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The Projects Physics Course

Text and Handbook

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The

Project Physics Course

Text and Handbook

UNIT I Concepts of Motion

HfHaiiBi Published by

A Component of the UhBH HOLT, RINEHART and WINSTON, Inc.
Project Physics Course ISQl New York, Toronto

The following is a partial list of the
contributors whose creative assistance
fashioned the basis for the Project
Physics Course material (the affilia-
tions indicated are those just prior to
or during their association with the
Project).

Directors of Harvard Project Physics

F. James Rutherford, Capuchino High School, San

Bruno, Calif.
Gerald Holton, Dept. of Physics, Harvard

University
Fletcher G. Watson, Harvard Graduate School of

Education

Special Consultant
to Project Physics

Andrew Ahlgren, Harvard Graduate School of
Education

Advisory Committee

E. G. Begle, Stanford University, Calif.

Paul F. Brandwein, Harcourt, Brace & World.

Inc., San Francisco, Calif.
Robert Brode, University of California, Berkeley
Erwin Hiebert, University of Wisconsin, Madison
Harry Kelly, North Carolina State College, Raleigh
William C. Kelly, National Research Council,

Washington, D.C.
PhUippe LeCorbeiller, New School for Social

Research, New York, N.Y.
Thomas Miner, Garden City High School, New

York, N.Y.
Philip Morrison, Massachusetts Institute of

Technology, Cambridge
Ernest Nagel, Columbia University, New York,

N.Y.
Leonard K. Nash, Harvard University
I. I. Rabi, Columbia University, New York. N.Y.

This Text-Handbook is one of the many instruc-
tional materials developed for the Project Physics
Course, including texts, laboratory experiments,
films, and teacher guides. Development of the
course has profited from the help of many col-
leagues listed at the front of the text units.

Cover Photograph,

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

Staff and Consultants

L. K. Akers, Oak Ridge Associated Universities,

Tenn.
Roger A. Albrecht. Osage Community Schools,

Iowa
David Anderson, Oberlin College, Ohio
Gary Anderson, Harvard University
Donald Armstrong, American Science Film

Association, Washington, D.C.
Arnold Arons. University of Washington
Sam Ascher, Henry Ford High School. Detroit,

Mich.
Ralph Atherton, Talawanda High School, Oxford,

Ohio
Albert V. Baez, UNESCO, Paris
William G. Banick. Fulton High School, Atlanta.

Ga.
Arthur Bardige, Nova High School. Fort

Lauderdale, Fla.
Rolland B. Bartholomew, Henry M. Gunn High

School, Palo Alto. Calif.
O. Theodor Benfey. Earlham College. Richmond.

Ind.
Richard Berendzen. Harvard College Observatory
Alfred M. Bork, Reed College, Portland, Ore.
F. David Boulanger, Mercer Island High School.

Washington
Alfred Brenner. Harvard University
Robert Bridgham. Harvard University
Richard Brinckerhoff. Phillips Exeter Academy,

Exeter. N.H.

Donald Brittain, National Film Board of Canada,

Montreal
Joan Bromberg, Harvard University
Vinson Bronson, Newton South High School,

Newton Centre, Mass.
Stephen G. Brush, Lawrence Radiation Laboratory.

University of California, Livermore
Michael Butler, CIASA Films Mundiales, S. A.,

Mexico
Leon Callihan, St. Mark's School of Texas, Dallas
Douglas Campbell, Harvard University
Dean R. Casperson, Harvard University
Bobby Chambers, Oak Ridge Associated

Universities, Tenn.
Robert Chesley, Thacher School, Ojai, Calif.
John Christensen, Oak Ridge Associated

Universities, Tenn.
Dora Clark, W. G. Enloe High School, Raleigh,

N.C.
David Clarke, Browne and Nichols School,

Cambridge, Mass.
Robert S. Cohen, Boston University, Mass.

Brother Columban Francis, F.S.C., Mater Christi

Diocesan High School, Long Island City, N.Y.
Arthur Compton, Phillips Exeter Academy,

Exeter, N.H.
David L. Cove, Los Altos High School, Calif.
William Cooley, University of Pittsburgh, Pa.
Ann Couch, Harvard University
Paul Cowan, Hardin-Simmons University,

Abilene, Tex.
Charles Davis, Fairfax County School Board,

Fairfax, Va.
Michael Dentamaro, Senn High School, Chicago,

111.
Raymond Dittman, Newton High School, Mass.
Elsa Dorfman, Educational Services Inc.,

Watertown, Mass.
Vadim Drozin, Bucknell University, Lewisburg,

Pa.
Neil F. Dunn, Burlington High School, Mass.
R. T. Ellickson, University of Oregon, Eugene
Thomas Embry, Nova High School, Fort

Lauderdale, Fla.
Walter Eppenstein, Rensselaer Polytechnic

Institute, Troy, N.Y.
Herman Epstein, Brandeis University, Waltham,

Mass.
Thomas F. B. Ferguson, National Film Board of

Canada, Montreal
Thomas von Foerster, Harvard University
Kenneth Ford, University of California, Irvine
Robert Gardner, Harvard University
Fred Geis, Jr., Harvard University
Nicholas J. Georgis, Staples High School,

Westport, Conn.
H. Richard Gerfin, Somers Middle School,

Somers, N.Y.
Owen Gingerich, Smithsonian Astrophysical

Observatory, Cambridge, Mass.

Stanley Goldberg, Antioch College, Yellow Springs,

Ohio
Leon Goutevenier, Paul D. Schreiber High School,

Port Washington, N.Y.
Albert Gregory, Harvard University
Julie A. Goetze, Weeks Jr. High School, Newton,

Mass.
Robert D. Haas, Clairemont High School, San

Diego, Calif.
Walter G. Hagenbuch, Plymouth-Whitemarsh

Senior High School, Plymouth Meeting, Pa.
John Harris, National Physical Laboratory of

Israel, Jerusalem
Jay Hauben, Harvard University
Peter Heller, Brandeis University, Waltham, Mass.
Robert K. Henrich, Kennewick High School,

Washington
Ervin H. HofFart, Raytheon Education Co., Boston
Banesh Hoffmann, Queens College, Flushing, N.Y.
Elisha R. Huggins, Dartmouth College, Hanover,

N.H.
Lloyd Ingraham, Grant High School, Portland,

Ore.
John Jared, John Rennie High School, Pointe

Claire, Quebec
Harald Jensen, Lake Forest College, 111.
John C. Johnson, Worcester Polytechnic Institute,

Mass.
Kenneth J. Jones, Harvard University
LeRoy Kallemeyn, Benson High School, Omaha,

Neb.
Irving Kaplan, Massachusetts Institute of

Technology, Cambridge
Benjamin Karp, South Philadelphia High School,

Pa.
Robert Katz, Kansas State University, Manhattan,

Kans.
Harry H. Kemp, Logan High School, Utah
Ashok Khosla, Harvard University
John Kemeny, National Film Board of Canada,

Montreal
Merritt E. Kimball, Capuchino High School, San

Bruno, Calif.
Walter D. Knight, University of California,

Berkeley
Donald Kreuter, Brooklyn Technical High School,

N.Y.
Karol A. Kunysz, Laguna Beach High School,

Calif.
Douglas M. Lapp, Harvard University
Leo Lavatelli, University of Illinois, Urbana
Joan Laws, American Academy of Arts and

Sciences, Boston
Alfred Leitner, Michigan State University, East

Lansing
Robert B. LUlich, Solon High School, Ohio
James Lindblad, Lowell High School, Whittier,

Calif.
Noel C. Little, Bowdoin College, Brunswick, Me.
Arthur L. Loeb, Ledgemont Laboratory, Lexington,

Mass.

Richard T. Mara, Gettysburg College, Pa.
Robert H. Maybury, UNESCO, Paris
John McClain, University of Beirut, Lebanon
E. Wesley McNair, W. Charlotte High School,

Charlotte, N.C.
William K. Mehlbach, Wheat Ridge High School,

Colo.
Priya N. Mehta, Harvard University
Glen Mervyn, West Vancouver Secondary School,

B.C., Canada
Franklin Miller, Jr., Kenyon College, Gambler,

Ohio
Jack C. Miller, Pomona College, Claremont, Calif.
Kent D. Miller, Claremont High School, Calif.
James A. Minstrell, Mercer Island High School,

Washington
James F. Moore, Canton High School, Mass.
Robert H. Mosteller, Princeton High School,

Cincinnati, Ohio
William Naison, Jamaica High School, N.Y.
Henry Nelson, Berkeley High School, Calif.
Joseph D. Novak, Purdue University, Lafayette,

Ind.
Thorir Olafsson, Menntaskolinn Ad, Laugarvatni,

Iceland
Jay Orear, Cornell University, Ithaca, N.Y.
Paul O'Toole, Dorchester High School, Mass.
Costas Papaliolios, Harvard University
Jacques Parent, National Film Board of Canada,

Montreal
Eugene A. Platten, San Diego High School, Calif.
L. Eugene Poorman, University High School,

Bloomington, Ind.
Gloria Poulos, Harvard University
Herbert Priestley, Knox College, Galesburg, 111.
Edward M. Purcell, Harvard University
Gerald M. Rees, Ann Arbor High School, Mich.
James M. Reid, J. W. Sexton High School,

Lansing, Mich.
Robert Resnick, Rensselaer Polytechnic Institute,

Troy, N.Y.
Paul I. Richards, Technical Operations, Inc.,

Burlington, Mass.
John Rigden, Eastern Nazarene College, Quincy,

Mass.
Thomas J. Ritzinger, Rice Lake High School, Wise.
Nickerson Rogers, The Loomis School, Windsor,

Conn.
Sidney Rosen, University of Illinois, Urbana
John J. Rosenbaum, Livermore High School,

Calif.
William Rosenfeld, Smith College, Northampton,

Mass.
Arthur Rothman, State University of New York,

Buffalo
Daniel Rufolo, Clairemont High School, San

Diego, Calif.

Bemhard A. Sachs, Brooklyn Technical High

School. N.Y.
Morton L. Schagrin, Denison University, Granville,

Ohio
Rudolph Schiller, Valley High School. Las Vegas,

Nev.
Myron O. Schneiderwent, Interlochen Arts

Academy, Mich.
Guenter Schwarz, Florida State University,

Tallahassee
Sherman D. Sheppard, Oak Ridge High School.

Tenn.
William E. Shortall, Lansdowne High School.

Baltimore, Md.
Devon Showley, Cypress Junior College. Calif.
William Shurcliff, Cambridge Electron

Accelerator, Mass.
George I. Squibb, Harvard University
Sister M. Suzanne Kelley, O.S.B., Monte Casino

High School, Tulsa. Okla.
Sister Mary Christine Martens, Convent of the

Visitation, St. Paul, Minn.
Sister M. Helen St. Paul, O.S.F., The Catholic

High School of Baltimore. Md.
M. Daniel Smith, Earlham College. Richmond.

Ind.
Sam Standring, Santa Fe High School. Santa Fe

Springs. Calif.
Albert B. Stewart, Antioch College, Yellow

Springs, Ohio
Robert T. Sullivan, Burnt Hills-Ballston Lake

Central School, N.Y.
Loyd S. Swenson. University of Houston. Texas
Thomas E. Thorpe, West High School. Phoenix.

Ariz.
June Goodfield Toulmin, Nuffield Foundation,

London. England
Stephen E. Toulmin. Nuffield Foundation, London,

England
Emily H. Van Zee, Harvard University
Ann Venable, Arthur D. Little, Inc., Cambridge,

Mass.
W. O. Viens, Nova High School, Fort Lauderdale,

Fla.
Herbert J. Walberg, Harvard University
Eleanor Webster, Wellesley College. Mass.
Wayne W. Welch, University of Wisconsin,

Madison
Richard Weller. Harvard University
Arthur Western, Melbourne High School, Fla.
Haven Whiteside, University of Maryland, College

Park
R. Brady Williamson, Massachusetts Institute of

Technology, Cambridge
Stephen S. Winter, State University of New York.

Buffalo

Science is an adventure of the whole human race to learn to live in and perhaps to
love the universe in which they are. To be a part of it is to understand, to understand
oneself, to begin to feel that there is a capacity within man far beyond what he felt
he had, of an infinite 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 historical understanding,
with a certain philosophical understanding, with a social understanding and a
human understanding in the sense of the biography, the nature of the people who
made this construction, the triumphs, the trials, the tribulations.

I. I. RABI

Nobel Laureate in Physics

Preface

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

These preliminary results led to the second phase of the
Project when a series of major grants were obtained from the U.S.
Office of Education and the National Science Foundation, starting
in 1964. Invaluable additional financial support was also provided
by the Ford Foundation, the Alfred P. Sloan Foundation, the
Carnegie Corporation, and Harvard University. A large number of
collaborators were brought together from all parts of the nation,
and the group worked together for over four years under the title
Harvard Project Physics. At the Project's center, located at
Harvard University, Cambridge, Massachusetts, the staff and
consultants included college and high school physics teachers,
astronomers, chemists, historians and philosophers of science,
science educators, psychologists, evaluation specialists, engineers,
film makers, artists and graphic designers. The teachers serving as
field consultants and the students in the trial classes were also of
vital importance to the success of Harvard Project Physics. As each
successive experimental version of the course was developed it was
tried out in schools throughout the United States and Canada. The
teachers and students in those schools reported their criticisms and
suggestions to the staflTin Cambridge. These reports became the
basis for the next year's revision. The number of participating

teachers during this period grew from 2 in 1962-63 to over 100 in
1967-68. In that year over five thousand students participated in a
large-scale formal research program to evaluate the results
achieved with the course materials.

During 1968, the last of the experimental course materials was
completed. With the culmination of course development and data
gathering activities, the final phase of Harvard Project Physics got
under way. During 1968-69 and 1969-70 the work of the Project
concentrated on developing and conducting special training
programs for teachers, disseminating information about the course
to physics teachers, science department heads, school administrators
and other interested persons, analyzing the large pool of final
evaluation data and writing a complete report on the results, and
trying to find out how the course might be reshaped to fit special
audiences.

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

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

About ten years ago it became clear that a new physics course,
having far wider appeal than the existing ones, was needed.
Students who plan to go to college to study the humanities or
social sciences, those already intent on scientific careers, and those
who may not wish to go to college at all, can all benefit from a good
introductory physics course. The challenge facing Harvard Project
Physics was to design a humanistic course that would be useful
and interesting to students with widely differing skills, backgrounds,
and career plans. In practice, this meant designing a course that
would have the following effect:

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

2. To help students see physics as the wonderfully many-sided
human activity that it really is. This meant presenting the subject
in historical and cultural perspective, and showing that the ideas
of physics have a tradition as well as ways of evolutionary
adaptation and change.

3. To increase the opportunity for each student to have
immediately rewarding experiences in science even while gaining
the knowledge and skill that will be useful in the long run.

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

5. To take into account the importance of the teacher in the
educational process, and the vast spectrum of teaching situations
that prevail.

How well did Harvard Project Physics meet the challenge? In a
sense each student who takes this course must answer that
question himself. It is a pleasure to report, however, that the large-
scale study of student achievement and student opinion in the
participating schools throughout the United States and Canada
showed gratifying results -ranging from the excellent scores on the
College Entrance Examination Board achievement test in physics
to the personal satisfaction of individual students. It is clear that
the diverse array of individual students in the experimental groups
responded well to the physics content, the humanistic emphasis of
the course, and to its flexible multimedia course materials.

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

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

Since their last use in experimental form, all of the instruc-
tional materials have been more closely integrated and rewritten in
final form. The course now consists of a large variety of coordinated
learning materials of which this textbook is only one; in addition
there are readers, handbooks, programmed instruction booklets,
film loops, documentary films, transparencies, apparatus and various
materials for teachers. With the aid of these materials and the
guidance of your teacher, with your own interest and effort, you can
look forward to a successful and worthwhile experience.

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

Contents

TEXT SECTION

Prologue 1

Chapter 1 : The Language of Motion

The motion of things 9

A motion experiment that does not quite work 1 1

A better experiment 1 3

Leshe's "50" and the meaning of average speed 15

Graphing motion and finding the slope 18

Time out for a warning 21

Instantaneous speed 23

Acceleration— by comparison 28

Chapter 2: Free Fall— Galileo Describes Motion

The Aristotelian theory of motion 37

Galileo and his time 43

Galileo's Two New Sciences 43

Why study the motion of freely falling bodies? 47

GalUeo chooses a definition of uniform acceleration 47

Galileo cannot test his hypothesis directly 49

Looking for logical consequences of Galileo's hypothesis 50

Galileo turns to an indirect test 53

Doubts about Galileo's procedure 56

Consequences of Galileo's work on motion 57

Chapter 3: The Birth of Dynamics— Newton Explains Motion

Explanation and the laws of motion 67
The Aristotelian explanation of motion 69
Forces in equilibrium 70
About vectors 73
Newton's first law of motion 75
The significance of the first law 78
Newton's second law of motion 79
Mass, weight, and free fall 83
Newton's third law of motion 86
Using Newton's laws of motion 88
Nature's basic forces 90
Chapter 4: Understanding Motion

A trip to the moon 99

Projectile motion 101

What is the path of a projectile? 103

Moving frames of reference 105

Circular motion 107

Centripetal acceleration and centripetal force 109

The motion of earth satellites 113

What about other motions? 116

Epilogue 118

Contents Handbook Section 127

Index 193

Answers to End of Section Questions 197

Brief Answers to Study Guide 199

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Physicist Enrico Fermi (1901-1954)
at different stages of his career in
Italy and America. Mrs. Laura Fermi
is shown in the photograph at the top
left of the page.

UNIT

Concepts of Motion

CHAPTERS

1 The Language of Motion

2 Free Fall -Galileo Describes Motion

3 The Birth of Dynamics -Newton Explains Motion

4 Understanding Motion

PROLOGUE It is January 1934, a dreary month in the city of Paris. A
husband and wife, working in a university laboratory, are exposing a
piece of ordinary aluminunn to a stream of tiny charged bits of matter
called alpha particles. Stated so simply, this certainly does not sound like
a momentous event. But let us look more closely, for it is momentous
indeed.

Never mind the technical details. Don't let them get in the way of
the story. It all began as something of a family affair. The husband and
wife are the French physicists Frederic Joliot and Irene Curie. The alpha
particles they are using in their experiment are shooting out of a piece
of naturally radioactive metal, polonium, discovered 36 years before
by Irene's parents, Pierre and Marie Curie, the famous discoverers of
radium. What Frederic and Irene have found is that when the aluminum
is bombarded by alpha particles, the commonplace bit of material
becomes radioactive for a while.

This is a surprise. Until this moment, nothing like this-a familiar,
everyday substance becoming artificially radioactive — has ever been
observed. But physicists in the laboratory cannot force new phenomena
on nature, they can only show more clearly what nature is like. We
know now that this sort of thing is a frequent occurrence. It happens,
for example, in stars and in our atmosphere when it is bombarded by
cosmic rays.

The news was exciting to scientists and traveled rapidly, though it
made few, if any, newspaper headlines. Enrico Fermi, a young physicist
on the staff of the University of Rome, became intrigued by the
possibility of repeating the experiment of Frederic and Irene- repeating
it with one significant alteration. The story is told in the book Atoms in
the Family by Enrico Fermi's wife, Laura. She writes:

... he decided he would try to produce artificial radioactivity
with neutrons [instead of alpha particles]. Having no electric

The Language of Motion

charge, neutrons are neither attracted by electrons nor
repelled by nuclei; their path inside matter is much longer
than that of alpha particles; their speed and energy remain
higher; their chances of hitting a nucleus with full impact are
much greater.

All quotations in the Prologue are
from Laura Fermi, Atoms in the
Family: IVIy Life Witt) Enrico Fermi,
University of Chicago Press, Chicago,
1954 (available as a paperback
book in the Phoenix Books series).
Fermi was one of the major
physicists of the twentieth century.

Usually a physicist has some theory to guide him in setting up an
experiment. This time, no good theory had yet been developed. Only
through actual experiment could one tell whether or not neutrons would
be good projectiles for triggering artificial radioactivity in the target
nuclei. Therefore, Fermi, already an outstanding theoretical physicist
at the age of 33, decided to design some experiments that could settle
the issue. His first task was to obtain instruments suitable for detecting
the particles emitted by radioactive materials. The best such laboratory
instruments by far were Geiger counters, but in 1934 Geiger counters
were still relatively new and not readily available. Therefore, Fermi built
his own.

The counters were soon in operation detecting the radiation from
radioactive materials. But Fermi also needed a source of neutrons. This
he made by enclosing beryllium powder and the radioactive gas radon
in a glass tube. Alpha particles from the radon, striking the beryllium,
caused it to emit neutrons, which passed freely through the glass tube.

Now Enrico was ready for the first experiments. Being a
man of method, he did not start by bombarding substances
at random, but proceeded in order, starting from the lightest
element, hydrogen, and following the periodic table of
elements. Hydrogen gave no results; when he bombarded
water with neutrons, nothing happened. He tried lithium next,
but again without luck. He went on to beryllium, then to
boron, to carbon, to nitrogen. None were activated. Enrico
wavered, discouraged, and was on the point of giving up his
researches, but his stubbornness made him refuse to yield. He
would try one more element. That oxygen would not become
radioactive he knew already, for his first bombardment had
been on water. So he irradiated fluorine. Hurrah! He was
rewarded. Fluorine was strongly activated, and so were other
elements that came after fluorine in the periodic table.

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

With the help of his colleagues, Fermi's work at the laboratory was
pursued with high spirit, as Laura Fermi's account shows:

Follow the story rather than worrying
about the techniques of the
experiment.

. . . Irradiated substances were tested for radioactivity with
Geiger counters. The radiation emitted by the neutron source
would have disturbed the measurements had it reached the

Prologue

counters. Therefore, the room where substances were
irradiated and the room with the counters were at the two
ends of a long corridor.

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

And then, on the morning of October 22, 1934, a fateful discovery
was made. Two of Fermi's co-workers were irradiating a hollow
cylinder of silver with neutrons from a source placed at the center of the
cylinder, to make it artificially radioactive. They found that the amount
of radioactivity induced in the silver depended on other objects that
happened to be present in the room!

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

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

They took a big block of paraffin, dug a cavity in it, put
the neutron source inside the cavity, irradiated the silver
cylinder, and brought it to a Geiger counter to measure its
activity. The counter clicked madly. The halls of the physics
building resounded with loud exclamations: "Fantastic!
Incredible! Black Magic! " Parrafin increased the artificially
induced radioactivity of silver up to one hundred times.

By the time Fermi came back from lunch, he had already formulated
a theory to account for the strange action of paraffin.

Paraffin contains a great deal of hydrogen. Hydrogen
nuclei are protons, particles having the same mass as
neutrons. When the source is enclosed in a paraffin block, the
neutrons hit the protons in the paraffin before reaching the
silver nuclei. In the collision with a proton, a neutron loses
part of its energy, in the same manner as a billiard ball is
slowed down when it hits a ball of its same size [whereas it
loses little speed if it is reflected off a much heavier ball, or a
solid wall]. Before emerging from the paraffin, a neutron will
have collided with many protons in succession, and its
velocity will be greatly reduced. This slow neutron will have

tjtufron Source

Silver cilindir-

paraffin block

Because of Fermi's earlier
experiments, they knew the water
would not become artifically
radioactive. However, they now
reasoned that it would slow down
neutrons and so allow silver to
become more strongly radioactive.

The Language of Motion

^m^^^^is^sm-

^^*^"*^ V f^^n

"SHHV^^^VH

^^^H

■KBIi^nHl

^^^^^^^^^1

The same process by which neutrons
were slowed down in the fountain is
used in todays large nuclear reactors.
An example is the "pool" research
reactor pictured above.

a much better chance of being captured by a silver nucleus
than a fast one, much as a slow golf ball has a better chance
of making a hole than one which zooms fast and may
bypass it.

If Enrico's explanations were correct, any other
substance containing a large proportion of hydrogen should
have the same effect as paraffin. "Let's try and see what a
considerable quantity of water does to the silver activity,"
Enrico said on the same afternoon.

There was no better place to find a "considerable
quantity of water " than the goldfish fountain ... in the garden
behind the laboratory . . .

In that fountain the physicists had sailed certain small
toy boats that had suddenly invaded the Italian market. Each
little craft bore a tiny candle on its deck. When the candles
were lighted, the boats sped and puffed on the water like
real motor-boats. They were delightful. And the young men,
who had never been able to resist the charm of a new toy,
had spent much time watching them run in the fountain.

it was natural that, when in need of a considerable
amount of water, Fermi and his friends should think of that
fountain. On that afternoon of October 22, they rushed their
source of neutrons and their silver cylinder to that fountain,
and they placed both under water. The goldfish, I am sure,
retained their calm and dignity, despite the neutron shower,
more than did the crowd outside. The men's excitement was
fed on the results of this experiment. It confirmed Fermi's
theory. Water also increased the artificial radioactivity of
silver by many times.

This discovery- that slowed-down neutrons can produce much
stronger effects in the transmutation of certain atoms than can fast
neutrons- turned out to be a crucial step toward further discoveries
that, years later, led Fermi and others to the controlled production of
atomic energy from uranium.

About this course: We will return to the study of nuclear physics later
in the course. The reason for presenting a description of Fermi's
discovery of slow neutrons here was not to instruct you now on the
details of the nucleus, but to present a quick, almost impressionistic,
view of scientists in action. Not every discovery in science is made in
just the way Fermi and his colleagues made this one. Nevertheless, the
episode does illustrate many of the major themes or characteristics of
modern science— some of which are discussed below. Look for these
themes as you read through this course; you will find them appearing
over and over again in many varied situations.

Progress in science over the years is the result of the work of many
people in many lands — whether working alone, in pairs or small groups,
or in large research teams. No matter how different the individual way
of working, no matter where he works, each scientist expects to share
his ideas and results with other scientists who will try independently to
confirm and add to his findings. As important as such cooperation is,
the most essential ingredient of science is individual thought and
creativity.

Prologue

Fermi and his associates showed stubborn perseverance in the face
of discouraging results, innagination in the invention of theories and
experiments, alertness to the appearance of unexpected results,
resourcefulness in exploiting the material resources at hand, and joy in
finding out something new and important. Traits we usually think of as
being distinctly humane are of value in pursuing scientific work no less
than elsewhere in life.

Scientists build on what has been found out and reported by other
scientists in the past. Yet, every advance in science raises new scientific
questions. The work of science is not to produce some day a finished
book that can be regarded as closed once and for all, but to carry
investigation and imagination on into fields whose importance and
interest had not been realized before.

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

All these are characteristics of science as a whole and not of
physics alone. This being a physics text, you may well wish to ask, "Yes,
but just what is physics?" The question is fair enough, yet there is no
simple answer. Physics can be thought of as an organized body of
tested ideas about the physical world. Information about this world is
accumulating ever more rapidly; the great achievement of physics has
been to find a fairly small number of basic principles which help to
organize and to make sense of certain parts of this flood of information.
This course will deal with some, but not nearly all, of the ideas that
together make up the content of physics. The purpose of this course is
to provide you with the opportunity to become familiar with some of
these ideas, to witness their birth and development, and to share in the
pleasure that comes from using them to view the world in a new light.

Physics is more than just a body of laws and an accumulation of
facts. Physics is what each physicist does in his own way: It is a
continuing activity- a process of search that sometimes leads to
discovery. Look in on different physicists at work and you will see
differences in problems being studied, in apparatus being used, in
individual style, and in much more. Fermi has provided us with one
example, but as the course proceeds, we will encounter other, sometimes
very different examples. By the end of this course, you will have dealt
with many of the ideas and activities which together comprise physics.
You will not just have learned about it-you will have actually done
some physics.

Science gives us no final answers. But it has come upon wondrous
things, and some of them may renew our childhood delight in the
miracle that is within us and around us. Take, for example, so basic a
thing as size ... or time.

The Project Physics Course has
made two documentary films that you
might like to see. One is called The
World of Enrico Fermi and includes
the discovery described here. The
other is entitled People and Particles
and shows what it is like to be
working now on a research problem
in elementary particle physics.

The Language of Motion

Our place in space

Physics deals with those laws of the universe that apply
everywhere -fronn the largest to the smallest.

ORDER OF MAGNITUDE

Distance to the furthest observed galaxy

Distance to the nearest galaxy

Distance to the nearest star

Distance to the sun

Diameter of the earth

One mile

Human height

Finger breadth

Paper thickness

Large bacteria

Small virus

Diameter of atom

Diameter of nucleus

10-*^ meters
■1022

10'^

10"

10'

103

10"

10--

10-^

10-5

io-»

10-10

10-'^

A globular star cluster

The estimated size of the universe
now is of the order of 100 million,
million, million, million times a man's
height (man's height x 10.000,000,
000,000,000,000,000,000).

The smallest known constituent
units of the universe are less in size
than a hundreth of a millionth of a
millionth of a man's height (mans
height x 0.000,000,000,000,01).

Prologue

Our place in time

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

ORDER OF MAGNITUDE

Age of universe
Precession of the earth's axis
Human life span
One year
One day
Light from sun to earth
Time between heartbeats
One beat of fly's wings
Duration of strobe flash
Short laser pulse
Time for light to cross an atom
Shortest-lived subatomic particles

10'^ seconds

109
10^
105

103
10«
10-3
10-5

io-»

10-'«
lO-^'

Particle tracks in a bubble chamber

Fossilized trilobites

•#^

The history of the universe has
been traced back as far into the past
as a hundred million times the length
of a man's life (man's life x 100,000,
000).

Events have been recorded that
last only a few millionths of a millionth
of a millionth of a millionth of a
man's heartbeat (man's heartbeat x
0.000,000,000,000,000,000,000,001).

It is hard to resist the temptation to say more about these intriguing
extremes; however, this is not where physics started. Physics started
with the human-sized world-the world of horse-drawn chariots, of
falling rain, and of flying arrows. It is with the physics of phenomena on
this scale that we shall begin.

1.1 The motion of things

1.2 A motion experiment that does not quite work

1.3 A better experiment

1.4 Leslie's "50" and the meaning of average speed

1.5 Graphing motion and finding the slope

1.6 Time out for a warning

1.7 Instantaneous speed

1.8 Acceleration -by comparison

9
11
13
15
18
21
23
28

1.1 The motion of things

CHAPTER ONE

The Language of Motion

The world is filled with things in motion: things as small as
dust and as large as galaxies, all continually moving. Your textbook
may seem to be lying quietly on the desk, but each of its atoms is
incessantly vibrating. The "still" air around it consists of molecules
tumbling chaotically, at various speeds, most of them moving as fast
as rifle bullets. Light beams dart constantly through the room,
covering the distance from wall to wall in about a hundred-millionth
of a second, and making about ten million vibrations during that
time. Even the whole earth, our majestic spaceship, is moving at
about 18 miles per second around the sun.

There is a very old maxim: "To be ignorant of motion is to be
ignorant of nature." Of course we cannot investigate all motions.
So, from this swirling, whirling, vibrating world of ours let us choose
just one moving object for our attention, something interesting and
typical, but above all, something manageable. Then let us describe
its motion.

But where shall we start? A machine, such as a rocket or a car?
Though made and controlled by man, they or their parts move in
fast and complicated ways. We really ought to start with something
simpler and slower, something that our eyes can follow in detail.
Then how about a bird in flight? Or a leaf falling from a tree?

Surely, in all of nature there is no motion more ordinary than
that of a leaf fluttering down from a branch. Can we describe how
it falls or explain why it falls? As we think about it we quickly
realize that, while the motion may be "natural," it is very
complicated. The leaf twists and turns, sails to the right and left,

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

-r

' y

Section 1.2

back and forth, as it floats down. Even a motion as ordinary as this
may turn out, on closer examination, to be more complicated than
the motion of machines. And even if we could describe it in detail,
what would we gain? No two leaves fall in quite the same way;
therefore, each leaf would seem to require its own detailed
description. Indeed, this individuality is typical of most events
occurring spontaneously on earth.

And so we are faced with a dilemma. We want to describe
motion, but the motions we encounter under ordinary circumstances
appear too complex. What shall we do? The answer is that we
should go, at least for a while, into the physics laboratory -because
the laboratory is the place to separate the simple ingredients that
make up all complex natural phenomena and to make those
phenomena more easily visible to our limited human senses.

1.2 A motion experiment that does not quite worit

A billiard ball, hit squarely in the center, speeds easily across a
tabletop in a straight line. An even simpler motion (simpler because
there is no rolling) can be obtained in this way: Take a disk of what
is called "dry ice" (really frozen carbon dioxide), put it on a
smooth floor, and give it a gentle push. It will move slowly and
with very little friction, supported on its own vapor. We did this in
front of a camera to get a photograph that would "freeze" the action
for easier measurement later. While the dry ice disk was moving,

Laboratory setup

Time exposure of the disk in motion

Close-up of
a dry ice disk

The Language of Motion

The speed of an object, is, of course,
how fast it moves from one place
to another. A more formal way to
say the same thing is: Speed is the
time rate of change of position.

From time to time you will be
referred to items in the Study Guide,
a few pages found at the end of
each chapter. Usually the letters SG
plus a number will indicate this. See
SG 1.1 on page 31 for more
information on how to study for this
course and on the use of the Study
Guide.

the shutter of the camera was kept open; the resulting time-
exposure shows the path taken by the disk.

What can we learn about the disk's motion by examining the
photographic record? Our question is easy enough to answer: as
nearly as we can judge by placing a ruler on the photograph, the
disk moved in a straight line. This is a very useful result, and we
shall see later that it is really quite surprising. It shows how
simplified the laboratory can be: the kinds of motion one ordinarily
sees are almost never that simple. But did it move steadUy, or did it
slow down? From this photograph we cannot tell. Let us improve
our experiment. Before we do so, however, we must be clear on
just how we might expect to measure the speed.

Why not use something like an automobile speedometer? A
speedometer is supposed to tell us directly the speed at which the
car is moving at any time. Everyone knows how to read that most
popular of all meters, even though few of us have a clear notion of
how it works. Think of how speeds are expressed. We say, for
example, that a car is moving at 60 miles per hour. This means
that if the car continues to move with the same speed it had at the
instant the speed reading was taken, the car would move a distance
of 60 miles in a time interval of 1.0 hour. Or we could say that the
car would move 1.0 mile in 1/60 of an hour, or 6.0 miles in 1/10 of
an hour — or any distance and time intervals for which the ratio of
distance to time is 60 miles per hour.

Unfortunately, an automobile speedometer cannot be hooked to
a disk of dry ice, or to a bullet, or to many other objects whose
speed we might wish to measure. (See SG 1.2.) However, there is a
way to measure speeds in most cases that would interest us.

As a clue, remember what you would have to do if the
speedometer in your car were broken and you still wanted to know
your speed as you moved along a turnpike. You would do one of two
things (the result is the same in either case): you would count the
number of mile markers passed in one hour (or some fraction of it)
and find the average speed by getting the ratio of miles and hours;
or, you would determine the fraction of an hour it takes to go from
one mile marker to the next (or to another marker a known number
of miles away) and find again the average speed as a ratio of miles
to hours.

Either method gives, of course, only the average speed for the
interval during which speed is measured. That is not the same as
the speed at any given instant, which a speedometer registers, but
it is good enough for a start. After we get average speeds clear, we
shall see a simple way of getting instantaneous speeds.

Therefore, to find the speed of an object, we measure the
distance it moves and the time it takes to move that distance. Then
we divide the distance by the time, and the speed comes out in
miles per hour, or feet per second, or meters per second, depending
upon the units used to measure distance and time. With this plan of
attack, we return to the experiment with the dry ice disk. Our task
now is to find the speed of the disk as it moves along its straight-line
path. If we can do it for the disk, we can do it for many other
objects as well.

Section 1.3

There will usually be one or more brief questions at the end of
each section in a text chapter. Ql below is the first. Use these to
check on your own progress. Answer the questions before continuing
to the next section. Check your answers to the end-of- section
questions at the back of this book (page 197); whenever you
find you did not get the correct answers, study through the section
again. And of course, if anything is still unclear after you have
tried to study it on your own or together with other students, then
ask your teacher!

Ql Why is it not possible to determine the speed of the dry ice
puck in the time-exposure photograph on page 11?

1.3 A better experiment

To find speed, we need to be able to measure both distance and
time. So let's repeat the experiment with the dry ice disk after first
placing a meter stick (100 cm) on the table parallel to the expected
path of the disk. This is the photograph we obtain:

We now have a way of measuring the distance traveled by the
disk, but we still need a way to measure the time it takes the disk
to travel a given distance.

This can be done in various ways but here is a fine trick that
you can try in the laboratory. The camera shutter is again kept open
and everything else is the same as before, except that the only
source of light in the darkened room comes from a stroboscopic
lamp. This lamp produces bright flashes of light at a frequency
which can be set as we please. Since each pulse or flash of light
lasts for only about 10 millionths of a second (10 microseconds), the
moving disk appears in a series of separate, sharp exposures, rather
than as a continuous blur. The photograph below was made by
using such a stroboscopic lamp flashing 10 times a second, after the
disk had been gently pushed as before.

The Language of Motion

See the articles "Motion in Words"
and "Representation of Movement"
in Project Physics Reader 1.

Now we're getting somewhere. Our special setup enables us to
record accurately a series of positions of the moving object. The
meter stick helps us to measure the distance moved by the front
edge of the disk between successive light flashes. The time interval
between images is, of course, equal to the time interval between
stroboscopic lamp flashes (which is 0.10 second in these photos).

We can now determine the speed of the disk at the beginning
and end of its photographed path. The front edge of the first clear
image of the disk at the left is 6 cm from the zero mark on the
meter stick. The front edge of the second image from the left is at
the 19-cm position. The distance traveled during that time was the
difl"erence between those two positions, or 13 cm. The corresponding
time interval was 0.01 second. Therefore, the speed at the start must
have been 13 cm/0.10 sec, or 130 cm/sec.

Turning now to the two images of the disk farthest to the right
in the photograph, we find that the distance traveled during 0.10
sec was 13 cm. Thus the speed at the right end was 13 cm/0.10 sec,
or 130 cm/sec.

The disk's motion was not measurably slower at the right end
than at the left end. Its speed was 130 cm/sec near the beginning
of the path — and 130 cm/sec near the end of the path. However,
that does not yet prove that the speed was constant all the way.
We might well suspect that it was, and you can easily check for
yourself that this suspicion is justified. Since the time intervals
between images are equal, the speeds will be equal if the distance
intervals are equal to one another. Is the distance between images
always 13 cm? Did the speed stay constant, as far as you can tell
from the measurements?

When you think about this result, there is something really
unusual in it. Cars, planes, and ships do not move in neat, straight
lines with precisely constant speed even when they go under
power. Yet this disk did it, coasting along on its own, without
anything to keep it moving. You might well think it was just a
rare event and it would not happen again. In any case, you should
try it. The equipment you will use for this study of physics will
include cameras, strobe lamps (or mechanical strobes, which work
just as well), and low-friction disks of one sort or another. Repeat the
experiment several times at diff"erent initial speeds, and then
compare your results with those we found above.

You may have a serious reservation about the experiment. If
you ask, "How do you know that the disk didn't slow down an
amount too small to be detected by your measurements?" we can
only answer that we don't know. All measurements involve some
uncertainty which one can usually estimate. With a meter stick we
can measure distances reliably to the nearest 0.1 cm. If we had
been able to measure to the nearest 0.01 cm or 0.001 cm. we might
have detected some slowing down. But if we again found no change
in speed, you could still raise the same objection. There is no way
out of this. We must simply admit that no physical measurements
are ever infinitely precise. Thus it is wise to leave open to question

Section 1.4

the results of any set of measurements and the findings based on
them if increased precision could reveal other results.

Let us briefly review the results of our experiment. We devised
a way to measure the successive positions of a moving dry ice disk
at known time intervals. From this we calculated first the distance
intervals and then the speed between selected positions. We soon
discovered that (within the limits of accuracy of our measurement)
the speed did not change. Objects that move in such a manner are
said to have uniform speed or constant speed. We know now how
to measure uniform speed. But, of course, actual motions are
seldom uniform. What about the more usual case of nonuniform
speed? That is our next concern.

Q2 Suppose the circles below represent the successive positions
of a moving object as photographed stroboscopically. Did the object
move with uniform speed? How do you know?

o o o o o o

Q3 Describe uniform speed without referring to dry ice pucks
and strobe photography or to any particular object or technique of
measurement.

Some practice problems dealing
with constant speed are given in
Study Guide 1.3 (a, b, c, and d).

1.4 Leslie's "50" and the meaning of average speed

Consider the situation at a swimming meet. At the end of each
race, the name of the winner is announced — the swimmer with the
shortest time; but since in a given race — say the 100-yard back-
stroke—every swimmer goes the same distance, the swimmer with
the shortest time is the one having the highest average speed while
covering the measured distance. The ratio of the distance traveled
to the elapsed time is the measure of average speed. This relation-
ship is expressed in the following equation:

distance traveled

average speed

elapsed time

What information does a knowledge of the average speed give us?
We shall answer this question by studying a real example.

Leslie is not the fastest girl freestyle swimmer in the world, but
Olympic speed is not necessary for our purposes. One day after
school, Leslie was timed while swimming two lengths of the
Cambridge High School pool. The pool is 25.0 yards long, and it took
her 56.1 seconds to swim the two lengths. Thus her average speed
over the whole 50-yard distance was

50.0 yd

56.1 sec

= 0.89 yd/sec, or nearly 2.7 ft/sec

Did Leslie swim the 50 yards at uniform (or constant) speed? If
not. which length did she cover more quickly? What was her
greatest speed? her least speed? How fast was she moving when
she passed the 10-yard, or 18-yard or 45-yard mark? These are

2.7 ft/sec is the equivalent of 1.8
miles per hour. No great speed! A
sailfish can do over 40 mph. But
man is a land animal. For short
distances he can run better than
20 mph.

The Language of Motion

useful things to know when training for a meet. But so far we do
not have a way to answer any of these questions. The value 0.89
yd/sec probably comes closer than any other one value to describing
the whole event.

To compare Leslie's speed at different parts of the swim, we
must observe the times and distances covered as we did in
experimenting with the dry ice disk. That is why we arranged the
event as shown on the photograph below.

Observers stationed at 5-yard intervals from the 0 mark along
the length of the pool started their stopwatches when the starting
signal was given. Each observer had two watches, one which he
stopped as Leslie passed his mark going down the pool, and another
which he stopped as she passed on her return trip. The data are
tabulated in the margin.

O.Oijci

O.OstJC

5.0

2.5

10.0

5.5

15.0

ll.O

20.0

/6.0

25.0

2].0

50.0

26.-J

35.0

52.0

40.0

39.5

45.0

47.5

50.0

56.1

From these data we can determine Leslie's average speed for
the first 25 yards and for the last 25 yards separately.

Average speed fov first 25 yards =

Average speed for the last 25 yards =

distance traveled
elapsed time
_ 25.0 yards

22.0 seconds
= LIO yd/sec

distance traveled

elapsed time
25.0 yards
56.0 sec - 22.0 sec
25.0 yd

34.0 sec

= 0.735 yd/sec

It is now clear that Leslie did not swim with uniform speed. She
swam the first length much faster (1.10 yd/sec) than the second
length (0.74 yd/sec). Notice that the overall average speed (0.89
yd/sec) does not describe either lap very well. Here and elsewhere

Section 1.4

in our study of motion, the more we refine our measurements to look
at detail, the more variation we find.

In a moment we shall continue our analysis of the data we have
obtained for Leslie's swim — mostly because the concepts we are
developing here, to discuss this everyday type of motion, will be
needed later to discuss other motions, ranging from that of planets
to that of atoms. First, we shall introduce some shorthand notation
with which the definition of average speed can be simplified from

average speed =

distance traveled

elapsed time

to the more concise statement that says exactly the same thing:

In this equation v,„. is the symbol for the average speed. Ad is the
symbol for change in position, and At is the symbol for an elapsed
interval of time. The symbol A is the fourth letter in the Greek
alphabet and is called delta. When A precedes another symbol, it
means "the change in. . . ." Thus, Ad does not mean "A multiplied
by d" but rather "the change in d" or "the distance interval."
Likewise, At stands for "the change in t" or "the time interval."

We can now go back to the data to see what we can learn about
Leslie's average speed for each 5-yard interval, from beginning to
end. This calculation is easily made, especially if we reorganize our
data as in the table on page 19. The values of v,„. calculated at
5-yard intervals for the first lap are entered in the right-hand column.
(The computations for the second lap are left for you to complete.)

Much more detail is emerging from the picture. Looking at the
speed column, we see that Leslie's speed was at its greatest right
near the beginning. Her racing jump into the water gave her extra
speed at the beginning. In the middle of her first length she was
swimming at a fairly steady rate, and she slowed down coming into
the turn. Use your own figures to see what happened after the turn.

Although we have determined Leslie's speeds at various
intervals along the path, we are still dealing with average speeds.
The intervals are smaller — 5 yards rather than the entire 50 — but
we do not know the details of what happened within any of the
5-yard intervals. Thus, Leslie's average speed between the 15- and
20-yard marks was 1.0 yd/sec, but we don't know yet how to
compute her speed at the very instant and point when she was, say,
18 yards or 20 yards from the start. Even so, we feel that the average
speed computed over the 5-yard interval between the 15- and
20-yard marks is probably a better estimate of her speed as she went
through the 18-yard mark than is the average speed computed
over the whole 50 yards, or over either 25-yard length. We shall
come back to this problem of the determination of "speed at a
particular instant and point" in Sec. 1.7.

Q4 Define average speed.

Practice problems on average speed
can be found in Study Guide 1.3
(e, f, g, and h.) Study Guide 1.4, 1.5,
1.6, and 1.7 offer somewhat more
challenging problems. Some
suggestions for average speeds to
measure are listed in Study Guide
1.8 and 1.9. An interesting activity
is suggested in Study Guide 1.10.

The Language of Motion

Q5 If you have not already completed the table on page 19, do
so now before going on to the next section.

1.5 Graphing motion and finding the slope

What can we learn about motion by graphing the data rather
than just tabulating them? Let us find out by preparing a distance-
versus-time graph, using the data from Leslie's 50-yard swim. As
shown in the first graph on the next page, all we really know are the
data points. Each point on the graph shows the time when Leslie
was at a particular position along her path. In the second graph,
dotted lines have been drawn to connect the points. We don't
actually know what the values are between the data points — the
straight-line connections are just a very simple way of suggesting
what the overall graph might look like. In fact, the straight lines are
not likely to be a very good approximation, because the resulting
broken-line graph would indicate very abrupt changes. If we believe
that Leslie changed speed only gradually, we can get a better
approximation by drawing the smoothest curve possible through the
data points. One experimenter's idea of a smooth curve is shown
in the last graph.

Now let us "read" the graph. Notice that the line is steepest at
the start. This means that there was a comparatively large change
in position during the first seconds — in other words, she got off to a
fast start! The steepness of the graph line is an indication of how
fast she was moving. From 10 yards to 20 yards the line appears to
be straight, becoming neither more nor less steep. This means that
her speed in this stretch was constant. Reading the graph further,
we see that she slowed down noticeably before she reached the
25-yard mark, but gained in speed right after the turn. The steepness
decreases gradually from the 30-yard mark to the finish as Leslie
was slowing down. There was no final spurt over the last 5 yards.
(She could barely drag herself out of the pool after the trial.)

