7 1978
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Nourse
Universe, earth, and atom
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UNIVERSE, EARTH, AND ATOM
DUE
1.
Books by Dr. Alan E. Nourse
Universe, Earth, and Atom
Nine Planets
Junior Intern
So You Want to Be a Doctor
So You Want to Be a Physicist
So You Want to Be a Scientist
So You Want to Be a Surgeon
So You Want to Be a Nurse (with Eleanore Halliday, R.N.)
So You Want to Be a Chemist (with James C Webbert)
So You Want to Be an Engineer (with James C. Webbert)
So You Want to Be a Lawyer (with William B. Nourse)
Universe,
Earth,
and Atom
The Story of Physics
by Alan E. Nourse, M. D,
HARPER & ROW, PUBLISHERS
New York and Evanston
1817
For Christopher, Jonathan, Rebecca and Benjamin
and their mother
UNIVERSE EARTH, AND ATOM: THE STORY OF PHYSICS. Copyright 1969
y
f K f America.l, rights
m USCd r re P roduced in any manner whatsoever
m
, Perm ! ssion ^^ in the case of brief quotations embodied
articles and reviews. For information address Harper & Row Pub
lishers, Incorporated, 49 East 33rd Street, New York, N.Y. 10016. '
FIRST EDITION
LIBRARY OF CONGRESS CATALOG CARD NUMBER; 69-13493
CONTENTS
Acknowledgments ix
Introduction xiii
Part I: Physics in Perspective
1 The Physics of Common Sense 3
2 The Origins of Physics 18
3 From Philosophy to Science 28
4 The Methods of Discovery 37
Part II: The Universe of Classical Physics
5 Asssumptions, Observations, and Measurements 57
6 The Riddle of Falling Objects 66
7 The Riddles of Friction and Inertia 82
8 Push and Push Back: The Riddle of Collisions 98
9 Motion, Momentum, and Universal Gravitation 122
10 The Forms and Shapes of Matter 146
11 The Manifestations of Energy 169
12 Electricity, Magnetism, and the Phenomena of Waves 192
13 The Baffling Enigma of Light 216
Part III: The Einstein Revolution
14 The Riddle of the Ether Wind 253
15 The House That Einstein Built: Special Relativity 274
16 The Puzzle of Time 305
17 The House That Einstein Built: General Relativity 333
Part IV: The Universe of the Inconceivably Large
18 Macro-Universe: The Problems of Observing 369
19 Macro-Universe: The Birth of Stars and Planets 389
20 Mega-Universe: The Puzzle of Distant Galaxies 403
21 The Puzzle of the Expanding Universe 421
22 The Riddle of the Quasars 441
vi Contents
Part V: The Universe of the Inconceivably Small
23 Micro-Universe: The Earliest Explorations 461
24 Micro-Universe: The Puzzle of Radioactivity 485
25 In Quest of the Atom: Measurements and Tools 508
26 The Puzzle of Energy Quanta 524
27 Into the Heart of Physical Matter 563
Part VI: Practicalities and Promises: The Impact of Modern Physics
28 Lasers, Transistors, and Other Practicalities 611
29 Hydrogen Fusion and Thermonuclear Energy 649
30 The Endless Investigation 670
Index 677
Acknowledgments
Any book of this nature inevitably is the result of the work, help, and
encouragement of many people other than the one whose name is on the
manuscript. The author wishes to express his indebtedness in particular to
Harold E. Grove, who provided the initial spark for the project; to Pro
fessor Louis Trimble, of the University of Washington in Seattle, for his
careful reading and annotation of early drafts of the manuscript; to Dr.
Hong-Yee Chiu, of the Goddard Institute for Space Science and the
State University of New York at Stonybrook, Long Island, for his pro
fessional criticism of the advanced draft of the book; to Jerry Jermann
for the diagrams and drawings; to George W. Jones and Carl D. Brandt
for their perseverance and encouragement in seeing the project through to
its present form; and to Elinor Busby, Doris Vinnedge, and Becky Nourse
for their help in preparation of the manuscript.
Little wheel spin and spin,
Big wheel turn around and 'round. . . .
Buffy Sainte-Marie
Introduction
This book is written about the world of modern physics and the work of
physicists today how they know what they know, what they are still trying
to find out, and what their discoveries mean to us all, both now and in the
future.
We are living today in an age of research and discovery more intense
and exciting than in any comparable period in history. Some of our scientists
are probing the mysteries of life in the nuclei of microscopic cells; others
are preparing to explore the outer reaches of our solar system. Discovery
eclipses discovery as we learn more and more about ourselves, our planet,
and our universe. But of all areas of research in science today there is none
more fascinating, and none more baffling to the nonscientist, than the work
of modern physics the study of the structure and function of the physical
universe of which we are a part.
On first thought it might seem odd that so many people are so com
pletely lost when it comes to understanding the ideas of modern physics.
Other areas of science medicine, space technology, psychology do not
seem so formidable and confusing. Yet when we hear about a new dis
covery in physics we suddenly find ourselves in deep and murky water. We
don't even understand what these discoveries are, much less what they may
mean to us in our modern technological society. And the knowledge that
these discoveries may have profound effects upon our lives perhaps even
on our survival makes such confusion not merely regrettable but down
right dangerous.
Such widespread public confusion ignorance, if you will is not the
fault of the physicists who are doing the research. Many of them have tried
diligently to explain to the nonphysicist just what, exactly, they were doing,
in the pages of well- written books literally hundreds of books. Why, then,
do we need another, and one written by a layman in the world of physics at
that?
The reason is simple and personal. It is my conviction that existing
books, however brilliantly written, have simply not done the job they set
out to do. Readers seeking a broad general understanding of what is hap
pening in physics just don't understand what they are reading. They find
x Introduction
themselves grappling with baffling or flatly incomprehensible ideas that
have no discernible relationship to anything they have ever experienced in
their lives. They become trapped in a quagmire of confusing terms, defini
tions, and abstract concepts, and then feel angry and somehow cheated
when they find that these concepts are, in fact, shaking the very ground
they walk on, whether they understand what is going on or not.
Obviously something is missing. Some comprehensible frame of reference
is needed, some way of drawing together a huge and confusing body of
information about the laws of nature and the discoveries of modern physics
so that they can be readily understood, visualized, and related in some way
to the everyday world we live in. No one can tackle totally unfamiliar ideas
and apparently fantastic and incomprehensible concepts without some
familiar frame of reference as a starting place, some place to stand. The
major goal of this book is to find such a frame of reference.
A secondary goal is to discuss what has happened throughout past cen
turies of research in physics, and what is happening today, as clearly as
possible, not in terms of exquisite technical detail, but by showing the
general direction that scientific thinking has followed in the past, and is
still following today. This means choosing repeatedly between what is
really significant and what is really not. It means drawing illustrative ex
amples and analogies freely, and using comprehensible generalizations,
even at the price of sacrificing some degree of precise scientific accuracy.
In this book the crying need to express general ideas clearly must neces
sarily override the scientist's absolute devotion to accuracy.
What frame of reference do we propose? For one thing, an historical
frame of reference, for there is no more fascinating approach to modern
concepts in physics than to see how physicists themselves came to their
own conclusions, changing and rejecting and modifying the things they
once thought to be true, step by painful step throughout the ages. An
other part of our frame of reference will be to consider in detail some of
the baffling riddles that physicists have struggled to solve the innumerable
perplexing contradictions, for example, between theoretical predictions
and actual laboratory observations that have been confounding scientists
since the earliest times, forcing them to reappraise what they thought was
right before and find out what new concepts may lie closer to the truth.
Still another part will be to see how the most complex and fantastic con
cepts of modern physics actually impinge on the way we live today and
will be living tomorrow. Finally, part will be concerned with the future
with the direction that today's search for clearer understanding of the uni
verse is taking us, and with what that direction will mean, or can mean, in
the world of tomorrow.
To accomplish this, we will first consider a way of looking at the uni
verse around us, or, more accurately, several quite different ways, for this
Introduction xi
is the only avenue to understanding what is happening in the universe in
comprehensible terms. For centuries, one very limited way of regarding
the universe completely dominated scientific thinking. Then, within the last
century, that one view was found inadequate to explain things which were
actually observed to be happening to the amazement and chagrin of
practically all scientists everywhere. Bit by bit in the last few decades that
whole "classical" view of the structure and function of the universe, as it
always had been understood, began to crumble before the challenge of a
few brilliant men, and a different way of regarding the universe in many
ways a wildly incredible way seemed to be needed.
Other views of the universe were needed, and were found. We will see
what these different ways of looking at things were, how they evolved in
the first place, and why they were so desperately needed to achieve a more
complete understanding of how the universe really worked. At first each of
these views may seem complete in itself, quite unrelated to any others, as
though we lived not in one universe but in several at once, with one set
of rules applying in one area and another set in another. But we will see
how physicists have discovered, bit by bit, that each of these views fits
together with each of the others, that each is no more than a view of the
same universe, in all its incredible complexity, regarded from a different
angle; and we will see how discoveries made from each of these angles
have proven consistent with discoveries from the other angles, and how
each has had a profound influence on the everyday lives of all of us.
Finally, we will consider where research in physics has taken us up to
the present time, just how it is affecting our lives, and where it may be
taking us tomorrow. Many physicists today believe that we have reached
an all-but-impenetrable barrier to more detailed understanding of the uni
verse . . . that only another totally new way of looking at the physical
universe, an approach even more fantastically different than any yet ex
plored, can possibly breach that barrier and lead us on to more detailed
knowledge. Others, equally well grounded in modern physics, insist that
the complex and esoteric mathematical techniques even now being used by
theoretical physicists have already provided that "new view" and that most
researchers simply have not yet succeeded in fully understanding these
techniques and their long-range implications. At the same time, multitudes
of new practical uses have been found for the new knowledge already
accumulated, and workers in modern physics have begun approaching
goals and achievements that have been dreamed of for centuries. We will
discuss these new frontiers as they stand today, and try to foresee some of
the directions in which current research may be leading us.
In approaching a book of this sort, certain things must be clearly under
stood from the beginning. Obviously, this is not a learned scientific trea-
xii Introduction
tise, not a textbook of physics. Rather, it is a book of general information
written for the intelligent but untrained layman who (like the author)
seeks to understand in clear and simple terms what the current work in
modern physics is doing, how it got where it is, and where it is now going.
Much technical detail and a certain degree of scientific precision will be
sacrificed, not because they are unimportant (obviously they are im
mensely important to the professional physicist) but because they are
unimportant to the goal of this book. They would hinder more than they
would help in presenting a clear, readable discussion of the general prin
ciples, exciting ideas, and revolutionary discoveries of modern physics
which today are affecting the lives of physicist and layman alike. Too much
public comprehension of physics has already been slaughtered on the altar
of scientific accuracy; this book, it is hoped, will help revive the victim.
This is not to say that accuracy will simply be thrown to the winds. It
does mean that we will walk a razor's edge between "writing down" to
more sophisticated readers, on the one hand, and outdistancing others
who are attempting once again to grapple with some basic ideas that have
eluded them in a dozen previous encounters. Here we must plead for
patience; many items that are "common knowledge" to some readers will
be new and difficult concepts to others. Certain technical terms must be
defined and used constantly as we go along; we will seek out the clearest
definitions and descriptions that we can find. In general, we will assume
that the reader has a certain basic acquaintance with high-school-level
mathematics, chemistry, and physics, and is aware of the common terminol
ogy associated with such everyday phenomena as electricity, magnetism,
radioactivity, and so forth but when in doubt, we will clarify funda
mentals before moving to the more complex. Wherever possible we will
avoid mathematical formulas in favor of verbal descriptions; too often,
formulas are nothing more than so many blank spaces on the page to
many readers a barrier to understanding rather than a help.
Finally, we will make certain arbitrary choices as to what to discuss
and what to pass over. A book ten times longer than this one could not
hope to consider all the staggering number of phenomena that modern
physics is concerned with, from the interaction of subatomic particles to
the cosmic explosion of distant galaxies, and from the behavior of light
and other radiant energy permeating the universe to the nature and be
havior of time. We can only touch the high spots, selecting topics that will
contribute the most to clear general understanding.
In short, we are setting ambitious goals for this book, and must accept
certain limitations in order to attain those goals. This is a book about how
things work and why they work the way they do, as far as is known today.
If we can describe, in understandable and nontechnical language, some of
the great discoveries that have been made in physics over the centuries,
and gain some understanding of what those discoveries really mean to us|
Introduction xiii
we will be approaching our goal. And if, in addition, we can convey a
sense of the excitement of those discoveries and stir a sense of wonder at
the mysteries that human minds are penetrating even today, and at the
frightening and promising enigmas that still remain unanswered in the
universe around us, then this book will have been worth the writing.
DR. ALAN E. NOURSE
North Bend, Washington
Part I
Physics in Perspective
CHAPTER 1
The Physics of Common Sense
On a cool dark night almost four hundred years ago a young man walked
up a hill in northern Italy with a lantern in his hand. This man was a
scientist whose name and fame would one day be known to every school
child; but on this particular night, unknown to him, he was setting out on
one of the most celebrated wild-goose chases in all the history of science.
No one knows exactly what night it was, nor whether it was summer or
winter. We might imagine that it was a balmy evening with no moon, for
this particular man was a very acute thinker and would have picked the
jime and circumstances best suited to the success of his experiment. At
distance across the valley his laboratory assistant climbed another hill
with another lantern, probably quite convinced that his master was afflicted
by demons. Yet the assistant knew that his master had performed other
experiments with rather surprising results, and who could say? perhaps
this would be another.
In any case, this event was more than a mere walk in the country. This
scientist was intent upon actually measuring the speed with which a beam
of light would travel from one point to another. He thought he had devised
a way to accomplish this feat. He had observed that other kinds of signals
traveled a given distance at a given, measurable speed. He knew, for ex
ample, that the disturbance created by a stone dropped in a quiet pond
could be followed and the velocity of its movement timed. He had also
observed that when a distant woodsman was seen cutting down a tree,
the sound of each blow required a measurable period of time to reach
the observer after the ax struck the time necessary for the sound to
travel from the tree to the ear. Thus, it seemed logical to our young scientist
that a beam of light would take a discrete period of time to travel from its
source on one hilltop to an observer on a distant hilltop. His goal was
to measure this time interval and then, knowing the distance between
hilltops, to calculate the velocity of the light beam.
The procedure planned was very simple. With his assistant watching
from a distant hilltop, the scientist would unmask his lantern. The instant
his assistant saw the light, he would unmask his lantern in turn. The
difference in time between opening the first lantern and observing the
4 Physics in Perspective
answering light from the second should then be equivalent to the time
required for a light beam to travel to the distant hilltop and back again.
