the currents in two 
wires of given length and separation? 



Experiment 4-6 
CURRENTS, MAGNETS, AND 
FORCES 



Insert another loop. Adjust it so that it is level 
with the fixed coil and so that the distance 
between loop and coil is just the same as you 
had before. This is important. The loop must 
cilso be parallel to the fixed coU, both as you 
look down at the wires from above and as you 
look at them from the side. Also reset the clip 
on the balance for maximum sensitivity. Check 
the zero position, and see that the currents cire 
still 4.5 A. 

Repeat your measurements for each balance 
loop. 



10. What is the relationship between the length / 
of the loop and the force F on it? One way to 
discover this is to find some function of / (such as 
/, /2, Ml, etc.) that gives a straight line when plotted 
against F. Another way is to find what happens to 
F when you double /. 

11. Can you convert the force, as measured in 
centimeters of wire hung on the pointer arm, into 
force in newtons? 

12. If the force on the balance loop is F, what is 
the force on the fixed coil? 



If you did Experiment 4-5, "Forces on Cur- 
rents," you discovered how the force between 
two wires depends on the current in them, 
their length, and the distance between them. 
You also know that a nearby magnet exerts a 
force on a current-carrying wire. In this exper- 
iment, you will use the current balance to study 
further the interaction between a magnet and a 
current-carrying wire. You may need to refer 



hooks 



■=Sh:> 



F <- 



oop 



Fig. 4-23 Side view of a balanced loop. Since the 
distance from the pivot to the wire hooks is the same as 
the distance to the horizontal section of the loop, the 
weight of the additional wire hooks is equal to the 
horizontal magnetic force on the loop. 



188 



UNIT 4 / LIGHT AND ELECTROMAGNETISM 



back to the notes on Elxperiment 4-5 for details 
on the equipment. 

In this experiment, you will not use the fixed 
coil. The frame on which the coil is wound will 
serve merely as a convenient support for the 
balance and the magnets. 

Attach the longest of the balance loops to the 
pivotal horizontal bar, and connect it through 
an ammeter to a variable source of current. 
Hang weights on the pointer notch until the 
pointer rod returns to the zero mark (see Fig. 
4-23). 

(a) How the Force between Current 
and Magnet Depends on the Current 

1. Place two small ceramic magnets on the 
inside of the iron yoke. Their orientation is 
important; they must be turned so that the two 
near faces attract each other when they are 
moved close together. (Caution: Ceramic mag- 
nets are brittle. They break if you drop them.) 
Place the yoke and magnet unit on the platform 
so that the balance loop passes through the 
center of the region between the ceramic 
magnets (Fig. 4-24). 



^balance loop 
^^^ 'po\e piece 



Fig. 4-24 Each magnet consists of a yoke and a pair of 
removable ceramic-magnet pole pieces. 

2. Check whether the horizontal pointer 
moves up when you turn on the current. If it 
moves down, change something (the current? 
the magnets?) so that the pointer swings up. 

3. With the current off, mark the zero position 
of the pointer ami v\'ith the indicator. Adjust 
the current in the coil to about 1 A. Hang wire 
weights in the notch of the balance arm until 
the pointer returns to the zero position. 

Record the current and the total lialancing 
weight. Rej)eat the measurements for at least 
four greater currents. Between each pair of 
readings, check the zero position of the pointer 
arm. 



1. What IS the relationship between the current /,, 
and the resulting force f that the magnet exerts 
on the wire? (Try plotting a graph.) 

2. If the magnet exerts a force on the current, do 
you think the current exerts a force on the 
magnet? How would you test this? 



3. How would a stronger or a weaker magnet 
affect the force on the current? If you have time, 
try the experiment with different magnets or by 
doubling the number of pole pieces. Then plot F 
against /,, on the same graph as in Question 1 
above. How do the plots compare? 



(b) How the Force between a 
Magnet and a Current Depends 
on the Length of the Region 
of Interaction 

1. Place two small ceramic magnets on the 
inside of the iron yoke to act as pole pieces (Fig. 
4-24). (Caution: Ceramic magnets are brittle. 
They break if you drop them.) Their orientation 
is important; they must be turned so that the 
two near faces attract each other when they ax^ 
moved close together Place the yoke and 
magnet unit on the platform so that the balance 
loop passes through the center of the region 
between the ceramic magnets (Fig. 4-241. 

Place the yoke so that the balance loop 
passes through the center of the magnet and 
the pointer moves up v\+ien you turn on the 
current. 

With the current off, mark the zero position 
of the pointer with the indicator. 

2. Hang 10 cm or 15 cm of wire on the notch in 
the balance rod, and adjust the current to 
return the pointer rod to its zero position. 
Record the current and the total length of wire, 
and set aside the magnet for later use 

3. Put a second yoke and pair of pole pieces in 
position and see if the balance is restored. You 
have changed neither the current nor the 
length of wire hanging on the pointer There- 
fore, if balance is restored, this magnet must be 
of the same strength as the first one If it is not. 
try other combinations of pole pieces until you 
have two magnets of the same strength You 
can produce small variations in strength by 
moving the pole pieces: ai into the yoke to make 
it strongt-r; l)i to th»« ends of the yoke to make it 
weaker. If possible, try to get three matched 
magnets. 

4. Now you are ready for the important test. 
Place two of the magnets on the platform at the 
same time iFig 4-251 To keep the magnets from 
afTectitig each others field appn^ciably. the>' 
should Im* at least 10 cm apart ()f course, each 
nuignet must l^e jK)sitioned so that the pointer 
IS ileflected upwanl With the current just what 
it was l>efore, haiig wire weights in the notch 
until the Iwlance is restored. 



EXPERIMENTS 



189 




Fig. 4-25 

If you have three magnet units, repeat the 
process using three units at a time. Again, keep 
the units well apart. 

Interpreting Your Data 

Your problem is to find a relationship between 
the length / of the region of interaction and the 
force F on the wire. 

You may not know the exact length of the 
region of interaction between magnet and wire 
for a single unit. It certcdnly extends beyond the 
region between the two pole pieces. But the 
force decreases rapidly with distance from the 
magnets. As long as the separate units are far 
from each other, neither wiU be influenced by 
the prescence of the other. You can then 
assume that the total length of interaction uath 
two units is double that for one unit. 

? 

4. How does F depend on /? 

(c) A Study of the Interaction 
Between the Earth and an Electric 
CXirrent 

The magnetic field of the earth is much weaker 
than the field near one of the ceramic mag- 
nets and the balance must be adjusted to its 
maximum sensitivity. The following sequence 
of detailed steps will make it easier for you to 
detect and measure the small forces on the 
loop. 

1. Set the balance, with the longest loop, to 
maximum sensitivity by sliding the sensitivity 
clip to the top of the vertical rod. The sensitivity 
can be increased further by adding a second 
clip; be careful not to make the balance 
top-heavy so that it falls over. 

2. With no current in the balance loop, align 
the zero mark with the end of the pointer arm. 

3. Turn on the current and adjust it to about 
5 A. Turn off the current £ind let the balance 
come to rest. 

4. Turn on the current, and observe carefully: 
Does the balance move when you turn the 
current on? Since there is no current in the 
fixed coil, and there are no magnets nearby, any 



force acting on the current in the loop must be 
due to an interaction between it and the earth's 
magnetic field. 

5. To make measurements of the force on the 
loop, you must set up the experiment so that 
the pointer swings up when you turn on the 
current. If the pointer moves down, try to find a 
way to make it go up. (If you have trouble, 
consult your instructor). Turn off the current, 
and bring the balance to rest. Mark the zero 
position with the indicator. 

6. Turn on the current. Hang weights on the 
notch, and adjust the current to restore 
balance. Record the current and the length of 
wire on the notch. Repeat the measurement of 
the force needed to restore balance for several 
different values of current. 

If you have time, repeat your measurements 
of force cind current for a shorter loop. 

Interpreting Yoiu* Data 

Try to find the relationship between the 
current /b in the balance loop and the force F 
on it. Make a plot of F against /b. 



5. How can you convert your weight unit (cen- 
timeters of wire) into newtons of force? 

6. What force (in newtons) does the earth's 
magnetic field in your laboratory exert on a 
current /b in the loop? 

For Class Discussion 

Diff"erent members of the class have investi- 
gated how the force F between a current and a 
magnet varies with current / and with the 
length of the region of interaction with the 
current /. It should also be clear that in any 
statement that describes the force on a current 
due to a magnet, you must include another 
term that takes into account the "strength" of 
the magnet. 

Be prepared to report to the class the results 
of your own investigations and to help formu- 
late an expression that includes till the relevant 
factors investigated by members of the class. 



7. The strength of a magnetic field can be 
expressed in terms of the force exerted on a wire 
carrying 1 A of current when the length of the wire 
interacting with the field is 1 m. Try to express the 
strength of the magnetic field of your magnet 
yoke or of the earth's magnetic field in these units, 
newtons per ampere-meter. (That is, what force 
would the fields exert on a horizontal wire 1 m 
long carrying a current of 1 A?) 



190 



UNIT 4 / LIGHT A.\D ELECTROMAGNETISM 





In using the current balance in this experi- 
ment, all measurements were made in the zero 
position when the loop was at the very bottom 
of the swing. In this position a vertical force will 
not affect the balance. Therefore, you have 
measured only horizontal forces on the bottom 
of the loop. 

However, since the force exerted on a current 
by a magnetic field is always at right angles to 
the field, you have therefore measured only 
the vertical component of the magnetic fields. 
From the symmetry of the magnet yoke, you 
might guess that the field is entirely vertical in 
the region directly between the pole pieces. But 
the earth's magnetic field is exactly vertical only 
at the magnetic poles. (See the drawing on page 
457 of the Text.) 



8. How would you have to change the experiment 
to measure the horizontal component of the 
earth's magnetic field? 



Experiment 4-7 

ELECri RON BEAM TL^BE. I 

If you did the experiment "Electric Forces. II: 
Coulomb's Law, " you found that the force on a 
test charge, in the vicinity of a second charged 
body, decreases rapidly as the distance be- 
tween the two charged bodies is increased. In 
other words, the electric field strength due to a 
single small chargrci body dei rcas«'.s with 
distance from the body. In many experiments it 
is useful to have a region wtiere the field is 
uniform, that is, a region where the force on a 
test charge is the same at all points. The field 
evervwhert" between two closely spared paral- 
lel. Hat, oppositely charged plates is verv nearly 
unifomi (Fig. 4-26). 

The nearly imifomi magnitude E depends 
upon the potential diffenMice Ix'tween \hv 
plates and upon their separation (/: 



In addition to electric forces on charged bodies, 
you found (if you did either of the pre\ious two 
experiments with the current balance) that 
there is a force on a current-carrying wire in a 
magnetic field. 




• a 



Fig. 4-26 The field t>etween two parallel flat plates is 

uniform £ = Vid where V is the potential dtHerence 
(volts) between the two plates 

Free Charges 

In this experiment, the chaises will not be 
confined to a Styrofoam ball or to a metallic 
conductor. Instead they will be free chaiiges, 
free to move through the field on their own in 
air at low pressure. 

You will build a sp>ecial tube for this experi- 
ment. The tube will contain a filament wire and 
a metal can with a small hole in one end. 
Electrons emitted from the heated filament are 
accelerated towartl the positi\ely rhar^jed can 
and some of them pass through the hole into 
the space l>eyond formmg a lM*am of electrons 
If the tube is carefully constructed, the air 
pumf>ed out to the right pressure, and the tube 
well sealed, it is possible to observe how the 
beam is affected by electric and magnetic fields. 

\Mien one of the air molecules remaining in 
the partially e\acuated tube is struck by an 
electron, the molecule emits some light. 
\fol(>cules of different gases emit light of 
different colors (Neon gas, for example, glows 
red 1 The bluish glow of the air left in the tube 
shows the path of the electron beam. 



EXPERIMENTS 



191 




lasv 



Building Your Electron Beam Tube 

Full instructions on how to build the tube are 
included with the parts. Note that one of the 
plates is connected to the can. The other plate 
must not touch the can. 

After you have assembled the filament and 
plates on the pins of the glass tube base, you 
Ccin see how good the alignment is if you look 
in through the narrow glass tube. You should 
be able to see the filament across the center of 
the hole in the can. Do not seal the header in 
the tube until you have checked this alignment. 
Then leave the tube undisturbed overnight 
wtiile the sealaint hairdens. It is not unusual that 
some of the tubes in a given class do not work 
well. In that case, try to share the tubes that 
function successfully. 

Operating the Tube 

With the power supply turned off, connect the 
tube £is showTi in Figs. 4-27 and 4-28. The 
low-voltage connection provides a current to 
heat the filament and make it emit electrons. 




-C: 



0-4Vo — /«kV 



0~C ftjnp 

Fig. 4-28 The pins to the two plates are connected so 
that they will be at the same potential and there will be 
no electric field between them. 

The ammeter in this circuit allows you to keep 
a close check on the current and avoid burning 
out the filament. Be sure the 0- 6 V control is 
turned down to 0. 

The high-voltage connection provides the 
field that accelerates these electrons toward 
the can. Let the instructor check the circuits 
before you proceed further. 

Turn on the vacuum pump and let it run for 
several minutes. If you have done a good job of 
putting the tube together, and if the vacuum 
pump is in good condition, you should not 
have much difficulty getting a glow in the 
area where the electron beam comes through 
the hole in the can. 

You should woiSc with the faintest glow that 
you can see clearly. Even then, it is important to 
keep a close watch on the brightness of the 
glow. There is an appreciable current from the 
filament wire to the can. As the residual gas 
gets hotter, it becomes a better conductor, thus 
increasing the current. The increased current 
will cause further heating, and the process can 
build up; the back end of the tube will glow 
intensely blue-white and the can wiil become 
red hot. You must immediately reduce the 
current to prevent the tube from being de- 
stroyed. If the glow in the back end of the tube 
begins to increase noticeably, turn down the 
filament current very quickly, or turn off the 
power supply altogether. 

Deflection by an Electric Field 

When you get an electron beam, try to deflect it 
in an electric field by connecting the deflecting 
plate to the ground terminal (see Fig. 4-29). You 




Rg. 4-27 



Rg. 4-29 Connecting one deflecting plate to ground will 
put a potential difference of 125 V between the plates. 



192 



UNIT 4 / LIGHT AND ELECTROMAGNETISM 



will put a potential diiference between the 
plates equal to the accelerating voltage. Other 
connections can be made to get other voltages, 
but check your ideas with your instructor 
before trying them. 



1. Make a sketch showing the direction of the 
electric field and of the force on the charged 
beam. Does the deflection in the electric field 
confirm that the beam consists of negatively 
charged particles? 



Deflection by a Magnetic Field 

Now tiy to deflect the beam in a magnetic field, 
using the yoke and magnets from the current 
balance experiments. 



2. Make a vector sketch showing the direction of 
the magnetic field, the velocity of the electrons, 
and the force on them. 



Balancing the Electric and 
Magnetic Effects 

Try to orient the magnets so as to cancel the 
effect of an electric field between the two 
plates, permitting the charges to travel straight 
through the tube. 



3. Make a sketch showing the orientation of the 
magnetic yoke relative to the plates. 

The Speed of the Charges 

As explained in Chapter 14 of the Text, the 
magnitude of the magnetic force is q\'B, where 
q is the electron charge, v is the speed, and B is 
the magnetic field strength. The magnitude of 
the electric force is qE, where E is the strength 
of the electric field. If you adjust the voltage on 
the plates until the electric force just balances 
the magnetic force, then qvB = qE and, there- 
fore, v =B/E. 



4. Show that fl/£ will be in speed units if B is 
expressed in newtons/ampere-meter and £ is 
expressed in newtons/coulomb. Hint: Remember 
that 1 A 1 C/sec. 



If you know the value of B and £. you can 
calculate the speed of the electrons The value 
of E is easy to find since, in a uniform field 
between parallel plates, £ = V/d, where V is the 



potential difference between the plate I in volts) 
and d is the separation of the plates (in meters I. 
(The unit volts/meter is equivalent to 
newtons/coulomb.) 

A rough value for the strength of the 
magnetic field between the poles of the magnet 
yoke can be obtained as described in Elxperi- 
ment 4-6, "Currents, Magnets, and Forces." 



5. What value do you get for E (in volts per 
meter)? 

6. What value did you get for B (in newtons per 
ampere-meter)? 

7. What value do you calculate for the speed of 
the electrons in the beam? 



An Important Question 

One of the problems facing phvsicists at the 
end of the nineteenth century was to decide on 
the nature of these cathode rays ' (so called 
because they are emitted from the negative 
electrode or cathode). One group of scientists 
(mostly German) thought that cathode rays 
wei^ a form of radiation, like light, v\t»ile others 
(mostly English) thought they v\*er^ streams of 
particles. J.J. Thomson at the Cavendish Labo- 
ratory in Cambridge, England, did exp>erimenls 
much like the one described here which 
showed that the cathode rays behaved like par- 
ticles; these particles are now called electrons 
These experiments \\'erv: of great imjxjrtance 
in the eariy dev-elopment of atomic phvsics In 
Unit 5, you will do an experiment to determine 
the ratio of the charge of an electix>n to its mass. 



Experiment 4-8 

ELECTRON BEAM TUBES. II' 

1. Focusing the Electron Beam 

A current in a wire coiled around the electron 
tube will produce inside the coil a magnetic 
field parallel to the axis of the tube (Ring- 
shaped magnets slipped oxT-r the tube will 
pniduce the same kind of field.) An electron 
mo\ing directly along the axis will experience 
no fon^e since its velocity' is parallel to the 
magnetic field .An electron mo\ing perjx'ndiru- 
lar to the axis, howvwr. will exf>enrn(v a forre 
IF = q\Bi at right artgles to Ixith wlority and 
field. If the curved path of the electron remains 



'\otr: This experiment is more romplfx than usual. 



EXPERIMENTS 



193 



in the uniform field, it will be a circle. The 
centripetal force F = mv^/R that keeps it in the 
circle is just the magnetic force qvB, so 



qvB = 



R 



where R is the radius of the orbit. In this simple 
case, therefore, 



R 



mv 



Suppose the electron is moving down the 
tube only slightly off axis, in the presence of a 
field parallel to the axis [Fig. 4-30la)]. The 
electron's velocity can be thought of as made 
up of two components: an axial portion of v^ 
and a transverse portion (perpendicular to the 
eixis) V, [Fig. 4-30(b)]. Consider these two 
components of the electron's velocity indepen- 
dently. You know that the axial component will 
be unaffected; the electron will continue to 
move down the tube with speed v^ [Fig. 4-30(c)]. 
The transverse component, however, is per- 
pendicular to the field, so the electron will also 
move in a circle [Fig. 4-30(d)]. In this case, 

qB 

The resultant motion (uniform speed down 
the cLxis plus circular motion perpendicular to 
the axis) is a helix, like the thread on a bolt [Fig. 
4-30(e)]. 



L-J , 
-/ J\- I' 

(Q) (b) Cc) 



B + 



(d) 



Rg. 4-30 



In the absence of any field, electrons travel- 
ing off-axis would continue toward the edge of 
the tube. In the presence of an axicil magnetic 
field, however, the electrons move down the 
tube in helixes; that is, they have been focused 
into a beam. The radius of this beam depends 
on the field strength B and the transverse 
velocity v,. 

Wrap heavy-gauge copper wire, such as #18, 
around the electron beam tube (about two 
turns per centimeter) and connect the tube to a 
low-voltage (3-6 V), high-current source to give 
a noticeable focusing effect. Observe the shape 
of the glow, using different coils and currents. 



(Alternatively, you can vary the number and 
spacing of ring magnets slipped over the tube 
to produce the axial field.) 

2. Reflecting the Electron Beam 

If the pole of a very strong magnet is brought 
near the tube (with great care being taken that 
it does not pull the iron mountings of the tube 
toward it), the beam glow will be seen to spiral 
more and more tightly as it enters stronger field 
regions. If the field lines diverge enough, the 
path of the beam may start to spiral back. The 
reason for this is suggested in SG 39 in Chap- 
ter 14. 

This kind of reflection operates on particles 
in the radiation belt around the earth as they 
approach the earth's magnetic poles. (See 
drawing at the end of Text Chapter 14.) Such 
reflection is what makes it possible to hold 
tremendously energetic, changed peirticles in 
magnetic "bottles." One kind of coil used to 
produce a "bottle" field is called a magnetic 
torus. 

3. Diode and Triode Characteristics 

This experiment gives suggestions for how you 
can explore some characteristics of electronic 
vacuum tubes with your electron beam tube 
materials. 

These experiments are performed at ac- 
celerating voltages below those that cause 
ionization (a visible glow) in the electron beam 
tube. 

Rectification 

Connect an ammeter between the can and 
high-voltage supply to show the direction of 
the current, and to show that there is a current 
only when the can is at a higher potential than 
the filament (Fig. 4-31). 

riil|ia»irtter 
0-30YC ^- 



0-tV^ 



-e 




Fig. 4-31 

Note that there is a measurable current at 
voltages far below those needed to give a visible 
glow in the tube. Then apply an alternating 
potential difference between the can and 



194 



UNIT 4 / LIGHT AND ELECTRO VIACiNETISM 



filament (for example, from a Variac). Use an 
oscilloscope to show that the can is alternately 
above and below filament potential. Then 
connect the oscilloscope across a resistor in 
the plate circuit to show that the current is only 
in one direction. (See Fig. 4-32.) 



oscilloscope 




J 



Fig. 4-32 The one-way valve (rectification) action of a 
diode can be shown by substituting an ac voltage 
source for the dc accelerating voltage, and connecting a 
resistor (about 1,000 ohms) in series with it. When an 
oscilloscope is connected, as shown by the solid lines 
above, it will indicate the current in the can circuit. When 
the one wire is changed to the connection shown by the 
dashed arrow, the oscilloscope will indicate the voltage 
on the can. 



Triode 

The "triode" in Fig. 4-33 was made with a 
thin aluminum sheet for the plate and ni- 
chrome wire for the grid. The filament is the 
original one from the electron beam tube kit, 
and thin aluminum tubing fixjm a hobby shop 
was used for the connections to plate and grid. 
(For reasons lost in the history of vacuum 
tubes, the can is usually called the plate.) It is 
interesting to plot graphs of plate current 
versus filament heating current, and plate 
current versus voltage. Note that these charac- 
teristic curves apply only to voltages too low to 
produce ionization. With such a triode, you can 
plot curves showing triode characteristics: 
plate current against grid voltage; plate current 
against plate voltage. 

You can also measure the voltage amplifica- 
tion factor, which descril>es how large a change 
in plate voltage is produced by a change in grid 
voltage. More precisely, the amplification factor 



M = 



- AV 




Change the grid voltage by a small amount, 
then adjust the plate \oltage until you have 
regained the original plate current TTie mag- 
nitude of the ratio of these two \'oltage chcinges 
is the amplification factor. (Commercial vacuum 
tubes commonly have amplification factors as 
high as 500.1 The tube gave noticeable ampli- 
fication in the circuit shown in Fig. 4-35 and 
Fig. 4-36. 



^A^PuomES 



affia Xtfo— 




Outpul Si^n«t In f^Jlt. Ci'(K>'f 



/nput »»^imI "tf ^rid 



Fig. 4-35 An amplifying circuit 



plait 



AV, 



I'K* 1 

\ 







<»IU^« 



when the plate current is kept constant. 



Fig. 4-M Schematic diagram of amplifyir>g circuit. 



EXPERIMENTS 



195 



Experiment 4-9 

WAVES AND COMMUNICATION 

Having studied many kinds and characteristics 
of waves in Units 3 and 4 of the Text, you are 
now in a position to see how they are used in 
communications. Here are some suggestions 
for investigations with equipment that you have 
probably already seen demonstrated. The fol- 
lowing notes assume that you understand how 
to use the equipment. If you do not, then do 
not go on until you ask for instructions. 
Although different groups of students may use 
different equipment, all the investigations are 
related to the same phenomenon: communica- 
tion by means of waves. 

A. Turntable Oscillators 

Turn on the oscillator with the pen attached to 
it. (See Unit 5, page 231.) Turn on the chart 
recorder, but do not turn on the osciUator on 
which the recorder is mounted. The pen vvQl 
trace out a sine curve as it goes back and forth 
over the moving paper. When you have re- 
corded about 7 cm- 8 cm, turn off the oscilla- 
tor and bring the pen to rest in the middle of 
the paper. Now turn on the second oscillator 
at the same rate that the first one was going. 
The pen will trace out a similar sine curve as 
the moving paper goes back and forth under it. 
The wavelengths of the two curves are probably 
very nearly, but not exactly, equal. 



1. What do you predict will happen if you turn 
on both oscillators? Try it. Look carefully at the 
pattern that is traced out with both oscillators on 
and compare it to the curves previously drawn by 
the two oscillators running alone. 



Chcmge the wavelength of one of the compo- 
nents slightly by putting weights on one of the 
platforms to slow it down a bit. Then make 
more traces firom other pairs of sine curves. 
Each trace should consist of three parts, as in 
Fig. 4-37: the sine curve from one oscillator; the 



sine curve fiDm the other oscillator; and the 
composite curve from both oscillators. 



2. According to a mathematical analysis of the 
addition of sine waves, the wavelength of the 
envelope {K in Fig. 4-37) will increase as the 
wavelengths of the two components (A.,, Xj 
become more nearly equal. Do your results 
confirm this? 

3. If the two wavelengths X, and X^ were exactly 
equal, what pattern would you get when both 
turntables were turned on; that is, when the two 
sine curves were superimposed? What else would 
the pattern depend on, in addition to X, and X2? 



As the difference between \, and X2 gets 
smaller, Xp gets bigger. You can thus detect a 
very small difference in the two wavelengths by 
examining the resultant wave for changes in 
amplitude that teike place over a relatively long 
distance. This method, called the method of 
beats, provides a sensitive way of comparing 
two oscillators, and of adjusting one until it has 
the same frequency as the other. 

This method of beats is also used for tuning 
musical instruments. If you play the same note 
on two instruments that are not quite in tune, 
you can hear the beats. The more ne<irly in tune 
the two instruments are, the lower the fre- 
quency of the beats. You might like to try this 
with two guitars or other musical instnjments 
(or two strings on the same instrument). 

In radio communication, a signal can be 
transmitted by using it to modulate a carrier 
wave of much higher frequency. (See Part E for 
further explanation of modulation.) A snapshot 
of the modulated wave looks similar to the 
beats you have been producing, but it results 
from one wave being used to control the 
amplitude of the other, not from simply adding 
the waves together. 

B. Resonant Circuits 

You have probably seen a demonstration of 
how a signal can be transmitted from one 



■liFwHlHHi 



Fig. 4-37 



196 



UNIT 4 / LIGHT AM) ELECTROMACNETISM 



circuit to another that is tuned to it. (If you have 
not seen the demonstration, you should set it 
up for yourself using the apparatus shown in 
Fig. 4-38.) 




Fig. 4-38 Two resonant circuit units. Each includes a wire 
coil and a variable capacitor. The unit on the right has an 
electric cell and ratchet to produce pulses of oscillation 
in its circuit. 

This setup is represented by the schematic 
drawing in Fig. 4-39. 




ratchet 



ha t tery 



5 



Fig. 4-39 



E3 



diode 
— H-— 



The two coils have to be quite close to each 
other for the receiver circuit to pick up the 
signal from the transmitter. The effect is due to 
the fluctuation of the magnetic field from one 
coil inducing a fluctuating currt^nt in the other 
coil. It works on the same principle as a 
transformer. 

Investigate the efi^ect of changing the position 
of one of the coils. Try turning one of them 
around, moving it farther away, etc. 



4. What happens when you put a sheet of metal, 
plastic, wood, cardboard, wet paper, or glass 
between the two coils? 

5. Why does an automobile always have an 
outside antenna, while a home radio does not? 



You liiur probably learned that to transmit a 
signal fn)m one circuit to another, the two 
circuits must be tuned to the same frequenc\'. 
To investigate the range of frequencies obtain- 
able with \()ur nvsonant circuit, connect an 
antenna (length of v\irel to the resonant 
receiving cinniit. in (»rder to increase its 
sensitivity, and n'place the speaker by an 
amplifier and oscilloscope (Fig. 4-401. Set the 




Rg. 4-40 

oscilloscope to Internal Svnc" and the sweep 
rate to about 100 kHz. 



6. Change the setting of the variable capacitor 
( ^ ) and see how the trace on the oscilloscope 
changes. Which setting of the capacitor gives the 
highest frequency? which setting the lowest? By 
how much can you change the frequency by 
adjusting the capacitor setting? 



When you tune a radio, you are usually, in 
the same way, changing the setting of a variable 
capacitor to tune the circuit to a different 
frequency. 

The coil also plays a part in determining the 
resonant frequency of the circuit If the coil has 
a different number of turns, a different setting 
of the capacitor would be needed to get the 
same frequency. 

C. Elementan- Properties of 
Micro\\a\'es 

With a microwave generator, you can in\-esti- 
gate some of the characteristics of short wa\«s 
in the radio part of the electromagnetic 
spectnim In Kxperiment 3-17 "Measuring 
Wavelength" and E.xperiment 3-18 "Sound, "ntju 
explored the behavior of sex-eral different kinds 
of waves These eariier experiments contained 
a number of ideas that will help \-ou conclude 
that the energ\' emitted by \-our microwave 
generator is in the form of wa\t»s 

Refer to your notes on these e.xj>eriments 
Then, using the arrangements suggested there 
or ideas of \our own explore the transmission 
of microwaws thnnigh various materials as 
well as micniwavv reflection and n^fraction Try 
to detect their diffraction aniunil obstacles and 
through narrow openings in some material that 
is opaque to them. Finally, if \x)u ha\« two 
transmitters available or a metal horn attach- 
ment with tw<) o|>«Miings see if vou can 
measure the wavvlength usirtg the interference 
method of K\|>«'riment 3-18 Discuss N-our 
results with students doii\g the following 



EXPERIMENTS 



197 




'k^iu-m. 



experiment (D) on the interference of reflected 
microwaves. 

D. Interference of Reflected 
Microuaves 

With microwaves it is easy to demonstrate 
interference between direct radiation ftxjm a 
source and radiation reflected from a flat 
surface, such as a metal sheet. At points where 
the direct and reflected waves arrive in phase, 
there wiU be maxima; at points where they 
arrive one-half cycle out of phase, there will be 
minima. The maxima and minima are readily 
found by moving the detector along a line 
perpendicular to the reflector. (Fig. 4-41) 



reflector 




\ 



\ 



/----^ 



tdetcctor 
i a long ♦♦"i 



Fig. 4-41 



again, you can sketch out lines of nnaxima and 
nninima. 

8. How is the interference pattern similar to 
what you have observed for two-source radia- 
tion? 



7. Can you state a rule with which you could 
predict the positions of nnaxima and minima? 
By moving the detector back and scanning 



Standing microwaves will be set up if the 
reflector is placed exactly perpendicular to the 
source (Fig. 4-42). (As with other standing 




Rg. 4-42 



198 



UIWIT 4 / LIC;HT AND ELECTROMAGNETISM 



waves, the nodes are one-half wavelength 
apart.) Locate several nodes by moving the 
detector along a line between the source and 
reflector, and from the node separation calcu- 
late the wavelength of microwaves. 



9. What is the wavelength of your microwaves? 
10. Microwaves, like light, propagate at 3 ^ 10* 
m/sec. What is the frequency of your microwaves? 
Check your answer against the chart of the 
electromagnetic spectrum given on page 511 of 
Text Chapter 16. 



The interference between direct and re- 
flected radio waves has important practical 
consequences. There are layers of partly 
ionized (and therefore electrically conducting) 
air, collectively called the ionosphere, that 
surround the earth roughly 30 km - 300 km 
above its surface. One of the layers at about 300 
km is a good reflector for radio waves; it is used 
to bounce radio messages to points that, 
because of the curvature of the earth, are too far 
away to be reached in a straight line. 

If the transmitting tower is 100 m high, then, 
as shown roughly in Fig. 4-43, point A, the 
farthest point that the signal can reach directly 
in flat country, is 35 km away. But by reflection 
from the ionosphere, a signal can reach points 
not in the line of sight like B and beyond. 



Fig. 4-43 



'^N^ 



Sometimes both a direct and a reflected 
signal will arrive at the same place and 
interference occurs; if the two signals are out of 
phase and have identical amfilitudes, the 
receiver will pick up nothing. Such destructive 
interference is responsible for radio fading. It is 
complicated by the fart that the height of the 
ionosphere and the intensity of reflection from 
it vary with tin* amoutit of sunlight and with 
time. 

The setup in Fig 4-44 is a model of this 
situation. Move the reflector (the "ionosphen>"l 
back and forth. What happens to the signal 

StHMlgtll? 



"»!«*» r 









'-V' / 
W 






\ 




Fig. 4-44 



There can also be multiple reflections; the 
radiation can bounce back and forth between 
earth and ionosphere sex'eral times on its way 
from, say, New York to Calcutta, India. Perhaps 
you can simulate this situation, too, with your 
microwave equipment. 



E. Signals and \Iicrou-a\'es 

Thus far, you have been learning about the 
behavior of microwaves of a single frequency' 
and constant amplitude. A signal can be added 
to these waves by changing their amplitude at 
the transmitter. The most ob\ious way to 
change the amplitude of the w^ves vwould be 
just to turn them on and off as represented in 
Fig. 4-45. Coded messages idots and dashes) 
can be transmitted in this primiti\-e fashion But 
the wave amplitude can be varied in a more 
elaborate way to earn,' music or voice signals. 
For example, a 1,000-Hz sine wax'e fed into part 
of the microwave transmitter will cause the 
amplitude of the microwaN-e to var>' smoothly at 
1,000 Hz. 



i'l — '' ' — i^r 



Rg 4-45 



Controlling the amplitude of the transmitted 
wav-e like this is called amplitude modulation, in 
Fig. 4-46, .A represents the unmodulated mi- 
cnnvave. B rrpn»sents a nuxlulating signal and 
C repre.sents the modulated mirrowaw The 
Hamon micnnvaw oscillator has an input for a 
inodulatiiTg signal. You can modulate the 
microvN-aNT output with a variet>- of signals, for 
example, with an audio frequency' oscillator or 
with a microphone and amplifier. 



EXPERIMENTS 



199 



mmm 
mmh 



n 



miz 



!jWw\Ajs|| 




• 2 



The microwave detector probe is a one-way 
device; it passes current in only one direction. 
If the microwave reaching the probe is repre- 
sented in C, then the electric signed finom the 
probe wiU be as shown in D. 

You can see this signal on the oscilloscope by 
connecting the oscilloscope to the microwave 
probe (through an amplifier if necessary). 

The detected modulated signal from the 
probe can be turned into sound by connecting 
an amplifier and loudspeaker to the probe. The 
speaker will not be able to respond to the 10^ 
individual pulses per second of the "carrier" 
wave, but only to their averaged effect, repre- 
sented by the broken line in E. Consequently, 
the sound output of the speaker wall corre- 
spond veiy nearly to the modulating signal. 






la 



Rg. 4-46 



11. Why must the carrier frequency be much 
greater than the signal frequency? 

12. Why is a higher frequency needed to transmit 
television signals than radio signals? (The highest 
frequency necessary to convey radio sound in- 
formation is about 12,000 Hz. The electron beam 
in a television tube completes one picture of 525 
lines in 0.03 sec, and the intensity of the beam 
should be able to vary several hundred times 
during a single line scan.) 



200 



UNIT 4 / LIGHT AND ELECTROMAGNETISM 



ACTIVITIES 



THIN FILM IIVTERFERENCE 

Press two clean microscope slides together. 
Look at the light they reflect from a source (like 
a mercury lamp or sodium flame) that emits 
light at only a few definite wavelengths. What 
you see is the result of interference between 
light waves reflected at the two inside surfaces 
that are almost, but not quite, touching. (The 
thin film is the layer of air between the slides.) 

The phenomenon can also be used to 
determine the flatness of surfaces. If the two 
inside surfaces are planes, the interference 
fringes are parallel bands. Bumps or depres- 
sions as small as a fraction of a wavelength of 
light can be detected as wiggles in the fringes. 
This method is used to measure very small 
distances in terms of the known wavelength of 
light of a particular color. If two very flat slides 
are placed at a slight angle to each other, an 
interference band appears for every wavelength 
of separation. (See Fig. 4-47.) 

How could this phenomenon be used to 
measure the thickness of a very fine hair or very 
thin plastic? 



^/> in+crftrsnce bo.iad« 



photos in Figs. 4-49 and 4-50 were produced 
with the setup diagrammed in Fig. 4-4S. 




Fig. 4-47 



HANDKERCHIEF DIFFRACTFION 
GRATING 

Stretch a linen or cotton handkerchief of good 
quality and look through it at a distant light 
source, such as a street light about one block 
away. You will see an interesting difTrartion 
pattern. (A window screen or cloth umbrella 
will also work.) 



PHOTOGRAPHING IMFFRACTFION 
PATTERNS 

Diffraction patterns like those pictured here 
can be produced in your lab or at home Tlie 




object 




It' 



Rg. 4-48 

To photograph the patterns, you must have a 
darkroom or a large, light-tight box. Figure 4-49 
was taken using a Polaroid 4x5 back on a 
Graphic press camera. The lens was remm-ed, 
and a single sheet of 3,000-ASA-speed Polaroid 
film was exposed for 10 sec; a piece of 
cardboard in front of the camera was used as a 
shutter. 




Fig 4-49 




Fig 4-50 



ACTIVITIES 



201 



As a light source, use a 1.5-volt flashlight bulb 
and AA cell. Turn the bulb so the end of the 
filament acts as a point source. A red (or blue) 
filter makes the fringes sharper. You can see the 
fringes by examining the shadow on the screen 
with the 10 X magnifier. Razor blades, needles, 
or wire screens make good objects. 



POISSON'S SPOT 

A bright spot can be observed in a photograph 
of the center of some shadows, like that shown 
in Fig. 4-51. To see this, set up a light source, 
obstacle, and screen as shown in Fig. 4-52. 
Satisfactory results require complete darkness. 
Tiy a 2-sec exposure with Polaroid 3,000-ASA 
fUm. 




Fig. 4-51 



)ight source 




SmOireB" cemented 
foglQSs slide 



Fig. 4-52 



PHOTOGRAPHIC ACTTIVITIES 

The number of photography activities is limit- 
less, so we shall not try to describe many in 
detail. Rather, this is a collection of suggestions 
to give you a "jumping-ofT' point for classroom 
displays, demonstrations, and creative work. 



History ctf Photography 

Life magazine, December 23, 1966, had an 
excellent special issue on photography. How 
the world's first trichromatic color photograph 
was made by James Clerk Maxwell in 1861 is 
described in the Science Study Series pa- 
perback, Latent Image, by Beaumont Newhall. 
Much of the early history of photography in the 
United States is discussed in Mathew Brady, by 
James D. Horan (Crown Publishers). 

Schlieren Photography 

For a description and instructions for 
equipment, see Scientific American, February, 
1964, pp. 132-133. 

Infrared Photography 

Try to make some photos like the one shown 
on page 513 of Unit 4 in the Text. Kodak infrared 
film is no more expensive than normal black 
and white film, and can be developed with nor- 
mal developers. If you have a 4 x 5 camera uath 
a Polaroid back, you can use 4x5 Polaroid 
infrared film sheets. You may find the Kodak 
Data Book M-3, "Infrared and Ultraviolet Pho- 
tography," very helpful. 