Looked at in this way, a graph provides us at a glance with a
visual representation of motion. But this way of representing
motion so far does not help us if we want to know actual values of
her speed at various times. For this, we need a way of measuring
the steepness of the graph line. Here we can turn to mathematics
for help, as we often shall. There is an old method in geometry for
solving just this problem. The steepness of a graph at any point is
related to the change in the vertical direction (Ai/) and the change
in the horizontal direction (Ax). By definition, the ratio of these two
changes (Ay I Ax) is the slope:

slope = - —

Slope is a widely-used mathematical concept, and can be used to
indicate the steepness of a line in any graph. In a distance-time
graph like the one for Leslie's swim, distance is usually plotted on

Section 1.5

Ad At

0.0 yd

O.Osec

2.0

2.5

10.0

5.5

15.0

II.O

20.0

16.0

25.0

22.0

30.0

26.^

55.0

52.0

-fO.O

39.5

45.0

47.5

50.0

56.1

5.0<fd

2.5 sec

5.0

3.0

5.0

5.5

5.0

6.0

5.0

4-J5

5.0

5.5

5.0

&ic.

5.0

2.0 i/o/scc

1.7
0.9

1.0
0.6

^ 301

"I 201

/0| ©
©

^% ?0 20 X W

time (seco/x/^)

50 60

501

j?f

^ /D 2D 30 ?D '^ 50

5 ^

^ 201

js-

10 20 30 ?D 50 60

time (.5&:onds)

the vertical axis (d replaces y) and time on the horizontal axis (t
replaces x). Therefore, in such a graph, the slope of a straight line
is given by

slope =

Ad
At

But this reminds us of the definition of average speed, Va,. = Ad/At.
Therefore, Va,- = slope! In other words, the slope of any straight-line
part of a graph of distance versus time gives a measure of the
average speed of the object during that interval. What we do when
we measure slope on a graph is basically the same thing that
highway engineers do when they specify the steepness of a road.
They simply measure the rise in the road and divide that rise by the
horizontal distance one must go in order to achieve the rise. The
only difference between this and what we have done is that the

Above are shown four ways of repre-
senting Leslie's swim: a table of
data, a plot of the data points, broken
straight-line segments that connect
the points, and a smooth curve that
connects the points.

If this concept is new to you or if
you wish to review it, turn now to
Study Guide 1.11 before continuing
here.

The Language of Motion

highway engineers are concerned with an actual physical slope: on
a graph of their data the vertical axis and horizontal axis would
both show distance. We, on the other hand, are using the
mathematical concept of slope as a way of expressing distance
measured against tim.e.

We can get a numerical value quickly and directly for the slope
of each straight-line segment in the graph on p. 19. so we will have
the value of the average speed for each of the 5-yard intervals
between data points. For example, we used our data table to
calculate Leslie's average speed between the 5- and 10-yard
markers as 1.4 yd/sec. She moved 5 yards on the vertical
(distance) axis during a lapse of 3.5 seconds on the horizontal
(time) axis. Therefore, the slope of the hne segment connecting the
5-yard and 10-yard points is equal to 5 yards divided by 3.5 seconds,
or 1.4 yd/sec.

The slope, as we have defined it here, is not exactly the same
thing as the steepness of the line on the graph paper. If we had
chosen a different scale for either the distance or time axis (making
the graph, say, twice as tall or twice as wide), then the apparent
steepness of the entire graph would be different. The slope, however,
is measured by the ratio of the distance and time units — a Ad of 10
meters in a At of 5 seconds gives a ratio of 2 meters/second, no
matter how much space is used for meters and seconds on the
graph.

But the graph is more useful than just leading us back again
to the values in the table. We can now ask questions that cannot
be answered directly from the original data: What was Leslie's
speed 10 seconds after the start? What was her speed as she
crossed the 37-yard mark? Questions like these can be answered
by finding the slope of a fairly straight portion of the graph line
around the point of interest. Two examples are worked out on the

MiSL_ = 0.7O yVjec
■1.0 sec "^

- 0.S5 i^lscc

20 30 40

timt (stconds)

Section 1.6

graph at the bottom of page 20. For each example, At was chosen The 4-sec value for f is just for
to be a 4-sec interval -from 2 sec before the point in question to 2 convenience; some other value

sec after it; then the Ad for that At was measured. ^°"'^ ^^"^ ^^^" ^,^^^- ^'' ^^ ^°"'^

have chosen a value for Ad and then
The reasonableness of usmg the graph in this way can be measured the corresponding At.

checked by comparing the results with the values listed in the table
on p. 19. For example, the speed near the 10-second mark is found
from the graph to be about 3.0 yd/4.0 sec = 0.75 yd/sec. This is
somewhat less than the value of 0.9 yd/sec given in the table for
the average speed between 6 and 11 seconds; and that is just what
we would expect, because the smooth-curve graph does become
momentarily less steep around the 10-second point. If the smooth
curve that was drawn really is a better description of Leslie's
swimming than the broken line is, then we can get more information
out of the graph than we put into it.

Q^ Turn back to p. 13 and draw a distance- time graph for the
motion of the dry ice disk.

Which of the two graphs below has the greater slope?

400

0/2545676
timt (I toon)

Q8 Where was Leslie swimming most rapidly? Where was she
swimming most slowly?

Q9 From the graph, find Leslie's speed at the 47-yard mark.
From the table on p. 19, calculate her average speed over the last
5 yards. How do the two values compare?

1.6 Time out for a warning

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

The placement of a line through a series of data points, as in
the graph on page 19, depends on personal judgment and
interpretation. The process of estimating values between data
points is called interpolation. That is essentially what you are
doing when you draw a line between data points. Even more risky
than interpolation is extrapolation, where the graph line is

The Language of Motion

19 minutes

17 minutes

27 minutes

The Language of Motion

These photographs show a stormy
outburst of incandescent gas at the
edge of the sun, a developing chive
plant and a glacier. From these pic-
tures and the time intervals given
between pictures, you can determine
the average speeds (1) of the growth
of the solar flare with respect to the
sun's surface (radius of sun is about
432,000 mi), (2) of the growth of one
of the chive shoots with respect to the
graph paper behind it (large squares
are one inch), (3) of the moving glacier
with respect to its "banks."

17 hours

33 hours

4 years

Section 1.7

extended to provide estimated points beyond the known data.

An example of a high-altitude balloon experiment carried out in
Lexington, Massachusetts, nicely illustrates the danger of
extrapolation. A cluster of gas-filled balloons carried cosmic ray
detectors high above the earth's surface, and from time to time a
measurement was made of the height of the cluster. The graph on
the right shows the data for the first hour and a half. After the first 20
minutes the balloons seem to be rising in a cluster with unchanging
speed. The average speed can be calculated from the slope: speed
of ascent = Ad/ At = 27,000 ft/30 min = 900 ft/min. If we were asked
how high the balloons would be at the very end of the experiment
(500 min), we might be tempted to extrapolate, either by extending
the graph or by computing from the speed. In either case we would
obtain the result 500 min x 900 ft/min = 450,000 ft, which is over
90 miles high! Would we be right? Turn to Study Guide 1.12 to see
for yourself. (The point is that mathematical aids, including graphs,
can be a splendid help, but only within the limits set by physical
realities.)

Q10 What is the difference between extrapolation and
interpolation?

Q11 Which estimate from the graph would you expect to be
less accurate: Leshe's speed as she crossed the 30-yard mark, or
her speed at the end of an additional lap?

Jimt (min)

SG 1.13

1.7 Instantaneous speed

Now let us wrap up the chief lessons of this first chapter. In
Sec. 1.5 we saw that distance-time graphs could be very helpful in
describing motion. When we reached the end of the section, we
were speaking of specific speeds at particular points along the path
(like "the 14-yard mark") and at particular instants of time (like "the
instant 10 seconds after the start"). You probably were bothered by
this manner of talking, since at the same time we admitted that the
only kind of speed we can actually measure is average speed. To
find average speed we need a ratio of distance and time intervals.
A particular point on the path, however, does not have any
interval. Nevertheless, it makes sense to speak about the speed at a
point. We will summarize what reasons there are for using "speed"
in this way, and see how well we can get away with it.

You remember that our answer to the question (page 20),
"How fast was Leslie swimming at time t = 10 sec?" was 0.85 yd/sec.
That answer was obtained by finding the slope of a small portion
of the curve encompassing the point P when t = 10 sec. That
section of the curve has been reproduced in the margin here. Notice
that the part of the curve we used appears to be nearly a straight
line. As the table under the graph shows, the value of the slope
for each interval changes very little as we decrease the time interval
At. Now imagine that we closed in on the point where t = 10 sec

^ 20

./me

At Ad

^tt

G.O sec 5A()d 0.90y'^/5C£
4.0 3.4 0.85
2.0 \n 0-95

The Language of Motion

until the amount of curve remaining became vanishingly small.
Could we not safely assume that the slope of that infinitesimal part
of the curve would be the same as that on the straight line of which
it seems to be a part? We think so. That is why we took the slope
of the straight line from t = 8 sec to t = 12 sec, and called it the
speed at the midpoint, the speed at t = 10 sec, or to use the correct
term, "the instantaneous speed" at t = 10 sec.

In estimating a value for Leslie's instantaneous speed at a
particular time, we actually measured the average speed over a
4.0-sec interval. We then made the conceptual leap that we have
described. We decided that the instantaneous speed at a particular
instant can be equated to an average speed Ad/At provided: 1) that
the particular instant is included in At, and 2) that the ratio Ad/At
is obtained for a small enough part of the curve, one which is
nearly a straight-line segment, so that it does not change appreciably
when we compute it over a still smaller time interval.

A second concrete example will help here. In the oldest known
study of its kind, the French scientist de MontbeOlard periodically
recorded the height of his son during the period 1759-1777. A graph
of height versus age for his son is shown in the margin.

From the graph, we can compute the average growth rate
(v,a^ over the entire 18-year interval or over any shorter time
interval within that period. Suppose, however, we wanted to know
how fast the boy was growing just as he reached his fifteenth
birthday. The answer becomes evident if we enlarge the graph in
the vicinity of the fifteenth year. (See the second graph.) His height
at age 15 is indicated as point P, and the other letters designate

100

170
/60
/50

150
120

100

1 —

i i

;

( ,

1 1

!■ -

, i

!
1

' — 1

80
70

60
50

1 i

\ i

/«0

f60-

Q^e ( {jrs )

0 I 2 3+ 56 1 S> 9 10 It IZ li If li lb IT Id

l^c (yr,5)

age (jfr-s)

Section 1.7

instants of time on either side of P. The boy's average growth rate
over a two-year interval is given by the slope of the line segment
AB in the enlarged figure in the margin. Over a one-year interval
this average growth rate is given by the slope of CD. (See the third
graph.) The slope of EF gives the average growth rate over six
months, etc. The four lines, AB, CD, EF, GH, are not parallel to each
other and so their slopes are different. However, the difference in
slope gets smaller and smaller. It is large when we compare AB and
CD, less if we compare CD and EF, less still between EF and GH.
For intervals less than At = 1 yr, the lines appear to be more nearly
parallel to each other and gradually to merge into the curve,
becoming nearly indistinguishable from it. For very small intervals,
you can find the slope by drawing a straight line tangent to this
curve at P, placing a ruler at P (approximately parallel to line GH),
and extending it on both sides as in Study Guide 1.11.

The values of the slopes of the straight-line segments in the
middle and lower graphs have been computed for the corresponding
time intervals and are tabulated at the right.

We note that values of Vav calculated for shorter and shorter time
intervals approach closer and closer to 6.0 cm/yr. In fact, for any
time interval less than 2 months, the average speed Va,- will be 6.0
cm/yr within the limits of accuracy of measuring height. Thus we
can say that, on his fifteenth birthday, young de Montbeillard was
growing at a rate of 6.0 cm/yr. At that instant in his life, t = 15.0 yr,
this was his instantaneous growth rate (or if you will, the
instantaneous speed of his head with respect to his feet!)

Average speed over a time interval At, we have said, is the ratio
of distance traveled to elapsed time, or in symbols,

Vav

We now have added the definition of instantaneous speed at an
instant as the final limiting value approached by the average
speeds when we compute Vgv for smaller and smaller time intervals
including the instant t. In almost all physical situations, such a
limiting value can be accurately and quickly estimated by the
method described on the previous page.

From now on we will use the letter v without the subscript m- to
mean the instantaneous speed defined in this way. You may wonder
why we have used the letter "z;" instead of "s" for speed. The
reason is that speed is closely related to velocity. We shall reserve
the term "velocity" for the concept of speed in a specified direction
(such as 50 mph to the north) and denote it by the symbol v. When
the direction is not specified and only the magnitude (50 mph) is of
interest, we remove the arrow and just use the letter v, calling the
magnitude of the velocity "speed." This crucial distinction between
speed and velocity, and why velocity is more important in physics,
will be discussed in more detail in later sections.

Q12 Define instantaneous speed, first in words and then in
symbols.

Line

Growth rate

between
points

^"' = aF

2 yr

19.0 cm

9.5 cm/year

1 yr

8.0

DEE

6 mo

3.5

7.0

4 mo

2.0

6.0

2 mo

1.0

6.0

SG 1.14

See SG 1.15, 1.16, and 1.17 for
problems that check your under-
standing of the chapter up to this
point.

1 . Paris street scene, 1 839. A daguerro-
type made by Louis Daguerre himself.

2. American street scene, 1859

3. Boys on skateboards

Photography 1839 to the Present

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

2. However, by 1859, due to improvements in photographic emulsions and
lenses, it was not only possible to photograph a person at rest, but one
could capture a bustling crowd of people, horses and carriages. Note the
slight blur of the jaywalker's legs.

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

4. A new medium-the motion picture. In 1873 a group of California
sportsmen called in the photographer Eadweard Muybridge to settle the
question, "Does a galloping horse ever have all four feet off the ground at
once?" Five years later he answered the question with these photos. The
five pictures were taken with five cameras lined up along the track, each
camera being triggered when the horse broke a string w'hich tripped the
shutter. The motion of the horse can be restructured by making a flip pad
of the pictures.

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

()'

•0

^^^^^^^

? - -^

Jj^J^SfT

4. Muybridge's series, 1878

5. Stroboscopic photo of golfer's
swing. (See the article "The Dynamics
of a Golf Club" in Project Physics
Reader 1 .)

5. A light can be flashed successfully at a controlled rate and a multiple
exposure (similar to the strobe photos in this text) can be made. In this
photo of a golfer, the light flashed 100 times each second.

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

7. An interesting offshoot of motion pictures is the high-speed motion
picture. In the frames of the milk drop series shown below, 1000 pictures
were taken each second (by Harold Edgerton). The film was whipped past
the open camera shutter while the milk was illuminated with a flashing
light (similar to the one used in photographing the golfer) synchronized
with the film. When the film is projected at the rate of 24 frames each
second, action which took place in 1 second is spread out over 42 seconds.

It is clear that the eye alone could not have seen the elegant details of
this event. This is precisely why photography of various kinds is used in
the laboratory.

6. Bullet cutting through a playing
card.

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

The Language of Motion

Q13 Explain the difference in meaning between average speed
and instantaneous speed.

Unless noted otherwise, "rate of
change" will always mean "rate of
change with respect to time."

1.8 Acceleration -by comparison

You can tell from the photograph below of a rolling baseball
that it was changing speed — accelerating. The increasing distances
between the instantaneous images of the ball give you this informa-
tion, but how can you tell how much acceleration the ball has?

To answer this question we have only one new thing to learn —
the definition of acceleration. The definition itself is simple; our
task is to learn how to use it in situations like the one above. For
the time being, we will define acceleration as rate of change of
speed. Later this definition will have to be modified somewhat when
we encounter motion in which change in direction becomes an
important additional factor. But for now, as long as we are dealing
only with straight-line motion, we can equate the rate of change
of speed with acceleration.

Some of the effects of acceleration are familiar to everyone. It
is acceleration, not speed, that you notice when an elevator
suddenly starts up or slows down. The flutter in one's stomach
comes only during the speeding up and slowing down, not during
most of the ride when the elevator is moving at a steady speed.
Likewise, much of the excitement of the roller coaster and other
rides at amusement parks is a result of their unexpected
accelerations. Speed by itself does not cause these sensations.
Otherwise they would occur during a smooth plane ride at 650 mph,
or even just during the continuous motion of the earth around the
sun at 65,000 mph.

Simply stated, speed is a relationship between two objects, one
of which is taken to be the reference object while the other moves
with respect to it. Some examples are the speed of the earth with
respect to the stars, the speed of the swimmer with respect to the
pool edge, the speed of the top of the growing boy's head with
respect to his feet. . . In a perfectly smooth-riding train, we could
tell that we were moving at a high speed only by seeing the
scenery whizzing by. We would have just the same experience if
the train were somehow fixed and the earth with rails, etc., were to
whiz by in the other direction. And if we "lost the reference object "
(by pulling down the shades, say) we might not know at all
whether we were moving or not. In contrast, we "feel" accelerations
and do not need to look out the train window to realize that the
engineer has suddenly started the train or has slammed on the

Section 1.8

brakes. We might be pushed against the seat, or the luggage might
fly from the rack.

All this suggests a profound physical diff^erence between
motion at constant speed and motion with acceleration. While it is
best to learn about acceleration at first hand (in the laboratory and
through the film loops), we can summarize the main ideas here. For
the moment let us focus on the similarities between the concepts
speed and acceleration; for motion in a straight line:

The rate of change of position
is called speed.

The rate of change of speed
is called acceleration.

This similarity of form will enable us to use what we have just
learned about the concept of speed as a guide for making use of the
concept of acceleration. For example, we have learned that the
slope of the line of a distance-time graph is a measure of the
instantaneous speed. The slope of a speed-time graph is a measure
of the instantaneous acceleration.

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

1. Speed is the rate of change of position. Acceleration is the
rate of change of speed.

2. Speed is expressed in units of distance/time. Acceleration is
expressed in units of speed/time.

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

Var =

Average acceleration over any time interval is the ratio of the
change of speed Ar and the time interval At :

At;

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

5. On a distance-time graph, the instantaneous speed at any
instant is the slope of the straight line tangent to the curve at the
point of interest. On a speed-time graph, the instantaneous
acceleration at any instant is the slope of the straight line tangent
to the curve at the point of interest.

6. For the particular case of constant speed, the distance-time
graph is a straight line; everywhere on it the instantaneous speed
has the same value, equal to the average speed computed for the
whole trip. For the particular case of constant acceleration, the
speed- time graph is a straight line; everywhere on it the

For example, if an airplane changes
its speed from 500 mph to 550 mph
in 10 minutes, its average
acceleration would be

Ay 550 mi/hr - 500 mi/hr

_ 50 mi/hr
10 min

_ 5 mi/hr
min

That is, its speed changed at a rate
of 5 mph per minute. (If the speed
was decreasing, the value of the
acceleration would be negative.)

A^-l

a =

_Ay_

Constant speed and constant
acceleration are often called
"uniform" speed and "uniform"
acceleration. In the rest of this
course, we will use the terms
interchangeably.

The Language of Motion

SG 1.18 provides an opportunity to
worl< with distance-time and
speed-time graphs and to see their
relationship to one another.
Transparencies T3 and T4 may be
helpful also.

SG 1.19 to 1.21 are review problems
for this chapter. Some of these will
test how thoroughly you grasp the
language used for describing
straight-line motion.

instantaneous acceleration has the same value, equal to the
average acceleration computed for the whole trip. When speed is
constant, its value can be found from any corresponding Ad and At.
When acceleration is constant, its value can be found from any
corresponding At; and At. (This is useful to remember because
constant acceleration is the kind of motion we shall encounter
most often in the following chapters.)

We now have most of the tools needed to get into some real
physics problems. The first of these is the accelerated motion of
bodies caused by gravitational attraction. It was by studying motion
of falling objects that Galileo, in the early 1600's, was first able to
shed light on the nature of accelerated motion. His work remains
to this day a wonderful example of how scientific theory,
mathematics, and actual measurements can be combined to develop
physical concepts. More than that. Galileo's work was one of the
early and most crucial battlegrounds of the scientific revolution.
The specific ideas he introduced are even now fundamental to the
science of mechanics, the study of bodies in motion.

Q14 What is the average acceleration of an airplane which
goes from 0 to 60 mph in 5 seconds?

Q15 What is your average acceleration if, while walking, you
change your speed from 4.0 miles per hour to 2.0 miles per hour in
an interval of 15 minutes? Is your answer affected by how your
change of speed is distributed over the 15 minutes?

STUDY GUIDE 1

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

1. Do not write in this book unless your
teacher gives you permission to do so. In many
schools the books must be used again next year by
other students. However, if you are fortunate
enough to be in a situation in which the teacher
can permit you to mark in the book, we encourage
you to do so. You will note that there are wide
margins. One of our reasons for leaving that
much space is to enable you to record questions
or statements as they occur to you when you are
studying the material. Mark passages that you do
not understand so that you can seek help from
your teacher.

2. If you may not write in the textbook itself,
try keeping a notebook keyed to the text chapters.
In this study notebook jot down the kinds of
remarks, questions and answers that you would
otherwise write in the textbook as suggested
above. Also, you ought to write down the questions
raised in your mind by the other learning
materials you will use, by the experiments

you do, by demonstrations or other observations,
and by discussions you may have with
fellow students and others with whom you talk
physics. Most students find such an informal
notebook to be enormously useful when studying,
or when seeking help from their teachers (or, for
that matter, from advanced students, parents,
scientists they may know, or anyone else whose
understanding of physics they have confidence in).

3. You will find answers to all of the end-of-
section review questions on page 197. Always try
to answer the questions yourself first and then
check your answers. If your answer agrees with
the one in the book, it is a good sign that you
understand the main ideas in that section —
although it is true that you can sometimes get the
right answer for the wrong reason, and also that
there may sometimes be other answers as good
(or better than!) those given in the book.

4. There are many different kinds of items in
the Study Guide at the end of each chapter.
Brief answers to some of them are given on page
199. It is not intended that you should do every
item. Sometimes we include material in the Study
Guide which we think will especially interest only
some students. Notice also that there are several
kinds of problems. Some are intended to give
practice in the use of a particular concept,
while others are designed to help you bring
together several related concepts. Still other
problems are intended to challenge those students
who particularly like to work with numbers.

5. This text is only one of the learning
materials of the Project Physics course. The
course includes several other materials such as
film loops, programmed instruction booklets, and
transparencies. Use those. Be sure to familiarize
yourself also with the Handbook, which de-
scribes outside activities and laboratory
experiments, and with the Reader, in which we

have collected interesting articles related to
physics. Each of these learning aids makes its
own contribution to an understanding of physics,
and all are designed to be used together.

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

Experiments (in the Handbook)
Naked Eye Astronomy
Regularity and Time
Variations in Data
Measuring Uniform Motion

Activities (in the Handbook)
Using the Electronic Stroboscope
Making Frictionless Pucks

Reader Articles

Motion in Words
Representation of Motion
Motion Dynamics of a Golf Club
Bad Physics in Athletic
Measurements

Transparencies

Analyzing a Stroboscopic Photograph
Stroboscopic Measurements
Graphs of Various Motions
Instantaneous Speed
Instantaneous Rate of Change

In addition the following Project Physics materials
can be used with Unit 1 in general:

Reader Articles

The Value of Science
Close Reasoning
How to Solve It
Four Pieces of Advice
to Young People
On Being the Right Size
The Vision of Our Age
Becoming a Physicist
Chart of the Future

1.2 One type of automobile speedometer is a
small electric generator driven by a flexible cable
run off" the drive shaft. The current produced
increases with the rate at which the generator is
turned by the drive shaft. The speedometer needle
indicates the current. Until the speedometer is
calibrated it cannot indicate actual speeds in

STUDY GUIDE 1

miles per hour. Try answering the questions
below. If you have trouble you may want to try
again after you have studied through Sec. 1.9.

(a) How would you calibrate the speedometer
in a car if the company had forgotten to
do the job?

(b) If you replaced the 24"-diameter rear
wheels with 28"-diameter wheels, what
would your actual speed be if the
speedometer read 50 mph?

(c) Would the speedometer read too high or
too low if you loaded down the rear end of
your car and had the tire pressure too low?

(d) Does the operation of the speedometer
itself affect the motion of the car?

(e) How would you test to see if a bicycle
speedometer affects the speed of a bike?

(f ) Can you invent a speedometer that has no
effect on the motion of the vehicle that
carried it?

1.3 Some practice problems:

SITUATION

FIND

Speed uniform, dis-
tance = 72 cm,
time= 12 sec

Speed

Speed uniform at
45 miles per hour

Distance traveled
in 20 minutes

Speed uniform at
36 ft/min

Time to move
9.0 feet

d, = 0 f, = 0

dz = 1 5 cm fa = 5.0 sec

d;, = 30 cm f3= 10 sec

Speed and posi-
tion at 8.0 sec

You drive 240 miles in
6.0 hr

Average speed

Same as e

Speed and
position after
3.0 hr

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

Time taken

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

Distance moved

1. 1 A tsunami caused by an earthquake occurring
near Alaska in 1946 consisted of several sea
waves which were found to travel at the average
speed of 490 mph. The first of the waves reached

Hawaii 4 hrs and 34 min after the earthquake
occurred. From these data, calculate how far the
origin of the tsunami was from Hawaii.

1.5 Light and radio waves travel through a
vacuum in a straight line at a speed of very nearly
3 X IQs m/sec.

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

(b) The nearest star, Alpha Centauri, is
4.06 X 10'« m distant from us. If this star
possesses planets on which highly
intelligent beings live, how soon, at the
earliest, could we expect to receive a reply
after sending them a radio or light signal
strong enough to be received there?

1.6 If you traveled one mile at a speed of 1000
miles per hour and another mile at a speed of 1
mile per hour, your average speed would not be
1000 mph + 1 mph/2 nr 500.5 mph. What would
be your average speed? (Hint: What is the total
distance and total time?)

1.7 What is your average speed in each of these
cases?

(a) You run 100 m at a speed of 5.0 m/sec
and then you walk 100 m at a speed of
1.0 m/sec.

(b) You run for 100 sec at a speed of 5.0 m/sec
and then you walk for 100 sec at a speed
of 1.0 m/sec?

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

(a) A baseball thrown from outfield to home
plate

(b) The wind

(d) A raindrop

(e) A hand moving back and forth as fast as
possible

(f ) The tip of a swinging baseball bat

(g) A person walking on level ground,
upstairs, downstairs

(h) A bird flying
(i) An ant walking

(j) A camera shutter opening and closing

(k) An eye blinking
( 1 ) A whisker growing
(m) The center of a vibrating guitar string

1-9 What problems arise when you attempt to
measure the speed of light? Can you design an
experiment to measure the speed of light?

1.10 Sometime, when you are a passenger in an
automobile, compare the speed as read from the
speedometer with the speed calculated from Id! At.
Explain any differences. Refer again to SGI. 2.
(For other activities see your Project Physics
Handbook.)

STUDY GUIDE 1

l.n Take a look at the graph of y versus x
shown below:

Although in this particular graph the
steepness of the line increases as x increases, the
method presented below would also hold for a
curve of any other shape. One way to indicate the
steepness of the line at a point P is by means of
its "slope." The numerical value of the slope at a
point P is obtained by the following procedure
(diagrammed above): At a very short distance
along the line from point P to either side of it,
mark 2 points, A and B. Choose these points so
close to P that although they also lie on the curve,
the line APB is a straight line as nearly as one
can determine with a ruler. Measure Ay Uhe
change in y) in going from A to B. In this example
y = 0.6. Measure Ax (the corresponding change in
x) in going from A to B. Ax here is 0.3. The slope
of the segment AB is defined as the ratio of Ay
to Ax of the short straight-line-segment APB. By
definition, the slope of the curve at point P is
taken to be equal to the slope of the straight-line-
segment APB.

Aw
slope = —

In this example,

Ay
slope = -T-^ =
Ax

0.6
0.3

Q. What are the dimensions or units for the
slope?

A. The dimensions are just those of y/x. For
example, if y represents a distance in meters and
X represents a time in seconds, then the units for
slope will be meters per second (or m/sec).

Q. In practice, how close must A and B be to
point P? (Close is not a very precise adjective.
Baltimore is close to Washington if you are flying

over both by jet. If you are walking, it is not close.)

A. Choose A and B near enough to point P so
that a straight line drawn carefully to connect A
and B also goes through point P.

Q. Suppose A and B are so close together that
you cannot adequately read Ax or Ay from your
graph. How would you try to calculate the slope?

A. Extend the straight line AB in both
directions, as shown in the figure, as far as you
wish, and compute its slope. What you are then
doing is putting a tangent line to the curve at the
chosen point between A and B. Notice that the
small triangle is similar to the large triangle, and,
therefore

Ay/ Ax = AY/ AX

Problem :

(a) Determine the slope of this graph of
distance versus time (y in meters, t in
seconds) at four different points or
instants, namely when t = 1, 2, 3, and 4
seconds.

(b) Find the instantaneous speed at these 4
points, and plot a graph of speeds vs. time.

1.12 (Answer to question in text, page 23.)

Indeed the prediction based upon the first
hour and a half would be vastly wrong. A
prediction based on an extrapolation from the first
labour's observation neglects all the factors
which limit the maximum height obtainable by
such a cluster of balloons, such as the bursting of
some of the balloons, the change in air pressure
and density with height and many others.
Actually, at the end of 500 minutes the cluster
was not 450,000 feet high but had come down
again, as the distance-time graph for the entire
experiment shows. See top of next page. For
another extrapolation problem, see SG 1.13.

STUDY GUIDE 1

100 zoo 300 400 500
iime (sec)

1.13 World's 400-nieter swimming records in
minutes and seconds for men and women
(numbers in parentheses are ages):

1926

1936

1946

1956

1966

4:57.0

5:53.2

4:46.4

28.5

46.4

00.1

33.3

47.2

4:11.1

4:38.0

Johnny Weissmuller (18)
Gertrude Ederle (17)
Syozo Makino (17)
Helene Madison (18)
(1936 record unbroken)
R. Hveger (18)
Hironoshin Furuhashi (23)
Lorraine Crapp (18)
Frank Weigand (23)
Martha Randall (18)
By about how many meters would Martha Randall
have beaten Johnny Weissmuller if they had
raced each other? Could you predict the 1976
records for the 400-meter race by extrapolating
the graphs of world's records vs. dates up to the
year 1976?

1.14 How can we justify defining instantaneous
speed as we have on p. 25? How can we be sure
the definition is right?

1.15 Using the graph on p. 20 find the
instantaneous speeds v at several points (0, 10,
20, 30, 40, and 50 sec, and near 0, or at other
points of your choice) by finding the slopes of lines
tangent to the curve at each of those points. Make
a graph of v vs. t. Use your graph to describe her
swim.

1.1 (i Turn back to p. 28. At the bottom of this page
there is a multiple-exposure photograph
of a baseball rolling to the right. The time interval
between successive flashes was 0.20 sec. The
distance between marks on the meter stick was
1 centimeter. You might tabulate your
measurements of the ball's progress between
flashes and construct a distance-time graph. From
the distance-time graph, you can determine the
instantaneous speed at several instants and

construct a speed-time graph. You can check your
results by referring to the answer page at the end
of this unit.

1.17 Careful analysis of a stroboscopic photograph
of a moving object yielded information which
was plotted on the graph below. By placing your
ruler tangent to the curve at appropriate points
estimate the following:

0-
7

■^

^_,^

■

^ 7

1 _^

u n

4 6

t\n^e (sec)

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

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

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

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

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

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

(b) What is the maximum acceleration?

Time (sec)

Speed (m/

sec)

Time (sec)

Speed (m/sec)

0.0

6.0

27.3

1.0

6.3

7.0

29.5

2.0

11.6

8.0

31.3

3.0

16.5

9.0

33.1

4.0

20.5

10.0

34.9

5.0

24.1

1.19 The electron beam in a typical TV set
sweeps out a complete picture in 1/30 sec and
each picture is composed of 525 lines. If the
width of the screen is 20 inches, what is the speed
of that beam over the surface of the screen?

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

STUDY GUIDE 1

1.21 Discuss the motion of the cat in the
following series of photographs, "Cat in trot
changing to gallop." The numbers on each

photograph indicate the number of inches
measured from the fixed line marked "0." The
time interval between exposures is 0.030 sec.

10 20 ^iP 20 I 10 I 20

^^^ ^^^ ^^^

20 1 ; 20 30 j 20 30 1 2l) 30

^ --fr /^y" 1^^^

50 : 40

50 40

♦*- vtlStr

2.1 The Aristotelian theory of motion 37

2.2 Galileo and his times 43

2.3 Galileo's Two New Sciences 43

2.4 Why study the motion of freely falling bodies? 47

2.5 Galileo chooses a definition of uniform acceleration 47

2.6 Galileo cannot test his hypothesis directly 49

2.7 Looking for logical consequences of Galileo's hypothesis 50

2.8 Galileo turns to an indirect test 53

2.9 Doubts about Galileo's procedure 56

2.10 Consequences of Galileo's work on motion 57

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

CHAPTER TWO

Free Fall-

Galileo Describes Motion

2.1 The Aristotelian theory of motion

In this chapter we shall follow the development of an important
piece of basic research: Galileo's study of freely falling bodies.
While the physical problem of free fall is interesting in itself, our
emphasis will be on the way Galileo, one of the first modern
scientists, presented his argument. His view of the world, his way
of thinking, his use of mathematics, and his reliance upon
experimental tests set the style for modem science. These aspects of
his work, therefore, are as important to us as the actual results of his
investigation.

To understand the nature of Galileo's work and to appreciate its
significance, we must first examine the previous system of
physical thought that it eventually replaced. In medieval physical
science, as Galileo learned it at the University of Pisa, a sharp
distinction was thought to exist between the objects on the earth
and those in the sky. All terrestrial matter, the matter within our
physical reach, was believed to contain a mixture of four "elements"
-Earth, Water, Air, and Fire. These elements were not thought of
as identical with the natural materials for which they were named.
Ordinary water, for example, was thought to be a mixture of all
four elements, but mostly the element Water. Each of the four
elements was thought to have a natural place in the terrestrial
region. The highest place was allotted to Fire. Beneath Fire was Air,
then Water, and finally, in the lowest position, Earth. Each was
thought to seek its own place. Thus Fire, if displaced below its
natural position, would tend to rise through Air. Similarly, Air would
tend to rise through Water, whereas Earth would tend to fall
through both Air and Water. The movement of any real object
depended on its particular mixture of these four elements, and on
where it was in relation to the natural places of these elements.

SG 2.1

A sketch of a medieval world-system.

Free Fall — Galileo Describes Motion

A good deal of common-sense
experience supports this natural-
place view. See SG 2.2

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

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

When water boiled, for example, the element Water would be joined
by the element Fire, whose higher natural place would cause the
mixture to rise as steam. A stone, on the other hand, being
composed primarily of the element Earth, would fall when released
and would pass through Fire, Air, and Water until it came to rest on
the ground, its natural place.

The medieval thinkers also believed that the stars, planets, and
other celestial bodies differed in composition and behavior from
objects on or near the earth. The celestial bodies were believed to
contain none of the four ordinary elements, but instead to consist
solely of a fifth element, the quintessence. The natural motion of
objects composed of this element was neither rising nor falling, but
endless revolution in circles around the center of the universe. That
center was considered to be identical with the center of the earth.
Heavenly bodies, although moving, were at all times in their
natural places. Thus heavenly bodies were altogether different from
terrestrial objects, which displayed natural motion only as they
returned to their natural places from which they had been displaced.

This theory, so widely held in Galileo's time, had originated
almost 2000 years before, in the fourth century B.C. We find it stated
clearly in the writings of the Greek philosopher Aristotle. This
physical science, built on order, class, place, and purpose, fits well
many facts of everyday observation. It seemed particularly plausible
in societies like those in which Aristotle and Galileo lived, where
rank and order were dominant in human experience. Moreover,
these conceptions of matter and motion were part of an all-
embracing universal scheme or "cosmology." In his cosmology
Aristotle sought to relate ideas which are nowadays discussed
separately under such headings as science, poetry, politics, ethics,
and theology.

Not very much is known of Aristotle's physical appearance or
life. It is thought that he was bom in 384 B.C. in the Greek
province of Macedonia. His father was the physician to the King of
Macedonia, and so Aristotle's early childhood was spent in an
environment of court life. He completed his education in Athens
and later returned to Macedonia to become the private tutor to
Alexander the Great. In 335 B.C., Aristotle came back to Athens
and founded the Lyceum, a school and center of research.

500 BC

400 BC 384 BC ^^^tt2 BC 300 BC

ARISTOTLE

XERXES

PERICLES

o
c/)

C
(0

a
o

■CONFUCIUS

ANAXAGORAS
ZENO OF ELEA
PARMENIDES OF ELEA
PROTAGORAS

DEMOCRITUS
SOCRATES

200 BC

m§

PTOLEMY I of Egypt
PHILIP II

ALEXANDER

HANNIBAL

MENCIUS (MENGTZU)
EPICURUS

ZENO OF CITIUM

HSUN TZU

ARCHIMEDES

ERASTOSTHENES

AESCHYLUS
I

PINDAR

SOPHOCLES

HERODOTUS
EURIPIDES

THUCYDIDES

ARISTOPHANES
XENOPHON

DEMOSTHENES

CHUANG TZU

PHIDIAS SCOPAS

MYRON
POLYGNOTUS

POLYCLITUS
ZEUXIS

TIMOTHEUS

40 Free Fall -Galileo Describes Motion

After the decline of the ancient Greek civilization, the writings
of Aristotle remained virtually unknown in Western Europe for
1500 years. They were rediscovered in the thirteenth century A.D.
and were later incorporated into the works of Christian scholars and
theologians. Aristotle became such a dominant influence in the late
Middle Ages that he was referred to simply as "The Philosopher."

The works of Aristotle make up almost an encyclopedia of
ancient Greek thought. Some of it was summarized from the work
of others, but much of it seems to have been created by Aristotle
himself. Today it is hard to believe that one man could have been
so well informed on such different subjects as logic, philosophy,
theology, physics, astronomy, biology, psychology, politics, and
literature. Some scholars doubt that it was all the work of one man.

Unfortunately, Aristotle's physical theories had serious
limitations. (This does not, of course, detract from his great
achievements in other fields.) According to Aristotle, the fall of a
heavy object toward the center of the earth is an example of
"natural" motion. He evidently thought that any object, after
release, quickly achieves some final speed of fall at which it
continues to move to the end of its path. What factors determine the
final speed of a falling object? It is a common observation that a
rock falls faster than a leaf. Therefore, he reasoned, weight is a
factor that governs the speed of fall. This fitted in well with his idea
that the cause of weight was the presence of the element Earth,
whose natural tendency was to the center of the earth. Thus a
heavier object, having a greater content of Earth, has a greater
tendency to fall to its natural place, and hence develops a greater
speed in falling.

The same object falls more slowly in water than in air, so it
seemed to Aristotle that the resistance of the medium must also be
a factor. Other factors, such as the color or temperature of the
falling object, could conceivably affect the rate of fall, but
Aristotle: rate of fall is proportional Aristotle decided that their influence could not be significant. He

to weight divided by resistance. concluded that the rate of fall must increase in proportion to the

weight of the object and decrease in proportion to the resisting
force of the medium. The actual rate of fall in any particular case
would be found by dividing the weight by the resistance.

Aristotle also discussed "violent" motion — that is. any motion
SG 2.3 of ^n object other than going freely toward its "natural place."

Such motion, he argued, must always be caused by a force, and the
speed of the motion will increase as the force increases. When the
force is removed, the motion must stop. This theory agrees with
our common experience, say in pushing a chair or a table across the
floor. It doesn't work quite so well for objects thrown through the
air, since such projectiles keep moving for a while even after we
have stopped exerting a force on them. To account for this kind of
motion, Aristotle proposed that the air itself somehow exerts a force
that keeps the object moving.

Later scientists proposed some modifications in Aristotle's

Section 2.1

theory of motion. For example, in the fifth century A.D. John
Philoponus of Alexandria argued that the speed of an object in
natural motion should be found by subtracting the resistance of
the medium from the weight of the object, rather than dividing by
the resistance. Philoponus claimed that his experimental work
supported his theory, though he did not report the details; he simply
said that he dropped two weights, one of which was twice as heavy
as the other, and observed that the heavy one did not reach the
ground in half the time taken by the light one.

There were still other difficulties with Aristotle's theory of
motion. However, the realization that his teachings concerning
motion had limitations did little to modify the importance given
to them in the universities of France and Italy during the fifteenth
and sixteenth centuries. Aristotle's theory of motion did, after all,
fit much of ordinary experience in a general -if qualitative- way.
Besides, the study of motion through space was of major interest to
only a few scholars, just as it had been only a very small part of
Aristotle's own work.

Two other influences stood in the way of radical changes in
the theory of motion. First, Aristotle believed that mathematics was
of little value in describing terrestrial phenomena. Second, he put
great emphasis upon direct, qualitative observation as the basis for
theorizing. Simple qualitative observation was very successful in
Aristotle's biological studies. But as it turned out, real progress in
physics began only when the value of mathematical prediction and
detailed measurement was recognized.

A number of scholars in the fifteenth and sixteenth centuries
had a part in this change to a new way of doing science. But of
all these, Galileo was by far the most eminent and successful. He
showed how to describe mathematically the motions of simple,
ordinary objects -falling stones and balls rolling on an incline. This
work not only paved the way for other men to describe and explain
the motions of everything from pebbles to planets, it also began an
intellectual revolution which led to what we now consider modem
science.

John Philoponus: rate of fall is
proportional to weight minus
resistance.

SG2.4

Qualitative refers to quality -
the sort of thing that happens.
Quantitative refers to quantity -
the measurement or prediction of
numerical values. This distinction
will appear often in the course.

Q1 Describe two ways in which, according to the Aristotelian
view, terrestrial and celestial bodies differ from each other.

Q2 Which of these statements would be accepted in the
fifteenth and sixteenth centuries by persons who believed in the
Aristotelian system of thought?

(a) Ideas of motion should fit in with poetry, politics, theology
and other aspects of human thought and activity.

(b) Heavy objects fall faster than light ones.

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

1500

Sections 2.2 and 2.3

2.2 Galileo and his times

Galileo Galilei was bom in Pisa in 1564 -the year of
Michelangelo's death and Shakespeare's birth. Galileo was the son
of a nobleman from Florence, and he acquired his father's active
interest in poetry, music, and the classics. His scientific inventive-
ness also began to show itself early. For example, as a young
medical student at the University of Pisa, he constructed a simple
pendulum-type timing device for the accurate measurement of
pulse rates.

Lured from medicine to physical science by reading Euclid and
Archimedes, Galileo quickly became known for his unusual ability
in science. At the age of 26, he was appointed Professor of
Mathematics at Pisa. There he showed an independence of spirit
unmellowed by tact or patience. Soon after his appointment, he
began to challenge the opinions of his older colleagues, many of
whom became his enemies. He left Pisa before his term was
completed, apparently forced out by financial difficulties and by his
enraged opponents. Later, at Padua in the Republic of Venice, he
began his work in astronomy. His support of the sun-centered theory
of the universe eventually brought him additional enemies, but it
also brought him immortal fame. We shall deal with that part of
his work in Unit 2.

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

SWITZEKLANB

AVSTRIfi,

hlUH<ffif(Y

Italy about 1600

2.3 Galileo's Two New Sciences

Galileo's early writings on mechanics (the study of the
behavior of matter under the influence of forces) were in the
tradition of the standard medieval theories of physics, although he
was aware of some of the shortcomings of those theories. During
his mature years his chief interest was in astronomy. However,
when his important astronomical book. Dialogue on the Two Great
World Systems (1632), was condemned by the Roman Catholic
Inquisition and he was forbidden to teach the "new" astronomy,
Galileo decided to concentrate again on mechanics. This work led to
his book Discourses and Mathematical Demonstrations Concerning
Two New Sciences Pertaining to Mechanics and Local Motion
(1638), usually referred to as Two New Sciences. This treatise
signaled the beginning of the end, not only of the medieval theory
of mechanics, but also of the entire Aristotehan cosmology which it
supported.

Galileo was old, sick, and nearly blind at the time he wrote
Two New Sciences. Yet, as in all his writings, his style is spritely

Title page of Dialogue on Two Great
World Systems (1632).

Free Fall -Galileo Describes Motion

D I S C O R S I

DIMOSTRAZIONI

MATEMATI CHE,

intorno a due nuoue fcicnzf

Ancncnci alia

MeCANICA & i MOVIMENTI LoCALI>

itlSigntr

GALILEO GALILEI LINCEO,

Filofofo e Matemacico primario del Screnilllmo

Granjd Duca di Tofcana.

Cm vnt Afftniice iclctntrt digrtuiti itUnni Stliii.

W^^Sl

IN L E I D A,
Apprcdb gli Eifcvicii. m. d. c. xxxviii.

Title page of Discourses and Mathe-
matical Demonstrations Concerning
Two New Sciences Pertaining to Me-
chanics and Local Motion (1638).

SG 2.5

DEL Galileo. tfj

ftcM Htafiftrtbhc il moto , Upcfizica del rtcu> dj/ituttmrnte
frtfi, entn in relticione tl mote , nan vien dtUrultt , mi per dire
qutlche ftt tu»entMr*folr<bher ri^ondcre (\itegii anlichi , tceti
mtglitfifiorga,autnlo (tndudi U dimfffrtzionc d' AriihieU.mi
fir chejifiirebht tndir ctniro i gli afunli di ijuttlo , nrgtridogli
tmindiu. E ijHtHit itprimi^io frtndtmenle dahiio, (he Arilfo-
tele Ktitjferimentiiffe mti ifatntefu vero , the due fieire -vnt fii
grille di/fulirt died velie Ufiitte net medefma inittnte c*der
dtvH tUet.t,t , V. gr. di eenio iraccid fujfer ttlmente differeali
we Be lor veliciii , the tit arriuc dellt mtggior iit tern [tUrtfitre-
Mtffe Hen htuere ni tneefeji died true ci*^

Simp. Si vede ftre dtUe fue f*rele,tlfei meHrt ^hmerlt fbe-
rimenino, perche ei dice: yeggiamt ilfiHgrtne : her quel' vederfi
tccennt I'hiuernefattt tefferienT^.

Sajir. MiieS.Sim». chen'hhfttteUpreiiiMtlpcurcehevH*
fttldttdrliglieri* yche pefi tenia, dugente,eineepilitihl)re,n<in
enliciperidi vnpilmo feUmenie ttrnm in terrt dellipelUdua
mefthetto , the nepefi -unt mez,i,t , venende tnta delt tltez,t.t di
dugento hretdi.

Salu. Mifent,' tltre e^erienT^ ten breue , e ccndudente dime-
Hriz,iene poffttme thitrtmente preutrentn ejfervere^thevn me-
tile piu greueji mucin piii veletemenie d'un'iltre men grtue , in -
tendende di mobiti dell" ittejft meierid ) cf injimmd MqteSi de i
qudlipdrld AriU elite. Peri dilemi S. Simp.fi vei dmmettete,the
di ddfthedtino terpo grdue tddintefid vnd dd netitrd determindtd
velecitk \(ithe I'dttreJierglieU ,ediminuirglieldnenfipejfdfe Hen
ten ffirgli vielent,d , i tpporgU qudlche impedimente.