It was a well-conceived experiment. One might have expected that;
this particular man was one of the most acute scientific observers in all
history, and a clever experimenter as well. Despite all that, the experiment
ended up as a spectacular failure. The answering light from the distant
hill was seen to appear at the very same instant that the scientist opened
his lantern. There was no time lag observed. What was more, the same
thing happened each time, no matter how many times the experiment was
repeated.
To Galileo Galilei, his hilltop experiment that night could mean only one
thing: that light had no measurable velocity, but rather, spread instantane
ously to all parts of the universe at the same time. Today, of course, we know
that that conclusion was wrong. Light does indeed have a measurable
velocity; it requires a definite interval of time to travel from one point to
any other point in the universe. But the fault did not lie in Galileo's
thinking. He had no way to guess that the distance between the hilltops
that he had chosen was so small compared to the enormously swift speed
of light that the time lag he was trying to measure was simply not per
ceptible to the human eye. Galileo's experiment was sound enough. His
instruments were simply too crude to measure the time lag; and if anyone
had told him that light (which, incidentally, is not a particle but a wave)
actually traveled 186,000 miles in a single second, he probably would
have laughed uproariously. After all, common sense said that nothing
could travel that fast.
Four centuries later, in the year 1886 in a completely different part
of the world, another scientist performed another famous experiment,
designed to settle, once and for all, a problem that had been perplexing
scientists for generations. As in the case of Galileo, Albert Michelson's
experiment was also concerned with measurement of the velocity of light.
And, like Galileo's earlier experience, Michelson's experiment turned
out to be a spectacular failure. There, however, the resemblance between
the two experiments ended.
Michelson's experiment took place, not on a windy hilltop, but in the
basement laboratory of Michelson's friend and fellow physicist Edward
Morley, near Cleveland, Ohio. The apparatus for the experiment was
ponderous. It involved a huge slab of stone five feet square and more
than a foot thick, floating in a container of liquid mercury with an elaborate
system of mirrors arranged at the four sides of the slab and a semireflecting
mirror set in the center. In the darkened room a spotlight sent a beam of
light from one side of the slab to the side opposite. The beam was reflected
back by the mirror there but part of the beam was diverted by the
semireflecting mirror at the center of the slab, and directed back and forth
The Physics of Common Sense 5
between the other sides in a direction perpendicular to the original beam.
Finally, the two beams of light reflected across the slab in opposite direc
tions were directed to the same white card "target" after each half of the split
light beam had traversed the slab many times, one half in one direction, the
other in the other (see Fig. 1).
Fig. 1 A simplified diagram of the Michelson-Morley experiment. A beam of
light from source A was divided by semi-reflective mirror B so that part of the
beam (unbroken arrows) went straight through to mirror C 2 while part (broken
arrows) was diverted to mirror Ci- Both beams traveled equal distances against
the "ether wind" to reach target card D, but one beam (broken arrows) had
to travel three times as far across the "ether wind" as the other, and was thus
expected to reach card D slightly later than the other beam, creating an inter
ference pattern.
Michelson was not directly concerned with measuring the speed of light
in this experiment. That had long since been accomplished with such a high
degree of accuracy that most scientists were in agreement about the
measurement. Rather, Michelson was determined to find out to what degree
a beam of light would be slowed down as it traveled head-on through
a mysterious substance known as the "universal ether," a weightless,
invisible substance which virtually all scientists of the day believed per
meated all the space surrounding the earth and all the space between
the stars.
The notion that all space was filled with this strange invisible ether
had a long and respectable history. True, no scientist had ever seen the
6 Physics in Perspective
ether, nor was there any direct evidence that it even existed. Yet its
existence was widely accepted as necessary in order to explain how certain
commonly observed phenomena could possibly occur. Physicists knew,
for example, that sound waves and water waves had to travel through
some medium. They also knew that light traveled at a great velocity from
distant stars to the earth. But there was clear evidence that light was
also a form of wave. Thus physicists argued that light also had to travel
through some medium, whether the medium could actually be detected
or not. The universal ether, although never actually detected, was assumed
to exist as a necessary medium through which light waves could travel
from one place to another.
At best, the idea of the ether had always been a little awkward. For
one thing, if the ether was assumed to exist, other things also had to be
assumed as a result. It was known, for example, that the earth was moving
through space in its orbit around the sun at a velocity of about 18 miles
per second. If all space were filled with ether, then it followed that the
surface of the earth should be subjected to a constant "wind" of ether
passing by as the earth plunged through it. But if so, why had this "ether
wind" never been detected? Numerous attempts to demonstrate it had
failed. Some scientists were even beginning to doubt that such a thing
existed. Others maintained that it had to exist, but remained undetected for
lack of instruments sufficiently sensitive to measure it.
This was the puzzle that Michelson and Morley had decided to tackle with
their elaborate apparatus. They were certain they had devised a way to prove
beyond question that an "ether wind" existed on earth. They reasoned that
light waves traveling with the ether wind ought to be carried along faster
than light waves traveling either directly into it or "across-wind," so to
speak. By splitting a beam of light in half and sending one half to and fro
in the direction of the ether wind by means of mirrors, and sending the
other half of the beam to and fro across-wind, they believed the velocity
of the crosswind light waves would lag behind slightly, and that this
time lag would have to show up in the form of interference between the
two portions of the light beam when they were brought together into one
beam again to strike the target card. They further reasoned that as their
apparatus was slowly rotated in its mercury bath to bring the mirrors in
exact alignment with the direction of the ether wind or perpendicular to it,
a familiar interference pattern of light and dark bands ought to appear on
the target card.
This experiment, like Galileo's, was well conceived. By all rights it should
have worked but it failed completely. To Michelson's incredulous dis
appointment, not the slightest evidence of any interference pattern appeared
on the target card when the apparatus was in operation, no matter in what
direction they rotated the mirrors. As a colossal flop, the Michelson-
Morley experiment rivaled Galileo's failure of almost four centuries before.
The Physics of Common Sense 7
Worst of all, Michelson couldn't understand how it could have failed. Was
the flaw in his apparatus? In his methods of measurement? Like any good
scientist he set about diligently to examine his own experiment, hoping
to devise more accurate equipment.
Nor were Michelson and Morley the only scientists thrown for a loss by
this experiment's failure; the whole world of science was nonplused. No
one, at first, suspected the truth: that the very failure of this experiment
to detect an ether wind actually made it one of the most spectacularly
successful and useful experiments in the history of science, for the very
failure of the experiment led directly and inevitably to the whole shattering
concept of relativity. It remained for an obscure little Swiss-German
mathematician, a few years later, to point out the only possible conclusion
that made sense: that the reason Michelson and Morley had failed to
detect an ether wind was simply because there was no ether wind. And
with that conclusion as a basic foundation, Albert Einstein then proceeded
to propose a strange new theory that was soon completely to revolutionize
scientists' way of looking at the universe and ultimately to revolutionize
the everyday world we live in.
The two stories related above have no direct connection with one another.
They occurred in different eras of scientific development, and they were
concerned with totally different scientific ideas. All the same, these stories
together provide an appropriate introduction to a book about the remark
able discoveries and the complex enigmas of the world of modem physics.
For one thing, both experiments were concerned, one way or another,
with the nature and propagation of light still today one of the most
puzzling and contradictory phenomena in the known universe. Even more
significant, the experiments took place during two different periods in
history that were critical to the development of our present-day under
standing of the physical universe in which we live. In the centuries before
Galileo, scientists and philosophers had made remarkably little progress
trying to explain everyday occurrences in nature. It was Galileo more
than any other man in history who single-handedly pried open the door
to the scientific method of observation, hypothesis, experiment, and con
clusion that has led us to virtually everything we know in modern physics
today. Slowly the men of science who followed after Galileo began to
reject mere philosophical guesswork about what things were and why
things happened, and painfully began piecing together a group of rules-
so-called laws of physics, or laws of nature that seemed to describe how
things worked in the universe. By the time of the Michelson-Morley
experiment these laws of nature had been codified, confirmed, modified,
tested, retested, built upon, and expanded to the point where they seemed,
taken together, to describe accurately all known natural phenomena. Many
worthy scientists of Michelson's day actually believed that the end of
8 Physics in Perspective
scientific discovery would soon be at hand, that all the really basic and
important laws of nature had already been discovered and defined, so that
all that remained for physicists was a sort of mopping-up exercise, a matter
of tying up a few loose ends here and there.
The Michelson-Morley experiment delivered a jarring blow to that
attitude of self-satisfied complacency. Einstein's relativity theories shattered
it completely. Within thirty short years the whole splendid edifice of
classical physics, built up over the centuries, was torn to shreds by the
work of a few brilliant men who dared to question the validity of the laws
of nature as they were then understood, and found them wanting. The
names of some of those men have become commonplace family words.
But just what they did is not always so clearly understood. Often they
have been thought of as "mad scientists" or at least decidedly odd ones
and their work has been considered too complicated for ordinary non-
scientists to comprehend. Often there has been a sort of a vague resentment
connected with these men: why was it necessary for them to upset the
clean-cut classical laws of nature and to leave things in such a muddle?
But those men were not mere protesters and iconoclasts. They had no
desire to destroy mankind's hard-won and comfortable picture of an orderly
universe functioning according to orderly rules, just for the joy of making
a mess. They were simply hard-nosed, sharp-minded, stubborn men who
were disturbed about things in the world of physics that they could not
explain and who insisted that a law of nature either had to do its job and
explain all cases that came within its scope, or else it had to be changed
until it did.
Contrary to popular opinion, these modern physicists and mathematicians
were not prophets, or gods, or magicians. They were human beings with
their own special abilities, their own failings, their own tempers and irritabil
ities and prejudices, just like all other human beings, including you and me.
They were living in the same world of sunlight and shadow, war and
politics and peace, in which you and I live. But somehow these people were
able to look beyond the everyday world of sunlight and shadow, and to
probe the puzzles and enigmas of new and different worlds of physics:
the incredibly tiny micro-universe of atoms and their nuclei; the incredibly
huge mega-universe of far-flung galaxies and cosmic expansion; still another
strange mathematical universe of time and space, dimension and light-speed.
To the outsider their work in these unfamiliar areas seemed puzzling, con
tradictory, paradoxical, even flatly incredible. To many it has seemed
incomprehensible. But nevertheless, like it or not, comprehend it or not,
the work of these men has molded the world that we live in and provoked
drastic changes in the very lives we are leading today.
The work of these pioneer physicists and others who followed them is
not yet over, not by a long shot. It continues to blossom today in thousands
of laboratories in countries all over the world. It continues to shape the
The Physics of Common Sense 9
world that we live in. It is enormously important work. To understand what,
exactly, it is, what it has achieved so far, what it seeks to accomplish
today and tomorrow, is to understand more clearly every aspect of our
bewildering lives as human beings today. It is to understand better how
to control the forces at work today in the physical universe around us.
No one in his right mind can seriously deny the overwhelming importance
of that work.
But recognizing the importance of modern physics is one thing, while
understanding its concepts is something else altogether. And unfortunately,
most people do not understand very much about modern physics or the work
of the physicist today. Of course, we all recognize vaguely that the solid,
material objects of the universe are allegedly composed of multitudes of tiny
particles arranged in certain peculiar ways and moving about with some
kind of invisible motion. We all know that rockets can go to the moon,
and that the spacemen inside them seem to float around with a jolly
indifference to gravity. We all know that relativity theory has something to
do with space and time and the speed of light (even if we don't know
precisely what it has to do with these things) and that matter and energy,
whatever they may be, are supposed to be the same thing (even if we don't
really understand how or why this might be so). We even know that the
conversion of matter into energy is somehow related to hydrogen bombs
whatever they are and to radioactive fallout whatever that may be.
But for many of us, our understanding of modern physics is uncom
fortably vague. We know that something is afoot but we don't know what.
We regard physicists as shadowy figures who are somehow suspect in the
mysterious work they are doing. But in all fairness, we cannot blame the
physicists for our lack of understanding. Very few of them care for the role
of mystery man that has been thrust upon them. Most would like nothing
better than to have more people understand more clearly what they were up
to. Above all, they would like to be regarded in the same way that most
other scientists are regarded: as fairly normal, mundane human beings who
happen to be engaged in a fascinating but complicated kind of scientific
work. Most physicists feel that they should not be blamed for "creating"
hydrogen bombs and radioactive fallout, even if people don't completely
understand what physics is all about. After all, most people don't under
stand the stock market either but they don't hold stockbrokers responsi
ble for economic crashes and depressions. Many physicists have even set
out to explain to the general reader, in print, just what they are doing yet
the average layman still finds himself at sea as far as the work of modern
physics is concerned.
10 Physics in Perspective
THE IMPEDIMENTS TO UNDERSTANDING
It is not just bad luck that this should be so. In fact, there are certain
factors that make it almost inevitable that the nonscientist should have
trouble understanding what is going on in physics.
One such stumbling block in his path is the matter of language and
terminology. Many of the words used by physicists to describe their work
lead to confusion because these words mean one thing to the physicist
and something quite different to the layman. Precision is the very keystone
of research in physics precision of measurement, precision of calculation,
precision of description. When the physicist uses common words such as
"speed," "velocity" or "momentum," "heat" or "temperature," he uses
these words in a very limited and specific sense. He gives such words
precise definitions. Most of us, however, understand these words in far
more general terms. When we use the terms "heat" and "temperature"
almost interchangeably, the physicist accuses us of sloppy, imprecise speech
while we accuse the physicist of unreasonable fussiness.
In addition to this and often because of it the physicist has been
forced to invent completely new terms which mean nothing whatever to
the reader who is not acquainted with them through scientific training.
When the physicist tries to explain to the layman what these terms mean,
he finds himself at a loss to "translate" them accurately and precisely.
Too often he gives up the struggle at this point, saying, "Well, / know what
I mean, but there isn't any way to explain it to youl" Anyone who doubts
that language and terminology create a real barrier to communication
between physicists and nonphysicists needs only to look at a modern journal
of physics to be convinced. To the untrained layman, nothing in it will be
comprehensible. The gulf of words is enormous.
Another stumbling block, perhaps even more formidable, is the fact that
so few nonscientists have any significant acquaintance with one of the phys
icist's most important working tools: the language of mathematics. Of
course, most of us have had some training, one way or another, in simple
arithmetic', algebra, plane geometry, perhaps trigonometry. Some may even
have studied calculus once. But for the physicists, a casual passing acquaint
ance with math some years back just isn't enough. The physicist uses mathe
matics as an automatic, constant, indispensable working tool, day after day.
Even more important, he uses it as a concise language. And many physicists
insist that a layman simply cannot understand the work of physics at all with
out a solid grounding in advanced algebra, calculus, and higher mathematics
that physics by its very nature has to remain a mystery to anyone who is
not so equipped.