COLOR 

You can easily carry out many intriguing 
experiments and activities related to the physi- 
cal, physiological, and psychological aspects of 
color. Some of these are suggested here. 

Scattered Light 

Add about one-quarter teaspoon of milk to a 
drinking glass full of water. Set a flashlight 
about 60 cm away so that it shines into the 
glass. When you look through the milky water 
toward the light, it has a pale orange color. As 
you move around the glass, the milky water 
appears to change color. Describe the change 
and explain what causes it. 

The Rainbow Effect 

The way in which rainbows are produced 
can be demonstrated by using a glass of water 
as a large cylindrical raindrop. Place the glass 
on a piece of white paper in the early morning 
or late afternoon sunlight. To make the rainbow 
more visible, place two books upright, leaving a 
space a little wider than the glass between 
them, so that the sun shines on the glass but 
the white paper is shaded (Fig. 4-53). The rain- 
bow will be seen on the backs of the books. 
What is the relationship between the arrange- 



202 



UNIT 4 / LIGHT AND ELECTROMAGMETISM 




-^ii^-— 



Fig. 4-53 



ment of colors of the rainbow and the side of 
the glass that the light entered? This and other 
interesting optical effects are described in 
Science from Your Airplane Window, by Eliza- 
beth A. Wood (Dover, 1975, paperback). 

Color Vision by Contr€ist 
(Land Effect) 

Hook up two small lamps as shown in Fig. 
4-54. Place an obstacle in front of the screen so 
that adjacent shadows are formed on the 
screen. Do the shadows have any tinge of color? 
Now cover one bulb with a red filter and notice 
that the other shadow appears green by 
contrast. Try this wath different colored filters 
and vary the light intensity by moving the 
lamps to various distances. 



SOU/CtS 




obstacle 








F]g. 4-54 



POLARIZED LIGHT 

The use of polarized light in basic research is 
spreading rapidly in many fields of science. The 
laser, the most intense laboratory source of 
polarized light, was invented by researchers in 
electronics and microwaves. Botanists have 
discovered that the direction of growlh of 
certain plants can be determined by controlling 
the polarization form of illumination, iuid 
zoologists have found that bees, ants, and 
various other creatures routinely use the 
polarization of sky light as a navigational 
"compass." High-enei^ physicists hax-e found 
that the most modem particle accelerator, the 
synchrotron, is a superb source of polarized X 
rays. Astronomers find that the polarization of 
radio waves fixim planets and from stars offers 
important clues to the dvnamics of those 
bodies. Chemists and mechanical engineers are 
finding new uses for polarized light as an 
analytical tool. Theoreticians hav-e discov-ered 
shortcut methods of dealing with polarized 
light algebraically. From all sides, the onrush of 
new ideas is imparting new vigor to this 
classical subject. 

A discussion of many of these asp)ects of the 
nature and application of polarized light, 
including activities such as those discussed 
below, can be found in Polarized Light, by W. A. 
Shurcliff and S.S. Ballard l\'an Nostrand 
Momentum Book #7, 1964). 

Detection 

Polarized light can be detected directly by 
the unaided human e\e provided you know 
what to look for. To develop this ability', begin 
by staring through a sheet of Polaroid film at 
the sky for about 10 sec. Then quickly turn the 
polarizer 90° and look for a pale wllow 
brush-shaped pattern similar to Fig 4-55. 



Land Two-Color Demonstrations 

A different and interesting activity is to 
demonstrate that a full-color picture can be 
created by simultaneously projecting two 
black-and-white transparencies taken thn)ugh 
a red and a gnuMi filter For more infomiation 
see Scientific American. May 1959; September 
1959; and January 1960. 

The eye itself responds to color in a wav that 
is neither obvious nor rt)inpletely undenitood, 
even today. It does not work like a camera in 
this respect For a discussion oi a ctintem- 
poriiiy tiu'oty, see Scientific American , Uwem- 
ber, 1977. p. 108. 



^-iv!^: 



Rg. 4-5S 



1 P(MC«( ^ 



The color will fade in a few seconds, but 
another pattern v\ill ap[)ear when the Polaroid 
is .igaiti rotated 90". A light-blue filter l>ehind 
the Polan»i(i may help. 



ACTIVITIES 



203 



B.C. 



By John Htrt 



see CCLCf267HE SAM^. . 

y .1^ 




..U<E,..6PeENJ 
TbYbU MlCrtr LCOK 

ThIOUGM I CALL W 

GReeM too! 




By peraisslon of John Hare and Field Encerprlses Inc. 



How is the axis of the brush related to the 
direction of polarization of light transmitted by 
the Polaroid? iTo determine the polarization 
direction of the filter, look at light reflected 
from a horizontal nonmetallic surface, such as 
a tabletop. Turn the Polaroid until the reflected 
light is brightest. Put tape on one edge of the 
Polaroid parallel to the floor to show the 
direction of polarization.) Does the axis of the 
yellow pattern always make the same angle 
with the axis of polarization? 

Some people see the brush most clearly 
when viewed with circularly polarized light. To 
make a circular polarizer, place a piece of 
Polaroid in contact with a piece of cellophane 
with its axis of polarization at a 45° angle to the 
fine stretch lines of the cellophane. 

Picket-Fence Analogy 

At some time, you may have hecird the 
polarization of light explained in terms of a 
rope tied to a fixed object at one end, and being 
shaken at the other end. In between, the rope 
passes through two picket fences (as in Fig. 
4-56), or through two slotted pieces of 
cardboard. This analogy' suggests that when the 
slots are parallel the waves pass through, but 
when the slots are perpendicular the waves are 
stopped. (You may want to use a rope and 
slotted boards to see if this really happens.) 



completely. Then place a third filter between 
the first two, and rotate it about the axis of all 
three. What happens? Does the picket-fence 
cinalogy still hold? 

A similar experiment can be done with 
microwaves using parallel strips of tinfoil on 
cardboard instead of Polaroid filters. The elec- 
tric field in the microwaves is "shorted out," 
however, when the pickets are parallel to the 
field. This is just the opposite of the rope-and- 
fence analogy. Prove this for yourself. 



MAKE AN ICE LEXS 

Dr. Clawbonny, in Jules Verne's The Adventures 
of Captain Hatteras, was able to light a fire in 
—48° weather (thereby sa\ing stranded travel- 
ers) by shaping a piece of ice into a lens and 
focusing it on some tinder. If ice is clear, the 
sun's rays pass through with little scattering. 
You can make an ice lens by freezing water in a 
round-bottomed bowl. Use boiled, distilled 
water, if possible, to minimize problems due 
to gcis bubbles in the ice. Measure the focal 
length of the lens and relate this length to the 
radius of the bowl. (Adapted from Physics for 
Entertainment, Y. Perelman, Foreign Languages 
Publishing House, Moscow, 1936.1 




Fig. 4-56 

Place two Polaroid filters parallel to each 
other and turn one so that it blacks out the light 



DETECTING ELECTRIC FIELDS 

Many methods can be used to explore the 
shape of electric fields. Two very simple ones 
are described here. 

Gilbert's Versorium 

A sensitive electric "compass " is easily con- 
structed from a toothpick, a needle, and a cork. 
An external electric field induces surface 
charges on the toothpick. The forces on these 



204 



UNIT 4 / LIGHT AND ELECTROMAGNETISM 



induced charges cause the toothpick to line up 
along the direction of the field. 

To construct the versorium, first bend a flat 
toothpick into a slight arc. When it is mounted 
horizontally, the downward curve at the ends 
will give the toothpick stability by lowering its 
center of gravity below the pivot point of the 
toothpick. With a small nail, drill a hole at the 
balance point almost all the way through the 
pick. Balance the pick horizontally on the 
needle, being sure it is free to swing like a 
compass needle. Try bringing chained objects 
near it. 

For details of Gilbert's and other experi- 
ments, see Holton and Roller, Foundations of 
Modern Physical Science, Chapter 26. 

Charged Ball 

A charged pithball (or conductor-coated 
Styrofoam ball) suspended from a stick on a 
thin insulating thread can be used as a rough 
indicator of fields around charged spheres, 
plates, and wires. 

Use a point source of light to project a 
shadow of the thread and ball. The angle 
between the thread and the vertical gives a 
rough measure of the forces. Use the charged 
pithball to explore the nearly uniform field necir 
a large, chai-ged plate suspended by tape strips, 
and the l/r drop-off of the field near a long 
charged wire. 

Plastic strips rubbed with cloth are adequate 
for charging well-insulated spheres, plates, or 
wires. (To prevent leakage of the chaise from 
the pointed ends of a charged wire, fit the ends 
with small metal spheres. Even a smooth small 
blob of solder at the ends should help.) 



AN 11^ BATTERY 

Using a penny (95% copper) and a silver dime 
(90% silver) you can make an l\C battery. Cut a 
2.5-cm square of filter paper or paper towel, dip 
it in salt solution, and place it between the 
penny and the dime. Connect the penny and 
the dime to the terminals of a galvanometer 
with two lengths of copper wire Does your 
meter indicate a current.' Will the batterv also 
produce a current with the penny and dime in 
direct dry contact? 



voltaic: pile 

CaH 20 or more disks each of two difTerent 
metals. Copper and zinc make a good combina- 



tion. (The round metal "slugs " from electrical 
outlet-box installations can be used for zinc 
disks because of their heavy zinc coating. i 
Pennies and nickels or dimes will work, but not 
as well. Cut pieces of filter paper or paf>er towel 
to fit in between each pair of two metals in 
contact. Make a pile of the metal disks and the 
salt-water soaked paper, as V'olta did. Keep the 
pile in order for example, copf>er-pap>er-zinc, 
copper-paper-zinc, etc. Connect copper wires 
to the top and bottom ends of the pile. Touch 
the fi^e ends of the wires with two fingers of 
one hand. What is the effect? Can you incr^ease 
the effect by moistening your fingei^? In w+iat 
other ways can you increase the effect? How 
many disks do you need in order to light a 
flashlight bulb? 

If you have metal fillings in your teeth, try 
biting a piece of aluminum foil. Can you explain 
the sensation? 



MEASllUNG MAGNETIC FIELD 
INTENSITY 

Many important devices used in phv'sics exp>er- 
iments make use of a uniform magnetic field of 
known intensity. Cyclotir>ns, bubble chambers, 
and mass spectrometers are examples. Use the 
current balance described in Experiments 4-4 
and 4-6. Measure the magnetic field intensit>' in 
the space between the pole faces of two 
ceramic disk magnets placed close together 
Then when you are learning about radioacti\it>' 
you can observe the deflection of beta particles 
as they pass through this space, and determine 
the average enei^' of the particles 

Bend two strips of thin sheet aluminum or 
copper (not ironi. and tape them to twx) disk 
magnets as shown in Fig. 4-57. 




N^Q^r*»i 



l.75on 



Fig 4-57 



Loop on 
Curre/>t 



ACTIVITIES 



205 



Be sure that the pole faces of the magnets tire 
parcdlel and are attracting each other (unlike 
poles facing each other). Suspend the movable 
loop of the current balance midway between 
the pole faces. Determine the force needed to 
restore the balance to its initial position when a 
measured current is passed through the loop. 
You learned in Experiment 4-5 that there is a 
simple relationship between the magnetic field 
intensity', the length of the part of the loop that 
is in the field, and the current in the loop. It is 
F = BII, where F is the force of the loop (in 
newtons), B is the magnetic field intensity (in 
newtons per ampere-meter), / is the current 
(in amperes), and / is the length (in meters) of 
that part of the current-carryang loop that is 
actually in the field. With your current balance, 
you can measure F, /, and /, and thus compute 
B 

For this activity, make two simplifying as- 
sumptions that are not strictly true but which 
enable you to obtain reasonably good values for 
B: (a) the field is fairly uniform throughout the 
space between the poles, and (b) the field drops 
to zeix) outside this space. Ulth these ap- 
proximations you can use the diameter of the 
magnets as the quantity / in the above expres- 
sion. 

Try the same experiment with two disk 
magnets above and two below the loop. How 
does B change? Bend metal strips of different 
shapes so you can vary the distance between 
pole faces. How does this affect B? 

An older unit of magnetic field intensity still 
often used is the gauss iGi. To convert from 
one unit to the other, use the conversion fac- 
tor, 1 N/A (1 tesla, T) = 10^ G. 




D— H S-!) 



^ A r ';)C^ 




/'V 



<^ 



Rg.4-58 

to eight levers and are securely hinged to wheel 
E at the point marked F. Each magnet is also 
provided with a roller wheel, G, to prevent 
friction as it rolls on the guide marked C. 

Guide C is supposed to push each magnet 
toward the hub of this mechanism as it is being 
carried upward on the left-hand side of the 
mechanism. As each mcignet rolls over the top, 
the fixed magnets facing it cause the magnet on 
the wheel to fall over. This creates an overbal- 
ance of weight on the right of wheel E and thus 
perpetually rotates the wheel in a clockwise 
direction. 

In Fig. 4-59, A represents a w+ieel in uliich 
eight hollow tubes marked E tire placed. In 
each of the tubes a magnet, B, is inserted so 



MORE PERPETUAL MOTION 
MACHINES 

The diagrams in Figs. 4-58 amd 4-59 show two 
more of the perpetual motion machines dis- 
cussed by R. Raymond Smedile in his book. 
Perpetual Motion and Modem Research for 
Cheap Power. (See also page 150 of Unit 3, 
Handbook .) What is the weakness of the cirgu- 
ment for each of them? (Also see "Perpetual 
Motion Machines," Stanley W. Angrist, Scientific 
American, January, 1968.) 

In Fig. 4-58, A represents a stationary wheel 
around which is a larger, movable wheel, E. 
Three magnets mariced B are placed on station- 
ary wheel A in the position shown in the 
drawing. On rotary w+ieel E are placed eight 
magnets marked D. The magnets are attached 




Fig. 4-59 



ao6 



UNIT 4 / LIGHT AND ELECTROMAGNETISM 



that it will slide back and forth. D represents a 
stationaiy rack in which five magnets are 
anchored as shown in the drawing. ELach 
magnet is placed so that it will repel the 
magnets in wheel A as it rotates in a clockwise 
direction. Since the magnets in stationary rack 
D will repel those in rotaiy wheel A, this will 
cause a perpetual overbalance of magnet 
weight on the right side of w+ieel A. 



cube with all the N sides facing outward. The 
pieces rep>el each other strongly and may be 
either glued (with rubber cement) or tied 
together with thread. 

Do you now have an isolated north jxile, that 
is, a north pole all over the outside land south 
pole on the inside)? 

Is there a magnetic field directed outward 
from all surfaces of the cube? 



TRANSISTOR AMPLIFIER 

The function of a PNP or NPN transistor is very 
similar to that of a triode vacuum tube 
(although its operation is not so easily de- 
scribed). The diagram in Fig. 4-60 show^ a 
schematic transistor circuit that is analogous to 
the vacuum tube circuit showoi in Experiment 
4-8. In both cases, a small input signal controls 
a large output current. 




input 
Sianal 



Some inexpensive transistors can be bought 
at almost any radio supply store, and almost 
any PNP or NPN will do. Such stores also 
usually carry a variety of paperback books that 
give simplified explanations of how transistors 
work and how you can use cheap components 
to build some simple electronic equipment. 



AN ISOLATED NORTH 
MAGNETIC POLE? 

Magnets made of a rather soft rubber-like 
substance are available in some hardware 
stores. Typical magnets an? flat pieces 20 mm x 
25 nun and about 5 iniu thick, \%ith a magnetic 
north pole on one 20 x 25 mm surface and a 
south pole on the other They may be cut with a 
shaq) knife. 

Cut six of these magnets so that you haw six 
square pieces, 20 mm on an edge Then lew! 
the edges on the S side of each piece so that the 
pieces can be fitted together to form a hollow 



FARADAY DISK DYNAMO 

You can easily build a disk dynamo similar to 
the one shown in Fig. 4-61. Cut a 20-cm 
diameter disk of sheet copp>er Drill a hole in 
the center of the disk and put a bolt through 
the hole. Run a nut tight up against the disk so 
the disk will not slip on the bolt. Insert the bolt 
in a hand drill and clamp the drill in a ring 
stand so that the disk passes through the 
region between the poles of a large magnet. 
Connect one wire of a 100-microamp l/iA) dc 
meter to the metal part of the drill that does not 
turn. Tap>e the other wire to the magnet so it 
brushes lightly against the copper disk as the 
disk is spun between the magnet p>oles. 




Fig. 4-61 



Frantic cranking can create a IO-/1A current 
with the magnetron magnet shown in Fig 4-61 
If vou use one of the metal vokes from the 
cum*nt iKilant^. with three ceramic magnets 
on each side of the yoke, \ou may Ix* able to get 
the needle to move from the zero position just 
noticeably 



ACTIVITIES 



207 



The braking effect of currents induced in the 
disk can also be noticed. Remove the meter, 
wires, and magnet. Have one person crank 
while another holds the magnet so that the disk 
is spinning between the magnet poles. Com- 
pare the efifort needed to turn the disk vvath and 
without the magnet over the disk. 

If the disk will coast, compare the coasting 
times with and without the magnet in place. (If 
there is too much friction in the hand drill for 
the disk to coast, loosen the nut and spin the 
disk by hand on the bolt.) 



GENERATOR JUMP ROPE 

With a piece of wire about twice the length of a 
room, and a sensitive galvanometer, you can 
generate an electric current using only the 
earth's magnetic field. Connect the ends of the 
wire to the meter. Pull the wire out into a long 
loop and twirl half the loop like a jump rope. As 
the wire cuts across the earth's meignetic field, 
a voltage is generated. If you do not have a 
sensitive meter, connect the input of one of the 
amplifiers, and connect the amplifier to a less 
sensitive meter. 



T 




rxiU 



How does the current generated when the 
axis of rotation is along a north- south line 
compeire with the current generated with the 
same motion along an east- west line? What 
does this tell you about the ecirth's magnetic 
field? Is there any effect if the people stand on 
two landings and hang the wire (while swing- 
ing it) down a stairwell? 

SIMPLE METERS AND MOTORS 

You can make workable current meters and 
motors from very simple parts: 

2 ceramic magnets 
1 steel yoke 
1 #7 cork 

1 metal rod, about 2 mm in diameter 
and 12 cm long (a piece of bicycle 



(from current 
bcilance kit) 



(for meter 
only) 



spoke, coat-hanger wire, or a large 
finishing nail will do) 

1 block of wood, about 10 cm x 5 cm x 1 cm 
about 2.7 m of insulated #30 copper magnet 

wire 

2 thumbtacks 
2 safety pins 

2 carpet tacks or small nails 

1 white card (10 cm x 12.5 cm) 

stiff black paper, for pointer 

electrical insulating tape (for motor only) 

Meter 

To build a meter, follow the steps below paying 
close attention to the diagremis. Push the rod 
through the cork. Mcike the rotating coil, or 
armature, by winding about 20 turns of wire 
around the cork, keeping the turns parallel to 
the rod. Leave about 30 cm of wire at both ends 
(Fig. 4-62). 



\- 




Fig. 4-62 

Use nails or ccirpet tacks to fix two safety pins 
firmly to the ends of the wooden-base block 
(Fig. 4-63). 




Fig. 4-63 



Make a pointer out of the black paper and 
push it onto the metal rod. Pin a piece of the 
white card to one end of the base. Then 



208 



UNIT 4 / LIGHT AND ELECTROMAGNETI8M 



suspend the armature between the two safety 
pins from the free ends of wire into two loose 
coils, and attach them to the base with 
thumbtacks. Put the two ceramic magnets on 
the yoke (unlike poles facing), and place the 
yoke around the armature (Fig. 4-64). Clean the 
insulation off the ends of the leads, and you are 
ready to connect your meter to a low-voltage dc 
source. 




-gAlS to 

voltage o:\irci. 



Fig. 4-64 



Calibrate a scale in volts on the white card 
using a variety of known voltages from dry cells 
or from a low-voltage power supply, and your 
meter is complete. Minimize the parallcix 
problem by having your pointer as close to the 
scale as possible. 

Motor 

To make a motor, wind an armature as you did 
for the meter. Leave about 6 cm of wire at each 
end; carefully scrape the insulation from the 
wire. Bend each into a loop and then twist into 
a tight pigtail. Tape the two pigtails along 
opposite sides of the metal rod (Fig. 4-65). 




Fig. 4-66 



Fix the two safety pins to the base as for the 
meter, and mount the coil between the safety 
pins. 



The leads into the motor iire made fhsm two 
pieces of wire attached to the baseboard with 
thumbtacks at points X (Fig. 4-66). 



=<=-H=--- 




I — 



No Coiitact 




Fig. 4-66 

Place the magnet yoke around the coil. The 
coil should spin freely (Fig. 4-67). 




FJg. 4-67 



Connect a l.S-x-olt batter>' to the leads Start 
the motor by spinning it with N-our finger If it 
does not start, check the contacts between 
leads and the contact wires on the rod. You 
may not have remo\'ed all the enamel from the 
wires Tr>' pressing lightly at points A iFig. 4-66) 
to miprow the contact .-Vlso check to see that 
the two contacts touch the armature wires at 
the same time (Fig. 4-67). 

SI.MPI.r MOTOR- GENERATOR 
DEMONSTRATION 

With two fairiy .stn)ng I'-magents and two coils, 
which vou wind \'ourself, vou can prepare a 
simple demonstration showing the principles 



ACTIVITIES 



209 



of a motor and a generator. Wind two flat coils 
of magnet wire 100 turns each. The cardboard 
tube from a roll of paper towels makes a good 
form. Leave about 0.5 m of wire free at each end 
of the coil. Tape the coil so it does not wind 
when you remove it from the cardboard tube. 
Hang the coils frtjm two supports as shown 
(Fig. 4-68) so the coils pass over the poles of two 
U-magnets set on the table about 1 m apart. 
Connect the coils. Pull one coil to one side and 
release it. What happens to the other coil? 
Why? Does the same thing happen if the coils 
are not connected to each other? What if the 
magnets are reversed? 



Connectors 




Fig. 4-68 



Try vcirious other changes, such as turning 
one of the magnets over while both coils £ire 
swinging, or starting with both coils at rest and 
then sliding one of the magnets back and forth. 

If you have a sensitive galvanometer, it is 
interesting to connect it between the two coils. 



PHYSICS CXILLAGE 
I33SZES] 



1 ■"> 




SITIVE 

^ Research 



• Strike 



IL 



>iOoa 

M33M S S 




Many of the words used in physics class enjoy 
wide usage in everyday language. Cut "physics 
words" out of magazines, newspapers, etc., and 
make your ov\ti collage. You may wish to take 
on a more challenging art problem by trying to 
give a visual representation of a physical 
concept, such as speed, light, or waves. 



BICYCLE 
GENERATOR 

The generator on a bicycle (Fig. 4-69) operates 
on the same basic principle as that described in 
the Text, but with a different, and extremely 
simple, design. Take apart such a generator and 
see if you can explain how it works. Note: You 
may not be able to reassemble it. 




Fig. 4-69 



LAPIS POLARIS, 
MAGNES 

The etching in Fig. 4-70 shows a philosopher in 
his study surrounded by the scientific equip- 
ment of his time. In the left foreground, in a 
basin of water, a natural magnet or lodestone 
floating on a piece of wood orients itself north 
£ind south. Traders from the great Mediterra- 
nean port of Amalfi probably introduced the 
floating compass, having learned of it from Arab 
mariners. An Amalfi historian, Flavius Blondus, 
writing about A.D. 1450, indicates the uncertJiin 
origin of the compass, but later historians in 
repeating this early reference warped it and 
gave credit for the discoveiy of the compass to 
Flavius. 



210 



UNIT 4 / LIGHT AND ELECTROMAGNETISM 




LAPIS POLARIS, M \ N K 

Lapis mlujU ifle Flauio abdttum ^oU Juum huru arutrm 



Fig. 4-70 

Can you identify the various devices lying 
around the study? When do you think the 
etching was made? (If you have some back- 
ground in art, you might consider whether 
your estimate on the basis of scientific clues is 
consistent with the style of the etching.) 

MICROWAVE TRANSMISSION 
SYSTEMS 

Microwaves of about 6-cm wavelength are used 
to transmit telephone conversations over long 
distances. Because microwave radiation has a 
limited range, a series of relay stations has been 
erected about 50 km apart. At each station the 
signal is detected anti amplified before being 
retransmitted to the ne.xt one. If you have 
several microwave generators that can be 
amplitude modulated, see if you can put 
together a demonstration of how this system 
works. You will need an audio frequency 
oscillator (or microphone), amplifier, mic- 
rowave generator and povve'r supply, detector 
another amplifier and a loudspeaker; another 
microwaw generator, and another detector, a 
third amplifier, and a loudspeaker 



GOOD READING 

Several good paperbacks in the Science Study 
Series (Anchor Books. Doubleday and Col are 
appropriate for Unit 4, including The Physics of 
Tele\ision, by Donald G. Fink and David M. 
Lulyens; Waves and Messages, by John R. 
Pierce; Quantum Electronics, by John R Pierce; 
Electrons and \\'a\-es, by John R Pierce: Com- 
puters and the Human Mind, by Donald G Fink. 
Throughout this course, you should make a 
point of checking ynur library' for books or 
articles on topics that interest \x)u 




FILM LOOP NOTES 



211 



FILM LOOP NOTES 



Film Loop 44 

STANDING ELECTROMAGNETIC 

WAVES 

Standing waves are not confined to mechanical 
waves in strings or in gas. It is only necessary to 
reflect the wave at the proper distance from a 
source so that two oppositely moving waves 
superpose in just the right way. In this film, 
standing electromagnetic waves are generated 
by a radio transmitter. 

The transmitter produces electromagnetic 
radiation at a frequency of 435 x lo^ Hz. Since 
all electromcignetic waves travel at the speed of 
light, the wavelength is A. = cff = 0.69 m. The 
output of the transmitter oscUlator (Fig. 4-71) 
passes through a power-indicating meter, then 
to an antenna of two rods each one-quarter 
wavelength (Vi \), or a total antenna of Vi k, 
long. 





1 


]_e_ 

Power 
Meter 


Trtinsmit+'ng 

Dipole 
Antenna 






1 (n) 


® 


mm 

\ (N) 


o 


05cilla1 
455" IC 


or 


Receiving 

Dipole 

Antenna 



Fig. 4-71 

The receiving antenna (Fig. 4-72) is also V2 \ 
long. The receiver is a flashlight bulb connected 
between two stiff wares each V4 X long. If the 
electric field of the incoming wave is parallel to 
the receiving antenna, the force on the elec- 
trons in the wire drives them back and forth 
through the bulb. The brightness of the bulb 
indicates the intensity of the electromagnetic 
radiation at the antenna. A rectangular 
aluminum cavity, open toward the camera, 
confines the waves to provide sufficient inten- 
sity. 

Initial scenes show how the intensity de- 
pends on the distance of the receiving antenna 
from the transmitting antenna. The radiated 




Rg. 4-72 

power is about 20 watts. Does the received 
intensity decrease as the distance increases? 
The radiation has vertical polarization, so the 
response falls to zero when the receiving 
antenna is rotated to the horizontal position. 

Standing waves are set up when a metal 
reflector is placed at the right end of the cavity. 
The reflector must be placed at a node. The 
distance from source to reflector must be an 
integral number of half-wavelengths plus one- 
quarter wavelength. The cavity length must be 
"tuned" to the wavelength. Nodes and an- 
tinodes are identified by moving a receiving 
antenna back and forth. Then a row of vertical 
receiving antennas is placed in the cavity, and 
the nodes and antinodes are shown by the 
pattern of brilliance of the lamp bulbs. How 
many nodes and antinodes can be seen in each 
trial? 

Standing waves of different types can aU have 
the same wavelength. In each case, a source is 
required (tuning fork, loudspeaker, or dipole 
antenna). A reflector is also necessaiy (support 
for string, wooden piston, or sheet aluminum 
mirror). If the frequencies are 72 Hz for the 
string, 505 Hz for the gas, and 435 x 10* Hz for 
the electromagnetic waves and they all have the 
same wavelength, what can you conclude 
about the speeds of these three kinds of waves? 
Discuss the similarities and differences among 
the three cases. What can you say about the 
"medium" in which the electromagnetic waves 
travel? 




Models of the Atom 

EXPERIMENTS 



212 



Experiment 5-1 
ELECTROLYwSIS 

Volta and Davy discovered that electric cur- 
rents create chemical changes never observed 
before. As you have already learned, these 
scientists were the first to use electricity to 
break down apparently stable compounds and 
to isolate certain chemical elements. 

Later, Faraday and other experimenters 
compared the amount of electric charge used 
with the amount of chemical products formed 
in such electrochemical reactions Their mea- 
surements fell into a regular pattern that hinted 
at some underlying link between electricity and 
matter. 

In this experiment, you will use an electric 
current just as Volta and Daw did to decom- 
pose a compound By comparing the charge 
used with the mass of one of the pn)ducts. you 
can compute the mass and volume of a single 
atom of the product. 

Tlioon Behind the Lxperiiiieiit 

A l)raker of copper sulfate (CuSO^I solution in 
wat«'r is suppoHed under one arm of a lialancr 



(Fig. 5-1). A negatively charged copper electrode 
is supported in the solution by the balance arm 
so that you can measure its mass without 
removing it from the solution .\ second 
positively charged copper electrode fits around 
the inside wall of the beaker. The beaker, its 
solution, and the positi\e electrode are not 
supported by the balance arm. 

If you have studied chemistiy. \x>u pmb>ably 
know that in solution copper sulfate breaks 
down into separate, char>;ed particles, called 
ions, of copper (Cu**) and sulfate (SO«"l, which 
move about freely in the solution. 

Utien a voltage is applied across the copper 
electrodes, the electric field causes the SC)4" 
ions to drift to the positiw electrode lor ancxlel 
and the Cu'* ions to drift to the negative 
electrode (or cathode). At the cathode, the Cu* * 
particles acquire enough negative charge to 
form neutral copper atoms that dejxisit on the 
cathode and add to its weight Hie motion of 
charged particles lowartl the electnxles is a 
continuation of the electric current in the wires 
and the rate of transfer of charge (coulombs per 
second) is equal to it in magnitude. The electric 
current is pm\ided bv a power suppK that 



EXPERIMENTS 



213 




power supply 



Fig. 5-1 



converts a lOO-V alternating current into a 
low-voltage direct current. The current is set by 
a variable control on the power supply (or by an 
external rheostat) and measured by an amme- 
ter in series with the electrolytic cell as shovvTi 
in Fig. 5-1. 

With a watch to measure the time the current 
flows, you can compute the electric charge that 
passed through the cell. By definition, the 
current / is the rate of transfer of charge, 
/ = AQ/Ar. It follows that the charge trans- 
ferred is the product of the current and the 
time. 



AQ = / X At 



(coulombs = 



coulombs 



X sec) 



Since the amount of charge carried by a 
single electron is known (qe = 1-6 x lO"'^ C), 
the number of electrons transferred, N^, is 

If n electrons are needed to neutralize each 
copper ion, then the number of copper atoms 
deposited, Ncu> is 

n 
If the mass of each copper atom is mcu, then 
the total mass of copper deposited, Mq^, is 

Mcu =A/c-u"icu 
Thus, if you measure /, Af, and Mqu and you 
know Qe and n, you can calculate a veilue for 
mcu/ the mass of a single copper atom! 



Combining the above equations algebraically 
gives 

/ Af 

Setup and Procedure 

Either an equal-arm or a triple-beam balemce 
C£m be used for this experiment. First arrange 
the cell cind the balance as shown in Fig. 5-1. 
The cathode cylinder must be supported far 
enough above the bottom of the beaker so that 
the balance arm can move up and down freely 
when the cell is full of the copper sulfate 
solution. 

Next connect the circuit as illustrated in the 
figure. Note that the electrical connection fipom 
the negative terminal of the power supply to 
the cathode is made through the balance beam. 
The knife-edge and its seat must he bypassed 
by a short piece of thin flexible wire, as shown 
in Fig. 5-1 for equal-arm balances, or in Fig. 5-2 
for triple-beam balances. The positive terminal 
of the power supply is connected directly to 
the anode in any convenient manner. 

Before any measurements are made, operate 
the cell long enough (10-15 min) to form a 
preliminary deposit on the cathode unless this 
has already been done. In any case, run the 
current long enough to set it at the value 
recommended by your instructor, probably 
about 5 A. 

When all is ready, adjust the balance and 
record its reading. Pass the current for the 
length of time recommended by your instruc- 



214 



UNIT 5 / MODELS OF THE ATOM 




copper ions, Cu**. At the cathode the copper 
ions are neutralized by electrons and neutral 
copper atoms are deposited: Cu** + 2e" = Cu. 



2. How many electrons were required to neu- 
tralize the total charge transferred? (Each electron 
carries - 1.6 ^ 10 '» C.) 

3. How many electrons (single negative charges) 
were required to neutralize each copper ion? 

4. How many copper atoms were deposited? 

5. What is the mass of each copper atom? 

6. The mass of a penny is about 3 g. If it were 
made of copper only, how many atoms would a 
penny contain? (In fact, modem pennies contain 
zinc as well as copper.) 

7. The volume of a penny is about 0.3 cm'. How 
much volume does each atom occupy? 



Fig. 5-2 This cutaway view shows how to bypass the 
knife-edge of a typical balance. The structure of other 
balances may differ. 



tor. Measure and record the current / and the 
time interval Ar during which the current 
passes. Check the ammeter occasionally and, if 
necessary, adjust the control in order to keep 
the current set at its original value. 

At the end of the run, record the new reading 
of the balance and find, by subtraction, the 
increase in mass of the cathode. 

Calculating Mass and 
Volume of an Atom 

Since the cathode is buoyed by a liquid, the 
masses you have measured are not the true 
masses. Because of the buoyant force exerted 
by the liquid, the mass of the cathode and its 
increase in mass will both appear to be less 
than they would be in air. To find the true mass 
increase, you must divide the observed mass 
increase by the factor (1 - DjDc). where D, is 
the density of the solution and D^ is the density 
of the copper. 

Your instructor will give you the values of 
these two densities if you cannot find values for 
them yourself, and also explain how the 
correction factor is derived. The important 
thing for you to understand here is why a 
coiTPrtion factor is necos.sarv. 



1. How much positive or negative charge was 
transferred to the cathode? 



In the solution, this positive charge is carried 

from anode to cathode b\ doubly charged 



Experiment 5-2 

THE CHARGE.TO-.\IASS RATIO 

FOR AN ELECTRON 

In this experiment, you will make mea- 
surements on cathode rays. A set of similar 
experiments by J.J. Thomson convinced physi- 
cists that these rays are not wav-es but streams 
of identical, charged particles, each with the 
same ratio of charge to mass. If you did the 
experiment on the "Electron Beam Tube," you 
have already worked with cathode rax-s and 
ha\'e seen how they can be deflected by electric 
and magnetic fields. 

Thomson's use of this deflection is described 
on page 543 of L'nit 5, Text. Read that section 
of the Text before beginning this experiment. 

Theory- of the Experiment 

The basic plan of the experiment is lo measure 
the bending of the electron beam by a known 
magnetic field. From these measurements and 
a knowledge of the voltage accelerating the 
electrons, you can calculate the electron 
charge-to-mass ratio The reasonitig behmd the 
calculation is illustrated in Fig. 5-3. The alge- 
braic steps are described belo%v 

Wlien the beam of electrons (each of mass m 
and charge q^i is bent into a circular arc of 
radius fl by a uniform magnetic field R. the 
centripetal force m\Vfl on each electron i> 
supplied by the magnetic force Bq,* Therefon- 

or, rearrariging to get v by itself. 



EXPERIMENTS 



218 




Fig. 5-3 The combination of two relationships, for 
centripetal and kinetic energy, with algebraic steps that 
eliminate velocity, v, lead to an equation for the 
charge-to-mass ratio of an electron. 



Fig. 5-4 



The electrons in the beam are accelerated by 
a voltage V, which gives them a kinetic energy 

If you replace v in this equation by the 
expression for v in the preceding equation, you 
get 

or, after simplifying, 

Qe - 2V 

You can measure with your apparatus all the 
quantities on the right-hand side of this 
expression, so you can use it to calculate the 
charge-to-mass ratio for an electron. 



Preparing the Apparatus 

You will need a tube that gives a beam at least 5 
cm long. If you kept the tube you made in 
Elxperiment 4-7, you may be able to use that. If 
your class did not have success with that 
experiment, you will have to use another 
method. 

In this experiment, you need to be able to 
adjust the strength of the magnetic field until 
the magnetic force on the charges just balances 
the force due to the electric field. To enable you 
to change the magnetic field, you will use a 
pair of coils instead of permanent magnets. A 
current in a pair of coils, which are separated 
by a distance equal to the coil radius, produces 
a nearly uniform magnetic field in the central 
region between the coils. You can vary the 
magnetic field by changing the current in the 
coils. 

Into a cardboard tube about 7.5 cm in 
diameter and 7.5 cm long, cut a slot 3 cm wide 
(Fig. 5-4). Your electron beam tube should fit 



into this slot as shown in the photograph of the 
completed setup (Fig. 5-5). A current in the pair 
of coils will create a magnetic field at right 
angles to the axis of the cathode rays. 




Fig. 5-5 The magnetic field is parallel to the axis of the 
coils; the electric and magnetic fields are perpendicular 
to each other and to the electron beam. 



Now wind the coUs, one on each side of the 
slot, using a single length of insulated copper 
wire (magnet wire). Wind about 20 turns of wire 
for each of the two coils, one coil on each side 
of the slot, leaving 25 cm of wire free at both 
ends of the coil. Do not cut the wire off the reel 
untQ you know how much you will need. Make 
the coils as neat as you can and keep them 
close to the slot. Wind both coUs in the same 
direction (for example, make both clockwise). 

When you have made your set of coils, you 
must calibrate it; that is, you must find out 
what magnetic field strength B corresponds to 
what value of current / in the coils. To do this, 
you can use the current balance, as you did in 
Experiment 4-5. Use the shortest of the balance 
"loops" so that it will fit inside the coils as 
shown in Fig. 5-6. 



^ 



216 



UNIT S I MODELS OF THE ATOM 



t«_J 




Fig. 5-6 



Connect the two leads from your coils to a 
power supply capable of giving up to 5 A of 
direct current. There must be a variable control 
on the power supply (or a rheostat in the 
circuit) to control the current; and an ammeter 
to measure it. 

Measure the force F for a current / in the 
loop. To calculate the magnetic field due to the 
current in the coils, use the relationship F - Bll 
w+iere / is the length of short section of the 
loop. Do this for several different values of 
current in the coil and plot a calibration graph 
of magnetic field B against coil current /. 

Set up your electron beam tube as in 
Experiment 4-7. Reread the instructions for 
operating the tube. 

Connect a shorting ware between the pins for 
the deflecting plates. This will insure that the 
two plates are at the same electric potential, so 
the electric field between them will l>e zero. 
Pump the tube out and adjust the filament 
current until you have an easily visible beam. 
Since there is no field between the plates, the 
electron beam should go straight up the center 
of the tube between the two plates (If it does 
not, it is probably because the filament and the 
hole in the anode are not properly aligned.) 