Simp. Nen/ipuidiibitdre,thel'iilejfemebite neWifteJfemez-
zt hdbbid vnd fldtuild , e dd ndturd determinttd veledti. Id qud-
U nen ft gli pejfd dttrefeere fe nen (en nueue impete tenferite-
gU , i diminuiiglieUfilM the ten qudlche impedimente the le ri-
tardi.

Salu. SsfiteU eUmqne lui htHejJimt Jin mtbili, U ntturdli

vele-

A page from the original Italian edition
of Two New Sciences, showing state-
ments that are translated in this text.

and delightful. He used the dialogue form to allow a lively
conversation among three 'speakers": Simplicio, who competently
represents the Aristotelian view; Salviati, who presents the new
views of Galileo; and Sagredo, the uncommitted man of good will
and open mind, eager to learn. Eventually, of course, Salviati leads
his companions to Galileo's views. Let us listen to Galileo's three
speakers as they discuss the problem of free fall:

Salviati: I greatly doubt that Aristotle ever tested by
experiment whether it is true that two stones, one
weighing ten times as much as the other, if allowed to
fall at the same instant from a height of, say, 100 cubits,
would so differ in speed that when the heavier had
reached the ground, the other would not have fallen
more than 10 cubits. [A "cubit" is equivalent to about 20
inches.]

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

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

Here, perhaps, one might have expected to find a detailed report
on an experiment done by Galileo or one of his colleagues. Instead,
Galileo uses a "thought experiment" -an analysis of what would
happen in an imaginary experiment -to cast grave doubt on
Aristotle's theory of motion:

Salviati: But, even without further experiment, it is
possible to prove clearly, by means of a short and
conclusive argument, that a heavier body does not move
more rapidly than a lighter one provided both bodies are
of the same material and in short such as those mentioned
by Aristotle. But tell me, Simplicio, whether you admit '
that each falling body acquires a definite speed fixed by
nature, a velocity which cannot be increased or
diminished except by the use of violence or resistance?

Simplicio: There can be no doubt but that one and the
same body moving in a single medium has a fixed velocity
which is determined by nature and which cannot be
increased except by the addition of impetus or diminished
except by some resistance which retards it.

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

Section 2.3

Simplicio: You are unquestionably right.

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

Simplicio: I am all at sea.
beyond my comprehension.

. This is, indeed, quite

SG 2.6

Simplicio retreats in confusion when Salviati shows that the
Aristotelian theory of fall is self-contradictory. But while Simplicio
cannot refute Galileo's logic, his own eyes tell him that a heavy
object does fall faster than a light object:

Simplicio: Your discussion is really admirable; yet I do
not find it easy to believe that a birdshot falls as swiftly as
a cannon ball.

Salviati: Why not say a grain of sand as rapidly as a
grindstone? But, Simplicio, I trust you will not follow the
example of many others who divert the discussion from
its main intent and fasten upon some statement of mine
that lacks a hairsbreadth of the truth, and under this hair
hide the fault of another that is as big as a ship's cable.
Aristotle says that "an iron ball of one hundred pounds
falling from a height of 100 cubits reaches the ground
before a one-pound ball has fallen a single cubit." I say
that they arrive at the same time. You find, on making
the experiment, that the larger outstrips the smaller by
two fingerbreadths. . . . Now you would not hide behind
these two fingers the 99 cubits of Aristotle, nor would you
mention my small error and at the same time pass over
in silence his very large one.

This is a clear statement of an important principle: even in
careful observation of a common natural event, the observer's
attention may be distracted by what is really a minor effect, with the
result that he fails to see a much more significant regularity.
Different bodies falling in air from the same height, it is true, do
not reach the ground at exactly the same time. However, the
important point is not that the times of arrival are slightly different,
but that they are very nearly the same\ Galileo regarded the
failure of the bodies to arrive at exactly the same time as a minor
effect which could be explained by a deeper understanding of motion
in free fall. Galileo himself correctly attributed the observed results
to differences in the effect of the resistance of the air on bodies of

A stroboscopic photograph of two
freely falling balls of unequal weight.
The balls were released simultane-
ously. The time interval between
images is 1/30 sec.

Free Fall — Galileo Describes Motion

The phrase "free fall" as now used
in physics generally refers to fall
when the only force acting is gravity;
that is, when air friction is
negligible.

different size and weight. A few years after Galileo's death, the
invention of the vacuum pump allowed others to show that Galileo
was right. Once the effect of air resistance was eliminated — for
example, when a feather and a heavy gold coin were dropped from
the same height at the same time inside an evacuated container—
the different bodies fell at the same rate and struck the bottom of
the container at the same instant. Long after Galileo, it became
possible to formulate the laws of air resistance, so one could
understand exactly why and by how much a light object falls
behind a heavier one.

Learning what to ignore has been almost as important in the
growth of science as learning what to take into account. In the case
of falling bodies, Galileo's explanation depended on his being able
to imagine how an object would fall if there were no air resistance.
This may be easy for us who know of vacuum pumps, but in
Galileo's time it was an explanation that was difficult to accept. For
most people, as for Aristotle, common sense said that air resistance
is always present in nature. Thus a feather and a coin could never
fall at the same rate. Why should one talk about hypothetical
motions in a vacuum, when a vacuum could not be shown to exist?
Physics, said Aristotle and his followers, should deal with the world
all around us that we can readily observe, not with some imaginary
world which might never be found.

Aristotle's physics had dominated Europe since the thirteenth
century, mainly because many intelligent scientists were convinced
that it offered the most rational method for describing natural
phenomena. To overthrow such a firmly established doctrine
required much more than writing reasonable arguments, or simply
dropping heavy and light objects from a tall building, as Galileo is
often said to have done (but probably did not) at the Leaning Tower
of Pisa. It demanded Galileo's unusual combination of mathematical
talent, experimental skill, literary style, and tireless campaigning
to discredit Aristotle's theories and to begin the era of modem
physics.

A chief reason for Galileo's success was that he exposed the
Aristotelian theory at its weakest point: he showed that physics can
deal better with the world around us if we realize that the world of
common observation is not the simple starting point the
Aristotelians thought it to be. On the contrary, the world as we
ordinarily observe it is usually quite complex. For example, in
observing the fall of bodies you see the effects of both the law of
fall and the law of resistance on objects moving through air. To
understand what you see, you should start from a simple case (such
as fall without resistance), even if this has to be "seen" only in your
mind or by a mathematical model. Or you may turn to an
experiment in the laboratory, where the usual conditions of
observation can be changed. Only after you understand each of the
different effects by itself should you go back to face the complexities
of the ordinary case.

Sections 2.4 and 2.5

Q3 If a nail and a toothpick are simultaneously dropped from
the same height, they do not reach the ground at exactly the same
instant. (Try it with these or similar objects.) How would Aristotelian
theory explain this? What was Galileo's explanation?

2.4 Why study the motion of freely falling bodies?

In Galileo's attack on the Aristotelian cosmology, few details
were actually new. However, his approach and his findings together
provided the first coherent presentation of the science of motion.
Galileo realized that, out of all the observable motions in nature,
free-fall motion is the key to the understanding of all motions of all
bodies. To decide which is the key phenomenon to study is the real
gift of genius. But Galileo is also in many ways typical of scientists
in general. His approach to the problem of motion makes a good
"case" to be used in the following sections as an opportunity to
discuss strategies of inquiry that are still used in science.

These are some of the reasons why we study in detail Galileo's
attack on the problem of free fall. Galileo himself recognized
another reason — that the study of motion which he proposed was
only the starting phase of a mighty field of discovery:

My purpose is to set forth a very new science dealing
with a very ancient subject. There is, in nature, perhaps
nothing older than motion, concerning which the books
written by philosophers are neither few nor small;
nevertheless, I have discovered some properties of it that
are worth knowing that have not hitherto been either
observed or demonstrated. Some superficial observations
have been made, as for instance, that the natural motion
of a heavy falling body is continuously accelerated; but to
just what extent this acceleration occurs has not yet
been announced. . . .

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

2.5 Galileo chooses a definition of uniform acceleration

Two New Sciences deals directly with the motion of freely
falling bodies. In studying the following paragraphs from it, we
must be alert to Galileo's overall plan. First, he discusses the
mathematics of a possible, simple type of motion (which we now
call uniform acceleration or constant acceleration). Then he
proposes that heavy bodies actually fall in just that way. N'ext, on
the basis of this proposal, he derives a prediction about balls rolling
down an incline. Finally, he shows that experiments bear out these
predictions.

By Aristotelian cosmology is meant
the whole interlocking set of ideas
about the structure of the physical
universe and the behavior of all the
objects in it. This was briefly
mentioned in Sec. 2.1. Other aspects
of it will be presented in Unit 2.

In fact, more than mere "superficial
observations" had been made long
before Galileo set to work. For
example, Nicolas Oresme and others
at the University of Paris had by
1330 discovered the same distance-
time relationship for falling bodies
that Galileo was to announce in
the Two New Sciences. Some of
their reasoning is discussed in
SG 2.7.

It will help you to have a plan clearly
in mind as you progress through the
rest of this chapter. As you study
each succeeding section, ask
yourself whether Galileo is

— presenting a definition

— stating an assumption (or
hypothesis)

-deducing predictions from his

hypothesis
-experimentally testing the

predictions

Free Fall — Galileo Describes Motion

This is sometimes l<nown as the
Rule of Parsimony: unless forced to
do otherwise, assume the simplest
possible hypothesis to explain
natural events.

Rephrasing Galileo and using our
symbols: for uniform speed v, the
ratio Ad/Af is constant. Similarly,
recall that for accelerated motion,
as we saw in Chapter 1, we defined
uniform acceleration as

a = -rr = constant
Af

Other ways of expressing this
relationship are discussed in SG 2.8
and 2.9.

The first part of Galileo's presentation is a thorough discussion
of motion with uniform speed, similar to our discussion in Chapter 1.
That leads to the second part, where we find Salviati saying:

We pass now to . . . naturally accelerated motion, such
as that generally experienced by heavy falling bodies.

... in the investigation of naturally accelerated
motion we were led, by hand as it were, in following the
habit and custom of nature herself, in all her various
other processes, to employ only those means which are
most common, simple and easy . . .

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

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

A motion is said to be uniformly accelerated when,
starting from rest, it acquires during equal time-
intervals, equal increments of speed.

Sagredo: Although I can offer no rational objection to
this or indeed to any other definition devised by any
author whosoever, since all definitions are arbitrary, I
may nevertheless without defense be allowed to doubt
whether such a definition as the foregoing, established in
an abstract manner, corresponds to and describes that
kind of accelerated motion which we meet in nature in
the case of freely falling bodies ....

Here Sagredo questions whether Galileo's arbitrary definition of
acceleration actually corresponds to the way real objects fall. Is
acceleration, as defined, really useful in describing their observed
change of motion? Sagredo wonders about a further point, so far
not raised by Galileo:

From these considerations perhaps we can obtain an
answer to a question that has been argued by philosophers,
namely, what is the cause of the acceleration of the
natural motion of heavy bodies ....

But Salviati. the spokesman of Galileo, rejects the ancient
tendency to investigate phenomena by looking first for their causes.
It is premature, he declares, to ask about the cause of any motion
until an accurate description of it exists:

Section 2.6

Salviati: The present does not seem to be the proper
time to investigate the cause of the acceleration of
natural motion concerning which various opinions have
been expressed by philosophers, some explaining it by
attraction to the center, others by repulsion between the
very small parts of the body, while still others attribute it
to a certain stress in the surrounding medium which
closes in behind the falling body and drives it from one
of its positions to another. Now, all these fantasies, and
others, too, ought to be examined; but it is not really worth
while. At present it is the purpose of our Author merely
to investigate and to demonstrate some of the properties
of accelerated motion, whatever the cause of this
acceleration may be.

Galileo has now introduced two distinct propositions: 1)
"uniform" acceleration means equal speed increments Ai; in equal
time intervals At; and 2) things actually fall that way. Let us first
look more closely at Galileo's proposed definition.

Is this the only possible way of defining uniform acceleration?
Not at all! Galileo says that at one time he thought a more useful
definition would be to use the term uniform acceleration for motion
in which speed increased in proportion to the distance traveled. Ad,
rather than to the time At. Notice that both definitions met Galileo's
requirement of simplicity. (In fact, both definitions had been
discussed since early in the fourteenth century.) Furthermore, both
definitions seem to match our common sense idea of acceleration
about equally well. When we say that a body is "accelerating," we
seem to imply "the farther it goes, the faster it goes," and also "the
longer time it goes, the faster it goes." How should we choose
between these two ways of putting it? Which definition will be more
useful in the description of nature?

This is where experimentation becomes important. Galileo chose
to define uniform acceleration as the motion in which the change
of speed Av is proportional to elapsed time At, and then demonstrate
that this matches the behavior of real moving bodies, in laboratory
situations as well as in ordinary, "un-arranged," experience. As you
will see later, he made the right choice. But he was not able to
prove his case by direct or obvious means, as you shall also see.

Q4 Describe uniform speed without referring to dry ice pucks
and strobe photography or to any particular object or technique of
measurement.

Q5 Express Galileo's definition of uniformly accelerated
motion in words and in the form of an equation.

Q6 What two conditions did Galileo want his definition of
uniform acceleration to meet?

Here Salviati refers to the
Aristotelian assumption that air
propels an object moving through it
(see Sec. 2.1).

2.6 Galileo cannot test his hypothesis directly

After Galileo defined uniform acceleration so that it would
match the way he believed freely falling objects behaved, his next

50 Free Fall -Galileo Describes Motion

task was to devise a way of showing that the definition for uniform
acceleration was useful for describing observed motions.

Suppose we drop a heavy object from several different heights —
say, from windows on different floors of a building. We want to
check whether the final speed increases in proportion to the time it
The symbol a: means "directly takes to fall — that is, whether At; cc At, or what amounts to the same

proportional to." thing, whether Az;/At is constant. In each trial we must observe the

time of fall and the speed just before the object strikes the ground.
But there's the rub. Practically, even today, it would be very
SG 2.10 difficult to make a direct measurement of the speed reached by an
object just before striking the ground. Furthermore, the entire
time intervals of fall (less than 3 seconds even from the top of a
10-story building) are shorter than Galileo could have measured
accurately with the clocks available to him. So a direct test of
whether Aiy/At is constant was not possible for Galileo.

Q7 Which of these are valid reasons why Galileo could not test
directly whether the final speed reached by a freely falling object is
proportional to the time of fall?

(a) His definition was wrong.

(b) He could not measure the speed attained by an object just
before it hit the ground.

(d) He could not measure ordinary distances accurately enough.

(e) Experimentation was not permitted in Italy.

2.7 Looking for logical consequences of Galileo's hypothesis

Galileo's inability to make direct measurements to test his
hypothesis -that Av/At is constant in free fall -did not stop him. He
turned to mathematics to derive from this hypothesis some other
relationship that could be checked by measurement with
equipment available to him. We shall see that in a few steps he
came much closer to a relationship he could use to check his
hypothesis.

Large distances of fall and large time intervals for fall are, of
course, easier to measure than the small values of Ad and At that
would be necessary to find the final speed just before the falling
body hits. So Galileo tried to find, by reasoning, how total fall
distance ought to increase with total fall time if objects did fall with
uniform acceleration. You already know how to find total distance
from total time for motion at constant speed. Now we will derive a
new equation that relates total fall distance to total time of fall for
motion at constant acceleration. In this we shall not be following
Galileo's own derivation exactly, but the results will be the same.
First, we recall the definition of average speed as the distance
traversed Ad divided by the elapsed time At :

Section 2.7 51

This is a general definition and can be used to compute the average
speed from measurement of Ad and At, no matter whether Ad and
At are small or large. We can rewrite the equation as

Ad = Vav X At

This equation, still being really a definition of !;„,., is always true.
For the special case of motion at a constant speed v, then Vav = v
and therefore, Ad = i; x At. When the value of v is known (as, for
example, when a car is driven with a steady reading of 60 mph on
the speedometer), this equation can be used to figure out how far
(Ad) the car would go in any given time interval (At). But in
uniformly accelerated motion the speed is continually changing — so
what value can we use for Vav"^

The answer involves just a bit of algebra and some plausible
assumptions. Galileo reasoned (as others had before) that for any
quantity that changes uniformly, the average value is just halfway
between the beginning value and the final value. For uniformly
accelerated motion starting from rest (where t^initiai ^ 0 and ending

at a speed Vanau this rule tells us that the average speed is halfway More generally the average speed

between 0 and rnnai - that is, t;„,. = y t'nnai. If this reasoning is would be

correct, it follows that w ^ yjninai + Vfinai

" av 2

Ad = Jl^final X ^t

for uniformly accelerated motion starting from rest. SG 2.11 and 2.12

This relation could not be directly tested either, because the last
equation still contains a speed factor. What we are trying to arrive
at is an equation relating total distance and total time, without any
need to measure speed.

Now we look at Galileo's definition of uniform acceleration:
a = Az;/At. We can rewrite this relationship in the form Av= aX At.
The value of Ai; is just L'finai - ^^initiai; and i^initiai = 0 for motion that
begins from rest. Therefore we can write

Az;= a X At

l^final ~ ^initial — « X At
X^final ^ a X At

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

Ad = iVfinal X ^t

= ^(a X At) X At

Or. regrouping terms,

Ad = ja(My

This is the kind of relation Galileo was seeking -it relates total
distance Ad to total time At, without involving any speed term.

Before finishing, though, we will simplify the symbols in the
equation to make it easier to use. If we measure distance and time
from the position and the instant that the motion starts (dmitiai ^ 0

Free Fall — Galileo Describes Motion

SG 2.13 and 2.14

SG 2.15

Because we will use the expression
t'fitu./f"fin;,i many times, it is simpler
to write it as d/r--it is understood
that d and f mean total distance and
time interval of motion, starting
from rest.

and tinitiai = 0), then the intervals Ad and At have the values given
by dfinai and tfinai- The equation above can therefore be written more
simply as

"final ~ 2"^^ final

Remember that this is a very specialized equation — it gives the
total distance fallen as a function of total time of fall but only if the
motion starts from rest (t'lniuai ^ 0), if the acceleration is uniform
(a = constant), and if time and distance are measured from the
start (tinitiai = 0 and di„i,iai = 0).

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

"final °~ t final

For example, if a uniformly accelerating car starting from rest
moves 10 m in the first second, in twice the time it would move
four times as far, or 40 m in the first two seconds. In the first 3
seconds it would move 9 times as far— or 90 m.

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

^^= constant

t final

Thus a logical result of Galileo's original proposal for defining
uniform acceleration can be expressed as follows: if an object
accelerates uniformly from rest, the ratio dlt' should be constant.
Conversely, any motion for which this ratio of d and t'^ is found to
be constant for different distances and their corresponding times,
we may well suppose to be a case of motion with uniform,
acceleration as defined by Galileo.

Of course, we still must test the hypothesis that freely falling
bodies actually do exhibit just such motion. Recall that earlier we
confessed we were unable to test directly whether Av/At has a
constant value. Galileo showed that a logical consequence of a
constant value of Av/At would be a constant ratio of dfi„a\ to t-fmai-
The values for total time and distance of fall would be easier to
measure than the values of short intervals Ad and At needed to find
All. However, measuring the time of fall still remained a difficult
task in Galileo's time. So, instead of a direct test of his hypothesis,
Galileo went one step further and deduced an ingenious, indirect
test.

Q8 Why was the equation d = ^at- more promising for Galileo
than a = Ai^/At in testing his hypothesis?

Q9 If you simply combined the two equations Ad = i;At and
Ai; = aAt it looks as if one might get the result Ad = aAt-. What is
wrong with doing this?

Section 2.8

2.8 Galileo turns to an Indirect test

Realizing that a direct quantitative test with a rapidly and
freely falling body would not be accurate, Galileo proposed to make
the test on an object that was moving less rapidly. He proposed a
new hypothesis: if a freely falling body has an acceleration that is
constant, then a perfectly round ball rolling down a perfectly
smooth inclined plane will also have a constant, though smaller,
acceleration. Thus Galileo claimed that if dit- is constant for a body
falling freely from rest, this ratio will also be constant, although
smaller, for a ball released from rest and rolling different distances
down a straight inclined plane.

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

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

Note the careful description of the
experimental apparatus. Today an
experimenter would add to his
verbal description any detailed
drawings, schematic layouts or
photographs needed to make it
possible for other competent
scientists to duplicate the
experiment.

This picture painted in 1841 by G.
Bezzuoii, attempts to reconstruct an
experiment Galileo is alleged to have
made during his time as lecturer at
Pisa. Off to the left and right are men
of ill will: the blase Prince Giovanni
de Medici (Galileo had shown a dredg-
ing-machine invented by the prince to
be unusable) and Galileo's scientific
opponents. These were leading men
of the universities; they are shown
here bending over a book of Aristotle,
where it is written in black and white
that bodies of unequal weight fall
with different speeds. Galileo, the
tallest figure left of center in the
picture, is surrounded by a group of
students and followers.

^«;

Free Fall-Galileo Describes Motion

For each angle, the acceleration is
found to be a constant.

Galileo's technique for measuring
time is discussed in the next section.

SG 2.16

Spheres rolling down planes of in-
creasingly steep inclination. At 90° the
inclined plane situation matches free
fall. (Actually, the ball will start slip-
ping instead of rolling long before the
angle has become that large.)

Galileo has packed a great deal of information into these lines.
He describes his procedures and apparatus clearly enough to allow
other investigators to repeat the experiment for themselves if they
wished. Also, he gives an indication that consistent measurements
can be made, and he restates the two chief experimental results
which he believes support his free-fall hypothesis. Let us examine
the results carefully.

(a) First, he found that when a ball rolled down an incline at a
fixed angle to the horizontal, the ratio of the distance covered to the
square of the corresponding time was always the same. For
example, if d^, d^, and d^ represent distances measured from the
same starting point on the inclined plane, and t,, tj, and tg the
corresponding times taken to roll down these distances, then

In general, for each angle of incline, the value of dlt^ was
constant. Galileo did not present his experimental data in the full
detail which has become the custom since. However, his experiment
has been repeated by others, and they have obtained results which
parallel his (see data in SG 2.16). This is an experiment which you
can perform yourself with the help of one or two other students.
(The photographs on the next page show students in the Project
Physics course doing this experiment and also show some of their
results.)

(b) Galileo's second experimental finding relates to what happens
when the angle of inclination of the plane is changed. He found
that whenever the angle changed, the ratio dit- took on a new value,
although for any one angle it remained constant regardless of
distance of roll. GalUeo confirmed this by repeating the experiment
"a full hundred times" for each of many different angles. After
finding that the ratio d/t- was constant for each angle of inclination
for which measurements of t could be carried out conveniently,
Galileo was willing to extrapolate. He concluded that the ratio dlt^
is a constant even for larger angles, where the motion of the ball is
too fast for accurate measurements of t to be made. Finally, Galileo
reasoned that in the particular case when the angle of inclination
became 90°, the ball would move straight down — and so becomes
the case of a falling object. By his reasoning, d/t- would still be
some constant in that extreme case (even though he couldn't say
what the numerical value was.)

Because Galileo had deduced that a constant value of dIt- was
characteristic of uniform acceleration, he could conclude at last
that free fall was uniformly accelerated motion.

Q10 In testing his hypothesis that free fall motion is uniformly
accelerated, Galileo made the unproved assumption that (check one
or more):

(a) dlt^ is constant.

Free Fall — Galileo Describes Motion

For problems that will check and
extend your understanding of
uniform acceleration. See SG 2.17
through 2.24.

(b) the acceleration has the same value for all angles of
inclination of the plane.

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

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

Q11 Which of the following statements best summarizes the
work of Galileo on free fall when air friction is negligible? (Be
prepared to defend your choice.) Galileo:

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

(b) proved that for any freely falling object the ratio dlt^ is
constant for any distance of fall.

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

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

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

2.9 Doubts about Galileo's procedure

This whole process of reasoning and experimentation looks long
and involved on first reading, and some doubts may well arise
concerning it. For example, was Galileo's measurement of time
precise enough to establish the constancy of dlt^ even for the case
of a slowly rolling object? In his book, Galileo tries to reassure
possible critics by providing a detailed description of his
experimental arrangement (thereby inviting any skeptics to try it
for themselves):

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

The water clock described by Galileo was not invented by him.
Indeed, there are references to water clocks in China as early as the

Section 2.9

sixth century B.C., and they were probably used in Babylonia and
India even earlier. In the early 16th century a good water clock was
the most accurate of the world's instruments for measuring short
time intervals. It remained so until shortly after Galileo's death,
when the work of Christian Huygens and others led to practical
pendulum clocks. When better clocks became available, Galileo's
results on inclined-plane motion were confirmed.

Another reason for questioning Galileo's results is related to the
great difference between free fall and rolling motion on a slight
incline. Galileo does not report what angles he used in his
experiment. However, as you may have found out from doing a
similar experiment, the angles must be kept rather small. As the
angle increases, the speed of the ball soon becomes so great that it
is difficult to measure the times involved. The largest usable angle
reported in a recent repetition of Galileo's experiment was only
6°. (See SG 2.15) It is not hkely that Galileo worked with much
larger angles. This means that the extrapolation to free fall (90°
incline) is a large one, perhaps much too large for a cautious
person — or for one not already convinced of Galileo's argument.

Still another reason for questioning Galileo's results is the
observation that, as the angle of incline is increased, there comes
a point where the ball starts to slide as well as roll. This change in
behavior could mean that the motion is very different at large
angles. Galileo does not discuss these cases. It is surprising that he
apparently did not repeat the experiment with blocks which would
slide, rather than roll, down a smooth incline. If he had, he would
have found that for accelerated sliding motion the ratio dit- is also
a constant, although the constant has a different numerical value
than for rolling at the same angle.

01 2 Which of the following statements could be regarded as
major reasons for doubting the validity of Galileo's procedure?

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

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

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

(e) dlt^ was not constant for a sliding object.

Early water clock

SG 2.25

2.10 Consequences of Galileo's work on motion

Galileo seems to have been well aware that one cannot get the
correct numerical value for the acceleration of a body in free fall
simply by extrapolating the results to increasingly large angles of
inclination. He did not attempt to calculate a numerical value for
the acceleration of freely falling bodies. But for his purposes it was
enough that he could support the hypothesis that the acceleration is
constant for any given body, whether rolling or falling. This is the

Free Fall — Galileo Describes Motion

We now know by measurement that
the magnitude of the acceleration of
gravity, symbol a^, is about 9.8
m/sec per sec, or 32 ft/sec per sec,
at the earth's surface. The Project
Physics Handbook contains five
different experiments for finding a
value of a^. (For many problems,
the approximate value 10 m/sec/sec
is satisfactory.)

SG 2.26

You can derive this equation. (See
SG 2.27)

SG 2.28 and 2.29

first consequence of Galileo's work, one that has been fully borne
out by all subsequent tests.

Second, if spheres of different weights are allowed to roll down
an inclined plane set at a given angle, they turn out to have the
same acceleration. We do not know how much experimental
evidence Galileo himself had for this conclusion, but it is consistent
with the observations for freely falling objects. It is consistent also
with his "thought experiment" by which he argued that bodies of
different weights fall at the same rate (aside from the comparatively
small effects of air resistance). His results provided a decisive
refutation of Aristotle's theory of motion.

Third, Galileo developed a mathematical theory of accelerated
motion from which other predictions about motion could be
derived. We will mention just one example here, which will turn
out to be very useful in Unit 3. Recall that Galileo chose to define
acceleration as the rate at which the speed changes with time. He
then found by experiment that falling bodies actually do experience
equal changes of speed in equal times, and not in equal distances
as some had supposed. Still, the idea of something changing by
equal amounts in equal distances has an appealing simplicity, too.
One might ask if there isn't something that does change in that way
during uniform acceleration. In fact, there is. It follows without
any new assumptions that, during uniform acceleration from rest,
the square of the speed changes by equal amounts in equal
distances. There is a mathematical equation which expresses this
result: If z/inuiai = 0, and a = constant, then

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

These consequences of Galileo's work, important as they are to
the development of physics, would scarcely have been enough to
bring about a revolution in science by themselves. No sensible
scholar in the seventeenth century would have given up his belief
in the Aristotelian cosmology only because some of its predictions
had been refuted in the case of falling (or rolling) bodies. But
Galileo's work on free-fall motion helped to prepare the way for the
development of a new kind of physics, and indeed a new cosmology,
by planting the seeds of doubt about the crucial assumptions of
Aristotelian science. For example, when it was recognized that all
bodies fall with equal acceleration if air friction is negligibly small,
then the whole Aristotelian explanation of falling motion (Section
2.1) broke down.

The most agitating scientific problem during Galileo's lifetime
was not in mechanics but in astronomy. A central question in
cosmology was whether the earth or the sun is the center of the
universe. Galileo supported the view that the earth and other
planets revolve around the sun, a view entirely contrary to

Section 2.10 59

Aristotelian cosmology. But to support such a view required a
physical theory of why and how the earth itself moved. Galileo's
work on free fall and other motions turned out to be just what was
needed to begin to construct such a theory. His work did not have its
full effect, however, until it had been combined with the
investigations of forces and motion by the English scientist Isaac
Newton. But as Newton acknowledged, Galileo was the pioneering
pathfinder. (In the next chapter we will consider Newton's work on
force and motion. In Chapter 8, after studying about motion in the
heavens, we will return to Newton's laws and the revolution they
began in science.)

Galileo's work on motion introduced a new and significant
method of doing scientific research, a method as applicable today
as when GalHeo demonstrated it. The basis of this procedure is a
cycle, repeated as often as necessary, entirely or in part, until a
satisfactory theory has emerged: general observation -^ hypothesis
-* mathematical analysis or deduction from hypothesis ->■
experimental test of deduction -* modification of hypothesis in light
of test, and so forth.

While the steps in the mathematics are often determined
mainly by "cold logic," this is not so for the other parts of the SG 2.30

process. A variety of paths of thought can lead to the hypothesis in
the first place. A new hypothesis can come from an inspired hunch
based on general knowledge of the experimental facts, or from a
desire for mathematically simple statements, or from modifying a
previous hypothesis that failed. Moreover, there are no general
rules about exactly how well the experimental data must agree
with the theoretical predictions. In some areas of science, a theory
is expected to be accurate to better than one 1/ 1000th of one
percent; in other areas, or at an early stage of any new work, one
might be delighted to find a theory from which he could make
predictions with an error of only 50 percent. Finally note that while
experiment has an important place in this process, it is not at all
the only or even the main element. On the contrary, experiments
are worthwhile only in conjunction with the other steps in the
process.

The general cycle of observation, hypothesis, deduction, test,
modification, etc., so skillfully demonstrated by Galileo in the
seventeenth century, commonly appears in the work of scientists
today. Though there is no such thing as the scientific method, some
form of this cycle is almost always present in scientific research. It
is used not out of respect for Galileo as a towering figure in the
history of science, but because it works so well so much of the
time.

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

Salviati: ... we may say the door is now opened, for the

60 Free Fall -Galileo Describes Motion

first time, to a new method fraught with numerous and
wonderful results which in future years will command
the attention of other minds.

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

During this long and laborious day, I have
enjoyed these simple theorems more than their proofs,
many of which, for their complete comprehension, would
require more than an hour each; this study, if you will
be good enough to leave the book in my hands, is one
which I mean to take up at my leisure after we have
read the remaining portion which deals with the motion
of projectiles; and this if agreeable to you we shall take
up tomorrow.

Salviati: I shall not fail to be with you.

Many details of physics, mathe- Q13 which one of the following was not a result of Galileo's

matics and history have appeared , ^. o

• ».-• u * .- ■ s .^ work on motion .''

in this chapter. For a review of the

most important ideas, see SG 2.31, ^^^ The correct numerical value of the acceleration in free fall

2.32, and 2.33. was obtained by extrapolating the results for larger and

larger angles of inclination.

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

(c) Bodies rolling on a smooth inclined plane are uniformly
accelerated (according to Galileo's definition of acceleration).

STUDY GUIDE 2

2.1 Note that at the beginning of each chapter in
this book there is a Hst of the section titles. This
is a sort of road map you can refer to from time to
time as you study the chapter. It is important,
expecially in a chapter such as this one, to know
how the part you are studying relates to what
preceded it and to have some idea of where it is
leading. For this same reason, you will find it very
helpful at first to skim through the entire chapter,
reading it rapidly and not stopping to puzzle out
parts that you do not quickly understand. Then
you should return to the beginning of the chapter
and work your way through it carefully, section
by section. Remember also to use the end-of-section
questions to check your progress.
The Project Physics learning materials particularly
appropriate for Chapter 2 include:

Experiments

A Seventeenth-Century Experiment

Twentieth Century Version of Galileo's

Experiment

Measuring the Acceleration Due to

Gravity, Og

Activities
When is Air Resistance Important?
Measuring Your Reaction Time
Falling Weights
Extrapolation

Reader Article
On the Scientific Method

Film Loops

Acceleration Due to Gravity — Method I
Acceleration Due to Gravity — Method II

Transparency

Derivation of d

Vit + jat^

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

2.3 Drop sheets of paper with various degrees of
"crumpling." Try to crumple a sheet of paper
tight enough that it will fall at the same rate as a
tennis ball. Can you explain the results with
Aristotle's theory?

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

2.5 Consider Aristotle's statement "A given
weight moves [falls] a given distance in a given
time; a weight which is as great and more moves
the same distances in less time, the times being
in inverse proportion to the weights. For instance.

if one weight is twice another, it will take half as
long over a given movement." (De Caelo)

Indicate what Simplicio and Salviati each
would predict for the falling motion in these
cases:

(a) A 2-pound rock falls from a cliff and,
whUe dropping, breaks into two equal
pieces.

(b) A hundred-pound rock is dropped at the
same time as one hundred 1-pound
pieces of the same type of rock.

(c) A hundred 1-pound pieces of rock, falling
from a height, drop into a draw-string
sack which closes, pulls loose and falls.

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

2.7 A good deal of work preceded that of Galileo
on the topic of motion. In the period 1280-1340,
mathematicians at Merton College, Oxford,
carefully considered different quantities that
change with the passage of time. One result that
had profound influence was a general theorem
known as the "Merton Theorem" or "Mean Speed
Rule."

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

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

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

STUDY GUIDE 2

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

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

(a) Av is proportional to At

(b) AvIAt = constant

(d) V is proportional to t

2.9 In the Two New Sciences Galileo states, ". . .
for so far as I know, no one has yet pointed out
that the distances traversed, during equal intervals
of time, by a body falling from rest, stand to one
another in the same ratio as the odd numbers
beginning with unity (namely 1:3:5:7 ...)...."

The area beneath the curve in a speed-time
graph represents the distance traveled during
some time interval. Using that idea, give a proof
that the distances an object falls in successive
equal time intervals will be in the ratios of the odd
numbers.

2.10 Using whatever modem equipment you
wish, describe how you could find an accurate
value for the speed of a falling object just before
striking the ground.

2.11 Show that the expression

,, _ ^Initial + ^final
Vav 2

is equivalent to the "Merton Rule" discussed in
SG 2.7.

2.12 For any quantity that changes uniformly,
the average is the sum of the initial and final
values divided by two. Try it out for any quantity
you may choose -for example: what is the
average age in a group of five people having
individually the ages of 15, 16, 17, 18, and 19
years? What is your average earning power over
five years if it grows steadily from $5000

per year at the start to $9000 per year at the end?

2.13 Several special assumptions have been
made in arriving at the equation d = jat^. What
is the "unwritten text" behind it?

2.1 I Lt. Col. John L. Stapp achieved a speed of
632 mph (284 m/sec) in an experimental rocket
sled at the Holloman Air Base Development
Center, Alamogordo, New Mexico, on March 19,
1954. Running on rails and propelled by nine
rockets, the sled reached its top speed within 5
seconds. Stapp survived a maximum acceleration
of 22 g's in slowing to rest during a time interval
of I7 seconds (one g is an acceleration equal in
magnitude to that due to gravity; 22 g's means
22 X Og.)

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

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

2.1.T Derive the expression dlt^ = constant from
the expression d = jat^.

2.1(i Table 2.1 reports results from a recent
repetition of Galileo's experiment in which the
angle of inclination was 3.73° (Science, 133, 19-23,
June 6, 1961). A water clock with a constant-level
reservoir was used.

TABLE 2.1

TIME (measured in

DISTANCE (ft)

liliters of water)

d/t-

0.00185

0.00183

0.00192

0.00182

0.00185

0.00187

23.5

0.00182

Do these data really support Galileo's
assertion that d/t^ is constant? Explain your
conclusion.

2.17 Indicate whether the following statements
are true or false when applied to the strobe photo
below :

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

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

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

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

STUDY GUIDE 2

2.18 (a) Show by means of equations that

Galileo's statement in SG 2.9 follows from
dlt^ = constant for free fall from rest,
(b) The time interval between strobe flashes
was 0.35 sec. Use this information to
make a rough graph of d vs. t, also one of
V vs. t, and find the acceleration of the
ball.

2.19 The photograph in the figure below is of a
ball thrown upward. The acceleration due to
gravity increases the speed of the ball as it goes
down from its highest point (like any free-falling
object), if air friction is negligible. But the
acceleration due to gravity, which does not change,
acts also whUe the ball is still on its way up, and
for that portion of the path causes the baU to slow
down as it rises.

Stroboscopic photograph of a ball
thrown into the air.

When there is both up and down motion, it
will help to adopt a sign convention, an arbitrary
but consistent set of rules, similar to designating
the height of a place with respect to sea level. To
identify distances measured above the point of
initial release, give them positive values, for
example, the distance at B or at D, measured from

the release level, is about +60 cm and +37 cm,
respectively. If measured below the release level,
give them negative values; for example, E is at
—23 cm. Also, assign a positive value to the speed
of an object on its way up to the top (about +3
m/sec at A) and a negative value to a speed a
body has on the way down after reaching the top
(about —2 m/sec at D and —6 m/sec at E).
(a) Fill in the table with + and — signs.

AT POSITION

SIGN GIVEN TO

VALUE OF

(b) Show that it foUows from this convention
and from the definition of a = Az;/At that
the value or sign given to the acceleration
due to gravity is negative, and for both
parts of the path.

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

2.20 Draw a set of points (as they would appear
in a strobe photo) to show the successive positions
of an object that by our convention in SG 2.19
had a positive acceleration, that is, "upward." Can
you think of any way to produce such an event
physically?

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

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

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

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

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

2.22 From the definition for a, show it follows
directly that t^nnai = ^'initial + at for motion with
constant acceleration. Using this relation, and the
sign convention in SG 2.19. answer the questions
below. (Assume Og = 10 m/sec/sec.) An object is
thrown straight upward with an initial speed of
20 m/sec.

(a) What is its speed after 1.0 sec?

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

STUDY GUIDE 2

(d) How high is this maximum height?

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

If you have no trouble with this, you may wish
to try problems SG 2.23 and 2.24.

2.23 A batter hits a pop fly that travels straight
upwards. The ball leaves his bat with an initial
speed of 40 m/sec. (Assume a„ = 10 m/sec/sec)

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

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

(d) How high is this highest point?

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

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

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

(a) What acceleration does the ball
experience?

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

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

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

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

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

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

(b) "An iron shot will be dropped from the
roof of a 4-story building. As the shot falls,
it passes a window at each story. At each
window there will be a student who starts
his stopwatch upon hearing a signal that
the shot has been released, and stops the
watch as the shot passes his window.
Also, each student records the speed of the

shot as it passes. From his own data, each
student will compute the ratio vlt. I
expect that all four students will obtain
the same numerical value of the ratio."
(c) "Galileo's inclined planes dilute motion
all right, but the trouble is that there is
no reason to suppose that a ball rolling
down a board is behaving like a ball
falling straight downward. A better way
to accomplish this is to use light, fluffy,
cotton balls. These will not fall as rapidly
as metal spheres, and therefore it would
be possible to measure the time of the
fall t for different distances. The ratio dlt^
could be determined for different distances
to see if it remained constant. The
compactness of the cotton ball could then
be changed to see if a different value was
obtained for the ratio."

2.26 A student on the planet Arret in another
solar system dropped an object in order to
determine the acceleration due to gravity at that
place. The following data are recorded (in local
units):

TIME

DISTANCE

TIME

DISTANCE

(in surgs)

(in welfs)

(in surgs)

(in welfs)

0.0

2.2

10.41

0.5

0.54

2.4

12.39

1.0

2.15

2.6

14.54

1.5

4.84

2.8

16.86

2.0

8.60

3.0

19.33

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

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

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

equations d = ^at^ and v = at. What
special conditions must be satisfied for the
relation to be true?
(b) Show that if a ball is thrown straight
upward with an initial speed v it will rise
to a height

^ = 2^

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

Vf^ = Vi^ + 2ad

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

STUDY GUIDE 2

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

d = v,t + jaf

can be summarized in the three equations listed
below.

i'lmt

2.30 List the steps by which Galileo progressed
from his first definition of uniformly accelerated
motion to his final confirmation that this definition
is useful in describing the motion of a freely
falling body. Identify each step as a hypothesis,
deduction, observation, or computation, etc. What
limitations and idealizations appear in the
argument?

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

Vax-

Ad ^v , I .

The last of these equations can be applied only
to those cases where the acceleration is constant.
Because these three equations are so useful, they
are worth remembering (together with the
limitation on their use).

(a) State each of the three equations in words.

(b) Make up a simple problem to demonstrate
the use of each equation. (For example:
How long will it take a jet plane to travel
3200 miles if it averages 400 mi/hr?)
Then work out the solution just to be sure
the problem can be solved.

2.32 Show to what extent the steps taken by
Galileo on the problem of free fall, as described
in Sections 2.5 through 2.8, follow the general
cycle in the scientific process.

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

3.1 "Explanation" and the laws of motion

3.2 The Aristotelian explanation of motion

3.3 Forces in equilibrium

3.4 About vectors

3.5 Newton's first law of motion

3.6 The significance of the first law

3.7 Newton's second law of motion

3.8 Mass, weight, and free fall

3.9 Newton's third law of motion

3.10 Using Newton's laws of motion

3.11 Nature's basic forces

67
69
70
73
75
78
79
83
86
88
90

*'^.^- '"^r^

CHAPTER THREE

The Birth of Dynamics —
Newton Explains Motion

3.1 "Explanation" and the laws of motion

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

When Isaac Newton began his studies of motion in the second
half of the seventeenth century, Galileo's earlier insistence that
"the present does not seem to be the proper time to investigate the
cause of the acceleration of natural motion . . . ." was no longer
appropriate. Indeed, because Galileo had been so effective in
describing motion, Newton could turn his attention to dynamics, the
study of why an object moves the way it does — why it starts to
move instead of remaining at rest, why it speeds up or moves on a
curved path, and why it comes to a stop.

How does dynamics differ from kinematics? As we have seen in
the two earlier chapters, kinematics deals with the description of
motion. For example, in describing the motion of a stone dropped
from a cliff, we can write an equation showing how the distance d
through which the stone has dropped is related to the time t the
stone has been falling. We can find the acceleration and the final
speed attained during any chosen time interval. But when we have
completed our description of the stone's motion, we are still not
satisfied. Why, we might ask, does the stone accelerate rather than
fall with a constant speed? Why does it accelerate uniformly as
long as air friction is negligible? To answer these questions, we
will have to add to our store of concepts those of force and mass;
and in answering, we are doing dynamics. Dynamics goes beyond
kinematics by taking into account the cause of the motion.

SG 3.1

Some kinematics concepts: position,
time, speed, acceleration.
Some dynamics concepts: mass,
force, momentum (Ch. 9), energy
(Ch. 10).

The Birth of Dynamics- Newton Explains Motion

In Chapter 4 we will take up motion
also along curved paths.

Newton's First Law: Every object
continues in its state of rest or of
uniform motion in a straight line
unless acted upon by an unbalanced
force.

Newton's Second Law: The
acceleration of an object is directly
proportional to, and in the same
direction as, the unbalanced force
acting on it, and inversely pro-
portional to the mass of the object.

Newton's Third Law: To every action
there is always opposed an equal
reaction; or, mutual actions of two
bodies upon each other are always
equal and in opposite directions.

In our study of kinematics in Chapters 1 and 2, we encountered
four situations: an object may:

(a) remain at rest; (b) move uniformly in a straight line; (c) speed
up during straight-line motion; (d) slow down during straight-line
motion.

Because the last two situations are examples of acceleration, the
list could really be reduced to:

(a) rest; (b) uniform motion; and (c) acceleration.

Rest, uniform motion, and acceleration are therefore the
phenomena we shall try to explain. But the word "explain" must
be used with care. To the physicist, an event is "explained" when he
can demonstrate that the event is a logical consequence of a law
he has reason to believe is true. In other words, a physicist with
faith in a general law "explains" an observation by showing that it
is consistent with the law. In a sense, the physicist's job is to show
that the infinite number of separate, different-looking occurrences
all around and within us are merely different manifestations or
consequences of some general rules which describe the way the
world operates. The reason this approach to "explanation" works
is still quite remarkable: the number of general rules or "laws" of
physics is astonishingly small. In this chapter we shall learn three
such laws. Taken together with the mathematical schemes of
Chapters 1 and 2 for describing motion, they will suffice for our
understanding of practically all motions that we can readily
observe. And in Unit 2 we shall have to add just one more law (the
law of universal gravitation), to explain the motions of stars,
planets, comets, and satellites. In fact, throughout physics one sees
again and again that nature has a marvelous simplicity.

To explain rest, uniform motion, and acceleration of any
object, we must be able to answer such questions as these: Why
does a vase placed on a table remain stationary? If a dry-ice disk
resting on a smooth, level surface is given a brief push, why does it
move with uniform speed in a straight line rather than slow down
noticeably or curve to the right or left? Answers to these (and
almost all other) specific questions about motion are contained
either directly or indirectly in the three general "Laws of Motion"
formulated by Isaac Newton. These laws appear in his famous
book, Philosophiae Naturalis Principia Mathematica (Mathe-
matical Principles of Natural Philosophy, 1687), usually referred
to simply as The Principia. They are among the most basic laws in
physics to this day.

We shall examine Newton's three laws of motion one by one.
If your Latin is fairly good, try to translate them from the
original. A modernized version of Newton's text of these laws, in
English, is reproduced in the margin at the left.

Before we look at Newton's contribution, it will be instructive
to find out how other scientists of Newton's time, or earlier, might
have answered questions about motion. One reason for doing this
now is that many people who have not studied physics still show
Intuitively a bit of the pre-Newtonian viewpoint! Let us look at
what we must overcome.

Section 3.2

[•»>

3=7

mL'Jj^:^

AXIOMATA

SIVE

LEGES MOT US

Cfrftu omae ftrftv<T<ire in ftatn fno jHiefcndi vcl movendi tmifof
miter in dirtQwH, Kifi fiatemts aviribnf imfrtffu ccgltur fialiim
iUwH tnHart.