From the physicist's point of view, this may be very true. It may indeed
be impossible for us, as laymen, to understand the minute details of physi
cal theory. It may be quite impossible for us to follow the intricate math-
The Physics of Common Sense 1 1
ematical reasoning that has been part and parcel of so many of the great
discoveries in physics without a great deal of skill and experience in higher
mathematics behind us. But such skill and experience is not necessary in
order for us to grasp, in broad perspective, the basic concepts and the great
general laws of physics as they are understood today. It may take an artist
years to learn the correct strokes and the application of great talent and
highly trained skill to render a painting on canvas, but his work can be ap
preciated at differing degrees of depth by laymen, fellow artists, learned
critics, and so on. Several levels of understanding may be possible. Similarly,
great skill and experience in higher mathematics is not necessary in order for
us to understand what the great discoveries of physics mean in terms of a
general description of the universe, or a general prediction of what we can
expect to see happen as a result of given circumstances.
To a physicist, of course, simply understanding the laws of physics in
general terms is not enough. For him actually to work in physics, either in
pure research or in developing practical applications for what has already
been discovered, a real expertise in higher mathematics is indispensable.
And often the physicist who is immersed in his work has difficulty under
standing how anything less than a full professional and scientific grasp of
all the detail can possibly be of any use to anyone. When he does try to
"summarize" the work he is doing in simple and nonmathematical terms, he
feels forced into scientific inaccuracies and distortions that make his skin
crawl. If he is really game, he may plow doggedly through to the end and
come up with descriptions of his work that he considers at least scientifically
tolerable; but they may well be less than crystal clear to the untrained lay
man. By and large it is far easier and more comfortable for the physicist to
say, "It just can't be done without the math" and to let it go at that.
Either way, we have the same result: a failure of communication, and a
failure of the average untrained person to understand what the physicist is
doing.
Formidable as these stumbling blocks may be, there is still another im
pediment to a clear understanding of the laws and work of physics, an
impediment so imposing that it completely overshadows all the rest. This
impediment, simple as it may seem, is nothing more nor less than the limita
tion of our everyday experience. And when it comes to understanding the
laws of nature as they are or may be, our everyday human experience is
very limited indeed.
THE LIMITATIONS OF EXPERIENCE
After all, what do we actually know of the world around us? How do we
know what we know? From infancy on, we know primarily what we have
experienced through our human senses. We know what we can see, hear,
smell, taste, and feel. From the evidence of these senses we have developed
1 2 Physics in Perspective
a "normal" view of the universe around us a picture, so to speak, that we
carry in clear focus in our minds. But this picture is severely limited by the
boundaries of our senses.
Over the centuries, by means of various clever devices, we have been able
to augment our human senses in a number of ways, up to a point. By using
the lenses of a microscope, for example, we are able to magnify up to a
point objects too tiny for us to see with our unaided eyes. The telescope
greatly expands our ability to observe heavenly bodies, up to a point.
Various devices can amplify sound waves so that we can hear things we
could not ordinarily hear, and certain drugs and chemicals can render us
hypersensitive to various sensations of touch, smell, or taste.
Even more cleverly, we have learned to extend the limits of our senses
by converting unavailable data into some more available form. For example,
sound waves that are beyond audibility can nevertheless be detected by con
verting them into visual images on the oscilloscope screen. Light which ap
pears monochromatic and homogeneous can be split into its varicolored
components by a crown glass or diamond crystal, or a spectrograph. In some
cases, artificial amplification of our senses in this way is so much an ordinary
everyday part of our experience that we don't even stop to consider that we
are not actually experiencing what we seem to be experiencing at all. During
a telephone conversation we know that sound waves at one end are really
being converted into electrical impulses which are transmitted along con
ducting wires and then reconverted into sound waves again. But most of
us find it easier to imagine that the voice we are hearing is the actual voice
of the person talking to us, transported across the miles by some sort of
miracle.
Yet for all our cleverness, most of us still picture the universe in the terms
that it is revealed to us by our senses, and in no other way. When we try
to understand the laws governing the behavior of the universe the laws
of physics we are really trying to understand those laws solely in terms of
the normal world of our senses. And this, unfortunately, cannot be done
for the simple reason that the universe extends far beyond the limits of
human sensory experience.
The physicist knows this and accepts it without qualms. It does not bother
him that the natural laws he is trying to define involve objects and forces
which he can neither see, hear, feel, taste, nor smell. Some of his work, of
course, involves tangible objects in the "real" or "normal" world but not
much of it. Far more of it involves investigations in one of several other
"worlds" in which it is not possible for him to measure or experience any
thing by means of his senses. But what may seem perfectly clear-cut, simple,
and quite understandable to the physicist seems obscure and incomprehen
sible to most other people because it does not seem to fit into any frame of
reference that they can understand.
What are these "worlds" of physics, these views of the universe, that
The Physics of Common Sense 13
physicists find so easy to deal with and the rest of us find so obscure? Need
they be so obscure? Not if we can find some comfortable and familiar place
to start, and then find ways to relate unfamiliar ideas to the things in our
experience that we already comprehend. To find such a frame of reference,
we must begin by looking for a moment at these different "worlds" in which
the physicist works, to see just how they are related to the "normal" universe
of our everyday experience.
THE FOUR WORLDS OF MODERN PHYSICS
Everyone is acquainted with the first and most obvious "world" which
physicists have explored: the "normal" universe that we see around us
every day.
There is nothing particularly mysterious or frightening about this familiar
and comfortable view of the universe. It is the universe of earth and sky,
fire and water, in which we see trees growing, animals bearing their young,
or the wind moving the grass. This is the universe our senses can explore
directly: a world of buildings and oceans and sounds, of tangible objects we
can grasp, of light and darkness. Beyond our immediate planet in this uni
verse are the sun, the moon, and the stars that we see on a dark night. This
is the world of ordinary sensory experience.
It is important to realize that throughout centuries and millenniums of
mankind's existence-^until very recently indeed this was the only view of
the universe that there was. It was with this world of physics that the classi
cal scientists grappled, trying to search out laws of nature that described the
phenomena that were observed by the senses. In this world of physics objects
had fixed masses and behaved according to simple laws of mechanics. Ob
jects in motion moved with finite, measurable speeds along paths that could
be predicted paths that were either straight or curved but not both at
once. Space was three-dimensional in this universe, described to perfection
by the geometry of Euclid that we all learned in high school. Gravity was a
downward pull. Light was a phenomenon that could be observed, studied,
measured, and manipulated with lenses. Electricity and magnetism were
rather mysterious forces, difficult to explain, but observed to behave ac
cording to certain consistent, logical, sharply defined rules.
In short, this was a universe in which the classical "laws" of mechanics,
heat, light, sound, gravitation, electricity, and magnetism all applied. A
long succession of brilliant scientists had labored to discover those classical
laws: Galileo, Copernicus, Kepler, Newton, Faraday, and Maxwell, to
name but a few. In a sense it was a comfortable and cozy picture of the
universe that these men painted over the centuries, and those classical laws
worked, as far as our human senses could detect. It was not surprising that
physicists, toward the end of the nineteenth century, were beginning to think
that the work of physics was almost done.
14 Physics in Perspective
But this comfortable view of the universe was not the whole picture. Not
that it was wrong, exactly. It just did not cover enough ground. It was nice
to have the jigsaw puzzle almost finished, but there were chunks of the
picture still unaccountably missing. In order to see even a glimpse of those
missing parts, physicists found it necessary to begin regarding the universe
in some very different ways than they had ever regarded it before. In fact,
they had to explore it as if it were really composed of several quite different
worlds all at the same time, with each world superimposed on the others,
and each with certain singular rules and regulations of its own.
One of these different views of the universe is the "microcosmic" view,
in which all matter in existence is regarded as being composed of incredibly
small bits and pieces, elementary particles and wavelets too tiny to imagine
and too numerous to mention. In this microcosmic universe, few if any of
the classical laws of physics that apply to the "normal" universe as we see
it seem to apply. The particles making up this microcosmic universe are far
too tiny ever to be observed directly. Some, in fact, are virtually impossible
to detect at all. Here the overriding force acting in the "normal" universe
(the force of gravity) seems to have little or no power at all; in the micro-
cosmic universe, other forces quite unheard of in the everyday world seem
to prevail: the nuclear binding forces that hold atoms together, for example,
and various "interactions" discovered to occur between elementary particles.
In this microcosmic universe the speeds with which particles or wavelets
move seem as incredibly great as the particles themselves are incredibly
small. The position and momentum of such particles cannot be accurately
measured at all at least, not at the same time! And the laws of mechanics,
those rules that physicists had used for centuries to enable them to predict
with (they thought) absolute accuracy what would happen to object A if
force B were applied to it, no longer seem even remotely relevant in this
microcosmic universe. The laws just don't cover the ground.
The microcosmic view of the universe was originally, in fact, so totally
different from our "normal" view that there seemed to be no real relation
ship between the two at all, at least to a layman. The microcosmic view
seemed more of an intellectual abstraction than a real part of our "normal"
universe until the development of hydrogen bombs, nuclear power re
actors, transistors, and laser beams made it more and more obvious that
the microcosmic universe was indeed "real" enough to change our lives
profoundly.
But still another strange and different view of the universe also affects our
lives: a "macrocosmic" view, in which our earth and our solar system are
themselves mere particles of matter too tiny to mention in a universe that is
incomprehensibly large. In this macrocosmic universe, laws of nature have
been discovered which cannot be a part of our sensory experience nor even
remotely understandable in terms of our "normal" universe. This macro-
universe is so huge that no one yet can really comprehend what relation our
The Physics of Common Sense 15
minuscule part of it our earth, our sun, even our galaxy may have to
the whole. Here physicists are concerned not only with the underlying struc
ture of matter but also with the structure of galaxies and clusters of galaxies,
with the birth and death of stars, and with the cosmological history of a
universe so vast and so crowded with matter, yet apparently so empty, that
human minds are at a loss to comprehend or define it. Here are forces at
work so subtle we do not even recognize them acting under our very noses,
yet powerful enough to form suns and planets out of swirling dust clouds.
These forces may be so great that whole galaxies are hurled away from each
other as if in a stupendous silent explosion yet even such forces as these
must have had a beginning somewhere at some point in time, and must at
some unimaginable future time come to an end.
In this macrocosmic universe this universe of the inconceivably large
the classical laws of physics again do not seem to apply. Even the Euclid
ian geometry scientists have used for centuries to describe our "normal"
universe seems unable to describe some basic aspects of the macrocosm.
The forces at work here are neither the forces of mechanical energy so
familiar to us in the "normal" world nor the nuclear binding forces that hold
the particles of the universe together. Here is a wholly different universe,
unexaminable by ordinary human senses and seemingly quite unrelated to
our "normal" universe of experience, yet a very real aspect of the universe"
as it exists, just the same.
Finally (and much to their own dismay) physicists have come to realize
that there is still another view of the universe that must somehow be taken
into account, apparently unrelated to the normal, the microcosmic, or the
macrocosmic, yet which applies to all three. This is the strange relativ-
istic view of the universe that was inexorably outlined, hypothesized, and
then proved valid by Einstein and other giants of twentieth-century physics.
In this view the universe is not merely a certain volume of space containing
various chunks of matter, but rather, a vast continuum of space and time,
It is a universe in which matter and energy must be regarded merely as two
different manifestations of the same thing, totally interchangeable from one
into the other. It is a universe in which there appears to be one previously
unsuspected but inalterable absolute, a single fixed physical limitation that
seems to confine the operation of all other forces in the universe: the limita
tion of the speed of light. Stranger than any other picture of the universe,
the relativistic view at first appeared to refute everything that had ever been
believed about the classical laws of physics. Yet the evidence that began to
accumulate, observed in the other worlds of physics, suggested more and
more emphatically that the relativistic view was indeed every bit as valid
and as necessary for understanding how things work as any of the others.
1 6 Physics in Perspective
AN APPROACH TO COMPREHENSION
Just as scientists realized, some hundred years ago, that the ordinary
human senses presented an incomplete picture of the universe, so physicists
today recognize that no single one of these differing views of the universe
alone is sufficient. All four views the "normal," the microcosmic, the
macrocosmic, and the relativistic must be taken into account if there is
to be any hope of understanding how our universe really works. Thanks to
their scientific training and experience, physicists can accept this notion,
make peace with it, and carry on from there. Those of us who have not
had such training and experience tend to balk and stumble. We stumble,
for instance, over the whole idea of infinite time and infinite space. True,
we cannot see the end of the sky, but our experience with other things
tells us that there must be an end to it somewhere. We cannot help but
think of our "normal" universe as a finite universe, and also a universe
with definite fixed limits and boundaries perhaps very wide ones, but
boundaries nonetheless. The idea of infinite extension of anything is an
awkward and uncomfortable abstraction. So is the idea of a universe with
out boundaries. In t^e microcosmic universe, the macrocosmic universe,
and the universe of relativity there are no such finite limits and no certain
boundaries known, as yet, so we inevitably try, without scientific training
and experience to help us, to cram these other views of the universe into the
finite limits of our own human experience.
And we find that it can't be done. At first, perhaps, we are confused; we
don't quite understand what's wrong. Then we tend to reject these other
awkward views of the universe. After all, it is easy to say, "It just can't be
understood!" and even easier to say, "What nonsense! These things don't
have any real meaning in my world anyway."
Unfortunately, in our world, we are confronted with growing evidence
that these views do indeed have real meaning in our world and very real
influence in our everyday lives. Furthermore, we discover that understanding
something about these strange views of the universe is becoming more and
more important to us personally every day. We need to know something
about nuclear physics, something about cosmology, and something about
space, time, and relativity nowadays just to keep up with what is happening
in our familiar "normal" universe!
But how can we avoid trying to cram these other views of the universe
into the limits of our own experience? One way would be first to look at
each of these four views of the universe separately and distinctly, to see how
they developed, where they came from, and where and why they seem to
contradict each other. Our modern knowledge of the laws of physics as they
are understood today did not appear by revelation overnight. It was ac
cumulated bit by bit over the centuries. Since the days of ancient Greece
men have been struggling with a long series of riddles, each seemingly more
The Physics of Common Sense 17
impenetrable than the last. One by one these riddles have been solved. In
the early days physicists and mathematicians were concerned with riddles
of the universe as we see it the normal universe of our everyday experi
ence. Later they began probing into other worlds which they could neither
see nor measure. They began thinking in terms of the ultimate atomic
structure of matter the riddles of the infinitely small; or in terms of the
ultimate size, shape, and limits (if any) of the universe of stars and galaxies
the riddles of the infinitely large. Finally they began to center in on other
phenomena that seemed closely associated with infinite quantities the
riddles of light, of time-and-space dimension, of mass-energy conversion,
and of forces of gravity and other stupendous forces operating on a cosmic
scale.
To understand where this work has led, we will follow the footsteps of the
multitudes of scientists who have tried to answer these riddles. We will try
to develop a clear basic understanding of the natural laws governing each
of these views of the universe in turn: the normal, the microcosmic, the
macrocosmic, and the relativistic. Then, with good fortune, we will try to
see how these views meet and join in an orderly, sensible, and understand
able picture of the universe around us, to understand how this knowledge is
already affecting our everyday lives, and to predict where it may be leading
us in the future.