Turn down the filament current and sv\ilch 
off the power supply. Now, without releasing 
the vacuum, mount the coils around the tube 
as shown in Fig. 5-6. 

Connect the coils as before to the power 
supply. Connect a voltmeter across the power 
supply tenninals that provide the accelerating 
voltage V. 

Your apparatus is now complete. 

Fert'oniiing the Experiment 

Turn on Ihe beam and make surt> it is travelling 
in a straight line Ihe electric field remains off 



hroughout the experiment, and the deflecting 
olates should still be connected. 

Turn on and slowly increase the current in 
the coils until the magnetic field is strong 
enough to deflect the electron beam noticeably. 

Record the current / in the coils. 

Using the calibration graph, find the mag- 
netic field B. 

Record the accelerating voltage V between 
the filament and the anode plate. 

Finally, you need to measure R , the radius of 
the arc into which the beam is bent by the 
magnetic field. The deflected beam is slightly 
fan-shaped because some electrons are slowed 
by collisions with air molecules and are bent 
into a curve of smaller radius fl. You need to 
know the largest value of fl (the "outside" edge 
of the curved beam I, which is the path of 
electrons that have made no collisions. You will 
not be able to measure fl directly, but you can 
find it from measurements that are easy to 
make (Fig. 5-7). 

You can measure x and d. It follows from 
Pythagoras' theorem that fl*=d* + (fl-Ac)*, so 

2ji 




\ 



n 



Al 






6 pofx*' 



R- 



Hg S-7 



EXPERIMENTS 



217 



1. What is your calculation of R on the basis of 
your measurements? 



Now that you have values for V, B, and fl, you 
can use the formula qjm = 2V/B^fl^ to 
calculate your value for the charge-to-mass 
ratio for an electron. 



2. What is your value for qJm, the charge-to- 
mass ratio for an electron? 



Experiment 5-3 

THE MEASUREMENT OF 

ELEMENTARY CHARGE 

In this experiment, you wall investigate the 
charge of the electron, which is a fundamental 
physical constant in electricity, electromag- 
netism, and nuclear physics. This experiment 
is substantially the same as MiUikan's famous 
oil-drop experiment, described on page 547 of 
Unit 5, Text. The following instructions assume 
that you have read that description. Like 
MiUikan, you are going to measure very small 
electric charges to see if there is a limit to how 
small an electric charge can be. Try to answer 
the following three questions before you begin 
to do the experiment in the lab. 



surements. Fortunately, you can now use 
suitable objects whose sizes are accurately 
known. You wall use tiny latex spheres (about 
10~^ cm diameter), which are almost identical 
in size in any given sample. In fact, these 
spheres, shown magnified (about 5,000 x) in 
Fig. 5-8, are used as a convenient way to find 
the magnifying power of electron microscopes. 
The spheres can be bought in a water suspen- 
sion, with their diameter recorded on the 
bottle. When the suspension is sprayed into the 
air, the water quickly evaporates and leaves a 
cloud of these particles, which have become 
charged by friction during the spraying. In the 
space between the plates of the Millikan 
apparatus, they appear through the 50-power 
microscope as bright points of light against a 
dark background. 




Fig. 5-8 Electron micrograph of latex spheres 1.1x10^ 
cm, silhouetted against diffraction grating of 11,340 
lines/cm. What magnification does this represent? 



1. What is the electric field between two parallel 
plates separated by a distance d meters if the 
potential difference between them is V volts? 

2. What is the electric force on a particle 
carrying a charge of q coulombs in an electric 
field of E volts/meter? 

3. What is the gravitational force on a particle of 
mass m in the earth's gravitational field? 



Background 

Electric charges are measured through the 
forces they experience and produce. The 
extremely small charges that you cire seeking 
require that you measure extremely small 
forces. Objects on which such small forces can 
have a visible efifect must also, in turn, be very 
small. 

Millikan used the electrically charged drop- 
lets produced in a fine spray of oil. The varying 
size of the droplets complicated his mea- 



You will find that an electric field between 
the plates can pull some of the particles 
upward against the force of gravity, so you will 
know that they are charged electrically. 

In your experiment, you will adjust the 
voltage producing the electric field until a 
particle hangs motionless. On a balanced 
particle carrying a charge q, the upward 
electric force Eq and the downward gravita- 
tional force mag cire equal; therefore, 
mag = Eq 

The field E = V/d, where V is the voltage 
between the plates (the voltmeter reading) and 
d is the separation of the plates. It follows that 

rnaM 

q = =— 

^ V 

Notice that magd is a constant for all 
measurements and need be found only once. 



218 



UNIT S / MODELS OF THE ATOM 



Each value of q will be this constant ma^ times 
1/V as the equation above shows. That is, the 
value of q for a particle is proportional to 1/V; 
the greater the voltage required to balance the 
weight of the particle, the smaller the chaise of 
the particle must be. 

Using the Apparatus 

If the apparatus is not already in operating 
condition, consult your teacher. Study Figs. 5-9 
and 5-10 until you can identify the \arious 
parts. Then switch on the light source and look 
through the microscope. You should see a 
series of lines in clear focus against a uniform 
gray background. 




Fig. 5-9 A typical set of apparatus. Details may vary 
considerably. 



Tlie lens of the light source may fog up as the 
heat from the lamp drives moisture out of the 
light-source tube. If this happens, remov-e the 
lens and wipe it on a clean tissue Wait for the 
tube to warm up thoroughly before replacing 
the lens. 

Squeeze the bottle of latex suspension two or 
three times until five or ten particles drift into 
view. You will see them as tiny bright spots af 
light. You may have to adjust the focus slightly 
to see a specific particle clearly Notice how the 
particle appears to move upward. The \iew is 
inverted by the microscojje; the particles are 
actucilly falling in the earth's gravitational field. 

Now switch on the high voltage across the 
plates by turning the switch up or dovMi. Notice 
the effect on the particles of varying the electric 
field by means of the voltage-control knob. 

Notice the effect when you reverse the 
electric field by re\'ersing the switch position. 
(When the switch is in its mid-position, there is 
zero field between the plates.) 



4. Do all the particles move in the same direc- 
tion when the field is on? 

5. How do you explain this? 

6. Some particles move much more rapidly in 
the field than others. Do the rapidly moving 
particles have larger or smaller charges than the 
slowly moving particles? 



To chomber 



To voltrnete 



(^versing 
s^vitch 



I I lo enamour 
Pi\ rl"^ yellow 




voltoge 
control 



To power supply 



Fig 5-10 A typical arrangement of connections to the 
high-voltage reversing switch. 



Sometimes a few particles cling together, 
making a clump that is easy to see. The clump 
falls more rapidly than single particles when 
the electric field is off. Do not tI^' to use these 
clumps for measuring q. 

Tr>' to balance a particle by adjusting the 
field until the particle hangs motionless Ob- 
serve it carefully to make sure it is not slowly 
drifting up or down. The smaller the chaiige, 
the greater the electric field must be to hold up 
the* particle. 

Taking Data 

It is n(»t worth working at voltages much t>elow 
50 \ Only highly charged particles can be 
balanced in these small fields and nou are 
interested in obtaining the least charge 
possible. 

Set the potential difTereni-e Iwtweon the 
plates to about 75 \' Rext^rse the field a few 
times so that the more quickly moving particles 
(those with the greater rhargei are swept out of 
the field of view Any particles that remain have 



EXPERIMENTS 



219 



low charges. If no particles remain, squeeze in 
some more and look again for some with small 
charges. 

When you have isolated one of these parti- 
cles carrying a low charge, adjust the voltage 
carefully until the particle hangs motionless. 
Observe it for some time to make sure that it is 
not moving up or down very slowly, and that 
the adjustment of voltage is as precise as 
possible. (Because of uneven bombardment by 
air molecules, there will be some slight, uneven 
drift of the particles.) 

Read the voltmeter. Then estimate the preci- 
sion of the voltage setting by seeing how little 
the voltage needs to be changed to cause the 
particle to start moving just perceptibly. This 
small change in voltage is the greatest amount 
by which your setting of the balancing voltage 
can be uncertain. 

When you have balanced a particle, make 
sure that the voltage setting is as precise as you 
can make it before you go on to another 
particle. The most useful range to work uathin 
is 75- 150 V, but try to find particles that can be 
brought to rest in the 200- 250 V range too, if 
the meter can be used in that range. Remember 
that the higher the balancing field the smaller 
the charge on the particle. 

In this kind of an experiment, it is helpful to 
have large amounts of data. This usually makes 
it easier to spot trends and to distinguish main 
effects fix)m the background scattering of data. 
Thus, you may wish to contribute your findings 
to a class data pool. Before doing that, however, 
arrange your values of V in a vertical column of 
increasing magnitude. 



7. Do the numbers seem to clump together in 
groups, or do they spread out more or less evenly 
from the lowest to the highest values? 



If you would like to make a more complete 
quantitative analysis of the cIeiss results, calcu- 
late an average vcilue for each of the highest 
three or four clumps of V values in the class 
histogram. Next change those values to values 
of 1/V and list them in order. Since q is 
proportional to l/V, these values represent the 
magnitude of the charges on the particles. 

To obtain actual values for the charges, the 
lA's must be multiplied by magd. The separa- 
tion d of the two plates, typically about 5 mm, 
or 5 X 10"^ m, is given in the specification 
sheets provided by the manufacturer. You 
should check this. 

The mass m of the spheres is worked out 
from a knowledge of their volume and the 
density D of the matericil they are made from. 
Mass = volume x density, or m = */3H x D. 
The sphere diameter (2r) h<is been previously 
measured and is given on the supply bottle. 
The density D is 1,077 kg/m' (found by 
measuring a large batch of latex before it is 
made into little spheres). 



9. What is the spacing between the observed 
average values of 1/1/, and what is the difference 
in charge that corresponds to this difference in 
MV? 

10. What is the smallest value of MV that you 
obtained? What is the corresponding value of q? 

11. Do your experimental results support the idea 
that electric charge is quantized? If so, what is 
your value for the quantum of charge? 

12. If you have already measured qjm in Experi- 
ment 4-9, compute the mass of an electron. Even 
if your value differs from the accepted value by a 
factor of 10, perhaps you will agree that its 
measurement is a considerable intellectual 
triumph. 



Now combine your data with that collected 
by your classmates. This can conveniently be 
done by placing your values of V on a class 
histogram. When the histogram is complete, 
the results can easily be transferred to a 
transparent sheet for use on an overhead 
projector. Alternatively, you may wish to take a 
Polaroid photograph of the completed histog- 
ram for inclusion in vour laboratory notebook. 



8. Does your histogram suggest that all values 
of q are possible and that electric charge is, 
therefore, endlessly divisible, or the converse? 



Experiment 5-4 

THE PHOTOELECTRIC EFFECT 

In this experiment, you will make observations 
of the effect of light on a metal surface; then you 
will compare the appropriateness of the wave 
model and the particle model of light for 
explaining what you observe. 

Before doing the experiment, read Text Sec. 
18.4 (Unit 5) on the photoelectric eflfect. 

How the Apparatus W'^orks 

Light that you shine through the window of the 
phototube falls on a hiilf-cylinder of metcil 



220 



UNIT S / MODELS OF THE ATOM 



K,.fc^-!B 



« 





LICHT SOURCE 



AMPUCitR 



PHOTOELECTRIC 



utite-\os«. 



TUBE 



Fig. 5-11 



called the emitter. The light drives electrons 
from the emitter surface. 

Along the axis of the emitter (the center of the 
tube) is a wire called the collector. When the 
collector is made a few volts positive with 
respect to the emitter, practically all the 
emitted electrons are drawn to it, and will 
return to the emitter through an external wire. 
Even if the collector is made slightly negative, 
some electrons wall reach it and there will be a 
measurable current in the external circuit. 

The small current cam be amplified several 
thousand times and detected in any of several 
different ways. One way is to use a small 
loudspeaker in which the amplified photoelec- 
tric current causes an audible hum; another is 
to use a cathode-ray oscilloscope. The follow- 
ing description assumes that the output cur- 
rent is read on a microammeter (Fig. 5-11). 

The voltage control knob on the phototube 
unit allows you to vary the voltage between 
emitter and collector. In its full coun- 
terclockwise position, the voltage is zero. As 
you turn the knob clockwise the photocurrent 
decreases. You are making the collector more 
and more negative and fewer and fewer elec- 
trons get to it. Finally the photorunvnt ceases 
altogether; all the electnins are turned bark 
before reaching the collector The voltage 
between emitter and collector that just stops all 
the electrons is called the stopping vnltaf^r. The 
value of this voltage indicates the maximum 
kinetic energy v\ith which the elertnnis lea\e 
the emitter. To iuul the value of the stopping 
voltage precisely, you \mI1 ha\e to be able to 







Fig. 5-12 However much the details may differ, any 
equipment for the photoelectric effect experiment will 
consist of these basic parts. 



determine precisely when the photocurrent is 
reduced to zero Because there is some drift of 
the amplifier output, the current indicated on 
the meter will drift around the zero point e\-en 
when the actual current remains exactly zero 
Therefore you will have to adjust the amplifier 
offset occasionally to be sure the zero level is 
really zero. .\n altemati\-e is to ignore the 
precise reading of the cumMit meter and adjust 
the collector voltage until turning the light off 
and on causes no detectable change in the 
current. Turn up the negati\-e collector x-oltage 
until blocking the light from the lube (with 
black paperi has no effect on the meter reading. 
The exact location of the meter pointer is not 
imporiant. 

rhe position of the \oltage-control knob at 
the curri'nt cutoff gi\vs_NX)u a rough measure of 
stopping voltage. To measure it mor« precisely. 
connect a voltmeter as shown in Fig 5-13. 

In the exfx'riment. yxtu will measure the 
stopping xdltaji^es as ligtit of diffenMit fmjuen- 
cies falls on the phototul>e CkkmI colored filters 
will allow light of only a certain range of 



EXPERIMENTS 



221 




to 
Vo Itroeteir^ 



Fig. 5-13 

frequencies to pass through. You can use a 
hand spectroscope to find the highest fre- 
quency line passed by each filter. The filters 
select frequencies from the mercury spectrum 
emitted by an intense mercury lamp. Useful 
frequencies of the mercury spectrum are: 

yellow 5.2 x lO'Vsec 

green 5.5 X lO'Vsec 

blue 6.9 X lO'Vsec 

violet 7.3 X lO'Vsec 

(ultraviolet) 8.2 x lO'Vsec 

Doing the Experiment 
PART I 

The first part of the experiment is qualitative. 
To see if there is a time delay between light 
falling on the emitter and the emission of 
photoelectrons, cover the phototube and then 
quickly remove the cover. Adjust the light 
source and filters to give the smallest photocur- 
rent that you can conveniently notice on the 
meter. 



1. Can you detect any time delay between the 
moment that light hits the phototube and the 
moment that the motion of the microammeter 
pointer (or a hum in the loudspeaker or deflection 
of the oscilloscope trace) signals the passage of 
photoelectrons through the phototube? 



To see if the current in the phototube 
depends on the intensity of incident light, vary 
the distance of the light source. 



To find out whether the kinetic energy of the 
photoelectrons depends on the intensity of the 
incident light, measure the stopping voltage 
with different intensities of light falling on the 
phototube. 

? 

3. Does the kinetic energy of the photoelectrons 
depend on intensity, that is, does the stopping 
voltage change with light intensity? 

Finally, determine how the kinetic ener^ of 
photoelectrons depends on the frequency of 
incident light. You will remember (Text Sec. 
18.5) that the maximum kinetic energy of the 
photoelectrons is Vstopf/e' where Vs,op is the 
stopping voltage, and q^ - 1.60 x 10"'^ C, the 
charge on an electron. Measure the stopping 
voltage wdth various filters over the uandow. 

*? 

4. How does the stopping voltage and, thus, the 
kinetic energy change as the light is changed from 
red through blue or ultraviolet (no filters)? 



PART U 

In the second part of the experiment you will 
make more precise measurements of stopping 
voltage. To do this, adjust the voltage-control 
knob to the cutoff (stopping voltage) position 
and then measure V with a voltmeter (Fig. 5-13). 
Connect the voltmeter only after the cutoff 
adjustment is made so that the voltmeter leads 
will not pick up any ac voltage induced from 
other conducting wares in the room. 

Measure the stopping voltage \^^^ for three 
or four different light frequencies, and plot the 
data on a graph. Along the vertical axis, plot 
electron energy Vs,opqie- When the stopping 
voltage V is in volts, and q^ is in coulombs, Vq^ 
wall be energy, in joules. 

Along the horizontal axis, plot the frequency 
of light/. 

Interpretation of Results 

As suggested in the opening paragraph, you 
can compare the wave model of light and the 
particle model in this experiment. Consider, 
then, how these models explain your observa- 
tions. 



2. Does the number of photoelectrons emitted 
from the sensitive surface vary with light inten- 
sity, that is, does the output current of the 
amplifier vary with the intensity of the light? 



5. If the light striking your phototube acts as 

waves, 

(a) can you explain why the stopping voltage 

should depend on the frequency of light? 



zzz 



UNIT S / MODELS OF THE ATOM 



(b) would you expect the stopping voltage to 
depend on the Intensity of the light? Why? 

(c) would you expect a delay between the tinne 
that light first strikes the ennitter and the emission 
of photoeiectrons? Why? 

6. If the light is acting as a stream of particles, 
what would be the answer to questions a, b, and c 
above? 



If you drew the graph suggested in Part II of 
the experiment, you should now be prepared 
to interpret the graph. It is interesting to recall 
that Einstein predicted its form in 1905, and by 
experiments similar to yours, Millikan verified 
Einstein's prediction in 1916. 

Einstein's photoelectric equation (Text Sec. 
18.5) describes the energy of the most energetic 
photoeiectrons (the last ones to be stopped as 
the voltage is increased), as 

= hf-W 

This equation has the form 
y =k}c — c 
In this equation c is a constant, the value of 
y at the point where the straight line cuts the 
vertical axis; and k is another constant, namely 
the slope of the line. (See Fig. 5-14.) Therefore, 
the slope of a graph of V,u,pqe against / should 
be/i. 




Fig 5-15 



With the equipment you used, the slop>e is 
unlikely to agree with the accepted value of h 
(6.6 X 10"** J/sec) more closely than an order of 
magnitude. Perhaps you can give a few reasons 
why your agreement cannot be closer. 

The lowest frequency at which any electrons 
are emitted from the cathode surface is called 
the threshold frequency, /q. At this frequency 
Vinrvmax - and hfo = W, where W is the work 
function . Your experimentally obtained value of 
W is not likely to be the same as that found for 
very clean cathode surfaces, more carefully 
filtered light, etc. The important thing to notice 
here is that there is a value of W, indicating that 
there is a minimum energy needed to release 
photoeiectrons from the emitter. 




Fig. 5-14 



7. What IS the value of the slope of your graph' 
How well does this value compare with the value 
of Planck's constant, h 6.6 • 10 ** J sec? (See 
Fig. 5-15.) 



? 

8. Einstein's equation was derived from the 
assumption of a particle (photon) model of light. If 
your results do not fully agree with Einstein's 
equation, does this mean that your experiment 
supports the wave theory? 



Experiment 5-5 
SPECTROSCOPY 

In Text Chapter 19 you learned of the immense 
importance of spectra to an understanding of 
nature. You are about to observT" the spectra of 
a variet>' of light sources to see for yourself how 
spectra differ from each other and to leam how- 
to measure the wavelengths of spectrum lines. 
In particular, vou will measure the wa\-elengths 
of the hydrogen spectrum and relate them to 
the stnictun* of the hydrogen atom. 

Before \'ou begin, re\iew carefully Sec. 19.1 of 
Text Chapter 19. 

Cre.itinj( Spectra 

Materials ran be made to gi\« ofT light (or be 
"excited"! in sewral different ways: by heating 



EXPERIMENTS 



223 



in a flame, by an electric spark between 
electrodes made of the material, or by an 
electric current through a gas at low pressure. 

The light emitted can be dispersed into a 
spectrum by either a prism or a diflraction 
grating. 

In this experiment, you will use a diflraction 
grating to examine light from various sources. A 
diffraction grating consists of many very fine 
parallel grooves on a piece of glass or plastic. 
The grooves can be seen under a 400-power 
microscope. 

In Experiment 4-2 (Young's Experiment), you 
saw how two narrow slits spread light of 
different wavelengths through different angles, 
and you used the double slit to make approxi- 
mate measurements of the wavelengths of light 
of different colors. The distance between the 
two slits was about 02 mm. The distance 
between the lines in a diffraction grating is 
about 0.002 mm. A diffraction grating may have 
about 10,000 grooves instead of just two. 
Because there are more lines and they are 
closer together, a grating diffracts more light 
and separates the different wavelengths more 
than a double slit, aind can be used to make 
veiy accurate measurements of wavelength. 

Observing Spectra 

You can observe diffraction when you look at 
light that is reflected from a phonograph 
record. Hold the record so that light from a 
distant source is eilmost parallel to the record's 
surface, as in Fig. 5-16. Like a diffraction grating, 
the grooved surface disperses light into a 
spectrum. 






Fig. 5-16 

Use a recil diffraction grating to see spectra 
simply by holding the grating close to your eye 
with the lines of the grating parallel to a 
distant light source. Better yet, arrange a slit 
about 25 cm in fhant of the grating, as shown in 
Fig. 5-17, or use a pocket spectroscope. 

Look through the pocket spectroscope at a 
fluorescent light, at an ordinary (incandescent) 
light bulb, at mercury-vapor and sodium-vapor 
street lamps, at neon signs, at light from the sky 
(but do not look directly at the sun), and at a 
flame into which various compounds are 




Fig. 5-17 

introduced (such as salts of sodium, potassium, 
strontium, barium, and calcium). 



1. Which color does the grating diffract into the 
widest angle and which into the narrowest? Are 
the long wavelengths diffracted at a wider angle 
than the short wavelengths, or vice versa? 

2. The spectra discussed in the Text are (a) either 
emission or absorption, and (b) either line or 
continuous. What different kinds of spectra have 
you observed? Make a table showing the type of 
spectra produced by each of the light sources you 
observed. Do you detect any relationship between 
the nature of the source and the kind of spectrum 
it produces? 

Photographing the Spectrum 

A photograph of a spectrum has several 
advantages over visual observation. A photo- 
graph reveals a greater range of wavelengths; 
also, it allows* greater convenience for your 
measurement of wavelengths . 

When you hold the grating up to your eye, 
the lens of your eye focuses the diffracted rays 
to form a series of colored images on the retina. 
If you put the grating in front of the camera lens 
(focused on the source), the lens will produce 
sharp images on the film. 

The spectrum of hydrogen is particularly 
interesting to measure because hydrogen is the 
simplest atom and its spectrum is feiirly easily 
related to a model of its structure. In this 
experiment, hydrogen gas in a glass tube is 
excited by an electric current. The electric 
discharge separates most of the Hj molecules 
into single hydrogen atoms. 

Set up a meter stick just behind the tube (Fig. 
5-18). This is a scale against which to observe 
and measure the position of the spectrum 
lines. The tube should be placed at about the 
70-cm mark since the spectrum viewed 
through the grating will appear nearly 70 cm 
long. 



224 



LNIT 5 / MODELS OF THE ATOM 




Fig. 5-18 

From the camera position, look through the 
grating at the glowing tube to locate the 
positions of the visible spectral lines cigainst the 
meter stick. Then, with the grating fastened 
over the camera lens, set up the camera with its 
lens in the same position your eye was. The 
lens should be aimed perpendicularly at the 
50-cm mark, and the grating lines must be 
parallel to the source. 

Now take a photograph that shows both the 
scale on the meter stick and the spectral lines. 
You may be able to take a single exposure for 
both, or you may have to mcike a double 
exposure: first the spectrum, and then, with 
more light in the room, the scale. It depends on 
the amount of light in the room. Consult your 
instructor. 

Analv'zing the Spectrum 

Count the number of spectral lines on the 
photograph, using a magnifier to help pick out 
the faint ones. 



3. Are there more lines than you can see when 
you hold the grating up to your eye? If you do see 
additional lines, are they located in the visible part 
of the spectrum (between red and violet) or in the 
infrared or ultraviolet part? 



The angle through which light is diffracted 
by a grating depends on the wavelength X of the 
light and the distance d between lines on the 
grating. The formula is a simple one: K = d sin 

e. 

To find 6, you need to find tan 6 = }c./l as 
shown in Fig. 5-19. Here )i. is the distance of 
the spectral line along the meter stick from the 
source, and / is the distance from the source to 
the grating. Use a magnifier to read ,< from your 
photograph. Calculate tan 6, and then look up 
the corresponding values of 6 and sin 6 in 
trigonometric tables. 

To find d, remember that the grating space is 
probably given as lines per centimeter. You 
must convert this to the distance between lines 
in meters. One centimeter is 1.00 x lo * m, so if 
there are 5,275 lines per centimeter, then d is 
11.00 X 10^1/(0.52 X 10^1 = 1.92 x lo • m. 

Calculate the values of X for the various 
spectral lines you ha\-e measured. 



4. How many of these lines are visible to the eye? 

5. What would you say is the shortest wavelengtti 
to which your eye is sensitive? 

6. What is the shortest wavelength that you can 
measure on the photograph? 






X < 



4* a^orrut po'^.fioO 



iA 






t»„ G 



imoae. at Sj^<.^ 



( fo*.< tA 







\ ( /»*w>j« of Co Jftt. 



Fig. 5-19 Different images of the source are formed on 
the film by different colors of diffracted light The angle 
of diffraction is equal to the apparent angular dis- 



placement angle of the source in the photograph, so 



EXPERIMENTS 



225 



Compare your values for the wavelengths 
with those given in the Text, or in a more 
complete list (for instance, in the Handbook of 
Chemistry and Physics). The differences be- 
tween your values and the published ones 
should be less than the experimental uncer- 
tainty of your measurement. Are they? 

This is not all that you can do with the 
results of this experiment. You could, for 
example, work out a value for the Rydberg 
constant for hydrogen (mentioned in Text Sec. 
19.2). 

More interesting perhaps is to calculate 
some of the energy levels for the excited 
hydrogen atom. Using Planck's constant ih — 
6.6 X 10"^), the speed of light in vacuum (c = 
3.0 X 10* m/sec), and your measured Vcilue of 
the wavelength X of the separate lines, you can 
calculate the energy of vcirious wavelengths of 
photons (£ — hf = hc/k) emitted when 
hydrogen atoms change from one state to 
another. The energy of the emitted photon is 
the difference in energy between the initial and 
final states of the atom. 

Make the assumption (which is correct) that 
for all lines of the series you have observed, the 
final energy state is the same. The energies that 
you have calculated represent the energy of 
various excited states above this final level. 



Draw an energy-level diagram something like 
the one shown in Fig. 5-20. Show on it the 
energy of the photon emitted in transition from 
each of the excited states to the final state. 



f. 



f, 



r 

r 
i 

; ''^rooncJ 5 We" for 3 
i HalMicr 'ttatjsitibDj 



n =5 



lou.es't' «oCr<jy strf+e. 
of hydrogen octom — 



n=( 



Fig. 5-20 



7. How much energy does an excited hydrogen 
atom lose when it emits red light? 



236 



UNIT S / MODELS OP THE ATOM 



ACTIVITIES 



DALTON^ PUZZLE 

Once Dalton had his theory to work with, the 
job of figuring out relative atomic masses and 
empirical formulas became nothing more than 
working through a series of puzzles. Here is a 
very similar kind of puzzle with which you can 
challenge your classmates. 

Choose three sets of objects, each having a 
different mass. Large ball bearings with masses 
of about 70, 160, and 200 g work well. Let the 
smallest one represent an atom of hydrogen, 
the middle-sized one an atom of nitrogen, and 
the large one an atom of oxygen. 

From these "atoms" construct various 
"molecules." For example, NH3 could be repre- 
sented by three small objects and one middle- 
sized one; NjO by two middle-sized ones and 
one large, and so forth. 

Conceal one molecule of your collection in 
each one of a series of covered Styrofoam cups 
(or other lightweight, opaque containers). Mark 
on each container the symbols (but not the 
formula!) of the elements contained in the 
compound. Dalton would have obtained this 
information by qualitative analysis. 

Give the covered cups to other students. 
Instruct them to measure the "molecular" mass 
of each compound and to deduce the relative 
atomic masses and empirical formulas from the 
set of masses, making Dalton's assumption of 
simplicity. If the objects you have used for 
"atoms ' are so light that the mass of the 
Styrofoam cups must be taken into account, 
you can either supply this information as part 
of the data or leave it as a complication in the 
problem. 

If the assumption of simplicity is relaxed, 
what other atomic masses and molecular 
formulas would be consistent with the data? 



ELECrrROLYSIS 
OF UA lEK 

The fact that electricity can decompose water 
was an amazing and exciting discovery, yet the 
process is one that you can easily demonstrate 
with materials at your disposal. Figure 5-21 
provides all thr necessary infonnation. Set up 
an electrolysis apparatus and demonstrate the 
pn)cess for your classmates 

In the sketch it looks as if al>out twice as 
many bubbles are coming from one electrode 



as from the other. Which electixxie is it? Does 
this happen in your apparatus? Would you 
expect it to happ>en? 

How would you collect the two gases that 
bubble off the electrodes? How could you prove 
their identity? 




Fig. 5-21 



If water is really just two gases "put together" 
chemically, you should be able to put the gases 
together again and get back the water with 
wtiich you started. Using your knowledge of 
physics, predict what must then happ>en to all 
the electrical energy you sent flowing through 
the water to sep>arate it. 



SIXGLE-ELECTTRODE 
PLATING 

A student asked if copper would "plate out" 
from a solution of copper sulfate if only a 
negati\'e electrode were placed in the solution. 
It was tried and no copper was obserx'ed even 
when the electrode was connected to the 
negative terminal of a high-voltage source for 5 
min. Another student suggested that only a 
very small ( invisible 1 amount of copper was 
deposited since copper ions should be at- 
tracted to a negative electrode. 

A more precise test was devised. A nickel 
sulfate solution wius made containing several 
micpociiries of radit)active nickel mo radiocop- 
per was available' A single cartxin electrode 
was immersed in the solution, and cx)nnected 
to the negative terminal of the high-vxillage 
source again for 5 min. The electrode was 
removed, dried, and tested with a Geiger 
counter The rod was slightly radioactive A 
control test was run using identical test 
contlitions exiepl that no eUMtrical connec- 
tion was made to the electrode. The control 
showed more radioactrvitv. 



J 



ACTIVITIES 



227 



Repeat these experiments and see if the 
effect is true generally. What explanation would 
you give for these effects? (Adapted from Ideas 
for Science Investigations, N.S.T.A., 1966). 



Complete instructions appeeir in the PSSC 
Physics Laboratory Guide, Second Edition, D.C. 
Heath Company, Elxperiment IV-12, "The Mass 
of the Electron," pp. 79- 81. 



ACTIVITIES FROM 
SCIENTIFIC AMERICAN 

Articles from the "Amateur Scientist" section of 
Scientific American often relate to Unit 5. These 
articles range widely in difficulty. In your 
libraiy, scan the index of Scientific American 
for the past few years for such topics as 
accelerators, cloud chambers, spectroscopy, 
etc. 



WRITINGS BY OR ABOUT 
EUVSTEIN 

In addition to his scientific works, Einstein 
wrote many perceptive essays on other areas of 
life that are easy to read and are still very 
timely. The chapter titles from Out of My Later 
Years (Citadel, 1973) indicate the scope of these 
essays: Convictions and Beliefs; Science; Public 
Affairs; Science and Life; Personalities; My 
People. This book includes his writings from 
1934 to 1950. The best (and most inexpensive) 
book of Einstein's writings is Albert Einstein, 
Ideas and Opinions (Dell Publishing Co., 1973). 
The most scholarly biography is Philip Frank, 
Einstein, His Life and Times (Knopf, 1947). 

Albert Einstein: Philosopher-Scientist , edited 
by P. Schilpp (Library of Li\ing Philosophers, 
Vol. 7, 1973) contains Einstein's autobiographi- 
Ccil notes, left-hand pages in German and 
right-hand pages in EngUsh, and essays by 12 
physicist contemporaries of Einstein's about 
various aspects of his woric. An informative and 
very readable book on Einstein is Albert 
Einstein: Creator and Rebel by Banesh 
Hofifinann in collaboration with Helen Dukas, 
Einstein's secretary for neariy 30 years (Viking 
Press, 1972). Also see Einstein (Penguin, 1976, 
paperback) by Jeremy Bernstein. 



MEASURING q/m FOR THE 
ELECTRON 

With the help of a 'tuning eye" tube such as 
you may have seen in radio sets, you can 
measure the charge-to-mass ratio of the elec- 
tron in a way that is veiy close to J.J. 
Thomson's original method. 



CATHODE RAYS IN A 
CROOKES TUBE 

A Crookes tube ha\ing a metal barrier inside it 
for demonstrating that cathode rays travel in 
straight lines may be available in your 
classroom. In use, the tube is excited by a Tesla 
coil or induction coU. 

Use a Crookes tube to demonstrate to the 
class the deflection of cathode rays in meignetic 
fields. To show how a magnet focuses cathode 
rays, bring one pole of a strong bar magnet 
toward the shadow of the cross-shaped obsta- 
cle near the end of the tube. Watch wtiat 
happens to the shadow as the magnet gets 
closer and closer to it. What happens when you 
switch the poles of the magnet? What do you 
think would happen if you had a stronger 
magnet? 

Can you demonstrate deflection by an elec- 
tric field? Try using static charges as in 
Experiment 4-3, 'Electric Forces. I," to create a 
deflecting field. Then, if you have an electro- 
static generator, such as a small Van de Gra<iff 
or a Wimshurst machine, try deflecting the rays 
using parallel plates connected to the 
generator. 

LIGHTING AN ELECTRIC LAMP 
WITH A MATCH 

Here is a trick with which you can challenge 
your friends. It illustrates one of the many 
cimusing and useful applications of the photo- 
electric effect in real life. You will need the 
phototube from Experiment 5-4 "The Photo- 
electric Effect," together with the Project 
Physics Amplifier and Power Supply. You will 
also need a 1.5-V diy cell or power supply and a 
6-V light source such as the one used in the 
MUlikan apparatus. (If you use this light source, 
remove the lens and cardboard tube and use 
only the 6-V lamp.) Mount the lamp on the 
photoelectric-effect apparatus and connect it 
to the 0.5-V, 5 A variable output on the power 
supply. Adjust the output to maximum. Set the 
transistor switch input switch to switch. 

Connect the photoelectric-effect apparatus 
to the amplifier as shown in Fig. 5-22. Notice 
that the poIarit\' of the 1.5-V cell is reversed and 



228 



UNIT 5 / MODELS OF THE ATOM 



, Amp/ifi'er | 



(» ^ 



? ° i 



J 






nil 






(,V bulo 



' ((0© 



n-f ■•' r,„. 



Fig. 5-22 

that the output of the amplifier is connected 
to the transistor switch input. 

Advance the gain control of the amplifier to 
maximum, then adjust the offset control in a 
positive direction until the filament of the 6-V 
lamp ceases to glow. Ignite a match near the 
apparatus (the wooden type works the best) 
and bring it quickly to the window of the 
phototube while the phosphor of the match is 
still glowing brightly (Fig. 5-23). The phosphor 
flare of the match head will be bright enough to 
cause sufficient photocurrent to operate the 
transistor swatch that turns on the bulb. Once 
the bulb is lit, it keeps the photocell activated 
by its own light; you can remove the match and 
the bulb will stay lit. 



pinch out the wick. When your fingers pass 
between the bulb and the photocell, the bulb 
turns off, although the filament may glow 
a little, just as the wick of a freshly snuffed 
candle does. You can also make a "candle 
snuffer" from a little cone of any reasonably 
opaque material and use this instead of your 
fingers. Or you can "blow out" the bulb. It will 
go out obediently if you take care to remove it 
from in front of the photocell as you blow it out. 



X RAYS FROM 

A CROOKES XrBE 

To demonstrate that X rays penetrate materials 
that stop visible light, place a sheet of 10 cm x 
12.5 cm 3,000-ASA-speed Polaroid Land film, 
still in its protective pap>er jacket, in contact 
with the end of the Crookes tube. (A film pack 
cannot be used, but any other photographic 
film in a light-tight paper en\'elop>e could be 
substituted.) Support the film on books or the 
table so that it does not move during the 
exposure. The photo in Fig. 5-24 is from a 1-min 
exposure using a hand-held Tesla coil to excite 
the Crookes tube. 




Fig. 5-23 



VVlien you an* demonstrating tlii.s »<fl»'(t. tell 
your audience thai the bulb is n'ally a candle 
and thai it sliould nol sur-|)rise Ihrm that vou 
can light il with a inalcli ()t course, one way to 
put out a candle is to moisten your fingers and 



sriEivnsTs on stamps 

Scienlisls an* often pictured on Ihe stamps of 
many countries, often b<»ing honored by coun- 
lrii«s other than their homelands ^ on ma\ 
want to visit a stamp shop and assemble a 
display for your classroom. 



ACTIVITIES 



229 




MEASURING IONIZATION: 
A QUANTUM EFFECT 

With an inexpensive thyratron 884 tube, you 
can demonstrate an effect that is closely related 
to the famous Franck- Hertz effect. 

Theory 

According to the Rutherford- Bohr model, an 
atom can absorb and emit energy only in 
certain amounts that correspond to permitted 
"jumps" between states. 

If you keep adding energy in lai^er and larger 
"packages," you will finally reach an amount 
large enough to separate an electron entirely 
firom its atom, that is, to ionize the atom. The 
energy needed to do this is called the ioniza- 
tion energy. 

Now imagine a beam of electrons being 
accelerated by an electric field through a region 



of space filled with argon atoms. This is the 
situation in a thyratron 884 tube with its grid 
and anode both connected to a source of 
variable voltage, as shown schematically in Fig. 
5-25. 



V 



-V 



/\k 



-f-^ 







grid . 

cathcxlt-\ 

filomir* 

Fig. 5-25 



In the form of its kinetic energy, each 
electron in the beam carries energy in a single 
"package." The electrons in the beam collide 
with argon atoms. As you increase the ac- 
celerating voltage, the electrons eventually 
become energetic enough to excite the atoms, 
as in the Franck- Hertz effect. However, your 
equipment is not sensitive enough to detect the 
resulting small energy absorptions, so nothing 
seems to happen. The electron current from 
cathode to anode appears to increase quite 
linearly with the voltage, as you would expect, 
until the electrons get up to the ionization 
energy of argon. This happens at the ionization 
potential Vj, which is related to the ionization 
energy Ej and to the charge qe on the electron 
as follows: 

£, = qeV, 

As soon as electrons begin to ionize argon 
atoms, the current increases sharply. The argon 
is now in a different state, called an ionized 
state, in which it conducts electric current 
much more easily than before. Because of this 
sudden decrease in electrical resistance, the 
thyratron tube can be used as an "electronic 
swatch" in such devices as stroboscopes. lA 
similar process ionizes the air so that it can 
conduct lightning.) As argon ions recapture 
electrons, they emit photons of ultraviolet and 
of visible violet light. When you see this violet 
glow, the argon gas is being ionized. 