PRo^iKi perfeverant in moribusfuis ni/i quatenus a refiHcn-
m. leris retardannir & vi gravitatis impelluntur dcoifum.
Trochui, cujus partes cohaerendo perpetuo r«nihunt Me
a motibus refiillneis , non Cfflae rotaii nil! quatcnus ab acre re-
tardatur. Ma^ra auton Planctarum & Cometarum corpora n»o-
tus fuos & progrcflllvos & circulares in fpatiis minus rcuftentibus
6£los ci»Krvant duitHK.

Lex. n. .

UKtiHuMem iMtti/ profartimdiem effevi mctrici imfreff'^, ^ fieri fe-
ewrdHM liaitm rtSiam qua vit ill* imfrimitur.

Si vb aliqua oiotum qucmvis gcnerct, dupla diiplum, tripla tri-
plum gcserabit, five fimul&fotml, fivcgradatim& fucccflivcim-
preila iilcrk. Et hie motus quontitn in eaodcm fempcr plagam
cumvigcneratiice dctcrminatur, ficofpiuantca movebatur, mo-
tu3e)iisvc!confpirantiaddltur, vcl contrario fubducimr, vcl oblk
^y> oblique ad;icitur, & cum (o i^uodumutriurqidetenninacio-
aaa compooitui. Lex. Ilt-

•1

Lex. UL

A&i>iu CMtTarUm femptr ^ itfiultm ejtertadkiuni •■fiuit ,
dtunim tHitnet in Je muino femfer ejje nputtt ^ nr fttUt etrtri
rittdingi.

Qlticquid pmnkvclcrafaic altenim, tanrundonab copmnitur
vcl trabinir. Siquis bpidem digito prnnit, ptewkur & lin)ui
^iginua lapide. Ste(]uuslap«kin fimi aliegatum trahit, ratrahe-
tur ctiam & equub xquaiiter in lapidein: nam funis iirri(t(}idHiciitus
eodciD rcbicandi fc conatu urgeSit Equum verfus lapidcsi, ac la-
pidem'ver&5 equum, tantumq, impedietprogrcfTumunius quan^
njm promovet progrcfTum aTtcrius. Si corpus afiquod in corpus
afiud inipiflgcns, niotume)usvi fua quomcxjocunqt aiutaverit, i-
deni quo^Ue vKifl^m in nioni proptk) eatid<uii inubtknnniii pat';
fern coBffariam vi iktam^th «qualitatnn pfcflto;iis anjr r ^
futnbit. Hisadion)buts<{uale5fiuiKuiuta(i6nc<DOO vcm 1..
iedmotuum, ( icilicct^in corporJbui non armndciiDfcJi^
tatiooet entna vclockanun, m contracus ttkkis part'et 60*
AMtus zqualiter mutalttur, funt corpodbus rtc^tfooe proi
nales.

cofo!. I. ; ;;•

Corpiu virAtu aeiJHitQif iitgoKaltm ftriBtUgnmm
defcriber*^ ^mo laHmftf^Hir.

Si corpus dato tempore, vifola M,
fcrretur ab /* ad B, & vi fola N, ab
^ ad C, complcatur paralielograni-
mum ABDC, tc vi utraoi feretur id
eodein tempore ab A ti D. Nam
quontim vb tf aeit fccunduin lineam
.^C ipfi B D parallcliin, hxc vis nihil mutab^t velotitatera a«ce-
dendi ad lineam illam B D a vi altera genitam. Accedet iginjr
«<f pus eodem tempore ad lineam B D five vis N imprimatur, five
afqi adeo in fine illius temporb xeperictur atcubi in linea

Q1 A baseball is thrown straight upward. Which of these
questions about the baseball's motion are kinematic and which
dynamic?

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

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

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

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

3.2 The Aristotelian explanation of motion

The idea of force played a central role in the dynamics of
Aristotle, twenty centuries before Newton. You will recall from
Chapter 2 that in Aristotle's physics there were two types of motion
— "natural" motion and "violent" motion. For example, a falling
stone was thought to be in "natural" motion (towards its natural
place), but a stone being steadily lifted was thought to be in
"violent" motion (away from its natural place). To maintain this
uniform violent motion, a force had to be continuously applied.
Anyone lifting a large stone is very much aware of this as he
strains to hoist the stone higher.

The Aristotelian ideas were consistent with many common-
sense observations. But there were also difficulties. Take a specific
example — an arrow shot into the air. It cannot be in violent motion
without a mover, or something pushing on it. Aristotelian physics

The Birth of Dynamics— Newton Explains Motion

Keeping an object in motion at uni-
form speed.

SG 3.2

required that the arrow be continually propelled by a force; if the
propelling force were removed, the arrow should immediately stop
its flight and fall directly to the ground in "natural" motion.

But of course the arrow does not fall to the ground as soon as
it loses direct contact with the bowstring. What then is the force
that propels the arrow? Here, the Aristotelians offered an ingenious
suggestion; the motion of the arrow through the air was maintained
by the air itself! A commotion is set up in the air by the initial
movement of the arrow. That is; as the arrow starts to move, the air
is pushed aside; the rush of air to fill the space being vacated by
the arrow maintains it in its flight.

More sophisticated ideas to explain motion were developed
before the mid-seventeenth century. But in every case, a force was
thought to be necessary to sustain uniform motion. The explanation
of uniform motion depended on finding the force, and that was not
always easy. There were also other problems. For example, a falling
acorn or stone does not move with uniform speed — it accelerates.
How is acceleration explained? Some Aristotelians thought the
speeding up of a falling object was associated with its approaching
anival at its natural place, the earth. In other words, a falling object
was thought to be like the tired horse that starts to gallop as it
approaches the barn. Others claimed that when an object falls, the
weight of the air above it increases while the column of air below
it decreases, thus offering less resistance to its fall.

When a falling object finally reaches the ground, as close to the
center of the earth as it can get, it stops. And there, in its "natural
place," it remains. Rest, being regarded as the natural state of
objects on earth, required no further explanation. The three
phenomena — rest, uniform motion, and acceleration — could thus be
explained in a more or less plausible fashion by an Aristotelian.
Now, let us examine the Newtonian explanation of the same
phenomena. The key to this approach is a clearer understanding of
the concept of force.

Q2 According to Aristotle, what is necessary to maintain
uniform motion?

Q3 Give an Aristotelian explanation of a dry-ice puck's uniform
motion across a table top.

3.3 Forces in equilibrium

Our common-sense idea of force is closely linked with our own
muscular activity. We know that a sustained effort is required to
lift and support a heavy stone. When we push a lawn mower, row a
boat, split a log, or knead bread dough, our muscles let us know
we are applying a force to some object. Force and motion and
muscular activity are naturally associated in our minds. In fact,
when we think of changing the shape of an object, or moving it or
changing its motion, we naturally think of the muscular sensation

Section 3.3

of applying a force to the object. We shall see that many -but not
all -of the everyday common-sense ideas about force are useful in
physics.

We know intuitively that forces can make things move, but
they can also hold things still. The cable supporting the main span
of the Golden Gate Bridge is under the influence of mighty forces,
yet it remains at rest. Apparently, more is required to start motion
than just the application of forces.

Of course, this is not surprising. We have all seen children
quarrelling over a toy. If each child pulls determinedly in his own
direction, the toy may go nowhere. On the other hand, the tide of
battle may shift if one of the children suddenly makes an extra
effort, or if two children cooperate and pull side by side against the
third.

Likewise, in the tug-of-war between the two teams shown
above, large forces were exerted on each side, but the rope remained
at rest: one may say the forces balanced, or they "cancelled." A
physicist would say that the rope was in equilibrium when the
sum of the forces on each side of it were equally large and acting
in opposite directions. Equally well, he might say the net force is
zero. Thus a body in equilibrium would not start to move until a
new, "unbalanced" force was added which destroyed the
equilibrium.

In all these examples, both the magnitude of the forces and
their directions are important. The effect of a force depends on the
direction in which it is applied. We can represent this directional
nature of forces in a sketch by using arrows: The direction the
arrow points represents the direction in which the force acts; the
length of the arrow represents how large the force is (for example,
a 10-lb force is shown by an arrow twice as long as a 5-lb force).

Now we discover a surprising result. If we know separately each
of the forces applied to any object at rest, we can predict whether
it will remain at rest. It is as simple as this: The object acted on by
forces will be in equilibrium under these forces and wUl remain at
rest only if the arrows representing the forces all "add up to zero."

How does one "add up" arrows? By a simple graphical trick.
Take the tug-of-war as an example. Let us call the force exerted by
the team pulling to the right ?,. (The httle arrow over the F

The Birth of Dynamics— Newton Explains Motion

(d-)

■Pore* F2

force

+ea»«2

"team

(b)

(c)

^•^?2'^

There are several ways of expressing
the idea of unbalanced force: net
force, resultant force, total force,
vector sum of forces. All mean the
same thing.

indicates that we are dealing with a quantity for which direction is
important.) The force of the second team is then called F-,. Figure
(a) in the margin shows the two arrows corresponding to the two
forces, each applied to the central part of the rope, but in opposite
directions. Let us assume that these forces, F, and F2, were
accurately and separately measured, for example, by letting each
team in turn pull on a spring balance as hard as it can. The arrows
for Ti and T2 are carefully drawn to a chosen scale, such as
1" = 1000 lb, so that 750 lb of force in either direction would be
represented by an arrow of 3/4" length. Next, in Figure (b), we take
the arrows F, and ¥2 and draw them again in the correct direction
and to the chosen scale, but this time we put them "head to tail."
Thus F] might be drawn first, and then To is drawn with the tail of
P2 starting from the head of Fj. (Since they would of course overlap
in this example, we have drawn them a little apart in Figure (b) to
show them both more clearly.) The trick is this: If the head end of
the second arrow falls exactly on the tail end of the first, then we
know that the effects of Fj and ¥2 balance each other. The two
forces, acting in opposite directions and equally large, add up to
zero. If they did not, the excess of one force over the other would
be the net force and the rope would accelerate instead of being at
rest.

To be sure, this was an obvious case, but the graphical
technique turns out to work also for cases that are not simple. For
example, apply the same procedure to the toy, or to a boat that is to
be secured by means of three ropes attached to different moorings.

TIT

jmrnrnwi

We are defining equilibrium without
worrying about whether the object
will rotate. For example: The sum of
the forces on the plank in the dia-
gram below is zero, but it is obvious
that the plank will rotate.

F/F.'^3-0

'■^■^J^.g^L,..: ■';

^T F>F>

Consider a situation where Ti is a force of 34 lb, ?, is 26 lb. f-^ is
28 lb, each in the direction shown. (The scale for the magnitude
of the forces here is 0.1 cm = 1 lb of force.) Is the boat in equilibrium
under the forces? Yes, if the forces add up to zero. Let's see. With
rule and protractor the arrows are drawn to scale and in exactly
the right direction. Then, adding f*,. F... and ?., head to tail, we see
that the head of the last arrow falls on the tail of the first. Yes. the
forces cancel; they add up to zero; the net force is zero. Therefore

Section 3.4

the object is in equilibrium. This method tells us when an object is
in equilibrium, no matter how many different forces are acting on it.

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

An interesting case of equilibrium, very different from the
disputed toy or rope, is part of the "free fall" of a sky-diver. In fact
his fall is "free" only at the beginning. The force of air friction
increases with speed, and soon the upward frictional force on the
sky-diver is great enough to balance the force of gravity
downward. Under those circumstances he falls with constant speed,
much like a badminton bird or falling leaf. The sensation is not of
falling but, except for the wind, the same as lying on a soft bed.
During part of a dive from an airplane you can be as much in
equilibrium as lying in bed! In both cases the net force acting on
you is zero.

Q4 A vase is standing at rest on a table. What forces would you
say are acting on the vase? Show how each force acts (to some
scale) by means of an arrow. Can you show that the sum of the
forces is zero?

Q5 In which of these cases are the forces balanced?

SG 3.3

■/

Q6 Does an object have to be at rest to be in equilibrium?

3.4 About vectors

Graphical construction with arrows really works. With it we can
predict whether the forces balance and will leave the object in
equilibrium or whether any net force is left over, causing the object
to accelerate. Why can we use arrows in this way? The reason
involves the precise mathematical definitions of displacement and
of force, but you can demonstrate for yourself the reasonableness
of the addition rule by trying a variety of experiments. For example,
you could attach three spring scales to a ring and have some
friends pull on the scales with forces that just balance, leaving the
ring at rest. While they are pulling, you read the magnitudes of the
forces on the scales and mark the directions of the pulls. You can
then make a graphical construction with arrows representing the
forces and see whether they add to zero. Many different experiments
of this kind ought all to show a net force of zero.

The Birth of Dynamics— Newton Explains Motion

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

In the example of the tug-of-war we talked about the effect of
equally large, opposing forces. If two forces act in the same
direction, the resultant force is found in essentially the same way,
as shown below.

If two forces act at some angle to each other, the same type of
construction is still useful. For example, if two forces of equal
magnitude, one directed due east and the other directed due north,
are applied to an object at rest but free to move, the object will
accelerate in the northeast direction, the direction of the resultant
force. The magnitude of the acceleration will be proportional to the
magnitude of the resultant force which is shown by the length of
the arrow representing the resultant.

You can equally well use a graphical
construction called the "parallelo-
gram method." It looks different
from the "head-to-tail" method, but
Is really exactly the same. In the
parallelogram construction, the
vectors to be added are represented
by arrows joined tail-to-tail instead
of head-to-tail, and the resultant is
obtained by completing the diagonal
of the parallelogram.

Cfr-

The same adding procedure is used if the forces are of any magni-
tude and act at any angles to each other. For example, if one force
were directed due east and a somewhat larger force were directed
northeast, the resultant vector sum could be found as shown below.

Section 3.5

To summarize, we can now define a vector quantity. It is a
quantity which has both direction and magnitude and which can
be added by the graphical construction of the head-to-tail
representation of arrows, or by the equivalent parallelogram method.
(It also has other properties which you will study if you take further
physics courses.) By this definition, many important physical
concepts are vectors — for example, displacement, velocity, and
acceleration. Some other physical concepts, including volume,
distance, and speed, do not require specification of direction, and so
are not vector quantities; these are called scalar quantities. When
you add 10 liters of water to 10 liters of water, the result is always
20 liters, and direction has nothing to do with the result. Similarly,
the term speed has no directional meaning; it is the magnitude of
the velocity vector, as given by the length of the arrow, without
regard to its direction. By contrast, when you add two forces of 10
lb each, the resultant force may be anywhere between zero and 20
lb, depending on the directions of the two individual forces.

We shall soon have to correct an oversimplification we had to
make in Sec. 1.8, where we defined acceleration as the rate of
change of speed. That was only partly correct, because it was
incomplete. We shall also want to consider changes in the direction
of motion as well. The more useful definition of acceleration is the
rate of change of velocity, where velocity is a vector having both
magnitude and direction. In symbols.

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

-* Ax;

where Ai; is the change in velocity. Velocity can change in two
ways: by changing its magnitude (speed), and by changing its
direction. In other words, an object is accelerating when it speeds
up, or slows down, or changes direction. We shall explore this
definition more fully in later sections.

We shall use vectors frequently. To
learn more about them you can use
the Project Physics Programmed In-
struction booklets on vectors. See
also Reader 1 article "Introduction to
Vectors."

Q7 List three properties of vector quantities.
Q8 How does the new definition of acceleration given above
differ from the one used in Chapter 1?

3.5 Newton's first law of motion

Were you surprised when you first watched a dry-ice disk or
some other nearly frictionless device? Remember how smoothly it
glides along after just the slightest nudge? How it shows no sign of
slowing down or speeding up? Although our intuition and everyday
experience tell us that some force is constantly needed to keep an
object moving, the disk fails to hve up to our Aristotelian
expectations. It is always surprising to see this for the first time.

Yet the disk is behaving quite naturally. If the forces of friction
were absent, a gentle, momentary push would make tables and

The Birth of Dynamics— Newton Explains Motion

chairs take off and glide across the floor just like a dry-ice disk.
Newton's first law directly challenges the Aristotelian notion of
what is "natural." It declares that the state of rest and the state of
uniform, unaccelerated motion in a straight line are equally
natural. Only the existence of some force, friction for example,
keeps a moving object from moving forever \ Newton's first law of
motion can be stated as follows in modem terminology:

Because constant velocity means
both constant speed and constant
direction, we can write Newton's
first law more concisely:

r= constant
if and only if

This statement includes the
condition of rest, since rest is a
special case of unchanging velocity
—the case where v'= 0.

SG 3.5

Every object continues in its state of rest or of uniform
rectilinear motion unless acted upon by an unbalanced
force. Conversely, if an object is at rest or in uniform
rectilinear motion, the unbalanced force acting upon it
must be zero.

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

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

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

It was from this observation that Galileo generated his thought
experiment. He predicted that a ball released from a height on a
frictionless ramp, would roll up to the same height on a similar
facing ramp, regardless of the actual path length. For example, in
the diagram at the top of the next page, as the ramp on the right is
changed from position (a) to (b) and then to (c). the ball must roll
further in each case to reach its original height. It slows down
more gradually as the angle of the incline decreases. If the second
ramp is exactly level as shown in (d). the ball can never reach its
original height. Therefore, Galileo believed, the ball on this
frictionless surface would roll on in a straight line and at an

Section 3.5

unchanged speed forever. This could be taken to be the same as
Newton's first law. and some historians of science do give credit to
Galileo for having come up with the law first. Other historians,
however, point out that, for Galileo, rolling on forever meant staying
at a constant height above the earth — not moving in a straight line
through space.

This tendency of objects to maintain their state of rest or of
uniform motion is sometimes called "the principle of inertia."
Newton's first law is therefore sometimes referred to as the "law of
inertia." Inertia is a property of all objects. Material bodies have, so
to speak, a stubborn streak so far as their state of motion is
concerned. Once in motion, they continue to move with unchanging
velocity (unchanging speed and direction) unless compelled by
some externally applied force to do otherwise. If at rest, they remain
at rest. This is why seat belts are so helpful when the car stops very
suddenly, and also why a car may not follow an icy road around a
turn, but travel a straighter path into a field or fence. The greater
the inertia of an object, the greater its resistance to a change in its
state of motion, and hence the greater is the force needed to
produce a desired change in the state of its motion. This is why it is
more difficult to start a train or a ship and to bring it up to speed
than it is to keep it going once it is moving at the desired speed. (In
the absence of friction, it would keep moving without any applied
force at all.) But for the same reason it is difficult to bring it to a
stop, and passengers and cargo keep going forward if the vehicle is
suddenly braked.

Newton's first law tells us that if we see an object moving with
a constant speed in a straight line, we know at once that the forces
acting on it must be balanced, that is, it is in equilibrium. In Sec.
3.4 we established that an object at rest is in equilibrium. Does this
mean that in Newtonian physics the state of rest and the state of
uniform motion are equivalent? It does indeed. When we know that
a body is in equilibrium, we know only that v = constant. Whether
the value of this constant is zero or not depends in any case on
which body is chosen as reference for measuring the magnitude of
V. We can decide whether to say that it is at rest or that it is moving
with constant t/ larger than zero only by reference to some other
body.

Take, for example, a tug-of-war. Suppose two teams were sitting
on the deck of a barge that was drifting with uniform velocity down
a lazy river. Two observers — one on the same barge and one on the
shore — would each give a report on the incident as viewed from his

Inside the laboratory there is no
detectable difference between a
straight (horizontal) line and a
constant height above the earth. But
on a larger scale, Galileo's eternal
rolling would become motion in a
circle around the earth. Newton
made clear what is really important:
that in the absence of the earth's
gravitational pull or other external
forces, the ball's undisturbed path
would extend straight out into space.

Galileo's idea of a straight
line.

Newton' s idea of a straight
line.

The Birth of Dynamics — Newton Explains Motion

own frame of reference. The observer on the barge would observe
that the forces on the rope were balanced and would report that it
was at rest. The observer on the shore would report that the forces
on the rope were balanced and that it was in uniform motion.
Which observer is right? They are both right; Newton's first law of
motion applies to both observations. Whether a body is at rest or in
uniform motion depends on which reference frame is used to
observe the event. In both cases the forces on the object involved
are balanced.

Q9 What is the net force on the body in each of the four cases
sketched in the margin of the opposite page?

Q10 What may have been a difference between Newton's
concept of inertia and GalOeo's?

3.6 The significance of the first law

Of course, the idea of inertia does
not explain why bodies resist change
in their state of motion. It is simply
a term that helps us to talk about
this basic, experimentally observed
fact of nature. (See SG 3.6 and 3.7.)

The correct reference frame to use
in our physics turns out to be any
reference frame that is at rest or
in uniform rectilinear motion with
respect to the stars. The rotating
earth is, therefore, strictly speaking
not allowable as a Newtonian
reference frame; but for most
purposes the earth rotates so little
during an experiment that the
rotation can be neglected. (See
SG 3.8.)

You may have found Galileo's thought experiment convincing.
But think how you might try to verify the law of inertia
experimentally. You could start an object moving (perhaps a dry-
ice disk) in a situation in which you believe there is no unbalanced
force acting on it. Then you could observe whether or not the object
continued to move uniformly in a straight line, as the first law
claims it should.

The experiment is not as simple as it sounds; in fact, Newton's
laws involve some profound philosophical content (see SG 3.7); but
we can see the significance of Newton's first law even without
going into all these subtleties. For convenience let us list the
important insights the first law provides.

1. It presents the idea of inertia as a basic property of all material
objects. Inertia is the tendency of an object to maintain its
state of rest or uniform motion.

2. It points up the equivalence of a state of rest for an object
and a state of uniform motion in a straight line. Both states
indicate that the net force is zero.

3. It raises the whole issue of frame of reference. An object
stationary for one observer might be in motion for another
observer; therefore, if the ideas of rest or uniform motion are
to have any significance, a frame of reference must be
specified from which the observations of events are to be
made.

4. It purports to be a universal law. It emphasizes that a single
scheme can deal with motion anywhere in the universe. For
the first time no distinction is made between terrestrial and
celestial domains. The same law applies to objects on earth as
well as on the moon and the planets and the stars. And it
applies to balls, dry-ice pucks, magnets, atomic nuclei,
electrons — everything !

Section 3.7

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

3.7 Newton's second law of motion

In Section 3.1 it was stated that a theory of dynamics must
account for rest, uniform motion, and acceleration. So far we have
met two of our three objectives: the explanation of rest and of
uniform motion. In terms of the first law, the states of rest and
uniform motion are equivalent; they are different ways of describing
the state of equilibrium — that state in which no unbalanced force
acts on an object.

The last section concluded with a list of insights provided by
the first law. You noticed that there was no quantitative relationship
established between force and inertia. Newton's second law of
motion enables us to reach our third objective — the explanation
of acceleration — and also provides a quantitative expression, an
equation for the relationship between force and inertia. We shall
study separately the way in which force and inertia enter into the
second law. Later in this section we will look more closely at how
force and inertia are measured. But first we will take some time to
be sure that Newton's statement is clear. First we consider the
situation in which different forces act on the same object, and then
the situation in which the same force acts on different objects.

Force and Acceleration. To emphasize the force aspect, Newton's
second law can be stated as follows :

The net, unbalanced force acting on an object is directly
proportional to, and in the same direction as, the acceleration
of the object.
More briefly, this can be written as: "acceleration is proportional
to net force." If we let F^et stand for net force and a stand for
acceleration, we can write this relationship precisely as:

a ^ f net

Both a and f^net are vectors; the statement that they are proportional
includes the understanding that they also point in the same
direction.

To say that one quantity is proportional to another is to make
a precise mathematical statement. Here it means that if a given net
force (Fnet) causes an object to move with a certain acceleration (a),
then a new force equal to twice the previous force (2Fnet) will cause
the same object to have a new acceleration equal to twice the
earlier acceleration (or 2a); three times the net force will cause
three times the acceleration; and so on. Using symbols, this
principle can be expressed by a statement like the following:

SG 3.9
SG 3.10

Apple falling- negligible friction

Feather falling at nearly constant
speed

l^ili^^T^:^^ ""vS

Kite held suspended in the wind

Man running against the wind

80 The Birth of Dynamics- Newton Explains Motion

If a force f^net will cause a, then a force equal to

2P^net will cause 2a
sfnei will cause 3a
2-Fnet will cause ja
5.2Fnet will cause 5.2a

and so on.

One can readily imagine a rough experiment to test the
validity of the law — more easily as a thought experiment than
as a real one. Take a nearly frictionless dry-ice puck on a flat table,
attach a spring balance, and pull with a steady force so that it
accelerates continuously. The pull registered by the balance will be
the net force since it is the only unbalanced force acting. Measure
the forces and the corresponding accelerations in various tries, then
compare the values of Fnet and a. We shall look into this method in
detail in the next section.

Mass and Acceleration. Now we can consider the inertia aspect of
the second law, the effect of the same net force acting on different
objects. In discussing the first law, we said inertia is the resistance
an object exhibits to any change in its velocity. We know from
experience and observation that some objects have greater inertia
than others. For instance, if you were to throw a baseball and then
put a shot with your full effort, you know that the baseball would
be accelerated more and hence would reach a greater speed than
the shot. Thus, the acceleration given a body depends as much on
the body as it does on the force applied to it. The concept of the
amount of inertia a body has is expressed by the term mass.

Mass is a familiar word, but it becomes useful in physics only
after it is disentangled from some aspects of its common sense
meaning. For example, mass is often used as a synonym for weight.
But although mass and weight are closely related, they are not
at all the same thing. Weight is a force, the force with which
gravity is acting on an object; mass, on the other hand, is a
measure of an object's resistance to acceleration. It is true that
on or near the surface of the earth, objects that are hard to
accelerate are also heavy, and we will return to this relationship in
SG 3.11 Sec. 3.8.

If you supply the same force to several different objects, their
What does it mean to say that mass accelerations will not be the same. Newton claimed that the
is a scalar quantity? resulting acceleration of each object is inversely proportional to its

mass. Using the symbol m for mass (a scalar quantity), and the
symbol a for the magnitude of the vector acceleration a, we can
write "a is inversely proportional to m," or what is mathematically

the same, "a is proportional to — , " or

a a. —

This means that if a certain force makes a given object have a
certain acceleration, then the same force will cause an object
having twice the mass to have one-half the acceleration, an object

Section 3.7 81

having three times the mass to have one-third the acceleration,
an object of one-fifth the mass to have five times the acceleration,
and so on. This is why, for example, a truck takes much longer to
reach the same cruising speed when it is full than when it is nearly
empty. Using symbols, we can express this as follows:

If a given force Fnet is applied, and an object

of mass m experiences a, then an object

of mass 2m will experience ^a,

of mass 3m will experience ^a,

of mass jm will experience 5a,

of mass 2.5m will experience 0.4a,

and so on.

This can be demonstrated by experiment. Can you suggest how it
might be done?

The roles played by force and mass in Newton's second law can
be combined in a single statement:

The acceleration of an object is directly proportional to, and
in the same direction as, the unbalanced force acting on it, and
inversely proportional to the mass of the object.

The ideas expressed in this long statement can be summarized by

the equation SG 3.12

Yn SG 3.13

We can regard this equation as one possible way of expressing
Newton's second law of motion. The same relation may of course
be equally well written in the form

FnPt —

In either form, this is probably the most fundamental single
equation in all of Newtonian mechanics. Like the first law, the
second has an incredible range of application: It holds no matter
whether the force is mechanical or electric or magnetic, whether
the mass is that of a star or a nuclear particle, whether the
acceleration is large or small. We can use the law in the easiest
problems and the most sophisticated ones. By measuring the
acceleration which an unknown force gives a body of known m.ass,
we can compute a numerical value for the force from the equation
Fnet ^ 'ma- Or, by measuring the acceleration that a known force
gives a body of unknown mass, we can compute a numerical value
for the mass from the equation (m = Fnet/a)- Clearly we must be
able to measure two of the three quantities in order to be able to
compute the other.

Units of mass and force. Even before we can make such measure-
ments, however, we must establish units for mass and force that
are consistent with the units for acceleration (which have already
been defined in terms of standards of length and time -for
example, meters per second per second).

The Birth of Dynamics- Newton Explains Motion

1 kg corresponds to the mass of
about 1 liter of water, or about
2.2 lb (more precisely 2.205 lb).
The 1/1000th part of 1 kg is 1 gram

(ig).

SG 3.14

One way to do this is to choose some convenient object, perhaps
a piece of corrosion-free metal, as the universal standard of mass,
just as a meter is a universal standard of length. We can
arbitrarily assign to this object a mass of one unit. Once this unit
has been selected we can proceed to develop a measure of force.

Although we are free to choose any object as a standard of
mass, ideally it should be exceedingly stable, easily reproducible,
and of reasonably convenient magnitude. Such a standard object
has, in fact, been agreed on by the scientific community. By
international agreement, the primary standard of mass is a
cylinder of platinum-iridium alloy, kept near Paris at the
International Bureau of Weights and Measures. The mass of this
platinum cylinder is defined as exactly 1 kilogram (abbreviated
1 kg). Accurately made copies of this international primary
standard of mass are kept in the various standards laboratories
throughout the world. Further copies have been made from these
for distribution to manufacturers and research laboratories.

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

SG 3.15, 3.16

SG 3.17,3.18

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

Now we can go on to answer the question of how much "push"
or "pull" should be regarded as one unit of force. We define 1 unit
of force as a force which, when acting alone, causes an object that
has a mass of 1 kilogram to accelerate at the rate of exactly
1 meter/second/per second.

Imagine an experiment in which the standard 1-kg object is
pulled with a spring balance in a horizontal direction across a level,
frictionless surface. The pull is regulated to make the 1-kg object
accelerate at exactly 1 m/secl The required force will by definition
be one unit in magnitude:

Fnei = 1 kg X 1 m/sec"^ = 1 kgm/sec-

Section 3.8

Thus, 1 kgm/sec^ of force is that quantity of force which causes a
mass of 1 kg to accelerate 1 m/sec-.

The unit kgm/sec^ has been given a shorter name, the newton
(abbreviated as N). The newton is therefore a derived unit, defined
in terms of a particular relationship between the meter, the
feilogram, and the second. Thus the newton is part of the "mks"
system of units, which is used almost universally in modem
scientific work.

The "hidden text" in Newton's second law involves both
definitions and experimental facts. There are several possible ways
of analyzing it: if you choose to define some part, you must prove
others experimentally— or vice- versa. Textbooks do not all agree on
how best to present the relation of definition and experiment in
Newton's second law, and Newton himself may have not thought it
through entirely. However, as a system of ideas (whichever way it
is analyzed), it was powerful in leading to many discoveries in
physics.

Newton did not "discover" the concepts of force and mass. But
he did recognize that these concepts were basic to an understanding
of motion. He clarified these concepts, and found a way to express
them in numerical values, and so made a science of dynamics
possible.

Q11 Which three fundamental units of distance, mass and
time are used to define the unit of force?

Q12 A net force of 10 N gives an object a constant acceleration
of 4 m/sec^. What is the mass of the object?

Q13 True or false? Newton's second law holds only when
frictional forces are absent.

Q14 A 2-kg object, shoved across the floor with a speed of 10
m/sec, slides to rest in 5 sec. What was the magnitude of the force
producing this acceleration?

Q15 Complete the table in the margin which lists some
accelerations resulting from applying equal forces to objects of
diff"erent mass.

The units of acceleration "m/sec
per second" can be written as
"m/sec/sec" or "m/sec-". The sec-
means that division by time units
occurs twice, not something like
"square time."

SG 3.19, 3.20, 3.21, 3.22, 3.23.

MASS ACCELERATION

30 m/sec^

15 m/sec^

3/77

^/5m

0.5m

45m

3 m/sec^

75 m/sec^

3.8 Mass, weight, and free fall

The idea of force has been generalized in physics to include
much more than muscular pushes and pulls. Whenever we observe
an acceleration, we infer that there is a force acting. Forces need
not be "mechanical" or exerted by contact only; they can be due to
gravitational, electric, magnetic, or other actions. Newton's laws
are valid for all of them.

The force of gravity acts without direct contact between objects
that are separated not only by a few feet of air, as is the case with
the earth and a falling stone, but also across empty space such as
separates the earth from an artificial satellite in orbit.

We shall use the symbol fg for gravitational force. The

The Birth of Dynamics— Newton Explains Motion

Is the boy weightless? Explain.
SG 3.24.)

(See

magnitude of the gravitational pull T„ is, roughly speaking, the
same anywhere on the surface of the earth for a particular object.
When we wish to be very precise, we must take into account the
facts that the earth is not exactly spherical, and that there are
irregularities in the composition of the earth's crust. These factors
cause slight differences — up to 1/2% — in the gravitational force
on the same object at different places. An object having a constant
mass of 1 kg will experience a gravitational force of 9.812 newtons
in London, but only 9.796 newtons in Denver, Colorado. Geologists
make use of these variations in locating oil and other mineral
deposits.

The term weight is often used in everyday conversation as if it
meant the same thing as bulk or mass. In physics, we define the
weight of an object as the gravitational force acting on the body.
Weight is a vector quantity, as are all forces. Your weight is the
downward force our planet exerts on you whether you stand or sit.
fly or fall, orbit the earth in a space vehicle or merely stand on a
scale to "weigh" yourself.

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

Now imagine for a moment a ridiculous but instructive thought
experiment: as you stand on the scale, the floor (which, sagging
slightly, has been pushing up on the scale) suddenly gives way, and
you and the scale are dropping into a deep well in free fall. At every
instant, your fall speed and the scale's fall speed will be equal,
since you started falling together and fall with the same
acceleration. Your feet would now touch the scale only barely (if at
all), and if you looked at the dial you would see that the scale
registers zero. This does not mean you have lost your weight -that
could only happen if the earth suddenly disappeared, or if you were
suddenly removed to far, interstellar space. No, Pg still acts on you
as before, accelerating you downward, but since the scale is
accelerating with you, you are no longer pushing down on it -nor
is it pushing up on you.

You can get a fairly good idea of the difference between the
properties of weight and mass by holding a big book: First, just lay
the book on your hand; you feel the weight of the book acting down.
Next, grasp the book and shake it back and forth sideways. You still
feel the weight downwards, but you also feel how hard the book
is to accelerate back and forth — its mass. You could make your
sensation of the book's weight disappear by hanging the book on a
string, but the sensation of its inertia as you shake it remains the
same. This is only a crude demonstration, and it isn't clear that the
shaking sensation doesn't still depend on the pull of the earth. More
elaborate experiments would show, however, that weight can

Section 3.8 85

change without changing mass. Thus when an astronaut on the Consider SG 3.14 again,

moon's surface uses a big camera, he finds it much easier to hold-
its weight is only 1/6 of its weight on earth. But its mass or
inertia is not less, and it is as hard to swing around suddenly
into a new position as it is on earth.

We can now understand the results of Galileo's experiment on
falling objects in a more profound way. Galileo's discussion of
falling objects showed that any given object (at a given locality)
falls with uniform acceleration, a,,. What is responsible for its
uniform acceleration? A constant net force — in this case of free fall,
just Fg. Now Newton's second law expresses the relationship
between this force and the resulting acceleration. Applying the
equation Fnet = ma to this case, where Fnet ^ r „ and a = ay, we can
write

fg = mag

We can, of course, rewrite this equation as

We conclude from Newton's second law that the reason why the
acceleration of a body in free fall is constant is that for an object of
given mass m the gravitational force Fg over normal distances of
fall is nearly constant.

Galileo, however, did more than claim that every object falls
with constant acceleration: he found that all objects fall with the
same uniform acceleration, which we now know has the value of
about 9.8 m/sec at the earth's surface. Regardless of the mass m or
weight Fg, all bodies in free fall (in the same locality) have the
same acceleration a^. Is this consistent with the relation ag= Fglm7
It is consistent only if for every object Fg is directly proportional to
mass m: that is, if m is doubled, Fg must double; if m is tripled,
Fg must triple. This is a significant result indeed. Weight and mass
are entirely different concepts. Weight is the gravitational force on
an object (hence weight is a vector). Mass is a measure of the
resistance of an object to change in its motion, a measure of inertia
(hence mass is a scalar). Yet the fact that different objects fall
freely with the same acceleration means that the magnitudes of
these two quite different quantities are proportional in any given
locality.

Q16 An astronaut is orbiting the earth in a space vehicle. The
acceleration due to gravity at that distance is half its value on the
surface of the earth. Which of the following are true? SG 3.25, 3.26, 3.27, 3.28

(a) His weight is zero.

(b) His mass is zero.

(d) His mass is half its original value.

(e) His weight remains the same.

(f ) His mass remains the same.

The Birth of Dynamics- Newton Explains Motion

Q17 A boy jumps from a table top. When he is halfway between
the table top and the floor, which of the statements in Q16 are true?

3.9 Newton's third law of motion

He is, to be sure, pushing against
the ground -but that is a force
acting on the ground.

In his first law, Newton described the behavior of objects when
they are in a state of equilibrium; that is, when the net force acting
on them is zero. His second law explained how their motion changes
when the net force is not zero. Newton's third law added a new
and surprising insight about forces.

Consider this problem: In a 100-meter dash, an athlete will go
from rest to nearly his top speed in less than a second. We could
measure his mass before he makes the dash, and we could use
high-speed photography to measure his initial acceleration. With his
mass and acceleration known, we could use F = ma to find the force
acting on him during the initial acceleration. But where does the
force come from? It must have something to do with the runner
himself. Is it possible for him to exert a force on himself as a
whole? Can he lift himself by his own bootstraps?

Newton's third law of motion helps us to understand just such
puzzling situations. First, let us see what the third law claims.
In Newton's words:

To every action there is always opposed an equal reaction: or,
mutual actions of two bodies upon each other are always equal
and directed to contrary parts.

SG 3.29

This is a word-for-word translation from the Principia. It is
generally agreed, however, that in Newton's statement the expression
force on one object may be substituted for the word action, and
the expression equally large force on another object for the words
equal reaction. Read it over with this change.

The most startling idea to come out of this statement is that
forces always exist in mirror-twin pairs, and on two different
objects. Indeed, the idea of a single force unaccompanied by another
force acting somewhere else is without any meaning whatsoever.
On this point Newton wrote: "Whatever draws or presses another
is as much drawn or pressed by that other. If you press a stone
with your finger, the finger is also pressed by the stone." This
suggests that forces always arise as a result of interactions between
objects: object A pushes or pulls on B. while at the same time
object B pushes or pulls with precisely equal amount on A. These
paired pulls and pushes are always equal in magnitude, opposite in
direction, and on two different objects.

Applying this idea to the athlete, we now see that his act of
pushing his feet on the earth (one may call it here the action) is
accompanied by a push of the earth on him (one can call it the
reaction) -and the latter is what propels him forward. In this and
all other cases it really makes no difference which we call the action

Section 3.9

and which the reaction, because they occur at exactly the same
time. The action does not "cause" the reaction -if the earth could
not "push back" on his feet, the athlete could not push on the earth
in the first place, but would slide around — as on slippery ice.
Action and reaction coexist. You can't have one without the other.
And most important, the two forces are not acting on the same
body. In a way, they are like debt and credit: one is impossible
without the other; they are equally large but of opposite sign; and
they happen to two different objects.

Any body A that affects body B must itself be affected by B —
equally and oppositely. We can use the efficient shorthand of algebra
to express the idea that whenever bodies A and B interact:

f^AB — ~^BA

This is the equivalent of Newton's explanatory statement:
Whenever two bodies interact, the forces they exert on each other
are equal in magnitude and opposite in direction.

A host of everyday observations illustrate Newton's third law:
A boat is propelled by the water that pushes forward on the oar
while the oar pushes back on the water. A car is set in motion by
the push of the ground on the tires as they push back on the
ground; when friction is not sufficient, the tires cannot start the car
forward. While accelerating a bullet forward, a rifle experiences a
recoil kick. A balloon jumps forward while the air spurts out the
opposite direction. Many such effects are not easily observed; for
example, when an apple falls, pulled down by its weight, the earth
accelerates upward, pulled up by the attraction to the apple.

Now note what the third law does not say — this, too, is
important. The third law speaks of forces, not of the effects these
forces produce. Thus in the last example, the earth accelerates
upward as the apple falls down; the forces on each are equally
large, but the accelerations produced by the forces are quite
different; owing to the enormous mass of the earth, the earth's
upward acceleration is insensibly small. The third law also
does not describe how the push or pull is applied, whether by
contact or by magnetic action or by electrical action. Nor does the
law require that the force be either an attraction or repulsion. The
third law really does not depend on any particular kind of force. It
applies equally to resting objects and to moving objects, to
accelerating objects as well as to objects in uniform motion. It
applies whether or not there is friction present. Indeed, the
universality of the third law makes it extremely valuable throughout
physics.

In the collision between the ball and
the club, the force the ball exerts on
the club is equal and opposite to the
force the club exerts on the ball. Both
the club and the ball get defornned by
the forces acting on them.

Force on
bail due
V- to club

is equal and
opposite to

Force on
club due
to ball

foy-ce, on earih

force on moon

The force on the moon due to the
earth is equal and opposite to the
force on the earth due to the moon.

88 The Birth of Dynamics — Newton Explains Motion

Q18 According to Newton's third law, what are the four
general characteristics of forces?

Q19 Identify the forces that act according to Newton's third
law when a horse accelerates; when a swimmer moves at constant
speed.

Q20 A piece of fishing line breaks if the force exerted on it is
greater than 500 N. Will the line break if two people at opposite
ends of the line pull on it, each with a force of 300 N?

Q21 State Newton's three laws of motion as clearly as you can
SG 3.30, 3.31, 3.32 in your own words.

3.10 Using Newton's laws of motion

We have discussed each of Newton's three laws of motion in
some detail. The first law emphasizes the modem point of view in
the study of motion: What requires explanation is not motion itself,
but change of motion. The first law stresses that one must account
for why an object speeds up or slows down or changes direction. The
second law asserts that the rate of change of velocity of an object
is related to both the mass of the object and the net force applied to
it. In fact, the very meanings of force and mass are shown by the
second law to be closely related to each other. The third law is a
statement of a force relationship between interacting objects.

Despite their individual importance, Newton's three laws are
most powerful when they are used together. So successful was the
mechanics based on Newton's laws that until the late nineteenth
century it seemed that all of creation must be understood as
"matter in motion." Let us examine a specific example that
illustrates the use of these laws.

Example 1

On September 12, 1966, a dramatic experiment based on
Newton's second law was carried out high over the earth. In this
experiment, the mass of an orbiting Agena rocket case was
determined by accelerating it with a push from a Gemini spacecraft.
After the Gemini spacecraft made contact with the Agena rocket
case, the aft thrusters on the Gemini, calibrated to give an
average thrusting force of 890 N, were fired for 7.0 sec. The change
in velocity of the spacecraft and rocket case was found to be 0.93
m/sec. The mass of the Gemini spacecraft was known to be about
3400 kg. The question to be answered was: What is the mass of the
Agena?

(Actually, the mass of the Agena had already been measured
independently. The purpose of the experiment was to develop a
technique to find the unknown mass of a foreign satellite in orbit.)

Section 3.10

In this case, a known force of magnitude 890 N was acting on
two objects in contact, with a total mass of m,otai, where

Wtotal ~ mr

+ ni;

= 3400 kg + rriAgena

The magnitude of the average acceleration produced by the thrust
is found as follows:

At;
At

_ 0.93 m/sec
7.0 sec

= 0.13 m/sec2

Newton's second law gives us the relation

F = mtotai X «

= (mAgena + 3400 kg) X a

Solving for mAgena gives

F ^,^^, 890 N

mAgena--- 3400 kg =^^3^^^^^

= 6900 kg - 3400 kg
= 3500 kg

3400 kg

The actual mass of the Agena, as previously determined, was
about 3660 kg. The technique of finding the mass by nudging the
Agena while in orbit therefore gave a result that was accurate to
within 5% — well within the margin of error expected in making
this measurement.

The Birth of Dynamics— Newton Explains Motion

Example 2

Imagine taking a ride on an elevator: (A) At first it is at rest on
the ground floor; (B) it accelerates upward uniformly at Im/sec/sec
for a few seconds; then (C) continues to go up at a constant speed
of 5m/sec.

floor

^3»V

iFn*

l-jrW

rfttM-

FjMV

SG 3.33 is an elaboration of a
similar example. For a difficult
worked-out example see SG 3.34.

If a 100-kg man (whose weight would therefore be about 1000
newtons) is standing in the elevator, with what force is the
elevator floor pushing up on him during (A), (B), and (C)?

Parts (A) and (C) are dynamically the same: Since he is not
accelerating, the net force on him must be zero. So the floor must
be pushing up on him just as hard as gravity is pulling him down.
The gravitational force on him, his weight, is 1000 N. So the floor
must be exerting an upward force of 1000 newtons.

Part (B): Since the man is accelerating upward, there must be
a net force upward on him; the unbalanced force is

^net ~ TnUyjp

= 100 kg X 2 m/sec/sec
= 200 N

So the floor must be pushing up on him with a force 200 N greater
than what is required just to balance his weight; therefore, the total
force upward on him is 1200 N,

3.11 Nature's basic forces

Our study of Newton's laws of motion has increased our
understanding of objects at rest, moving uniformly, and
accelerating. However, we have accomplished much more in the
process. Newton's first law alerted us to the importance of frames

Section 3.11

of reference. A critical analysis of the relationship between
descriptions of the same event seen from different frames of
reference was in fact the necessary first step toward the theory of
relativity.

Newton's second law shows the fundamental importance of the
concept of force. In fact, it presents us with a mandate: when you
observe acceleration, find the force! This is how we were first
directed to the gravitational force as an explanation of Galileo's
kinematics: For all objects, at a given place, a*g is constant for all
objects; since Ug = Tglm by Newton's second law, we must conclude
that the magnitude of Tg is always proportional to m.

But this is only a halfway solution. Now we want to know why
Fg is proportional to m for all bodies at a given place and how Fg
changes for a given body as it is moved to places more distant from
the earth. Is there a law connecting Fg, m, and distance -a "force
law"? As Unit 2 will show, there is indeed. Knowing that force
law, we shall be able to claim to understand all gravitational
interactions among objects.

Gravitational attraction is not the only basic force by which
objects interact. However, it is satisfying to realize that there
appear to be very few such basic forces. In fact, physicists now
believe that everything we observe in nature is the consequence of
just four basic interactions. In terms of our present understanding,
all the events of nature — subnuclear and nuclear, atomic and
molecular, terrestrial and solar, galactic and extragalactic — are the
manifestations of one or more of these few types of forces.

There is, of course, nothing sacred about the number four. New
discoveries or theoretical insights might increase or reduce the
number. For example, two (or more) of the basic interactions might
some day be seen as consequences of something even more basic.

The first of the interactions is the gravitational force, which
becomes important only on a relatively large scale, that is, when
tremendous numbers of atoms of matter are involved. Between
individual atoms, gravitational force is so weak so to be
insignificant, but it is this weak force that literally holds the parts
of the universe together. The second interaction involves electric
and magnetic processes and is most important on the atomic and
molecular scale. It is electromagnetic force that holds together
objects in the range between the atom and the earth.

We know the force laws governing gravitational and electro-
magnetic interactions; therefore these interactions are fairly well
"understood." The situation changes completely when we consider
the two remaining basic interactions. They are the subject of
vigorous research today. The third interaction (the so-called
"strong" interaction) somehow holds the particles of the nucleus
together. The fourth interaction (the so-called "weak" interaction)
governs certain reactions among subnuclear particles.