CHAPTER 2
The Origins of Physics
Every day we encounter multitudes of things which ought by rights to give
us pause, but which we rarely even notice at all. Some of these things, which
we accept as perfectly commonplace, would seem little short of miraculous
if we stopped to think about them at all. Others would merely seem re
markable or extraordinary. In truth, however, all of these things are nothing
more than simple manifestations of a few basic rules that limit the behavior
of objects and forces in the world around us.
Consider a few simple examples. In the morning before we awake on a
cold day, a device on the wall goes "click" and somewhere at a distance
the furnace turns on, pouring out heat to take the chill off the house before
we arise for the day. At about the same time another device on the bed
side stand goes "click" and coffee water begins heating in an automatic pot.
After a carefully regulated period of perking hot water through coffee
grounds, the pot turns itself off, but still maintains a constant temperature
so that hot coffee is waiting ready-made at whatever hour we decide that we
must face the day. Then, when that hour arrives, still another device sets off
an alarm to wake us up so that we can drink the coffee and arise to enjoy
a warm house.
Do we marvel at these things? Of course we do not. They are nothing
more than common household routine.
But the day's miracles have hardly even begun. Upon arising we walk
into the bathroom to shower and shave; the water is already heated for us,
and the electric razor doesn't even need to be plugged into a wall socket.
But even more remarkable, if we stop to think about it, is the simple and
extraordinary fact that we were able to transport ourselves from the bed
room to the bathroom at all, thanks to forces of gravity, inertia, and me
chanical principles of leverage, without either having a rope to tow us, falling
flat on our faces, or inadvertently leaping out of the bedroom window at
the first step.
Later, at breakfast, we listen to the morning news on the radio, broadcast
and transmitted from a station perhaps thirty miles away, while a device
on the table heats up a network of thin wires to prepare our toast. Both
these things are miraculous enough, but no more so than the fact that an
18
The Origins of Physics 19
hour later we climb into a chrome-plated transportation device weighing
almost two tons, and by manipulating a few levers manage to induce it to
carry its own weight and ours down the road at a velocity of sixty miles
per hour, and then deposit us at the office door without flattening our faces
against the windshield. We think nothing of this, but then we seem to be
hard to surprise; on the way to work we saw a jet airliner weighing four
hundred tons climb gracefully into the air, and we thought nothing of that
either. It was an ordinary day.
Leaving the car, we ride on an elevator which lifts us twenty flights up
to our office in less than half a minute. Nothing remarkable there. Once at
work, we hardly look at the small machine that takes the sound of our voice,
converts it into altered molecular patterns on a magnetized tape, and then
hours (even years!) later reconverts it on command into the sound of a
human voice for transcription. At home again after a day's work we use
a somewhat similar magnetized tape to create further magic: By pushing
a few buttons we record a television show we especially enjoy, or make an
original videotape record of some family event, and then later see and hear
this fragment of freshly recorded history any time we desire merely by feed
ing the tape through the television receiver again. Our eight-year-old son
brings in a new toy for us to see, a gyroscopic top which balances at a rakish
angle on the sharpened tip of a lead pencil as its flywheel spins madly
about. We start to explain the principle of the gyroscope, but the boy cuts us
off, blandly remarking that he knows all about it, that it's what keeps our
rocket ships from wobbling too badly during takeoff.
We encounter the laws of physics in action constantly, wherever we turn,
and think nothing about it at all. This is not really surprising, of course.
Throughout our lives we have been both confined and liberated by these
natural laws; we are accustomed to their effects even if we don't know what
the laws are. We know from experience that certain things always happen
in certain ways. Such things we take for granted. We also have learned that
nature imposes certain limits on the way things will behave, and that we
get in trouble if we try to exceed those limits. Odd as it may seem, one of
the earliest of all human insights was man's realization that when he co
operated with nature things generally went well, whereas when he tried to
thwart or alter natural patterns, things almost always went badly. Each
individual has to learn this fact for himself, to some degree, just as a child
learns the limits of his environment. But it is also man's nature that the
lesson has to be learned over and over again; throughout history men have
tried repeatedly to thwart the laws of nature and get away with it. And
we continue to try. The fact that the laws of nature always win in the
end has not dampened our enthusiasm a bit!
How have we learned these basic rules and limits of nature which we so
easily take for granted in our everyday life? Certainly not from a textbook
of physics (although the rules can be found there). Rather, ever since in-
20 Physics in Perspective
fancy we have been learning them, in a multitude of amazingly simple ways,
in the school of practical experience.
THE SCHOOL OF EXPERIENCE
Not long ago I spent an instructive half-hour watching a four-year-old
neighbor boy learn a fundamental law of nature the hard way and fly into
a rage in the process as he tried in vain to make a wagonload of rocks
behave the way he wanted it to.
The child obviously wanted to move the rocks from a nearby gravel bank
to the site of some architectural marvel he was working on a hundred
yards down the sidewalk. Getting the rocks into the wagon was no problem.
He loaded them in one at a time: clunk, clunk, clunk. But getting the loaded
wagon rolling was something else again. The child fought and strained and
tugged and pulled; then finally, grudgingly, the wagon full of rocks began
to move.
Once it started rolling, everything was splendid. The child pulled the
wagon along faster and faster. But then he stopped to inspect a bug on the
sidewalk, and whackl the wagon caught up with him and knocked him
sprawling. He got up and looked at the wagon, which once again had
stopped moving. Once again he strained and tugged and pulled to get it
under way; once again it caught up with him, knocked him sprawling
again and, of course, stopped. And once again one furious little boy started
tugging at the stalled wagon.
In the course of that single hundred-yard trip, that hapless child got
himself knocked down no fewer than six different times as his determina
tion dissolved into wailing frustration. When I stepped outside to ask him
what the trouble was, he said it was a bad wagon; when he wanted it to go
it wouldn't go, and then once it got going it wouldn't stop. The experience
was sad indeed, but no college professor could have found a better way to
teach that child one of the most basic of all the laws of nature!
Of course, the child had never heard of Newton's laws of inertia. He might
well live to be eighty without ever learning how to express those laws in
words, or in terms of mathematical formulas. But by the age of four he had
already learned what those laws meant, as far as getting along in the world
was concerned.
Obviously the behavior of the wagonload of rocks was a frustrating
conundrum to that four-year-old, a riddle that seemed to defy understand
ing. Presently he recognized that he had to adjust to the way that load of
rocks behaved whether it made any sense or not. That was how it was with
a wagonload of rocks. But the riddle still remained. Adjusting to it didn't
explain it. Possibly the very existence of this unsolved riddle became a
challenge to that boy's ingenuity. We can imagine him later coming back
to the riddle time and again, trying to puzzle out why that load of rocks
The Origins of Physics 2 1
behaved the way it did. We can even imagine him discovering, one day,
that the rules that applied to wagonloads of rocks also applied to baseball
bats and automobiles and rifle shells and a myriad other things.
There is nothing remarkable about the episode of the boy and the load
of rocks. Each of us has encountered the same problem one time or another,
and puzzled over the same riddle. In fact, simple as it may seem, this minor
episode is a perfect example of the way that men from earliest times have
grappled with the mystifying riddles of how things work in our universe and
slowly pieced together the basic rules which we know today as u the laws
of physics."
THE EARLIEST PHYSICISTS
Some time long before the dawn of written history, Ug the caveman had
a problem.
To us, it might seem a simple problem, hardly worth a second thought.
But in the primitive world of Ug the caveman it was a matter of life or
death. Early one morning Ug had emerged from his cool, damp cave and
set out to hunt for his dinner. While he was away, a downpour loosened a
boulder from the hillside above Ug's cave, and it rolled down to block the
cave's doorway completely. When Ug returned toward dusk with a fine
haunch of hippopotamus that he had rescued from the jackals, he couldn't
get back into his cave. Try as he would, he couldn't budge the boulder that
stood in his way.
Now brains were not Ug the caveman's long suit, but he knew certain
things with terrible clarity. He knew it grew cold at night, so a caveman
needed the shelter of a warm secure cave. He also knew that when dark
ness came, the saber-toothed tiger began to roam in search of a better meal
than a mere haunch of hippopotamus. Ug the caveman knew quite clearly
that unless he could move that boulder away from the doorway by night
fall, he might never live to see the dawn.
Whether from instinct or from some half -remembered earlier experience,
Ug the caveman had a sudden bright idea. He searched for a bough from
a nearby tree, broke it off, and wedged one end of it in between the boulder
and the doorway. Although he could not budge the boulder before, with
the aid of the stick he could move it a little. When he moved closer to
the boulder and pushed on the bough, he found it was much harder to
move. When he moved farther out to push on the distant end of the bough,
the rock moved more easily but the bough snapped off. Finally, as day
light faded, Ug found a stronger, longer pole, wedged one end in behind
the boulder, pushed on the far end and miraculously, the boulder moved
easily away from the cave mouth and bounded down the hillside.
Ug the caveman never thought to wonder why he could move that
boulder with the aid of a long pole when he couldn't even budge it otherwise.
22 Physics in Perspective
For Ug, there were more important things to think about: fire and warmth
and feast and protection and sleep, for instance. Yet under pressure of
urgent necessity this simple caveman had discovered and used, perhaps for
the first time in man's history, a simple machine operating according to
fixed mechanical principles. It would not be the last time that Ug the cave
man would use this new machine a simple lever and fulcrum to save his
own life, to get things done that he could never otherwise do, and generally
to elevate himself from the status of low-class caveman to upper-middle-
class caveman.
Of course no one knows who Ug the caveman might have been, nor
where he lived, nor when he made his discovery. Like so many other simple
machines the roller, the wheel, the pulley, the inclined plane, the bolus,
the bow and arrow, or the siphon, to name a few the origin of the lever
and fulcrum is lost in the mists of antiquity. All these machines were known
long before the first human records were kept. Some early physicist dis
covered each of them. No one knows who. Yet these machines formed the
first link between the world of practical experience and the scientific study
of physics.
THE PRACTICAL ENGINEERS
We are so accustomed today to the whole idea of scientific study that we
forget that research has not always been a normal part of human activity.
It is easy for us to assume without thinking that organized scientific in
vestigation came about as a natural result of man's insatiable curiosity and
intellectual vigor. In truth, it is far more probable that the earliest explora
tions in physics arose either from desperation (as in the case of Ug the
caveman) or from man's age-old, insatiable desire to get something for
nothing. It was only when practical advantages began to appear that early
physicists began searching out the reasons and principles underlying things
around them that had always been taken for granted.
Thus, the discovery and development of simple machines almost certainly
came about as a result of men trying to get more work done with less effort.
Consider the Egyptian pharaoh who wanted to build a huge tomb for him
self out on the desert. Naturally, he wanted it as big as possible but it had
to be finished before he died if it was to do him any good. Getting it built
raised real problems of engineering and logistics. The stone had to be cut
miles away and dragged to the construction site by slaves. Still more slaves
had to lift the stone blocks into place. The pharaoh had plenty of slaves,
but it still took days and weeks to move a single stone to the tomb and lift
it into place, and the harder he worked the slaves the faster they dropped
dead on him. To cut smaller stones would take more time and reduce the
magnificence of the tomb. But the pharaoh was getting old, and so many
slaves could do only so much work and no more.
The Origins of Physics 23
Or so it seemed. Then some bright young underling came up with an
idea. By dragging a rock along on rollers instead of sliding it across the
desert, he found that the same number of slaves could transport twice as
many stones of twice the size in the same length of time as without the rollers.
He found that if the rollers could be moved swiftly from the rear to the front
as the rock moved, things went faster than when the load was constantly
stopped and started again. Finally, he discovered that by hauling the stones
up an inclined plane to the level where they were needed, heavier stones
could be lifted higher and faster by fewer slaves than before. The pharaoh
had never heard of "mechanical advantage" before in his life, but he could
tell a good idea when he saw it; this was practically instant pyramids! So
he rewarded his bright young engineer, and rollers and inclined planes
became standard operating equipment for pyramid-building.
Later, other pyramid builders discovered other things about the simple
machines they were using. They learned, for instance, that a ramp with a
long, gentle incline worked better than a short, steep ramp. The stones had to
be pulled farther horizontally to get them to the required height, but up to
a point the ease and speed with which they could be lifted far outweighed the
additional distance. When a stone was lifted with lever and fulcrum, they
noticed that for some reason the job was easier if the lever arm was long
than if it was short. Even the water boys discovered that unless the water
bucket was suspended at the exact center of the pole between two carriers,
the one closest to the bucket did most of the work.
These were practical observations which led to useful refinements of the
earlier simple machines. Inevitably, one day, somebody scratched his head
and said, "Now, wait a minute how much easier is the long gentle incline
than the short steep one? What makes it easier? And why can't we figure
out the exact length and slope of the ramp that we need so that the fewest
men can raise the heaviest rock to the greatest height the fastest?"
It was when questions of this sort began to arise that the first scientific
study of the physical universe really began.
We know today that some of these primitive observations resulted in
amazingly accurate predictions, and certain primitive techniques proved
remarkably useful and durable. As early as 3000 B.C. Egyptian astronomers
had learned enough from the cyclic movements of the sun and moon to
establish a year as a unit of time 365V4 days long a far more reliable
measure than the annual flooding of the Nile. They even knew that this
time measurement was sufficiently inaccurate that a correction of three
days had to be made every four hundred years, and the search was on
for a device to clock hours and minutes. In matters of calculation, early men
gave up counting on their toes in favor of chalk marks on the wall.
Presently they invented symbols to stand for various numbers. In ancient
Sumeria and Babylon these symbols were used to develop a method of
calculation we know today as arithmetic. Still later, in Arabia, it was dis-
24 Physics in Perspective
covered that when special symbols were used to represent unknown
quantities and simple rules of logic were applied, certain kinds of problems
could be solved which arithmetic could not handle. Thus the basic tech
niques of algebra were developed. In those primitive days, too, the abacus
was invented as mankind's first mechanical computer a device so simple
yet so efficient that it is still in widespread use in the world today.
Such observations and discoveries did not arise from any philosophic
search for the meaning of it all. Those were violent and hazardous days in
which to live. A man's average life expectancy was about 26 years the day
he was born; he was likely to be too preoccupied with feeding, sheltering,
and protecting himself to have much spare time for philosophic ruminations.
He needed to understand simple machines in order to build his cities, draw
his water, plough his fields, or build monuments to his kings or gods. He
needed to know when to plant and when to harvest. He needed arithmetic
and algebra in order to hold his own in an era of cutthroat commercial
dealings. He did not often ask why things around him happened the way
they did.
But people in those days did learn one very important thing about the
way things worked. They learned that whatever the reason things worked
the way they did, they always seemed to -work the same way one time as
another. Whatever laws of nature might be at work, those laws were orderly.