For theoretical purposes, the important 
point is that ionization takes place in any gas at 
a particular energy that is characteristic of that 
gas. This is easily observed evidence of one 
special case of Bohr's postulated discrete 
energy states. 



230 



UNIT 6 / MODELS OF THE ATOM 



Equipment 

thyratron 884 tube 

octal socket to hold the tube (not essential but 

convenient) 
voltmeter (0- 30 V dc) 
ammeter (0- 100 mA) 
potentiometer (10,000 f), 2 W or lai^er) or 

variable transformer, 0- 120 V ac 
power supply, capable of delivering 50- 60 mA 

at 200 V dc 
Connect the apparatus as shown schematically 
in Fig. 5-25. 

Procedure 

With the potentiometer set for the lowest 
available anode voltage, turn on the power and 
wait a few seconds for the filament to heat. Now 
increase the voltage by small steps. At each new 
voltage, call out to your partner the voltmeter 
reading. Pause only long enough to permit your 
partner to read the ammeter and to note both 
readings in your data table. Take data as rapidly 
as accuracy permits: Your potentiometer will 
heat up quickly, especially at high currents. If it 
gets too hot to touch, turn the power off and 
wait for it to cool before beginning again. 

Watch for the onset of the violet glow. Note in 
your data table the voltage at which you first 
observe the glow, and then note what happens 
to the glow at higher voltages. 

Plot current versus voltage, and mark the 
point on your graph where the glow first 
appeared. From your graph, determine the first 
ionization potential of argon. Compare your 
experimental value with published values, such 
as the one in the Handbook of Chemistry and 
Physics . 

What is the amount of energy an electron 
must have in order to ionize an argon atom? 



MODELING ATOiMS HITH 
MAGIVETS 

Here is one easy way to demonstrate some 
of the important differences between the 
Thomson "raisin-pudding" atom model and 
the Rullu'rford nuclear nuidcl. 

To show how alpha particles wouiil l)c 
expected to Iwliave in collisions with a Ihoin- 
son atom, repn«sent the spivad-out "pudding 
of positive change by a roughly circular ar- 
rangement of small disk magnets, spaced 
10- 12.5 cm apart, under the center of a 
smooth tray, as shown in Fig 5-26 I'se tape or 
putty to fasten the magnets to the undei-side of 




Fig. 5-26 The arrangement of the magnets tor a 
"Thomson atom " 



the tray. Put the large magnet (representing the 
alpha particle) down on top of the tray in such 
a way that the large magnet is repelled by the 
small magnets. Sprinkle onto the tray enough 
tiny plastic beads to make the large magnet 
slide freely. Now push the alpha particle" from 
the edge of the tray toward the "atom." As long 
as the "alpha particle" has enough momentum 
to reach the other side, its deflection by the 
small magnets under the tray will be quite 
small: never more than a few degrees. 

For the Rutherford model, on the other hand, 
gather all the small magnets into a vertical stack 
under the center of the tray, as shown in Fig. 
5-27. Turn the stack so that it repels "alpha 
particles" as before This tiucleus of positKe 
charge" now has a much greater effect on the 
path of the "alpha pjuticle." 






Fig. 5-27 The arrangement o< the magnets (or a 
"Rutherford atom " 



Haw a fiartner ta(M> an unknown array of 
magnets to the bottom of the tray: can nyiu 



ACTIVITIES 



231 



determine the arrangement just by scattering 
the large magnet? 

With this magnet analogue you can do some 
quantitative work with the scattering relation- 
ships that Rutherford investigated. (See Text 
Sec. 19.3 and the notes on Film Loop 47, 
"Rutherford Scattering" in this Handbook.) Try 
again with different sizes of magnets. Devise a 
launcher so that you can control the velocity of 
your projectile magnets and the distance of 
closest approach. 

(1) Keep the initicil projectile velocity v con- 
stant and vary the distance b (see Fig. 5-28); 
then plot the scattering angle </> versus b. 

nucleus 



alpha. 

porticle 



Rg. 5-28 



(2) Hold b constant and vary the speed of the 
projectile, then plot </> versus v. 

(3) Try scattering hard, nonmagnetized disks 
off of each other. Plot (/) versus b and 4> versus v 
as before. Contrast the two kinds of scattering- 
angle distributions. 



"BLACK-BOX" ATOMS 

Place two or three different objects, such as a 
battery, a small block of wood, a bar magnet, or 
a ball bearing, in a small box. Seal the box, and 
have one of your fellow students tiy to tell you 
as much about the contents as possible, 
without opening the box. For example, sizes 
might be determined by tilting the box, relative 
masses by balancing the box on a support, or 
w+iether or not the contents are magnetic by 
checking with a compass. 

The object of all this is to get a feeling for 
what you can or cannot infer about the 
structure of an atom purely on the basis of 
secondary evidence. It may help you to write a 
report on your investigation in the form you 
may have used for writing a proof in plane 
geometry, with the property of the box in one 
column and your reason for asserting that 
the property is present in the other column. 



The analogy can be made even better if you 
are exceptionally imaginative: Do not let the 
guesser open the box, ever, to find out what 
is really inside. 



STANDING WAVES ON A 
BAND-SAW BLADE 

Standing waves on a ring can be shown by 
shaking a band-saw blade with your hand. 
Wrap tape around the blade for about 15 cm to 
protect your hand. Then gently sh£ike the blade 
up and down until you have a feeling for the 
lowest vibration rate that produces reinforce- 
ment of the vibration. Double the rate of 
shaking, and continue to increase the rate of 
shaking, watching for standing waves. You 
should be able to maintain five or six nodes. 



TURNTABLE OSOLLATOR 
PATTERNS RESEMBLING 
DE BROGLIE WAVES 

If you set up two turntable osciUators and a 
Variac as shown in Fig. 5-29, you can draw 
pictures resembling de Broglie waves, like those 
shown in Chapter 20 of the Text. 




Fig. 5-29 

Place a paper disk on the turntable. Set both 
turntables at their lowest speeds. Before start- 
ing to draw, check the back-and-forth motion of 
the second turntable to be sure the pen stays 
on the paper. Turn both turntables on and use 
the Variac as a precise speed control on the 
second turntable. Your goal is to get the pen to 
follow exactly the same path each time the 
paper disk goes around. Try higher frequencies 
of back-and-forth motion to get more 
wavelengths around the circle. For each sta- 
tionary pattern that you get, check whether the 
back-and-forth frequency is an integral multi- 
ple of the circular frequency. 



232 



UNIT S / MODELS OF THE ATOM 



STANDING WAVES IN A 
WIRE RING 

With the apparatus described here, you can set 
up circular waves that somewhat resemble the 
de Broglie wave models of certain electron 
orbits. You will need a strong magnet, a fairly 
stiff wire loop, a frequency oscillator, and a 
power supply with a transistor chopping 
switch. 

The output current of the oscillator is much 
too small to interact with the magnetic field 
enough to set up visible standing waves in the 
wire ring. However, the oscillator current can 
operate the transistor switch to control 
("chop") a much larger current from the power 
supply (see Fig. 5-30). 




Fig. 5-30 The signal from the oscillator controls the 
transistor switch, causing it to turn the current from the 
power supply on and off. The "chopped" current in the 
wire ring interacts with the magnetic field to produce a 
pulsating force on the wire. 

The wire ring must be of nonmagnetic metal. 
Insulated copper magnet wire works well. 
Twist the ends together and support the ring at 
the twisted portion by means of a binding post, 
Fahnestock clip, thumbtack, or ringstand 
clamp. Remove a little insulation from each end 
for electrical connections. 

A ring 10- 15 cm in diameter made of 
22-guage enameled copper wire has its lowest 
rate of vibration at about 20 Hz. StifTer wire or a 
smaller ring will have higher characteristic 
vibrations that are more difficult to see. 

Position the ring as shovMi. with a section of 
the wiiv passing between the poles of the 
magnet VVlien tlie piilseti current passes 
thn>ugh the ring, the cunvnt interacts with the 
magnetic field, piDducing alternating forces 



that cause the wire to vibrate. In Fig. 5-30, the 
magnetic field is vertical, and the vibrations are 
in the plane of the ring. You can turn the 
magnet so that the vibrations ar ep>erp>endicular 
to the ring. 

Because the ring is clamped at one point, it 
can support standing waves that have any 
integral number of half-wavelengths. In this 
respect they are different from waves on a free 
wire ring, which are restricted to integral 
numbers of whole wavelengths. Such waves are 
more appropriate for comparison to an atom. 

When you are looking for a certain mode of 
vibration, position the magnet between ex- 
pected nodes (at antinodesl. The first "charac- 
teristic state," or "mode of vibration," that the 
ring can support in its plane is the first 
harmonic, having two nodes: one at the point of 
support and the other opposite it. In the 
second mode, three nodes are spaced evenly 
around the loop, and the best position for the 
magnet is directly opposite the support, as 
shown in Fig. 5-31. 



f 




Fig. 5-31 

You can demonstrate the various modes of 
vibration to the class by setting up the magnet, 
ring, and support on the platform of an 
overhead projector. Be careful not to break the 
glass with the magnet, especially if the frame of 
the pnijector hapj>ens to be made of a mag- 
netic material 

Ihe Project I'hysics Film Loop "Vibrations of 
a Wire" also shows this type of vibration. 



FILM LOOP (VOTES 



233 



FILM LOOP NOTES 



Film Loop 45 

PRODUCTION OF SODIUM BY 

ELECTROLYSIS 

In 1807, Humphry Day>' produced metallic 
sodium by the electrolysis of molten lye 
(sodium hydroxide). 

In the film, sodium hydroxide (NaOH) is 
placed in an iron crucible and heated until it 
melts at a temperature of 318°C. A rectifier 
connected to a power transformer supplies a 
steady current through the liquid NaOH 
through iron rods inserted in the liquid. 
Sodium ions are positive and are therefore 
attracted to the negative electrode; there they 
pick up electrons and become metallic sodium, 
as indicated symbolically in this reaction: 

Na" + e" = Na 

The sodium accumulates in a thin, shiny layer 
floating on the surface of the molten sodium 
hydroxide. 



Sodium is a dangerous material that com- 
bines explosively with water. The experimenter 
in the film scoops out a little of the metal and 
places it in water (Fig. 5-32). Energy is released 
rapidly, as you can see from the violence of the 
reaction. Some of the sodium is vaporized and 
the hot vapor emits the yellow light characteris- 
tic of the spectrum of sodium. The same yellow 
emission is easily seen if common salt, sodium 
chloride, or some other sodium compound is 
sprinkled into an open flame. 



Film Loop 46 

THOMSON MODEL OF THE ATOM 

Before the development of the Bohr theory, a 
popular model for atomic structure was the 
"raisin-pudding" model of J.J. Thomson. Ac- 
cording to this model, the atom was supposed 




Fig. 5-32 



234 



UNIT 5 / MODELS OF THE ATOM 



to be a uniform sphere of positive charge in 
which were embedded small negative "corpus- 
cles" (electrons). Under certain conditions, the 
electrons could be detached and obser\'ed 
separately, as in Thomson's historic experi- 
ment to measure the charge/mass ratio. 

The Thomson model did not satisfactorily 
explain the stability of the electrons and 
especially their arrangement in "rings," as 
suggested by the periodic table of the elements 
In 1904, Thomson performed experiments 
which to iiim showed the possihilitv of a ring 
structure within the broad outline of the 
raisin-pudding model. Thomson also made 
mathematical calculations of the various ar- 
rangements of electrons in his model. 

In the Thomson model of the atom, the cloud 
of positive charge cit'ated an electric field 
directed along radii, strongest at the surface of 
the sphere of charge and decreasing to zero at 
the center. You are familiar with a gra\ntational 
example of such a field. The earth's downward 
gravitational field is strongest at the surface and 
it decreases toward the center of the earth. 

For his model-of-a-model, Thomson used 
still another type of field: a magnetic field 
caused by a strong electromagnet above a tub 
of water. Along the water surface the field is 
"radial, " as shown by the pattern of ircjn tilings 
sprinkled on the glass bottom of the tub. 
Thomson used vertical magnetized steel nee- 
dles to represent the electrons; these were 
stuck through corks and floated on the surface 
of the water. The needles were oriented with 
like poh's pointing upwartl: their mutual re- 
pulsion tended to c:ause the magnets to sprt?ad 
apart. Tin; outward repulsion was rounterart- 
ed by the radial magnetic field directed inwaixl 
toward the center. When the floating magnets 
were placed in the tub of water, they came to 
eciuilibriiim configurations under the com- 
bined action of all the forc(!s. Thomson .saw in 
this experiment a partial verification of his 
calculation of how electrons (raisins) might 
come to equilibrium in a spherical blob of 
positive fluid. 

In the film the fioating magnets avf 3.8 cm 
long, supported b\ ping-pong balls (Fig. 5-331 
K(]uilibrium configurations aiv shown for \ari- 
ous numlxM's of i)ails. fn)in 1 to 12 IVrliaps sou 
can interjinU tlie patlcnis in tcnus of rings, as 
did Thomson. 

Thomson was unable to make an exact 
con-elation with the facts of chemistiy. For 
exampit*, he knew that the eleventh electron is 
easily r-ernou-d (corresponding to sotliuin. tiie 




Fig 5-33 



eleventh atom of the periodic tabiei, yet his 
floating magnet model failed to show this. 
Instead, the patterns for 10, 11. and 12 floating 
magnets are rather similar. 

Thomson's work with this apparatus illus- 
trates how physical theories may be tested 
with the aid of analogies. He was disappointed 
by the failure of the model to account for the 
details of atomic structure. A few years later, 
the Rutherford model of a nuclear atom made 
the Thomson model obsolete, but in its day the 
Thomson model did receive some support 
from experiments such as those shown in the 
film. 



Film IxNip 47 

RiTiiEKioKi) sr^irrFRixc; 

This film simulates the scattering of alpha 
fiarticles b\ a heavy nucleus, such as gold, as in 
F.rnest Rutherford s famous experiment. Tlie 
film was matle with a digital computer 

The computer program was a slight 
modification of that used in Film Loops 13 and 
14. on program oriiits. concerned v\ith plane- 
tarv orbits The onl\ difference is that the 
()p«Tator selected an inverse- squan* law of 
rrpiilsion instead of a law of attraction such as 
that of gnuit>'. The results of the computer 
calculation were displayed on a cathode-ray 
tul>e and then photographed (Fig. 5-341. Points 
ari» shown at ecjual time intervals \erif\ the law 
of arjMs for the motion of the alpha particles by 
pnijerting the film for measurements Why 
vvduUI \t)u expect etjual areas to \te s\>vpt out 
in ('(iii.il limes? 



FILM LOOP NOTES 



235 




Fig. 5-34 

All the scattering particles shown are near a 
nucleus. If the image from your projector is 30 
cm high, the nearest adjacent nucleus would 
be about 150 m above the nucleus shown. Any 
alpha particles mo\ing through this large area 
between nuclei would show no appreciable 
deflection. 



The computer and a mathematical model are 
used to tell what the result will be if particles 
are shot at a nucleus. The computer does not 
know" about Rutherford scattering. What it 
does is determined by a program placed in the 
computer's memory, written, in this particular 
instance, in a language called Fortran. The 
programmer has used Newton's laws of motion 
and has assumed an inverse-square repulsive 
force. It would be easy to change the program 
to test another force law, for example F = K/r^. 
The scattering would be computed and dis- 
played; the angle of deflection for the same 
distance of closest approach would be different 
than for inverse-square force. 

Working backward from the observed scatter- 
ing data, Rutherford deduced that the inverse- 
square Coulomb force law is correct for all 
motions taking place at distances greater than 
about 10 '^ m from the scattering center, but he 
found deviations from Coulomb's law for closer 
distances. This suggested a new type of force, 
called nuclear force. Rutherford's scattering 
experiment showed the size of the nucleus 
(supposedly the same as the range of the 
nuclear forces I to be about 10"'* m, which is 
about 1/10,000 the distance between the nuclei 
in solid bodies. 




The Nucleus 

EXPERIMENTS 



236 



Experiment 6-1 
RANDOM EVENTS 

In Unit 6, after having explored the random 
behavior of gas molecules in Unit 3, you are 
learning that some atomic and nuclear events 
occur in a random manner. The purjjose of this 
experiment is to give you some firsthand 
experience with random events. 

What Is a Random Event? 

Dice are useful for studying random beha\ior. 
You cannot predict with certainty how many 
spots will show on a single throw. But you are 
about to discover that you can make usefiil 
pn'diclions about a large number of thn)ws If 
the beha\ior of the diet* is tnily random, you 
can use probability theoiy to make predictions 
When, for example, you shake a box of 100 dice, 
you can predict with some confidence how 
many will fall with one spot up, how many with 
two spots up. and so on Probability theorv' has 
many appliiations For example, it is used in 
the study of automobile tratVic flow, the in- 
teii)i-f>tation of faint radar echoes fiDrn the 
planets, the pretliction of biilh. death, and 



accident rates, and the study of the breakup of 
nuclei. 

The theory' of probabilit\' proxides wavs of 
determining whether a set of events is random. 
An important characteristic of all truly random 
e\ents is that each e\ent is independent of the 
others. For example, if you throw a legitimate 
die four times in a row and find that a single 
spot turns up each time, your chance of 
observing a single spot on the fifth throw is no 
greater or smaller than it was on the first throw 

If events are to be independent, the circum- 
stances under which the observations are 
made must never favor one outcome over 
another. This condition is met in each of the 
following parts of this experiment. You are 
expected to do only one of these parts. The 
section ' Reconiing Your Data ' that follows the 
descriptions applies to all parts of the experi- 
ment Read this section in pn*paring to do any 
part of the experiment 

(a) Kaiicloni Event Disks 

\ini will u.se 100 random »'\ent disks and a large 
piece of graph paper Wlien the disk^ are 
spread around on the graph paper, even into 



EXPERIMENTS 



237 



the comers, what is the chance that a cross of 
the heavy grid marks of the graph paper will be 
covered by a disk (a "hit")? 

On your graph paper, spread the disks 
around fairiy evenly and count the number of 
"hits" for that trial. Record your results in a 
table like that on p. 238. Then spread the disks 
again and make another count. The counting 
wall go faster if you divide the graph paper into 
sections and have helpers count in each area. 
Repeat the process until you have counted 100 
trials. 

As described below, calculate the mean 
number of hits per trial, then divide by 100 to 
obtain the fraction of one grid area covered by a 
disk. 

From your distribution table, estimate the 
spread of vcilues around the mean that include 
two-thirds of the values. That number, called 
the standard deviation (s.d.), is a characteristic 
of the distribution and, like the mean, should 
be nearly the same value for each set of trials. 

An estimate of the uncertainty of the mean, 
called the standard error (s.e.), is found from 
s.d./V/V where N is the number of trials used 
to obtain the mean. As N increases, the s.e. 
decretises. 



1. What is the s.e. for your set of 100 trials? 

2. If you combine several sets of trials, what is 
the new mean and its s.e.? 



(b) Twenty-Sided Dice 

A tray containing 120 dice is used for this 
experiment. Each die h£is 20 identical faces (the 
name for a solid with this shape is icosahed- 
ron). One of the 20 faces on each die should be 
marked; if it is not, mark one face on each die 
with a felt-tip pen. 



3. What is the probability that the marked face 
will appear at the top for any one throw of one 
die? To put it another way, on the average, how 
many marked faces would you expect to see face 
up if you roll all 120 dice? 




Fig. 6-1 Icosahedral dice in use. 

The counting vvdll go faster if the floor area or 
tabletop is divided into three or four sections, 
with a different person counting each section 
and another person recording the total count. 
Work rapidly, taking turns vvath others in your 
group if you get tired, so that you can count at 
least 100 trials. 

(c) Diffusion Cloud Chamber 

A cloud chamber is a device that makes visible 
the trail left by the particles emitted by 
radioactive atoms. One version is a transparent 
box filled with supercooled alcohol vapor. 
When an a partical passes through, it leaves a 
trail of ionized air molecules. The alcohol 
molecules are attracted to these ions and they 
condense into tiny droplets which mark the 
trail. 

Your purpose in this experiment is not to 
leam about the operation of the chamber, but 
simply to study the randomness with which 
the a particles are emitted. A barrier with a 
narrow opening is placed in the chamber near 
a radioactive source that emits a particles. 
Count the number of tracks you observe 
coming through the opening in a convenient 
time interval, such as 10 sec. Continue counting 
for as many intervals as you can during the 
class period. (See Fig. 6-2.) 



■ rad 



laCLcJlve. £/£•■ 



Now try it, and see how well your prediction 
holds. Record as mciny trials as you Ccin in the 
time available, shaking the dice, pouring them 
out onto the floor or a large tabletop, and 
counting the number of marked faces showing 
face up. (See Fig. 6-1.) 



r 



iOJ.a.1 iirr.e. ir+ervo.'s 

Fig. 6-2 A convenient method of counting events in 
successive time intervals is to mark them in one slot of 
the "dragstrip" recorder, while marking seconds (or 
10-sec intervals) in the other slot. 



238 



UNIT 6 / THE NUCLEUS 



(d) Geiger Counter 

A Geiger counter is another device that detects 
the passage of invisible particles. A potential 
difference of several hundred volts is main- 
tained between the two electrodes of the Geiger 
tube. When a /3 particle or a y ray ionizes the 
gas in the tube, a short pulse of electricity 
passes through the tube. The pulse may be 
heard as an audible click in an earphone, seen 
as a "blip" on an oscilloscope screen, or read as 
a change in a number on an electronic scaling 
device. When a radioactive source is brought 
near the tube, the pulse rate goes up rapidly. 
But even without the source, an occasional 
pulse still occurs. These pulses are called 
background and are caused by cosmic radia- 
tion and by a slight amount of radiactivity 
always present in objects around the tube. 

Use the Geiger counter to determine the rate 
of background radiation, counting over and 
over again the number of pulses in a conven- 
ient time interval, such as 10 sec. 

Recording Your Data 

Whichever of the experiments you do, prepare 
your data record in the following way: 

Down the left-hand edge of your paper write 
a column of numbers frxjm to the highest 



of ei/enfi 





2 
30 

SJ. 

65 
/Jib 



Numbtr of 

C¥(ntS obserygd 

in one 

time 'mterrtLL 


(fre<i(jenoj) 


Lnj 


if) 



1 


1 


2 


1 


3 


Htt 44H- 


4 


■Hff 4Ut III 


5" 
6 


ftrf / 


7 


^^yf- 4*H- U4-t- 1 


8 


Hff Hff It ' 


^ 


//// 


10 


nil 


II 


nil 


ISL 


1 


13 


1 




number you ever expect to observe in one 
count. For example, if your Geiger counts seem 
to range from 3 to 20 counts in each time 
interval, record numbers finom to 20 or 25. 

To record your data, put a tally mark 
opposite each number in the column for each 
time this number occurs. Continue making 
tally marks for as many tried observations as you 
can make during the time you have. When you 
are through, add another column in w+iich you 
multiply each number in the first column by 
the number of tallies opposite it. Whichever 
experiment you do, your data sheet will look 
something like the sample in Fig 6-3. The third 
column shows that a total of 623 marked faces 
(or pulses or tracks) were observed in the 100 
trials. The average is 623 divided by 100, or 
about 623. You can see that most of the counts 
cluster around the mean. 

This arrangement of data is called a distribu- 
tion table. The distribution shown was ob- 
tained by shaking the tray of 20-sided dice 100 
times. Its shape is also typical of any random 
events such as random disks, Geiger-counter, 
and cloud-chamber results. 

A Graph of Random Data 

The pattern of your results is easier to \isualize 
if you display your data in the form of a bar 
graph, or histogram , as in Fig. 6-4. 



20 



IS 



? 10 



tLd 



4 * 't lo la. 
number" of event* (n) 



^ 



Fig. 6-3 A typical data page. 



Fig. 6-4 The results obtained when a tray of 20-sided 
dice (one side marked) was shaken 100 times 



If \i)u wen» to shake the dice another set of 
100 times, your distribution would not be 
exactly the same as the first one Howe\^r, if 
sets of 100 trials were repeated sr\rral times, 
the combined n*sull.s wDuid begin to form a 
smoother histogram Figiin' 6-5 shows the kind 
of result you could expect if wu did 1.000 trials 



EXPERIMENTS 



239 



zoor 
180 
160 
140 

> 
I loof- 

3 

I eo 

401- 



20 



m 




O 2 4 6 8 10 12 

number of evcni-s (n) 

Fig. 6-5 The predicted results of shaking the dice 1,000 
times. Notice that the vertical scale is different from that 
in Fig. 6-4. Do you see why? 

Compare this with the results for only 10 
trials shown in Fig. 6-6. As the number of trials 
increases, the distribution generally becomes 
smoother and more like the distribution in Fig. 
6-5. 



6 

>- 

C2 - 



I • 



h n 



1 2 3 4 3 t 7 8 9 10 II IZ 13 K 
numberof events (n) 

Fig. 6-6 Results of shaking the dice 10 times. 



Predicting Random Events 

How can data like these be used to make 
predictions? 

On the basis of Fig. 6-5, the best prediction of 
the number of marked faces turning up would 
be 5 or 6 out of 120 rolls. Apparently the chance 
of a die having its marked face up is about 1 in 
20, that is, the probability is V20. 

But not all trials had 5 or 6 marked faces 
showing. In addition to the average of a 
distribution, you also need to know something 
about how the data spread out around the 
average. Examine the histogrcun and answer 
the following questions. 



4. How many of the trials in Fig. 6-5 had from 5 
to 7 counts? 

5. What fraction is this of the total number of 
observations? 

6. How far, going equally to the left and right of 
the average, must you go to include one-half of all 
the observations? to include two-thirds? 

For a theoretical distribution like this (which 
your own results will closely approximate as 
you increase the number of trials), it turns out 
that there is a simple rule for expressing the 
spread: If the average count is A, then two- 
thirds of the counts will be between A - WA 
and A -f- \A. Putting it another way, about 
two-thirds of the vcilues will be in the range of 

A ±y/A. 

Another example may help make this clear. 
For example, suppose you have been counting 
cloud-chamber tracks and find that the average 
of a lai^e number of 1-min counts is 100 tracks. 
Since the square root of 100 is 10, you would 
find that about two-thirds of your counts 
would lie between 90 and 110. 

Check this prediction in Fig. 6-5. The average 
is 6. The square root of 6 is about 2.4. The points 
along the base of the histogram corresponding 
to 6 ± 2.4 are between 3.6 and 8.4. (Of course, it 
does not really make sense to talk about a 
fraction of a marked side. You would need to 
round off to the nearest whole numbers, 4 cmd 
8.) Therefore, the chances are about two out of 
three that the number of mariced sides showing 
after any shake of the tray will be in the range of 
4 to 8 out of 120 throws. 



7. How many of the trials did give results in the 
range 4 to 8? What fraction is this of the total 
number of trials? 

8. Whether you used the disks, rolled dice, 
counted tracks, or used the Geiger counter, 
inspect your results to see if two-thirds of your 
counts do lie in the range A ± V47 



If you counted for only a single 1-min trial, 
the chances are about two out of three that 
your single count C will be in the range 
A ± wA, where A is the true average count 
(which you would find over many trialsl. This 
implies that you can predict the true average 
value fairiy well e\'en if you have made only a 
single 1-min count. The chances are about two 
out of three that the single count C will be 
within "NM of the true average A. If you assume 
C is a fairly good estimate of A, you can use Vc 
as an estimate of Va and conclude that the 



240 



UNIT 6 / THE NUCLEUS 



TABLE 6-1 SAMPLE RESULTS AND ESTIMATED "TWO-THIRDS RANGES " 



Time 

min 


Total Count 


Expected 
Uncertainty 


Average 
Count 
per mm 


Expected 

Uncertainty 
per mm 


1 

10 

100 


100 

1000 

10000 


±10 

±32 

±100 


100 
100 
100 


±10 
±3.2 
±1.0 



chances are two out of three that the value 
obtained for C is within ± \C of the true 
average. 

You can decrease the uncertainty in predict- 
ing a true average like this by counting for a 
longer period. Suppose you continued the 
count for 10 min. If you counted 1,000 tracks, 
the expected "two- thirds range" would be 
about 1000 ± VlOOO or 1000 ± 32. The result is 
1000 ± 32 counts in 10 min, which gives an 
average of 100 ±3.2 counts per minute. If you 
counted for still longe r, say 10 min, the range 
would be 10,000 ± VlO.OOO or 10,000 ± 100 
counts in 100 min. Your estimate of the average 
count rate would be 100 ± 1 counts per minute. 
Table 6-1 lists these sample results. 

Notice that although the expected uncer- 
ttunty in the total count increases as the count 
goes up, it becomes a smaller /racfion of the 
total count. Therefore, the uncertainty in the 
average count rate (number of counts per 
minute) decreases. 

(The percentage of uncertainty can be ex- 

Vc" 1 
pressed as , which is equal to . In thi.s 

expression, you can see clearly that the per- 
centage of uncertainty goes down as C in- 
creases.) 

You can see from these examples that the 
higlier the total count (the longer you count or 
more trials you do), the mort» precisely you can 
estimate the true average. This becomes impor- 
tant in the measurement of the arti\ity of 
radioactive samples and many other kinds of 
random events. To get a precise measure of the 
activity (the average count rate), you must work 
with large numbers of counts. 



9. If you have time, take more data to increase 
the precision of your estimate of the mean. 
10. If you count 10 cosmic-ray tracks in a cloud 
chamber during 1 min, for how long would you 
expect to have to go on counting to get an 
estimate of the average with a "two-thirds range" 
that is only 1% of the average value? 



This technique of counting o\er a longer 
period to get better estimates is fine as long as 
the true count rate remains constant. But it 
does not always remain constant. If you were 
measuring the half-life of a short-lKed radioac- 
tive isotope, the acti\ity rate would change 
appreciably during a 10-min period. In such a 
case, the way to increase precision is still to 
increase the number of observations, by ha\ing 
a larger sample of material or by putting the 
Geiger tube closer to it, so that you can record a 
lai^e number of counts during a short time. 



11. In a small town it is impossible to predict 
whether there will be a fire next week. But in a 
large metropolitan area, firefighters know with 
remarkable accuracy how many fires there will be. 
How is this possible? What assumption must the 
firefighters make? 



Experiment 6-2 

RANGE Of a AXD /i PARTICLES 

An important property of particles from 
radioactKe sources is their ability to penetrate 
solid matter. In this experiment, you mil 
determine the distances a and fi particles can 
travel in \arious materials. 

Alpha (a) particles are most easily studied in 
a cloud chamber, a transparent box ix)ntaining 
super-cooled alcohol vapor. Since the a parti- 
cles are relatively massi\e and ha\e a double 
positive charge, they leave a thick trail of 
ionized air molecules behind them a.s they 
m()\«' along The ions then ser\e as centers 
about which alcohol condenses to form tracks 
of \isible droplets. 

Beta (/3i particliw also ionize air molecules as 
they mcne But because of their smaller mass 
and smaller chaiTge. they form n'latively few 
ions, and they an* farther apart than those 
fonned by a particles .Vs a n^ult. the trail of 
droplets in the chaml>er from /3 particles is 
much harder to see. 



EXPERIMBIVTS 



241 



A Geiger counter, on the other hand, detects 
P particles better than a particles. This is 
because a particles, in fonming a heavy trail, 
lose all their energy long before they get 
through e\en the thin window of cin ordinary 
Geiger tube. Beta particles encounter the atoms 
in the tube uindow also, but they give up 
relatively less energy so that their chances of 
getting through the wall are fairly good. 

For these reasons, you count a particles 
using a cloud chamber and fi particles with a 
Geiger counter. 

Observing a Particles 

Mark off a distance scale on the bottom of the 
cloud chamber so that you will be able to 
estimate, at least to the nearest 0.5 cm, the 
lengths of the tracks formed iFig. 6-7). Insert a 
source of a radiation and a barrier (as in the 
preceding experiment on random events) with 
a small slot opening at such a hei^t that the 
tracks form a fairly narrow beam moving 
parallel to the bottom of the chamber. Put the 
cloud chamber into operation according to the 
instructions supplied with it. 

Practice watching the tracks until you can 
report the length of any of the tracks you see. 




Rg. 6-7 

When you are ready to take data, count and 
record the number of as that come through 
the opening in the barrier in 1 min. Measure 
the opening and calculate its area. Measure 
and record the distance from the source to 
the barrier. 

Actually, you have probably not seen aU the 
peirticles coming through the opening since the 
sensitive region in which tracks are visible is 
rather shallow and close to the chamber floor. 
You will probably miss the as above this layer. 

The Range and Energy- of a Particles 

The maximum range of radioactive particles as 
they travel through an absorbing material 



depends on several factors, including the 
density and the atomic number of the absorber. 
The graph (Fig. 6-8) summarizes the results of 
many measurements of the range of a particles 
travelling through air. The range- energy curve 
for particles in air saturated with alcohol vapor, 
as the air is in your chamber, does not differ 
significantly from the curve shown. You are, 
therefore, justified in using Fig. 6-8 to get a fiiir 
estimate of the kinetic energy of the a particles 
vou observed. 



^60 



I" 









'" 










1 















J 


^ 


1 


1 






/ 


/ 


' 






y 


/ 


1 


1 
1 


y 


/ 


! 


/ 








\ ; 


1 




> 




1.0 JO 


3 


.0 i 


iO t 


-^0 d^ 



Fig. 6-8 Range of a particles in air as a function of their 
energy. 



1. Was there a wide variation in Q-particle ener- 
gies, or did most of the particles appear to have 
about the same energy? What was the energy of 
the a particle that caused the longest track you 
observed? 



Now calculate the rate at which enei^gy is 
being carried away from the radioactive source. 
Assume that the source is a point. From the 
number of a particles per minute passing 
through an opening of known area at a known 
distance from the source, estimate the number 
of a particles per minute leaving the source in 
all directions. 

For this estimate, imagine a sphere with the 
source at its center and a radius r equal to the 
distcince from the source to the barrier (Fig. 
6-91. From geometry, the surface area of the 
entire sphere is known to be 47rr^. You know 
the approximate rate c at which particles are 
emerging through the small opening, whose 
area a you have ccilculated. By proportion, you 



342 



UNIT e / THE NUCLEUS 




Fig. 6-9 

can find the rate C at which the particles must 
be penetrating the total area of the sphere: 
C ^ 4ffr' 
c a 

(The a-particle source is not a point, but 
probably part of a cylinder. This discrepancy, 
combined wdth a failure to count those parti- 
cles that pass above the active layer, will 
introduce an error of as much as a factor of 10.) 

The total number of particles leaving the 
source per minute, multiplied by the average 
energy of the particles, is the total energy lost 
per minute. 

To answer the following questions, use the 
relationships 

IMeV = 1.60 X 10 '^ J 
leal = 4.18 J 



2. How many joules of energy are leaving the 
source per minute? 

3. How many calories per minute does this 
equal? 

4. If the source were placed in 1 g of water in a 
perfectly insulated container, how long would it 
take to heat the water from DC to 100 C? 

5. How many joules per second are leaving the 
source? What is the power output in watts? 

Observing /3 Particles 

After removing all radioactive sources from 
near the Geiger lube, count the number of 
pulses caused by background radiation in 
several minutes. Calcuiatt- the* avenige back- 
ground radiation in counts per minute Then 
place a source of /3 radiation near the Geiger 
tube, and determine the new count rate. (Make 
sure that the sourro and Geiger tube are not 
moved during the n\st of the experiment.! Since 
you arv concemetl only with tin* partic-U»s from 
the source, subtract the average backgix)und 
count rate. 



Next, place a piece of absorbing material 
(such as a sheet of cardboard or thin sheet 
metal) between the source and the tube and 
count again. Place a second, equally thick sheet 
of the same material in front of the first and ■ 
count a third time. Keep adding absort)ers and 
recording counts until the count rate ha 
dropped nearly to the level of background 
radiation. 

Plot a graph on wtiich the horizontal scale is 
the total thickness (number) of absorbers and 
the vertical scale is the number of /3's getting 
through the absorber per minute. 

In addition to plotting single points, show 
the uncertainty in your estimate of the count 
rate for each point plotted. You know that 
because of the random nature of radioacthity, 
the count rate actually fluctuates around some 
average value. You do not know wtiat the trve 
average value is; it would ideally take an infinite 
number of 1-min counts to determine the 
"true" average. But you know that the distribu- 
tion of a great number of 1-min counts will 
have the property that two-thirds of them will 
differ from the average by less than the square 
root of the average. (See Experiment 6-1.) 

For example, suppose you ha\e observed 100 
counts in a given 1-min interval. The chances 
are two out of three that, if you counted for a 
veiy long time, the mean count rate would be 
between 90 and 110 counts (between 100 - 
VlOO and 100 + VlOO counts) For this reason, 
you would mark a vertical line on your graph 
extending from 90 counts up to 110. In this way 
you avoid the pitfall of making a single 
measurement and assuming you know the 
"correct" value. (For an example of this kind of 
graph, see notes for Film Loop 9 in L'nit l.i 

If other kinds of absorbing material are 
available, repeat the experiment with the same 
source and another set of absorbers. For 
sources that emit very low-energv- /3 ravs, it may 
be necessary to use very thin materials, such as 
paper or household aluminum foil. 

Range and Absorption 
of ;3 Particles 

Examine your graph of the absorption of 
particles. 



6. Is It a straight Ime^ 

7. What would the graph look like if (as is the case 
for II particles) all fi particles from the source were 
able to penetrate the same thickness of a given 
absorbing matenal before giving up all their 
energy? 



EXPERIMENTS 



243 



8. If you were able to use different absorbing 
materials, how did the absorption curves com- 
pare? 

9. What might you conclude about the kinetic 
energy of /3 particles? 



groups give a more uniform decay curve and a 
more reliable value for the half-life? 
3. Use your best decay curve to estimate the 
number of trials needed to reduce the decay 
activity to one-eighth the initial value. How many 
trials would be needed before you were sure that 
there was no more radioactivity in the sample? 



Experiment 6-3 
HALF-LIFE. I 

The more people there are in the worid, the 
more people die each day. The less water there 
is in a tank, the more slowly water leaks out of a 
hole in the bottom. 

In this experiment, you will observe three 
other examples of quantities that change at a 
rate that depends on the total amount of the 
quantity present. The objective is to find a 
common principle of change. Your conclusions 
will apply to many familiar growth and decay 
processes in nature. 

If you experimented earlier with random 
disks, rolling dice, and radioactive decay IEjc- 
periment 6-1), you were studying random 
events you could observe one at a time. You 
found that the fluctuations in such small 
numbers of random events were relatively 
large. But this time you will deal with a large 
number of events, and you will find that the 
outcome of your experiments is therefore more 
precisely predictable. 

A. Random Disks 

Use the random disks and graph paper cis in 
Experiment 6-1, Part (a), as an analogue of 
radioactive decay. Spread the 100 disks, which 
represent radioactive nuclei, over the graph 
paper. When a disk covers a heavy cross on the 
graph paper, consider it a radioactive decay. 
For each trial, remove the disks that "decayed" 
and record their number. When removed, these 
disks can be arranged like a bar graph in a 
series of pUes. Make up to 20 trials, or until you 
have less than three decays per trial. Graph the 
decay curve, which shows for each trial the 
number of nuclei that decayed. 



1. What was the initial decay rate for your first 
trial? After how many trials had the decay rate 
decreased to half the original rate? to one-fourth 
the original rate? 