We do, of course, have other names for forces, but each of these
belongs to one of the basic types. One of the most common is the
"frictional" force; it is thought to be an electrical interaction — that is,

Refer to K. Ford's: The World of
Elementary Particles lor brief
discussion of four forces.

Einstein spent most of the latter
half of his life seeking a theory that
would express gravitational and
electromagnetic effects in a unified
way. A satisfactory "unified field
theory" is still being sought.

The Birth of Dynamics- Newton Explains Motion

"The Starry Night, " by Vincent Van

Gogh.

The intuitive feeling that all of nature's

phenomena are interlinked on a grand

scale is shared by scientists as well

as artists.

the atoms on the surfaces of the objects shding or rubbing against
each other interact electrically.

We shall be encountering these ideas again. We shall deal with
the gravitational force in Unit 2, the electrical and magnetic forces
in Units 4 and 5, and the forces between nuclear particles in Unit 6.
In all these cases, an object subjected to the force will behave in
accordance with Newton's laws of motion.

The knowledge that there are so few basic interactions is both
surprising and encouraging. It is surprising because at first glance
the events all around us seem so varied and complex. It is
encouraging because our elusive goal — an understanding of the
events of nature — looks more attainable.

^^^

.<^.^v.

4P SW*^

STUDY GUIDE 3

3.1 The Project Physics learning materials
particularly appropriate for Chapter 3 include the
following :

Experiments

Newton's Second Law
Mass and Weight

Activities

Checker Snapping

Beaker and Hammer

Pulls and Jerks

Experiencing Newton's Second Law

Make One of These Accelerometers

Reader Articles

Introduction to Vectors

Newton's Laws of Dynamics

The Scientific Revolution

How the Scientific Revolution of the 17th

Century Affected Other Branches of

Thought

Film Loops

Vector Addition-

■ Velocity of a Boat

3.2 The Aristotelian explanation of motion should
not be dismissed lightly. Great intellects of the
Renaissance period, such as Leonardo da Vinci,
who among other things designed devices for
launching projectiles, did not challenge such
explanations. One reason for the longevity of
these ideas is that they are so closely aligned with
our common sense ideas.

In what ways do your common sense notions
of motion agree with the Aristotelian ones?

3.3 Three ants are struggling with a crumb. One
ant pulls toward the east with a force of 8 units.
Another pulls toward the north with a force of 6
units, and the third pulls in a direction 30° south
of west with a force of 12 units.

(a) Using the "head-to-taU" construction of
arrows, find whether the forces balance,
or whether there is a net (unbalanced)
force on the crumb.

(b) If there is a net force, you can find its
direction and magnitude by measuring
the line drawn from the tail of the first
arrow to the head of the last arrow. What
is its magnitude and direction?

3.4 Show why the parallelogram method of adding
arrows is geometrically equivalent to the head-to-
tail method.

3.5 There are many famihar situations in which
the net force on a body is zero, and yet the body
moves with a constant velocity. One example of
such "dynamic equilibrium" is an automobile
traveling at constant speed on a straight road: the
force the road exerts on the tires is just balanced
by the force of air friction. If the gas pedal is
depressed further, the tires will push against the
road harder and the road will push against the
tires harder; so the car will accelerate forward —
until the air friction builds up enough to balance

the greater drive force. Give another example of a
body moving with constant velocity under
balanced forces. Specify the source of each force
on the body and, as in the automobile example,
explain how these forces could be changed to
affect the body's motion.

3.6 (a) You exert a force on a box, but it does not

move. How would you explain this? How
might an Aristotelian explain it?

(b) Suppose now that you exert a greater
force and the box moves. Explain this
from your (Newtonian) point of view and
from an Aristotelian point of view.

(c) You stop pushing on the box and it
quickly comes to rest. Explain this from
both the Newtonian and the Aristotelian
points of view.

3.7 There are at least two drawbacks to an
experimental test of Newton's law of inertia.

(a) How can you really be sure that there is
no unbalanced force acting on the object,
even if you see that the object moves
uniformly in a straight line? We can
answer that we are sure because the
object does continue to move uniformly in
a straight line. But this answer is merely
a restatement of the first law, which we
wanted to prove by experiment. Surely we
cannot use the first law to verify the first
law! But we are not really caught in a
circular argument. Practically, we can
expect to find forces on an object only
when other objects are in contact with it,
or somewhere near it. The influences may
be of unfamiliar kinds, and we may have
to stretch what we mean by "near"; but
whenever a force is detected we look for
the source of the influence. If aU known
influences on an object were balanced,
and yet it didn't move uniformly, we
would suspect an unknown influence and
track it down — and we would find it. At
least, that's how it has always turned out
so far. As a practical example, consider
the demonstration involving low friction
pucks on a level surface. Without using
Newton's first law, how could you be sure
the surface was level?

(b) What is meant by a straight line?

3.8 (a) Assume that the floor of a laboratory

could be made perfectly horizontal and
perfectly smooth. A dry ice puck is placed
on the floor and given a small push.
Predict the way in which the puck would
move. How would this motion differ if the
whole laboratory were moving uniformly
during the experiment? How would it
differ if the whole laboratory were
accelerating along a straight line? If the
puck were seen to move in a curved path
along the floor, how would you explain this?
(b) A man gently starts a dry ice puck in
motion while both are on a rotating

STUDY GUIDE 3

platform. What will he report to be the
motion he observes as the platform keeps
rotating? How will he explain what he
sees if he believes he can use Newton's
first law to understand observations made
in a rotating reference frame? Will he
be right or wrong?

3.9 In terms of Newton's first law, explain:

(a) Why people in a moving car lurch forward
when the car suddenly slows down;

(b) What happens to the passengers of a car
that makes a sharp, quick turn;

(c) When a coin is put on a phonograph
turntable and the motor started, does the
coin fly off when the turntable reaches a
certain speed? Why doesn't it fly off
sooner?

3.10 A balloon-like object stands before you,
unmoving, suspended in mid-air. What can you
say about the forces that may be acting on it?
Suddenly it moves off in a curved path. Give two
diff'erent explanations. How can you test which is
right?

3. 11 In an actual experiment on applying the
same force to different masses, how would you
know it was the "same force"?

3.12 Several proportionalities can be combined
into an equation only if care is taken about the
units in which the factors are expressed. When
we wrote Ad = ii x At in Chapter 1, we chose
meters as units for d, seconds as units for t, and
then made sure that the equation came out right
by using meters/second as units for v. In other
words, we let the equation define the unit for v.
If we had already chosen some other units for v,
say miles per hour, then we would have had to
write instead something like

Ad= fe X vAt

where fe is a constant factor that matches up the
units of d, t, and v.

What value would k have if d were measured in
miles, t in seconds, and v in miles per hour?
Writing a = T„Jm before we have defined
units of F and m is not the very best mathematical
procedure. To be perfectly correct in expressing
Newton's law, we would have had to write:

- f

where fe is a constant factor that would match up
whatever units we choose for a, F, and m. In fact,
we will take the easiest way out and let the
equation define the units of F in terms of the
units we choose for a and m, so the equation
comes out right without using k. (Or if you prefer
to say it that way, we choose units so that k = 1.)

3.13 A body is being accelerated by an
unbalanced force. If the magnitude of the net
force is doubled and the mass of the body is
reduced to one-third of the original value, what

wUl be the ratio of the second acceleration to the
first?

3.14 What does a laboratory balance measure-
mass or weight? What about a spring balance?
(Hint: consider what would happen to readings on
each if they were on the moon instead of the
earth.) You might want to consider this question
again after reading Sec. 3.8.

3.1.) Describe as a thought experiment how you
could calibrate a spring balance in force units. If
you actually tried to do the experiments, what
practical difficulties would you expect?

3.16 "Hooke's law" says that the force exerted by
a stretched or compressed spring is directly
proportional to the amount of the compression or
extension. As Robert Hooke put it in announcing
his discovery:

. . . the power of any spring is in the same
proportion with the tension thereof: that
is, if one power stretch or bend it one
space, two will bend it two, three will
bend it three, and so forward. Now as the
theory is very short, so the way of trying
it is very easie.

If Hooke says it's "easie," then it might well
be so. You can probably think immediately of how
to test this law using springs and weights, (a) Try
designing such an experiment; then after
checking with your instructor, carry it out. What
limitations do you find to Hooke's law? (b) How
could you use Hooke's law to simplify the
calibration procedure asked for in SG 3.15?

3.17 Refer to the discussion in SG 3.12. Show
that fe = 1 when we define a newton as we do on
p. 83.

3.18 When units for different terms in a relation
are defined completely independently from one
another, the numerical value of the constant
must be found experimentally. (Later in this
course you will see how finding the value of k in
certain relations was very important in the
development of physics.) Say. for example, that
we had decided to measure force in "tugs."
defining a tug as the force required to stretch a
standard rubber band one inch. How could we go
about finding k?

STUDY GUIDE 3

3.19 Complete this table:

RESULTING

NET FORCE

MASS

ACCELERATION

a 1.0 N

1.0

1 .0 m/sec^

b 24.0

2.0

12.0

3.0

8.0

74.0

0.2

0.0066

130.0

f 72.0

8.0

g 3.6

12.0

h 1.3

6.4

i 30.0

10.0

j 0.5

0.20

k 120.0

48.0

being applied each time. Record the
position of the cart at equal time intervals
by means of stroboscopic photography.

(c) Repeat the last step in all details, but use
two carts hooked together. Repeat again
using all three carts hooked together. In
all three cases it is crucial that the applied
force be essentially the same.

(d) Determine the value of acceleration for
masses of m (1 cart), 2m (2 carts), and
3m (3 carts).

(e) Prepare a graph of a vs. m, of a vs. llm,
and of 1/a vs. m. Comment on your
results.

3.22 Describe in detail the steps you would take
in an idealized experiment to determine the
unknown mass m of a certain object (in
kilograms) if you were given nothing but a
frictionless horizontal plane, a 1-kg standard, an
uncalibrated spring balance, a meter stick, and a
stopwatch.

3.23 A block is dragged with constant velocity
along a rough horizontal table top by means of a
spring balance horizontally attached to the block.
The balance shows a reading of 0.40 N at this
and any other constant velocity. This means that
the retarding frictional force between block and
table is 0.40 N, and is not dependent on speed.

Now the block is pulled harder and given a
constant acceleration of 0.85 m/sec^; the balance
is found to read 2.1 N. Compute the mass of the
block.

3.20 A rocket-sled has a mass of 4440 kg and is
propelled by a solid-propellent rocket motor of
890,000-N thrust which bums for 3.9 seconds.

(a) What is the sled's average acceleration
and maximum speed?

(b) This sled has a maximum acceleration of
30 g (= 30 Ug). How can that be, considering
the data given?

(c) If the sled travels a distance of 1530 m
while attaining a top speed of 860 m/sec
(how did it attain that high a speed?),
what is its average acceleration?

3.21 If you have "dynamics carts" available,
here is one way of doing an experiment to
demonstrate the inverse proportionality between
acceleration and mass:

(a) Add load blocks to one or the other of two
carts until the carts balance when placed
on opposite platforms of a laboratory
balance. Balance a third cart with one of
the first pair. Each cart now has the
same mass m. (State two main
assumptions involved here.)

(b) Accelerate one cart on a level surface,
using a rubber band; that is, pull the cart
with the rubber band, keeping the rubber
band stretched a known constant amount
so that it supplies a constant force. Any
other method can be used that will assure
you that, within reason, the same force is

3.24 We have claimed that any body in free fall
is "weightless" because any weight-measuring
device falling with it would read zero. This is not
an entirely satisfactory explanation, because you
feel a definite sensation during free fall that is
exactly the same sensation you would feel if you
were truly without weight — say deep in space far
from any star or planet. (The sensation you feel
on jumping off a roof or a diving board, or when
someone pulls a chair out from under you.) Can
you explain why your insides react in the same
way to lack of weight and to free fall?

3.25 Explain the statement that while the mass
of an object is the same everywhere, its weight
may vary from place to place.

3.26 (a) A replica of the standard kilogram is

constructed in Paris and then sent to the
National Bureau of Standards near
Washington. D.C. Assuming that this
secondary standard is not damaged in
transit, what is

(i) its mass in Washington?
(ii) its weight in Paris and in Washington?
(In Paris, a, = 9.81 m/sec^; in
Washington, a„= 9.80 m/sec*.)

(b) What is the change in your own weight as
you go from Paris to Washington?

STUDY GUIDE 3

3.27 (a) Find your mass in kg, and your weight

in newtons.
(b) How much force is needed to accelerate
you 1 m/sec^? How many kilograms can
you lift? How many newtons of force
must you exert to do this?

3.28 Why is it often said that astronauts in orbit
are weightless?

3.29 When a runner pushes on the earth with the
sole of his shoe, the earth pushes with an equal
and opposite force on the sole of the shoe. This
latter force has an accelerating effect on the
runner, but what does the force acting on the
earth do to the earth? From Newton's second law
we would conclude that such an unbcdanced
force would accelerate the earth. The mass of the
earth is very great, however, so the acceleration
caused by the runner is very small. A reasonable
value for the average acceleration of a runner
when he starts is 5 m/sec/sec, and a reasonable
value for his mass would be 60 kg. The mass of
the earth is approximately 60 x 10" kg.

(a) What acceleration of the earth would the
runner cause?

(b) If the acceleration lasts for 2 seconds,
what speed will the runner have reached?

3.30 In terms of Newton's third law, assess the
following statements:

(a) You are standing perfectly still on the
ground; therefore you and the earth exert
equal and opposite forces on each other.

(b) The reason that a propeller airplane
cannot fly above the atmosphere is that
there is no air to push one way while the
plane goes the other.

(c) Object A rests on object B. The mass of
object A is 100 times as great as that of
object B, but even so, the force A exerts
on B is no greater than the force of B
on A.

3.31 Consider a tractor pulling a heavy log in a
straight line. On the basis of Newton's third law,
one might argue that the log pulls back on the
tractor just as strongly as the tractor pulls the
log. But why, then, does the tractor move? (Make
a large drawing of the tractor, rope, log, and earth,
and enter the forces.)

^~^^^^Hh

^^^mmn^i^^^;^^^

Bk-^;^-

HHHHHHHJIHJR

ii^^^Bk-.* " '* -*

3.32 Consider the system consisting of a 1.0-kg
ball and the earth. The ball is dropped from a
short distance above the ground and falls freely.
Assuming that the mass of the earth is
approximately 6.0 x 10" kg,

(a) make a vector diagram illustrating the
important forces acting on each member
of the system.

(b) calculate the acceleration of the earth in
this interaction.

(d) make a vector diagram as in (a) but
showing the situation when the ball has
come to rest after hitting the ground.

3.33 (a) A 75-kg man stands in an elevator. What

force does the floor exert on him when the
elevator

(i) starts moving upward with an

acceleration of 1.5 m/sec^?
(ii) moves upward with a constant speed

of 2.0 m/sec?
(iii) starts accelerating downward at 1.5
m/sec^?

(b) If the man were standing on a bathroom
(spring) scale during his ride, what
readings would the scale have under
conditions (i), (ii), and (iii) above?

(c) It is sometimes said that the "apparent
weight" changes when the elevator
accelerates. What could this mean? Does
the weight really change?

3.34 Useful hints for solving problems about the
motion of an object and the forces acting on it.

(a) make a light sketch of the physical
situation.

(b) in heavy line, indicate the limits of

the particular object you are interested in,
and draw all the forces acting on that
object. (For each force acting on it, it will
be exerting an opposite force on something
else — but we don't care about those.)

(d) using Newton's second law, set this sum,
F„e„ equal to ma.

(e) solve the equation for the unknown
quantity.

(f ) put in the numerical values you know and
calculate the answer.

Example:

A ketchup bottle whose mass is 1.0 kg rests
on a table. If the friction force between the table
and the bottle is a constant 6 newtons, what
horizontal pull is required to accelerate the bottle
from rest to a speed of 6 m/sec in 2 sec?

First, sketch the situation:

STUDY GUIDE 3

Second, draw in arrows to represent all the forces
acting on the object of interest. There will be
the horizontal pull Fp, the friction Pf, the
gravitational pull Fg (the bottle's weight), and the
upward force Ft exerted by the table. (There is,
of course, also a force acting down on the table,
but we don't care about that — we're interested
only in the forces acting on the bottle.)

Next, draw the arrows alone. In this sketch all
the forces can be considered to be acting on the
center of mass of the object.

The mass m is given as 1.0 kg. The acceleration
involved in going from rest to 6.0 m/sec in 2
seconds is

Ai;

" At

6.0 m/sec

2 sec

= 3.0 m/sec/sec

So the

net

force required is

Fne, = 1.0 kg X 3.0 m/sec/sec

= 3.0 kg m/sec/sec

= 3.0 newtons

If we consider toward the right to be the
positive direction, Fnet is 3.0 newtons and Ff, which
is directed to the left, is —3.0 newtons.

Fnet = Fp + F/

3.0N = Fp + (-3.0N)
fp = 3.0N + 3.0N
fp = 6.0N

If you prefer not to use + and — signs, you can
work directly from your final diagram and use
only the magnitudes of the forces:

3.0>; 3. ON

Because the bottle is not accelerating up or
down, we know there is no net force up or down —
so fr must just balance fg. So the net force
acting on the bottle is just the vector sum of fp
and ff. Using the usual tip- to- tail addition:

from which the magnitude of Fp is obviously 6. ON.

As the last arrow diagram shows, the horizontal
pull must be greater than the force required for
acceleration by an amount equal to the friction.
We already know ff. We can find Fnet from
Newton's second law if we know the mass and
acceleration of the bottle, since Fnet = ^«- The net
force required to accelerate the case is found
from Newton's second law:

4.1 A trip to the moon

4.2 Projectile motion

4.3 What is the path of a projectile?

4.4 Moving frames of reference

4.5 Circular motion

4.6 Centripetal acceleration and centripetal force

4.7 The motion of earth satellites

4.8 What about other motions?
Epilogue

99
101
103
105
107
109
113
116
118

". . . the greater the velocity . . . with
which [a stone] is projected, the
farther it goes before it falls to the
earth. We may therefore suppose the
velocity to be so increased, that it
would describe an arc of 1, 2, 5, 10,
100, 1000 miles before it arrived at the
earth, till at last, exceeding the limits
of the earth, it should pass into space
without touching it." — Newton's Sys-
tem of the World

CHAPTER FOUR

Understanding Motion

4.1 A trip to the moon

Imagine a Saturn rocket taking off from its launching pad at
Cape Kennedy. It climbs above the earth, passing through the
atmosphere and beyond. Successive stages of the rocket shut off
leaving finally a capsule hurtling through the near-vacuum of space
toward its destination 240,000 miles away. Approximately 65 hours
after take-off, the capsule circles the moon and descends to its
target— the center of the lunar crater Copernicus.

The complexity of such a voyage is enormous. To direct and
guide the flight, a great number and variety of factors must be
taken into account. The atmospheric drag in the early part of the
flight depends upon the rocket's speed and altitude. The engine
thrust changes with time. The gravitational pulls of the sun, the
earth, and the moon change as the capsule changes its position
relative to them. The rocket's mass is changing. Moreover, it is
launched from a spinning earth, which in turn is circling the sun,
and the target — the moon — is moving around the earth at a speed
of about 2,300 miles per hour.

Yet, as for almost any complex motion, the flight can be broken
down into small portions, each of which is relatively simple to
describe. What we have learned in earlier chapters will be useful in
this task.

In simplified form, the earth-moon trip can be divided into
these eight parts:

Part 1. The rocket accelerates vertically upward from the

surface of the earth. The force acting on the rocket is
not really constant, and the mass of the rocket
decreases as the propellent escapes. The value of the
acceleration at any instant can be computed using
Newton's second law; it is given by the ratio of net
force (thrust minus weight) at that instant to the mass
at that instant.

Part 2. The rocket, still accelerating, follows a curved path as
it is "injected" into an orbit about the earth.

In his science-fiction novels of more
than a hundred years ago, the French
author Jules Verne (1828-1905)
launched three spacemen to the
moon by means of a gigantic charge
fixed in a steel pipe deep in the earth.

SG4.1

100

Understanding Motion

"^1 ^n ■ ■■^.^^

'^■'K-'^

Part 3. In an orbit 115 miles above the earth's surface, the
capsule moves in a nearly circular arc at a constant
speed of 17,380 miles/hr.

Part 4. The rocket engines are fired again, increasing the

capsule's speed so that it follows a much less curved
path into space. (The minimum speed necessary to
escape the earth completely is 24,670 miles/hr.)

Part 5. In the flight between earth and moon, only occasional
bursts from the capsule's rockets are required to
keep it precisely on course. Between these
correction thrusts, the capsule moves under the
influence of the gravitational forces of earth, moon,
and sun; we know from Newton's first law that the
capsule would move with constant velocity if it were
not for these forces.

Part 6. On nearing the moon, the rocket engines are fired again
to give the capsule the correct velocity to "inject" into
a circular orbit around the moon.

Part 7. The capsule is moving with a constant speed of about 1
mile/sec in a nearly circular path 50 miles above the
moon's surface.

Part 8. After its rockets are fired in the direction of motion to
reduce the speed, the capsule accelerates downward
as it falls toward the surface of the moon. It follows
an arcing path before it lands in the crater Copernicus.
(Just before impact, the rocket engines fire a final time
to reduce speed of fall and prevent a hard landing.)

SG 4.2 Motion along a straight line (as in Parts 1 and 5) is easy

enough to describe. But let us analyze in greater detail other parts
of this trip: moving on a circular arc, as in Parts 3 and 7, and
projectile motion, as in Part 8, are two important cases.

How shall we go about making this analysis? Following the
example of Galileo and Newton, we can try to learn about the
behavior or moving objects beyond our reach, even on the moon or
in the farthest parts of the universe, by studying the motion of
objects near at hand. If we believe that physics is the same
everywhere, then the path of a lunar capsule moving as in Part 8
can be understood by studying a marble rolling off" the edge of a
table or a bullet fired from a horizontal rifle.

Section 4.2

4.2 Projectile motion

101

Consider this experiment: a rifle is mounted on a tower with its
barrel parallel to the ground; the ground over which the bullet will
travel is level for a very great distance. At the instant a bullet
leaves the rifle, an identical bullet is dropped from the height of the
barrel of the rifle. The second bullet has no horizontal motion
relative to the ground; it goes only straight down. Which bullet will
reach the ground first?

We do not need to know anything about the speed of the bullet
or the height of the tower in order to answer this question.
Consider first the motion of the second bullet, the one that is dropped.
As a freely falling object, it accelerates toward the ground with
constant acceleration. As it falls, the time t and the corresponding
downward displacement y are related by

y = iagt^

where Ug is the acceleration due to gravity at that location.

Now consider the bullet that is fired horizontally from the rifle.
When the gun is fired, the bullet is driven by the force of expanding
gases and accelerates very rapidly until it reaches the muzzle of the
rifle. On reaching the muzzle these gases escape and no longer
push the bullet. At that moment, however, the bullet has a large
horizontal speed, Vj,. The air will slow the bullet slightly, but we
shall ignore that fact and imagine an ideal case with no air
friction. As long as air friction is ignored, there is no force acting
on the projectile in the horizontal direction. Therefore, we expect

the horizontal speed will remain constant. From the instant the
bullet leaves the muzzle, we would expect its horizontal motion to
be described by the equation

X = Vj.t

So much for the forward part of the motion. There is, however,
another part that becomes more and more important as t increases.
From the moment the bullet leaves the gun, it falls toward the
earth while it moves forward, like any other unsupported body.
Can we use the same equation to describe its fall that we used to
describe the fall of the dropped bullets? And how will falling aff"ect
the bullet's horizontal motion? These doubts raise a more

J L

-riJSl^l!;.

102

Understanding Motion

The two balls in this strobscopic
photograph were released simultane-
ously. The one on the left was simply
dropped from rest position; the one
on the right was given an initial veloc-
ity in the horizontal direction.

SG 4.3

SG 4.4

fundamental question that goes beyond just the behavior of the
bullets; namely, is the vertical motion of an object affected by its
horizontal motion? Or vice versa?

To answer these questions, we can carry out a real experiment
similar to our thought experiment. We can use a special laboratory
device designed to fire a ball in a horizontal direction at the
moment that a second ball is released to fall freely from the same
height. We set up our apparatus so that both balls are the same
height above a level floor. The balls are released and, although the
motions of the balls may be too rapid for us to follow with the eye,
we will hear that they reach the floor at the same time. This result
suggests that the vertical motion of the projected ball is unaffected
by its horizontal velocity.

In the margin is a stroboscopic photograph of this experiment.
Equally spaced horizontal lines aid our examination. Look first at
the ball on the left, which was released without any horizontal
motion. You see that it accelerates because it moves greater
distances between successive flashes. Careful measurement of the
photograph shows that the acceleration is constant, within the
uncertainty of our measurements.

Now compare the vertical positions of the second ball, fired to
the right, with the vertical positions of the ball which is falling
freely. The horizontal lines show that the distances of fall are the
same for corresponding time intervals. The two balls obey the same
law for motion in a vertical direction. That is, at every instant they
both have the same constant acceleration a^,, the same downward
velocity and the same vertical displacement. The experiment
therefore supports the idea that the vertical motion is the same
whether or not the ball has a horizontal motion also. The horizontal
motion does not aff'ect the vertical motion.

We can also use the strobe photo to see if the vertical motion of
the projectile aff"ects its horizontal velocity, by measuring the
horizontal distance between successive images. We find that the
horizontal distances are practically equal. Since the time intervals
between images are equal, we can conclude that the horizontal
velocity v^ is constant. So we can conclude that the vertical motion
doesn't aff'ect the horizontal motion.

The experiment shows that the vertical and horizontal
componerits of the motion are independent of each other. This
experiment can be repeated from diff"erent heights, and with
diff"erent horizontal velocities, but the results lead to the same
conclusion.

The independence of motions at right angles has important
consequences. For example, it is easy to predict the displacement
and the velocity of a projectile at any time during its flight. We need
merely to consider the horizontal and vertical aspects of the
motion separately, and then add the results -vectorially. We can
predict the magnitude of the components of displacement (x and y)
and of the components of velocity (Vj. and v„) at any instant by
application of the appropriate equations. For the horizontal

Section 4.3

103

component of motion, the equations are

Vx = constant
and

X= Vjct

and for the vertical component of motion,

Vy^ Ugt

and

y = ^agt^

Q1 If a body falls from rest with acceleration Og, with what
acceleration will it fall if it has an initial horizontal speed Vx?

4.3 What is the path of a projectile?

It is easy to see that a thrown object, such as a rock, follows a
curved path, but it is not so easy to see just what kind of curve it
traces. For example, arcs of circles, ellipses, parabolas, hyperbolas,
and cycloids (to name only a few geometric figures) all provide
likely-looking curved paths.

Better knowledge about the path of a projectile was gained
when mathematics was applied to the problem. This was done by
deriving the equation that expresses the shape of the path. Only a
few steps are involved. First let us hst equations we already know
for a projectile launched horizontally:

X = v„t

and

y = Jttgt^

We would know the shape of the trajectory if we had an equation
that gave the value of y for each value of x. We can find the fall
distance y for any horizontal distance x by combining these two
equations in a way that eliminates the time variable. Solving the
equation x = v^t for t we get

_ ^

Because t means the same in both equations, we can substitute
xlvx for t in the equation for y:

— 20.gt

and thus

^"^(t)

In this last equation there are two variables of interest, x and y,
and three constant quantities: the number y, the uniform
acceleration of free fall a«, and the horizontal speed Vx which we

Specialized equations such as
these need not be memorized.

104

Understanding Motion

* t

r c i

b A

^.--^

5
[

tlr

Drawing of a parabolic trajectory from
Galileo's Two New Sciences.

SG 4.5
SG 4.6
SG4.7

"Philosophy is written in this grand
book, the universe, which stands
continually open to our gaze. But
the book cannot be understood
unless one first learns to
comprehend the language and read
the letters in which it is composed.
It is written in the language of
mathematics, and its characters
are triangles, circles, and other
geometric figures, without which it
is humanly impossible to understand
a single word of it." (Galileo, cited
in Discoveries and Opinions of
Galileo, translated by Stillman Drake,
Anchor Books, pp. 237-238.)

take to be constant for any one flight from launching to the end.
Bringing these constants together between one set of parentheses, we
can write the equation as

or, letting k stand for constant (agl2v/)

y = kx^

This equation shows a fairly simple relationship between x and y
for the trajectory. We can translate it as: the distance a projectile
falls away from a straight path is proportional to the square of the
distance it moves sideways. For example, when the projectile goes
twice as far horizontally from the launching point, it drops
vertically four times as far.

The mathematical curve represented by this relationship
between x and y is called a parabola. Galileo deduced the parabolic
shape of trajectories by an argument similar to the one we used.
(Even projectiles not launched horizontally — as in the photographs
on p. 103 and 123 — have parabolic trajectories.) With this
discovery, the study of projectile motion became much simpler,
because the geometric properties of the parabola had been
established centuries earlier by Greek mathematicians.

Here we find a clue to one of the important strategies in
modem science. When we express the features of a phenomenon
quantitatively and cast the relations between them into equation
form, we can use the rules of mathematics to manipulate the
equations, and so open the way to unexpected insights.

Galileo insisted that ''the proper language of nature is
mathematics," and that an understanding of natural phenomena is
aided by translating our qualitative experiences into quantitative
terms. If, for example, we find that trajectories have a parabolic
shape, we can apply all we know about the mathematics of
parabolas to describe — and predict — trajectories. Physicists have
often drawn on the previously developed parts of pure mathematics
to express (or to extend) their conceptions of natural
phenomena. Sometimes, as in the case of Newton's inventing
calculus, they have to develop new parts of mathematics. The
physical scientist often tries to use methods from another branch of
science, in addition to mathematics, to find a solution for his
particular problem. For example, just as Galileo used the already-
known mathematics of parabolas to deal with actual projectile
motions, so the modern sound engineer solves problems in acoustics
using ideas and mathematical techniques developed independently
by electrical engineers. Whatever the methods of science may be,
many ideas and concepts can often be extended from one specialty
to another, with fruitful results.

We can now apply our theory of projectile motion to the case
mentioned earlier, the free motion of a space capsule toward the

Section 4.4

105

moon's surface. Let us assume that the orbit is a low one, so that
the acceleration due to gravity is almost constant between the
orbit and the surface. If the rocket engines are fired forward, in
the direction of motion, the capsule's speed will be reduced and
it will begin to fall closer to the surface. After firing, the reduced
horizontal speed remains constant, so the capsule falls toward
the surface on a parabolic path. Spaceflight engineers apply
ideas like these to land a space capsule on a desired moon target.
(See SG 4.23).

Q2 Which of the conditions below must hold in order for the
relationship y = kx- to describe the path of a projectile?

(a) Ug is a constant

(b) Ug depends on t

(d) Vj. depends on t

(e) air friction is negligible

4.4 Moving frames of reference

stationary
earth

moving
earth

The critics of Galileo claimed that if
the earth moved, a dropped stone
would be left behind and land beyond
the foot of the tower.

Galileo's work on projectiles leads to thinking about reference
frames. As you will see in Unit 2, Galileo ardently supported the
idea that the preferred reference frame for discussing motions in
our planetary system is one fixed to the sun, not the earth. From
that point of view, the earth both revolves around the sun and
rotates on its own axis. For many scientists of Galileo's time, this
idea was impossible to accept, and they thought they could prove
their case. If the earth rotated, they said, a stone dropped from a
tower would not land directly at its base. For if the earth rotates
once a day, the tower would move on for hundreds of feet for every
second the stone is falling; hence, the stone would be left behind
while falling through the air and consequently would land far
behind the base of the tower. But this is not what happens. As near
as one can tell, the stone lands directly under where it was
released. Therefore, many of Galileo's critics believed that the tower
and the earth could not be considered to be in motion.

To answer these arguments, Galileo showed the same
observation can support his view that, during the time of fall, the
tower and the ground supporting it were moving forward together
with the same uniform velocity. While the stone was being held at
the top of the tower, it had the same horizontal velocity as the tower.
Releasing the stone allows it to gain vertical speed, but by the
principle of independence of Vj. and Vy discussed in Section 4.3, this
does not diminish any horizontal speed it had initially on being
released. In other words, the falling stone behaves like any other
projectile: the horizontal and vertical components of its motion are
independent of each other. Since the stone and tower continue to
have the same Vj. throughout, the stone will not be left behind as it
falls. Therefore, no matter what the speed of the earth, the stone

stationary
earth

moving
earth

Galileo argues that the falling stone
continued to share the motion of the
earth, so that an observer on earth
could not tell whether or not the earth
moved by watching the stone.

106

Understanding Motion

At high speeds, air drag will
affect the results considerably. The
situation is still indistinguishable
from a car at rest— but in a high
wind!

When relative speeds become a
noticeable fraction of the speed
of light (almost a billion mph),
some deviations from this simple
relativity principle begin to appear.
We will consider some of them in
Unit 5.

SG 4.8, 4.9, 4.10

will land at the foot of the tower. The fact that falling stones are
not left behind is not a proof that the earth is standing stiU.

Similarly, Galileo said, an object released from a crow's nest at
the top of a ship's perpendicular mast will land at the foot of the
mast, whether the ship is standing still in the harbor or moving
with constant velocity through quiet water. This was actually tested
by experiment in 1642 (and is also the subject of three Project
Physics film loops). We know this to be the case from everyday
observation: when you drop or throw a book in a bus or train or
plane that is moving with constant velocity, you will see it moving
just as it would if the vehicle were standing still. Or again, if an
object is projected vertically upward from inside an open car that is
moving at constant velocity, it will fall back into the car. A person
in the car will see the same thing happen whether the car has
been continuously moving at constant velocity or has been standing
still.

From these and other observations has come a valuable
generalization: If there is any one laboratory in which Newton's
laws hold, then these laws will hold equally well in any other lab
(or "reference frame") that moves at constant velocity with
respect to the first. This generalization is called the Galilean
relativity principle. It holds true for all "classical" mechanical
phenomena — that is, phenomena involving a tremendous range of
relative velocities, up to millions of miles per hour.

If the laws of mechanics are found to be the same for all
reference frames moving with constant velocity with respect to
each other, then there is no way to find the speed of one's own
reference frame from any mechanical experiment done in the
reference frame, nor can one pick out any one reference frame as
the "true" frame — the one that is, say, at absolute rest. Thus
there can be no such thing as the "absolute" velocity of a body —
all measured velocities are only relative.

What about observations of phenomena outside of one's own
frame of reference? Certainly some outside phenomena can appear
differently to observers in different reference frames — for example,
the velocity of an airplane will have a different value when seen
from the earth and from a moving ship. But other measurables such
as mass, acceleration, and time interval will have the same values
when a phenomenon is observed from different reference frames
that move with constant velocity with respect to one another. More-
over, certain relationships among such measurements will be
found to be the same for these different reference frames. Newton's
laws of motion are examples of such "invariant" relationships, and
so are all the laws of mechanics that follow from them.

Notice that the relativity principle, even in this restricted
classical form, does not say "everything is relative." On the
contrary, it asks us to look for relationships that do not change
when reference systems are changed.

Section 4.5

107

Q3 If the laws of mechanics are found to be the same in two
reference frames, what must be true of their motions?

4.5 Circular motion

A projectile launched horizontally from a tall tower strikes the
earth at a point determined by the speed of the projectile, the
height of the tower, and the acceleration due to the force of gravity.
As the projectile's launch speed is increased, it strikes the earth at
points farther and farther from the tower's base, and we would
have to take into account that the earth is not flat but curved. If we
suppose the launch speed to be increased even more, the projectile
would strike the earth at points even farther from the tower, till at
last it would rush around the earth in a nearly circular orbit. At this
orbiting speed, the fall of the projectile away from the forward,
straight line motion is matched by the curvature of the surface, and
it stays at a constant distance above the surface.

What horizontal launch speed is required to put an object into
a circular orbit about the earth or the moon? We shall be able to
answer this question quite easily after we have learned about
circular motion.

The simplest kind of circular motion is uniform circular motion,
that is, motion in a circle at constant speed. If you are in a car or
train that goes around a perfectly circular track so that at every
instant the speedometer reading is forty miles per hour, you are
executing uniform circular motion. But this is not the case if the
track is any shape other than circular, or if your speed changes at
any point.

How can we find out if an object in circular motion is moving
at constant speed? The answer is to apply the same test we used in
deciding whether or not an object traveling in a straight line does
so with constant speed: we measure the instantaneous speed at
many different moments and see whether the values are the same.
If the speed is constant, we can describe the circular motion of the
object by means of two numbers: the radius R of the circle and
the speed v along the path. For regularly repeated circular motion,
we can use a quantity more easily measured than speed: either the
time required by an object to make one complete revolution, or the
number of revolutions the object completes in a unit of time. The
time required for an object to complete one revolution in a circular
path is called the period of the motion. The period is usuaUy
denoted by the capital letter T. The number of revolutions completed
by the same object in a unit time interval is called the frequency
of the motion. Frequency will be denoted by the letter/.

As an example, we will use these terms to describe a car moving
with uniform speed on a circular track. Let us suppose the car takes
20 seconds to make one lap around the track. Thus, T= 20 seconds.
Alternatively, we might say that the car makes 3 laps in a minute.

In discussing circular motion it is
useful to keep clearly in mind a dis-
tinction between revolution and
rotation. We define these terms dif-
ferently: revolution is the act of
traveling along a circular or elliptical
path; rotation is the act of spinning
rather than traveling. A point on the
rim of a phonograph turntable travels
a long way; it is revolving about the
axis of the turntable. But the turn-
table as a unit does not move from
place to place: it merely rotates. In
some situations both processes
occur simultaneously; for example,
the earth rotates about its own axis,
while it also revolves (in a nearly
circular path) around the sun.

108

Understanding Motion

SG 4.11

The term "revolutions" is not
assigned any units because it is a
pure number, a count. There is no
need for a standard as there is for
distance, mass, and time. So, the
unit for frequency is usually given
without "rev." This looks strange,
but one gets used to it— and it is not
very important, because it is merely
a matter of terminology, not a fact
of physics.

Thus/= 3 revolutions per minute, or/= 1/20 revolution per second.
The relationship between frequency and period (when the same
time unit is used) is/= 1/T. If the period of the car is 20 sec/rev,
then the frequency is

1 1 rev

20 sec

sec
rev

All units are a matter of convenience. Radius may be expressed
in terms of centimeters, kilometers, miles, or any other distance
unit. Period may be expressed in seconds, minutes, years, or any
other time unit. Correspondingly, the frequency may be expressed
as "per second," "per minute," or "per year." The most widely used
units of radius, period, and frequency in scientific work are meter,
second, and per second.

Table 4.1 Comparison of the frequency and period for various kinds of
circular motion. Note the differences between units.

PHENOMENA

PERIOD

FREOUENCY

Electron in circular accelerator

io-«

sec

10«

per sec

Ultra-centrifuge

0.00033

sec

3000

per sec

Hoover Dam turbine

.33

sec

per sec

Rotation of earth

hours

0.0007

per min

Moon around the earth

days

0.0015

per hour

Earth about the sun

365

days

0.0027

per day

If an object is in uniform circular motion, and if we know the
frequency of revolution / and the radius R of the path, we can
compute the speed v of the object without difficulty. The distance
traveled in one revolution is simply the perimeter of the circular
path, that is, 2ttR. The time for one revolution is by definition the
period T. Since for uniform motion it is always true that

speed =
by substitution we get

distance traveled
time elapsed

27rR

To express this equation for circular motion in terms of the
frequency/, wejrewrite it as

V = 2ttR X j^

now, since by definition

J i-p

we can write

V = 2ttR X /
If the body is in uniform, circular motion, the speed computed

Section 4.6

109

with the aid of this equation is both its instantaneous speed and its
average speed. If the motion is not uniform, the formula gives only
the average speed; the instantaneous speed for any point on the
circle can be determined if we find Ad/At from measurements of
very small segments of the path.

Let us now see how the last equation can be used. We can, for
example, calculate the speed of the tip of a helicopter rotor blade in
its motion around the central shaft. On one model, the main rotor
has a diameter of 7.50 m and a frequency of 480 revolutions/minute
under standard conditions. Thus/= 480 per minute = 8.00 per
second and R = 3.75 m, and

V = 27rRf

v = 2 (3.14)(3.75)(8.00) meters/second

V = 189 m/sec

or about 420 miles/hr.

Q4 If a phonograph turntable is running at 45 rev.olutions per
minute,

(a) What is its period (in minutes)?

(b) What is its period (in seconds)?

Q5 What is the period of the minute hand of an ordinary clock?
If the hand is 3.0 cm long, what is the linear speed of the tip of the
minute hand?

Q6 The terms frequency and period can also be used for any
other periodic, repetitive phenomenon. For example, if your heart
beats 80 times per minute, what are the frequency and period for
your pulse?

4.6 Centripetal acceleration and centripetal force

Let us assume that a stone on a string is moving with uniform
circular motion, for example in a horizontal plane as the stone is
whirled overhead. The speed of the stone is constant. The velocity,
however, is always changing. Velocity is a vector quantity, which
includes both speed and direction. Up to this point we have dealt
with accelerations in which only the speed was changing. In
uniform circular motion the speed of the revolving object remains
the same, while the direction of motion changes continually. The
figure shows the whirling stone at three successive moments in its
revolution. At any instant, the direction of the velocity vector is
tangent to the curving path. Notice that its speed, represented by
the length of the velocity arrow, does not vary; but its direction
changes from moment to moment. Since acceleration is defined
as a change in velocity, the stone is in fact accelerating.

But to produce an acceleration a net force is needed. In the case
of the whirling stone, a force is exerted on the stone by the string,
and if we neglect the weight of the stone or air resistance, that

SG 4.12 a to f

a'r and K are parallel, but iTis perpen-
dicular to a^. and T,.. Note that usually
one should not draw different kinds of
vector quantities on the same drawing.

110

Understanding Motion

The adjective centripetal means
literally "moving, or directed,
toward the center."

In uniform circular motion, the
instantaneous velocity and the
centripetal force at any instant of
time are perpendicular, one being
along the tangent, the other along
the radius. So instantaneous velocity
and the acceleration are also
always at right angles.

will be the net force. If the string were suddenly cut, the stone
would go flying off on a tangent with the velocity it had at the
instant the string was cut — on a tangent to the circular path. As
long as the string holds, the stone is forced into a circular path.

The direction of this force acting on the stone is along the
string. Thus the force vector is always pointing toward the center of
rotation. This kind of force — always directed toward the center of
rotation — is called centripetal force.

From Newton's second law we know that force and
acceleration are in the same direction, so the acceleration vector is
also directed toward the center. We shall call this acceleration
centripetal acceleration, and give it the symbol ap. Any object
moving along a circular path has a centripetal acceleration.

We know now the direction of centripetal acceleration. What is
its magnitude? An expression for Op can be derived from the
definition of acceleration Op = Ax;/At. The details of such a derivation
are given on the next page. The result shows that a<, depends on v
and R, and in fact the magnitude of ac is given by

^^=R
Let us verify this relationship with a numerical example. If, as
sketched in the diagram, a car goes around a circular curve of
radius R = 100 m at a uniform speed of i; = 20 m/sec, what is its
centripetal acceleration a^ toward the center of curvature? By the
equation derived on the gray page:

(2oi^r

_ V sec/
100 m

jnf_
400 sec'
~ 100 m

This is about 4/10 of a„, and could
be called an acceleration of "0.4g."

= 4.0

m
sec^

Derivation of the equation Sc = -^

Assume the stone is moving uniformly in a circle of radius R.
We can find what the relationship between ac, v, and R is by treating
a small part of the circular path as the combination of a tangential
motion and an acceleration toward the center. To follow the circular
path, the stone must accelerate toward the center through a
distance h in the same time that it would move through a tangential
distance d. The stone, with speed v, would travel a tangential
distance d given by d = vM. In the same time At, the stone, with
acceleration a^ would travel toward the center through a distance h
given hy h = ^UcM^. (We can use this last equation because at
t = 0, the stone's velocity toward the center is zero.)

We can now apply the Pythagorean Theorem to the triangle in
the figure at the right.

R2 + ^2 = (R + h7

= R^ + 2Rh + h^

When we subtract R^ from each side of the equation we are left
with

d2 = 2Rh + h^

We can simphfy this expression by making an approximation: since
h is very small compared to R, h^ will be very small compared to Rh.
If we choose At to be vanishingly small (as we must to get the
instantaneous acceleration), h^ will become vanishingly small
compared to Rh; so we shall neglect h^ and write

d^ = 2Rh

Also, we know d = t;Af and h = ia^At^, so we can substitute for d^
and for h accordingly. Thus

(vMy = 2R • jUciMy
vKMy = RadMY

ar = -7r

The approximation becomes better and better as At becomes
smaller and smaller. In other words. v^lR is the magnitude of the
instantaneous centripetal acceleration for a body moving on a
circular arc of radius R. For uniform circular motion, V'^IR is the
magnitude of the centripetal acceleration at every point of the path.
(Of course it does not have to be a stone on a string. It can be a
point particle on the rim of a rotating wheel, or a house on the
rotating earth, or a coin sitting on a rotating phonograph disk, or a
car in a curve on the road.)

112 Understanding Motion

^^- -^-^^ Does this make sense? We can check the result by going back

'^ ^■^<\2.0m to the basic vector definition of acceleration: a„,. = AvIAt. We will

/ V make a scale drawing of the car's velocity vector at two instants a

\ short time At apart, measure the change in velocity Av between

.,-,i„, i.,, o,.n 1^ '' \ them, and divide the magnitude of Av by At to get a„,. over the

1fnm = Imiy'hr ' ^ interval.

! Consider a time interval of At = 1 second. Since the car is

moving at 20 m/sec, its position will change 20 m during At. Two
positions P and P', separated by 20 m, are marked in diagram B.
/ Now draw arrows representing velocity vectors. If we choose a

/ scale of 1 cm = 10 m/sec, the velocity vector for the car will be

represented by an arrow 2 cm long. These are drawn at P and P' in
diagram C.

If we put these two arrows together tail to tail as in diagram
D, it is easy to see what the change in the velocity vector has been
during At. Notice that if Ai; were drawn halfway between P and P',
it would point directly toward the center of the curve; so the
average acceleration between P and P' is indeed directed
centripetally. Measurement of the Az; arrow in the diagram shows
that it has a magnitude of 0.40 cm; so it represents a velocity
change of 4.0 m/sec. This change occurred during At = 1 second, so
the rate of change is 4.0 m/sec/sec — the same value we found using
the relation Uc = v^/Rl
/ The best way of showing that a^ = z/^/R is entirely consistent

with the mechanics we have developed in Unit 1 is to do some
experiments to measure the centripetal force required to keep an
object moving in a circle. If, for example, the mass of the car were
1000 kg, there would have to be a centripetal force acting on the
car:

^ m

\ = 1000 kg X 4.0 -^

\ se&

' m

' = 4000 kg = 4000 N (or about 1800 pounds).

I sec^

/ This force would be directed toward the center of curvature of

the road — that is, it would always be sideways to the direction the

car is moving. This force is exerted on the tires by the road. If the

road is wet or icy, and can not exert a force of 4000 N sideways on

SG 4.13 the tires, the centripetal acceleration will be less than 4.0 m/sec — so

SG 4.14 the car will follow a less curved path as sketched in the margin on

the next page. In situations where the car's path is less curved

than the road, we would say the car "left the road" -although it

might be just as appropriate to say the road left the car.