If a pulley worked one way one time, it would work the same way the next
time and the next. When two and two were added up, the result was always
four. It remained for a later and more sophisticated civilization to begin
questioning just what earth, air, and water were really made of, and why
things worked the way they did. The search for answers continues to this
day but the questions were first asked by the philosophers of ancient
Greece.
THE BIRTH OF MODERN PHYSICS
Most historians of science today concede that the first serious scientific
questioning began in the civilization of ancient Greece. In many ways this
is strange, because the early Greeks were anything but scientists. They did
not regard themselves as "investigators of nature" in the sense of modern
scientists, using observation and experiment as their tools. If anything,
they considered something as coarse as mundane experimentation to be far
beneath their dignity. Rather, the Greeks thought of themselves as natural
philosophers, seeking to penetrate the secrets of nature by means of reason
and logic. Many notions that could have been proved demonstrably wrong
by the simplest of experiments were accepted as true without question,
simply because they were philosophically and esthetically satisfying. Debate
and logical dialogue were the accepted methods of investigation; great men
would argue for months about some point of "natural philosophy" which a
The Origins of Physics 25
modern scholar could have resolved in one minute flat with a good slide
rule.
Even so, ancient Greece and her philosophers built an absolutely critical
groundwork for the organized body of scientific knowledge about the
physical world which was to come later. The Greeks did make certain dis
coveries about the ways the laws of nature could be investigated. They also
showed the world some of the ways those laws could not be investigated. It
was upon their ideas, discoveries, and errors that the whole structure of
modern scientific exploration first arose.
For one thing, the Greeks recognized philosophically that there was
order in the universe. Things that happened in nature happened consistently.
To them, this indicated that some kind of absolute "natural law" governed
the behavior of things. The movements of the stars, the operation of simple
machines, the phenomena of heat, light, and sound were not things that
occurred capriciously at the will of the gods. There was, the Greeks con
cluded, a definite cause-and-effect relationship between things that occurred
in nature. One thing happened because something else had happened first,
and this led to something else with such consistency and regularity that it
was actually possible to predict what was going to happen next before it
occurred, on the basis of what had already happened before.
The Greeks also believed that certain truths about nature could be ac
cepted as obviously true without proof, and then be used as basic axioms
from which other truths could be deduced by means of logic and reason.
These so-called intuitive truths were very fundamental things, so clearly
and self -evidently true that they were considered proofs unto themselves
things that "any fool could plainly see." For example, it seemed self-evident
that the material from which the earth was made had to be composed of
certain tiny, indivisible units. The idea of an infinitely divisible chunk of
rock simply defied reason. Break it up into smaller and smaller pieces and
sooner or later you must reach some small basic unit which could not again
be divided. Accepting this as an axiom, it followed logically that all forms
of matter must be built up from an assortment of such individual units. It
is hardly surprising that our modern word "atom" was first used by the
Greeks to denote a tiny particle of matter which itself could not be further
divided.
Again, the Greeks realized that certain shapes and patterns (such as
straight lines, triangles, or circles) occurred repeatedy in nature, and that
the concept of number or quantity was suggested in nature by collections and
sizes of objects. Certain facts about these geometric patterns appeared to be
self-evident without proof. Two circles drawn with the same radius had
to be equal in size. One right angle, by its very nature and definition, had
to be the same shape as any other right angle.
These conclusions did not arise from careful experiment or measurement.
They arose from somewhat casual common-sense observation. Yet on this
26 Physics in Perspective
basis the ancient Greeks began to collect a volume of basic scientific data,
and then started to build upon those data by means of logic and deduction.
The handful of basic axioms which form the basis for Euclid's system of
geometry (a system regarded as the only possible system of geometry for
almost two thousand years) were never considered subject to proof. They
were accepted as self-evidently true. But with these unproven axioms as a
foundation, each succeeding proposition in Euclid's system was then sub
jected to rigorous logical proof. Each new proposition, once proven, then
became the basis for still further propositions, until a whole series of rules
had been built up which consistently applied to any and all cases within
geometric experience.
Such a system of reliable rules was, of course, highly useful. Even more
astounding, it was discovered that by using these rules one could actually
discover physically meaningful information which could not possibly have
been discovered any other way. For example, it was impossible to prove by
observation or experiment just what shape of rectangle enclosed by a piece
of string of a given length would have the greatest area. One might guess,
but one could not prove. But by means of geometry it was possible to demon
strate beyond any doubt that a perfect square would have the greatest area
of any rectangle that could be enclosed by a string of given length. It could
be proven geometrically that a perfect pentagon formed by the same string
would enclose a greater area than a perfect square, and that the string laid
in a perfect circle would enclose the greatest area of all.
Thus the early development of plane geometry resulted in a discovery
with staggering implications. By applying logic and reasoning to situations
taken from nature it was possible to produce new and hitherto unsuspected
knowledge.
Simple as this idea was, it was vital to the growth of physical science.
For one thing, it encouraged men to begin observing "taken-for-granted"
natural phenomena more closely. The sun, the moon, and the planets moved
in the heavens. If one observed closely and then applied reason and logic,
surely it should be possible to determine the exact orbits of these heavenly
bodies, and thereafter to predict accurately where they would be found
at any given moment in the future. Of course, this did not prove to be as
simple as it seemed. In the second century A.D. an Alexandrian Greek
astronomer named Ptolemy undertook the job in the traditional Greek fash
ion, and created a misconception that took a thousand years to clear up.
Ptolemy assumed as self-evident that the earth itself stood still in the heavens
while the planets and the sun pivoted around it. He also assumed that all
the heavenly bodies moved in perfect circles, since the circle was obviously
the most perfect form of motion for a heavenly body (philosophically
speaking). Fitting his observations of the planetary movements into these
axioms, he developed a theory to explain the motion of the sun, the moon,
and the other planets around the earth.
The Origins of Physics 27
Unfortunately, later observations of planetary movements never quite
fitted into this "Ptolemaic system" he had devised; so Ptolemy and his
followers had to refine and modify his theory over and over again through
the years. Finally, fifteen centuries later, somebody proved that both of
Ptolemy's "self-evident" axioms were wrong, but so great was the stature
and authority of those early Greek philosophers that it often took millen
niums finally to replace some of the inconvenient theories they propounded.
For all their shortcomings, however, the ancient Greeks' intellectual and
logical approach to the study of nature did bear some useful fruit. The
Greeks examined an enormous number of natural phenomena and devel
oped logical theories to explain the nature of heat, light, and sound, the
operation of the lever and the inclined plane, the factors and forces acting
upon fixed or moving bodies, and the nature of work and energy. They
catalogued the heavens and created astronomical theories which, incorrect
as they were, still provided future astronomers with a solid foundation on
which to work. They recognized the three physical states of matter solid,
liquid, and gaseous even though they completely missed the relationship
that existed between those three states.
Above all else, the ancient Greeks proved that man could learn how nature
behaved, and thus could hope to predict nature's future behavior. As we
will see later, physicists today seriously challenge even this idea, and not
without reason. But it is pertinent to note that without that concept to guide
scientists throughout the centuries in searching out answers to the riddles of
the universe, there would be no modern physics today. By proving to their
own satisfaction that nature behaved in an orderly manner, and that
the truth about nature could be uncovered by human intellect, these
ancient explorers opened the door to a two-thousand-year-long assault upon
the riddles and conundrums of nature that men had to face in the world
about them.
CHAPTER 3
From Philosophy to Science
By the close of the ancient Greek era of intellectual achievement, the ground
work had been laid for a giant step forward in scientific discovery. But over
fifteen hundred years were to pass before that forward step was begun, and
another four hundred years before the classical laws of physics governing
the "normal" universe of everyday experience were finally outlined.
We should not forget that the physical world those early Greek philos
opher-scientists were seeking to explore was the world they saw about them.
It was the world they knew from the experience of their own senses, a
universe they could see, hear, touch, smell, and feel. Objects in that world
had measurable size and weight, and moved at measurable speeds in dis
cernible directions. Everything on the surface of the earth was influenced
by a mysterious, undefined force which tended to pull everything in one
direction and that direction was down the force we know today as gravity.
Aristotle explained gravity very simply; Every object on earth tended to
seek its "natural place," he said, and the "natural place" of all objects was
on the ground. Ergo, any object not resting on the ground tended to fall to
the surface. It sounded good, but as for really explaining anything, even
Aristotle must have realized, on occasion, that it was fatuous nonsense. The
Greeks knew that when an irresistible force (such as a warrior's battle-ax)
struck an immovable object (another warrior's skull, for instance) the battle-
ax came suddenly to rest and the skull got crunched. They even knew that
the harder the battle-ax hit, the more satisfying the crunch. They were not
aware that the total momentum of the ax-skull system remained unchanged
by the encounter, nor that the kinetic energy of the ax was largely absorbed
by the elasticity of the skull, nor that a certain amount of heat was generated
in the process. These things came later.
Again, when the ancient Greeks discussed the atom as the "ultimate
indivisible unit" of matter, they were discussing intellectual abstractions that
had no real meaning to them in terms of their experience. They knew per
fectly well that in the real world a bit of sand could be crushed into a fine
powder, but that was as close to an "ultimate indivisible unit" of sand as any
one could hope to approach or needed to, for that matter. They were,
28
From Philosophy to Science 29
perhaps, able to imagine very large or very small objects or distances, but
no real concept of infinity was possible from their observation of nature
around them. They simply had no toe hold for such a concept. Whenever
they encountered it inadvertently (as in Zeno's paradox about the runner
who could never finish a race because he would first have to run halfway to
the goal, and then run half of the remaining distance, and then half of that
half, etc., ad infinitum, and thus could never quite get to the goal line) the
concept was regarded as precisely what it was called: a paradox or conun
drum of mutual exclusives which simply did not admit of a solution.
What is more, as the centuries went by, as experimental methods were
developed and as devices were found to extend the range of human senses,
the study of the natural laws of the universe still remained a study of the
"real" world that could be seen, measured, and experienced. Not that there
were not clues to the existence of other and unsuspected worlds of physics
beyond sensory measurement and experience. The phenomenon of gravity
and the phenomenon of light were two such major clues but they were
either ignored completely or examined only in terms of the real world of
solid objects and measurable forces. Gravity was studied as a constant force
that made things fall to the ground when you let go of them. Light was an
unexplained and apparently unexplainable something which no one pre
tended to understand, but which could be manipulated by lenses in a useful
manner. When mathematicians began coming up with concepts that had no
relationship to the world of experience the concept of imaginary numbers,
for instance scientists almost invariably tended to distrust the mathematics
and the mathematicians, rather than to consider seriously that there might
be some aspect of the universe that was completely out of reach of human
experience.
Even within such limitations, a long succession of scientists beginning
with the Greeks and ending with the nineteenth century physicists actually
learned an amazing amount about the nature of the universe and the natural
laws that prevailed or at least, about the universe of human experience.
This knowledge was not accumulated suddenly, nor in any steady progres
sion. It pursued no particularly logical course of development; in fact, it
developed in a succession of torpid pauses, staggers, and lurches, assisted by
a few perfectly incredible coincidences about the most disorderly history
of discovery imaginable.
Along with a solid groundwork of observation, the ancient Greeks had
provided such basic tools as a highly developed system of plane geometry
and an increased skill in the use of algebra. They also provided a tradition
of inquiry. At least they recognized that things were going on in nature that
they did not understand, and that these things were worth wondering about.
They did not, however, have any workable concept of a scientific method o]
investigation, as we think of it today. They disdained experimentation, and
3O Physics in Perspective
considered their speculations and hypotheses "proven" if they were logically
and philosophically pleasing even when new observations flatly contra
dicted those hypotheses!
In addition, the Greeks overlooked or ignored many natural phenomena
simply because they didn't seem to admit of philosophical explanations. For
example, they were perfectly aware of the existence of certain kinds of
natural stones to which bits of iron mysteriously seemed to cling, even in
opposition to the "downward" force of gravity. The Greeks did not under
stand why an iron swordblade should be "drawn to the lodestone rock" in
this manner, nor did they understand why a piece of soft iron rubbed on a
lodestone took on some of this curious iron-attracting quality itself. But
they never investigated this phenomenon, nor did they ever discover that an
iron rod rubbed on a piece of lodestone and then suspended from a string
would always assume a north-south orientation with respect to earth.
Similarly, the Greeks were aware that a piece of silk rubbed on a lump
of amber tended to repel another piece of silk, and to crackle and spark in
the darkness when it was shaken, but this fact seemed to arouse no excite
ment or curiosity. These people were just not emotionally or intellectually
equipped to investigate such phenomena in any kind of orderly fashion. If
they had been, the world might be quite a different place from what it is
today. What would have happened had the magnetic compass been avail
able to mariners from the time of ancient Greece on? What if Archimedes or
Aristotle had begun a systematic investigation of electricity and magnetism?
It is useless to speculate; they did not. Nor did they attempt a study of the
curious properties of light, although they had certainly observed rainbows
in the sky, and most assuredly knew of the brilliant play of colored light
in a natural quartz crystal.
THE RENAISSANCE GIANTS
With such a varied foundation built by the Greeks, we might have ex
pected a steady growth of scientific investigation in the centuries that fol
lowed. But in fact, after the decline of Greek civilization progress simply
ground to a halt. Practically nothing of scientific significance happened at all
for over a thousand years.
Historians have a variety of explanations for this long period of scientific
stagnation. Certainly a number of factors contribated. The Roman empire
rose to power as Greek civilization declined, and the Romans were neither
philosophers nor scientists. Preoccupied as they were with expansion, com
merce, politics, and warfare (and, later, with living the good life at the
expense of all else) the Romans simply accepted and copied what the Greeks
had worked so hard to achieve. They made no effort to investigate or expand
Greek ideas about the nature of the universe; they bought them wholesale
and passed them on as revealed truth. Later, the Church also played an im-
From Philosophy to Science 3 1
portant role in discouraging new directions of thinking and scientific ex
ploration. Threatened by any unorthodox concepts, the Church would accept
only those scientific ideas and hypotheses which seemed consistent with
Christian teachings and for centuries the Church had the power of life or
death over anyone within its realm who deviated from these accepted prin
ciples.
But perhaps the main reason for the long stagnant period in scientific
investigation was the simple fact that the Greeks had painted themselves
into a corner with their philosophical and speculative approach to science.
The ideas they had developed from their casual observations of nature had
already been expanded as far as possible by means of reasoning and logic
alone, and they had no alternative approaches to offer. Further investigation
along those lines could only lead to further refinement of the same ideas,
as the gulf between the "proven" conclusions of those early philosophers
and new, more accurate observations of nature grew wider and wider.
Then, when Rome fell, the Church became custodian of what scientific
knowledge did exist, and most of the thinkers and philosophers of the time
were far more concerned with questions of theology than with new or
challenging ideas about the nature of the universe. Indeed, for over a
thousand years the closest approach to science was the pseudoscience of
alchemy, that strange mixture of philosophy, scientific investigation, and
mumbo-jumbo whose practitioners sought in vain for the mystical "philoso
pher's stone" that could turn base metals into gold.