2. If other groups are making similar experi- 
ments, how well do the results agree? Are the 
differences within the range to be expected from 
sampling? Consider your results for Experiment 
6-1. Does a combination of results from several 



A variation that is more representative of 
actual radioactive disintegration series can be 
made by using a small disk representing a 
daughter nucleus to replace each of the larger 
disks that decays. Then the number of nuclei 
remains constant; jione has vanished. Because 
the daughter nuclei are represented by small- 
er disks, their decay rate will be slower. Make 
20 to 25 trials using both size disks. For each 
trial, tally the number of decays of lai^e and 
of small disks. Also record their sum, the 
total activity of the sample. A plot of the total 
activity wUl show the pattern of activity for a 
sample that is a mixture of two radioactive 
nuclei. 



4. What is the half-life of the daughter nucleus? 



You could even add a granddaughter nu- 
cleus by using pennies or cardboard pieces as a 
third size disk. 

B. Twenty-Sided Dice 

Mark any two sides of each 20-sided die with a 
(washable) marking pen. The chances will 
therefore be 1 in 10 that a marked surface 
will be face up on any one die when you shake 
and roll the dice. When you have rolled the 120 
dice, remove all the dice that have a marked 
surface face up. Record the number of dice you 
removed lor line them up in a column). With 
the remaining dice, continue this process by 
shaking, rolling, and removing the marked dice 
at least 20 times. Record the number you 
remove each time (or line them up in a series of 
columns). 

Plot a graph in which each roll is represented 
by one unit on the horizontal axis, and the 
number of dice removed after each roll is 
plotted on the vertical axis. (If you have lined 
up columns of removed dice, you already have 
a graph.) 

Plot a second graph with the same horizontal 
scale, but with the vertical scale representing 
the number of dice remaining in the tray after 
each roll. 



244 



UNIT 6 / THE NUCLEUS 



You may find that the numbers you have 
recorded are too erratic to produce smooth 
curves. Modify the procedure as follows: Roll 
the dice and count the dice v^th marked 
surfaces face up. Record this number but do 
not remove the dice. Shake and count again. Do 
this five times. Now find the mean of the five 
numbers, and remove that number of dice. The 
effect will be the same cis if you had actually 
started with 120 x 5 or 600 dice. Continue this 
procedure as before, and you will find that it is 
easier to draw smooth curves that pass very 
nearly through all the points on your two 
graphs. 



5. How do the shapes of the two curves com- 
pare? 

6. What is the ratio of the number of dice 
removed after each shake to the number of dice 
shaken in the tray? 

7. How many shakes were required to reduce 
the number of dice in the tray from 120 to 60? 
from 60 to 30? from 100 to 50? 



C. Electric Circuit 

A capacitor is a device that stores electric 
charge. It consists of two conducting surfaces 
placed very close together, but separated by a 
thin sheet of insulating material. When the two 
surfaces are connected to a battery, negative 
charge is removed from one plate and added 
to the other so that a potential difference is 
established between the two surfaces. (See Sec. 
14.6 of Unit 4, Text.) If the conductors are 
disconnected from the battery and connected 
together through a resistor, the charge will 
begin to flow back from one side to the other. 
The chaise will continue to flow as long as 
there is a potential difference between the sides 
of the capacitor. As you learned in I'nit 4, the 
rate of How of char-ge Ithe current) thniugh a 
conducting path depends both on the resis- 
tance of the path and the potential difference 
across it. 

To picture this situation, think of two partly 
filled tank.s of water connected by a pipe 
running from the bottom of one tank to the 
bottom of the other (Fig. 6-10). When water is 
transfem'd from one tank to the other, the 
additional potential energ\' of the water is given 
by tlie (liffenMice in hiMght, just as the potential 
diffen»nce between the sides of a chargetl 
capacitor is proportional to the potential 
energy stonul in the capacitor. Water flows 
through the pipe at the bottom until the water 




Fig. 6-10 An analogy: The rate of flow of water depends I 
upon the difference in height of the water in the two ' 
tanks and upon the resistance the pipe offers to the flow 
of water. 



levels are the same in the two tanks Similarly, 
charge flows through the conducting path 
connecting the sides of the capacitor until 
there is no potential difference between the 
two plates. 

Connect the circuit as in Fig. 6-11, close the j 
switch, and record the reading on the I 
voltmeter. Now open the switch and take a 
series of voltmeter readings at regular intervals. 
Plot a graph using time intervals for the 
horizontal axis and voltmeter readings for the 
vertic2il. 



rr 



0^) Chare^inf the capacitor 
-V rf-— 



Cb) discharging Through the renifor ' 

Fig 6-11 



8. How long does it take for the voltage to drop 
to one-half of its initial value? from one-half to 
one-fourth? from one-third to one-sixth? 



Repeat the experiment with a different resis- 
tor in the circuit. Find the time required for the 
voltage to drop to one-half its initial value Do 
this for sjneral resistors 



9. How does the time required for the voltage to m 
drop to half its initial value change as the 1 
resistance in the circuit is changed? 



EXPERIMENTS 



245 



D. Short-Lived Radioisotope 

Whenever you measure the radioactivity of a 
sample with a Geiger counter, you must first 
determine the level of background radiation. 
With no radioactive material near the Geiger 
tube, take a count for several minutes and 
calculate the average number of counts per 
minute caused by background radiation. This 
number must be subtracted from any count 
rates you observe with a sample near the tube, 
to obtain what is called the net count rate of the 
sample. 

The measurement of background rate can be 
carried on by one member of your group while 
another prepares the sample according to the 
directions given below. Use this measurement 
of background rate to become familiar with the 
operation of the counting equipment. You will 
have to woric quite quickly when you begin 
counting radiation from the sample itself. 

First, a sample of a short-lived radioisotope 
must be isolated from its radioactive parent 
material and prepared for the measurmeent of 
its radioactivitv'. 

Although the amount of radioactive material 
in this experiment is too small to be considered 
at all dangerous (unless you drink large quan- 
tities of it), it is a very good idea to practice 
caution in dealing with the material. Respect 
for radioactivity is cm important attitude in our 
increasingly complicated worid. 

The basic plan is to 111 prepare a solution 
containing several radioactive substances, 
(21 add a chemical that absorbs only one of the 
radioisotopes, 13) wash most of the solution 
away leaving the absorbing chemical on a piece 
of filter paper, (4) mount the filter paper close to 
the end of the Geiger counter. 

1. Prepare a funnel — filter assembly by 
placing a small filter paper in the funnel and 
wetting it with water. 

Pour 12 mL of thorium nitrate solution into 
one graduated cylinder, and 15 mL of dilute 
nitric acid into another cylinder. 

2. Take these materials to the filter flask set 
up in your laboratory. Your teacher will con- 
nect your funnel to the filter flask and pour in 
a quantity of ammonium phosphomolybdate 
precipitate, (NH4)3PMo,2O40. The phospho- 
molybdate precipitate adsorbs the radioiso- 
tope's radioactive elements present in the 
thorium nitrate solution. 

3. Wash the precipitate by sprinkling several 
milliliters of distilled water over it, and then 
slowly pour the thorium nitrate solution onto 
the precipitate (Fig. 6-12). Distribute the solu- 




Fig. 6-12 

tion over the whole surface of the precipitate. 
Wash the precipitate again with 15 mL of dilute 
nitric acid and wait a few moments while the 
pump attached to the filter flask dries the 
sample. By the time the sample is diy, the nitric 
acid should have carried all the thorium nitrate 
solution through the filter. Left behind on the 
phosphomolybdate precipitate should be the 
short-lived daughter product whose radioactiv- 
ity you wish to measure. 

4. As soon as the sample is dry, remove the 
upper part of the funnel from the filter flask and 
take it to the Geiger counter. Make sure that the 
Geiger tube is protected with a layer of thin 
plastic food wrapping. Then lower it into the 
funnel carefully until the end of the tube eilmost 
touches the precipitate (Fig. 6-13). 




Fig. 6-13 



You will probably find it convenient to count 
for one period of 30 sec in each minute. This 
will give you 30 sec to record the count, reset 
the counter, and so on, before beginning the 



246 



UNIT 6 / THE NUCLEUS 



next count. Record your results in a table like 
Fig. 6-14. Try to make about 10 trials. 

background - 12 ccxjiits per minute^ 
- 6 countb pfrJ^ ivinufc 



(mms) 


CO u nt 




0- 'A 


^C3 


7^7 


i.i'^ 


^ji 7 


i^/ 


^ -;z/i 




• 


3 -3'/x 






H - y<3 







Fig. 6-14 

Plot a graph of nef count rate as a function of 
time. Draw the best curve you can through all 
the points. From the curve, find the time 
required for the net count rate to decrease to 
half its initial vfdue. 



10. How long does it take for the net count rate to 
decrease from one-half to one-fourth its initial 
value? one-third to one-sixth? one-fourth to one- 
eighth? 

11. The half-life of a radioisotope is one of the 
important characteristics used to identify it. Using 
the Handbook of Chemistry and Physics, or 
another reference source, identify which of the 
decay products of thorium is present in your 
sample. 

12. Can you tell from the curve you drew whether 
your sample contains only one radioisotope or a 
mixture of isotopes? 



The relationship between the half-life of a 
process and the decay constant X is discussed 
in the special page "Mathematics of Decay" in 
Chapter 21 of the Text. There you learned that 
for a large number of truly random e%'ents, the 
half-life T.^ is related to the decay constant K 
by the equation: 



T^ = 



0.693 



13. From the known decay constant of the dice, 
calculate the half-life of the dice and compare it 
with the experimental value found by you or your 
classmates. 

14. If you measured the half-life of capacitor 
discharge or of radioactive decay, calculate the 
decay constant for that process. 



Experiment 6-4 
HALF-LIFE. II 

Look at the thorium decay series in Table 6-2. 
One of the members of the series, radon-220. is 
a gas. In a setiled bottle containing thorium or 
one of its salts, some radon gas always gathers 
in the air space abo\-e the thorium. Radon-220 
has a very short half-life (51-5 sec). The sub- 
sequent members of the series (poIonium-214. 
lead-210, etc.) are solids. Therefore, as the 
radon-220 decays, it forms a solid deposit of 



TABLE ft-2. THE THORIUM DECAY SERIES 



Discussion 

It should be clear from your graphs and those 
of your classmates that the three kinds of 
quantities you observed all have a common 
property: It takes the same time (or numl)er of 
rolls of the dice) to reduce the quantity to 
one-half its initial value as it does to reduce 
from one-half to one-fourth, from one-third to 
one-sixth, from one-fourth to one-eighth, etc. 
This quantity is the hnlj'life. 

In the experiments on the "decay" of 20- 
sided dice uith two marked faces, you knew 
beforehand that the "decay rate" was one- 
tenth. That is, over a large number of throws, an 
average of one-tenth of the dice would be 
removed for each shake of the trav. 







Mod* 




Nam* 


Symbol 


o< D*cay 


HaM-Life 


Thorlum-232 


•oTh»*» 


a 


139 X 10'«yr 


Radium-228 


MRa»* 


fi 


6.7 yr 


Actinium-228 


..Ac*" 


fi 


6.13 hr 


Thorium-228 


.oTh»» 


a 


191yr 


Radium-224 


..R8»»* 


a 


3G4d 


Radon-220 


..Rn»» 


a 


51.5 SM 


Polonium-216 


mPo»'« 


a 


0.16 »ac 


Lead 212 


.iPb»'» 





10.6 hr 


Bismuth-212 


.>Bi»'» 


aorfi* 


60.5 min 


Polonium-212 


mPo»'» 


a 


3 * 10 'aec 


Thallium-208 


..Tl»<" 





3.10 mm 


Lead 208 


.jPb*- 


Stable 


3 10 min 



•Bismulh-212 can decay m two ways: 34% decays by 
n emission to thallium 208: 66% decays by emission 
to polonium-212 Both thallium-20e and polonium-212 
decay to lead-206 



EXPERIMENTS 



247 



radioactive material in the bottle. In this 
experiment, you will measure the half-life of 
this radioactive deposit. 

Although the amount of radioactive material 
in this experiment is too small to be considered 
at all dcingerous (unless you drink lai^e quan- 
tities of it), it is a very good idea to practice 
caution in dealing with the material. Respect 
for radioactivity is an important attitude in our 
increasingly complicated worid. 

The setup is illustrated in Fig. 6-15. The 
thorium nitrate is spread on the bottom of a 
seeded container. (The air inside should be kept 
damp by moistening the sponge with water.) 
Radon gas escapes into the air of the container, 
and some of its decay products are deposited 
on the upper foil. 



ta aroW<d \ 



thorium 
nitrate- 



to + HSO 



Fig. 6-15 




P^tr 






/; \ •^"spq'^g'f ■ • .' ■• '. ■^\ zj\\ \ 



When radon disintegrates in the nuclear 
reaction 

86Rn22«^g,Po2'6-^2He^ 

the polonium atoms formed are ionized, ap- 
parently because they recoil fast enough to lose 
an electron by inelastic collision with air 
molecules. 

Because the atoms of the first daughter 
element of radon are ionized (positively 
charged), you can increase the amount of 
deposit collected on the upper foil by charging 
it negatively to several hundred volts. Although 
the electric field helps, it is not essential; you 
will get some deposit on the upper foil even if 
you do not set up an electric field in the 
container. 

After two days, so much deposit has accumu- 
lated that it is decaying nearly as rapidly as the 
constant rate at which it is being formed. 
Therefore, to collect a sample of maximum 
activity, your apparatus should stand for about 
two days. 



Before beginning to count the activity of the 
sample, you should take a count of the 
background rate. Do this far away from the 
vessel containing the thorium. Remove the 
cover, place your Geiger counter about 1 mm 
above the foil, and begin to count. Make sure, 
by adjusting the distance between the sample 
and the window of the Geiger tube, that the 
initial count rate is high (several hundred per 
minute). Fix both the counter and the foil in 
position so that the distance will not change. 
To get fairly high precision, take a count over a 
period of at least 10 min (see Experiment 6-1). 
Because the deposit decays rather slowly, you 
can afford to wait several hours between 
counts, but you will need to continue tciking 
counts for several days. Make sure that the 
distance between the sample and the Geiger 
tube stays constant. 

Record the net count rate and its uncertainty 
(the "two-thirds range" discussed in Experi- 
ment 6-1). Plot the net count rate against time. 

Remember that the deposit contains several 
radioactive isotopes and each is decaying. The 
net count rate that you measure is the sum of 
the contributions of all the active isotopes. The 
situation is not as simple as it was in Experi- 
ment 6-2, in which the single radioactive 
isotope decayed into a stable isotope. 



1. Does your graph show a constant half-life or a 
changing half-life? 



Look again at the thorium series and, in 
particular, at the half-lives of the decay prod- 
ucts of radon. Try to interpret your observa- 
tions of the variation of count rate with time. 



2. Which isotope is present in the greatest 
amount in your sample? Can you explain why this 
is so? Make a sketch to show approximately how 
the relative amounts of the different isotopes 
in your sample vary with time. Ignore the iso- 
topes with half-lives of less than 1 min. 



You can use your me<isurement of count rate 
and half-life to get an estimate of the tmiount 

An 

of deposit on the foil. The activity, — — , de- 
pends on the number of atoms present N: 

AN 



At 



= AN. 



248 



UNIT 6 / THE NUCLEUS 



The decay constant X is related to the half-life 
T.. by 

0.693 



X = 



T^ 



Use your values of counting rate and half-life 
to estimate N, the number of atoms present in 
the deposit. What mass does this represent? il 
amu = 1.7 X 10'^^ kg.) The smallest amount of 
material that can be detected with a chemical 
balance is on the order of 10~* g. 

Discussion 

it is not too difficult to calculate the speed and 
therefore the kinetic energy of the polonium 
atom. In the disintegration 

„«Rn"«^H4Po^'« + 2He\ 

the a particle is emitted with kinetic energy 6.8 
MeV. Combining this with the value of its mass, 
you can calculate v^ and, therefore, v. What is 
the momentum of the a particle? Momentum is 
of course conserved in the disintegration. So 
what is the momentum of the polonium atom? 
What is its speed? What is its kinetic energy? 

The ionization energy (the energy required to 
remove an outer electron from the atom) is 
typically a few electron volts. How does your 
value for the polonium atom's kinetic energy 
compare with the ionization enepgv'? Does it 
seem likely that most of the recoiling polonium 
atoms would ionize? 



Experiment 6>o 
R^\IIIOACTI\ E TRACERS 

In this group of experiments, you have the 
opportunity to invent your procedures yourself 
and to draw your own conclusions. Most of the 
experiments will take more than one class 
period and will require careful planning in 
advance. 

Caution 

All these experiments take cooperation from 
the biologN' or the chemistrv department, and 
require that safety pn»rautions be obserx-ed 
very carefully so that neither you nor other 
students will be exposed to radiation. 

For example, handle radioisotopes as you 
would a stn)ng arid: wear ili.sposabie plastic 
gloN'es and goggles, and work with all contain- 
ers in a Iray lined with paper to s(»ak up an\ 
spills. Never draw radioactive liquids into a 



pipette by mouth; use a mechanical pipette or a 
rubber bulb. Your teacher will discuss other 
safety precautions with you before you begin. 
None of these acthities is suggested just for 
the sake of doing tricks with isotopes. You 
should have a question cleariy in mind before 
you start, and should plan carefully so that you 
can complete your experiment in the time you 
have available. 

Tagged Atoms 

Radioactive isotopes have been called tagged 
atoms because e\en when they are mixed with 
stable atoms of the same element, they can still 
be detected. To see how tagged atoms are used, 
consider the following example. 

A green plant absorbs carbon dioxide (COj) 
from the air, and by a series of complex 
chemical reactions, builds the carbon dioxide 
(and water) into the material of w+iich the plant 
is made. SuppKJse you tried to follow the steps 
in the series of reactions. You can separate each 
compound from the mixture by using ordinary 
chemical methods. But how can you trace the 
chemical steps by w+iich each compound is 
transformed into the next w+ien they are all 
jumbled together in the same place? Tagged 
atoms proxide an answer. 

Put the growing green plant into an atmos- 
phere containing carbon dioxide. A tiny quan- 
tity of COj molecules containing the radioactive 
isotope carbon-14 in place of normal carbon-12 
should be added to this atmosphere. Less than 
1 min later, the radioacti\it>' can be detected 
within some, but not all, of the molecules of 
complex sugars and amino acids being svnthe- 
sized in the leaves. As time goes on. the 
radioactiv-e carbon enters step b\' step into each 
of the carbon compounds in the leaves. 

With a Geiger counter, one can, in effect. 
watch each compound in turn to detect the 
moment when radioacti\-p molecules begin to 
be added to it In this way the mixture of 
compounds in a plant cjtn be arranged in the 
order of their formation, which is obviously a 
useful clue to chemists studying the reactions 
Photos\Tithesis, long a mv-ster^'. has been 
studied in detail in this way 

Radioactiw isotopes used in this manner are 
called tracers. The quantit>- of tracer material 
needed to do an experiment is astonishingly 
small For example, compare the amount of 
carbon that ran l»e detected by an anal\1ical 
balance with the amount needinl to do a tracer 
experiment ^our (i««iger counter may, t>pirally. 
need 100 net counts per minute to distinguish 



EXPERIMENTS 



249 



the signal from background radiation. If only 1% 
of the particles emitted by the sample are 
detected, then, in the smallest detectable 
sample, 10,000 (or 10^) atoms are decaying each 
minute. This is the number of atoms that decay 
each minute in a sample of only 4 x io~^ /xg of 
carbon-14. Under ideal conditions, a chemical 
balance might detect 1 /xg. 

Thus, in this particular case, measurement 
by radioacti\ity is over 10,000 times more 
sensitive than by the balance. 

In addition, tracers give you the ability to find 
the precise location of a tagged substance 
inside an undisturbed plant or animal. Radia- 
tion from thin sections of a sample placed on 
photographic film produces a xisible spot I Fig. 
6-16). This method can be made so precise that 
scientists can tell not only which cells of an 
or^ganism have absorbed the tracer, but also 
which parts of the cell (nucleus, mitochondria, 
etc.). 

Choice of Isotope 

The choice of which radioactive isotope to use 
in an experiment depends on many factors, 
only a few of which are suggested here. 

Carbon-14, for example, has several proper- 
ties that make it a useful tracer. Carbon 
compounds are a major constituent of all living 
organisms. It is usually impossible to follow the 
fate of any one carbon compound that you 
inject into cm organism, since the added 
molecules and their products are immediately 
lost in the sea of identical molecules of which 
the organism is made. Carbon-14 atoms, how- 
ever, can be used to tag the carbon compounds, 
which can then be followed step by step 
through complex chains of chemical processes 
in plants and animals. On the other hand, the 
carbon-14 atom emits only (3 particles of rather 
low energy. This low energy makes it impracti- 
cal to use carbon-14 inside a Icirge liquid or 
solid sample since all the emitted particles 
would be stopped inside the sample. 

The half-life of carbon-14 is about 6,000 years, 
which means that the activity of a sample will 
remain practically constant for the duration of 
an experiment. But sometimes the experi- 
menter prefers to use a short-lived isotope so 
that it will rapidly drop to negligibly low activity 
in the sample or on the laboratoiy table if it gets 
spilled. 

Some isotopes have chemical properties that 
make them especially useful for a specific kind 
of experiment. Phosphorus-32 (half-life: 14.3 
days) is especially good for studying the grow^ 



of plants because phosphorus is used by the 
plant in many steps of the growth process. 
Practically cdl the iodine in the human body is 
used for just one specific process, the manufac- 
ture of a hormone in the thyroid gland that 
regulates metabolic rate. Radioactive iodine-131 
(half-life: 8.1 days) has been immensely useful 
as a tracer in unravelling the steps in that 
complex process. 

The amount of tracer to be used is deter- 
mined by its activity, by how much it wiU be 
diluted during the experiment, and by how 
much radiation can be safely allowed in the 
laboratory. Since even very small amounts of 
radiation are potentially harmful to people, 
safety precautions and regulations must be 
carefully followed. The Department of Energy 
has established licensing procedures and regu- 
lations governing the use of radioisotopes. As a 
student you are permitted to use only limited 
quantities of certain isotopes under carefully 
controlled conditions. However, the variety of 
experiments you can do is still so great that 
these regulations need not discourage you 
from using radioactive isotopes as tracers. 

One unit used to measure the radioactivity of 
a source is called the curie. When 3.7 x io'° 
atoms within a source disintegrate or decay in 
1 sec, its activity is said to be 1 curie (ci. (This 
number was chosen because it is the approxi- 
mate average activity of 1 g of pure radium-226.) 
A more practical unit for tracer experiments is 
the microcurie ifxc), which is 3.7 x iff* disinteg- 
rations per second or 2.2 x lO® per minute. The 
quantity of radioisotope that students may 
safely use in experiments, without special 
license, varies from 0.1 /xc to 50 /xc depending 
on the type and energy of radiation. 

Notice that even when you are restricted to 
0.1 /xc for your experiments, you may still 
expect 3,700 disintegrations per second, which 
would cause 37 counts a second in a Geiger 
counter that recorded only 1% of them. 



1. What would be the "two-thirds range" in the 
activity (disintegrations per minute) of a 1 /xc 
source? 

2. What would be the "two-thirds range" in 
counts per minute for such a source measured 
with a Geiger counter that detects only 1% of the 
disintegrations? 

3. Why does a Geiger tube detect such a small 
percentage of the /3 particles that leave the 
sample? (Review that part of Experiment 6-2 on 
the range of /3 particles.) 



250 



UNIT 6 / THE NUCLEUS 



A. Autoradiography 

One rather simple experiment you can almost 
certainly do is to reenact Becquerel's original 
discovery of radioactivity. Place a radioactive 
object (lump of uranium ore, luminous watch 
dial with the glass removed, etc.) on a Polaroid 
film packet or on a sheet of X-ray film in a 
light-tight envelope. A strong source of radia- 
tion wall produce a visible image on the fUm 
within an hour, even through the paper 
wrapping. If the source is not so strong, leave it 
in place overnight. To get a very sharp picture, 
you must use unwrapped film in a completely 
dark room and expose it wath the radioactive 
source pressed firmly against the film. 

(Most Polaroid film can be developed by 
placing the packet on a Hat surface and passing 
a metal or hard-rubber roller firmly over the 
pod of chemicals and across the film. Other 
kinds of film are processed in a darkroom 
according to the directions on the developer 
package.) 

This photographic process has grown into an 
important experimental technique called au- 
toradiography. The materials needed are rela- 
tively inexpensive and easy to use, and there 
are many interesting applications of the 
method. For example, you can grow plants in 
soil treated with phosphoms-32, or in water to 
which some phosph()rus-32 has been added, 
and make an autoradiograph of the roots, stem, 
and leaves (Fig. 6-16). Or each day take a leaf 
from a fast-growing young plant and show how 
the phosphorus moves from the roots to the 
growing tips of the leaves. 




B. Chemical Reactions and 
Separations 

Tracers are used as sensitive indicators in 
chemical reactions. You may w^ant to try a 
tracer experiment using iodine-131 to study the 
reaction between lead acetate and potassium 
iodide solutions. Does the radioacti\it\' remain 
in the solute or is it carried down with the 
precipitate? How complete is the reaction? 

When you do experiments like this one with 
liquids containing /3 sources, transfer them 
carefully (with a special mechanical pipette or a 
disposable plastic s\Tingei to a small, dispos- 
able container called a planchet, and evaporate 
them so that you count the diy sample. This is 
important when you are using /3 sources since, 
otherwise, much of the radiation would be 
absorbed in the liquid before it reached the 
Geiger tube. 

You may want to try more elaborate exj>eri- 
ments involving the movement of tracers 
through chemical or biological systems. Stu- 
dents have grown plants under bell jars in an 
atmosphere containing radioactive carbon 
dioxide, fed radioactive phosphorus to earth- 
worms and goldfish, and studied the metab- 
olism of rats with iodine-131. Be sure to re- 
view safety and humane guidelines for the 
use of animals in resefirch before attempting 
any of these exp>eriments. 



Fig 6-16 Autoradiograph made by a student to show 
uptake of phosphorus-32 in Coleus leaves. 



Experiment 6-6 

MKASl'RIXC; THE ENERGY OF 

(i RADIATION 

With a device called a ^ ray spectrometer, you 
can sort out the /3 particles emitted by a 
radioartiw source according to their energV' 
just as a grating or prism spertroscxipe spreads 
t)iit the colors of the \isible spectrum You can 
make a simple /3-ray spectrometer with two 
disk magnets and a packet of Polaroid film. 
With it \ou can make a fairly good estimate of 
the average energv of the /3 particles emitted 
fnim \arious soun-es b\- observiivi how much 
the\ arr deflected by a magnetic field of known 
mtt'nsity 

Mount twtj disk magnets as shown in Fig. 
6-17. Be sure the faces of the magnets are 
parallel and opposite poles are facing each 
other. 



EXPERIMENTS 



251 




Fig. 6-17 



Bend a piece of sheet meted into a curve so 
that it will hold a Polaroid film packet snugly 
around the mcignets. Place a /3 source behind a 
barrier made of thin sheet lead containing two 
narrow slits that will allow a beam of /3 particles 
to enter the magnetic field as shown in Fig. 
6-18. Expose the film to the )8 radiation for two 
days. Then carefully remove the magnets 
without changing the relative positions of the 
film and P source. Expose the film for two more 
days. The long exposure is necessary because 
the coUimated beam contains only a small 
fraction of the /3 radiation given off by the 
source, and because Polaroid film is not very 
sensitive to /3 radiation. (You can shorten the 
exposure time to a few hours if you use X-ray 
film.) 




t= 



-Ptsa of developer I-3000 ASA 
chemicols "^ Pbiaroid 

Film packet 

to source 



^Sheet /€ad collimator 
Disc magnet 

Sheet metal bocliing 



Fig. 6-18 



When developed, your film will have two 
blurred spots on it; the distance between their 
centers will be the arc length a in Fig. 6-19. 




An interesting mathematical problem is to 
find a relationship between the angle of 
deflection, as indicated by a, and the average 
energy of the particles. You can Ccilculate the 
momentum of the particle fairly easily. Unfor- 
tunately, since the /3 particles ftxjm radioactive 
sources are travelling at nearly the speed of 
light, the simple relationships between 
momentum, velocity, and kinetic energy (which 
you learned in Unit 3) cannot be used. Instead, 
you need to use equations derived from the 
special theory of relativity which, although not 
at all mysterious, are a little beyond the scope 
of this course. (The necessary relations are 
developed in the supplemental unit, "Elemen- 
tary Particles.") A graph (Fig. 6-20) that gives the 
values of kinetic energy for various values of 
momentum is provided. 



ii" 




0.0 

z 4 6 e /o /2 l'^ 

mv = qBB 

(^IO'"'kj-n^/Si.<.l 

Fig. 6-20 Kinetic energy versus momentum for electrons 
{m c^ = 0.511 MeV). 



First, you need an expression that will relate 
the deflection to the momentum of the particle. 
The relationship between the force on a 
charged particle in a magnetic field and the 
radius of the circular path is derived on page 
542 of Unit 5, Text. Setting the magnetic force 
equal to the centripetal force gives 



Bgv 



mv' 



Fig. 6-19 



which simplifies to 

mv = Bqfl 

If you know the magnetic field intensity B 
(measured with the current balance as de- 
scribed in the Unit 4 Handbook), the charge on 
the electron, and can find R, you can compute 
the momentum. A little geometry will enable 
you to calculate B from the arc length a and 
the radius r of the magnets. A detailed solu- 



2S2 



UNIT 6 / THE NUCLEUS 



lion will not be given here, but a hint is shown 
in Fig. 6-21. 




Fig. 6-21 



The angle B is 

d = -^x 360° 
Zvr 

You should be able to prove that if tangents are 
drawn from the center of curvature O to the 
points where the particles enter and leave the 
field, the angle between the tangents at O is 
also 6. With this as a start, see if you can 
calculate R . 



The relationship between momentum and 
kinetic energy for objects travelling at nearly 
the speed of light 

E = y/p^c^ -hmo'c* 

is discussed in most college physics texts. The 
graph in Fig. 6-21 was plotted using data 
calculated from this relationship. 

From the graph, find the average kinetic 
energy of the fi particles whose momentum 
you have measured. Compare this with values 
given in the Handbook of Chemistry and 
Physics, or another reference book, for the 
particles emitted by the source you used. 

You will probably find a value listed VN+iich is 
two to three times higher than the value you 
found. The value in the reference book is the 
maximum energy that any one /3 particle from 
the source can have, whereas the value you 
found was the average of all the /3 particles 
reaching the film. Ttiis discrepancy' between 
the maximum energy 'which all the /3's should 
theoretically have) and the average energy 
puzzled physicists for a long time The explana- 
tion, suggested by Enrico Fermi in the mid- 
1930s, led to the discovery of a strange new 
particle called the neutrino, wtiich you will 
want to find out about. 



FILM LOOP NOTES 



253 



FILM LOOP NOTES 



Film Loop 48 

COLLISIONS WITH AN OmECTT 

OF UNKNOWN MASS 

In 1932, Chadwick discovered the neutron by 
analyzing collision experiments. This film al- 
lows a measurement similar to Chadwick's, 
using the laws of motion to deduce the mass of 
an unknown object. The film uses balls rather 
than elementary particles and nuclei, but the 
finalysis, based on conservation laws, is re- 
markably similar. 

The first scene shows collisions of a small 
ball with stationary target balls, one of similar 
mass and one of larger mass (Rg. 6-22). The 
incoming ball always has the same velocity, as 
you can see. 




Fig. 6-22 

The slow-motion scenes allow you to mea- 
sure the velocity acquired by the targets. The 



problem is to find the mass and velocity of the 
incoming ball without measuring them di- 
rectly. The masses of the targets are M, = 352 g, 
M2 = 4,260 g. 

Chadwack used hydrogen and nitrogen nuc- 
lei as targets and measured their recoil vel- 
ocities. The target balls in the film do not have 
the same mass ratio, but the idea is the same. 

The analysis is shown in detail on a special 
page in Chapter 23 of the Text. For each of the 
two collisions, equations can be written ex- 
pressing conservation of energy and conserva- 
tion of momentum. These four equations 
contain three quantities that Chadwick could 
not measure: the initial neutron velocity and 
the two final neutron velocities. Some algebraic 
manipulation results in the elimination of these 
quantities, leaving a single equation that can be 
solved for the neutron mass. If v,' and Vj' are 
the speeds of targets 1 and 2 after collision, and 
M, andM2 the masses, the neutron mass m can 
be found from 

m(v,' — Vj') - M2V2' — M,v,' 



MoV,' — M,v,' 

m = — ^-^, j—^ 

V, -V, 



Make measurements only on the targets, as 
the incoming ball (representing the neutron) is 
supposed to be unobservable both before and 
after the collisions. Measure v,' and v^' in any 
convenient unit, such as divisions per second. 
(Why is the choice of units not important here?) 
Calculate the mass m of the invisible, unknown 
particle. In what ways might your results differ 
from Chadwick's? 



254 



ANSWERS TO END-OF-SECTION QUESTIONS 



Answers to End-of-Section Questions 



Chapter 1 

1. We have no way of knowing the lengths of time 
involved in going the observed distances. 

2. No. The time between stroboscope flashes is 
constant and the distance intervals shoviai are 
not equal. 

3. An object has a uniform speed if it travels equal 
distances in equal time intervals; or, if the 
distance traveled divided by time taken = 
constant, regardless of the particular distances 
and times chosen. 

4. V =^ 

_ 720 m 



8. 







540 sec 




= 


1 .3 m/sec 




Ad = 


(20 m/sec) (IS sec) 


At 


Ad/Af 


300 m 


(5.0) 


(1.0) 




(6.01 
(4.5) 
(5.5) 


(0.8) 

1.1 

0.9 


(Entries in parentheses 
are those already given 
in the text.) 


7.5 


0.67 




8.0 


0.62 




8.6 


0.58 





Hint: To determine the location of the left edge 
of the puck relative to readings on the meter 
stick, line up a straight edge with the edge of the 
puck and both marks on the meter stick 
corresponding to a given reading. 




10. 



11. 



d(cm) r(sec) 


13 01 

26 02 

39 03 

52 0.4 

65 0^ 

78 0.6 

92 0.7 

The one on the left has the lai^er slope 
mathematically; it corresponds to' 160 km/hr, 
whereas the one on the right corresponds to 80 
km/hr. 

most rapidly at the beginning when the slope is 
steefjest; most slowly toward the end w+jere the 
slope is most shallow 
Ad _ 2m 
Af 4 sec 

= 03 m/sec (&x)m table) 

_ 4.5 m 
83 sec 

= OSm (from graph) 
The two results are the same. 

(a) 1.7 cm at 42 min (interpolation) 

(b) 3 cm at 72 min (extrapolation) 



S - 



i 



f 

fmin) 


(cm) 



19 
36 
63 


0.2 
O.M 
1.6 
2.4 




y> AC so 

Ttm» (min) 



70» 60 



13. an estimate for an additional lap (extrapolation) 
*3- *'in»i — Ad/Af as the time interval Al becomes 

veiy small Instantanpou!> speed is the limit of 
the average speeds computed for ver> small 
time intervals at a given point 

14. (a) alMiut 5 nv/sor 

(b) about 25 m/sec 

(c) zero 

Average speed is equal to the distance traxfled 
dividjHl by the elapsed lime Instantaneous 
sjieed is the sjxmxI at a particular moment of 
time 

_ final sti oc<l ini tial speed 
*' ela(isf>d time 

_ 100 km/hr- km/hr 

■ 20km/hr/sec 



18 



ANSWERS TO END-OF-SECTIO\ QUESTIONS 



255 



16. 



17, 



_ 2 km/hr - 5 km/hr 
^* 15 min 

_ - 3 km/hr 

15 min 
= —02 km/hr/min 
No, since average acceleration is specified. 
Av 
At 



aav = 



At = 



Av 

aav 

12 m/sec 



18. 



4 m/secj 
= 3 sec 
Av =aav Af 

= (4m/sec2)(6 sec) 
= 24 m/sec 



Chapter 2 

1. shape, size, and weight 

Drop several different objects, being sure to 
change only one quality with each object, and 
observe the results. 

2. Composition: Terrestrial objects are composed 
of combinations of earth, water, air, and fire, 
celestial objects of nothing but a unique fifth 
element. 

Motion: Terrestricil objects seek their natural 
positions of rest depending on their relative 
contents of earth (heaviest), water, air, and fire 
(lightest); celestiid objects moved endlessly in 
circles. 

3. (a), (b), and (c) 

4. Aristotle: The nail is heavier than the toothpick 
so it falls faster. 

Gcilileo: Air resistance slows down the toothpick 
more than the nail. 

5. (1) The bag, since it is lighter than the rock, 

slows the rock down. Therefore, the rock and 
bag together fall slower than the rock alone. 
(2) The bag and the rock together are heavier 
than the rock alone. Therefore, the bag and 
rock together fall^asfer thcin the rock alone. 

6. See question 3 of Chapter 1. 

7. An object is uniformly accelerated if its speed 
increases by equcd amounts during equal time 
intervals; Av/ At = constant. 

^ At 

_ 32 m/sec - 22 m/sec 

5 sec 
= 2 m/sec, 
. _ 40 m/sec - 32 m/sec 
2 m/sec.2 
= 4 sec 
9. The definition should (1) be mathematically 
simple and (21 correspond to actual free-fall 
motion. 

10. (b) 

11. Distances are relatively easy to measure as 
compared with speeds; measuring short time 
intervals remained a problem, however. 



12. The expression d =vt can only be used if v is 
constant. The second equation refers to acceler- 
ated motion in which v is not constant. 
Therefore, the two equations cannot be applied 
to the same event. 

13. (c) and (e) 

14. (d) 

15. (a), (c), and (d) 

16. (a) 



Chapter 3 



1. kinematic: (a), (b), (d) dynamic: (c), (e) 

2. a continuously applied force. 

3. The air pushed aside by the puck moves around 
to fill the space left behind the puck as it moves 
along and so provides the propelling force 
needed. 

4. the force of gravity downward and an upward 
force of equal size exerted by the table 

The sum of the forces must be zero, because the 
vase is not accelerating. 

5. the first three 

6. 6; zero; 6; 2 up; 6 in 

7. no 

When an object is in equilibrium, the net force 

acting on it equals zero. 

no 

8. (a) vector (d) vector 

(b) scalar (e) scal2ir 

(c) scalar (f) vector 

9. Vector quantities (1) have magnitude and direc- 
tion, (2) can be represented graphically by 
arrows, and (3) can be combined to form a single 
resultant vector by using either the head-to-tail 
or the parallelogram method. (Note: Only vec- 
tors of the same kind are combined in this way; 
that is, we add force vectors to force vectors, not 
force vectors to velocity vectors, for example.) 

10. Direction is now taken into account. (We must 
now consider a change of direction to be as valid 
a case of acceleration as speeding up or slowing 
down.) 

11. Aristotle: A continuous force is required to 
sustain motion. 

Newton: Since a force is not needed to sustain 
motion, the pedaling force must exactly oppose 
air resistance, and the net force is zero. 
Aristotle's ideas represent a "common sense" 
interpretation of natural events, which we all 
tend to accept before studving the events 
carefully. 