The sideways force exerted on tires by a road is not easy to

measure. But in Project Physics Handbook 1 there are a number of

ways suggested for you to check experimentally whether Fc = muc

or Fr^m vVR.

For uniform motion in repeated cycles, it is often easier to

measure the frequency / or period T than it is to measure v directly.

Section 4.7

113

We can substitute the relations v = 27rR/or v = 2itRIT into the
equation equation for a^ to get alternative and equivalent ways of
calculating a^:

i^TTRff

/27rR\2
\ T /

_ 47r2R2/2

47r2R2

= 4rrmp

47r2R

Q7 In which of the following cases can a body have an
acceleration?

(a) moves with constant speed

(b) moves in a circle with constant radius

Q8 In what direction would a piece from a rapidly spinning
fly-wheel go if it suddenly shattered?

Q9 If a car of mass m going at speed v enters a curve of radius
R, what is the force required to keep the car curving with the road?

Q10 If a rock of mass m is being whirled overhead at 1
revolution/second on a string of length R, what is the force which
the string must be exerting?

4.7 The motion of earth satellites

Nature and technology provide many examples of the type of
motion where an object is in uniform circular motion. The wheel
has been a main characteristic of our civilization, first as it
appeared on crude carts and then later as an essential part of
complex machines. The historical importance of rotary motion in
the development of modern technology has been described by the
historian V. Gordon Childe:

Rotating machines for performing repetitive
operations, driven by water, by thermal power, or by
electrical energy, were the most decisive factors of the
industrial revolution, and, from the first steamship till the
invention of the jet plane, it is the application of rotary
motion to transport that has revolutionized communica-
tions. The use of rotary machines, as of any other human
tools, has been cumulative and progressive. The inventors
of the eighteenth and nineteenth centuries were merely
extending the applications of rotary motion that had
been devised in previous generations, reaching back
thousands of years into the prehistoric past. . . .
[The History of Technology.]

SG 4.12 g, h
SG4.15
SG 4.16
SG 4.17
SG 4.18

Chariot. Alberto Giacometti, 1950.

114

Understanding Motion

SG 4.19

As you will see in Unit 2, there is another rotational motion that
has also been one of the central concerns of man throughout
recorded history: the orbiting of planets around the sun and of the
moon around the earth.

Since the kinematics and dynamics for any uniform circular
motion are the same, we can apply what you have learned so far to
the motion of artificial earth satellites in circular (or nearly
circular) paths. As an illustration, we will select the satellite
Alouette I, Canada's first satellite, which was launched into a
nearly circular orbit on September 29, 1962.

Tracking stations located in many places around the world
maintain a record of any satellite's position in the sky. From the
position data, the satellite's distance above the earth at any time
and its period of revolution are found. By means of such tracking,
we know that Alouette I moves at an average height of 630 miles
above sea level, and takes 105.4 minutes to complete one revolution.

We can now quickly calculate the orbital speed and the
centripetal acceleration of Alouette I. The relationship v = IttRIT
allows us to find the speed of any object moving uniformly in a
circle if we know its period T and its distance R from the center
of its path (in this case, the center of the earth). Adding 630 miles
to the earth's radius of 3963 miles, we get R = 4594 miles, and

IttR

277 X 4593 mi
105.4 min

28, 860 mi
105.4 min

= 274 mi/min

or roughly 16,400 mi/hr.

To calculate the centripetal acceleration of Alouette I, we can
use this value of v along with the relationship a^ = v'^IR. Thus

SG 4.20
SG 4.21

_ (274 mi/min)^
4,594 mi

= 16.3 mi/min^

which is equivalent to 7.3 m/secl (To get the same result, we could
just as well have used the values of R and T directly in the
relationship a^ = ^tt'^RIT^)

What is the origin of the force that gives rise to this
acceleration? Although we will not make a good case for it until
Chapter 8, you surely know already that it is due to the earth's
attraction. Evidently the centripetal acceleration a<. of the
satellite is just the gravitational acceleration a„ at that height,
which has a value 25% less than Qg very near the earth's surface.

Section 4.7

115

Earlier we asked the question, "What speed is required for an
object to stay in a circular orbit about the earth?" You can answer
this question now for an orbit 630 miles above the surface of the
earth. To get a general answer, you need to know how the
acceleration due to gravity changes with distance. In Chapter 8 we
will come back to the problem of injection speeds for orbits.

The same kind of analysis applies to an orbit around the moon.
For example, on the first manned orbit of the moon (Apollo 8, in
1968), the mission control group wanted to put the capsule into a
circular orbit 70 miles above the lunar surface. They believed that
the acceleration due to the moon's gravity at that height would be
ttg = 1.43 m/sec^. What direction and speed would they give the
capsule to "inject" it into lunar orbit?

The direction problem is fairly easy — to stay at a constant
height above the surface, the capsule would have to be moving
horizontally at the instant the orbit correction was completed. So
injection would have to occur just when the capsule was moving on
a tangent, 70 miles up, as shown in the sketch in the margin. What
speed (relative to the moon, of course) would the capsule have to
be given? The circular orbit has a radius 70 miles greater than the
radius of the moon, which is 1080 miles; so R = 1080 mi + 70 mi
= 1150 mi; this is equal to 1.85 x 10^ meters. The centripetal
acceleration is just the acceleration caused by gravity, which was
supposed to be 1.43 m/sec^, so

= x/(1.85x 10«m)x 1.43

sec'

= A/2.65mX 10«

m
sec^

= 1.63x 103

sec

The necessary speed for an orbit at 70 miles above the surface is
therefore 1630 m/sec (about 3600 mi/hr). Knowing the capsule's
speed, ground control could calculate the necessary speed changes
to reach 1630 m/sec. Knowing the thrust force of the engines and
the mass of the capsule, they could calculate the time of thrust
required to make this speed change.

SG 4.22
SG 4.23
SG 4.24

Q1 1 What information was necessary to calculate the speed for
an orbit 70 miles above the moon's surface?

116

Understanding Motion

Table 4.2 Some information on selected artificial satellites.

NAME LAUNCH DATE WEIGHT (lb) PERIOD (min)

Sputnik 1
1957 (USSR)

Oct. 4, 1957

184

Explorer 7
1958 (USA)

Jan. 31, 1958

30.8

Lunik 3
1959 (USSR)

Oct. 4, 1959

959

Vostok 1
1961 (USSR)

Apr. 12, 1961

10,416

Midas 3
1961 (USA)

July 12, 1961

3,500

Telestar 1
1962 (USA)

July 10, 1962

170

Alouette 1
1962 (USA-
Canada)

Sept. 29, 1962

319

Luna 4

Apr. 2, 1963

3,135

1963-08 (USSR)

Vostok 6 June 16, 1963 "about 5

1963-23 (USSR) tons"

Syncom 2 July 26, 1963

1963-31 (USA)

HEIGHT (miles)

RIOD (min)

Perigee- Apogee

96.2

142-588

114.8

224-1573

22.300

30.000-291,000

89.34

109-188

161.5

2,129-2,153

157.8

593-3,503

REMARKS ('"eliding
purpose)

105.4

620-640

42,000

56,000-435,000

88.34

106-134

1,460.4

22,187-22,192

First earth satellite. Internal
temperature, pressure inside satellite.

Cosmic rays, micrometeorites,
internal and shell temperatures,
discovery of first Van Allen belts.

Transmitted photographs of far side of
moon.

First manned orbital flight (Major Yuri
Gagarin; one orbit)

Almost circular orbit.

Successful transmission across the
Atlantic: telephony, phototelegraphy,
and television.

Joint project between NASA and
Canadian Defense Research Board;
measurement in ionosphere.

Passed 5,300 miles from moon; very
large orbit.

First orbital flight by a woman;
(Valentina Terishkive; 48 orbits)

Successfully placed in near-
Synchronous orbit (stays above same
spot on earth).

4.8 What about other motions?

So far we have described straight-line motion, projectile motion,
and uniform circular motion. In all these cases we considered only
examples where the acceleration was constant -at least in
magnitude if not in direction -or very nearly constant. There is
another basic kind of motion that is equally common and important
in physics, where the acceleration is always changing. A common
example of this type of motion is that seen in playground swings, or
in vibrating guitar strings. Such back and forth motion, or
oscillation, about a center position occurs when there is a force
always directed toward the center position. When a guitar string is
pulled aside, for example, a force arises which tends to restore the
string to its undisturbed center position. If it is pulled to the other
side, a similar restoring force arises in the opposite direction.

A very common type of such motion is one for which the
restoring force is proportional, or nearly proportional, to how far the
object is displaced. This is true for the guitar string, if the
displacements are not too large; pulling the string aside 2 mm will
produce twice the restoring force that pulling it aside 1 mm will.
Oscillation with a restoring force proportional to the displacement

Section 4.8

117

is called simple harmonic motion. The mathematics for describing
simple harmonic motion is relatively simple, and many phenomena,
from pendulum motion to the vibration of atoms, have aspects that
are very close to simple harmonic motion. Consequently, the
analysis of simple harmonic motion is used very widely in physics.
The Project Physics Handbook 1 describes a variety of activities you
can do to become familiar with oscillations and their description.

Either simply or in combination, the dynamics discussed in this
chapter will cover most of the motions that will interest us, and is a
good start toward understanding apparently very complicated
motions, whether those of water ripples on a pond, a person running,
the swaying of a tall building or bridge in the wind, a small
particle zig-zagging through still air, an amoeba seen under a
microscope, or a high-speed nuclear particle moving in the field of a
magnet. The methods we have developed in this and the preceding
chapters give us means for dealing with any kind of motion
whatsoever, on earth or anywhere in the universe.

When we considered the forces needed to produce motion,
Newton's laws supplied us with the answers. Later, when we shall
discuss other motions ranging from the elliptical motion of planets
to the hyperbolic motion of an alpha particle passing near a nucleus,
we shall continue to find in Newton's laws the tool for inferring the
magnitude and direction of the forces acting in each case.

Conversely, if we know the magnitude and direction of the
forces acting on an object, we can determine what its change in
motion will be. If in addition we know also the present position,
velocity and mass of an object, we can reconstruct how it moved
in the past, and we can predict how it will move in the future under
these forces. Thus Newton's laws provide a comprehensive view of
forces and motion. It is not surprising that Newtonian mechanics
became a model for many other sciences: here seemed to be a
method for understanding all motions, no matter how mysterious
they previously may have appeared to be.

SG 4.25
SG 4.26

118 Understanding Motion

EPILOGUE The purpose of this Unit was to deal with the
fundamental concepts of nnotion. We decided to start by analyzing
particularly simple kinds of motion in the expectation that they are
indeed the "ABC's" of physics. These ideas would allow us to turn our
attention back to some of the more complex features of the world. To
what extent were these expectations fulfilled?

We did find that a relatively few basic concepts allowed us to gain
a considerable understanding of motion. First of all, we found that
useful descriptions of the motion of objects can be given using the
concepts of distance, displacement, time, speed, velocity, and
acceleration. If to these we add force and mass and the relationships
expressed in Newton's three laws of motion, it becomes possible to
account for observed motion in an effective way. The surprising thing is
that these concepts of motion, which were developed in extraordinarily
restricted circumstances, can in fact be so widely applied. For example,
our work in the laboratory centered around the use of sliding dry ice
pucks and steel balls rolling down inclined planes. These are not
objects found moving around ordinarily in the everyday "natural" world.
Even so, we found that the ideas obtained from those specialized
experiments could lead us to an understanding of objects falling near
the earth's surface, of projectiles, and of objects moving in circular
paths. We started by analyzing the motion of a disk of dry ice moving
across a smooth surface and ended up analyzing the motion of a space
capsule as it circles the moon and descends to its surface.

Thus, we have made substantial progress in analyzing complex
motions. On the other hand, we cannot be satisfied that we have here
all the intellectual tools needed to understand all of the phenomena that
interest us. In Unit 3 we shall add to our stock of fundamental concepts
a few additional ones, particularly those of momentum, work, and
energy. They will help us when we turn our attention away from
interactions involving a relatively few objects of easily discernible size,
and to interactions involving countless numbers of submicroscopic
objects-molecules and atoms.

In this Unit we have dealt primarily with concepts that owe their
greatest debts to Galileo, Newton, and their followers. If space had
permitted, we should also have included the contributions of Ren^
Descartes and the Dutch scientist Christian Huyghens. The
mathematician and philosopher, A. N. Whitehead has summarized the
role of these four men and the significance of the concepts we have
been dealing with in the following words:

This subject of the formation of the three laws of motion
and of the law of gravitation [which we shall take up in Unit 2]
deserves critical attention. The whole development of
thought occupied exactly two generations. It commenced
with Galileo and ended with Newton's Principia: and
Newton was born in the year that Galileo died. Also the
lives of Descartes and Huyghens fall within the period
occupied by these great terminal figures. The issue of the
combined labours of these four men has some right to be

Epilogue

119

considered as the greatest single intellectual success which
mankind has achieved. (Science and the Modern World)

The laws of motion Whitehead speaks of, the subject of this Unit,
were important most of all because they suddenly allowed a new
understanding of celestial motion. For at least twenty centuries man
had been trying to reduce the complex motions of the stars, sun, moon,
and planets to an orderly system. The genius of Galileo and Newton
was in studying the nature of motion of objects as it occurs on earth,
and then to assume the same laws would apply to objects in the
heavens beyond man's reach.

Unit 2 is an account of the immense success of this idea. We shall
trace the line of thought, starting with the formulation of the problem of
planetary motion by the ancient Greeks, through the work of
Copernicus, Tycho Brahe, Kepler, and Galileo to provide a planetary
model and the laws for planetary motion, and finally to Newton's
magnificent synthesis of terrestrial and celestial physics in his Law of
Universal Gravitation.

STUDY GUIDE 4

4.1 The Project Physics learning materials
particularly appropriate for Chapter 4 include
the following:
Experiments

Curves of Trajectories

Prediction of Trajectories

Centripetal Force

Centripetal Force on a Turntable

Activities

Projectile Motion Demonstration

Speed of a Stream of Water

Photographing a Waterdrop Parabola

Ballistic Cart Projectiles

Motion in a Rotating Reference Frame

Penny and Coat Hanger

Measuring Unknown Frequencies

Reader Articles

Galileo's Discussion of Projectile Motion
Newton's Laws of Dynamics
Rigid Body
Fun in Space

Film Loops

A Matter of Relative Motion

Galilean Relativity- Ball Dropped from Mast

of Ship

Galilean Relativity -Object Dropped

from Aircraft

Galilean Relativity — Projectile Fired

vertically

Analysis of Hurdle Race I

Analysis of Hurdle Race II

4.2 The thrust developed by a Saturn Apollo
rocket is 7,370,000 newtons (approximately
1,650,000 lbs.) and its mass is 540,000 kg. What is
the acceleration of the vehicle relative to the
earth's surface at lift off"? How long would it take
for the vehicle to rise 50 meters?

The acceleration of the vehicle increases
greatly with time (it is 47 m/sec^ at first stage
burnout) even though the thrust force does not
increase appreciably. Explain why the acceleration
increases.

4.H A hunter points his gun barrel directly at a
monkey in a distant palm tree. Will the bullet
follow the line of sight along the barrel? If the
animal, startled by the flash, drops out of the
branches at the very instant of firing, will it then
be hit by the bullet? Explain.

4.4 The displacement d'of an object is a vector
giving the straightline distance from the
beginning to the end of an actual path; ?can
be thought of as made up of a horizontal (x) and
a vesical (y) component of displacement; that is,
d = x + y (added vectorially).

In a trajectory, x, y, and the total
displacement d can be thought of as the
magnitudes of the sides of right triangles. So can
Vx, v^ and the magnitude of the velocity v.

(a) Find an expression for d in terms of jc
and y.

120

(b) Find an expression for v in terms of Vj
and Vu

4.5 If you like algebra, try this general proof.
If a body is launched with speed v at some

angle other than 0°. it will initially have both a
horizontal speed v^. and a vertical speed v^. The
equation for its horizontal displacement is x = v^t,
as before. But the equation for its vertical
displacement has an additional term: y = Vyt +
jayt'\ Show that the trajectory is still parabolic
in shape.

4.6 A lunch pail is accidently kicked off a steel
beam on a skyscraper under construction. Suppose
the initial horizontal speed v^ is 1.0 m/sec. Where
is the pail (displacement), and what is its speed
and direction (velocity) 0.5 sec after launching?

4.7 In Galileo's drawing on page 104. the
distances be, cd. de, etc. are equal. What is the
relationship among the distances ho. oq. ql, and
/n?

4.cS You are inside a van that is moving with a
constant velocity. You drop a ball.

(a) What would be the ball's path relative to
the van?

(b) Sketch its path relative to a person driving
past the van at a high uniform speed.

You are inside a moving van that is
accelerating uniformly in a straight line. When
the van is traveling at lOmph (and still
accelerating) you drop a ball from near the roof of
the van onto the floor.

(d) What would be the ball's path relative to
the van?

(e) Sketch its path relative to a person driving
past the van at a high uniform speed.

(f ) Sketch its path relative to a person
standing on the road.

4.9 Two persons watch the same object move.
One says it accelerates straight downward, but
the other claims it falls along a curved path.
Describe conditions under which each would be
reporting correctly what he sees.

4.10 An airplane has a gun that fires bullets
straight ahead at the speed of 600 mph when
tested on the ground while the plane is stationary.

STUDY GUIDE 4

The plane takes off and flies due east at 600 mph.
Which of the following describes what the pilot
of the plane will see? In defending your answers,
refer to the Galilean relativity principle:

(a) When fired directly ahead the bullets
move eastward at a speed of 1200 mph.

(b) When fired in the opposite direction, the
bullets dropped vertically downward.

Specify the frames of reference from which (a),
(b), and (c) are the correct observations.

1.11 Many commercial record turntables are
designed to rotate at frequencies of 16 2/3 rpm
(called transcription speed). 33 1/3 rpm (long
playing). 45 rpm (pop singles), and 78 rpm (old
fashioned). What is the period corresponding to
each of these frequencies?

4.12 Two blinkies are resting on a rotating
turntable and are photographed in a setup as
shown in the figure below. The outer blinky has a
frequency of 9.4 flashes/sec and is located 15.0
cm from the center. For the inner blinky, the
values are 9.1 flashes/sec and 10.6 cm.

(a) What is the period of the turntable?

(b) What is the frequency of rotation of the
turntable? Is this a standard phonograph
speed?

(d) What is the speed of the turntable at the
position of the inner blinky?

(e) What is the speed of the turntable at the
very center?

(f ) What is the angular speed of each
blinky — that is. the rate of rotation
measured in degrees/sec? Are they equal?

(g) What is the centripetal acceleration
experienced by the inner blinky?

(h) What is the centripetal acceleration
experienced by the outer blinky?

(i) If the turntable went faster and faster,
which would leave the turntable first, and
why?

4.1.3 Passengers on the right side of the car in
a left turn have the sensation of being "thrown

against the door." Explain what actually happens
to the passengers in terms of force and
acceleration.

4.14 The tires of the turning car in the example
on page 112 were being pushed sideways by the
road with a total force of 1800 lb. Of course the
tires would be pushing on the road with a total
force of 1800 lb also, (a) What happens if the road
is covered with loose sand or gravel? (b) How
would softer (lower pressure) tires help? (c) How
would banking the road (that is. tilting the
surface toward the center of the curve) help?
(Hint: consider the extreme case of banking in
the bob-sled photo on p. 110.)

4.15 Using a full sheet of paper, make and
complete a table like the one below.

NAME OF
CONCEPT

SYMBOL

DEFINITION

EXAMPLE

Length of a
path between
any two points,
as measured
along the path.

Straight line
distance and
direction from
Detroit to
Chicago.

Instan-
taneous
speed

An airplane
flying west
at 400 mph at
constant
altitude.

Time rate of
change of
velocity.

a«

Centripetal
acceler-
ation

The drive shaft
of some
automobiles
turns 600 rpm
in low gear.

The time it
takes to make
one complete
revolution.

121

STUDY GUIDE 4

4.16 Our sun is located at a point in our galaxy
about 30,000 light years (1 light year= 9.46 x
10'^ km) from the galactic center. It is thought
to be revolving around the center at a linear
speed of approximately 250 km/sec.

(a) What is the sun's centripetal acceleration
with respect to the center of the galaxy?

(b) The sun's mass can be taken to be 1.98
X 10*" kg; what centripetal force is
required to keep the sun moving in a
circular orbit about the galactic center?

(c) Compare the centripetal force in (b) with
that necessary to keep the earth in orbit
about the sun. (The earth's mass is

5.98 X 10^^ kg and its average distance
from the sun is 1.495 x 10* km.)

4.17 The hammer thrower in the photograph
below is exerting a large centripetal force to keep
the hammer moving fast in a circle, and applies
it to the hammer through a connecting wire. The
mass of the 16-pound hammer is 7.27 kg. (a)
Estimate the radius of the circle and the period,
and calculate a rough value for the amount of
force required just to keep it moving in a circle.
(b) What other components are there to the total
force he exerts on the hammer?

4.18 Contrast rectilinear motion, projectile
motion, and uniform circular motion by

(a) defining each

(b) giving examples.

122

4.19 These questions are asked with reference to
Table 4.2 on page 116.

(a) Which satellite has the most nearly
circular orbit?

(b) Which satellite has the most eccentric
orbit? How did you arrive at your answer?

(d) How does the position of Syncom 2
relative to a point on earth change over
one day?

4.20 If the earth had no atmosphere, what would
be the period of a satellite skimming just above
the earth's surface? What would its speed be?

4.21 Explain why it is impossible to have an earth
satelhte orbit the earth in 80 minutes. Does this
mean that it is impossible for any object to go
around the earth in less than 80 minutes?

4.22 What was the period of the "70 mi" Apollo 8
lunar orbit?

4.23 Knowing Ug near the moon's surface, and
the orbital speed in an orbit near the moon's
surface, we can now work an example of Part 8
of the earth-moon trip described in Sec 4.1. The
Apollo 8 capsule was orbiting about 100 kilometers
above the surface. The value of a^ near the
moon's surface is about 1.5 m/sec^.

If the capsule's rocket engines are fired in the
direction of its motion, it will slow down.
Consider the situation in which the rockets fire
long enough to reduce the capsule's horizontal
speed to 100 m/secl

(a) About how long will the fall to the moon's
surface take?

(b) About how far will it have moved
horizontally during the fall?

4.24 Assume that a capsule is approaching the
moon along the right trajectory, so that it will be
moving tangent to the desired orbit. Given the
speed v„ necessary for orbit and the current speed
V, how long should the engine with thrust F fire to
give the capsule of mass m the right speed?

4.25 The intention of the first four chapters has
been to describe "simple" motions and to progress
to the description of more "complex" motions. Put
each of the following examples under the heading
"simplest motion." "more complex." or "very
complex." Be prepared to say why you place any
one example as you did and state any assumptions
you made.

(a) helicopter shown on p. 109

(b) "human cannon ball" in flight

(d) tree growing

(e) child riding a Ferris wheel

(f ) rock dropped 3 mi.

(g) person standing on a moving escalator
(h) climber ascending Mt. Everest

(i) person walking

( j ) leaf falling from a tree

4.26 Write a short essay on the physics involved
in the motions shown in one of the four pictures
on the opposite page, using the ideas on motion
from Unit 1.

STUDY GUIDE 4

123

Acknowledgments

Prologue

Pp. 1-4 Fermi, Laura, Atoms in the Family,
U. of Chicago Press, pp. 83-100 not inclusive.
Chapter Two

P. 6 Aristotle, De Caelo, trans. J. L. Stokes, Book
I, Chapter 6, Oxford University Press, p. 273b.

Pp. 44-60 Galilei, Galileo, Two New Sciences,
trans. Crew and DeSalvio, Dover Publications, pp.
62-243 not inclusive.
Chapter Three

P. 86 Newton, Sir Isaac, The Principia, Vol. I,
Mott's translation revised by Florian Cajori, U. of
Calif. Press, pp. 13-14.

Pp. 86-87 Ibid., pp. XIII-XV.

P. 88 Magie, W. P., A Source Book in Physics,
McGraw-Hill, p. 94.
Chapter Four

P. 92 Newton, Sir Isaac, op. cit.. Vol. II, p. 251.

P. Ill Childe, V. Gordon, "Rotary Motion," A
History of Technology, Vol. I, Oxford University
Press, p. 187.

P. 117 Pope, Alexander, Epitaph Intended for
Sir Isaac Newton (1732).

P. 127 Whitehead, A. N., Science and the
Modern World, a Mentor Book published by The
New American Library, pp. 46-47.

Picture Credits

Prologue

P. 4 U.S. Atomic Energy Commission.

P. 6 (left) Mt. Wilson and Palomar Observa-
tories: (right) Professor Erwin W. Mueller, The
Pennsylvania State University.

P. 7 (left) Museum of Comparative Zoology,
Harvard University; (right) Brookhaven National
Laboratory.
Chapter 1

P. 8 Yale University Art Gallery, Collection
Societe Anonyme.

P. 10 United Press International, LIFE
Magazine, © Time Inc.

P. 21 (solar flare) reproduced from Sydney
Chapman's IGY: Year of Discovery, by courtesy of
The University of Michigan Press; (glacier) from
the film strip "Investigating a Glacier" © 1966,
Encyclopaedia Britannica Educational Corpora-
tion, Chicago; (plants) Dr. Leland Earnest, Dept.
of Biology, Eastern Nazarene College.

P. 26 (1) Bayerisches Nationalmuseum, Munich;

(2) (4) George Eastman House, Rochester, N.Y.;

(3) Bill Eppridge, LIFE MAGAZINE, © Time Inc.

P. 27 (5) (6) (7) Dr. Harold E. Edgerton.
Massachusetts Institute of Technology,
Cambridge, Mass.

P. 30 George Silk, LIFE MAGAZINE, © Time
Inc.

P. 35 George Eastman House, Rochester, N.Y.
Chapter 2

P. 36 Cabinet des Dessins, Louvre Museum.

P. 38 Vatican Museum. Rome.

P. 43 (signature) Smith Collection, Columbia
University Libraries.

P. 44 Houghton Library, Harvard University.

P. 45 Courtesy of Educational Development
Center, Newton, Mass.

P. 53 Alinari-Art Reference Bureau.
Chapter 3

P. 66 A. G. Mill, © Time Inc.

P. 72 G. Kew, © Time Inc.

P. 74 C. T. Polumbaum, © Time Inc.

P. 85 (balance) Collection of Historical
Scientific Instruments, Harvard University.

P. 87 Dr. Harold E. Edgerton, MIT.

P. 89 National Aeronautics and Space
Administration.

P. 92 The Museum of Modem Art, New York.

P. 95 U.S. Air Force.
Chapter 4

P. 99 National Aeronautics and Space
Administration; Verne, Jules, De la terre a la lune,
Paris, 1866.

P. 102 from PSSC Physics, D.C. Heath & Co.,
Boston, 1965.

P. 103 (skater) National Film Board of Canada;
(fireworks) Stan Wayman, LIFE MAGAZINE, ©
Time Inc.

P. 108 (carousel) Ernst Haas, Magnum Photos,
Inc.

P. 109 (helicopter) Andreas Feininger,
LIFE MAGAZINE, ©Time Inc.

P. 117 (guitar) Photo by Albert B. Gregory, Jr.

P. 117 (runner) Associated Newspapers,
Pictorial Parade, Inc., New York City.
Facing p. 1 (Fermi at the blackboard) University
of Chicago

P. 96 Caterpillar Tractor.

P. 110 (train) P. Stackpole, LIFE MAGAZINE, ©
Time Inc.

P. 123 (Bouncing ball) Dr. Harold E. Edgerton.
MIT.; (Bicyclists) Walker Art Center, Minneapolis:
(acrobats) A. E. Clar|c, LIFE MAGAZINE, ©
Time Inc.
(Cathedral spires)

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

124

The Projects Physics Course

Concepts of Motion

Picture Credits

Cover: (top left) Cartoon by Charles Gary Solin and
reproduced by his permission only; (top right) from
the film loop Galilean Relativity l-Ball Dropped from
Mast of Ship.

P. 135 Isogonic chart through the courtesy of the En-
vironmental Sciences Services Administration, Coast
and Geodetic Survey.

Pp. 135, 138, 143, 165, 170, 171, 183. 187 (cartoons).
By permission of Johnny Hart and Field Enterprises
Inc.

P. 152 Photography unlimited by Ron Church from
Rapho Guillumette Pictures. New York.
P.184 (water drop parabola) Courtesy of Mr. Harold M.
Waage, Palmer Physical Laboratory, Princeton Uni-
versity.

P. 185 (water drop parabola -train) Courtesy of Edu-
cational Development Center, Newton. Mass.
All photographs used with film loops courtesy of
National Film Board of Canada.
Photographs of laboratory equipment and of stu-
dents using laboratory equipment were supplied with
the cooperation of the Project Physics staff and
Damon Corporation.

Contents

HANDBOOK SECTION

introduction

Keeping Records 1 29

Using the Polaroid Camera 132

The Physics Reader 133

Experiments

1. Naked Eye Astronomy 134

2. Regularity and Time 142

3. Variations in Data 144

Cliapter 1 The Language of IVIotion

Experiment

4. Measuring Uniform Motion 145
Activities

Using the Electronic Stroboscope 151
Making Frictionless Pucks 151

Ciiapter 2 Free Fall— Galileo Describes IVIotion

Experiments

5. A Seventeenth-Century Experiment 153

6. Twentieth-Century Version of Galileo's
Experiment 157

7. Measuring the Acceleration of Gravity a, 158

(A) a, by Direct Fall 158

(B) Ug from a Pendulum 159

(D) a, from Falling Water Drops 161

(E) a, with Falling Ball and Turntable 162

(F) Ug with Strobe Photography 162
Activities

When is Air Resistance Important? 163
Measuring Your Reaction Time 163
Falling Weights 163
Extrapolation 1 63
Fihn Loops 164

1. Acceleration Due to Gravity-I 164

2. Acceleration Due to Gravity-II 165

Chapter 4 Understanding Motion

Experiments

10. Curves of Trajectories 176

11. Prediction of Trajectories 179

12. Centripetal Force 181

13. Centripetal Force on a Turntable 182
Activities

Projectile Motion Demonstration 184
Speed of a Stream of Water 184
Photographing a Waterdrop Parabola 184
Ballistic Cart Projectiles 185
Motion in a Rotating Reference Frame 185
Penny and Coat Hanger 186
Measuring Unknown Frequencies 186
Film Loops

4. A Matter of Relative Motion 187

5. Galilean Relativity— Ball Dropped from
Mast of Ship 188

6. Galilean Relativity— Object Dropped
from Aircraft 189

7. Galilean Relativity— Projectile Fired
Vertically 190

8. Analysis of a Hurdle Race I 190

9. Analysis of a Hurdle Race II 191

160

Chapter 3 The Birth of Dynamics-Newton Explains Motion

Experiments

8. Newton's Second Law 166

9. Mass and weight 169
Activities

Checker Snapping 170

Beaker and Hammer 1 70

Pulls and Jerks 170

Experiencing Newton's Second Law

Make One of These Accelerometers

(A) The Liquid-Surface Accelerometer-I 170

(B) Automobile Accelerometer-I 172

(D) Damped-Pendulum Accelerometer 173
Film Loop

3. Vector Addition-Velocity of a Boat 174

170
170

This Handbook is your guide to observa-
tions, experiments, activities, and explorations,
far and wide, in the realms of physics.

Prepare for challenging work, fun and
some surprises. One of the best ways to learn
physics is by doing physics, in the laboratory
and out. Do not rely on reading alone. Also,
this Handbook is different from laboratory
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 addi-
tional ones that interest you. Also, if an activity
occurs to you that is not described here, dis-
cuss with your teacher the possibility of doing
it. Some of the most interesting science you
will experience in this course will be the result
of the activities which you choose to pursue
beyond the regular assignments of the school
laboratory.

This Handbook contains a section corre-
sponding to each chapter of the Text. Usually
each section is divided further in the following
way:

The Experiments contain full in-
structions for the investigations you will
be doing with your class.

The Activities contain many sugges-
tions for construction projects, demon-
strations, and other activities you can do
by yourself.

The Film Loop notes give instruc-
tions for the use of the variety of film
loops that have been specially prepared
for the course.

In each section, do as many of these things
as you can. With each, you will gain a better
grasp of the physical principles and relation-
ships involved.

Introduction 129

Keeping Records

Your records of observations made in the lab-
oratory or at home can be kept in many ways.
Your teacher will show you how to write up
your records of observations. But regardless of
the procedure followed, the key question for
deciding what kind of record you need is this:
"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 written without being fussy.
You should organize all numerical readings in
tables, if possible, as in the sample lab write up
on pages 130 and 131. You should always iden-
tify the units (centimeters, kilograms, seconds,
etc.) for each set of data you record. Also, iden-
tify 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 analy-
sis of the results.

If you have some reason to suspect that a
particular datum may be less reliable than

other data— perhaps you had to make the read-
ing very hurriedly, or a line on a photograph
was very faint— make a note of the fact. But
don't erase a reading. When you think an entry
in your notes is in error, draw a single line
through it— don't scratch it out completely or
erase it. You may find it was significant after
all.

There is no "wrong" result in an experi-
ment, although results may be in considerable
error. If your observations and measurements
were carefully made, then your result will be
correct. What ever happens in nature, includ-
ing 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 use-
ful. 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 this up pretty

well:

In subjects of which one has no l<nowledge,
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.

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On these two pages is shown an example of a student's lab notebook report. The table
is used to record both observed quantities (mass, scale position) and calculated quan-
tities (force, extension of rubber band). The graph shows at a glance how the extension
of the rubber band changes as the force acting on it is increased.

Introduction

131

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132

Introduction

Using the Polaroid Camera

You will find the Polaroid camera is a very use-
ful device for recording many of your labora-
tory observations. Section 1.3 of 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 with a mechanical
disk stroboscope (a slotted disk rotating in front
of the camera lens). Thi setup of the rotating
disk stroboscope with a Polaroid camera is
shown below.

Camera Cable Release

a LaoieKeiease -^
,ft'** X — ^ Camera

Rotating Disc

^4S)

Electric Motor

Strobe Mounting
Platform

Tripod

In the opposite column is a check list of
operations to help you use the modified Polaroid
Land camera model 210. For other models,
your teacher will provide instructions. Check
list of operations for Polaroid Land camera
model 002

1. Make sure that there is film in the camera.
If no white tab shows in 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.

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. Look through viewer to be sure that what-
ever part 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.)

7. Make sure the shutter is cocked (by depress-
ing the number 3 button).

8. Run through the experiment a couple of
times without taking a photograph, to accus-
tom yourself to the timing needed to photo-
graph the event.

9. Take the picture: keep the cable release
depressed only as long as necessary to record
the event itself. Don't keep the shutter open
longer than necessary.

10. Pull the white tab all the way out of the
camera. Don't 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 de-
velopment.

12. Wait 10 to 15 seconds (for 3000-speed
black-and-white film).

13. Ten to 15 seconds after removing 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 this dry
thoroughly, label it on the back for identifica-
tion and mount the picture in your (or a part-
ner's) 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 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 batter-
ies inside the camera will run down quickly.

Introduction 133

The Physics Readers

Your teacher probably will not often assign
reading in the Project Physics Reader, but you
are encouraged to look through it for articles
of interest to you. In the Unit 1 Reader most
students enjoy the chapter from Fred Hoyle's
science fiction novel, The Black Cloud. This
chapter, "Close Reasoning," is fictional, but
nevertheless accurately reflects the real ex-
citement of scientists at work on a new and
important problem.

Since different people have very different
interests, nobody can tell you which articles
you will most enjoy. Those with interests in art
or the humanities will probably like Gyorgy
Kepes' article "Representation of Movement."
If you are interested in history and in the role
science plays in historical development, you
are urged to read the Butterfield and Willey
articles.

The Reader provides several alternative
treatments of mechanics which either supple-
ment or go beyond the Unit 1 Text. Thus Sawyer
gives a discussion of the concept of speed dif-
ferent from that used in the Text. Clifford's
approach is interesting because it uses geom-
etry rather than algebra in explaining funda-
mental ideas. For those seeking a deeper un-
derstanding of mechanics, we particularly
recommend the article from the Feynman
Lectures on Physics. For articles that deal
with applications of physics, you can turn to
Strong on "The Dynamics of the Golf Club,"

Kirkpatrick on "Bad Physics in Athletic Mea-
surements," and DuBridge on "Fun in Space."
Practice the art of browsing! Don't decide
from the titles alone whether you are inter-
ested, but read portions of articles here and
there, and you may well discover something
new and interesting.

Project Physics Reader

An Introduction to Physics

Concepts of Motion

134 Experiment 1

EXPERIMENTS

EXPERIMENT 1
ASTRONOMY

NAKED EYE

The purpose of this first experiment is to fa-
miliarize you with the continually changing
appearance of the sky. By watching the heav-
enly bodies closely day and night over a period
of time, you wOl begin to understand what is
going on up there and gain the experience you
will need in working with 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? How do
the sun and planets move in relation to the
stars?

The Babylonians and Egyptians knew the
answers to these questions over 5000 years
ago. They found them simply by watching the
everchanging 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.
Our 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 from season to season.

Some of these changes occur very slowly.
In fact, this is why you may not have noticed
them. You will need to watch the motions in
the sky carefully, measuring them against
fixed points of reference that you establish.
You will need to keep a record of your obser-
vations 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 longi-
tude 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 position 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 (180°) and west (270°) and around
to north again (360°or 0°).

To measure the height of an object in the
sky, you can measure the angle between the
object and a horizontal plane, such as the
horizon, for your second reference. This plane
can be used even when the true horizon is
hidden by trees or other obstructions. The
angle between the horizontal plane and the
line to an object in the sky is called the altitude
of the object.

\ Altitude30°

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Establishing References

You can establish your north-south line in
several different ways. The easiest is to use a
compass to establish magnetic north but this
may not be the same as true north. A magnetic
compass responds to the total magnetic effect
of all parts of the earth, and in most localities
the compass does not point true north. The
angle between magnetic north and true north
is called the angle of magnetic declination. At
some places the magnetic declination is zero,
and the compass points toward true north.

At places east of the line where the de-
clination is zero, the compass points west of
true north; at places west of the line, the com-
pass points east of true north. You can find the

angle of decimation and its rate of change per
year for your area from the map below.

At night you can use the North Star (Po-
laris) 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 observer
anywhere in the Northern Hemisphere.

To locate Polaris, first find the "Big Dip-
per" which on a September evening is low in
the sky and a little west of north. (See the star
map, Fig. 1-1 page 136.) 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
and extended upward 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. On September 15 at 8:30 P.M.
these constellations are arranged about as
shown in the diagram below.

Experiment 1 135

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136 Experiment 1

5 November 20 a

Fig. 1-1.

02 ^^H

This chart of the stars will help you locate some of the bright stars and the constel-
lations. 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 hour ear//er than 8 p.m., rotate the chart 15 degrees (one sector) clock-
wise. For each hour later than 8 p.m., rotate the chart counter-clockwise. 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.

Experiment 1 137

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

Now that you have established a north-
south line, either with a compass or from the
North Star, 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, 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 the Unit 2 Handbook, in the ex-
periment dealing with 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 tell you when the
table is level.

Turn the base of the astrolabe on the table

until the north-south line on the base points
along your north-south line. Or you can 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 astrolabes, you can
also obtain the azimuth of the objects from the
base of the astrolabe.

To follow the position of the sun with the
astrolabe, slip a large 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 inches from the sighting
end of the tube. Move the tube about until the
bright image of the sun appears through the
tube on the paper. Then read the altitude of
the sun from the astrolabe, and the sun's azi-
muth, if your instrument permits.

Observations

Now that you know how to establish your ref-
erences for locating objects in the sky, here are
suggestions for observations you can make on
the sun, the moon, the stars, and the planets.
Choose at least one of these objects to observe.
Record the date and time of all your observa-

138

Experiment 1

tions. Later compare notes with classmates
who concentrated on other objects.

A. Sun

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

1. Observe the direction in which the sun sets.
Always make your observation from the same
observing position. If you don't have an un-
obstructed 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
on the horizon and the position of the setting
sun.

4. Repeat the observation 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
reasonably sure that the 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
the times of sunrise and sunset against the
times as they are often reported in newspapers.
Also if the weather does not permit you to
observe the sun, the newspaper reports may
help you to complete your observations.

7. During a single day, observe the sun's azi-
muth at various times. Keep a record — of the
azimuth and the time of observation. Deter-
mine whether the azimuth changes at a con-
stant rate during the day, or whether the 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 azi-
muth.

Similarly, 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 even an
entire year — observe the altitude of the sun at
noon — or some other convenient hour. (Don't
worry if you miss some observations.) Deter-
mine the date on which the noon altitude of
the sun is a minimum. On what date would the
sun's altitude be a maximum?

B.C. by John Hart

By permission of John Hart and Field Enterprises, Inc.

B. Moon

1. Observe and record the altitude and azi-
muth of the moon and draw its shape on suc-
cessive 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 night each week, make a
sketch showing the appearance of the moon
and another "overhead" sketch of the relative
positions of the earth, moon, and sun. If the
sun is below the horizon when you observe the
moon, you will have to estimate the sun's po-
sition.

2. Locate the moon against the background of

Experiment 1

139

the stars, and plot its position and phase on a
sky map suppHed by your teacher.

3. Find the full moon's maximum altitude.
Find how this compares with the sun's maxi-
mum 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 on page 140, or the
Celestial Calendar and Handbook, for the
dates of lunar eclipses. Observe one if you
possibly can.

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 iden-
tify some of these groups with constellations
that are named on the star map in Fig. 1-1,
which shows the constellations around the
North Star, or on another star map furnished
by your teacher. Record how much the stars
have changed their positions compared to your
horizon after an hour; after two 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 won't move
during the time exposures of an hour or more.
Use a small camera lens opening (large f-
number) to reduce fogging of your film by
stray light.

3. Viewing at the same time each night, find

This multiple exposure picture of the moon was taken
with a Polaroid Land camera by Rick Pearce, a twelfth-
grader in Wheat Ridge, Colorado. The time intervals
between successive exposures were 15 min, 30 min,
30 min, and 30 min. Each exposure was for 30 sec using
2000-speed film. Which way was the moon moving in
the sky?

A time exposure photograph of Ursa Major (The Big
Dipper) taken with a Polaroid Land camera on an au-
tumn evening in Cambridge, Massachusetts.

whether the positions of the star groups are
constant in the sky from month to month. Find
if any new constellations appear 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
how much the positions of the stars shift per
week and per month.

D. Planets and meteors

1. The planets are located within a rather
narrow band across the sky (called the ecliptic)
along which the sun and the moon also move.
For details on the location of planets, consult
Table 1 on page 140, or the Celestial Calendar
and Handbook, or the magazine Sky and Tele-
scope. Identify a planet and record its position
in the sky relative to the stars at two-week
intervals for several months.

2. On almost any clear, moonless night, go
outdoors away from bright lights and scan as
much of the sky as you can see for meteors.
Probably you will glimpse a number of fairly
bright streaks of meteors in an hour's time.
Note how many meteors you see. Try to locate
on a star map like Fig. 1-1 where you see them
in the sky.

Look for meteor showers each year around
November 5 and November 16, beginning
around midnight. Dates of other meteor show-
ers are given in Table 2 on page 141. Remem-
ber that bright moonlight will interfere with
meteor observation.

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

140

Experiment 1

TABLE 1

A GUIDE FOR PLANET AND ECLIPSE OBSERVATIONS

Check your local newspaper for eclipse times and extent of eclipse in your locality.

Mercury

Venus

Mars

Jupiter

Saturn

Lunar

Solar

Visible for about one

Visible for several months

Very bright for

Especially

Eclipses

week around

lated

around stated time.

one month on

bright for sev-

bright for two

time.

each side of
given lime.

en months be-
yond stated

months on
each side of

Mercury and Venus arc

best viewed the hour

Observable for

time.

given time.

before dawn v

when indicated as a.m. and the

16 months sur.

Visible for 13

hour after sunset when

indicated as p.m.

rounding given

months.

time.

mid Feb.:

a.m.

Mar. 7: total

1 late Apr.:

p.m.

late May:

early Dec:

in Fla., par-

9 early June

: a.m.

early Nov.: p.m.

overhead

Feb. 21

tial in east-

7 mid Aug.:

p.m.

mid Dec: a.m.

Aug. 17

em and

0 late Sept.:

a.m.

midnight

southern

early Dec.

p.m.

U. S.

mid Jan.:

a.m.

1 late Mar.:

p.m.

early Sept.:

late June:

late Dec:

9 mid May:

a.m.

overhead

Feb. 10

7 late July:

p.m.

1 mid Sept.:

a.m.

midnight

late Nov.:

p.m.

early Jan.:

a.m.

1 mid Mar.:

p.m.

late July:

9 early May:

a.m.

overhead

July 10

7 mid July:

p.m.

mid May: p.m.

Jan. 30

partial in

2 late Aug.:

a.m.

early Aug.: a.m.

midnight

July 26

northern U.S.

early Nov.

p.m.

mid Dec:

a.m.

late Feb.:

p.m.

1 late Apr.:

a.m.

late Nov.:

early Sept.:

early Jan.:

9 late June :

p.m.

late Dec: p.m.

overhead

Dec. 10

7 early Aug.

a.m.

3 mid Oct.:

p.m.

midnight

early Dec.

a.m.

mid Feb.:

p.m.

1 late Mar.:

a.m.

mid Oct.:

late Jan.:

9 early June

p.m.

early Mar.: a.m.

overhead

June 4

7 mid July:

a.m.

Nov. 29

4 late Sept.:

p.m.

midnight

early Nov.

a.m.

late Jan.:

p.m.

1 early Mar.

a.m.

early Nov.:

early Feb.:

9 mid May:

p.m.

mid-late July: p.m.

overhead

May 25

7 early July:

a.m.

early Oct.: a.m.

Nov. 18

5 mid Sept.:

p.m.

midnight

late Oct.:

a.m.

mid Jan.:

p.m.

1 late Feb.:

a.m.

late Jan.:

early Dec:

late Feb.:

9 early May:

p.m.

overhead

7 mid June:

a.m.

6 late Aug.:

p.m.

midnight

early Oct.:

a.m.

mid Dec:

p.m.

early Feb.:

a.m.

1 early Apr.:

p.m.

early Mar.: p.m.

9 late May:

a.m.

mid Apr.: a.m.

Apr. 4

7 mid Aug.:

p.m.

7 late Sept.:

a.m.

Experiment 1

141

TABLE 2

FAVORABILITY OF OBSERVING METEOR SHOWERS

THE BEST TIME FOR VIEWING METEOR SHOWERS IS BETWEEN MIDNIGHT AND 6 A.M., IN PARTICULAR

DURING THE HOUR DIRECTLY PRECEDING DAWN.