Above all else, no one had found the tools necessary for true scientific
investigation, and without the tools there was no place for science to go.
This long, sleepy period did not last forever. In the late 1400s, quite
suddenly and for no clear-cut reason, some giants began to appear men
who were to jolt the world of science to its very foundations in the course
of less than two hundred years. The names of these men are household words
today: Copernicus, Galileo, Tycho Brahe, Kepler, Isaac Newton, Faraday,
and Maxwell, to name but a few. Incredibly, after a thousand years' sleep,
five of these men were born within the span of 170 years, and four of them
lived and worked within the span of a single century.* All were physicists
in the broadest sense of the word investigators and explorers of the nature
of the physcal world. Among them, in two centuries, they changed the course
of history.
Later we will see in more detail just what discoveries each of these men
made and how they made them. Among them, they built up the first orderly
and sensible explanation of how things worked in the universe of human
experience. Theirs was the world of classical physics. They found answers
and then proved them, insofar as they could be proved by the senses.
At long last they overthrew the ancient Greek tradition of investigation by
* Galileo died in the same year that Isaac Newton was bora.
32 Physics in Perspective
debate and philosophy and established a new tradition of investigation by
experiment a tradition that has persisted to this day.
As we shall see, a great many conclusions of these giants of classical
physics have since been found to be incomplete. Many of the "laws of
nature" they outlined have been shown to be valid only under special or
limited circumstances, not universally valid always, under any circumstance.
Some of their conclusions have proved to be flatly wrong. But if their work
was incomplete, or limited, or flawed, it provided a solid and scientific
basis for the work of others. Above all else, these men performed one
service of staggering and overriding value to humanity: They forged the
missing tool by which men could study and hope to understand the nature
of this universe, the tool without which modern physicists would have re
mained as helpless as the ancient Greeks. Today that working tool is known
as the scientific method of investigation.
Nicolaus Copernicus, a Polish astronomer born in 1473, made the first
and probably most revolutionary break with the ancient Greek tradition.
Ever since Ptolemy had assumed that the earth was the center of the universe
and that everything in the heavens revolved around it in perfect circles,
astronomers had been trying to fit the observed movement of moon and
planets into the increasingly awkward Procrustean bed Ptolemy had pro
vided for them. When the guest did not fit the bed, they whittled off his legs
until he did. They even tampered with the bed, so to speak. Repeated efforts
were made to revise the Ptolemaic theory slightly, and each new revision
seemed to straighten things out for a while, but always new observations
came into conflict with the theory again. The repeatedly modified Ptolemaic
system became progressively more clumsy to use as time went on, but no
one ever dared question the basic assumptions that one could use only
the earth as the center of coordinates in the universe, and that heavenly
bodies had to move in circles.
Copernicus not only challenged the first of those assumptions, he de
vised clear-cut scientific proof that it had to be wrong. Drawing from a
lifetime of his own careful observation he concluded that the sun had to be
the center of our solar system, not the earth, and that the earth and all the
other planets revolved around the sun. True, our own moon revolved around
the earth; but on the other hand it was the earth and not a "celestial sphere"
of fixed stars that turned on its axis every twenty-hour hours, producing the
apparent motion of the stars, and the movement of the sun across the sky.
It was such a revolutionary concept that Copernicus himself withheld its
publication until the very end of his long life. But the Copernican system
had one very good thing going for it: It happened to agree splendidly with
what had actually been observed and recorded of the motions of the various
known planets, while the Ptolemaic earth-centered system failed to do so
even after centuries of refinement and modification. For a while it even ap
peared that the system of Copernicus was the final and ultimate answer.
From Philosophy to Science 33
But then, a century later, other astronomers began finding some new dis
crepancies between theory and observation. The Danish astronomer Tycho
Brahe spent decades between 1570 and 1600 in a patient study of planetary
motions, using better instruments and more astute observation than Coper
nicus could command. He accumulated a gold mine of data about move
ments of the planets, much of which just didn't quite fit the Copernican
theory. It remained for a young assistant of Tycho Brahe, a German
astronomer named Johannes Kepler, to study Brahe's data in the early 1600s
and discover what was wrong.
Copernicus had challenged one of Ptolemy's basic assumptions, that the
earth was the center of the solar system. But he had failed to question the
other assumption: that the heavenly bodies moved in perfect circles. Kepler
realized that the notion that a circle was the perfect path for a planet to
follow, while philosphically tidy, did not actually have to be true. He began
searching for some other path of motion for the planets which might explain
the discrepancies between the theory of Copernicus and the things that
Brahe had observed. Finally Kepler discovered the truth: that the planets
traveled in elliptical orbits, with the sun always located at one of the foci
of the ellipse. He also found a relationship between the speed with which a
planet moved and the distance it lay from the sun. As a planet moved closer
to the sun in its elliptical orbit, Kepler found, its velocity increased; when
it swung away from the sun its velocity decreased.*
Kepler also noted that planets that lay close to the sun sped around it
faster than those far distant, and these differences in the periods of revolu
tion of the various planets could be described in a fixed mathematical ratio
to their mean distances from the sun.
It was heady stuff, Kepler's contribution a real bonanza of new and
enormously important information for astronomers. But the work of these
men had far deeper significance for the whole world of physics. They had
plowed through a roadblock to scientific investigation which had persisted
for centuries. The Greek technique of investigation was simple: Apply
philosophy, reason, and speculation to casual observations of nature; then
arrive at a theory; then somehow cram any newly observed facts into the
theory, difficult as that might be. It was not that the Greeks were fools;
they simply assigned importance to the wrong things. Copernicus, Tycho
Brahe, and Johannes Kepler for the first time demonstrated that observation
and measurement were the real keys to scientific discovery. The Procrustean-
bed technique was no good; theory could be accepted only as long as all ob-
* Kepler described this much more accurately by saying that an imaginary line
drawn from a planet to the sun would always sweep across the same area of the
ellipse in the same unit of time; thus a planet near the sun (i.e., near perihelion m
its orbit) would move faster in order to sweep the same area of the ellipse each
second as it did when moving more slowly far from the sun (i.e., near aphelion).
See Figure 2.
34 Physics in Perspective
servations substantiated it, and not a moment longer. If careful observations
failed to substantiate a theory, then it was the theory that was wrong and
not the universe.
THE RENEGADE OF PADUA
Of all the other giants of those days, it was Galileo Galilei (1564-1642)
more than anyone else who established this simple idea once and for all,
and forced scientists all over Europe to throw out the accepted conclusions
of centuries. Galileo was the father of experiment, repeatable experiment
which anyone else could duplicate if he wanted to take the time to bother.
Fig. 2 Kepler's elliptical orbits. According to Kepler's laws, a planet at peri
helion moves faster in its orbit than at aphelion, covering distance from P x to P 2
in the same time required to cover the shorter distance P 3 to P 4 , and Area A
of the ellipse equals Area B. The elliptical orbit illustrated is, of course, greatly
exaggerated.
He was the father of the orderly statement of principles or conclusions
derived from his experiments, setting these conclusions down in simple, flat
statements which could then be subjected to further experiment either to
prove them or disprove them. Today Galileo's principles are usually en
countered as algebraic formulas in textbooks of physics, and thus in
timidate all but the brave and determined. This is unfortunate, because these
principles, for the most part, are actually nothing more than simple state
ments of how things change in relation to other things.
For example, we could record a commonplace observation by saying,
"Downtown traffic becomes lighter at a constant rate the longer after rush
hour that you measure it." We could state the same thing as a simple
formula;
From Philosophy to Science 35
Both statements express a constant (K) relationship between two meas-
urables: the volume of downtown traffic (D) and the time interval since
rush hour (t). Expressed either way, the principle could be a useful addi
tion to our knowledge, provided it was confirmed by repeated, independent
observations. With it we could predict something we might not otherwise
have any way to know: that driving through downtown will be easier (and
conceivably, safer) the longer we wait after rush hour. To the scientist the
principle expressed in a mathematical form is more useful than in words;
but to the layman, the words convey the meaning better.
Galileo used both means of expression. Certainly he was a genius at the
detailed study of mechanical things that happened in the world around him,
and at discovering relationships between one occurrence and another.
Above all, he was a master at generalizing from one specific case to other
similar but different specific cases. He was not content merely to observe
that an object dropped from his hand fell with increasing velocity until it
struck the earth, nor even just to record accurately the rate of acceleration of
that object as it fell. He did not merely conclude that that particular object
accelerated at a constant rate throughout the time of its fall. He went a step
further; by expressing his observation as a simple law or rule, he then
reasoned that he was really describing the acceleration of any freely falling
body anywhere in relation to the height from which it fell and the length
of time of the fall.
Galileo's study of falling objects is probably more familiar to more people
than anything else he did. But his work did not stop here. He also studied
the behavior of objects rolling down inclined planes, and worked out the
mechanical principles of the pendulum the groundwork for study of all
kinds of cyclic or oscillating mechanical motion. He established the basic
laws of work and energy, and studied the effects of friction and the various
phenomena of inertia in fact, the whole range of principles of mechanical
motion. He investigated such physical phenomena as light and sound, de
veloped and improved the first practical working telescope, and made
dozens of other basic contributions to the knowledge of classical physics.
In some areas Galileo failed completely, as in his attempt to measure the
velocity of light. He was in almost constant conflict with the Church of his
day, and was considered a renegade among scientists but his two great
and indispensable contributions to the future development of physics could
not be denied or suppressed. First, he established once and for all that
experiment, measurement, and observation were more valid ways of dis
covering the truth about nature than intuition and speculation indeed, that
intuition simply could not be trusted at all. Second, he established firmly
the idea that the workings of nature throughout the universe were uniform.
When something happened in one situation it could be counted upon always
to happen the same way in the same situation at another time or anywhere
else in the universe. To put it differently, Galileo demonstrated that any
36 Physics in Perspective
valid laws of nature were indeed laws', if nature later were observed not to
be obeying those laws, this then had to mean that the laws were not quite
valid and needed to be redefined, not that nature had turned fickle.
In fact, Galileo had stumbled upon a powerful tool for the investigation
of natural occurrences: the tool of scientific method. He had tried it and
found that it worked. And close behind him, other men went on to learn
just how powerful this tool could really be, properly used. With an orderly
scientific method at their command, scientists for the first time could brush
aside the cobwebs of ignorance, misinformation, and superstition to begin
searching out nature's most closely guarded secrets: the natural laws de
scribing the interactions of matter and energy in the universe.
But what, precisely, was this scientific method of investigation? And
what, precisely, are these "natural laws" that scientists have been searching
out for so long and continue to search out? Much general misunderstanding
and confusion about the discoveries of modern physics arises directly from
confusion about the scientific method and the nature of "natural law."
Before we go further, we would be wise to define these things as simply and
specifically as we can, not only to help us understand how things work in
the world of our everyday experience, but to better comprehend the other
strange and exciting worlds of modern physics as well.
CHAPTER 4
The Methods of Discovery
To the mind of a child, the universe is narrow and sharply defined. Things
which seem enormously complex to us are perfectly simple and obvious
to him. Because of this, strange as it may seem, any child knows what "the
scientific method" is, although he may not know it by that name. In fact,
he uses it constantly in his everyday life. He also understands what "natural
law" is, within the confines of his limited universe, with perfectly amazing
insight.
For some of us, it may seem unnecessary to discuss the scientific method
in simple and specific terms. But simple as the concept really is, it is often
widely misunderstood. Because our whole modern concept of natural
law' is, within the confines of his limited universe, with perfectly amazing
wise to review what it is and how it arose in the most fundamental terms
possible. A clear understanding of both the concepts of natural law and
scientific method is vital in any discussion of the ideas of modern physics.
How does a child use the scientific method? Consider the case of Johnny,
a three-year-old boy with a normal, healthy appetite for chocolate cake
frosting. One afternoon Johnny stood watching his mother frost a cake
with this brown, sticky stuff that tasted so good. From Johnny's limited
view of the universe, it seemed obvious that Mother was spreading that
frosting out solely for him, so he reached out and took a large sticky
handful off the top of the cake just as Mother finished spreading the last
spoonful.
Unfortunately, Johnny's view of the universe was too limited. His mother
shrieked, whacked him soundly, and said, "Keep out of the cake frosting."
Then to top it off, she washed his hand off before he even got a taste.
To Johnny this was a hard lesson a new and unnecessary complication
in an already overcomplicated world. Obviously, something was wrong with
his assumption that the cake frosting was there for him to grab. Yet he
remembered distincly that Mother had given him cake and frosting to eat
at dinner the last time a cake had been baked. What was going on here?
What rules applied? After mulling it over for a while, Johnny threw out his
previous assumption in favor of a new one that seemed to fit the facts
37
38 Physics in Perspective
better: The frosting was there for him to grab provided Mother wasn't
standing right beside him at the time.
A few days later when his mother baked another cake, Johnny tested
this new hypothesis. This time he waited until Mother had left the kitchen
and gone into the dining room to set the table. Then he grabbed his fistful
of frosting. This time he did get a taste before the reaction came, but
Mother, of course, returned too soon. Again she shrieked. Again Johnny
got whacked. Again he was admonished, "Keep out of that cake frosting or
you won't get any cake at all!"
Now this was really a conundrum. Johnny retired to nurse his wounds
and reconsider the data. The frosting was good. Since he liked it, it must
be made for him. But whenever Mother saw him reach for it, he got
whacked. Maybe the determining factor was whether his Mother saw him
take it or not. Well and good; the next time a cake was made he waited
until Mother was outside gardening. Then he took some frosting from the
top, with a little cake thrown in for good measure. Of course, he left a hole
in the top of the cake, but if Mother didn't see him take it, how could she
know? Johnny ate his frosting and then went on about his business, the
mystery of the cake frosting finally solved until Mother came in half an
hour later and found the violated cake. Another whacking, another scold
ing, and no cake for supper that night.
So it was back to the drawing board again for Johnny. Clearly some
thing was wrong with his whole approach. The frosting was there for him
to eat, but only at certain times (i.e., when served at dinner) . At other times
it was sharply proscribed. To Johnny this made no sense whatever but that
seemed to be how things worked. Even more puzzling, when he took the
frosting at the forbidden times his punishment seemed completely un
related to whether Mother could see him or not. The business had him
baffled but baffled or not, there was obviously some method of detection
at work, and that, too, was a part of the way things worked in the world
around him.
Johnny might have tossed in the towel, if he had been a little less stub
born and persistent. He might just have accepted things the way they were.
But Johnny was a stubborn little boy who really liked chocolate frosting.
He was not about to accept a law that made no sense to him u cake frosting
is forbidden at some times and permitted at others" nor to stop trying to
outwit the mysterious method of detection that was thwarting him. The situa
tion challenged his young mind, and presently a new approach came to
mind. Perhaps he could take the frosting at the forbidden times // he could
do it without being detected. Hard to achieve? Well, maybe not so hard.