12. W downward, 0,0,0 

13. Galileo's "straight line forever " motion may have 
meant motion at a constant height above the 
earth, whereas Newlon's meant moving in a 
straight line through empty space. 

14. 1 newton = 1 kg • m/sec^ 
ION 



15 



a 4m/sec- 



= 2.5 kg 



256 



ANSWERS TO END-OF-SECTION QUESTIONS 



16. false IFrictional forces must be taken into 
account in determining the actual net force 
exerted.) 

Ar 
_ 10 m/sec 

5 sec 
= 2 m/sec* 
IS. the magnitude and direction of the applied 

force, and the mass of the object 
19. Mass Acceleration 
(kg) (m/sec*) 



F =ma 

= (2kg)(2m/8ec*) 

= 4N 



1 

2 

3 

1/5 

0.5 

45 

10 

0.4 



30 
15 
10 
150 
«0 

0.67 

3 
75 



20. Weight (m x a,) is the force acting on a mass at 
the earth's surface, caused by the earth's 
gravitational attraction. Weight is a vector 
quantity. 

Mass (m) is the unchanging quality of a body 
that describes its resistance to any acceleration. 
Mass is a scalar quantity. 

21. Because the hammer is 20 times more massive 
than the nail, it also requires 20 times more force 
to accelerate the hammer. Thus, the accelera- 
tions are nearly equal. 

22. F =ma 

a=^ 
m 

_ 30N 

3kg 

= 10 m/sec* 
The acceleration on the moon or in deep space 
would be the same since the mass of the object 
remains the same. 

23. Ic) and (0 

24. (1) The second force is equal in magnitude 13 N). 

(2) The second force acts in the opposite 
direction (to the lef^). 

(3) The second force acts on the first object 

25. The horse pushes against the earth; the earth 
pushes aigainst the horse causing the horse to 
accelerate forward ll'he earth accelerates also, 
but can you measure it?) The swimmer pushes 
backward against the water; the water, accord- 
ing to the third law, pushes forward against the 
swimmer; however, there is also a backward 
fhrtional force of drag exerted by the water on 
the sv\-inimer The two fones acting on the 
swimmer add up to zen), since he is not 
accelerating 

26. No 1'he force pulling the string apart Ls slill 
only 300 N. the 500 N would haw to be exerted 
ul both ends to bnrak the line 

27. See /r,<r p BH 



28. 



29. 



F =ma 

= (70kg)(3m/»ec*) 
= 210N 

a = '' 



m 

_ 29.4 N 
3Kg 
= 9.8 m/»ec* 
The weight of the object is equal to its mass 
(3 kgj times the acceleration of gravity 19.8 
m/sec^l: that is, 3 kg x 9.8 m/sec^ = 29 4 .N 

Chapter 4 

1. It agrees with the exp>eriment. Also, the horizon- 
tal motion agrees with Newton's first law, v\'hile 
the vertical motion agrees with Newlon s second 
law. 

2. a,, because the two motions are indeptendent 

3. Ad =v, Ar ■»- Vxailti^ 

vertical (v, = 01 horizontal (a = 0) 

Ay =0 + Via(Af)* A^t = v, Al 

Ay = Mia( A;t/Av,)* ^t = A;c/Av, 

4. (a), Id, and (e) 

5. Aac = V, Al 

= (4 m/secMlOsec) 
= 40m 
Ay = Mia,( Aj>* 

= Vi(10m/»ec*»(10sec)* 
= 500 m 

6. They must be moving with a uniform speed 
relative to each other. 

7. A person in the stands sees a p2uvbolic path 
Two different observers see different motions. 
The moxing observer is in the plane of the 
motion and does not observe a sideways 
motion. 

8. la) T = l/f = 1/45 = 22 X 10 ' min 

Ibl 22 X 10* min x 60 sec/min = 132 sec 
Ic) / = 45 rpm x 1/60 min/sec = 0.75 rps 



9. 



2irR _ 2 x 3 14 x 3 



= Jl cm/min 



T 60 

10. / =80 vibrations/min = 13 vib/sec 
T = l/f = 1/13 = 0.75 sec 

la) and Ibl 



I 



11 
12 

13. 



mv 
R 



'■-(t) 



_ m(4iT*W) 

T* 
^ 4iH(l)(05) 

1 
= 19.7 N (to center) 
It would move in a straight path 

= 19 7 N (from center) 

14. !)«• \ahie of the gra\itatioi\al acceleration and 
ihv radius of the mmin (to wtiuh 110 km is 
added to determine HI 

15. 1'he force \w calculate to be holding it in orbit is 
the earth s gravity at that height 



ANSWERS TO END-OF-SECTION QUESTIONS 



257 



16. 



f = Unnifl^aaiL. 



h) 



= 1,660 N 



Chapter 5 



1. The sun would set 4 min later each day. 

2. Ceilendars were needed to schedule agricultural 
activities and religious rites. 

3. The sun has a westward motion each day, an 
eastward motion v\ith respect to the fixed stars, 
cind a north— south variation. 



follfl 




light 

— from 
Sun 



5. Eclipses do not occur each month because the 
moon and the earth do not have the same 
planes of orbit. 

6. (a) The moon continuously moves in an easterly 

direction through the slcy, exhibits changing 
phases, and rises and sets each day. 

(b) Throughout one year, the motion is similar 
month by month. 

(c) The words "moon" and "month" have the 
same origin 

7. Mercuj>' and Venus are always fciirly near the 
sun: either east of the sun (evening sky) or west 
of the sun (morning skyl. 

8. When in opposition, a planet is opposite the 
sun: therefore, the planet would rise at sunset 
and be on the north- south line at midnight. 

9. after they have been farthest east of the sun and 
are visible in the evening sk\' 

10. when they are near opposition 

11. No. They are always close to the ecliptic. 

12. Mercury and Venus remain near the sun. Mars, 
Jupiter, and Saturn (as well as Uranus, Neptune, 
and Plutol ccin be found in different locations 
along the zodiac. When they are nearly opposite 
the sun. Mars, Jupiter, and Saturn cejise their 
eastwcird motion and move westward (in retro- 
grade motion) for several months. They then 
return to their eastward motion. 

13. How may the irregular motions of the planets be 
accounted for by combinations of constant 
sjjeeds along circles? 

14. Many of their written records have been 
destroyed by fire, weathering, and decay. 

15. Only perfect circles and uniform sp)eeds were 
suitable for the perfect and changeless heavenly 
bodies. 

16. A geocentric system is an earth-centered sys- 
tem. The yearly motion of the sun is accounted 
for by assuming that it is attached to a separate 
sphere that moves contrary to the motion of the 
stars. 



17. The first solution, as proposed by Eudoxus, 
consisted of a system of transparent crystalline 
spheres that turned at various rates around 
various axes. 

18. Aristarchus assumed that the earth rotated 
daily, which accounted for all the daily motions 
observed in the sky. He also assumed that the 
e£irth revolved around the sun, which ac- 
counted for the many annual changes observed 
in the sky. 

19. When the earth moved between one of these 
planets and the sun (with the planet being 
observed in opposition), the earth would be 
moving faster than the planet. So the planet 
would appear to us to be moving westward. 

20. The direction to the steirs should show an 
£innual shift: the annucil parallax. (This involves a 
very smcill angle and so could not be observed 
with instruments available to the Greeks. It was 
first observed in ax) 1836.) 

21. Aristarchus was considered to be impious 
because he suggested that the earth, the abode 
of human life, might not be at the center of the 
universe. 

His system was neglected for a number of 
reasons: 

(1) "Religious": It displaced humanity &x)m the 
center of the universe. 

(2) Scientific: Stellar parallax was not observed. 

(3) Practical: It predicted celestial events no 
better than other, less offensive, theories. 

Chapter 6 

1. The lack of uniform velocity associated with 
equants was (1) not sufficiently absolute and (2) 
not sufficiently pleasing to the mind. 

2. (a) P, C (d) C 

(b) P, C (e) P, C 

(c) P (f) C 

3. the relative size of the planetary orbits as 
compared with the distance between the earth 
and the sun; these were related to the calculated 
periods of revolution about the sun. 

4. (b) and (d) 

5. 2° in both cases 

6. No. E'recise computations required more smcdl 
motions than in the system of Rolemy. 

7. Both systems were about equally successful in 
explaining observed phenomena. 

8. The position of humanity and its abode, the 
earth, were important in interpreting the divine 
plan of the universe. 

9. They are equally valid; for practical purposes we 
prefer the Copemican because of its simplicity. 

10. He challenged the earth-centered worid outlook 
of his time and opened the way for later 
modifications and improvements by Kepler, 
Galileo, and Newton. 

11. the appearance in 1572 of a "new star" of vaiying 
brightness 

12. It included expensive equipment and facilities 



258 



ANSWERS TO END-OF-SECTION QUESTIONS 



and involved the coordinated work of a staff of 
people. 

13. They showed that comets were distant as- 
tronomical objects, not local phenomena as had 
been believed. 

14. Pie made them lai^er and sturdier and devised 
scales with which angle measurements could be 
read more precisely. 

18. He analyzed the probable errors inherent in 
each piece of his equipment; also, he made 
corrections for the effects of atmospheric refrac- 
tion. 

16. He kept the earth fixed, as did Ptolemy, and he 
had the planets going around the sun, as did 
Copernicus. 

Chapter 7 

1. finding out the correct motion of Mars through 
the heavens 

2. Epicycles were only a convenient computational 
device for describing some of the cyclical 
motions. Kepler began to seek objects and forces 
that caused the motions. 

3. By means of circular motion, Kepler could not 
make the position of Mars agree with TVcho 
Brahe's observations. (There was a discrepancy 
of 8 min of arc in latitude.) 

4. By means of triangulation, based on observa- 
tions of the directions of Mars and the sun 687 
days apart, he was able to plot the orbit of the 
earth. 

8. A line drawn from the sun to a planet sweeps 
out equal areas during equal time intervals. 

6. where it is closest to the sun 

7. You would probably guess that he went to the 
store every day at 8 am An empirical law is a 
general statement based on observations. Em- 
pirical laws can result in over-generalizations 

8. Mars has the largest eccentricity of the planets 
Kepler could study. 

9. (a) Law of elliptical orbits 

(b) Law of areas 

(c) Both (plus date of passage of perihelion, for 
examplel 

10. The square of the period of any planet is 
pn)portional to the cube of its average distance 
to the sun. 

11. Jupiter: —, = '" ^'* = 1 

' R' (52)* 



Saturn: 



(29.46)» 



= 1 



fl' (9.54)» 

All objects orbiting the sun have T*/R* = 1 in 
units of years and astronomical imits. 

12. R^ = (5)' 
fl' = 25 

n = 2.9 All 

13. Keplcrs |wriod» drsrribed motion around the 
sun The complex motions of an earth-riMitereti 
system did not iii('liid«> such periods 

14. Kepler t>as(<(l his laws upon ol)servalions. and 
expressed them in a mathematical form 



18. popular language, concise mathematical ex- 
pression 

16. The common beliefe contradicted by GalUeo's 
observations were: 

(a) Changes do not occur in the stany heavens. 

(b) Stars are meant to provide light at night; 
therefore, no "invisible" stars can exist. 

(c) The moon is p»erfectly spherical 

(d) Only the earth is the center of revolutions. 

(e) The sun, like other heavenly bodies, is 
perfect. 

(f) Venus is always between the sun and the 
earth, and therefore cannot show a full" 
phase. 

17. both the heliocentric and TVchonic theories 

18. 7he sunspots and the mountains on the moon 
refuted the Ptolemaic assertion that all heavenly 
bodies were perfect spheres 

19. No They only supported a belief that he alread\' 
held 

20. Some believed that distortions in the telescopie 
(which M^re plentiful) could have caused the 
peculiar observations Otfiers believed that 
established physics, religion, and philosophy far 
outweighed a few odd observations 

21. b, c (d is not an unreasonable answer, since it 
was by writing in Italian that he stirred up many 
people.) 

Chapter 8 

1. (1) Scholars published journals and formed 

societies where they presented reports 

(2) Time and money were available. 

(3) Interest in science was growing. 

(4) Capable scientists and arii&ans were avail- 
able. 

(5) Experimental and mathematical tools were 
improving 

(6) Interestir\g problems were clearly stated 

2. "From the phenomena of motions to investigate 
(induce) the forces of nature, and then from 
these forces to demonstrate (deduce) other 
phenomena" 

3. (II Do not use more hy|X)theses than necessary- 
(21 To the same effrrts. assign the same causes 
(3) Properties of neartiv Ixnlies are assumed to 

be true of distant tx>dies also 
(41 H>iKJtheses (propxisilionsi based on observa- 
tions are accepted until refined by new 
observations 

4. Evei%' btxly in the universe attracts evwy other 
lx)dv 

8. The same set of rules (laws) applies on the earth 

and in the heawns 
e. The fones rxertetl on \\re planets are ahvavs 

directed lovvanl the single jHiini wl>erT the sun 

is located 

7. the fomuila for centripetal acceleration 

8. that the orbit was rin'ular 

9. No. he includetl the more general caae of all 
conic sections. 



ANSWERS TO END-OF-SECTION QUESTIONS 



259 



10. 

11. 

12. 
13. 
14. 
15. 
16. 



17. 

18. 
19. 



20. 



21. 
22. 



23. 



24. 
25. 
26. 



27. 



28. 
29. 



30. 
31. 



32, 
33. 



4X, 25X, 16X 

1/(60)- = 1/3,600 = 0.00028 = 0.028% 

9.8 m/sec- X 0.00028 = 0.0027 m/sec- 

That one law would be sufficient to account for 

both. 

He thought it was magnetic and acted tangen- 

tially. 

the physics of motion on earth and in the 

heavens under one universal law of gravitation 

No. He thought it was sufficient to simply 

describe and apply it. 

An all pervasive ether transmitted the force 

through larger distances. He did not wish to use 

a hypothesis that could not be tested. 

phenomenological and thematic 

discussion 

(a) The forces are equal. 

(b) The accelerations are inversely proportional 
to the masses. 

(a) 2F 

(b) 3F 

(c) 6F 

(b) Fab = 4Fcu 

Otherwise, calculation of the force would 
require considering all parts of the bodies and 
their distances separately. Experimental evi- 
dence and mathematical analysis justify the use 
of point masses" for rigid bodies. Only simple 
systems can really be studied. The conclusion 
that a simple law worked was very important, 
the values of the constant in Kepler's third law 
T'-/R^ =k as applied to satellites of each of the 
two planets to be compared 
the numerical value of G 
Fgrav. m,, m-i. R 

the period of the moon and the distance 
between the centers of the earth and the moon, 
or the ratio T'/R^ 

similar information about Saturn and at least 
one of its satellites 

1/1,000; inversely proportional to the masses 
On the near side, the water is pulled away from 
the solid earth; on the far side, the solid earth is 
pulled away firom the water. Since F oc l/fl ^, the 
larger R is, the smaller the corresponding F. 
all of them 

As the moon orbits, its distance to the sun is 
continually changing, thus affecting the net 
force on the moon due to the sun and the earth. 
Also, the earth is not a perfect sphere. 
Comets travel on very elongated ellipses, 
no 



diapter 9 

false 



No. Don't confuse mass with volume or mass 

wath weight. 

Answer C 

The total mass is 15 g. 

No. Change speed to velocity and perform 

additions by vector techniques. 



8. 



10. 



11. 



12. 



13. 



14. 



15. 



16. 



17. 

18. 



(a), (c), and (d) (Their momenta before collision 
are equal in magnitude and opposite in direc- 
tion.) 

(a) The total momentum does not change. 

(b) The total initial momentum is equal to the 
total final momentum. 

(c) Nothing can be said about the individual 
parts of the system. 

Least momentum: a pitched baseball (small 
mass and fairly small speed) 
Greatest momentum: a jet plane in flight (veiy 
large mass and high speed) 

(a) about 4 cm/sec; faster ball delivers more 
momentum to girl. 

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

(c) about 1 cm/sec; uath same gain in momen- 
tum, more massive girl gains less speed. 

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

(These answers assume the mass of the ball is 
much less than the mass of the girl.) 
It can be applied to situations where only 
masses and speeds can be determined. 
FAf = Sp' 

Al 

= 50 

15 

= 3.3N 
Conservation of mass: No substances are added 
or allowed to escape. 

Conservation of momentum: No net force from 
outside the system acts upon any body consi- 
dered to be part of the system. 
None of these is em isolated system. In cases (a) 
and (b), the earth exerts a net force on the 
system. In case (c), the sun exerts a net force on 
the system. 

You know that the total mass of the system will 
be 22 kg and the total momentum of the system 
udll be 30 kg ■ m/sec up. You cannot tell what the 
individual masses or momenta will be. 
Answer (c) (Perfectly elastic collisions can only 
occur between atoms or subatomic particles.) 
Answer (d) (This assumes mass is always 
positive.) 
Answer (c) 

(a) It becomes stored as the object rises. 

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



Chapter 10 



1. Answer (b) 

2. Answer (b) 

3. Answer (c) 

4. Answer (c); the increase in potential enei^ 
equals the work done on the spring. 

6. Answer (e); you must do work on the objects to 

push them closer together. 
6. Answer (e); kinetic energy increases as gravita- 



260 



ANSWERS TO END-OF-SECTION QUESTIONS 



tional potential enei^ decreases. Their sum 
remains the same (if air resistance is negligible). 

7. Potential energy is greatest at extreme position 
where the speed of the string is zero. Kinetic 
energy is greatest at midpoint where the string 
is unstretched. 

8. The less massive treble string will gain more 
speed although both gain the same amount of 
kinetic energy (equal to elastic potential ener^ 
given by guitarist). 

9. Multiply the weight of the boulder (estimated 
from density and volume) by the approximate 
distance above ground level. 

10. None. Centripetal force is directed inward along 
the radius, which is always perpendicular to the 
direction of motion for a circuleir orbit. 

11. same, if initial and final positions are identical 

12. same, if fnctional forces are negligible; less if 
frictional forces between skis and snow are 
taken into account. 

13. Answer (c) 

14. Answer (c) 

15. Answer (b) 

16. Nearly all. A small amount was transformed into 
kinetic energy of the slowly descending weights; 
the water container would also have been 
warmed. 

17. Answer (b) 

18. Answer (d) 

19. Answer (a) 

20. Answer (bl 

21. Answer (a) 

22. Answer (e) 

23. The statement means that the energ\' that the 
lion obtains from eating comes ultimately from 
sunlight. The lion eats animals, which eat 
plants, whiih grow by absorlwKl sunlight 

24. Answer (c) 

25. Answer (a) 

26. Answer (c) 

27. Answer (c) 

28. A£ is the change in the total ener^' of the 
system; AW is the net work (the work done on 
the system minus the work done by the system); 
AH is the net heat exchange (heat added to the 
system minus heat lost by the system). 

29. 1. heating (or cooling) it 

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

30. Answer (bl 

Chapter 11 

1. Answer (c) 

2. true 

3. false 

4. a working model; a theoretical model; discus- 
sion 

8. 10'" particU's; 10 '" m in tlianieler; 10' ni/sr«-: 
disonliMfd v»'lncities and clinM-tion.s distnhuted 
unifonnU': lln'sc particles might !>«• atoms, 
rnolcciile.s. or dust particles 

8. Answi-r (b) 



7. In gases, the molecules are far enough apart so 
thta the rather complicated intermolecular 
forces can safely be neglected. 

8. Answer (b) 

9. Answer (b) 

10. Answer (d) 

11. The atom had been described formerly in vague 
terms. His estimate put some real numbers into 
the theory and showed its usefulness in making 
predictions. 

12. Answer (c) 

13. When the piston is pushe?d in. work is done on 
the individual particles, thus increasing their 
kinetic energv' Since the temperature is propor- 
tional to this kinetic ener^gy, the temperature 
must rise 

14. Low-density gases not near a phase change to a 
liquid or solid. 

18. Answer (a) 

16. An irreversible process is one in which order 
decreases and therefore the entropy increases. 

17. Answers a, b, and c are correct 

18. The stove could warm up ewn more as the 
water molecules passed their kinetic enengv to 
it. The second law of thermodv-namics afFirms 
that heat v\ill not flow from a cool body to a hot 
body by itself The second law (a statistical law) 
does not apply to indKidual molecules. 

19. (a) unbroken egg 

(bl a glass of ice and warm water 

20. la) true 
(bl false 
Ic) false 

21. .Answer (hi 

Chapter 12 

1. transverse, longitudinal, and torsional 

2. longitudinal: fluids can be compressed but they 
are not stiff enough to be bent or twisted 

3. transverse 

4. No The movement of the bump in the rug 
depends on the movement of the mouse; it does 
not go on bv itst»lf 

8. energv' (Particles of the mediimi are not transfer- 
red alot^ the direction of the wave motion i 

6. the stiffness and the density 

7. less stiff: slower propagation 
less mass: faster pro^iagation 

8. (II wavelength, amplitude. |Milarization 
(21 frequeiKA |x*ri(Kl 

the distance between any two succ'essive points 
that have identical ^xisitions in the v\-ave pattern 
(1) 100 Hz 

1 _ 1 



9 



10, 



(21 T = i = 

/ 100 Hz 



= 0.01 sec 



(31 X = 



= 0.1 m 



^ V _ lOm/sec ^ 
/ 100 Hz 

11. Answer (bi 

12. A, + A, 

13. Yes. The resulting displaremrni v\-nul«l In- 
5 + (-6) = -1 cm: superposition 



ANSWERS TO END-OF-SECTION QUESTIONS 



261 



14. cancellation 

15. Antinodal lines are formed by a series of 
antinodal points. Antinodal points cire places 
where waves arrive in phase and maximum 
reinforcement occurs. (The amplitude there is 
greatest.) 

16. Answer (a) 

17. when the difference in path lengths to the two 
sources is an odd number of half wavelengths 

[-K-k,-k.etc.) 
2 2 2 

18. (a) no motion at the nodes 

(b) oscillates with maximum amplitude 

19. - 

2 

ao. 2L, so that one-half wavelength just fits on the 
string 

21. No. Only frequencies that are whole-number 
multiples of the fundamental frequency are 
pKJSsible. 

22. All points on a wave front have the same phase; 
that is, they all correspond to crests or troughs 
(or any other set of similar parts of the 
wavelength pattern). 

23. Every point on a wave front may be considered 
to behave as a point source for waves generated 
in the direction of the wave's propagation. 

24. If the opening is less than one-half a wavelength 
wide, the difference in distance to a point P from 
the two edges of the opening cannot be equal to 
k/2. 

25. As the wavelength increases, the diffraction 
pattern becomes more spread out cind the 
number of nodal lines decreases until the 
pattern resembles one-half of that produced by 
a p>oint source oscillator. 

26. yes to both (Final photograph shows diffraction 
without interference; interference occurs 
whenever waves pass each other.) 

27. A ray is a line drawn perpendicular to a wave 
front and indicates the direction of propagation 
of the wave. 

28. The angles are equal. 

29. parabolic 

30. The reflected wave fronts are parallel wave 
fronts. 

31. (1) stays the same 

(2) becomes smaller 

(3) changes so that the wave fronts are more 
nearly parallel to the boundcuy (Or its 
direction of propagation becomes closer to 
the perpendicular between the media.) 

32. 



33. 



34. 



(1) /X = V relationship 

(2) reflection 

(3) refraction 

(4) diffraction 

(5) interference 

Sound waves are longitudinal. 




Chapter 13 

1. No. Eventually diffraction begins to widen the 
beam. This property is called superposition. 

2. Romer based his prediction on the extra time he 
had calculated it would require light to cross the 
orbit of the earth. 

3. Romer showed that light has a finite sp)eed. 

4. Experiments carried out by Foucault and Fizeau 
showed that light has a lower speed in water 
than in air, whereas the particle model required 
that light have a higher speed in water. 

6. When light enters a more dense medium, its 
wavelength and speed decrease, but its fre- 
quency remains unchanged. 

6. Young's experiments showed that light could be 
made to form an interference pattern; such a 
pattern could be explained only by assuming a 
wave model for light. 

7. It was diffraction that spread out the light 
beyond the two pinholes so that overlapping 
occurred and interference took place between 
the two beams. 

8. Poisson applied Fresnel's wave equations to the 
shadow of a circular obstacle and found that 
there should be a bright spot in the center of the 
shadow. 

9. Newton passed a beam of white light through a 
prism and found that the white light was 
somehow replaced by a diverging beam of 
colored light. Further experiments proved that 
the colors could be recombined to form white 
light. 

10. Newton cut a hole in the screen on which the 
sjjectrum was projected and allowed a single 
color to pass through the hole and through a 
second prism; he found that the light was again 
refracted, but no further sepiiration took place. 

11. A shirt appeiirs blue if it reflects mainly blue 
light and absorbs most of the other colors that 
make up white light. 

12. The "nature philosophers " were apt to postulate 
unifying principles regardless of experimental 
evidence to the contrary, and were very un- 
happy with the idea that something they had 
regarded as unquestionably pure had many 
components. 

13. The amount of scattering of light by tiny 
obstacles is greater for shorter wavelengths than 
for longer wavelengths 

14. The 'sky' is sunlight scattered by the atmos- 
phere. Light of short wavelength (the blue end of 
the sf>ectrum) is scattered most. On the moon, 
the sky looks dark because there is no atmos- 
phere to scatter the light to the observer. 



262 



ANSWERS TO END-OF-SECTION QUESTIONS 



15. Hooke and Huygens had proposed that light 
waves are similar to sound waves. Newton 
objected to this view because the familiar 
straight-line propagation of light was so diffe- 
rent from the behavior of sound. In addition, 
Newlon realized that polarization phenomena 
could not be accounted for in terms of spherical 
pressure waves. 

16. reflection, refraction, diffraction, interference, 
polarization, color, finite speed, and straight- 
line propagation. I This last would be associated 
with plane waves.) 

17. No. 

18. Light had been shown to have wave properties, 
and all other knov^n wave motions required a 
physical medium to transmit them, so it was 
assumed that an "ether" must exist to transmit 
light waves. 

19. Because light is a transverse wave and propa- 
gates at such a high speed, the ether must be a 
very stiff solid. 



Chapter 14 

1. Lodestone continuously attracts or repels only 
lodestone or iron. It has two poles IN and SI and 
orients itself in a north- south direction. The N 
and S poles cannot be isolated. The effects of 
lodestone are magnetic. 

After beiiig rubbed, amber will attract or repel 
many different types of material; the effect 
depends on what type of material is used to loib 
the amber. These effects are electrical. 

2. He showed that the earth and the lodestone 
affect a magnetized needle in similar ways 

3. Amber attracts many substances: lodestone only 
a few Amber needs to be rubbed to attract; 
l()dest(jne always attracts Amber attracts to- 
ward its center; lodestone attracts toward «'ilher 
of its poles. 

4. 1. Like charges repel each other. A both- that has 

a fief positive charge i-epels an\' bod\' that has 
a net fyositivr charge Thai is, two glass nxls 
that have both been rubbed will tent! to n'p«'l 
each other A bod\' that has a net negaliw 
charge repels an\' other bod\ that has a net 
negative charge. 
2. I 'nhke charges attract eacli other A bods that 
has a net fuysitiw charge attracts any body 
that has a net negative charge and \ice wrsa 

5. A cork hung inside a chatted silver can was not 
attracted to the sides of the can. (This implied 
thai Ihrre was n(» net eliMtnc force on the cork: a 
n-Mill similar to that pni\ed b\ Newlon for 
graNitalionai forte inside a hollow sphere.) 

8. /•■,,, CI 1//J-' and/",,, ^^ q\qn 

7. /•■,., will lie one-(|uarter as large 

H. No. the ampen" is the unit of cunt-nl 



9. ''=*(^^) 'toward each otherl 

/I: = 9 X 10* N mVC* 
Therefore, 



F = 9 X 10* N mVC- ( ^-^ 
= 9X10* N 



IC X ic, 



m' 



10. Each point in a scalar field is gi\«n b\' a number 
only, whereas each point in a x-ector field is 
represented by a number and a direction. 
Examples of scalar fields: sound field near a 
horn, light intensity- near a bulb, temperature 
near a heater. Examples of vector fields: gravita- 
tional field of earth, electric fields near chained 
bodies, magnetic fields near magnets. 

11. To find the gravitational field at a (X)int place a 
known mass at the p)oint, and measure both the 
direction and magnitude of the force on it The 
direction of the force is the direction of the field: 
the ratio of the magnitude of force and the ma&s 
is the magnitude of the field 

To find the electric field, place a known 
positiN-e charge at the p>oint, and measure the 
direction and magnitude of the force on the 
charge. The direction of the force is the 
direction of the electric field The ratio of the 
magnitude of the force and the charge is the 
magnitude of the field 

Note: To determine the force in either case, 
one could observe the acceleration of a 
known mass or determine vs+ial addi- 
tional force must be introduced to 
balance the original force. 

12. The corresponding forces would also be dou- 
bled and therefore the ratios of force to mass 
and force to charge would be unchanged 

13. The negatix'e test bod>- will experience a force 
upward. 



14. 



9 

1 X 10 *N 



3 X 10 *C 
_j= 33 X 10* N/C up 

15. 'f'= qE 

= (5 X 10 *Clt33 X 10* N/CI 
= 1 7N up 

16. I'he field concept axwids the necessity- of using 
the Coulomb force law to find the force t»et%\tH*n 
ex-etA fviir of charged jwrticles w+ien \-ou want to 
find the fon~e on a charge at a particular |X)int 

17. If the droplets or spheres are charge*! nrgafnrA 
they will exjx'rience an electric for»"e in the 
<lin*ction op|>osite to the field direction 

IH. C^harge comes in Iwsic units: the charge of the 
electn»n 

19. A negalixT' charge (-> must also appear s<ime 
when* inside the same closed s\-stem iFor 
example, an electnm se|varates fr«im an atom 
leaving the atom ixisitiwK charg«*d i 

20. It pniduced a steadv curnMil for a \onf^ period of 
time 



ANSWERS TO END-OF-SECTION QUESTIONS 



2«3 



21. The voltage between two points is the work 
done in moving a charge from one point to the 
other divided by the magnitude of the chaise. 

22. No. The potential difference is independent of 
both the path taken and the magnitude of the 
charge moved. 

23. An electron volt is a unit of enei^gy. 

24. If the voltage is doubled the current is also 
doubled. 

25. It means that when a voltage is applied to the 
ends of the resistor and a current flows through 
it, the ratio of voltage to current will be 5 x lO*. 

26. Apply several voltages to its ends ^lnd measure 
the current produced in each case. Then find 
the ratios V// for each case. If the ratios are the 
same, Ohm's law applies. 

27. The electrical energy is changed into heat 
energy and possibly light energy. (If the current 
is changing, additional enei^ transformations 
occur; this topic will be discussed in Chapter 
16.) 

28. Doubling the current results in four times the 
heat production (assuming the resistance is 
constant). 

29. The charges must be moving relative to the 
magnet. (They must in fact be moving across the 
field of the magnet.) 

30. It was found to be a "sideways" force! 

31. Forces act on a magnetized (but uncharged) 
compass needle placed near the current. The 
magnetic field at any point near a straight 
conductor lies in a plane perpendicular to the 
wire and is tangent to a circle that is in that 
plane and has its center at the wire. The general 
shape of the magnetic field is circular. 

32. Ampere susp>ected that two currents should 
exert forces on each other. 

33. (b), (c), (d) 

34. (b), (c), (e) 

35. The mcignetic force is not in the direction of 
motion of the particle: It is directed off to the 
side, at an angle of 90° to the direction of motion. 
The magnetic force does not do any work on the 
particle, since the force is always perpendicular 
to the direction of motion. 

36. Gravity always acts toward the center of the 
ecirth and is proportional to the mass. (It is 
independent of the velocity.) 

The electric field acts in the direction of the 
field (or opposite to that direction for negative 
charges), is proportional to the chai^ge on the 
object, and is independent of the velocity of the 
object. 

The magnetic field acts perpendicularly to 
both the field direction and the direction of 
motion, is proportional to both the charge and 
the velocity, and depends on the direction in 
which the object is moving. 

Chapter 15 

1. The single magnetic pole is free to move, and it 



follows a circular line of magnetic force around 
the current-carrying wire. 

2. Faraday is considered the discoverer of elec- 
tromagnetic induction because he was the first 
to publish the discovery, and because he did a 
series of exhaustive experiments on it. 

3. the production of a current by magnetism 

4. The loop is horizontal for maximum current, 
vertical for minimum. The reason is that the coil 
is cutting lines of force most rapidly when 
horizontal, and least rapidly when vertical. 

5. It reverses the connection of the generator to the 
outside circuit at every half-turn of the loop. 

6. It comes from the mechanical device that is 
turning the coil in the magnetic field. 

7. Use a battery to drive current through the coU. 

8. Batteries were weak and expensive. 

9. An unknown workmcin showed that the dynamo 
could run as a motor. 

10. too glaring, too expensive, too inconvenient 

11. an improved vacuum pump 

12. A small current v^ have a large heating effect if 
the resistance is high enough. 

13. Cities becemie larger, since easy transportation 
from one part to another was now possible; 
buildings became taller, since elevators could 
carry people to upper floors; the hours available 
for work in factories, stores, and offices became 
much longer. 

14. There is less heating loss in the transmission 
wires. 

15. A current is induced in the secondary coil only 
when there is a changing current in the primary 
coil. 

Chapter 16 

1. a magnetic field 

2. the small displacement of charges that accom- 
panies a changing electric field 

3. The four principles are: 

(1) An electric current in a conductor produces 
magnetic lines of force that circle the 
conductor. 

(2) When a conductor moves across externally 
set up magnetic lines of force, a current is 
induced in the conductor. 

(3) A changing electric field in space produces a 
magnetic field. 

(4) A changing magnetic field in space produces 
an electric field. 

4. It was practicaUy the s<ime as the speed of light 
determined by Fizeau; they differed by only a 
little more than 1%! 

5. "Maxwell's synthesis" is his electromagnetic 
theory in which he showed the relationship 
between electricity, magnetism, and light. 

6. that electromagnetic waves exist, that they travel 
at the speed of light, and that they all have the 
ordinary properties of light, such as reflection, 
refraction, ability to form standing wa\'es, etc. 

7. a loop of wire 



ANSWERS TO E\D-OF-SECTION QL'ESTIO\8 



8. They have very great wavelengths (&X}m tens to 
thousands of meters! 

9. The signals travel in nearly straight lines and 
would otherwise pass into space instead of 
following the earth's curvature. 

10. the higher the frequency, the greater is their 
penetration of matter 

11. A radar wavelength of 1 m is about 2 x lO" (2 
million) times that of green light, which is about 
5 X 10 ^ m. 

12. X rays are produced by the sudden deflection or 
stopping of electrons; gamma radiation is 
emitted by unstable nuclei of radioactive mate- 
rials. 

13. It was almost unthinkable that there could be 
waves without a medium to transmit them. 

14. Albert Kinsteins (in his theoiy of relativity) 

Chapter 17 

1. The combining cap>acity (or valence) of sulfur in 
HjS is -2. In SO3, sulfur has a combining 
capacity of +6. 

2. In Al,203, aluminum has a combining capacity of 
+3. 

3. When elements combine, the atomic massy4 las 
compared to hydrogen, A = II is related to the 
reacting mass m (in grams) and the combining 
capacity v as follows: 

A = mv 
In CH^, the combining capacity of carbon is 
V = 4. Therefore, the atomic mass of carbon is 
given by 
A = mv 
= 3gx4 
= 12 (compared to one atom of hydrogen) 

4. Atomic Combining 

Mass Capacity Mass 

Element \A) (v) (grams) 

1 
1 
2 
2 
2 
3 



Hydrogen 


1.008 


Chlorine 


35 45 


Oxygen 


16.00 


Copper 


63.54 


Zinc 


65.37 


Aluminum 


26.98 



Chapter 18 



1. They could be deflected by magnetic and 
electric fields. 

2. The mass of an electron is about 1,800 times 
smaller than the mass of a hvdn)gen ion 

3. (1) Identical electrons vven> emitted by a x-ariety 
of materials: and (21 the mass of an electron was 
much smaller than that of an atom 

4. All other values of charge he found ui'n' 
multiples of thai lowest value 

5. Kewer eU'ctnins aiv emiltetl. hut with the same 
avi«rage enei^- as iM'fonv 

6. rhe awr.ige kinetic iMierj^ of the emiltetl 
eleclnins dinn'a.ses until below some fre<jiienr> 
value, none an* einillcd at all 



ght soorc* 



E\oc'ja*ed "ube 




"^^y—k £ 



Bottery 



8. The energy of .the quantum is proportional to 
the frequencv' of the wave, £ = hf. 

9. The electron loses some kinetic energy in 
escaping from the surface. 

10. The maximum kinetic enetgv' of emitted elec- 
trons is 2,0 eV. 

11. VXtien X rays passed through material, say air 
they caused electrons to be ejected from 
molecules, and so produced -♦- ions 

12. (II not deflected b> magnetic field: i2isho\v 
diffraction patterns when passing through crys- 
tals; (3) produce a pronounced photoelectric 
effect 

13. Ill diffraction pattern formed b\' "slils" v\ilh 
atomic sp>acing (that is. crystalsl; (21 energy of 
quantum in photoelectric effect: I3I their gr^al 
fjenet rating |X)wer 

14. For atoms to be electrically neutral, they must 
contain enough positive chai^ge to balance the 
negative charge of the electrons the^' contain 
but electrons ar^ thousands of times lighter 
than atoms 

15. There are at least two reasons: FirstK the facts 
never arc all in, so models cannot wail that long 
SecondU . it is one of the main functions of a 
model to suggest what some of the facts las \el 
undiscovei^) might be. 

Chapter 19 

1. Ihe source emits light of onK certain frequen- 
cies and is therefore pmfwbK an excittnl gas 

2. 1 he source is pnibabK made up of two parts an 
inside part that pnxluces a continuous spec- 
trum: and an outer la\T>r that absorbs onl\ 
certain frequencies 

3. I.ighl fix>m wry distant stars produces spectra 
that arr identical with those piDducoti b> 
elements and com|xiiinds here on earth 

4. None He predirletl I hat they would exLst 
Iwcause Ihe malhemalirs was so neat 

5. carefxil measurement and tabulation of data on 
spectral lines, together with a liking for 
nialhematical games 

8. .M this |Hiinl 111 Ihe dewlopment of the book, 
one cannot say what s|HMifM-all\ acrounls for 
Ihe corn*clness of Balmer s fomuila iThe expla- 
nation requirT»» atomic theory, which is >vt to 



ANSWERS TO END-OF-SECTION QUESTIONS 



265 



come.) But the success of the formula does 
indicate that there must be something about the 
structure of the atom that makes it emit only 
discrete frequencies of light. 

7. They have a positiv-e electric charge and are 
repelled by the positive electric charge in atoms. 
The angle of scattering is usually small because 
the nuclei cire so tiny that the a particle rarely 
gets near enough to be deflected much. How- 
ever, once in a while there is a close approach, 
and then the forces of repulsion are great 
enough to deflect the a peirticle through a large 
angle. 

8. Rutherford's model located the positively 
charged bulk of the atom in a tiny nucleus: in 
Thomson s model the positive bulk filled the 
entire atom. 

9. It is the number iZi of positive units of charge 
found in the nucleus, or the number of electrons 
around the nucleus. 