Quadrantids

Jan. 3-5

Virgo

Lyrids

Apr. 19-23

Lyra

Perseids

July 27-Aug. 17

Perseus

Orionids
Oct. 15-25
Orion

Leonids

Nov. 14-18

Leo

Geminids

Dec. 9-14

Gemini

Rises in the east
around 2 a.m.,
upper eastern sky
at 5 a.m.

Rises in the east
around 10 p.m.,
western sky at
5 a.m.

Rises in the east
at 10 p.m.,
towards the west
at 5 a.m.

Rises in the east
at midnight, di-
rectly overhead
at 5 a.m.

Rises in the east
at 2 a.m., upper
eastern sky at
5 a.m.

Rises in the east
at 8 p.m.,
towards the far
west at 5 a.m.

Good

Through early
August, after
Aug. 10

Good

Good 1
9
6
8

Poor

Good

Aug. 3-17

After
Oct. 20

Good

Good 1
9
6

Good

Poor

July 27-Aug. 11

Oct. 18-25

Poor

Poor 1
9
7
0

Good

July 27-Aug. 2
Aug. 7-17

Good

Good 1
9

7
1

Good

Aug. 2-17

Oct. 15-20

Nov. 14-16

Good 1
9

7
2

Good

Apr. 21-23

July 27-Aug. 9

Good

Poor 1
9

7
3

Poor

Good

Aug. 7-17

Good

Good 1
9
7

Good

Oct. 21-25

Poor

Dec. 9-12 1
9

7
5

Good

July 27-Aug. 5
Aug. 12-17

Good

Good 1
9
7
6

Poor

Good

Aug. 3-17

Oct. 15-21

Good

Good 1
9

7
7

142

Experiment 2

EXPERIMENT 2 REGULARITY AND TIME

You will often encounter regularity in your
study of science. Many natural events occur
regularly— that is, over and over again at equal
time intervals. But if you had no clock, how
would you decide how regularly an event re-
curs? In fact, how can you decide how regular
a clock is?

The first part of this exercise is intended
merely to show you the regularity of a few
natural events. In the second part, you will try
to measure the regularity of an event against
a standard and to decide what is really meant
by the word "regularity."

Part A

You work with a partner in this part. Find
several recurring events that you can time in
the laboratory. You might use such events as a
dripping faucet, a human pulse, or the beat of
recorded music. Choose one of these events as
a "standard event." All the others are to be
compared to the standard by means of the
strip chart recorder.

One lab partner marks each "tick" of the
standard on one side of the strip chart recorder
tape while the other lab partner marks each
"tick" of the event being tested. After a long

run has been taken, inspect the tape to see how
the regularities of the two events compare.
Run for about 300 ticks of the standard. For
each 50 ticks of the standard, find on the tape
the number of ticks of the other phenomenon,
estimating to ^ of a tick. Record your results
in a table something like this:

STANDARD EVENT

TEST EVENT

First 50 ticks

Hrks

Second 50 ticks

ticks

Third 50 ticks

ticks

Fourth 50 ticks

ticks

The test event's frequency is almost certain to
be different from test to test. The difference
could be a real difference in regularity, or it
could come from your error in measuring.
Ql If you think that the difference is larger
than you would expect from human error, then
which of the two events is not regular?

Part B

In this part of the lab, you will compare the
regularity of some devices specifically designed
to be regular. The standard here will be the
time recording provided by the telephone com-
pany or Western Union. To measure two peri-

ods of time, you will have to make three calls
to the time station, for example, 7 p.m., 7 a.m.,
and 7 p.m. again. Agreement should be reached
in class the day before on who will check wall
clocks, who will check wristwatches, and so
on. Watch your clock and wait for the record-
ing to announce the exact hour. Tabulate your
results something like this:

TIME STATION

Time

"7 P.M. exactly"!
"7 A.M. exactly"^
"7 P.M. exactly"

Period
12:00:00 hr
12:00:00 hr

ELECTRIC WALL CLOCK

In Part I, you found that to test regularity you
need a standard that is consistent, varying as
little as possible. The standard is understood,
by definition, to be regular.
Q2 What is the standard against which the
time station signal is compared? Call to find
out what this standard is. Try to find the final
standard that is used to define regularity— the
time standard against which all other recur-
ring events are tested. How can we be sure of
the regularity of this standard?

Experiment 2 143

By Johnny Hart

'-#--*'v»»>.4-

144

Experiment 3

EXPERIMENT 3 VARIATIONS 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 with a ruler, your an-
swer will have a small margin of uncertainty.
That is, numbers read from measuring in-
struments do not give the exact measure-
ments in the sense that one or two is exact
when you count objects. Every measurement
is to some extent uncertain.

Moreover, if your lab partner measures the
length of this page, he will probably get a dif-
ferent answer from yours. Does this mean that
the length of the page has changed? Hardly!
Then can you possibly find the length of the
page without any uncertainty in your measure-
ment? 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 measure-
ment in a table like the one shown here. When
you have completed the series, write your
measurements on the 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.

TYPE OF
MEASUREMENT

REMARKS MEASUREMENT

Experiment 4

145

Chapter I The Language of Motion

EXPERIMENT 4 MEASURING
UNIFORM MOTION

If you roll a ball along a level floor or table,
eventually it stops. Wasn't it 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 dry-ice disk pic-
tured in Sec. 1-3 of the Text really be in uni-
form motion, even if the disk is called "friction-
less"? Would the disk just move on forever?
Doesn't everything eventually come to a stop?

In this experiment you check the answers
to these questions for yourself. You observe
very simple motion, like that pictured below,
and make a photo record of it, or work with
similar photos. You measure the speed of the
object as precisely as you can and 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. There 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 or due entirely to uncertainty in your
measurements.

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

Doing the Experiment

Various setups for the experiment are shown
on pages 145 and 146. It takes two people to
photograph a disk sliding on a table, or a glider

146

Experiment 4

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

on an air track, or a steadily flashing light
(called a blinky) mounted on a small box which
is pushed by a toy tractor. Your teacher will
explain how to work with the set up you are
using. Excellent photographs can be made of
any of them.

If you do not use a camera at all, or if you
work alone, then you may measure a trans-
parency or a movie film projected on the chalk
board or a large piece of paper.

Or you may simply work from a previously
prepared photograph such as Fig. 1-1, above.
If there is time, you might try several of these
methods.

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 for the disk
to slide with almost no friction. Make sure the
surface is quite level, too, so that the disk will
not start to move once it is at rest.

Set up the Polaroid camera and the strobo-
scope equipment according to your teacher's
instructions. Instructions for operating the
Polaroid model 210, and a diagram for mount-
ing this camera with a rotating disk strobo-
scope is shown on page 132. A ruler need not be
included in your photograph as in the photo-
graph above. Instead, you can use a magnifier
with a scale that is more accurate than a ruler
for measuring the photograph.

Either your teacher 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 launch-
es the disk while his companion operates the
camera. A "dry run" or two without taking a

picture will probably be needed for practice
before 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.

•A(borr\

\ 0

"T~T"r ! ' ' '

cro

Fig, 1-2. Estimating to tenths of a scale division.

Making Measurements

Whatever method you have used, your next
step is to measure the spaces between succes-
sive images of your moving object. For this,
use a ruler with millimeter divisions and esti-
mate the distances to the nearest tenth of a
millimeter, as shown in Fig. 1-2 above. If
you use a magnifier with a scale, rather than
a ruler, you may be able to estimate these quite
precisely. List each measurement in a table
like Table 1.

Since the intervals of time between one
image and the next are equal, you can use that
interval as a unit of time for analyzing the
event. If the speed is constant, the distances
of travel would turn out to be all the same, and
the motion would be uniform.
Ql How would you recognize motion that is
not uniform?

Q2 Why is it unnecessary for you to know the
time interval in seconds?

Experiment 4 147

TIME
INTERVAL

1st

2nd

3rd

4th

5th

6th

TABLE 1

DISTANCE TRAVELED IN
EACH TIME INTERVAL

0.48 cm

0.48

Table 1 has data that indicate uniform
motion. Since the object traveled 0.48 cm dur-
ing 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 2, particularly if
you measure with a ruler.

TABLE 2

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

Q3 Is the speed constant in this case? Since
the distances are not all the same, you might
well say, "No, it isn't." Or perhaps you looked
again at a couple of the more extreme data in
Table 2, such as 0.46 and 0.50 cm, checked
these measurements, and found them doubt-
ful. Then you might say, "The ups and downs
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 0.1
cm (the nearest whole millimeter) and esti-
mate the next digit between the marks for the
nearest tenth of a millimeter (0.01 cm), as
illustrated in Fig. 1-2 at the left.

In the same way, whenever you read the
divisions of any measuring device you should

read accurately to the nearest division or mark
and then estimate the next digit in the mea-
surement. Then probably your measurement,
including your estimate of a digit between
divisions, is not more than half of a division
in error. It is not likely, for example, that in
Fig. 1-2 on page 146 you would read more
than half a millimeter away from where the
edge being measured comes between the
divisions. In this case, in which the divisions
on the ruler are millimeters, 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 be-
tween distance measurements are due only to
the uncertainty in reading the ruler. What is
then the best estimate of the constant dis-
tance the object traveled between flashes?

Usually, to find the "best" value of dis-
tance you must average the values. The aver-
age for Table 2 is 0.48 cm, but the 8 is an un-
certain measurement.

If the motion recorded in Table 2 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
measurement. The uncertainty for a single
measurement is commonly taken to be half a
scale division. With many measurements, this
uncertainty may be less, but you can use it to
be on the safe side.

Now you can return to the big 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 2 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 down 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.

Our 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, we simply cannot
reliably detect it with our ruler.

148 Experiment 4

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Experiment 4 149

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150

Experiment 4

Study your own data in the same way.
Q4 Do they lead you to the same conclusion?
If your data vary as in Table 2, can you think
of anything in your setup that could have been
making 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 our example was not con-
stant. For example, if we used a measuring
microscope whose divisions are 0.001 cm apart
to measure the same picture again more pre-
cisely, we might arrive at the data in Table 3.
Such precise measurement reduces the uncer-
tainty greatly from ±0.05 cm to ±0.0005 cm.

TABLE 3

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

Just as in the example on Text page 19, lay
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.

Then lay 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.

The data of Table 2 on page 147 are plotted
as an example on the graph of the sample
write up of Experiment 4 on pages 148 and 149.
Q6 In what way does the graph on page 149
show uniform motion? Does your own graph
show uniform motion too?

If the motion in your experiment was not
uniform, review Sec. 1.9of the Text. Then from
your graph find the average speed of your
object over the whole trip.
Q7 Is the average speed for the whole trip
the same as the average of the speeds between
successive measurements?

Q5 Is the speed constant when we measure to
such high precision as this?

The average of these numbers is 0.4804,
and they are all presumably correct within
half a division which is 0.0005 cm. Thus our
best estimate of the true value is 0.4804 ±
0.0005 cm.

Drawing 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.

Additional Questions

Q8 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.)
Q9 The divisions on the speedometer scale
of many cars are 5 mi/hr in size. You can
estimate the reading to the nearest 1 mi/hr.

(a) What is the uncertainty in a speed
measurement by such a speedometer?

(b) Could you measure reliably speed
changes as small as 2 mi/hr? 1 mi/hr?
0.5 mi/hr? 0.3 mi/hr?

ACTIVITIES

USING THE ELECTRONIC STROBOSCOPE

Examine some moving objects illuminated by
an electronic stroboscope. Put a piece of tape
on a fan blade or mark it with chalk and watch
the pattern as you turn the fan on and off.
How can you tell when there is exactly one
flash of light for each rotation of the fan blade?
Observe a stream of water from a faucet,
objects tossed into the air, or the needle of a
running sewing machine. If you can darken
the room completely, try catching a thrown
ball lighted only by a stroboscope. How many
flashes do you need during the flight of the ball
to be able to catch it reliably?

MAKING FRICTIONLESS PUCKS
Method 1. Use a flat piece of dry ice on a very
smooth surface, like glass or Formica. When
you push the piece of dry ice (frozen carbon
dioxide), it moves in a frictionless manner
because as the carbon dioxide changes to a
vapor it creates a layer of CO2 gas between
the solid and the glass. (CAUTION: Don't
touch dry ice with your bare hands; it can
give you a severe frost bite!)

Method 2. Make a balloon puck if your lab does
not have a supply. First cut a 4-inch diameter
disk of 1-inch-thick Masonite. Drill a y" diam-
eter hole part way through the center of the
disk so it will hold a rubber stopper. Then
drill a ■^" diameter hole on the same center
the rest of the way through the disk. Drill a
iV" hole through the center of a stopper in the
hole in the masonite disk. Place the disk on
glass or Formica.

Method 3. Make a pressure pump puck. Make
a disk as described in Method 2. Instead of
using a balloon, attach a piece of flexible tub-
ing, attached at the other end to the exhaust
of a vacuum pump as shown in the diagram.
Run the tubing over an overhead support so

*ilth ViG* hole

Masonite I* thicJc

it does not interfere with the motion of the
puck.

Method 4. Drill a ■^" hole in the bottom of a
smooth-bottomed cylindrical can, such as one
for a typewriter ribbon. Break up dry ice
(DON'T touch it with bare hands) and place
the pieces inside the can. Seal the can with
tape, and place it on a very smooth surface.

152 Experiments

Chapter £m Free Fall— Galileo Describes Motion

Accelerated motion goes on all around you
every day. You experience many accelerations
yourself, although not always as exciting as
those shown in the photographs. What accel-
erations have you experienced today?

When you get up from a chair, or start to

walk from a standstill, hundreds of sensations
are gathered from all over your body in your
brain, and you are aware of these normal ac-
celerations. Taking off in a jet or riding on an
express elevator, you experience much sharper
accelerations. Often this feeling is in the pit
of your stomach. These are very complex
motions.

Note how stripped down and simple the
accelerations are in the following experiments,
film loops, activities. As you do these, you will
learn to measure accelerations in a variety of
ways, both old and new, and become more
familiar with the fundamentals of accelera-
tion.

If you do either of the first two experiments
of this chapter, that is, numbers 5 and 6. you
will try to find, as Galileo did, whether dlt'
is a constant for motion down an inclined
plane. The remaining experiments are mea-
surements of the value of the acceleration due
to gravity which is represented by the sym-
bol a„.

Experiment 5

153

EXPERIMENT 5 A SEVENTEENTH-
CENTURY EXPERIMENT

This experiment is similar to the one discussed
by Galileo in the Two New Sciences. It will
give you firsthand experience in working with
tools similar to those of a seventeenth-century
scientist. You will make quantitative measure-
ments of the motion of a ball rolling down 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. Then you should be able to tell whether
it was Aristotle or Galileo who was correct
about his thinking concerning the acceleration
of objects of different sizes.

Reasoning Behind the Experiment

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

But even a ball rolling down a low incline
still moved too fast to measure the speed for
different parts of the descent accurately. So
Galileo worked out the relationship d oo t^ (or
dlt^ = 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 mea-
sure, seventeenth-century scientists now had
a secondary hypothesis they could test by
experiment. And so have you. Sec. 2.8 of the
Text discusses much of this.

Apparatus

The apparatus that you will use is shown in
Fig. 2-1 below. It is similar to that described
by Galileo.

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

You use a water clock to time this experi-

Water clock, operat-ed by
openincj ai^ closing phe
top or the tobe wit^h

Stopping bbck

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CJhscM. Sibra^gHtneSS of

along lb ar)ci adju-^iri^
Support- "Stt^nd^

paper clip to
'l /adjusL flow to

1^ a. convenient^

Fig. 2-1

154 Experiments

Experiment 5 155

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156

Experiment 5

ment because that was the best timing device
available in Galileo's time. The way your own
clock works is very simple. Since the volume
of water is proportional to the time of flow, you
can measure time in milliliters of water. Start
and stop the flow with your fingers over the
upper end of the tube inside the funnel. When-
ever you refill the clock, let a little water run
through the tube to clear out the bubbles.

Compare your water clock with a stop
watch when the clock is full and when it is
nearly empty to determine how accurate it is.
Ql Does the clock's timing change? If so, by
how much?

It is almost impossible to release the ball
with your fingers without giving it a slight
push or pufl. Therefore, dam the ball up, with
a ruler or pencil, and release it by quickly
moving this dam away from it down the in-
clined plane. The end of the run is best marked
by the sound of the ball hitting the stopping
block.

Brief Comment on Recording Data

A good example of a way to record your data
appears on page 154. We should emphasize
again the need for neat, orderly work. Orderly
work looks better and is more pleasing to you
and everyone else. It may also save you from
extra work and confusion. If you have an or-
ganized 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 measure-
ment or not. A few minutes' preparation before
you start work will often save you an hour or
two of checking in books and with friends.

Operating Suggestions

You should measure times of descent for
several diff"erent distances, keeping the in-
clination of the plane constant and using
the same bah. 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 greater inclinations, the ball tends to slide
as well as to roll.

From Data to Calculations

Galileo's definition of uniform acceleration
(Text, page 49) was "equal increases in speed
in equal times." Galileo showed that if an
object actually moved in this way, the total
distance of travel should be directly propor-
tional to the square of the total time of fall,
or d » t^.

Q2 Show how this follows from Galileo's defi-
nition. (See Sec. 2.7 in the Text if you cannot
answer this.)

If two quantities are proportional, a graph
of one plotted against the other will be a
straight line. Thus, making a graph is a good
way to check whether two quantities are pro-
portional. Make a graph of d plotted against
t\

Q3 Does your graph support the hypothesis?
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 or, better
yet, repeat several trials of your experiment
using a stopwatch for timing.
Q4 How many seconds is one milliliter of time
for your water clock? Can the inaccuracy of
your water clock explain the conclusion you
arrived at in Q2 above?

Going Further

1. In Sec. 2.7 of the Text you learned that
a = 2dlt'\ Use this relation to calculate the
actual acceleration of the ball in one of your
runs.

2. If you have time, go on to see whether Gali-
leo or Aristotle was right about the accelera-
tion of objects of various sizes. Measure dlt'^
for several diff"erent sizes of balls, all rolling
the same distance down a plane of the same
inclination.

Q5 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.

Q6 Galileo claimed his results were accurate
to Jo 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?

Experiment 6 157

EXPERIMENT 6 TWENTIETH-CENTURY
VERSION OF GALILEO'S EXPERIMENT

Galileo's seventeenth-century experiment had
its limitations, as you read in the Text, Sec. 2.9.
The measurement of time with a water clock
was imprecise and the extrapolation from ac-
celeration at a small angle of inclination to
that at a verticle angle (90°) was extreme.

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

Determine Ug as carefully as you can. This
is a fundamental measured value in modern
science. It is used in many 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 spacecrafts in today's impor-
tant 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 pro-
cedure is the same as that used in Experiment
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 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 mag-
nifier with a scale attached to measure the
glider's motion recorded on the photograph.

Here the values of d will be millimeters on
the photograph and t will be measured in an
arbitrary unit, the "blink" of the stroboscope,
or the "slot" of the strobe disk.

Plot your data as before on a graph of d vs.
t^. Compare your plotted lines with graphs
of the preceding cruder seventeenth-century
experiment, if they are available. Explain
the differences between them.
Ql Is d/t^ constant for an air track glider?
Q2 What is the significance of your answer to
the question above?

As further challenge, if time permits, try
to predict the value of Ug, which the glider
approaches as the air track becomes vertical.
To do this, of course, you must express d and
t in famOiar units such as meters or feet, and
seconds. The accepted value of Ug is 9.8 m/sec^
or 32 ft/sec^ near the earth's surface.
Q3 What is the percentage error in your cal-
culated value? That is, what percent is your
error of the accepted value?
Percentage error

accepted value — calculated value
= X 100

accepted value
so that if your value of Ug is 30 ft/sec^ your
percentage error

32 ft/sec^ - 30 ft/sec^

32 ft/sec^
= -^x 100 =6%

X 100

Notice that you cannot carry this 6% out
to 6.25% because you only know the 2 in the

fraction 32" to one digit. Hence, you can only
know one digit in the answer, 6%. A calculated
value like this is said to have one significant
digit. You cannot know the second digit in the
answer untO you know the digit following the
2. To be significant, this digit would require
a third digit in the calculated values of 30
and 32.
Q4 What are some of the sources of your error?

158

Experiment 7

EXPERIMENT 7 MEASURING THE
ACCELERATION OF GRAVITY a«

Aristotle's idea that falling bodies on earth
are seeking out their natural places sounds
strange to us today. After all, we know the
answer: It's gravity that 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 gravita-
tional "attraction" of the earth. But 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 an-
swered satisfactorily.

Whether you do one or several parts of
this experiment, you will become more famil-
iar with the effects of gravity— you find the
acceleration of bodies in free fall yourself—
and you will learn more about gravity in later
chapters.

Part A: a^ by Direct Fall*

In this experiment you measure the accel-
eration of a falling object. Since the distance
and hence 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 Ug.

Doing the Experiment

The falling object is an ordinary laboratory
hooked weight of at least 200 g mass. (The
drag on the paper strip has too great an effect
on the fall of lighter weights.) The weight
is suspended from about a meter of paper
tape as shown in the photograph. Reinforce
the tape by doubling a strip of masking tape
over one end and punch a hole in the rein-
forcement one centimeter from the end. With
careful handling, this can support at least
a kilogram weight.

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

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

The tuning fork must have a frequency
between about 100 vibrations/sec and about
400 vibrations/sec. In order to mark the tape,
the fork 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 vibration/sec. 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 most con-
veniently 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 vi-
brating tip against it and said "Go." After a
few practice runs, you will become expert
enough to mark several feet 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 corner of the vibrating tuning fork

gently against a 1-inch square of carbon paper
which the thumbtacks hold ink surface in-
wards over the falling tape. With some prac-
tice, 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 begin-
ning of the pattern. Count 10 intervals be-
tween 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 in a time
t, a distance we shall call dj. The distance
di is described by the equation of free fall:

di = -yot + 2

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

dz = 2vot +

4a„t2

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

da — 3Vot H —

d^ = 4vot +

16a,t^

and so on.

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

Experiment

= Vot +

^Vot +

= v.t. +

7a,t'

159

AB = Vot +

and

From these equations you can see that the
weight falls farther during each time interval.
Moreover, when you subtract each of these
distances, AB, BC, CD, . . . from the subsequent
distance, you find that the increase in dis-
tance fallen is a constant. That is, each dif-
ference BC - AB = CD - BC = DE - CD = agt\
This quantity is the increase in the distance
fallen in each successive 10-wave interval
and hence is an acceleration. Our formula
shows that a body falls with a constant ac-
celeration.

From your measurements of AB, AC, AD,
etc., make a column of AB, BC, CD, ED, etc.,
and in the next column record the resulting
values of Ugt^. The values of Ugt^ should aU be
equal (within the accuracy of your measure-
ments). Why? Make all your measurements as
precisely as you can with the equipment you
are using.

Find the average of all your values of
Ugt^, the acceleration in centimeters/( 10-wave
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 interval t is 10/n seconds. Replacing
t of 10 waves by 10/n seconds gives you the
acceleration, Ug in cm/sec^.

The ideal value of Ug is close to 9.8 m/sec^
but a small force of friction impeding a falling
object is sufficient to reduce the observed value
by several percent.

Ql What errors would be introduced by using
a tuning fork whose vibrations are slower than
about 100 vibrations per second? higher than
about 400 vibrations per second?

Part B: a^, from a Pendulum

You can easily measure the acceleration due to
gravity by timing the swinging of a pendulum.

160

Experiment 7

Of course the pendulum is not falling straight
down, but the time it takes for a round-trip
swing still depends on Ug. The time T it takes
for a round-trip swing is

T = 27rxff

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

You may learn in a later physics course
how to derive the formula. Scientists often use
formulas they have not derived themselves,
as long as they are confident of their validity.

Making the Measurements

The formula is derived for a pendulum with
all the mass concentrated in the weight at the
bottom, called the bob. Hence the best pen-
dulum 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, I, 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, either in feet or meters.

Set the pendulum swinging with small
swings. The formula doesn't 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 start-
ing 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
will be only jo as large as if you had measured
only one.)

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 quan-
tities into the formula and solve it for a^.

If you measured I in meters, the accepted
value of ttg is 9.80 meters/sec^.

If you measured I in feet, the accepted
value of ttg is 32.1 ft/sec^.

Finding Errors

You probably did not get the accepted value.
Find your percentage error by dividing your
error by the accepted value and multiplying by
100:

Percentage error

_ accepted value — your value

accepted value

X 100

your error
accepted value

X 100

With care, your value of Op should agree within
about 1%.

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

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

If your possible error in measuring is not
enough to explain your difference in Oy try
changing your total time by a few tenths of a
second— a possible error in timing. Then you
must recalculate T and hence a,.

If neither of these attempts work (nor
both taken together in the appropriate direc-
tion) then you almost certainly have made an
error in arithmetic or in reading your measur-
ing instruments. It is most unlikely that a„ in
your school differs from the accepted value by
more than one unit in the third digit.
Q2 How does the length of the pendulum af-
fect your value of T? of ag?

Q3 How long is a pendulum for which T = 2
seconds? This is a useful timekeeper.

Part C: a,, with Slow-Motion
Photography (Film Loop)

With a high speed movie camera you could
photograph an object falling along the edge of
a vertical measuring stick. Then you could

Experiment 7

161

determine a^ 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 1 and 2. Detailed directions are
given for their use in the Film Loop notes on
pages 164-165.

Part D: a^ from Falling Water
Drops

You can measure the acceleration due to grav-
ity Ug simply with drops of water falling on a
pie plate.

Put the pie plate or a metal dish or tray on
the floor and set up a glass tube with a stop-
cock, valve, or spigot so that drops of water
from the valve will fall at least a meter to the
plate. Support the plate on three or four pen-
cils to hear each drop 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, now find the
time interval t between drops. For greater
accuracy, you m.ay want to count the number
of drops that fall in half a minute or a minute,
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.

Now you have all the data you need. You
know the time t it takes a drop to fall a dis-
tance d from rest. From these you can calcu-
late Ug, since you know that d = 2^gt'^ for ob-
jects falling from rest. Rewrite this relationship
in the form ag=. . .

Q4 When you have calculated Ug by this meth-
od, what is your percentage error? How does
this compare with your percentage error by any
other methods you have used? What do you
think led to your error? Could it be leaking
connections, allowing more water to escape
sometimes? How would this affect your an-
swer?

Distance of fall lessened by a puddle form-
ing in the plate: How would this change your
results?

Less pressure of water in the tube after a
period of dripping: Would this increase or
decrease 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
counting against the watch or clock affect your
answer? What other things may have shown
up in your error?

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?

162 Experiment 7

or if the water fell a greater distance, such as
down a stairwell?

Part E: a^ with Falling Ball and
Turntable

You can measure a^ with a record-player turn-
table, 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.

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. Calculate your percentage error and
suggest its probable source.

White l=^per

Turntable

can be graphed and analyzed to give an aver-
age value of ttg. The 12-slot strobe disk gives
a very accurate 60 slots per second. (Or, a
neon bulb can be connected to the ac line out-
let in such a way that it will flash a precise 60
times per second, as determined by the line
frequency. Your teacher has a description of
the approximate circuit for doing this.)

Part F: a,, with Strobe Photography

Photographing a falling light source with the
Polaroid Land camera provides a record that

ACTIVITIES

WHEN IS AIR RESISTANCE IMPORTANT?

By taking strobe photos of various falling ob-
jects, you can find when air resistance begins
to play an important role. You can find the
actual value of the terminal speed for less
dense objects such as a Ping-Pong or styro-
foam ball by dropping them from greater and
greater heights until the measured speeds do
not change with further increases in height.
(A Ping-Pong ball achieves terminal speed
within 2 m.) Similarly, ball bearings and mar-
bles can be dropped in containers of liquid
shampoo or cooking oil to determine factors
affecting terminal speed in a liquid as shown
in the adjoining photograph.

-^v^IZrs

A magnet is a
handy aid in rais-
ing the steel ball
to the top of the
container.

MEASURING YOUR REACTION TIME

Your knowledge of physics can help you cal-
culate your reaction time. Have someone hold
the top of a wooden ruler while you space your
thumb and forefinger around the bottom (zero)
end of the ruler. As soon as the other person
releases the ruler, you catch it. You can com-
pute your reaction time from the relation

d = Ugt^
by solving for t. Compare your reaction time
with that of other people, both older and young-
er than yourself. Also try it under different con-

ditions—lighting, state of fatigue, distracting
noise, etc. Time can be saved by computing d
for Jo sec or shorter intervals, and then taping
reaction-time marks on the ruler.

A challenge is to try this with a one-dollar
bill, telling the other person that he can have
the dollar if he can catch it.

FALLING WEIGHTS

This demonstration shows that the time it
takes a body to fall is proportional to the square
root of the vertical distance (d °^ t^). Suspend
a string, down a stairwell or out of a window,
on which metal weights are attached at the
following heights above the ground: 3", 1',
2'3", 4', 6'3", 9', 12'3", 16'. Place a metal tray
or ashcan cover under the string and then drop
or cut the string at the point of suspension. The
weights will strike the tray at equal intervals
of time— about ,8 second.

Compare this result with that obtained
using a string on which the weights are sus-
pended at equal distance intervals.

EXTRAPOLATION

Many arguments regarding private and public
policies depend on how people choose to ex-
trapolate from data they have gathered. From
magazines, make a report on the problems of
extrapolating in various cases. For example:

1. The population explosion

2. The number of students in your high
school ten years from now

3. The number of people who will die in
traffic accidents over next holiday
weekend

4. The number of lung cancer cases that
will occur next year among cigarette
smokers

5. How many gallons of punch you
should order for your school's Junior
prom

To become more proficient in making statis-
tics support your pet theory— and more cautious
about common mistakes— read How to Lie
with Statistics by Darrell Huff, published by
W. W. Norton and Company.

FILM LOOPS

FILM LOOP 1 ACCELERATION DUE
TO GRAVITY - 1

A bowling ball in free fall was filmed in real
time and in slow motion. Using the slow-
motion sequence, you can measure the ac-
celeration of the ball due to gravity. This film
was exposed at 3900 frames/sec and is pro-
jected at about 18 frames/sec; therefore, the
slow-motion factor is 3900/18, or about 217.
However, your projector may not run at ex-
actly 18 frames/sec. To calibrate your pro-
jector, time the length of the entire film which
contains 3331 frames. (Use the yellow circle
as the zero frame.)

To find the acceleration of the falling body
using the definition

acceleration =

change in speed
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 first half of any time interval, the
instantaneous speed is less than the average
speed; during the second half of the interval,
the speed is greater than average. Therefore,
for uniformly accelerated motion, the average
speed Vf,,. for the interval is the same as the
instantaneous speed at the mid-time of the
interval.

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

a =

V2-V,

tj ti
where v, and v.2 are the average speeds during

the two intervals, and where t, and ^2 are the
midtimes of these intervals.

Two intervals 0.5 meter in length are
shown in the film. The ball falls 1 meter be-
fore reaching the first marked interval, so it
has some initial speed when it crosses the first
line. Using a watch with a sweep second hand,
time the ball's motion and record the times at
which the ball crosses 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 determine the time
(in apparent seconds) between the midtimes
of the two intervals and the time required for
the ball to move through each j-meter inter-
val. 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
Va,,. Finally, calculate the acceleration a.

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

Film Loops 165

FILM LOOP 2 ACCELERATION DUE
TO GRAVITY -II

A bowling ball in free fall was filmed in slow
motion. The film was exposed at 3415 frames/
sec and it is projected at about 18 frames/
sec. You can calibrate your projector by timing
the length of the entire film, 3753 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 t; is x; — 0, or

V, and the average speed is Vgv =

O + i;

= -9V. The

time required to fall this distance is given by

The acceleration a is given by

a =

change of speed _ At;
time interval At

2dlv

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

instantaneous speed but only average speed
over the interval. For a small interval, how-
ever, you 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 mid-
time, 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 Im, 2m, 3m, and 4m
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 average out errors of mea-
surement. Convert your measured times into
real times using the slow-motion factor. Com-
pute the speeds, in m/sec, and then compute
the value of v^l2d 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 values?
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 judg-
ment based on the consistency of your four
measured values of the acceleration.

B.C. by John Hart

By permission of John Hart and Field Enterprises, Inc.

166 Experiments

Chapter O

The Birth of Dynamics — Newton Explains Motion

EXPERIMENT 8 NEWTON'S SECOND
LAW

Newton's second law of motion is one of the
most important and useful laws of physics.
Review Text Sec. 3.7 on Newton'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 proved by any
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 behavior of
objects in terms of force (F), mass (m), and
acceleration (a). You do this in the first part
of the experiment.

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

You will apply different forces to carts of
different masses and measure the accelera-
tion.

Fig. 3-1

How the Apparatus Works

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

Arrange the apparatus as shown in Fig.
3-1. A spring scale is firmly taped to a dynam-
ics cart. The cart, carrying a blinky, is pulled
along by a cord attached to the hook of the
spring scale. The scale therefore measures
the force exerted on the cart.

The cord runs over a pulley at the edge of
the lab table and from its end hangs a weight.

Fig. 3-2

Experiment 8 167

(Fig. 3-2.) The hanging weight can be changed
so as to produce various tensions in the cord
and hence 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
may want to include with it to vary the mass.
This is the mass m 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
midpoint of this range is a fairly good measure-
ment of the average force Fav producing the
acceleration.

Record Fav in newtons.

Our faith in Newton's law is such that we
assume the acceleration is the same and is
constant every time this particular Fav acts on
the mass m.

Use Newton's law to predict what the aver-
age acceleration Uav was during the run.

Then find a directly to see how accurate
your prediction was.

To measure the average acceleration Uav
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 170, or a blinky. Analyze your results just
as in the experiments on uniform and accel-
erated motion 4, 5, and 6 to find Uav

This time, however, you must know the
distance traveled in meters and the time in-
terval in seconds, not just in blinks, flashes or
other arbitrary time units.
Ql Does Fav (as measured) equal maav (as com-
puted from measured values)?

You may wish to observe the following
effects without actually making numerical
measurements.

1. Keep the mass of the cart constant and ob-
serve how various forces affect the accelera-
tion.

2. Keep the force constant and observe how
various masses of the cart affect the accelera-
tion.

Q2 Do your observations support Newton's
second law? Explain.

Experimental Errors

It is unlikely that your values of Fav and maav
were equal.

Does this mean that you have 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 in-
equality. One may be that you have not yet
measured everything necessary in order to
get an accurate value for each of your three
quantities.

In particular, the force used in the calcu-
lation 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
measure it by reading the spring scale as you
tow the cart by hand at constant speed. Do it
several times and take an average, Ff. Since
Ff acts in a direction opposite to the towing
force Ft,

Piiel — Ft — Ff

If Ff is too small to measure, then F„e, = Ft.
which is simply the towing force that you
wrote as Far in the beginning of the experi-
ment.

Another reason for the inequality of Fav
and niav may be that your value for each of
these quantities is based on measurements
and every measurement is uncertain to some
extent.

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

Uncertainty in average force F„,. Your uncer-
tainty in the measurement of F„,. 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 1.4N the average is 1.2N, and the range of
uncertainty is 0.2N. The value of F„,. would be
reported as 1.2 ± 0.2N.

Q3 What is your value of F„,. and its uncer-
tainty?

Uncertainty in mass m Your uncertainty in m
is roughly half the smallest scale reading of

168

Experiment 8

the balance with which you measured it. The
mass consisted of a cart, a bhnky, and a spring
scale (and 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:

m cart = 0.90 ± 0.05 kg
m blinky = 0.30 ± 0.05 kg
w scale =0.10 ±0.05 kg

The total mass being accelerated is the sum
of these masses. The uncertainty in the total
mass is the sum of the three uncertainties.
Thus, in our example, m — 1.30 ± 0.15 kg.
Q4 What is your value of m and its uncer-
tainty?

Uncertainty in average acceleration a„,. Finally,
consider a„^,. You found this by measuring
Ad/At for each of the intervals between the
points on your blinky photograph.

Adj — »|< Ld^

|^Ad.-4^ Ad-
Fig. 3-3

Suppose the points in Fig. 3-3 represent
images of a light source photographed through
a single slot— giving 5 images per second.
Calculate Ad/At for several intervals.

If you assume the time between blinks to
have been measured very accurately, the
uncertainty in each value of Ad/At is due pri-
marily 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 the Table below.

Average speeds Average accelerations

Ad,/At = 2.5 ± 0.1 cm/sec
AdJM = 3.4 ± 0.1 cm/sec
AdJAt = 4.0 ± 0.1 cm/sec
AdJAt = 4.8 ± 0.1 cm/sec

Aj/,/At = 0.9 ± 0.2 cm/sec^
AyJM = 0.6 ± 0.2 cm/sec^
Ay,/ At = 0.8 ± 0.2 cm/sec^
Average = 0.8 ± 0.2 cm/sec^

When you take the differences between
successive values of the speeds, Ad/At, you
get the accelerations, At;/At, which are re-
corded in the second column. When a differ-
ence in two measurements is involved, you

find the uncertainty of the differences (in this
case, AvIM) by adding the uncertainties of the
two measurements. This results in an uncer-
tainty in acceleration of (±0.1) + (±0.1) or ±0.2
cm/sec^ as recorded in the table.
Q5 What is your value of aav and its uncer-
tainty?

Comparing Your Results

You now have values of Far, Tn and aav, their
uncertainties, and you consider the uncer-
tainty of ma„,. When you have a value for the
uncertainty of this product of two quantities,
you will then compare the value of ma„,. with
the value of Fav and draw your final conclu-
sions. For convenience, we have dropped the
"av" 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 fac-
tors. In our example, m x a = 1.30 kg x 0.8
cm/sec^ = 1.04 newtons. 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% and we
can write our product as ma = 1.04 N +0.36%
which is, to two significant figures,

ma = 1.04 ±0.36 N
(The error is so large here that it really isn't
appropriate to use the two decimal places;
we could round off to 1.0 ± 0.4 N.) In our ex-
ample we found from direct measurement
that Fnet = 1.2 ± 0.2 N. Are these the same
quantity?

Although 1.0 does not equal 1.2, the range
of 1.0 ± 0.4 overlaps the range of 1.2 + 0.2, so
we can say that "the two numbers agree within
the range of uncertainty of measurement. '

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

In a similar way, work out your own values
of F„e, and maa,-

Q6 Do your own values agree within the range
of uncertainty of your measurement?
Q7 Is the relationship F„p, ^maav consistent
with your observations?

Experiment 9 169

EXPERIMENT 9 MASS AND WEIGHT

You know from your own experience that an
object that is pulled strongly toward the earth
(like, say, an automobile) is difficult to ac-
celerate by pushing. In other words, objects
with great weight also have great inertia. But
is there some simple, exact relationship be-
tween the masses of objects and the gravita-
tional 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 observing the accelerations they each ex-
perience when acted on by the same force.
Accelerating an object in one direction with a
constant force for long enough to take mea-
surements is often not practical in the labora-
tory. Fortunately there is an easier way. If
you rig up a puck and springs between two
rigid supports as shown in the diagram, you

■nnnnrwnnnnmnrdnnr^-q

^-cTrirsirsinnnsTnrsTr^

jMiiatmanjJl'noCToot

can attach objects to the puck and have the
springs accelerate the object back and forth.
The greater the 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 5 com-
plete round trips is a convenient measure.
Tape pucks on top of the first one, and time
the period for each new mass. (The units of
mass are not essential here, for we 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.

Ql Does there seem to be a simple relation-
ship between mass and period? Could you
write an algebraic expression for the relation-
ship?

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 we are interested
only in the ratio of the weights.

Comparing Mass and Weight

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

Q2 How does the ratio of the gravitational
forces compare to the ratio of the masses?
Q3 Describe a similar experiment that would
compare the masses of two iron objects to the
magnetic forces exerted on them by a large
magnet.

You probably will not be surprised to find
that, to within your uncertainty of measure-
ment, the ratio of gravitational forces is the
same as the ratio of masses. Is this really worth
doing an experiment to find out, or is the an-
swer obvious to begin with? Newton didn't
think it was obvious. He did a series of very
precise experiments using many different
substances to find out whether gravitational
force was always proportional to inertial mass.
To the limits of his precision, he found the
proportionality to hold exactly. (Newton's
results have been confirmed to a precision of
±0.000000001%, and extended to gravitational
attraction to bodies other than the earth).

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 inertia. 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. (See "Outside and
Inside the Elevator" in the Unit 5 Reader.)
Even before Einstein, Ernst Mach made the
ingenious suggestion that inertia is not the
property of an object by itself, but is the re-
sult of the gravitational forces exerted on an
object by everything else in the universe.

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

One teacher suggests placing 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 leave the beaker
sitting on the table.

PULLS AND JERKS

Hang a weight (such as a
heavy wooden block 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 along at such a speed that the

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

The acceleration will 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 accelero-
meter the last time you lurched forward in the
seat of your car as the brakes were applied.
With a knowledge of Newton's laws and cer-
tain information about you, anybody who
measured how far you leaned forward and how
tense your muscles were would get a good
idea of the magnitude and direction of the
acceleration that you were undergoing. But
it would be complicated.

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

A. The Liquid-Surface
Accelerometer

This device is a hollow, fiat plastic container

B.C.

By John Hart

By permission of John Hart and Fteld Enterprises, Inc.

Activities

171

partly filled with a colored liquid. When it is
not being accelerated, the liquid surface is
horizontal, as shown by the dotted line in Fig.
3-4. But when it is accelerated toward the left
(as shown) with a uniform acceleration a, the
surface becomes tilted, with the level of the
liquid rising a distance h above its normal
position at one end of the accelerometer and
falling 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.

acceleration

Fig. 3-4

The length of the accelerometer is 11, as
shown in Fig. 3-4 above. So the slope of the
surface may be found by

_ vertical distance
horizontal distance

I
Theory gives you a very simple relation-

ship between this slope and the acceleration

h a
slope ^ 7 = —

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, h = I and the sur-
face makes a 45° angle with its normal, hori-
zontal direction. If it is accelerating with j
flg, then the slope will be 2"; that is fi = |- /. In
the same way, if h = j I, then a'=\ Uy^ 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. 3-5 below. Then stick a piece
of white paper or tape to the back of the in-
strument to make it easier to read the level
of the liquid. Solving the equation above for
a gives

a^ UaX

wWit^ p&pe.r on
back, of c^eJI y
\ 'M^

Onacc-eieratcei liquid Level

Accelerate licjoid sur-face
Fig. 3-5

cm soale on
front of eel

B.C.

By John Hart

By permission of John Hart and Field Enterprises, Inc.

172

Activities

This shows that if you place a scale 10 scale
units away from the center you can read accel-
arations directly in joth's of "G's." Since ay
is very close to 9.8m/sec^ at the earth's surface
if you place the scale 9.8 scale units from the
center you can read accelerations directly in
m/sec^. For example, if you stick a centimeter
tape just 9.8 cm from the center of the liquid
surface, one cm on the scale is equivalent to
an acceleration of one m/sec^.

Calibration of the Accelerometer

You do not have to trust blindly the theory
mentioned above. You can test it for yourself.
Does the accelerometer really measure accel-
erations directly in m/sec^? Stroboscopic meth-
ods give you an independent check on the
correctness of the theoretical prediction.

Set the accelerometer on a dynamics cart
and arrange strings, pulleys, and masses as
you did in Experiment 9 to give the cart a uni-
form acceleration on a long tabletop. Don't
forget to put a block of wood at the end of the
cart's path to stop it. Make sure that the ac-
celerometer is fastened firmly enough so that
it will not fly off" the cart when it stops sud-
denly. 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
measure the accelerations from your strobe
records, plot t'^ against d, as you did in Experi-
ment 5. (What relationship did Galileo dis-
cover between dlt^ and the acceleration?) Or
use the method of analysis you need in Experi-
ment 9.

Compare your stroboscopic measurements
with the readings on the accelerometer during
each motion. It takes some cleverness to read
the accelerometer accurately, particularly
near the end of a high-acceleration run. One
way is to have several students along the table
observe the reading as the cart goes by; use
the average 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 stroboscopically measured accelerations.
This graph is called a "calibration curve." If
the two methods agree perfectly, the graph
will be a straight line through the origin at a
45° angle to each axis. If your curve turns out
to have some other shape, you can use it to
convert "accelerometer readings" to "accel-
erations"—if you are willing 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-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. 3-6 below.

0-4- i

O'i

ac/z^e

1 0-2

j o- \

Ini

-O- 1 ,

j —

-o-2>_:;: 1

■

Iw -■

.f-'v-^--.

Fig. 3-6

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 jq g = (about) 1 m/sec^.
by the same reasoning as before. The two
vertical arms should be at least three-fourths
as long as the horizontal arm (to avoid splash-
ing out the liquid during a quick stop). Attach
a scale to one of the vertical arms, as shown.
Holding the long arm horizontal, 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

Activities

173

the scale (and the arm by it) with cloth or card-
board, but leave both ends open. It is essential
that the accelerometer be horizontal if its
readings are 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 in-
troduce error in the accelerometer readings.
Can you think of a way to avoid this kind of
error?

C. Automobile Accelerometer-ll

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 in the sketch below. If the frame is
kept level, the acceleration of the system can

sc<xle

U)ood J*;
-Frame

Ikg cart \0^^

:-^;wv^::^-^^:rs^X

be read directly from the spring scale, since
one newton of force on the 1-kg mass indicates
an acceleration of one m/sec'. (Instead of a
cart, any 1-kg object can be used on a layer
of low-friction plastic beads.)

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 pre-
vious instruments do.

Hang a smaU metal pendulum bob by a
short string fastened to the middle of the lid
of a one-quart mason jar as shown on the left
hand side of the sketch at the bottom of the
page. 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 de-
pends 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 question is trickier than it looks.

To make a fascinating variation on the
damped-pendulum accelerometer, simply re-
place the pendulum bob with a cork and turn
the bottle upside down as shown on the right
hand side of the sketch at the bottom of the
page. If you have punched a hole in the bottle
lid to fasten the string, you can prevent leakage
with the use of sealing wax, parafin, or tape.

This accelerometer will do just the opposite
from what you would expect. The explanation
of this odd behavior is a little beyond the scope
of this course: it is thoroughly explained in
The Physics Teacher, vol. 2, no. 4 (April 1964)
page 176.

D. Damped-Pendulum Accelerometer

One advantage of liquid-surface acceler-
ometers is that it is easy to put a scale on them
and read accelerations directly from the instru-
ment. They have a drawback, though; they
give only the component of acceleration that
is parallel to their horizontal side. If you ac-
celerate one at right angles to its axis, it doesn't
register any acceleration at all. And if you don't
know the direction of the acceleration, you
have to use trial-and-error methods to find it
with the accelerometers we have discussed up
to this point.

FILM LOOP

FILM LOOP 3 VECTOR ADDITION-
VELOCITY 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 con-
stant setting to maintain a steady speed rela-
tive to the water. The task before you is to
find out if he succeeded.

This photograph was taken 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.

First project the film on graph paper and
mark the lines along which the boat's image
moves. You may need to use the reference
crosses on the markers. 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 cm" where cm refers to mea-
sured separations between marks on the mov-
ing 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?

Ar-*-s

'.^

Fig. 3-7

The head-to-tail method of adding vectors. For a review
of vector addition see Project Physics Programmed
instruction Booklet entitled Vectors II.