First, try taking the frosting from a part of the cake where Mother wouldn't
notice. If that didn't work, then take it off the top but smear some on the
cat's whiskers in order to get Mother confused
And so the struggle went on.
The Methods of Discovery 39
THE SCIENTIFIC METHOD
Foolish as this fanciful story may seem to us, we must realize that it was
anything but foolish to Johnny. Within his limited view of the universe,
Johnny was facing precisely the same sort of problem that adult men have
been facing for centuries: finding ways to supply their needs and wants, and
to obtain for themselves more comforts and satisfactions in life in return for
less effort.
In reaching for these goals men have always found themselves thwarted
by a series of inexplicable, baffling, and seemingly senseless rules governing
the way things happen in the world, just as Johnny did. Often these rules
have seemed to exist for no other reason than to annoy and hamper man
kind in the fulfillment of his needs. The man who tried to carry water home
from the well in his hands found that he couldn't do it. The stuff ran through
his fingers and was lost before he could get ten paces away from the well.
Sometimes the rules seemed completely arbitrary and confusing: In the
winter a man could carry water home in his hands in solid blocks of ice
while in the summer he could not. There was no sense to it, but it was
one of the facts of life he simply had to deal with if he wanted water. Only
later was it discovered that it was possible (1) to preserve water in its
hard, "carryable" state in the summertime by burying it in deep pits under
ground in the winter; or (2) to carry the liquid water more efficiently in an
earthenware jug than in your hands.
So men, faced with such awkward rules, set out to find ways to get around
the obstacles in their way, just as Johnny did. Bit by bit they learned which
rules could be by-passed easily and which could not. They learned how to
change and control conditions around them. Above all, painstakingly, they
learned to define what the rules actually were, in hopes of sometime dis
covering why they existed.
In his battle for the cake frosting Johnny was doing exactly what men
have been doing for centuries in their fight for survival. Johnny was explor
ing the "natural laws" governing his universe the narrowly limited uni
verse of his own experience as seen through his eyes. Instinctively, he used
a method of exploration which, bit by bit, provided him with useful knowl
edge and useful results.
Today we call that method of problem-solving the scientific method. It
involves four critical steps, each taken in turn and each equally important
in reaching a satisfactory solution. Those steps, in order, are observation,
hypothesis, experiment, and finally, conclusion-drawing.
How does it work in the hands of a modern scientist? First there must
be a question to be answered, some riddle to be solved. The scientist en
counters some question which he cannot answer, some phenomenon of
nature which he does not understand. He then begins to gather all of the
data about this riddle that he can find simply by observing it as closely and
Physics in Perspective
carefully as possible. This period of acute observation is absolutely vital.
Without it the scientist has no basis even to guess what the solution might
be.
Second, on the basis of what he observes, the scientist will think of one
or two, perhaps several, plausible and possible explanations for the phe
nomenon in question. These possible explanations are called hypotheses.
Often one or another hypothesis can be discarded right from the start.
Perhaps certain observations obviously don't fit one hypothesis; another
may seem highly improbable, even though the scientist can't pinpoint
exactly why he thinks so. After eliminating these, he will choose the one
remaining hypothesis that seems the most likely of all. Accepting this
at least tentatively as his working hypothesis, the scientist then devises
experiments to test whether it really does explain the phenomenon or not.
If he is a good scientist, this will not be any half-hearted gesture; he
will submit the hypothesis to a real trial by fire. Of course, he will seek out
experiments that seem likely to support it but he will also rack his
brains for any possible experiment that might prove it wrong. Indeed, the
most important step of all in this method of exploration is the cold, delib
erate attempt to poke holes in a possible explanation, to knock it to pieces
if that can possibly be done. Nor does the scientist trust himself to be
sufficiently objective. He knows from long experience how easy it is to fool
oneself, to select only favorably loaded experiments, and to try to prove
what one wants to believe, whether it is actually true or not. For this
reason, the scientist records his experiments for others to criticize. He
devises experiments which are repeatable, so that other scientists can
also do them and compare results. Because, as scientists know, any
experiment that cannot be repeated successfully by others with the same
results is of no value in proving anything.
Finally, from his experiments, the scientist gathers a large quantity of new
data which is then matched up with his working hypothesis. From this,
conclusions can be drawn. If all his experiments seem to bear out the
hypothesis in all respects, and if others come up with the same results,
the hypothesis begins to look really promising as the true explanation for the
phenomenon. If it survives new tests by others, it takes on new strength.
This is not to say that it is proven, by any means; at best it may be
considered "true until proven otherwise." The scientist is fully aware
that new data may be discovered at any time which might raise questions
about it or even disprove it altogether. But when sufficient corroborative
evidence has been gathered, with no evidence at all to contradict it, the
hypothesis attains a state of general conditional acceptance among scientists.
It is then considered a theory. If the theory stands up to multiple exper
iments carefully designed to try to disprove it, if it holds true time and
again no matter who may test it or in what way, and if no new observations
The Methods of Discovery 41
come along to challenge it, it may in time come to be considered a proven
law of nature.
Notice that the scientist did not start with a conclusion and then
attempt to bend the facts to fit it. Notice that from the start he accepts
that his hypothesis may prove to be wrong even though all of his
experiments seem to substantiate it. Given a single bit of contradictory
evidence, one single experiment which doesn't fit in even though a dozen
others do, and the scientist knows that something is wrong. Somewhere
there is a flaw in the conclusions that were drawn, something that is
missing, something that has been misinterpreted. When that happens
(and it almost invariably does happen) he must then revise his hypothesis
to fit the observed facts, changing it again and again as he goes along
until it fully and reliably explains every part of the phenomenon he is
trying to explain. Even a hypothesis that has been corroborated to the
extent of becoming a theory, and then substantiated to the point that
it is considered a proven law of nature is still vulnerable. New discoveries,
new observations, new methods of measurement may at any time cast
doubt upon it. Even the best-established laws of nature must be revised
if newly discovered data demand it. No natural law can ever be considered
finally and irrevocably proven.
There is no magic in such a method of finding an answer to a problem.
Indeed, it is so simple and logical that all of us, scientists or not, use it
to some degree or other every day of our lives in solving everyday problems.
It is the time-tested method of telling truth from nonsense and proving
it. As such, it is the method that has been used in discovering virtually
everything we know about our universe and the way in which it works.
But if the scientific method is so simple and logical, why was it such
a staggering idea when it first appeared? Probably because it was such
a complete reversal of the way ancient scientists and philosophers had
done things for centuries. Before the scientific method was devised, these
men started with conclusions they had come up with on the basis of
meditation, casual observation, and sheer guesswork. Then they wrenched
and twisted newly observed facts to fit the conclusions. Up to a point,
they got results, too; that ancient method worked splendidly as long as
men could manage to ignore the facts that didn't fit in with their conclusions.
It took centuries to discover that this was a blind alley, producing more
and more wrong answers all the time. It took centuries to recognize that
an unbroken chain of cause and effect ran throughout nature governing its
happenings. It was not until the scientific method became firmly established
that the knowledge of science began to grow and that our understanding
of the laws of nature began to expand.
42 Physics in Perspective
THE LAWS AND THE LAWYERS
Obviously, as this scientific method came into use, scientists everywhere
began to revise their ideas about what, exactly, a "law of nature" was.
Clearly it was not something that could be determined on the basis
of intuitive or self-evident "truths," philosophical dialogues, logic, or
reasoning. More and more, as the scientific method began to develop,
scientists began observing nature more closely to see what was going on
that they couldn't understand. Freed from the idea that observed events
had to fit into arbitrary molds, they began questioning everyday things
that were happening all around them.
A ball, when dropped from the hand, fell to the earth with increasing
speed, taking a measurable time to reach the ground. A phenomenon: what
was happening here? The "self-evident fact 1 ' that a heavy ball fell faster
than a lighter ball came under scrutiny, and was found to be neither
self-evident nor a fact. A ball rolling down an inclined plane also was
seen to roll faster and faster until it reached the ground.
On the other hand, a ball thrown straight up into the air appeared to
slow to a stop at a certain point in the air, then reverse its direction and
begin accelerating downward again. A pendulum swinging freely back
and forth seemed to do something strangely similar, accelerating from
one extreme of its swing down to the lowest point in its arc, then decelerat
ing to a stop at the other extreme, then reversing direction and accelerating
down to the low point again. Another phenomenon, something that always
happened in the same way but could there be a connection between
the way the ball tossed in the air behaved and the way a pendulum
behaved? If so, what was it? The two things seemed similar but not identical.
Could it be possible that both behaved according to the same general
principle? If so, then what were the rules? Could the same principle also
apply to the behavior of other kinds of moving objects; for example, to
the behavior of a ball thrown horizontally in an arc? And what about
the size of the ball? After all, the moon was a ball moving through space
but it was not moving freely. Something seemed to bind it to the earth
around which it revolved. Was there, conceivably, some similarity in the
behavior of that celestial ball and the behavior of a handball thrown in
an arc (which also seemed, in a different way, to be irrevocably bound
to the earth)? Or was such an idea merely a wild reach? If it wasn't a wild
reach, if there really was some similarity in the behavior of these moving
bodies, shouldn't the moon also be included in the ever-broadening
general principle that governed the behavior of the falling ball, the thrown
ball, and the pendulum?
It was a slow process, this vast exploration of utterly unknown territory.
But little by little certain broad general principles were identified ways
of describing things that happened, ways of comparing one phenomenon
The Methods of Discovery 43
with another. Whenever such broad principles (or "laws of nature," as we
call them today) began to emerge, they were subjected to relentless
testing. Generation upon generation of scientists experimented repeatedly
to see if these principles really did describe widespread or universal
phenomena, or whether they applied only in a few special or limited
situations, and even then, perhaps, only sometimes.
Under such ruthless scrutiny, many apparently valid "laws" fell by the
wayside, disproven by repeated experiment and testing. Others were
corroborated again and again sometimes modified, sometimes broadened,
but still holding up under the careful scrutiny of the scientific method.
Some of the strongest, best-proven of these principles are still being
challenged today; they are still accepted as valid only until proven other
wise. To many nonscientists this persistent effort to disprove things which
have been shown to be valid in thousands upon thousands of cases may
seem ridiculous. But we know today that many of the laws of nature which
seemed to apply to all areas of the universe in the early days of physics
have since been shown to apply only to one limited area: the world of
everyday experience. They have been proven to be incomplete or even
flatly invalid in describing events in the microcosmic universe of elemen
tary particles or in the macrocosmic universe of far-flung galaxies. So
those "proven" laws of nature have had to be discarded, or severely
modified. Indeed, over a period of two thousand years of painstaking
observation and experimentation, only the barest handful of basic, universal
laws of nature have survived unscathed to this day.
What, exactly, are these few basic laws? What do they say? More
important, what do they mean! Most of us have only vague memories
of these laws as complicated mathemical formulas encountered in high
school or college, never very meaningful at best, and now long forgotten.
Is it possible for us really to understand those laws now without getting
involved in complex calculations and pages of mathematical reasoning?
Perhaps so, because the real, incredible beauty of these basic laws of
nature is their splendid simplicity. Basically they are nothing more than
clear, simple statements of relationships simple quantitative descriptions
of things that have been observed to occur in the universe. They are state
ments that describe how things work, nothing more.
The few basic laws we are discussing are powerful as well as simple.
Evidence collected over centuries supports them. They have weathered
innumerable challenges, and face new ones every day. Scientists today,
as throughout the last eight hundred years, challenge them by use of the
scientific method. To investigate the law of gravitation, for example, a
physicist studies the motions of objects of all sorts: planets moving in
orbit, balls rolling down inclined planes, feathers set free to fall in a
vacuum tube. First the physicist observes and measures. How fast does
a given object move? In what direction? Is its speed constant, or changing?
44 Physics in Perspective
What about its direction? With his observations made, he then tries to
think of some rule or principle that can explain all the things he had
observed. With such a rule as hypothesis, he tries applying it to other
objects in motion under other circumstances. To be useful, the rules must
describe how any object will move under the influence of gravity. So again
and again he tests his rule, measuring it against the actual behavior of a
great variety of moving objects.
If he can find such a rule, and if neither he nor other physicists can find
even a single experiment that seems to disprove it, eventually the rule or
hypothesis becomes a theory, then later is considered a law of nature.
Thereupon, the physicist seeks to use this rule to predict how any object
in any gravitational field anywhere, will move. He knows the rule may not
hold up. Many early "laws of nature" proved to be nothing more than
descriptions of isolated, individual events with no application to other sim
ilar events. Other such "laws" described a broader range of phenomena but
still did not cover all phenomena of a similar nature. Such "laws" may well
be useful indeed to help make certain kinds of predictions or to solve certain
kinds of problems, but they are not really good laws, not because they are
untrue (as far as they go) but because they are too limited.
GOOD LAWS AND POOR
What, then distinguishes a good law of nature from a poor or limited
one? First, a good law of nature deals with situations in general, not with
specific cases. It does not, for example, describe the movement of one
particular kind, size, or shape of object under certain limited circum
stances. This would be little more than a description of a single limited
occurrence, perhaps even a single experiment. Rather, a good law pro
vides a general description of the movements of objects of any kind, size, or
shape under a wide variety of circumstances.
Second, a good law of nature should apply universally. If it describes
the motion of objects, it should apply not just in any one place or at any
one time, but anywhere in the universe at any time, whether in the world
of our sensory experience, in the microcosm, in the macrocosm, or wher
ever; and it should apply to any and all objects, no matter how large or
small, no matter where they are moving or how fast or slow, or in what
direction.
Third, a good law of nature should not contain too many exceptions.
If a law is hedged with ifs, ands, buts, and whereases, specifying various
exceptions in special cases, and applying only when everybody in the
world happens to be wearing green neckties, it becomes too complex for
any real usefulness. Whenever too much work and effort are required to
figure out when a law is supposed to be applying and when it isn't, this is
The Methods of Discovery 45
usually a pretty good indication that the law is not really a general, uni
versal description of anything, even though it may seem to have wide
application. It becomes just one more of a multitude of relatively insig
nificant rules of the road, and probably highly vulnerable to challenge and
testing anyway, rather than a good, useful law of nature.
Finally, a good law of nature is a complete and quantitative statement
or description, not just a hazy, indefinite expression of generalities. A law
describing the interaction between an object and a force acting upon it
must do more than just to state vaguely that something influences some
thing else. It must state in what specific way something influences some
thing else, how strongly, when, and in what direction. It must state
these things in exact quantitative terms that do not omit anything sig
nificant. This, of course, is why good laws of nature are so often expressed
in the form of mathematical equations; that is one of the most reliable
ways of being sure that the law describes some relationship completely
and quantitatively. The terms of an equation are explicit; a given factor is
either included or it is not.