10. three positive units of change 

11. Atoms of a gas emit light of only certain 
frequencies, which implies that each atoms 
enei-gj' can change onl\' by certain amounts. 

12. None. He assumed that electron orbits could 
have only certain values of angular momentum, 
which implied only certain enei^' states. 

13. All hydrogen atoms have the same size because 
in all unexcited atoms the electron is in the 
innermost allowable orbit. 

14. The quantization of the orbits prevents them 
from having other arbitrary sizes. 

15. Bohr derived his prediction from a physical 
model. Balmer onl\- followed a mathematical 
analog\- 

16. According to Bohr s model, an absorption line 
would result from a transition within the atom 
from a lower to a higher energ\' state ithe energv' 
being absorbed from the radiation passing 
through the matericill. 

17. (a) 4.0 eV (bl 0.1 eV (cl 2.1 eV 

18. The electron arrangements in noble gases are 
very- stable. When an additional nuclear charge 
and an additional electron are added, the added 
electron is bound very weakly to the atom. 

19. Period I contains the elements with electrons in 
the K shell only. Since only two electrons can 
exist in the K shell, Period I will contain only the 
two elements with one electron and two 
electrons, respectively. Period II elements have 
electrons in the K (full I and L shells. The L shell 
can accommodate eight electrons, so those 
elements with only one through eight electrons 
in the L shell will be in Period II: and so forth. 

20. It predicted some results that disagreed with 
experiment: and it predicted others that could 
not be tested in any known way. It did, however, 
give a satisfactory explanation of the observed 
frequency of the hydrogen spectral lines, and it 
provided a first physical picture of the quantum 
states of atoms. 



Chapter 20 

1. It increases, without limit. 

2. It increases, approaching ever nearer to a 
limiting value, the speed of light. 

3. Photon momentum is directly proportional to 
the frequency of the associated wave. 

4. The Compton effect is the scattering of light (or 
X ray) photons from electrons in such a way that 
the photons transfer a part of their energv' and 
momentum to the electrons, and thus emerge as 
lower-frequenc>' radiation. It demonstrated that 
photons resemble matericd particles in posses- 
sing momentum as well as energ>'; both energy 
and momentum are conserved in collisions 
involving photons and electrons. 

5. by analogv' with the same relation for photons 

6. The regular spacing of atoms in crystals is about 
the same as the wavelength of low-enei^ 
electrons. 

7. Bohr invented his postulate just for the purpose. 
Schrodinger's equation was derived from the 
wave nature of electrons and explained many 
phenomena other than hv'drogen spectra. 

8. It is almost entirely mathematical: no physical 
picture or models can be made of it. 

9. It can. But less energetic photons have longer 
associated wavelengths, so that the location of 
the particle becomes less precise. 

10. It can. But the more ener-getic photons will 
disturb the particle more and make measure- 
ment of velocity less precise. 

11. They are regions where there is a high probabil- 
ity of quanta arriving. 

12. As with all probabilitv' laws, the average behavior 
of a large collection of particles can be predicted 
v\ith great precision. 

Chapter 21 

1. It was phosphorescent. Becquerel wrapped a 
photographic plate in thick black paper to keep 
light out. Then he placed a small piece of the 
uranium compound on top of the black paper 
and allowed sunlight to fall on it. L'pon 
developing the plate he found the silhouette of 
the mineral sample recorded on the plate. When 
he tried putting metcillic objects between the 
sample and the plate, he found their outlines 
recorded even when a layer of glass was also 
introduced to eliminate possible chemical ac- 
tion. 

2. No treatment was needed: the emission was 
spontaneous. 

3. They were puzzling because they needed 
nothing to start them, and there was nothing 
that could stop them. They were similar to X 
rays in that both were very penetrating radia- 
tions, and both could ionize. 

4. It is not, although slight differences might be 
observed because of the other element absorb- 
ing some of the radiation. 



ANSWERS TO END-OF-8ECTION QUESTIONS 



5. The radioactivity was much greater than ex- 
pected for the amount of uranium in the ore. 

6. separating it from harium, which is almost 
identical chemically 

7. From most to least penetrating: y, /3, a. 
Penetrating power is inversely related to ioniz- 
ing power hecause rays that are easily stopped 
(have low penetrating power) are so because 
they are expending their energy ionizing many 
atoms of the slopping material (high ionizing 
power), and vice versa. 

8. Beta particles were found to have the same q/m 
ratio as electrons. 

9. Alpha rays were deflected much less than /3 rays 
by a magnetic field. 

10. Its emission spectrum, when caused to glow by 
an electric discharge, was the same as helium's. 

11. It occurs when only a single pure element is 
present, and is not affected by chemical 
combinations of that element. 

12. An example would be the decay of radon into 
polonium with the emission of an a particle (Rn 
— » Po + He). It was contrary to the ideas of 
indivisibility of atoms held by nineteenth cen- 
tury chemists. 

13. (1) Many of the substances in a series have 

similar chemical properties. 

(2) There are only small percentage differences 
in atomic mass. 

(3) Many of the substances decayed veiy rapidly 
into something else; all three kinds of rays 
are given off by the mixture. 

14. At the start, the emission will be relativ'ely slow 
and will consist entirely of a particles. Later, the 
emission will bo greater and will contain, 
besides alpha particles, /3 and y rays. 

15. The law of radioactivi* decay is a statistical law: it 
says nothing about how long it will take any 
given atom to decay. To s|>ecih' a "life time " 
would be to predict when the last atom would 
decay. Scientists do not know any way of doing 
that. 

16. 1/16 of it. 

17. We do not know. The statistical half-life laws do 
not apply to small numbers of atoms, and no 
other laws make pnnlictions al)uut individual 
atoms, or e\iMi about small numbers of atoms 

Chapter 22 

1. 1 hey wri^ chfniiriilly the siiini- as pn'\iousl\ 
known elements. 

2. The atomic mass ecjuals 12 amu It occupies 
|H)sition B in the list of I'leinents 

3. decreases 4 units: stays essentially the same 

4. DecnMses b\' 2 + charges: incnMses In 1 -»- 
charge 

5. The niles anv 

(1) In a decay, the mass number decn-asi's In- 4. 
and the atomic number ilecn-ases b\ 2 

(2) In fi dtH-ay, the mass number remains the 
sjune. and the atomic number incn<aM>s by 1. 



(3) In y decay, both the mass number and the 

atomic number remain the same. 
In the Rutherford - Bohr model of the atom, the 
entire positive chaise and almost the entire 
mass are contained in the nucleus Since a, fi. 
and y rays are ejected from the nucleus, they 
will carry away from it both mass and chaise 
The a particle carries 2 positive charges and 4 
amu: hence rule H) The /3 panicle carries 1 
negative chaise and negligible mass; hence rule 
(2). The y ray has no mass and is uncharged; 
hence rule (3). 

6. by subtracting a p>article masses from the mass 
of the parent of the decay series 

7. It must have a "velocity selector" that v*iU allow 
only ions of a single speed to enter the ma^etic 
field This can be done with crossed electric and 
magnetic fields. 

8. (1) faint second line in mass spectrum of pure 

neon 

(2) different atomic masses of samples of neon 
separated by diffusion 

(3) more intense second line in mass spectrum 
of one of the samples separated b> diffusion 

9. .More massive atoms have a lower average speed 
and so diffuse more slowly than the less massive 
ones. 

10. tbPi"*^; platinum 

11. \A — 4). The rule is: Emission of an a particle 
results in a decrease in A of 4 units 

12. (Z -t- II The rule is: Emission of a negative ft 
particle results in an increase in Z of 1 unit 

13. an isotope of hvdrogen VNilh twice the atomic 
mass of ortlinan- hvdrogen 

14. Heavy water is the compound D.O In other 
words, it is made with heavy hvdrogen 
uleuteriumi rathc-r than ortlinar^ hvclntgen. 

15. rin' third isolj)|)«' has a \rr\- low abundance 

16. «(!'-' is the current standard It was chosen 
mainlv because it readilv forms manv com- 
(lounds and so is available fur measuring other 
masses by mass spectrograph techniques, 
which are much more accurate than chemical 
methods 

Chapter 23 

1. S«'veral atomic masses (which wen' not recog- 
nized as the average of several isoto|Jcsi were 
not close to whole multiples of the atomic mass 
of hydn>gen 

2. 12 protons and 6 electrons 

3. Nes. roughly .He' would contain 4 pn)ton» and 
2 elettrons inside tlw nucleus lit does nnl wi»rk 
out. howwvr, when vvr> carefiil mass mea- 
surements an- made I 

4. Ihe numlM'r of tracks obwrvvtl in a cloud 
chamlM*r did not include any that would 
com-sjHind to the original a j>article breaking 
up into fragments 

H. Kor y ravs the wa\ it kni>rk«Ml protons out of 
(MrafTin would l>»" a violation of the pnnciples of 
enei|^ and momentum conservation 



BRIEF ANSWERS TO STL'DY GUIDE QUESTIONS 



267 



6. A neutron has no charge, and so is not deflected 
by magnetic or electric fields; nor does it leave a 
track in a cloud chamber. 

7. The laws of conservation of momentum and 
kinetic energy were applied to neutron - proton 
and neutron- nitrogen head-on collisions. This 
yielded four equations in the four vciriables: rrin, 
^n' *'n (proton collision I, and Vn' (nitrogen 
collision). The latter three were eliminated, and 
m„ found. 

8. 7 protons and 7 neutrons 

9. a nucleus of 2 protons and 2 neutrons, 
surrounded by 2 electrons 

10. A neutron in the nucleus changes into a proton 
and a /3 particle, which immediately escapes. 

11. V\ithout the extra particle, there was no way to 
explain the disappearance of energy in /3 decay. 

12. The repulsive electric force exerted by the lai-ge 
charge of the hea\y nucleus on an a particle 
prevents it ftxjm reaching the nucleus. 

13. Protons have only a single charge. 

14. Some devices for producing projectiles are: Van 
de Graaff generators, linear accelerators, cyclot- 
rons, synchrotrons. Devices that detect nuclear 
reactions cire: cloud chambers, spark chambers, 
photographic emulsions, and bubble chambers. 

15. They have no electric chai-ge and so are not 
repelled by nuclei. 

16. uSi-* 

17. gC; 7 protons, 6 neutrons before; 6 protons, 7 
neutrons after 



Chapter 24 

1. No. In some nuclear reactions energy' is ab- 
sorbed. 

2. It can go off as y rays or as the kinetic energy of 
the product particles. 

3. The binding energy of the deuteron nucleus is 
the enei^ that would be required to breaik up 



10. 



11 



the nucleus into its constituent particles: a 
proton and a neutron. 

4. A nuclide with a high average binding energy is 
more stable. 

5. No. Light nuclei are lower on the curve than 
heavy nuclei. 

6. capture of a neutron by a uranium nucleus, 
followed by the /3 decay of the new nucleus 

7. neutrons 

8. a substance that slows down neutrons 

9. It slows dov%Ti neutrons well (because of the 
abundance of H atomsl, but it cdso absorbs many 
(to form "heavA'" waterl. 

by control rods made of a material that absorbs 
neutrons: the farther in the rods, the slower the 
reaction. 

The positively charged nuclei repel each other; 
high speeds cire necessary for the nuclei to come 
near enough in collisions to fuse. 

12. Since at very high temperatures the gas is 
ionized, a properly shaped magnetic field could 
deflect the chained particles away fix)m the 
walls. 

13. decreasing 

14. The protons in a nucleus repel each other with 
intense electric forces. 

15. The average binding energy curve suggests that 
each particle in the nucleus is bound only by its 
immediate neighbors. 

16. An excited nucleus becomes distorted in shape; 
electric repulsion between bulges then forces 
them apart. 

17. In the case of LT-^", the excitation ener^ due to 
neutron capture alone is less than the activation 
energy required for fission. For U-^"', the excita- 
tion energy is greater than the activation energy. 
They correspond to completed shells (or sets of 
energy statesi of protons and neutrons in the 
nucleus. 

Neither. They each have different strengths and 
weaknesses. 



18 



19 



Brief Answers to 
Study Guide Questions 



Chapter 1 

1. information 

2. speed = v = Id/ It 

Uniform speed occurs when the ratio distance/ 
time is constant. 

average speed = v^^ = total distance/total time 
slope = Ay/ \^ 

Instantaneous speed (the speed at a given mo- 
ment) is shown by the slope of the speed line at 
that moment. 

average acceleration = a^y = change in speed/ 
time interval for the change 



average speed Vgv = 20 m/sec 

(a) 6 cm/sec (b) 24 km (ci 025 min (d) 3 cm/sec, 24 

cm (e) 64 km/hr (f) 64 km/hr; 192 km (g) 5.5 sec (h) 

8.8 m 

She falls 228 m in 19sec.Afterafurther25sec,she 

has fallen a total of 528 m. 

(ai 1.7 m/sec (bi 3.0 m/sec 

The rabbit wins by 368 sec. 

3.6 X 10^ km 

discussion 

(a) 9.5 X 10'-' m (b) 2.7 x 10" sec or 8.5 vt (c)-(gl 

discussion 

For the blue bicycle, v'a^ = 16.7 m/sec. 



BRIEF ANSWERS TO STUDY QUIDE QUESTIONS 



12. 



13. 



a^v for 5 sec isGm/sec. Afteran additional 5 sec (a 
total of 10 sec), v = 60 m/sec. 

(a) (1) 5 m/sec, (2) 5 m/sec, 13) 5 m/sec, (4) 5 m/sec 

(b) The straight line for distance versu* time 
shows that the speeds were equal at all times (ci 
The instantaneous speed is 5 m/sec The slope is 
the same at all times. 

14. la) fastest in section CD; slowest in section BC (b) 
v^H = 44 m/sec; Vf^ = 3.6 m/sec; v, „ = 136 m/sec; 
^Ai> - 37.9 m/sec (c) Instantaneous sp>eed at point 
/ = 136 m/sec. 

'a' *'inji - 0.5 m/sec (at 10-sec mark); v,n„ = 1.5 
m/sec (at 25-sec mark) (b) a^^ = 0.06 m/sec^ 
25.6 m 
graph 
discussion 

19. Between 1 and 4.5 sec, 1.3 m/sec (b) 0.13 m/sec 

(c) 0.75 m/sec (d) 1.0 m/sec (e) 0.4 m (approx.) 

20. (a) DE was covered fastest; BC was covered 
slowest, (b) The line repnisenting tF (a resting 
interval) should be parallel to the horizontal axis. 
(c) Vaj for 8 weeks is 75 km/week, (d) The instan- 
taneous speeds at points P and Q are: Vp = 60 
km/week, Vy = 200 km/week. 

21. (a) graph lb) graph 

22. 795,454 cm/sec 

23. graphs; d versus f : 0, 9, 22, 39.5, 60.5, 86 cm (ap- 
prox.) at intervals of 02 sec; v versus / : 45, 65, 87.5, 
105, 127 cm/sec (approx.) at intervals of 0.2 sec 

24. discussion 

26. discussion ' 

Chapter 2 



IS. 

16. 
17. 

18. 



20. (a) 20 m/sec (b) -20 m/sec (c) 4 sec (dl 80 m (e) 
(f) -40 m/sec 

21. la) -2 m/sec' (b) 2 m/sec (c) 2 m/sec (d) 4 m (el -2 
m/sec If) 4 sec 

22. discussion 

23. (a) 4.3 welfe/suiig' (b) 9.8 m/sec* 

24. derivation 

25. derivation 

26. derivation 

27. discussion 

28. discussion 

29. discussion 

30. (a) a = 2.5 m/sec* (b) d = 397 m (if a, = 10 m/sec*, 
d = 405 m) (c) The block slides for 43 sec. and its 
final velocity is 9 m/sec. (di The distance is 330 m 
north and the final velocity is 58 m/sec north 
le) The final speed is 4 m/sec if) The final speed 
is 8 m/sec. ig) The acceleration is 15 m/sec^ 
Ih) The initial sjjeed of the ball thrown upward is 
31 m/sec. 

31. (a) The average speed v^,, = 2-5 m/sec and v, ^ = 
7.5 m/sec. lb) The average speed v^, = 2-5 m/sec. 
The average acceleration a = 2S m/sec*. (c) dis- 
cussion 

32. la) graph A: constant velocity: graph B: accelera- 
tion: graph C: acceleration: graph D: negatKe ac- 
celeration (deceleration) (bi graph A: backward: 
graph B: forward: graph C: backward: graph D: 
backward 



Chapter 3 



1. 


iiitorination 


* 


2. 


discussion 


3 


3. 


discussion 




4. 


discussion 




S. 


discussion 




6. 


la) Distance is 36 m, average speed is 6 m/sec, and 
distance for uniform acceleration is 36 m. (b) The 






equations used assume uniform acceleration 


4 


7. 


derivation 




8. 


(a), lb), Ic) 




9. 


discussion 


S 


10. 


discussion 


6 


11. 


derivation 


7 


12. 


17 years; $10,000 




13. 


(a) 57 m/sec' lb) 710 m (c) -190 m/sec* 


8 


14. 


la) true lb) true (based on measurements of six 
lower positions) Ic) true (d) true (e) true 


9. 


IS. 


derivation 




16. 


(a) Position d v 


10 




A + + 


•> 




B + + 


12 




C + 


13 




D + - 


14. 




E - - 






(b) derivation (c) discussi<}n 




17. 


discussion 




18. 


(a) -5 m lb) -10 m/sec (cl - 15 m 


IS 


19. 


la) 120 m/sec lb) 15 m (cl 2 sec (dl 20 m 


16 




(e) -20 m/»cc 


17 



infonnation 
discussion 

(a) Mechanics deals with forces D^Tiamics deals 
with forces producing motions Kinematics deals 
with motion with no concern for the forces act- 
ing, (b) (1) d, scalar i2) /. N'ector i3) v, scalar i4)/. 
scalar (5) m. scalar (6) JTveclor (7) (, scalar (8) T! 
vector (9) v, \-ector 

(a) 3 blocks east from the starting [K)inl (b) IS 
blocks (c) Part lai deals with wcturs: part lb) deals 
with scalars 

la) constrviction lb) 2 4 units west 
pnH)f 

Air n\sistance equals the earth s graNilalional at- 
traction of 750 N. 
discussion 

la) In equilibrium, the net force on a bod>' is zero 
lb) In equilibrium, a b(xly ma> be ID at rest or 
12) in uniform motion 
discussion 
discussion 
discussion 
discussion 

la) Acceleratum is lit pn)|>onional to the force 
acting. 121 in the dinn'lion of the forrr and 13) 
inviTseh pni|M>rtional to the nta^s (bi The accel- 
eration IS 15 m/sec^ east 
28 X 10 • hr/sec 
discussion 
discussion 



BRIEF ANSWERS TO STUDY GUIDE QUESTIONS 



269 



18. discussion 

19. (c) 24 N out (d) 14.8 N left (e) 0.86 N north (f) 9.0 kg 
(g) 0.30 kg (h) 0.20 kg (i) 3 m/secr east (j) 2.5 m/sec- 
left (k) 2.50 m/sec- down 

20. (a) 2.0 X 10- m/sec-; 7.8 x 10- m/sec (b) discussion 

(c) 2.4 X 10- m/sec- 

21. discussion 

22. 2.0 kg 

23. discussion 

24. (ai 1 kg, 9.81 \ in Paris, 9.80 N in Washington (bl 
indixidual calculations 

25. (al Since the pound is a unit of force (weight! and 
the kilogram is a unit of mass, they cannot be 
directly con\erted. V\'eight is a measure of the 
earth's gra\itational attraction at its surface and 
therefore comparisons can only be made on 
earth. Ibl Student answers will varv'. (cl Student 
answers will vary . (For each 1 kg of mass lifted, 9.8 
N of force are required.) 

26. discussion 

27. (al -5 X 10-3 m/sec- (b) 10 m/sec (c) 10 x 10 -' 
m/sec 

28. The acceleration of the girl is 2 m/sec-. The force 
on the boy is 80 N, and his acceleration is 1.14 
m/sec- . 

29. discussion 

30. discussion 

31. (a) diagram (b) 1.7 x lO" m/sec- (c) (6 x 10")/1 

(d) diagram 

32. (al 862 N, 750 N, 638 \ (bl The same as in (a) for a 
scale calibrated in nev\1ons. (cl discussion 

33. hints for sohing motion problems 

34. (a) The object will mo\e 150 m to the right, (bl The 
speed will be 40 m/sec, (c) The net force is 16 X. 
(dl The mass is 5 kg. (el 16 m/sec- (f) 5 sec (g) zero 

Chapter 4 

1. information 

2. 3.8 m/sec*, 5.1 sec, mass decreases 

3. discussion 



4. (a) 



8 



10 
5 



20 



15 



25 



10 sec 
40 m 



30 m 



50 m 
5 m/sec 



Horizontal distance 

Iv, = 4 m/sec) 

Vertical distance 

(Vy = 3 m/sec) 

(b) Total distance 

(c) Total velocity 

5. derivation 

6. 1.3 m, at an angle of 67° below the horizontal; 5.1 
m/sec, 79° below the horizontal 

7. (a) fx = 2.5 sec (b) dj = 30.6 m ic) v^ 12.5 m/sec 

8. f = 3.1 sec; time of fall does not depend on hori- 
zontal velocity. 

9. (a) t = 4.1 sec (b) The time of fall is independent of 
the horizontal velocity, (c) Vj = 40 m/sec (dl v, 
remains at 8 m/sec. 

10. discussion 

11. discussion 

12. discussion 

13. discussion 



14. 6.0 X 10"-min, 3.0 x 10 -min,2.2 x lo -min,1.3 
X 10 - min 

15. discussion 

16. discussion 

17. table completion 

18. (a) 2.2 X 10 '" m/sec- (bl 4 x 10-" N (c) 
approximatelyl/100 

19. approximately 10^ N 

20. discussion 

21. (a) V = 6J2 m/sec (b) a, = 19.2 m/sec- (c) F,. = 38.4 
N 

22. V = 3.2 m/sec 

23. T = 3.4 sec; because no force is considered, no 
mass is involved. 

24. a = 1,970 m/sec- 

25. (al Syncom 2 (bl Lunik 3 (c) Luna 4 (dl does not 
change 

26. V = 7,690 m/sec; the orbital speed does not de- 
pend on mass. 

27. F = 683 \ 

28. The accelerating force is 114 N. T = 931 min 

29. 5.1 X 10^ sec or 85 min, 7.9 X 10^ m/sec 

30. discussion 

31. 7.1 X 10* sec or 120 min 

32. (a) 3.6 X 10- sec (b) 36 km (c) discussion 

33. Al = im/F) (vo - v) 

34. discussion 

35. essav 



Chapter 5 



1. intormation 

2. discussion 

3. (al 674 sec (bl 0.0021% 

4. table 

5. discussion 

6. discussion 

7. discussion 

8. 102°, 78°, 78° 102° starting with the upper right 
quadremt 

9. (a) 15° (bl geometric proof and calculation; about 
12,100 km 

10. a, b, c, d, e, f 

11. discussion 

12. discussion 



Chapter 6 

1. information 

2. diagram construction 

3. discussion 

4. 11 times; derivation 

5. Copernicus calculated the distances of the 
planetcuy orbits from the sun and the periods of 
planetary' motion around the sun. The Copemi- 
can system was more simple and harmonious 
than that of Ptolemy. In addition, the orbits 
began to seem like the paths of real planets, 
rather than mathematical combinations of cir- 
cles used to compute positions. 

6. The Copemican system led to a change in the 
order of importance of the earth and sun. The 



270 



BRIEF ANSWERS TO STUDY QUIDE QUESTIONS 



sun became dominant while the earth became 
"just another planet " These philosophical re- 
sults were more imfxirtant than the geometrical 
change. 

7. discussion 

8. discussion 

9. 2.8 X 10'' AU 

10. discussion 

11. discussion 

12. discussion 

13. discussion 

14. discussion 

Chapter 7 

information 

about 1/8 of a degree; about 1/4 of the moon's 

diameter 

discussion 



discussion 
8. a + c 

7. If the two foci (tacks) are placed at the same point, 
you will draw a circle. As the foci are separated, 
the ellipse becomes longer and thinner. 

8. e = da = 5/9 = 0.556 

9. la) same as sketch in question 6 (b) c = A - P, a = 
A + P,A = (a +c)/2, P = (a -c)/2(c)fl,v =a/2 = (A 
+ P)/2 

10. The second focus is empty. 

11. discussion 

12. (a) 43 (b) P = 2.5 cm; A = 7.5 cm (c) A = 45 cm, c 
= 40 cm, a = 50 cm 

13. 0J209 

14. 0.594/1 

15. (a) 17.9 AU (b) 35.3 AU (c) 0.54 AU (d) 66/1 

16. T = 249 years 

17. k = 1.0 for all three planets 

18. discussion 

19. 7Vfl' = 8.9 X 10 '* secVm' 

20. Tj = 8 T, 

21. n = 4.03 X 10" m 

22. 8.9 X 10 '* secVm-^ for each satellite 

23. (a) student sketches lb) student calculations Id 
Yes, Kepler's law of [leriods applies. 

24. discussion 

25. (lisciission 

26. Kepler expected the lhi't)r\' lo Irad to pn'cli( lions 
agn*ring closely with new observations Kepler 
sought algebraic patterns rather than geometric 
patterns Kepler sought physical causes for ob- 
servetl motions. 

27. Kmpirical laws are generalizations based on ob- 
servations. They an* n'liablo bases for theonMical 
speculations 

C'hapter 8 

1. infonnation 

2. la) a straight line at uniform speed lb) caused the 
planets to de\iali> tntiii a straight line lii dinn-ted 
lowartl a center (the suni (d I varies inwrwly with 
the squan> of the distance 11//!') 



6. 
7. 
8. 
9. 
10. 

11. 



12. 



lal It did not fall down, " but toward the center 

of the earth lb) 1/60^ of the earth's attraction on 

the apple; 2 7 = 10 * m/sec* (c) a, = 2.71 x 10* 

m/sec^. The two values almost agree. 

The Newlonian question is really "What holds 

the moon down? " 

Every object in the universe attracts eveiy other 

object with a gravitational force F = C(Afm/fl*l, 

where F is the force between objects, M and m 

are the masses of the objects. R is the distance 

between the centers of the objects, and G = 6£7 

X 10 " Nm'/kg' Available evidence show^ no 

change in G with time or p>osition. 

yes, to about 1% agreement 

discussion 

discussion 

derivation 

J-; = /47r'v n' 



13. 

14. 
15. 
16. 
17. 



18. 



19. 
20. 
21. 

22. 
23. 
24. 
25. 
26. 
27. 
2M. 
29. 
30. 



\ G ) m 



F = 6.67 X 10-*N;a,.ooo = 6 67 x 10 " m/»ec*;a,» 
= 6.67 X 10"'* m/sec*; these accelerations are 
veiy small; they might be measurable with a tor- 
sion p>endulum or a modem optical device. 
Unexplained irregularities in the motion of 
Uranus led to the prediction that one or more 
outer planets existed Calculations led to pre- 
dicted positions that were rather accurate for 
Neptune and approximate for Pluto The belief 
that such planets existed led to the svslematic 
search for them. 

lal 1.05 X 10^ daysVAU* lb) discussion Id discus- 
sion 
42.600 km 

5 98 X 10^* kg 

6 04 X 10^« kg 

The moons tidal force on the water on the dis- 
tant side of the earth is less than on the solid 
earth. As a result, the earth is pulled awav from 
the water. 
The ineriial motion is along a straight line. ,k = 

vr. The accelerated motion is toward the earths 
1. 



center, v 



-at' 



(a) 5.52 X 10* kg/m* Ibl discussion 

730 X 10*» kg 

(a) 5 99 X 10* sec, or 1 66 h (bl 355 km/sec (d 

collisions 

table 

alxiut 170 times as great 

17 7 AU: 60 AU; 34.8 AU 

tierivations 

discussion 

discussion: no 

discussion 

II IS useful today. 

tliscussion 



Chapter 9 

1 infonnation 
2. discussion 



BRIEF ANSWERS TO STUDY GUIDE QUESTIONS 



271 



10. 



11. 
12. 
13. 
14. 
15. 
16. 



17. 

18. 
19. 

20. 
21. 
22. 



23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 

31. 
32. 



33. 
34. 



(a) yes (b) the solar system 
discussion 

no 

discussion 

(ai 2202 g (b) The mass of the solids before and 

after the reactions cire equal. 

(al The mass is 60 g on the earth and on the moon. 

(b) Mass is an attribute of material. Weight de- 
scribes a gra\itational attraction, (cl Nothing is 
reported about masses. 

la) no change in total momentum 

(b) Disk A 40 kgm/sec west 
Disk B 150 kgm/sec north 
Disk C 20 kgm/sec east 
Disk D 20 kgm/sec east 

(c) The final velocit\' v., = 6 m/sec north, (di 
Momentum is a vector quantity', and opposing 
vectors can cancel, as in part (b) where A cancels 
IC + D). 

(al all except Va' (which =Vb') 

Id 0.8 m/sec 

dictionaiy comment 

3.3 X 10'' kg 

discussion 

discussion 

yes 

(al Af = 4 sec lb) Final velocit\' = 20 m/sec by 

both solutions. Ic) Some problems are easier to 

solve with the momentum fonnula, but it is not 

more basic. 

discussion 

12 X l(f kgm/sec; 4 x lO- N; 30 m 

lal about 100 m/sec lb) about 4.6 kg m/sec (c) less 

than 0.003 sec id) at least 1-5 x lO^ N 

yes 

derivation 



Kinetic 



la) 


At = 


m(Vo - 
F 


L^ 


(b) m(Vo 


-V) 




(r) 


m(Vo 


-V) 





deri\'ation 

10 m/sec 

10.5 X 10" kg m/sec 

discussion 

discussion 

discussion 

discussion 

(a) 0.8 X mass of ball (bl -0.8 x mass of ball (c) 1.6 

X mass of ball (dl depends on system considered 

discussion 

(a) The total kinetic energ>' of an isolated system 

involving onl\' elastic collisions is constant. Ibi 

The total kinetic energy does not change. Ic) KE = 

85 J (dl The totcil kinetic ener^' does not change. 

table 

(a) The total mass is 18 g. The total momentum is 

18 g • cm/sec east . The total kinetic energy is 1 .227 

J. 



(b) 


Mass 


Momentum 


Energy 


l.open, 


18 g 


unknown 


unknown 


elastic 








2. open, 


18 g 


unknown 


unknown 


inelastic 








3. closed, 


18 g 


18 g- cm/sec 


1227 J 


elastic 




east 




4. closed, 


18 g 


18 g- cm/sec 


1,227 J 


inelastic 




east 




35. derivation 








36. Both speeds equal v/2. 


)ut in opposite directions 



Chapter 10 



12. 
13. 



- u (blno(c)no(d)nole)yes(f)iii 
10'^ electrons 



1. infonnation 

2. discussion 

3. (a) V, — uandvj 
(gl discussion 
5 X lO"'-^ J, 2 X 
d = 10 m 

(a)673J(b)4.5 X 10*J(c)3.75 x ltf»J(d)2.7 X ia"J 
(a) 2 m/sec^; 30 sec; 60 m/sec (b) 60 m/sec 
(a) -90 J lb) 90 J (c) 18 x lO- N 
23 X 10- J 

10. (a) 22 X 10"' J lb) 5.4 x lO"- J 

11. (a) 02 m (b) 7 X lO' J 
discussion 

la) PE = mgh, where h is stretch ibi A(PE) = 
— AiKE) Id The energv is chcinging between KE, 
gravitational PE, and elastic PE. 

14. la) 1.1 X 10'- sec labout 3 x lO\vears) Ibl 1.6 x 
10"-' m (d The PE relates the rock to the earth. 
Potentiiil energ\' is always relative to a frame of 
reference, 
discussion 
discussion 

17. derivation 

18. discussion 

sketch 

proof 

(a) 96 X 10" J (b) 8.8 x itf^ m Id 48 x lO^ \ (dl 
discussion (e) discussion 
discussion 

23. (al d = 102 m (bl 93 W lapprox.l 

24. Engine A: 50^ efficient, 5 W power; Engine B: 40% 
efficient. 8 W power 

25. discussion 

26. discussion 

27. discussion 

28. lal 3.5% (bl 0.75 metric ton/sec 

29. (a) 15.5 Ibl 7.3 Id 5.8 

30. no 

31. c, a, e, b, d. f 

32. no heat loss; reversible; 100% efficient 

33. To be 100% efficient, the cold side of a heat engine 
must be at absolute zero. 

34. lal >1000 (bl discussion 

35. 1/8°C: no 

36. lal fossil fuel: 0.60: nuclear: 0.50 Ibl fossil fuel: 0.67 
M\\: nuclear: 1.00 MW 



15. 
16. 



19. 
20. 
21. 

22. 



272 BRIEF ANSWERS TO STUDY QUIDE QUESTIONS 



37. 1/4 kg 
3H. 21 5 days 

39. discussion 

40. discussion 

41. discussion 

42. discussion 

43. HIT, - TjjT.Tj 

44. (a) discussion lb) greater in lower orbit Ic) less (dl 
less (e) discussion 

la) discussion lb) i: all three; ii: all three; iii: A/Y; iv: 
AH; v: all three; vi: AH 
discussion 
derivation 

48. ice: 12 kJ/°K; water -12 kJ/IC; no change in the 
universe 



45. 



46 
47 



Chapter 11 



1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 

14. 
IS. 



16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
2S. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 
33. 
34. 



35. 
36. 
37. 

3H. 



information 

discussion 

discussion 

A distribution is a statistical description. 

discussion 

no 

discussion 

la) 10" m lb) 10 » m 

la) 10^' lb) 10'" 

Zero meters 

10.5 km 

la) P = 66 N/m^ lb) T = lOQ-C 

shoes: about 1/7 atm; skis: about 1/60 atm; skates: 

about 3 atm 

derivation 

P = VJrr, therefore, P oc D, P oc T, and D a 1/T 

when the other property is constant. The ideal 

gas law does not apply to veiy dense gases, or to 

gases when they liquil^. 

discussion 

discussion 

discussion 

derivation 

no change 

pressure, mass, volume, temperature 

discussion 

disi'ussion 

di.scussion 

discussion 

discussion 

derivation 

discussion 

Temperature will rise. 

no 

discussion 

discussion 

discussion 

The mclling of ice is an im'xiTsilile pn)crss Ix*- 

«"ause thr onlni'd arrangi'iiifnt of iii()li*('ulf*s in 

the ice <r\stals is lost and entropy increases 

discussion 

(lis<-ussion 

(li.s( ii.ssion 

(li.scii.ssion 



39. discussion 

40. derivation 

41. derivation 



Chapter 12 



1. 


information 


2. 


discussion 


3. 


discussion 


4. 


construction 


5. 


construction 


6. 


discussion 


7. 


discussion 


8. 


construction; F Ibeal&l = Fj - F, 


9. 


discussion 


10. 


discussion 


11. 


construction 


12. 


derivation 


13. 


no; discussion 


14. 


la) V = 343 cm/sec lb) 5.86 m let T = 33 x 10* sec 




everywhere 


15. 


la) 3/4 L lb) 2/3 L Ic) 1/2 L 


16. 


f = 80 Hz 


17, 


la) X = 4L 




lb) X - *^ 
2n + 1 




Ic) X = 2L, X = -^ In = 0, 1, 2, 3, etc.) 
n + 1 


18. 


discussion 


19. 


maximum 


20. 


100 and 1.000 Hz; yes 


21. 


discussion 


22. 


discussion 


23. 


X = 625 X 10 * cm 


24. 


construction 


25. 


straight line 


26. 


2 


27. 


construction 


28. 


1 
4k 


29. 


discussion 



30. Speed is independent of frequency. Speed in a 
medium depends on stiffness and density of the 
medium. 

31. Two straight-line waves inclined toward each 
other 

33. la)*;^ = ^BAD(b)flB = --CDAlclX^^ =BD(d»X, = 
AC le) deri>'ation If) deriv-ation 

33. Xp = 035 m; X. = 025 m 

34. V = 2-5 cm/sec 

35. discussion 

36. la) 127 X 10 " W lb) 8 X 10" mosquitoes Id 
subway train 

37. 2d = »1 

38. I air 03375 m 

1 000 H/ < sea water 1 44 m 

( steel 4 8 m 
One tenth of each of these values (or 10O00 Hi; 
discussion 

39. 3 X lO^Hz; 23 x 10' lU 



BRIEF ANSWERS TO STUDY GUIDE QUESTIONS 



273 



Chapter 13 



1. information 

2. 7.5 cm 

3. discussion 
discussion 

(a) 4.4 X 10" m (b) 3.0 X 10" m/sec (c» positive 
deviations; conjunction cycle 
1.8 X 10" m 

(a) 9.5 X 10'^ m (b) 4,300 years (c) 30 times as great 
derivation 

9. discussion 

10. 0.9 m; no, no 

11. discussion 

12. diagrams 

13. (a) diagram (b) discussion 

14. proof 

15. (a) (m + 1/2) when m = 0, 1, 2 (b) greater (c) 
increased separation of fringes (d) increased sep- 
aration of fringes le) fainter but more extensive 

for violet,/ = 7.5 x lO'Vsec; for red,/ = 4.2 x 

lO'Vsec 

d = 1.6 X 10"^ m 

18. d = 1 X 10"' m 

19. discussion 
discussion 
discussion 

6 X 10'^ Hz; 10" times AM frequencies; 10' times 
FM frequencies 

23. verticfd 

24. discussion 



16 



17 



20 
21 
22 



Chapter 14 



1. information 

2. (a) tripled (b) halved (c) no change 

3. 95 km 

4. discussion 

5. yes; discussion; sketches 

6. (a)1.6N/kg(b)4.2 x 10'* N/kg (c) directly propor- 
tional to Vr 

7. discussion 

8. vector diagraims 

9. (a) F = k(Q,Q2)/fl2 

WhenQ2= Sqe-/' = 2 76 x 10 " N east 
Q2 = 6qe. F = 5.5 X 10" N east 
Q2 = 10 fle. F = 92 X 10" N east 
Q2 = 34q^,F = 3.1 X 10- '» N east 
(b) 5.76 X 10' N/C. The same values for the force F 
are found as given in (a) for the various charges 
Qj. (c) The field concept specifies the field at emy 
point that will interact with a charge at that point, 
as in (b). 

10. (a) 10« C (b) 10-« C/m^ 

11. sketch (normal to surfaces) 

12. help 



13. 
14. 
15. 

16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 



25. 



26. 
27. 



28. 

29. 
30. 
31. 
32. 
33. 
34. 
35. 
36. 



37. 
38. 
39. 
40. 



625 X 10'* electrons 
3.4 X 10*^ 

(a)l/2mv2 = i/2/cq7fl(b)1.2 x l0-"'J(c)1.5 x 10" 
m/sec 

Metals are conductors. 
30 V 

same or zero 
derivation 
3 X lO*' V/m 
10' V/m 

(a) 12 V (b) zero (c) 12 V 
(a) 1.6 X 10" J (b) 5.7 X 10* m/sec 
(a) / = E/R = 12/3 = 4 A; if the voltage is doubled, 
the current / is doubled to 8 A. (b) Voltage de- 
scribes a potential difference between two 
points. Current describes a flow of charge 
through a conductor. 

fl = 25 ft; when / = 2 A, = £ 50 V. If Ohms law 
does not apply, and you do not know another 
relation between voltage, current, and resistance, 
you cannot relate current to voltage. 
P = 150 W; fl = 16.7 n 

Power before the cut is 5,825 W, current is 48.5 A. 
Power after the 5% cut is 5,534 W, current is 48.5 A. 
You are not cheated, since you pay for power in 
watts. 