The head-to-tail method of adding vectors
is illustrated in Fig. 3-7. 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^E velocity of boat relative to earth

Vb» velocity of boat relative to water

VnE velocity of water relative to earth
Then^ _^ _^

For each heading of the boat, a vector
diagram can be drawn by laying off the ve-
locities 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 over-
board. Time the blocks. Find the speed of the
river, the magnitude of v,, £•

Film Loops 175

Scene 2: The boat heads upstream. Measure
Vbe^ then find Vbw using a vector diagram simi-
lar to Fig. 3-8.

WE.

Fig. 3-8

Scene 3: The boat heads downstream. Measure
Vbe^ then find Vb» using a vector diagram.

90*

270"

Scene 4: The boat heads across stream and
drifts downstream. Measure the speed of the
boat and the direction of its path to find v^be-
Also measure the direction of Vbw, the direc-
tion the boat points. One way to record data is
to use a set of axes with the 0° - 180° axis pass-
ing through the markers anchored in the
river. A diagram, such as Fig. 3-9, will help
you record and analyze your measurements.
(Note that the numbers in the diagram are
deliberately not correct.) Your vector diagram
should be something like Fig. 3-10.

Fig. 3-10

Scene 5: The boat heads upstream at an angle,
but moves directly across stream. Again find
a value for Vbw-

Checking your work: (a) How well do the four
values of the magnitude of Vbh agree with
each other? Can you suggest reasons for any
discrepancies? (b) From scene 4, you can cal-
culate the heading of the boat. How well does
this angle agree with the observed boat head-
ing? (c) In scene 5. you determine a direction
for 7^1,. Does this angle agree with the ob-
served boat heading?

Fig. 3-9

176

Experiment 10

Chapter 4 Understanding Motion

EXPERIMENT 10 CURVES OF
TRAJECTORIES

Imagine you are a ski-jumper. You lean for-
ward at the top of the sHde, grasp the raihng
on each side, and yank yourself out into the
track. You streak down the trestle, crouch and
give a mighty leap at the takeoff lip, and soar
up and out, looking down at tiny fields far
below. The hill flashes into view and you thump
on its steep incline, bobbing to absorb the
impact.

This exciting experience involves a more
complex set of forces and motions than you
can deal with in the laboratory at one time.
Let's concentrate therefore on just one aspect:
your flight through the air. What kind of a path,
or trajectory, would your flight follow?

At the moment of projection into the air a
skier has a certain velocity (that is, a certain
speed in a given direction), and throughout
his flight he must experience the downward
acceleration due to gravity. These are circum-
stances that we can duplicate in the laboratory.
To be sure, the flight path of an actual ski-
jumper is probably aff"ected by other factors,
such as air, velocity and friction; but we now
know that it usually pays to begin experiments
with a simplified approximation that allows
us 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 manu-
facturer's instructions.

The apparatus being used by the students
in the photograph on page 177 consists of two
ramps down which you can roll a steel ball.
Adjust one of the ramps (perhaps with the help
of a level) so that the ball leaves it horizontally.

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 all across the
graph paper, release the ball from various
points up the ramp until you find one from
which the ball falls close to the bottom right-
hand corner 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 im-
pact 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 leaves a mark that you
can see through the onionskin paper, auto-
matically recording the point of impact be-
tween ball and board. (Make sure that the
impact board doesn't move 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 re-
lease it — it is impossible to let go of it in the
same way every time. Instead, dam it up with

Experiment 10 177

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 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 dis-
tances 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 ob-
serving the path the ball follows, you have
completed the first phase of the experiment.

If you have time, you will find it worth
while to go further and explore some of the
properties of your trajectory.

Analyzing 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 curve through the points
as shown in the figure at the bottom of the
page.

You already know that a moving object
on which there is no net force acting will move
at constant speed. There is no appreciable
horizontal force acting on the ball during its
fall, so we can make an assumption that its
horizontal progress is at a constant speed.
Then equally spaced hnes will indicate equal
time intervals.

Draw vertical hnes through the points on
your graph. Make the first line coincide with
the end of the launching ramp. Because of
your plotting procedure these lines should
be equally spaced. If the horizontal speed of
the ball is uniform, these vertical lines are
drawn through positions of the ball separated
by equal time intervals.

Now consider the vertical distances fallen
in each time interval. Measure down from your
horizontal line the vertical fall to each of your

178 Experiment 10

plotted points. Record your measurements in a
column. Alongside them record the corre-
sponding horizontal distances measured from
the first vertical line. A sample of results as
recorded in a student notebook is shown on
the right.

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

Earlier in your work with accelerated
motion you learned how to recognize uniform
acceleration (see Sees. 2.5-2.8 in the Text and
Experiment 5). Use the data you have just
collected to decide whether the vertical motion
of the ball was uniformly accelerated motion.
Q2 What do you find?

Q3 Do the horizontal and the vertical motions
affect each other in any way?
Q4 Write an equation that describes the hori-
zontal motion in terms of horizontal speed v,
the horizontal distance, Ax, and the time of
travel, At.

Q5 What is the equation that describes the
vertical motion in terms of the distance fallen
vertically, At/, the vertical acceleration, Uy,
and the time of travel, At?

7 Vtxluti .

«^

Try These Yourself

There are many other things you can do with

this apparatus. Some of them are suggested

by the following questions.

Q6 What do you expect would happen if you

repeated the experiment with a glass marble

of the same size instead of a steel ball?

Q7 What will happen if you next try to repeat

the experiment starting the ball from a dif-
ferent point on the ramp?
Q8 What do you expect if you use a smaller or
larger ball starting always from the same
reference point on the ramp?
Q9 Plot the trajectory 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 11

179

EXPERIMENT 11 PREDICTION OF
TRAJECTORIES

You can predict the landing point of a ball
launched horizontally from a tabletop at any
speed. If you know the speed, Vq, of the ball as
it leaves the table, the height of the table above
the floor 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 mo-
tion:

^x = v At

and you know an equation for free-fall from
rest:

The time interval is difficult to measure. Be-
sides, in talking about the shape of the path,
all we really need to know is how Ay relates to
Ax. Since, as you found in the previous experi-
ment, these two equations still work when an
object is moving horizontally and falling at
the same time, we can combine them to get an
equation relating Ay and Ax, without At ap-
pearing at all. We can rewrite the equation for
horizontal motion as:

Then we can substitute this expression for t
into the equation for fall:

At =

_ 1

Ay = ^a

(Axy

Thus the equation we have derived should
describe how Ay changes with Ax— that is,
it should give us the shape of the trajectory.
If we want to know how far out from the edge
of the table the ball will land (Ax), we can
calculate if from the height of the table (Ay),
Ug, and the ball's 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. 4-1 below.) 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 equation for Ai/
to calculate Ax. Place a target, a paper cup,
perhaps, on the floor at your predicted landing
spot as shown below. How confident are you of
your prediction? Since it is based on measure-
ment, some uncertainty is involved. Mark an
area around the spot to indicate your uncer-
tainty.

bal I mu^'t' be
e)tiTl in air

Support
Stand

Fig. 4-1

180 Experiment 11

^• — -^

—

-^ - -_ -

:---- ,t-- -— TT.^^

1-".""^ - .^--

/
/

/ Thread.

'<i

/
/

V7 -p ^*

-rziz^

Now release the ball once more. This time,
let it roll off the table and land, hopefully, on
the target as shown in the figure above.

If the ball actually does fall within the
range of values of x you have estimated, then
you have supported the assumption on which
your calculation was based, that vertical and
horizontal motion are not affected by each
other.

QJ How could you determine the range of a
ball launched horizontally by a slingshot?
Q2 Assume you can throw a baseball 40 meters
on the earth's surface. How far could you
throw that same ball on the surface of the
moon, where the acceleration of gravity is
one-sixth what it is at the surface of the earth?
Q^ Will the assumptions made in the equations
l^=vbA and Ai/ = jagC At)^ hold for a Ping-Pong
ball? If the table were 1000 meters above the
floor, could you still use these equations? Why
or why not?

(Tyial.y C^ .j fy-

^iH^ "'^^

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. Gun-
ners had previously found this by experience. What angles give the maximum range?

Experiment 12

181

EXPERIMENT 12 CENTRIPETAL FORCE

The motion of an earth sateUite and of a weight
swung around your head on the end of a string
are described by the same laws of motion.
Both are accelerating toward the center of
their orbit due to the action of an unbalanced
force.

In the following 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.

How the Apparatus Works

Your "satellite" is one or more rubber stop-
pers. When you hold the apparatus in both
hands, as shown in the photo above, 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 read
in newtons or else its readings should be con-
verted to newtons.

You can change the length of the string
so as to vary the radius R of the circular orbit,
and you can tie on more stoppers to vary the
satellite mass m.

The best way to set the frequency / is to
swing the apparatus in time with some peri-
odic sound from a metronome or an earphone
attachment to a blinky. You keep the rate con-
stant by adjusting the swinging until you see
the stopper cross the same point in the room at
every tick.

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 marker (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. It's easiest to 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.

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

Q2 How do changes in / affect F when m and
R are kept constant? Write a formula to ex-
press this too.

Q3 What is the effect of R on F?
Q4 Can you put m, f, and R all together in a
single formula for centripetal force, R?

How does your formula compare with the
expression derived in Sec. 4.7 of the Text.

182 Experiment 13

EXPERIMENT 13 CENTRIPETAL FORCE
ON A TURNTABLE

You may have had the experience of spinning
around on an amusement park contraption
known as the Whirhng Platter. The riders seat
themselves at various places on a large flat
polished wooden turntable about 40 feet 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 you 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 slid-
ing off.

If you do this under a variety of conditions,
you will see for yourself how forces act in cir-
cular 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-jR.

Studying Centripetal Force

For these experiments it is more convenient
to write the formula F = mvVR in terms of the
frequency/. This is because/can be measured
more easily than v. We can rewrite the form-
ula as follows:

^ ^ distance traveled ^ number of revolu-
in one revolution tions per sec

= 2ttR xf

Substituting this expression for v in the form-
ula gives:

P _7nx {2Tr Rfy
R

Friction on a Rotating Disk

For objects on a rotating disk, the centripetal
force is provided by friction. On a frictionless
disk there could be no such centripetal force.
As you can see from the equation we have
just derived, the centripetal acceleration is
proportional to R and to/"-^. Since the frequency
/ is the same for any object moving around
with a turntable, the centripetal 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 force must be to keep it in a
circular path.

You can measure the maximum force
F,„„j, that friction can provide on the object,
measure the mass of the object, and then cal-
culate the maximum distance from the center
R,„ax that the object can be without sliding off.
Solving the centripetal force equation for R
gives

_ 4ir^mR^P
R

= AttZ

4Tr^mRp

You can measure all the quantities in this
equation.

4n^mP

Use a spring scale to measure the force needed
to make some object (of mass m from 0.2 to
1.0 kg) start to slide across the motionless

Experiment 13

183

disk. This will be a measure of the maximum
friction force that the disk can exert on the
object.

Then make a chalk mark on the turntable
and time it (say, for 100 sec)— or accept the
marked value of rpm— and calculate the fre-
quency in rev/sec.

Make your predictions of R,„ax for turn-
table frequencies of 33 revolutions per minute
(rpm), 45 rpm, and 78 rpm.

Then try it!
Ql How great is the percentage difference

between prediction and experiment for each
turntable frequency? Is this reasonable agree-
ment?

Q2 What efTect would decreasing the mass
have on the predicted value of R? Careful!
Decreasing the mass has an effect on F also.
Check your answer by doing an experiment.
Q3 What is the smallest radius in which you
can turn a car if you are moving 60 miles an
hour and the friction force between tires and
road is one-third the weight of the car? (Care-
ful! Remember that weight is equal to a, x m.)

B.C. by John Hart

NO,MO, srURP... THE
OTHER ENP.

By permission of John Hart and Field Enterprises, Inc.

ACTIVITIES

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? Try different ones.

SPEED OF A STREAM 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 Ax from
the nozzle to where the water hits the ground.

Use the equation relating Ax and Ay that
was derived in Experiment 11, solving it for
v:

and

V = Ax

QAy

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

PHOTOGRAPHING A WATERDROP
PARABOLA

Using an electronic strobe light, a doorbell
timer, and water from a faucet, you can photo-
graph a water drop parabola. The principle of
independence of vertical and horizontal mo-
tions 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 without the bell.) Place the tube
through which the water runs under the clap-
per 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 photo on previous page).

To get more striking power, run the vibra-
tor from a variable transformer (Variac) con-
nected to the 110 volt a.c, 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 drop-
per nozzle. By viewing the drops with 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 in-
stead of the 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). The photo above was made in that

Activities

185

way. With front 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.

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

BALLISTIC CART PROJECTILES

Fire a projectile straight up from a cart or toy
locomotive as shown in the photo below 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.

MOTION IN A ROTATING REFERENCE
FRAME

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

Method I Attach a piece of paper to a phono-
graph turntable. Draw a line across the paper
as a turntable is turning (see Fig. 4-2 below),
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.

'tutrnfai^^

Fig. 4-2

186

Activities

Method II Place a Polaroid camera on the turn-
table on the floor and let a tractor run along
the edge of a table, with a flashlight bulb on a
pencil taped to the tractor so that it sticks out
over the edge of the table.

I j'flljt source

^ — Couv»"te»r- uitiqKt

V^OtSIb pf=** ^tov -tractor

caiTMerck

furnfftble"^

Method III How would an elliptical path appear
if you were to view it from a rotating reference
system? You can find out by placing a Polaroid
camera on a turntable on the floor, with the
camera aimed upwards. (See Fig. 4-3 below.)
For a pendulum, hang a flashhght bulb and
an AA dry cell. Make the pendulum long enough
so that the light is about 4 feet from the cam-
era lens.

/i^Vit Source

on 5*"i

different points in its swing by using a motor
strobe in front of the camera, or by hanging
a blinky.

PENNY AND COAT HANGER

Bend a coat hanger into the shape shown in
the sketch below in this right-hand column.
Bend the end of the hook slightly with a pair of
pliers so that it points to where the finger sup-
ports 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 (and bal-
anced penny) starts swinging Uke a pendulum.
Some practice will enable you to swing the
hanger in a vertical circle, or around your head
and still keep the penny on the hook. The cen-
tripetal 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.

■Turns On -firiacr- here.

Camtra.

turntoUe

Fig. 4-3

With the hghts out, give the pendulum a
swing so that it swings in an elliptical path.
Hold the shutter open while the turntable
makes one revolution. You can get an indi-
cation of how fast the pendulum moves at

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 108 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 our Milky Way galaxy.

Activities

187

FILM LOOPS

FILM LOOP 4 A MATTER OF RELATIVE
MOTION

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

Even though Events A, B, and C are visibly
different to the observer, in each the carts
interact similarly. The laws of motion apply
for each case. Thus, these events could be the
same event observed from different reference
frames. They are closely similar events photo-
graphed from different frames of reference, as
you see after the initial sequence of the film.

The three events are photographed by a
camera on a cart which is on a second ramp
parallel to the one on which the colliding carts
move. The camera is your frame of reference,
your coordinate system. This frame of refer-
ence 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 inde-
pendently 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 different
frames of reference?

You might think that the question of which
cart is in motion is resolved by sequences at

B.C. by John Hart

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 dif-
ferent when this reference object is present.
But is this fixed frame of reference any more
fundamental than one of the moving frames
of reference? fixed relative to what? Or is
there a "completely" fixed frame of reference?
If you have studied the concept of momen-
tum, you can also consider each of these three
events from the standpoint of momentum
conservation. Does the total momentum de-
pend on the frame of reference? Does it seem
reasonable to assume that the carts would
have the same mass in all the frames of refer-
ence used in the film?

By permission of John Hart and Field Enterprises, Inc.

188

Film Loops

FILM LOOP 5 GALILEAN RELATIVITY-
BALL DROPPED FROM MAST OF SHIP

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

If it be true that the impetus with which
the ship moves remains indelibly im-
pressed in the stone after it is let fall from
the mast; and if it be further true that
this motion brings no impediment or
retardment to the motion directly down-
wards natural to the stone, then there
ought to ensue an effect of a very won-
drous 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. Ac-
cording to what has been premised, it
shall take up the time of two pulses in its
fall, in which time the ship will 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 first
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 des-
cribe its transverse hnes still longer and
longer and yet shall pass them all in those
selfsame two pulses.
In the film a ball is dropped three times:

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.

Scene 2: The 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: The ball is picked up and held
briefly before being released.

The 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 par-
abola and it strikes a point below and to the
left of the starting point."

To test your understanding of Galilean
relativity, you should describe the following:
Scene 2 from the boat frame; Scene 2 in earth
frame; Scene 3 from the boat frame; Scene 3
from the earth frame.

Film Loops 189

FILM LOOP 6 GALILEAN RELATIVITY-
OBJECT DROPPED FROM AIRCRAFT

A Cessna 150 aircraft 23 feet long is moving
about 100 ft/sec at an altitude of about 200
feet. The action is filmed from the ground as
a flare is dropped from the aircraft. Scene 1
shows part of the flare's motion; Scene 2, shot
from a greater distance, shows several flares
dropping into a lake; Scene 3 shows the ver-
tical motion viewed head-on. Certain frames
of the film are "frozen" to allow measure-
ments. The time interval between freeze
frames is always the same.

Seen from the earth's frame of reference,
the motion is that of a projectile whose original
velocity is the plane's velocity. If gravity is the
only force acting on the flare, its motion should
be a parabola. (Can you check this?) Relative
to the airplane, the motion is that of a body
falling freely from rest. In the frame of refer-
ence of the plane, the motion is vertically
downward.

The plane is flying approximately at uni-
form 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 = -g-at^. But the trouble is that you cannot
be sure that the first freeze frame occurs at

the very instant the flare is dropped. However,
there is a way of getting around this difficulty.
Suppose a time B has elapsed between 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 = ^a(t + By

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

Vd = (constant) (t + B)

Now if we plot Vd against t, we expect a
straight line. Moreover, if B = 0, this straight
line will also pass through the origin.

Suggested Measurements

(a) Vertical motion. Project Scene 1 on paper.
At each freeze frame, when the motion on the
screen is stopped briefly, mark the positions
of the flare and of the aircraft cockpit. Measure
the displacement d of the flare below the plane.
Use any convenient units. The times can be
taken as integers, t = 0, 1, 2, . . ., designating
successive 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 origin?

(b) Analyze Scene 2 In the same way.

(c) Horizontal motion. Use another piece of
graph paper with time (in intervals) plotted
horizontally and displacements (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 in the horizontal
motion? the vertical motion? Explain your
findings between the effect of air friction on
the horizontal and vertical motions.

(d) Acceleration due to gravity. The "constant"
in your equation, d = (constant) (t + B), is
ja; this is the slope of the straight-line graph
obtained in part (a). The square of the slope
gives 2^ so the acceleration is twice the

190

Film Loops

square of the slope. In this way you can obtain
the acceleration in squares/(interval)^ To
convert your acceleration into ft/sec- or m/
sec^, you can estimate the size of a "square"
from the fact that the length of the plane is
23 ft (7 m). The time interval in seconds be-
tween freeze frames can be found from the
slow-motion factor.

FILM LOOP 7 GALILEAN RELATIVITY-
PROJECTILE FIRED VERTICALLY

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 tra-
jectories.

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
relative to the earth. Describe the motion of

the flare relative to the earth; describe the
motion of the flare relative to the ski-doo.

Scenes 3 and 4: The ski-doo's speed changes
after 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?

How do the events shown in this film illus-
trate 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 knowing 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 regula-
tion hurdle race, with 1-meter-high hurdles
spaced 9 meters apart. (Judging from the
number of hurdles knocked over, the com-
petitors were of something less than Olympic
caliber!) Next, a runner, Frank White, a 75-
kg student at McGill University, is shown in
medium slow-motion (slow-motion factor 3)
during a 50-meter run. His time was 8.1 sec-
onds. Finally, the beginning of the run is
shown in extreme slow motion (slow-motion
factor of 80). "Analysis of a Hurdle Race 11"
has two more extreme slow-motion sequences.

To study the runner's motion, measure
the average speed for each of the 1-meter
intervals in the slow-motion scene. A "drag-
strip" chart recorder is particularly convenient
for recording the data on a single viewing of
the loop. Whatever method you use for measur-
ing time, the small but significant variations
in speed will be lost in experimental uncer-
tainty unless you work very carefully. Repeat
each measurement several times.

The extreme slow-motion sequence shows
the runner from 0 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 ref-

Film Loops

191

rr^"^

3JMdJ.

erence 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 measure-
ments several times, viewing the film over
again, and average your results for each in-
terval. Your accuracy might be improved by
forming a grand average that combines your
average with others in the class. (Should you
use all the measurements in the class?) Cal-
culate 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 directly
beneath his hip and the time when that foot
is off the ground. Is there any relationship
between your graph of speed and the way the
runner's feet push on the track?

The initial acceleration of the runner can
be estimated from the time to move from the
starting point to the 1 -meter mark. You can
use a watch with a sweep second hand. Calcu-
late the average acceleration, in m/sec^ during
this initial interval. How does this forward
acceleration compare with the magnitude 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?

FILM LOOP 9 ANALYSIS OF A HURDLE
RACE-II

This film loop, which is a continuation of "An-

alysis of a Hurdle Race I," shows two scenes
of a hurdle race which was 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. (See photo-
graph.) 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 runner's pants fur-
nishes a convenient reference point for mea-
surements. (See the film-notes about the
"Analysis of a Hurdle Race I" for further de-
tails.)

No measurement is entirely precise; mea-
surement error is always present, and it cannot
be ignored. Thus it may be difficult to teU 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 experimen-
tal uncertainty graphically, along with the
measured or computed values.

For example, say that the dragstrip timer
was used to make three different measure-
ments of the time required for the first meter
of the run: 13.7 units, 12.9 units, and 13.5
units, which give an average time of 13.28

"2 I-

^ 7

J.
.0

— -^

J L

I I I

0 \ 2 3 ^ S

clfsp\acc.rrefrr ^^eje^)

192 Film Loops

units. (If you wish to convert the dragstrip units
to seconds, it will be easier to wait until the
graph has been plotted using just units, and
then add a seconds scale to the graph.) The
lowest and highest values are about 0.4 units
on either side of the average, so we could report
the time as 13.3 + 0.4 units. The uncertainty
0.4 is about 3% of 13.3, therefore the percent-
age uncertainty in the time is 3%. If we assume
that the distance was exactly one meter, so
that all the uncertainty 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 units, which equals
7.53 cm/unit. Since 3% of 7.53 is 0.23, the
speed can be reported as 7.53 + 0.23 cm/unit.
In graphing this speed value, you plot a point
at 7.53 and draw an "error bar" extending
0.23 above and below the point. 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 commonly obtained in scientific
research. For example, in the figure at the right
a research team has plotted its experimental
data; they published their results in spite of

3.:5- 3.\i, S.17 3.)g 3.19 B.2C 5.2.1 3.2Z

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 signifi-
cant errors here also?

Acceleration, 28-30, 68, 70

as a vector, 75

average, 29

centripetal, 109-113, 114-115

constant (uniform), 29-30, 47-49,
50. 51-52, 54, 58

defined, 75

direction and, 28, 29, 75

due to gravity, 80

force and, 79-80

instantaneous, 29

linear, 28

mass and, 80-81
Aether (see quintessence)
Agena rocket, 80-89
Alouette I (satellite), 114,. 116
Angle of inclination, 54, 57
Aristotelian cosmology, 47, 58-59
Aristotle, 38-42, 46, 58, 59, 69-70

air resistance, 40

his theory of motion, 38, 40, 69-70

his theory of motion attacked, 41,
46, 47

his theory of motion refuted, 58

time line chart, 39
Atom, diameter of, 6
Atoms in the Family: My Life with
Enrico Fermi (Laura Fermi),
1-4

Brahe, Tycho, 119

Centripetal acceleration

force, 109-110
Circular motion, 107-115
Copernicus, Nicolaus, 119
Cosmology

Aristotelian, 38

medieval, 38
Curie, Irene, 1
Curie, Marie, 1
Curie, Pierre, 1

De Medici, Prince Giovanni, 53

De Montbeillard, 24-25

Delta (A), 17

Dialogue on the Two Great World

Systems (GalUeo), 43
Direction

acceleration and, 28, 29, 75
constant, 76
of vectors, 74
velocity and, 75
Discourses and Mathematical
Demonstrations Concerning
Two New Sciences Pertaining
to Mechanics and Local Motion
(Galileo), 43-49, 53, 56-57, 60,
104
Discoveries and Opinions of Galileo
(tr. Drake), 104

INDEX /TEXT

Distance, measuring, 13-15
Dry ice, 11-15, 75-76
Dynamics, 67

"Dynamism of a Cyclist" (Boccioni),
9

Earth

diameter of, 6

motion of, 9

precession of the axis of, 7
Earth satellites, 113-116
Einstein, Albert, 107
Elements, Aristotle's four, 37-38
Equilibrium, 73, 77

forces in, 70-73
Ether (see quintessence)
Extrapolation. 22-23

Fermi, Enrico, 1-5
Fermi, Laura, 1-4
Force(s), 67, 69-70, 81-83, 86-88

acceleration and, 79-80

centripetal, 109-113

directional nature of, 71

equilibrium in, 70-73

friction al, 91

nature's basic, 90-92

net, 71, 72, 73

resultant (see net force)

total (see net force)

vector sum of, 72
Free fall, all Chapter 2, 83-86

defined, 45
Frame(s) of reference, 77-78, 105-

106
Frequency of circular motion. 107,

108
Friction, 91

Galaxies, distance to, 6
Galilean relativity principle, 106-107
Galileo, 30, 36, all Chapter 2, 76-77,
85, 100, 104. 105-107. 118. 119

consequences of works of, 58-60

Dialogue on the Two Great World
Systems, 43

idea of a straight line, 77

time line chart, 42

Two New Sciences, 43
Gemini spacecraft, 88-89
Graphs, 18-23, 24-25, 29

extrapolations of, 22-23

interpolations of, 22
Gravity, 30, 83-86, 91-92

Huygens, Christian, 57,

Hypothesis, 59
direct test of, 59-50
explanations, 67-68
indirect test of, 53-54,
of Galileo, 49-50

118

proven, 58
Inclined plane, 54, 57
Inertia, 77, 78, 80, 85

and Newton's second law, 79-80

law of, 77

measured, 81-82

principle of, 77
International Bureau of Weights

and Measures. 82
Interpolation, 22
Instantaneous acceleration, 29
Instantaneous speed, 23
Instantaneous velocity, 25

Kepler, Johannes, 1 19
Kilogram (unit of mass), 82-83
Kinematics

concepts of, 67

defined, 67

of uniform circular motion, 114

Laws of motion,
Newton's 1st (inertia), 75-78
2nd (force). 79-83
3rd (reaction), 86-88

Mach, E., 83, 85
Magnitude of vectors, 74
Mass, 67, 81-86

acceleration and, 80-81

compared with weight. 80. 84-85

defined, 82

force and, 81

standard of, 82
Mathematical Principles of Natu-
ral Philosophy (Newton), 68, 69,
86
Measurement, 5, 6, 7

accuracy of, 14

distance, 13-15

mass, 82

speed, 12-15

time, 13-15. 56-57

weight, 84
Medieval world system, 37-38
Meter, 82-83
Midpoint, speed at, 24
Moon, a trip to, 99-100
Motion pictures, 26. 27

Natural motion, 69
Newton, (unit of force), 83
Newton, Isaac, 59, 67-92, 98, 100,
117, 118, 119
first law of motion (inertia), 75-78,

88-92, 90-91
idea of a straight line, 77
The Principia, 68
second law of motion, (force). 79-

83. 85, 88-92, 1 10
third law of motion, (reaction), 86-
92

Nucleus, diameter of, 6

Orbit(s)

of Earth satellites, 113-116
Oresme, Nicolas, 47
Oscillation, 116-117

Parabola, 104

Parallelogram method of adding

vectors, 74
Parsimony, rule of, 48
Particles
alpha, 1-2

tracks in a bubble chamber, 7
People and Particles (documentary

film), 5
Period of circular motion, 107, 108
Philoponus, John, 41
Philosophiae Naturalis Principia
Mathematica (Newton), 68, 69,
86
Photography, 11-12, 26-27
Physics, definitions of, 5
Plato, 38

Projectiles, 101-107
defined, 101

trajectory of, 101, 103-105
Pythagorean Theorem, 1 1 1

Quintessence, 38

Radioactivity, 1-4

Raphael, 38

Reference frame(s), 77-78, 105-106

Relativity principle, Gahlean, 106-

107
Relativity theory, 81, 85
Rest, 68, 70, 73, 76, 77, 78
Revolution, 107-109

defined, 107

frequency of, 107

period of, 107

Rockets, 88-89, 99-100
Rotation, defined, 107
Rule of Parsimony, 48

Sagredo, 48

Salviati, 48-49

Simplicio, 47-48

Satelhtes of the earth, 113-116
Scalar quantities, 75
Simple harmonic motion, 117
Slope, of graph, 24, 25

finding, 18-22
Speed
absolute, 106
acceleration and, 28-30
average, 12, 15-17, 23, 24, 29, 108

defined, 17
constant (uniform), 15, 29, 47, 48,

76, 77
defined, 12

distinguished from velocity, 25
expressions of, 12
instantaneous, 12, 23-27, 29, 108
measuring, 12-15
nonuniform, 15
point, at a, 23
relative, 106
Speedometer(s), 12, 25
Stars, distance to, 6
Stroboscopic, 13-14

photography, 27
Sun
distance to, 6
radius of, 21
System of the World (Newton), 98

Tangent of graph line, 29
Time, 7

measuring, 13-15, 56-57
Time Line chart

Aristotle, 39

Galileo, 42

Thought Experiment (Galileo's),

53-54
Trajectory of a projectile, 101, 103-

105
Two New Sciences (Galileo), 43-49,

53, 56-57, 60, 104

Unbalanced force, 79
Uniform motion, 68. 70
U.S. Bureau of Standards. 82

Vacuum. 45-46

Van Goph, Vincent, 42

Vector, 73-75

defined, 75

direction of, 74

displacement, 73

magnitude of, 74

resultant, 74

sum of forces, 72
Velocity, 98, 106

average, 108

circular motion, 107-108

constant. 76

distinguished from speed, 25

frames of reference, 105-106

instantaneous, 25

two ways of changing. 75-78

unchanging. 76, 77

uniform, 70
Verne, Jules, 99
Violent motion, 69

Water clock, 56-57
Weight, 83-86

compared with mass, 80, 84-85

defined, 84

measuring. 84
Weightlessness. 84
World of Enrico Fermi, The (docu-
mentary film), 5

194

INDEX /HANDBOOK

Accelerated motion, 152
Acceleration
centripetal, 181

due to gravity— I (film loop), 164
due to gravity— II (film loop), 165
Acceleration (of gravity)
from falling water drops, 161
from a pendulum, 159-60
measurement by direct fall,

158-59
measurement by slow-motion

photography, 160-61
with falling ball and turntable,

162
with strobe photography, 162
Accelerometers (activity), 170-73
automobile, 172-73
calibration of , 172
damped-pendulum, 173
liquid-surface, 170-71
Activities
ballistic cart projectile, 185-86
beaker and hammer, 170
checker snapping, 170
experiencing Newton's second

law, 170
extrapolation, 163
falling weights, 163
make one of these accelerometers,

170-73
making a frictionless puck, 151
measuring unknown frequencies,

186
measuring your reaction time,

163
motion in a rotating reference

frame, 185-86
penny and coat hanger, 186
photographing a waterdrop

parabola, 184-85
projectile motion demonstration,

184
pulls and jerks, 170
speed of a stream of water, 184
using the electronic stroboscope,

151
when is air resistance important,
163
Air resistance

importance of (activity), 163
Altitude

of object, 134
Archytas, 129
Astrolabe, 137
Astronomy
naked eye (experiment), 134-41
references in 6-7, 134-35
Azimuth, 134, 137

Ballistic cart projectiles (activity),

185
Beaker and hammer (activity), 170
Black Cloud, The, 133
Big Dipper, 135

Camera, Polaroid, 132

Celestial Calendar and Handbook,

139
Centripetal acceleration, 182
Centripetal force (experiment), 181
on a turntable (experiment),

182-83
Checker snapping (activity), 170
Compass, magnetic, 134
Constant speed, 167
Constellations, 135, 136

Data

recording of, 156

variations in (experiment), 144
Direct fall

acceleration by, 158-59

Earth satellite, 181
Einstein, Albert, 169
Experimental errors, 167
Experiments
a seventeenth century experiment

153-56
centripetal force, 181
centripetal force on a turntable,

182-83
curves of trajectories, 176-78
mass and weight, 169
measuring the acceleration of

gravity, 158-62
measuring uniform motion, 145-

150
naked eye astronomy, 134-41
Newton's second law, 166-68
prediction of trajectories, 179-80
regularity and time, 142-43
twentieth-century version of
Galileo's experiment, 157
variations in data, 144
Extrapolation (activity), 163

Falling weights (activity), 163
Film loops

a matter of relative motion, 187

acceleration due to gravity I and
II, 164-65
analysis of a hurdle race I and II,
190-92

Galilean relativity, 188-90

vector addition, 174-75
Free Fall

approximation of, 152-56
Frequency

measuring unknown(activity), 186

of test event, 142
Friction

on a rotating disc, 182-83

Galilean relativity (film loop), 189-91

Galileo, 153, 156, 157
his relativity (film loops), 189-91
his Two New Sciences, 153, 189

Graphs
drawing, 150

195

Gravity
acceleration of , 158-62
measuring acceleration of
(experiment), 158-62

How to Lie with Statistics, 163
Hurdle race
analysis of (film loops), 190-192

Inertia, 169
Instantaneous speed, 164

Laboratory exercises

keeping records of, 129, 130-31
Little Dipper, 135

Mach, Ernst, 169
Magentic declination

angle of, 134
Mass

and weight (comparison), 169

and weight (experiment), 169

measuring, 169
Measurement, precise, 150
Meteors

observation of, 139
Meteor showers

observation of (table), 141
Moon

eclipse of, 139

observation of, 138-39
Motion

accelerated, 152

in rotating reference frame
(activity), 185-86

relative (film loop), 187

uniform measurement of
(experiment), 145-50

Newton, Isaac
experiencing his second law

(activity), 170
his second law of motion
(experiment), 166-68
North-south line, 134-35, 137
North Star (Polaris), 134-35

Parabola, waterdrop

photograph of (activity), 184-85
Pendulum

acceleration from a, 159-60
Penny and coat hanger (activity),

186
Photography

of waterdrop parabola, 184-85

slow-motion, 160-61

stroboscopic, 132, 146, 162
Physics Teacher, The, 173
Planets

and eclipse observations (table),
140

observation of, 139
Polaris (North Star), 134-35
Polaroid camera

use of, 132
Project Physics Reader, 133

Projectiles
ballistic cart (activity), 185-86
motion demonstration (activity),
184
Puck
Making a frictionless (activity),
151
Pulls and jerks (activity), 170

Reaction time

measurement of (activity), 163
References

in astronomy, 134-35

North-south line, 134-135, 137
Regularity

and time (experiment), 142-43

of an event, 142

Satellite, earth, 181
Seventeenth-century experiment,

153-56
Sky and Telescope, 139

Speed

and measurement of motion,
146-147

constant, 166

instantaneous, 164
Standard event, 142
Stars

chart of, 136

observation of, 139
Stroboscope, electronic (activity),

151
Stroboscopic photography, 132,

146, 162
Sun

observation of, 138

Table(s)
f avorability of observing meteor

showers, 141
guide for planet and eclipse

observations, 140

Time

and regularity (experiment),
142-43
Trajectories

curves of (experiment), 176-78

prediction of (experiment),
179-80
Twentieth-century version

of Galileo's experiment, 157
Two New Sciences, 153, 188

Ufano
drawing by, 180

Vectors
addition of (film loop), 174-75
diagrams, 174, 175

Water clock, 153-56
Weight
and mass (experiment), 169

196

Answers to End of Section Question

Chapter 1

Q1 We have no way of knowing the lengths of time

involved in going the observed distances.

Q2 No; the time between stroboscope flashes is

constant and the distance intervals shown are not

equal.

Q3 An object has a uniform speed if it travels

equal distances in equal time intervals; or, if the

distance traveled = constant, regardless of the

particular distances and times chosen.

Q4 Average speed is equal to the distance travelled

divided by the elapsed time while going that

distance.

(entries in brackets are those
already given in the text)

Q6 Hint: to determine location of left edge of puck
relative to readings on the meter stick, line up a
straight edge with the edge of puck and both marks
on meter stick corresponding to a given reading.

Ad /At

(5.0)

(1.0)

(6.0)

(0.8)

(4.5)

1.1

(5.5)

0.9

7.5

0.67

8.0

0.62

8.6

0.58

d(cm)

t(sec)

5 vds
— = 0.6 yd/sec from the table

Q7 The one on the left has the larger slope

mathematically; it corresponds to 100 miles/hr

whereas the one on the right corresponds to 50

miles/hr.

Q8 Most rapidly at the beginning when the slope

is steepest; most slowly toward the end where the

slope is most shallow.

Ad 2.5 yds.

aF " 4 sec " yd/sec from the graph

At 8.6 sec
Q10 Interpolation means estimating values
between data points; extrapolation means estimating
values beyond data points.

Q11 An estimate for an additional lap (extrapola-
tion).

Q12 Instantaneous speed means the limit
approached by the average speed as the time interval
involved gets smaller and smaller.

V = limit ^r- as At approaches zero.

iAt

Q13 Instantaneous speed is just a special case of
average speed in which the ratio Ad/Af does not
change as Af is made smaller and smaller. However,
Ad/ At always gives average speed no matter how
large or how small Af is.

_ final speed - initial speed _ 60 — 0 mph
time elapsed 5 sec

= 12 mph/sec

Q14

Q15

2 mph — 4 mph _

-8 mph/hr, or -0.13

1/4 hr
mph/min.
No, not since average is specified.

Chapter 2

Q1 Composition: terrestrial objects are composed

of combinations of earth, water, air and fire; celestial

objects of nothing but a unique fifth element.

Motion: terrestrial objects seek their natural positions

of rest depending on their relative contents of

earth (heaviest), water, air and fire (lightest);

celestial objects moved endlessly in circles.

Q2 (a), (b), and (c)

Q3 Aristotle: the nail is heavier than the toothpick

so it falls faster.

Galileo: air resistance slows down the toothpick

more than the nail.

04 See Q3 of Chapter 1 p. 15

197

Q5 An object is uniformly accelerated if its speed

increases by equal amounts during equal time

intervals. Av/At = constant

Q6 The definition should (1) be mathematically

simple and (2) correspond to actual free fall motion.

07 (b)

Q8 Distances are relatively easy to measure as

compared with speeds; measuring short time

intervals remained a problem, however.

Q9 The expression d = v t can only be used if v is

constant. The second equation refers to accelerated

motion in which v is not constant. Therefore the two

equations cannot be applied to the same event.

Q10 (c) and (e)

Oil (d)

Q12 (a), (c) and (d)

Chapter 3

Q1 kinematic — (a), (b), (d)

dynamic — (c), (e)

Q2 A continuously applied force

Q3 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.

Q4 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.

Q5 The first three.

Q6 No, in many cases equilibrium involves

frictional forces which depend on the fact that the

object is in motion.

Q7 Vector quantities (1) have magnitude and

direction

(2) can be represented graphically by arrows

(3) can be combined to form a single resultant vector
by using either the head to tail or the parallelogram
method. (Note: only vectors 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.)

Q8 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.)

Q9 W downward, 0,0,0

Q10 Galileo's "straight line forever " motion may

have meant at a constant height above the earth

whereas Newton's meant moving in a straight line

through empty space.

Q11 Meter, Kilogram and Second

Q12

10 N

= L =
a 4m/sec^

2.5 kg

Q13 False; (frictional forces must be taken into
account in determining the actual net force
exerted.)

Q14 Acceleration =

0 — 10 m/sec
5 sec

= —2 m/sec'^

Force = ma = 2 kg x (-2 m/sec^) = -4 Newtons

198

(the minus sign arises because the force and the

acceleration are opposite in direction to the original

motion. Since the question asks only for the

magnitude of the force it may be disregarded.)

Q15 10 m/sec^

150 m/sec-

60 m/sec^

0.67 m/seC

10 m

0.4 m

Q16 (c) and (f)

Q17 (e) and (f)

Q18 (1) appear in pairs

(2) are equal in magnitude

(3) opposite in direction

(4) act on two different objects

Q19 The horse pushes against the earth, the earth
pushes against the horse causing the horse to
accelerate forward. (The earth accelerates also but
can you measure it?) The swimmer pushes backward
against the water; the water, according to the third
law, pushes forward against the swimmer; however,
there is also a backward frictional force of drag
exerted by the water on the swimmer. The two
forces acting on the swimmer add up to zero, since
he is not accelerating.

Q20 No, the force "pulling the string apart " is still
only 300 N; the 500 N would have to be exerted at
both ends to break the line.
Q21 See text p. 68

Chapter 4

Q1 The same acceleration a^, its initial horizontal

speed has no effect on its vertical accelerated

motion.

Q2 (a), (c) and (e)

Q3 They must be moving with a uniform speed

relative to each other.

Q4 (a) T = 1/f = 1/45 = 2.2 X 10-= minutes

(b) 2.2 X ^0~^ minutes x 60 seconds/minute
= 1.32 sec.

277-R _2 X 3.14 X3
T 60

= .31 cm/minute

Q6 f = 80 vibrations/minute = 1.3 vib/sec

T = 1/f = 1/1.3 = .75 sec
Q7 (a) and (b)

Q8 Along a tangent to the wheel at the point where
the piece broke loose.

Q10 A-rrmR

Q11 The value of the gravitational acceleration and
the radius of the moon (to which 70 miles is added
to determine R).

Brief Answers to Study Guide

Chapter 1

1.1 Information

1.2 (a) discussion (b) 58.3 mph (c)
discussion (d) discussion (e)
discussion

1.3 (a) 6 cm/sec (b) 15 mi. (c) 0.25
min. (d) 3 cm/sec 24 cm (e) 30 mi/hr
(f) 30 mi/hr? 120 mi? (g) 5.5 sec (h)
8.8 m

1.4 22xl03mi

1.5 (a) 9.5 X 10'-^ m (b) 2.7 x 10" sec
or 8.5 years

1.6 1.988 mph or 2 mph

1.7 (a) 1.7 m/sec (b) 3.0 m/sec

1.8 discussion

1.9 discussion
.10 discussion

1.11 (a) 0.5, 1.0, 1.5, and 2.0 (b)
graph

1.12 Answer

1.13 25.6 meters; 4:00 for men, 4:30
for women

1.14 discussion

1.15 graph

1.16 graphs

d vs f: d = 0,9,22,39,5,60.5,86cm
(approx) at intervals of 0.2 sec

vwst:v = 45,65,87.5,105,127 cm/sec
(approx) at intervals of 0.2 sec

1.17 (a) 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)

1.18 (a) 14.1 m/sec (b) 6.3 m/sec^

1.19 315,000 in/sec

1.20 discussion

1.21 discussion

Chapter 2

2.1 Information

2.2 discussion

2.3 discussion

2.4 discussion

2.5 discussion

2.6 discussion

2.7 proof

2.8 (a), (b), (c)

2.9 discussion

2.10 discussion

2.11 proof

2.12 17 years $7000

2.13 discussion

2.14 (a) 57 m/sec- (b) 710 m (c)
-190 m/sec^

2.15 proof

2.16 discussion

2.17 (a) true (b) true (based on
measurements of 6 lower positions)
(c) true (d) true (e) true

2.18 proof

2.19 (a) Position

(b) proof (c) discussion

2.20 discussion

2.21 (a) 5.0 m (b) 10 m/sec (c) 15 m

2.22 (a) 10 m/sec (b) 15 m (c) 2 sec
(d) 20 m (e) -20 m/sec

2.23 (a) 20 m/sec (b) -20 m/sec (c)
4 sec (d) 80 m (e) 0 mi/sec (f) -40
m/sec

2.24 (a) -2 m/sec- (b) 2 m/sec (c)
2 m/sec (d) 4 m (e) —2 m/sec

(f) 4 sec

2.25 discussion

2.26 (a) 4.3 welfs/surg2 (b) 9.8
m/sec^

2.27 proof

2.28 proof

2.29 proof

2.30 discussion

2.31 discussion

2.32 discussion

2.33 discussion

Chapter 3

3.1 information

3.2 discussion

3.3 (a) construction (b) 2.4 units,
West

3.4 proof

3.5 discussion

3.6 discussion

3.7 discussion

3.8 discussion

3.9 discussion

3.10 discussion

3.11 discussion
3.12- 2.8 X 10-^ hr/sec

3.13 6/1

3.14 discussion

3.15 discussion

3.16 discussion

3.17 proof

3.18 discussion

3.19 (c) 24N (d) 14.8N (e) 0.86N (f)
9.0 Kg (g) 0.30 Kg (h) 0.20 Kg (i) 3
m/sec^ (j) 2.5 m/sec- (k) 2.50 m/sec-

3.20 (a) 2.0 X 10^ m/sec^ 7.8 x 10'^
m/sec (b) discussion (c) 2.4 x 10-
m/sec^

3.21 discussion

3.22 discussion

3.23 2.0 Kg

3.24 discussion

3.25 discussion

3.26 (a) 1 Kg. 9.81 N in Paris, 9.80N
in Washington (b) individual
calculation

3.27 individual calculation

3.28 discussion

3.29 (a) -5 X lO-" m/sec^ (b) 10
m/sec (c) 1 X 10"" m/sec

3.30 discussion

3.31 discussion

3.32 (a) diagram (b) 1.7 X lO-^-"

m/sec^ (c) ^ ^^^'' (d) diagram

3.33 (a) 862N, 750N, 638N (b) The
same as in (a) for scale calibrated in
Newtons (c) discussion

3.34 hints for solving motion
problems

Chapter 4

4.1 Information

4.2 13.6 m/sec-; 2.71 sec; mass
decreases

4.3 discussion

4.4 derivation

4.5 proof

4.6 1 .3 m; at an angle of 67° below
the horizontal; 5.1 m/sec, 78° below
the horizontal

4.7 discussion

4.8 discussion

4.9 discussion

4.10 discussion

4.11 6.0 X 10-2 min, 3.0 x lO"'^ min,
1.3 X 10"- min

4.12 (a) 1.9 sec (b) 32 rpm (c) 50
cm/sec (d) 35 cm/sec (e) 0 (f)
190Vsec, yes (g) 120 cm/sec^ (h) 160
cm/sec'^ (i) discussion

4.13 discussion

4.14 discussion

4.15 table completion

4.16 (a) 2.2 X 10"'" m/secMb) 4 X
10-" N (c) approximately 1/100

4.17 approximately 10'* N

4.18 discussion

4.19 (a) Syncom 2 (b) Lunik 3 (c)
Luna 4 (d) dosent change

4.20 5.1 X 10' sec or 85 min
7.9 X 10^ m/sec

4.21 discussion

4.22 7.1 X 10^ sec or 120 min

4.23 (a) 3.6 x 10^ sec (b) 36 Km (c)
discussion

4.24 t = (m/F)(Vo - V)

4.25 discussion

4.26 essay

199

HOLT, RINEHART AND WINSTON, INC.

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