The power limit of the circuit breaker is 1,200 W. 
You can add four lamps drawong 150 W each. 
(a) 4 A (b) 5 n (c) 15 V 
(a) 10' V (b) 5 X 10* J 
discussion 
20 W 

(a) 8 W (b) 20 W (c) 45 W 
magnetic field vertical at surface 
(a) north (b) 1 A, north 

(a) An ampere is defined as the amount of current 
in each of two long, straight parallel wires, set 1 m 
apart, that causes a force of exactly 2 x 10 ' N to 
act on each meter of each ware. The unit of force 
between the wires is N/mA^(b)F =4.8 x 10"^ N 
(a) derivation (b) v, B, and fl 
derivation 
west 
discussion 



Chapter 15 



1. information 

2. discussion 

3. yes 

4. all except (d) 

5. sketch 

6. (a) exercise (b) upward (d) downward 

7. Lenz's law 

8. outside magnet 

9. opposite 

10. discussion 

11. (a) the series circuit (b) the series circuit (c) the 
parallel circuit (d) In a parellel circuit, added 
resistors decrease the total effective resistance 
and the total current and power increase. 



274 



BHIEF ANSWERS TO STUDY QLIDE QUESTIONS 



12. In series, each resistor carrries 5 A. In series, the 
total resistance is 8 il and the total current is 1^ 
A. For each resistor, the current is 1 5 A at 6 V. In 
parallel, the total resistance is 2 il and the cur- 
rent is 6 A. For each resistor, the voltage is 6 V and 
the current is 4 A. 

13. In series, the total resistance is 15 O and the 
current is 3.3 A in each resistor. The voltage ac- 
ross the 5 fi resistor is 1G.7 V at 33 A. The voltage 
across the 10nresistoris33.3V and the current is 
3.3 A. In parallel, the total voltage is 50 Vand the 
total current is 15 A, The voltage is 50 V across 
both resistors. The current through the 5 il resis- 
tor is 10 A and through the 10 il resistor it is 5 A. 

14. (a) 1 A (b) 10 n (c) bum out 

15. (a) 1/12 A (b) 1,440 il, the same 

16. (a) 1 A (b) 1/5 W (c) 1/2 A; 1/20 W (d) 0.97 A; 0.19 W; 
5.6 W, 0.50 A, 0.05 W, 6 W 

17. 5 A 

18. derivation 

19. The constantly changing magnetic field of the 
primary coils induces a constantly changing cur- 
rent in the transformer core and the coils of the 
secondary circuit by electromagnetic induction. 

20. low voltage coil 

21. discussion 

22. discussion 

23. report 

24. discussion 

25. The efficiency of electric power plants is limited 
by the second law of thermodynamics. Modem 
power plants can achieve about 38-40?i effi- 
ciency for fossil fuel plants and 30?o for nuclear 
plants (the maximum possible efficiency is 60% 
for fossil fuel plants and 50% for nuclear plants). 

Chapter 16 

1. information 

2. symmetry 

3. no 

4. accelerating charge, mutual induction 

5. (a I height (bl pressure (c) field strength 

6. deflector orientation 

7. light properties 

8. discussion 

9. discussion 

10. 5 X 10* m; 600 m and 193 m; 11 m 

11. 10 m to 100 m 

12. discussion 

13. discussion 

14. (a)TVor FM: 10" Hz (frequency), 1 m (wawlength); 
r«(l light: lO'Mlz. 10 *m; infrared: lO'Uiz, 10 'm: 
electric wires: 10' Hz, 10* m (b) 1\' or VM: little 
diffraction; red light: shaqi shadow; infrared: 
shaqi shadow: electric wires: gn>at diffraction 

15. discussion 

16. ionospheric r«»flection of shorter wavelength 
radiation 

17. 42.400 km 

IH. phase differtMue betwiHMi dinnt and reflected 
waves 



19. 2 6 sec 

20. absorption 

21. e\'olution 

22. ultraviolet and infrared 

23. discussion 

24. unnecessary 

25. discussion 

26. discussion 

27. discussion 

28. essay 

29. essay 

Chapter 17 

1. information 

2. 80.3% zinc; 19.7% OJ^gen 

3. 47.9% zinc 

4.13 9 times mass of H atom; same 

5. 986 g nitrogen; 214 g hydrogen 

6. (a) 14.1 (b) 282 (c) 7S) 

7. Na;l M3 P;5 Ca2 Sn;4 

8. graph; discussion 

9. 8.0 g: 0.895 g 

10. (a) 0.05 g Zn (b) 030 g Zn (c) 12 g Zn 

11. (a) 0.88 g CI (b) 3 14 g I (c) discussion (d) discus- 
sion 

12. 35 45 g 

13. discussion 

14. discussion 



Chapter 18 



1. infonnation 

2. (a) 2 X 10' m/sec (b) 18 x 10" C/kg 

3. proof 

4. discussion 

5. discussion 

6. 2000 A; ultraviolet 

7. 4 X 10 ''J; 4 X 10 "J 

8. 2.6 X 10 '»; 1.6 eV 

9. 4.9 X iQ'Vsec 

10. (a) 6 X lO'Vsec (b) 4 x lO '• J (c) 25 x io*» 
photons (d) 2-5 photons/sec le) 04 sec (f) 2-5 x 
10'" photon (g) 6 25 x 10'" electrons/sec; 
1 A 

11. 13 x 10" photons 

12. (a) 6 X 10" elections (b) 54 x 10*' copper 
atoms/cm* (c) 12 x lO " cm* (d) 23 x lo * 

13. (a) 2;k = nX (b) 2;c = any odd number of half 
wavelengths (c) cos 6 = Zd/k for f\rs\ order 

14. 12 X lO'Vsec 

15. discussion 

16. 12 X 10* \: 19 X 10 '« J; 12 x lO* e\ 

17. glossary 

18. discussion 



Chapter 19 



1. information 

2. <ILs(-iis.sion 

3. Fiw are listed in the 7>.t(. but theoretically an 
infinite number. Four lines in visible region 



BRIEF ANSWERS TO STUDY GLIDE QUESTIONS 



275 



4. n X 8; X = 3880 A 
n X 10; X = 3790 A 
n X 12; X = 3740 A 

5. (a) yes (b) n, x x (c) Lyman series 910 A; Balmer 
series 3650 A; Paschen series 8200 A (dl 21.8 x 
10'* J, 13.6 eV 

6. discussion 

7. discussion 

8. 2.6 x lO-'« m 

9. (al discussion (b) 10"Vl 

10. 3.5 m 

11. derivation 

12. discussion 

13. list 

14. diagram 
lS-21. discussion 
22. essay 



Chapter 22 



1. information 

2. discussion 

3. discussion 

4. (a) discussion (b) discussion 

5. (a) fl = ^, Rvm (b) 5.4 cm (c) 5.640 m 

Bq 

Id) 0.004« m 

6. equations 

7. chart 

8. diagram 

9. diagram 

10. 4,000 years; 23,000 years 

11. (a) 12.011 amu (b) 6.941 amu (c) 2072 amu 

12. 4.0015 amu 

13. la) X 1/4 (b) about x 1/2 

(c) about 225 x 10* years Id) yes 



Chapter 20 

1. information 

2. 0.14 c or 42 X lo" m/sec 

3. 3.7 X 10-'^ N 

4. p = m„v and KE = m„v-/2 

5. la) changes are too small lb) 1.1 x lO"'^ kg 

6. la) 2.7 X ltf» J lb) 3.0 x io'« kg Ic) 5 x 10"^% Id) rest 
mass 

7. (a) 12 X 10--- kg m/sec lb) 1.1 x lO"" kg m/sec 
(c)2.4 X 10---kgm/secld)l.l x lO"" kg m/sec 

8. p = 1.7 X 10--' kg m/sec; v = 1.9 x 10^ m/sec 

9. discussion 

10. diagram 

11. 6.6 X lO*' m/sec 

12. 3.3 X 10"^ m 

13. X becomes larger 

14. la) 3.3 X 10"" m/sec lb) 5.0 x lO"' m/sec Ic) 3.3 x 
10* m/sec Id) 3.3 x 10* m/sec 

15. discussion 

16. 3 x 10-3' j„ 
17 — 24. discussion 



Chapter 21 

1. information 

2. discussion 

3. la) 12 X 10" J lb) 0.75 MeV 

4. la) 5.7 X 10"^ m lb) 210 m Ic) R = 3,700 R^ 

5. Charges are positive; field is into the page. 

6. la) 1.8 X 10^ N/C lb) 1.8 x lO^ V Ic) undeflected 

7. la) y lb) a Ic) a Id) y le) -y If) a or -y Ig) /3 Ih) a 
li) a Ij) /3 

8. discussion 

9. la) one-half lb) three-quarters (c) discussion 

10. 10% 

11. (a) graph lb) proof Ic) 5.0 x lo^" atoms 

12. (a) 5.7 xiO'^ J/disintegration lb) 45 W 

13. 3.70 x 10' disintegration/sec 

14. 2The number remains constant. 

15. la) about 4 days (b) discussion 
16-19. activity 



Chapter 23 

1. information 

2. discussion 

3. 235 protons; 143 electrons 

4. equations 

5. equations 

6. la) y lb) Al^"* Ic) Mg--" Id) Mg-^ 

7. la) discussion lb) in Unit 3 under conservation 
laws. 

8. 1.10 amu, 52% 

9. table 

10. la) 78 lb) 79 Ic) 80 Id) 80 

11. la) iiNa-"* lb) ,,Na-^ Ic) ..Na^'' Id) nNa^-* 
12-15. discussion 

16. Less by 0.02758 amu 

17. acti\ity 

Chapter 24 

1. information 

2. 4.95 MeV 

3. 7.07 MeV/nucleon 

4. opposite directions, each 8.65 MeV 

5. absorbed, 1.19 MeV 

6. 0.56 MeV 

7. 8.61 meV 

8. neutron capture, /3-decay, /3-decay 

9. Ba'*' is 1180 MeV; Kr^ is 800 MeV; 
U"* is 1790 MeV. discussion 

10. 208 MeV 

11. diagram 
12—14. discussion 

15. 26.7 MeV 

16. lal 4 33 X 10* kg/sec (b) 523 x lO" horsepower 

17. equations 

18. 1.59 MeV released 

19. IP^ is fissionable 

20. Pu-^' is fissionable 

21. discussion 

22. essay 

23. activity 



INDEX 



AccalaratkMi. cau— d In cra%itv 
Ifilm loop) 41- -U 
rentriprtal. X 
in tnr tsU 18-21 
of graviK '«r.\prnnwnll. 21-24 
me— uromfni of iFxpcrinwnt). 

tl-24. 25-28 
(flkn loopi. 47 
me aim idotnn Sprrd 
Accaiaranwtan tactK'irki. 35—38 
auhMnoUfe. 37- 38 
caMtarationat 3S-37 
fHBi|lMt-p0tMluluni. 38 
Kquid-Mvfarr 35-37 
AiKancv of penhrlwn. 101 
Air, standing mviv in iSm loop' 

iro-in 
Air track, m bullH speed cxpni- 
mmt. 122 
in roHiuon experioiante. 103. 
U8-121 
AircnA takaoA and conMnralion of 
«nBf|^ mm loo|A ICT- IC8 
Alpha ta> partidm ranp of (coqpn^ 

uiipni) im ao 

Euthcrfcrds a c aW fring experi- 
■Mnt witK 230-231 2»- 

tss 

Afeftudik in aatnmocnN 9 
drcu*. 194 
modulation 19» 
ploltinfl of >.arti%«> • 88 
And^yaia of a hutdfe rare (ttai loa|ik 

47-48 
Amkomeda. G(«bi \v6uU n. CS 
An^afB> of mndmcv and rHrar- 

tian.lTS 
Angular m«aaurtmBWH. aaakingof 

(artn-irr). 83- 84 
Aniarad tablet, in conscff>ratian of 
actn^y 147 
141 

AraHlm. conalvuLliuu oC 2V 
Aacanang noaifc 71 
ft^8S 
9-iaS5 
Aamaiuinkal unit .\l »9 
Aatronomv nak«d-r>r r\p«n- 
manlsJi 7- 12. 50- 54 
obaan r atjpm ir&prnment' 
l»-U 
Almoapliarh p raaa u ta lartiviiyt. 

149- 139 
Aliimlil hlack boa tactk^ 231 
aCZIS 



37-38 

AuloradiQgrapIrk «expcrinicnl>. 250 
AaanutK 7-8 

Balann«sJ 214 
co n alf uction and use of lezpev^ 
immt' 181-183 
cart proiectiks iacti«il>i 



'IwimIiAbh. mbuBeC apaad 

emperimmt ICO 
Band sa«« bladp standmg 

actnil>' 231 
Bating construction at kactkilyi 

204 
c4F%m-cml actiikil> • 2M 
Brokrr and hammer laclMityl 35 
Beam protector con at r u ct i on oC 

175- 17« 
Beta \fit p a f tkl e a > ranpp of teipcn- 

menli 240-242 
Beta (^> radiation aMiaauriag 

energy of «expaffiaaami. 

£50-252 
BrU ifi m spactttaaal 
Bir%Tie |SPnmfMv (artM^y*. IH 
B4|C Dipper 9 
Billiard ball d>naaDUCs of ( 

loop' 1C3-IM 
B aj y ^p fiicaf CncydBpadH af Set- 



C eh— icat i p ad rt acaJa. 127 
Canttff a da acair 127 
Central faecaa >fikn loop^ 98-99 
Centnprtal arcekratKxv 32-33 
Centripetal force leipenmenlt. 

32-33 
on a turntable tt 

33-34 
ChadMtck and neutron. 2S3 
Chaai at least -e i iei g t- 

U4-U5 



217-219 
speed of 192 



i«ctik«> rr7 
equaoon for Z\S 

214-217 

actn«>r SS 
IdMniBr and circtnc ri»> 

2l;-214 us 




laapA. 158- 159 
CoBaiP- ph^ aka tactik^- 20 
CofcOor tm 

knpi 44-45 
riiMlitii^ dmk i^iMi eipcn- 
111-112 
aei^il cars <■■ 
M2-ICS 

:« 111 
far #»» V 




1^238-231 

Buftbarfard nucfaar 

238-231 
Buthaifaid scattering eaper»- 




CancKMK lamd valocil^i of 1C2 
Cannon baA, path oC 32 
Chi Time Go 
ncr< IC8 
C^McHor 244 
Car. iwatg Nn g «k«h tae 
kactM«>* 15B 
14 in ma. 249 

195 
«invr actM«>l t4»- 138 
ruTke. LS5 
192 
of ^anar^y^ 227 
in a Oosftaa !«*» tactti^jil Hr 



134- &3S 



af «acia««^ 281- 



IVDEX XT7 



'i'vtafac 




t-r ^jTi^ '•at=--tf g~i~ 




278 



INDEX 



Faraday, and electrochemical reac- 
tions, 212 
Faraday disk dynamo (activity), 

206-207 
Forc;e, between magnet and cur- 
rent (experiment), 187- 190 
central (film loop), 98-99 
centripetal, 32-33 
on currents (experiment), 183- 

187 
electric (experiments), 179-183 
inverse-square, 99 
magnetic, 192 
nuclear, 235 

on a pendulum (activity), 92- 93 
variation in wire length and, 196 
FORTRAN, 97 

Foundations of Modern Physical 
Science (Holton and Roller), 
204 
Frames of reference (activity), 
89-90 
fixed and moving, 44-45 
Franck- Hertz effect, 229 
Free fall, 42-43 

acceleration in, 18-21 
from aircraft, 46- 47 
from mast of ship, 45-46 
Frequencies, measuring unknown 
(activity), 42 
of waves, 139 
Friction, on rotating disk, 34 

Galilean relativity (film loops). 

45-47 
Galileo (activity), 90 

inclined plane experiment. 
18-21 
Galileo (Brecht), 90 
"Galileo: Antagonist," Physics 

Teacher, 90 
"Galileo Galilei: An Outline of His 

Life," Physics Teacher, 90 
Galileo Quadricenlennial Supplc- 
mcrnt , Sky and Irli'scoiw . 90 
Galileo and the Scientific llevolu- 

tion (Femii), 90 
Gas(e8), behavior of (experiment), 
137- 138 
pn^ssun^ of, 137- 138 
standing waves in (film loop). 

170-171 
tcmpeiatiire of, 138 
volume and pressure (experi- 
ment), 137- 138 
volume and temperalun* (exper- 
imenll, 138 
(Jas Ihennonu'ler 128 
Gauss. 205 

(iay Lussac 's law. nMalioii between 
tempernlun' and volmne 
137 
Geiger counter 238 
(ient'ralor demonstration of larii\- 
ity), 209 
bicy«le (activity). 209 
(ieneialor jump rope (aclivilyi 207 



Geocentric model of universe, 94 

Gilbert's versorium (activity), 203- 
204 

Gliders, collisions of, 103, 120-121 

Graph, drawing of 18 

Gravitational potential energy 'film 
loop), 164-165 

Gravity, acceleration caused by 
(film loop), 42-43 
measuring acceleration of (ex- 
periment). 21-24 

Great Nebula, in Andromeda and 
Orion. 66 

Half-life (experiments). 243-248 

Halleys comet, plotting orbit of 
(experiment), 79-82 

Handkerchief diffraction grating 
(activity), 200 

Heat, conversion to mechanical 
energy (activity), 148-149 
exchange and transfer of (exper- 
iment), 132 
latent, of melting ice (experi- 
ment), 132 
mechanical equivalent of (activ- 
ity), 149-150 

Heat capacity, measurement of, 
130-131 

Heat energy, conversion of (activ- 
ity). 148-149 
measuring of. 128- 132 

Height of Piton. a mountain on the 
moon (experiment). 57-60 

H«'liocentric model of universe, 
94-95 

Heliocentric system, 63 

Histogram. 238-239 

Horizontal motion, measurement 
of. 47 

Horsepower, student (activity), 148 

Hurdle race, analysis of. I and II 
(film loops). 47- 48 

Hvadt's. observations of 66 

Hvclrogen atom, calculating eneq^ 
levc!ls for. 225 



Ice, calorimetry (experiment). 
131-132 
latent heal of melting. 132 
Ice lens, construction of lai ii\iivi 

203 
Icosahedral dice. 237 
Images, n-al. 63-64 
IncidcMice. angle of. 176 
Inclination, of Mars c>r<)it. 70-72 
Inclined air track in enei^ffk cc»n- 
servation experiment, 
120- t21 
Inclined-plane exjx'rimeni 18-21 
Inelastic collisions 106 

onc'-dimensional ililm IcKipl. 157 
tv\(i-climc>nsional (film Icxipl. 158 
Inertia and gravitation. 28 
Intrarecl pholographv (actixitv I 201 
Instanlaiieoiis s|>ec-cl. 42 



Interference, ultra&ound (experi- 
ment). 146 
wave. 139: ^experiment). 140, 144 

Interference pattern (experiment), 
141-142 
of light lexperiment), 178 

Inverse-square force, 99 

Ionization, measurement of (actK-- 
ity), 229-230 

Ionization enei^. 229. 248 

Ionosphere. 198 

Irregular areas, measurement of 
(activityi. 91 

Isolated north magnetic p>oIe (activ- 
ityi. 206 

Iterated blows. 98-99 

Iteration of orbits, 76 

Julian Day. 53 
Jupiter mass of, 97 

observations of 65-66 

positions of 60-61 

satellite orbit (film loop), 95-97 

Kepler's laws (film loop), 99-100 

satellite orbit and. 90 
Kinetic ener^gv- (film loop), 165- 166 

calculation of (film loop), 160 

see also Enei^' 

LaboratoPk' notebook report, 2-3 

L^nd effect. 202 

Land two-color demonstrations 

(activityi. 202 
Lapis Polaris Magnes (activityi. 

209-210 
Latent heat, measurement of (ex- 
periment!, 130- 131 
Latent Image iNewhalli 201 
Latex squares, electron microg- 
raph of 217 
Lead nitrate, in conservation of 

mass experiment, 147 
Least enei^^k' (experiment). 124- 

125 
Lenses, in telescope (experimenll, 

63-67 
Light, angles of incidence and re- 
fraction 176 
diffraction of 222 
dis|M>rsiun into s|)(>ctra. 222- 223 
effiHt on metal surface (experi- 
ment i. 219-222 
interfen-nc-e patterns of (experi- 
menll. 178 
pholcH'lcMtric eflFecl of (experi- 
ment!. 219-222 
|)olanA4Hi. 202-203 
rainbow effe<t (activity!. 201 - 202 
scattered lactKit> i. 201 
wave and particle models of. 

13- 16 
wavelength of (experimenll, 

177-179 
st'c alstt ("cilcjr \\avr(»l 
Ijghl tM*am n>traction of (experi- 
nu-nli 175-176 



INDEX 



279 



Line of nodes, 80 

Liquids, mixing hot and cold (ex- 
periment), 129 

Liquid-surface accelerometer, 
35-37 

Literature, Elizabethan world view 
in (activity), 89 

Longitudinal wave pulse, 139 

Lunar eclipses (table), 12 

Mach, Ernst, and inertia, 28 
Magnesium flashbulb, in conserva- 
tion of mass activity, 147 
Magnet(s), interactions of, 119- 120 
modeling atoms with (activity), 
230-231 
Magnetic field, deflection of elec- 
tron beam by, 192; (experi- 
ment), 214-217 
measuring intensity of (activity), 
204-205 
Magnetic pole (activity), 206 
Marbles, collision probability for a 
gas of (experiment), 133- 
134 
inferring size of, 133-134 
Mars, inclination of orbit (experi- 
ment), 70-72 
orbit of (experiment), 67- 70 
positions of, 60-61 
Mass, conservation of (activity), 147 
inertial, 28 
measuring of (experiment), 

27-28 
neutron, 253 

weight and (experiment), 28 
Mathew Brady (Horan), 201 
Mean free path, between collision 
squares (experiment), 
134-135 
Measurement(s), of acceleration 
(experiment), 25 — 28 
of acceleration in free fall, 18-21 
of acceleration of gravity, 21-24 
angular (activity), 83- 84 
of elementary charge (experi- 
ment), 217-219 
of enei^ (film loop), 164 
of energy of beta (/3) radiation 

(experiment), 250-252 
of irregular areas (activity), 91 
of magnetic field intensity (activ- 
ity), 204-205 
of mass and weight (experiment), 

27-28 
of momentum, 156-157 
precision in, 16—18 
of speed of sound (experiment), 

154-155 
of uniform motion (experiment), 

14-18 
of unknown frequencies (activ- 
ity), 42 
of wavelength (experiment), 
178-179 
Mechanical energy, conversion of 
heat to (activity), 148-149 



Mechanical wave machines (expjer- 

iment), 155 
Melting, 130 
Mercury, elongations of, 73 

orbit of (experiment), 72- 75 
Metal plate, vibrations of (film 

loop), 173-174 
Meters, construction of (activity), 

207-208 
Method of beats, 195 
Microwaves, interference of re- 
flected, 197-198 
properties of (experiment), 

196-197 
reflected, 197-198 
signals and (experiment), 198- 
199 
Microwave transmission systems 

(activity), 210 
Millikan, oU drop experiment, 217 
Modeling atoms with magnets (ac- 
tivity), 230-231 
Model of the orbit of Halley's comet 

(experiment), 79- 82 
"Moire Patterns" (Oster and 
Nishijima), Scientific 

American, 154 
Moire wave patterns (experiment), 

153-154 
Molecular collisions, Monte Carlo 

experiment on, 132-135 
Momentum, conservation of, 
110- 112, 157- 158; (activity), 
147- 148 
measurement of, 156-157 
Momentum devices, exchange of 

(activity), 147-148 
Monte Carlo method in moleculiir 
collisions experiment, 
132-135 
Moon, crater names of (activity), 89 
distance to (experiment), 57 
height of mountain on (experi- 
ment), 57-60 
observations of, 66; (experiment), 

10-11, 51 
phases of, 51 
surface of, 58-59 
"Moon Illusion, The," Scientific 

American, 84 
Motion, on inclined plane (experi- 
ment), 18-20, 20-21 
Neuron's second law of (experi- 
ment), 25-28 
perpetual (activity), 150—151 
relative (film loop), 44-45 
retrograde (experiment), 60-61 
in a rotating reference frame (ac- 
tivity), 40-41 
uniform (experiment), 14-18 
see also Acceleration; Speed; 
Velocity 
Motor, construction of (activity), 
208 
demonstration of (activity), 209 
Motor- generator demonstration 
(activity), 208-209 



Music, and speech (experiment), 
154 
wave patterns of (experiment), 
154 

Nails, in measurement of kinetic 
energy experiment, 165- 
166 

Naked-eye astronomy (experi- 
ment), 7-12, 50-54 

Net count rate, 245 

Neutron, calculating mass of, 253 

New Handbook of the Heavens, 65 

Newton's second law (experiment), 
25-28 

Nodal lines, 141 

Nodes, 141, 172 

North magnetic pole (activity), 206 

North star (Polaris), 9 

Nuclear force, 235 



Objective lens, 64 
Occultation, 96 

One-dimensional collisions (exper- 
iment), 102-109 
(film loop), 156-157 
stroboscopic photographs of, 
104-110 
Orbital eccentricity, Ccilculation of, 

74-75 
Orbits, comet (activity), 91 
computer program of, 97- 98 
earth's, 79 
five elements of, 72 
of Halley's comet (experiment), 

79-82 
of Jupiter satellite (film loop), 

95-97 
of Mars (experiment), 67-70 
of Merciory (experiment), 72-75 
parabolic (activity), 91- 92 
pendulum, 92- 93 
of planets, 53-54, 65-75 
satellite (activity), 90 
stepwise approximation to (ex- 
periment), 75-79 
unusual (film loop), 100-101 
Orion, Great Nebula in, 66 
Oscillator, calibration of, 28 
Out of My Later Years (Einstein), 227 

Parabola, in least energy experi- 
ment, 124-126 
waterdrop, photograph of (activ- 
ity), 39 

Parabolic orbit, drawing of (activ- 
ity), 91-92 

Parallax, 57, 186 

Partially elastic collisions (film 
loop), 162-163 

Particle and wave models of light, 
13-16 

Pendulum, ballistic, 122- 123 
forces on (activity), 92- 93 
measuring gravity by (experi- 
ment), 22-23 



280 



INDEX 



Pendulum accelerometer (activity), 

38 
Pendulum swing, energ>' analysis 

of lexptifiment), 124-126 
Penny and coathanger laciivity), 42 
Perfectly inelastic collision, 106 
Perihelion, advance of, 101 
Periodic wave, 139 
Perpetual Motion and Modern Re- 
search for Cheap Power 
(Smedile), 205 
Perpetual motion machines (activ- 
ity), 150-151, 205-206 
"Perpetual Motion Machines," Sci- 

entific American, 205 
Perturbation, 100-101 
Photoelectric effect (activity), 
227-228 
(experiment), 219-222 
Photoelectric equation, 222 
Photoelectric tube, 219-220 
Photographic activities, 201 
Photography, history of (activity), 
201 
infrai-ed, 201 
measuring gravity by, 23 
Polaroid, 4-5 
Schlieren, 201 
of spectrum (experiment), 223- 

224 
stroboscopic, measuring gravity 

by (experim«mt), 24 
of waterdi-op parabola, 23-24 
Physics collage (activity), ?tf9 
"Physics and Music," Scientific 

American, 153, 154 
Physics for Entertainment (Perel- 

man), 203 
Physics of Television (Fink and Lu- 

tyens), 210 
"Physics of Violins, The," Scientific 

American, 154 
"Physics of Woodwinds, The," Sci- 
entific American, 154 
Picket fence analog\', and polarized 

light. 203 
Piton, height of (experiment I, 

57-60 
Planck's constant, 222 
Planet(s), locating and graphing 
(experiment), 53-54 
observ'ati{)ns of lexperiment). 11 
Planetaiy longitudes Itablel, 53 
Planetary notes Itablel, 12 
Pleides, observations of. 66 
Poissons spot (activity), 201 
Polaris (North Star), 9 
»»olari/,e«l light (activity). 202-203 
Poliirir^'d Lifiht (Shurx-liflf and Hal- 

lartl), 202 
PolaiDid l,.ind camera, use of. 4-5 
Pole vault itllm loop). 166-167 
IH)lonium. disintegration of, 247- 

248 
IV)stag«' stamps honoring scientists 

(activity). 228-229 
Polenlial eiieiXV. 124-126 



Power, output of (activity), 148 

Pressure, atomospheric (activity), 
149-150 
volume of gas and (experiment), 
137-138 

Principia (Newton), 91 

Program orbit, I and II (film loops), 
97-98 

Proiectile(s), ballistic cart (activity), 
39-40 
fired vertically (film loop), 47 
speed of, 122- 123 

Projectile motion demonstration 
(activity), 38 

Projectile trajectories, photograph- 
ing of (activity), 39- 40 

Pucks, in collision experiment, 111 

Pulls and jerks (activity), 35 

Pulses, 139 
see also Wave(s) 

Quantum Electronics (Pierce), 210 

Radiation, heal exchange by. 132 
elect rxjmagnetic and microwave, 
197-198 
Radio transmitter, generation of 
electromagnetic waves by, 
211 
Radioactive decay, .see Half-life 
Radioactive isotopes, half-life of, 
245-246 
as tracers. 248-250 
Radioactive tracers (experiment). 

248-250 
Rainbow effect (activity). 201-202 
Raisin pudding" atom model. 230 
Random event)s) (experiment). 

236-240 
Random event disks. 236-237. 243 
Random two-digit nirmbers (table). 

136 
Reading suggestions (activit>'). 210 
Real images, 63-64 
Recoil (film loop). 162 
Recoi-d-kee|)ing. 4-5 
Rectification, of diode. 193-194 
Refer-erue. in astronomy. 7-10 
Reflection, soirnd iex|>erimentl. 143 
irltrasound (experiment). 145 
wave (experiment). 140 152- 153 
Refraction, angle of. 176 
of colors. 176- 177 
of a light beam (experiment), 
175-177 
Refraction, sound (experiment). 
143 
wavv, 139. 140 
Regirlarity. aird time (experiment), 

12 
Relativflv prirniple. 46 
Resonant cirxuitsiexiJorimenti. 15HJ 
Retnigrade motion (ex|>eriment), 
60-61 
epicycles and (activity). 84-86 
giMicentric model of (film loop). 
94 



heliocentric model of (film loop), 

94-95 
Right ascension. 86 
Ripple tank. wav«s in (exp>eriment), 

140 
Role of Music in Galileo's Experi- 
ments. The, " Scientific 

American, 20 
Rotating disk, friction on. 34 
Rotating reference frame, moving 

objfH-t in (activity). 40-41 
Rubber hose, vibrations of (film 

loop), 172-173 
Rubber tubing and welding rod 

wave machine. 155 
RutherfortI nuclear model. 230- 

231 
scattering experiment and (film 

loop). 234-235 

Satellite orbits, demonstrating of 
(activity). 90 
of Jupiter. 95-96 

"Satellite Ortiit Simulator." Scien- 
tific .American 90 

Saturn, observations of. 65 

Scale model of the solar svstem (ac- 
tivity), 88 

Scattered light (activity). 201 

Scattering of a cluster of objects 
(film loop). 158-159 

Schlieren photography (activityl, 
201 

Science of Moire Patterns, The (0»- 
terl. 154 
Science of Sounds, The ' Bell 
Telephone Laboratories . 154 

Science from Your Airplane Win- 
dow (V\'ood). 202 

.S'cienf//ic American, activities from. 
227 

Scientific method, 4-S 

Scientists on stamps (activityl, 
228-229 

Seventeenth Centur\- Background 
(Willev ). 89 

Sha|M* of the earths orbit (experi- 
ment), 61-63 

Siden'al day lactKityi 87 

Signals and micnnvavvs (exfieri- 
menti, 198- 199 

Similarities in Wave Behavior 
iShivv), 155 

Single-elcvtnKle plating (actKit>i, 
226-227 

Sinusoidal curvvs. 168- 170 

Size of the earth (experiment). 
54-56 

Sodiirm pnHhrction b\- elet-troK'sis 
(film liMip) 233 

Solar iTlrpsrs (table) 12 

Solar svstem. scale model of lactiv- 
it>), 88 

Sound (ex|»erimpnl), 142-144 
calculating s|)«>«'<l of 144 

Sounil vvavrs tliffr-action of lexjirr- 
iment), 143 



INDEX 



281 



reflection of, 143 

refraction of (experiment), 143 

speed of (experiment), 144, 

154-155 
transmission of, 143 
see also Ultrasound; Wave(s) 
Specific heat capacity, 131 
Spectra, creating and analyzing, 

222-225 
Spectroscopy (experiment), 222- 

225 
Speech, and music (experiment), 

154 
Speech wave patterns (experi- 
ment), 154 
Speed, of bullet (experiment), 
122-124; (film loop), 160- 
162 
constant, 26 
electron charge, 192 
instantaneous, 42 
of sound (experiment), 144, 
154-155 
of a stream of water (activity), 
38-39 
ultrasound (experiment), 146 
Stamps, scientists on (activity), 

228-229 
Standard deviation, 237 
Standard error, 237 
Standing wave(sl, on a band saw 
blade (activity), 231 
on a drum and xaolin (experi- 
ment), 152, 153 
electromagnetic (film loop), 211 
in a gas (film loop), 170-171 
on a string (film loop), 170 
in a wire ring, 232 
Stars, chart of, 8 

observations of (experiment), 11 
Steel balls, in collision experiment, 

104-109, 157-159 
Stepwise approximation of an orbit 

(experiment), 75-79 
Stonehenge (activity), 88-89 
Stonehenge Decoded ( Hawkins and 

White), 89 
"Stonehenge Physics," Physics To- 
day, 89 
Stopping voltage, 220 
String, standing waves on (film 

loop), 170 
Stroboscopic photography, mea- 
suring gravity by, 24 
of one-dimensional collision, 

104-110 
of two-dimensional collision, 
113-118, 121-122 
Sun, earths orbit around, 63 
observations of (experiment), 10, 
50-51 
Sundial, building of (activity), 88 
Sundials (Mayall and Mayall), 88 
Sunspots, observation of, 66-67 
Superposition (film loop), 168- 170 

Tagged atoms, 248-249 



Telescope," aiming and focusing of, 
64-65 
making of (experiment), 63- 67 
observations with, 64-67 
Temperature, of gas (exp>eriment), 
138 
thermometers and (experiment), 
126-128 
Temperature scale, defined, 126- 

127 
Terminator, 58 

Thermometers, comparison of, 
127-128 
constant pressure gas, 128 
making of, 126-127 
temperature and (experiment), 
126-128 
Thin film interference (activity), 200 
Thomson, J. J., and cathode rays, 
192 
model of the atom and (film 

loop), 233-234 
"raisin pudding" atomic model 
and, 230 
Thorium decay series, 246-247 
Threshold frequency, 222 
Time, and regularity (experiment), 
12 
reversibility of (film loop), 168 
Tire pressure gauge, weighing a car 

with (activity), 150 
Total internal reflection, 177 
Tracers, in chemical reactions, 250 
radioactive (experiment), 248- 
250 
Trajectories, curves of (experi- 
ment), 28-30 
prediction of (experiment), 
30-32 
Transistor amplifier (activity), 206 
Transit, 96 

Transmission, of sound (experi- 
ment), 143 
ultrasound (experiment), 145 
Transverse wave, 139 
Trial of Copernicus (activity), 93 
Triode, characteristics (experi- 
ment), 193-194 
Turntable, centripetal force on (ac- 
tivity), 33-34 
measuring gravity by (experi- 
ment), 24 
Turntable oscillator patterns, and 
de Broglie waves (activity), 
231 
Turntable oscillators in wave- 
communication experi- 
ment, 195 
Twentieth-century version of 
Galileo's experiment (exper- 
iment), 20-21 
Two-dimensional collisions, I and 
II (experiment), 110-118; 
(film loop), 157-158 
stroboscopic photographs of, 
113-118, 121-122 
TVvo Mew Sciences (Galileo), 45 



Ultrasound (experiment), 144-146 

speed of (experiment), 146 
Uniform motion, measuring of (ex- 
periment), 14-18 
Unusual orbits (film loop), 100- 101 

Vacuum tubes, characteristics of, 

193-194 
Vector addition (film loop), 43-44 
Velocity, of a boat (film loop), 43-44 
recoil, 162 
see also Speed 
Velocity time graph, 163 
Venus, and earth- sun distance 
(activity), 91 
observations of, 65 
Vernal equinox, 67, 86 
Versorium, Gilbert's (activity), 

203-204 
Vertical motion, measurement of, 

46 
Vibrations, of a drum (film loop), 
173 
of a metal plate (film loop), 173- 

174 
of a rubber hose (film loop), 

172-173 
of a uire, 232; (film loop), 171- 

172 
see also Wave patterns 
Violin, standing waves on, 152, 153 

wave patterns of, 170 
Volta, and electrochemical reac- 
tions, 212 
Voltaic pile, construction of (activ- 
ity), 204 
Volume, and pressure of gas (exper- 
iment), 137-138 
temperature of a gas and (expei^ 
iment), 138 



Water electrolysis of (activity), 226 

interference pattern in, 141-142 

speed of stream of (activity), 
38-39 
VVaterdrop( s), measuring gravity by, 

23-24 
V\'aterdrop parabola, photograph- 
ing of (activity), 39 
Water wave(s), reflection of (exper- 
iment), 152-153 
Wave(s), and communication (ex- 
periment), 195-199 

de Broglie, 231 

diffraction (experiment), 140 

electromagnetic, 211 

frequency, 139 

interference (experiment), 140, 
144 

interference pattern (experi- 
ment), 141-142 

longitudinal. 139 

microwaves, interference of, 
197-198 

periodic, 139 

properties (experiment), 139 



282 



INDEX 



reflection (experiment), 140, 

197-198 
refraction (experiment I, 140 
sinusoidal, 168-170 
sound, sec Sound waves 
standing, 141, 143- 144, 152; (film 

loop), 211 
transverse, 139 
water, 152- 153 
see also Light; Sound; Standing 

waves 
Wave fniquencies, 139 
Wavelength, 139 
of light, 177-179 



measuring of (experiment), 141, 

145, 178-179 
of sound (experiment), 143-144 
Wave machines (experiment), 155 
Waves and Messages (Pierce), 210 
Wave and particle models of light, 

13-16 
Wave patterns («;xperiment), 152, 
153 
moire (experiment), 153- 154 
music and sjjeech (experiment), 

154 
violin. 170 
Weight, and mass, 28 



measuring of (experiment), 
27-28 

Wire, vibrations of. 232; (film loop), 
171-172 

Wire ring, standing waves in (activ- 
ity), 232 

Work, output of (activity), 148 

X rays, fixjm a Crookes lube (activ- 
ity). 228 

Young's experiment: the 
wavelength of light (experi- 
nienli 177- 179 




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