This is also why good laws of nature are often so difficult to express
correctly in words: Words can be very slippery indeed when it comes to
pinning down exact meanings! In this respect, it is inescapable that laws of
nature be expressed in mathematical terms if they are to be stated com
pletely and quantitatively but at worst (in the case of the most basic and
fundamental laws of nature) they are expressed in the form of simple
equations which require little more than a rudimentary command of algebra
or calculus to interpret.*
To understand more clearly exactly what a good law of nature is and
how it fulfills the criteria we have discussed above, let us look at one of
these basic universal laws more closely. Many of the phenomena that
occur continuously all around us are described by one very familiar law of
nature the law of universal gravitation. A baseball arcs toward the earth
when we throw it. A coconut drops from a palm tree. Water runs downhill
when left to its own devices. The moon and planets move in their orbits
according to a reliable, repetitive pattern.
According to legend, the law of gravitation was discovered by Sir Isaac
Newton when an apple fell from a tree and hit him on the head. Whether
any such thing actually occurred is a moot question, but the idea that
Newton "discovered" this law in any blinding flash of revelation is pure
nonsense. Strictly speaking, Newton did not "discover" the law of gravi
tation at all; it had been obvious for centuries that some orderly principle
* For all of this, the meaning of these basic laws can usually be conveyed without
recourse even to this simple level of mathematics. In this book we will venture onto
the high seas of simple algebra on occasion but the author is convinced that many
readers tend to reject and pass over even simple mathematical equations they can
understand perfectly well, so we will seek to avoid them wherever possible.
46 Physics in Perspective
lay behind the behavior of moving objects near the surface of the earth.
Galileo had long before made accurate measurements of how fast objects
fell to the ground when released, and how their speed kept increasing
steadily as they fell. Copernicus had already observed that the moon re
volved around the earth at a certain velocity without flying off into space.
What Newton did achieve was to demonstrate that the same natural law
that described the movement of falling bodies on earth also described the
movement of the moon around the earth, or of the planets around the sun.
Newton showed that these objects all moved as they did because of a
simple, universal relationship between every object in the universe and
every other object: that every object in the universe attracts every other
object In the universe; that the force of attraction between any two given
objects is always dependent upon the masses of the objects and the distance
between them; and that this attractive force between any two objects in
the universe could be calculated according to a precise mathematical
equation.
In other words, the things that happen "according to the law of gravita
tion" actually do nothing of the sort. They are natural occurrences, which
happen. The law itself is nothing more than a generalized description of
all these events and phenomena put together the words (or mathematical
formulations) that we use to describe with extreme precision the attrac
tion that demonstrably exists between any two objects. The description
also states precisely how strong the force of attraction is, in what direction
it acts, and how it may change from place to place.
It is important that we realize clearly that this "law of gravitation" that
we are talking about is a simple description of events, so to speak. It does
not say why every object in the universe attracts every other object. It does
not say from whence the attraction arose, how long it has been there, nor
for that matter whether or not it will still be there tomorrow. AH it says is
this is the way it is, this is how things work as far as anyone has been able to
observe and measure so far,
The same thing exactly can be said about all the other great universal
natural laws. The "law of inertia," for example, simply describes certain
observable characteristics of objects in motion and objects at rest. In brief,
this law says: Any object in motion tends to remain in motion in a straight
line at a constant velocity, and any object at rest tends to remain at rest,
unless acted upon by an outside force.
Such a "law of nature" is a very useful thing to have around. It allows
us to know what behavior we can expect of objects at rest and of objects in
motion. It allows us to make accurate predictions about how objects,
whether at rest or in motion, will behave ( 1 ) if an outside force is applied;
or (2) if no outside force is applied. The law of inertia is indeed useful
but it does not even attempt to explain why an object at rest remains at
rest, nor why an object in motion remains in motion.
The Methods of Discovery 47
Well, then, why do they?
Nobody knows why.
Nobody knows why every object in the universe attracts every other
object, either. Nobody knows why the force of gravity exists, nor even
what it is, nor what inertial force is, nor why it exists. What is more, it took
the scientists of the world centuries of banging their heads on the wall
before they began to realize that it simply didn't matter whyoi, rather,
that to ask why is to ask a fruitless question.
And here we discover a final reason that the ordinary man in the street
finds it so hard to understand what the modern physicist is doing. Most
people who have thought about it at all have assumed without any ques
tion that physicists were trying to find out why matter is made up the way it
is and why forces act the way they do, and it just isn't true. Scientists gave
up asking why centuries ago. Indeed, they regard such questions as stum
bling blocks and blind alleys, rather than the proper and legitimate business
of the scientist.
Admittedly, this is a rather startling and unsettling observation when we
first encounter it. Isn't that what any study of science, and above all the
study of physics, is supposed to be all about? Isn't its whole purpose really
to explain why things happen the way they do in the universe? Certainly
the ancient Greeks thought so. They devoted an incredible amount of time
and energy trying to dream up plausible reasons why, and then trying to
prove them. And as an intellectual exercise, this was great. It stimulated
the mind no end. It developed the rules of logic and argument to a razor
edge. But as far as finding out what was happening was concerned, the
search for reasons why drove the ancient Greeks into a blind alley. For a
thousand years after the Greeks, scientists kept falling into the very same
trap. In their search for reasons why things happened they got nowhere.
They didn't even learn too much about what was happening. Only when
they decided to shelve the question of why things were happening did they
begin to forge ahead in accurate, useful observation and description of what
was going on in the universe in what way, whatever the reason why might
be
This is not to say that the question of why is somehow forbidden and
improper. A great many fine scientists today are still very much con
cerned about the why of things as well as the how. Perhaps this is indeed
the ultimate goal of science, to find out why. If so, then scientists still have a
long way to go because the one cannot come before the other. There is
little hope of ever finding out the why until what is actually happening
has been fully and accurately described until we know how matter is
constructed, what energy is, how matter and energy interact not just m our
universe of everyday experience but in all the microcosm, in all the
macrocosm, in all space, throughout all time, always, under all circum
stances conceivable or even inconceivable. And even if physicists were to
48 Physics in Perspective
begin right now with all the knowledge that has been already been ac
cumulated, this would even today be a whale of a large oider.
Of course over the centuries great progress has been made. We cer
tainly know more of what is going on (or of what probably is going on,
or at the very least, of what seems to be going on) today than the ancient
Greeks did. We have better tools to work with more accurate instruments
for observation and measurement, more fruitful methods of investigation,
newer and more useful mathematical techniques. And thousands of scien
tists all over the world today are working to refine these tools even
further.
Furthermore, what we do understand today of how the universe works
has paid almost unbelievable practical dividends. This knowledge has al
lowed us to take giant strides in fulfilling our everyday human needs better.
It has allowed human beings everywhere to live more comfortable, varied,
useful, and interesting lives. It has also shown us what men can poten
tially do with matter and energy in order to improve living conditions and
control environment. The application of our knowledge of how the uni
verse works has closely followed the discovery of how things work, revo
lutionizing the world we live in. And the promise held forth by modern
physics for even more useful and revolutionary applications of new knowl
edge in the future is staggering.
Unfortunately, all parts of the picture are not so bright. If our knowl
edge of physics has paid great dividends, it has also created frightening
liabilities. Man has found the means to destroy himself and all his works
in one great bloody ball of fire. Nature is not fooling around; raw energy
can be found or produced in nature in perfectly staggering supply for
Man to use for survival or for self-destruction, whichever he elects. If men
are clever enough to snoop out her secrets and discover how to use them,
nature seems to say, they had better learn quickly how to harness and con
trol the energies they are capable of releasing, if they hope to survive.
But nature doesn't care. There is a great deal of universe available, and
what men do or fail to do in this particular little corner of it will hardly
cause any cosmic upheaval.
So we search out information about how things work, not why. All of
the great, basic laws of nature which have survived all challenges over the
centuries are nothing more than simple, general, universal, quantitative
descriptions of how things happen in the universe descriptions of the
built-in characteristics of matter and energy, and of the way objects and
forces interact with each other. Many laws once thought to be "basic"
have been shown to apply perfectly only within certain limits; such laws
have had to be altered, or discarded, or modified to take into account new
observations. There is no law of nature known today, no matter how "good"
or "strong," that can be considered to be finally and absolutely proven. At
best it can be considered "unlikely to be proven wrong."
The Methods of Discovery 49
Of course, it is always a shock when a long-established, apparently basic,
and immutable law of nature is overthrown. That was the reason Albert
Einstein's work was so profoundly disturbing and challenging to the
world of science. He challenged laws of nature long believed to be basic
and universal, established beyond possible doubt, and he made his chal
lenges stick. As a result, like it or not, physicists were forced to revise their
outlook about what, exactly, those laws were really describing and how
things actually did work. In later chapters we will see just which of these
laws Einstein challenged and just why his challenges could not be brushed
off. Certainly that one man, more than any other in history, forced sci
entists to make an agonizing reappraisal of how things really worked in
the universe a reappraisal which is still going on today.
Einstein also combined certain other categories of "basic," "universal"
laws into one, thus showing that they were far more "basic" and "uni
versal" than anyone had ever imagined. For example, he combined space
and time into one entity, and he demonstrated that the two of the most
basic and universal of all known laws of nature the law of conservation
of matter and the law of conservation of energy were in fact two aspects
of the same law, an insight into the true nature of matter and energy that
opened up profound and frightening vistas. Ironically, even Einstein him
self was chagrined by some of the logical implications of his own work,
and refused to the end of his life to accept them, even though he could
not find a way to disprove them a heartening indication that one of the
greatest genius minds to appear in all history was still a human mind, shar
ing the same human emotions and human frailties that apply to you and
to me.
A CHECKLIST OF NATURAL LAWS
When we come down to final cases, we discover that the basic, universal
laws of nature that have held up over the centuries in the face of all chal
lenges (even if modified or altered somewhat) are relatively few in number.
There is nothing mysterious about them. They can be listed and stated very
simply, and understood without resort to great technical knowledge or the
skills of higher mathematics. In every case, these laws are "rules of the
road" and nothing more merely descriptions of -what happens, descriptions
of characteristics of matter and energy. To provide a clear focus for later
chapters, we can list these laws in a simple chart, as follows :
I. The laws of motion
a. The law of universal gravitation is a quantitative description of
the force of attraction that exists between any two objects any
where in the universe, and of how that force affects the objects.
It states that every object in the universe attracts every other
50 Physics in Perspective
object, and describes exactly how this attractive force can be calcu
lated in the case of any two objects, whether on earth or in the far-
flung reaches of space. It also provides a precise statement of how
powerful that force will be in any given case.
b. The law of inertia describes certain built-in characteristics of all
material objects in the universe. It states that any object at rest
tends to remain at rest, and any object in motion tends to remain
in motion in a straight line at a constant velocity, unless acted
upon by some outside force. This law originally served to explain
a variety of characteristics of objects at rest and objects in motion
that seemed to have nothing to do with the force of gravity. But
today, as we shall see later, there is at least good reason to sus
pect that these two "basic" laws the law of universal gravita
tion and the law of inertia may really be no more than two
different ways of describing the same phenomena. Unquestion
ably there are certain ways in which they are similar and inti
mately related; and there are certain circumstances in which it is
completely impossible to be certain which of the two is actually
the proper law to use to describe a given phenomenon.
II. The laws of conservation describe other characteristics of matter
and energy, and apply to certain other phenomena associated with
the motion of objects.
a. The law of conservation of matter-energy states that matter or
energy may be changed from one physical state or form to an
other, and may be converted reversibly from one to the other,
but that the combined total of matter and energy existing in the
universe can neither be increased nor decreased. Here, again,
two original conservation laws (the law of conservation of
matter and the law of conservation of energy) have been com
bined into a single, all*encompassing principle, since it was
found both theoretically and experimentally that matter and
energy are two totally interconvertible forms of the same thing.
b. The law of conservation of momentum states that the total
momentum of any interacting system is always conserved. This
law, together with its "twin sister" corollary, the law of con
servation of angular momentum, is probably the single most
firmly established and basic of all known laws of nature, yet it is
quite unfamiliar to most nonscientists, and is not even always
clearly understood by scientists. To make sense of it, we must
understand precisely what such terms as "momentum," "angu
lar momentum/' and "interacting system" mean to a physicist.
Thus our discussion of this law will be deferred to a later chapter.
The Methods of Discovery 5 1
c. Other conservation laws, including laws of conservation of
electrical charge, conservation of nucleon charge, mirror sym
metry, and electrical charge symmetry, as well as others, are all
essentially statements of properties or characteristics of matter
or energy which ultimately remain unchanged, always, under all
circumstances, no matter what happens as far as physicists
know today.
III. The laws of thermodynamics are intimately related to the macro
scopic or statistical behavior of molecules of matter in motion, and
describe how heat or thermal energy is transferred from object to
object or particle to particle. These are essentially nothing more
than the law of probabilities as applied to the behavior of molecules
en masse. In brief summary, the laws of thermodynamics state:
a. That the total entropy of any closed system must always either
remain unchanged or increase. From this we can deduce that:
1. The natural direction of heat flow is from a hot area to a
cold area, and this direction of flow cannot be reversed with
out the aid of some outside source of energy; and that
2. The natural direction of energy transformation is from me
chanical energy into heat energy. Again, our discussion of the
puzzling concept of entropy, related both to heat content and
molecular disorder, must be deferred to a later chapter.
In addition to these laws of classical physics, we must add to our list
the concepts of two great theories of modern physics, considered "laws of
nature" by many physicists, and four great forces which are known to act
upon objects at a distance and to create "fields of force" through which
they work:
IV. The theories (laws) of relativity, worked out by Albert Einstein and
others, are based on the notion that there is no such thing as
absolute motion in the universe, that all natural phenomena ob
served in the universe can be described only relative to the observer
and may be described differently by another observer in a different
location. The distinctions between the special theory of relativity,
and the general theory of relativity, and the overwhelming signifi
cance of these revolutionary theories in the world of physics, will
be discussed in detail in Part III of this book.
V. The theories (laws) of quantum mechanics deal with the behavior
of subatomic particles making up atoms of matter, and are based
on the concept that energy always occurs in tiny but finite packets
52 Physics in Perspective
or "quanta" which represent the smallest quantities of energy that
can be interchanged between particles. Part V of this book will deal
with quantum theory in considerable detail.
VI. The laws of forces and fields are concerned with four universal
forces capable of acting upon objects or particles at a distance (i.e.,
without physical contact) through the medium of energy fields:
a. Gravitational forces, acting most strongly between very massive
objects in the universe;
b. Electromagnetic forces, encompassing the attractive or repulsive
forces of magnets and of electrically charged particles;
c. Weak nuclear binding forces, arising from certain "weak" or
"relatively improbable" interactions between subatomic particles
in very close proximity to each other;
d. Strong nuclear binding forces, arising from certain "strong" or
"highly probable" interactions between subatomic particles in
very close proximity to each other.
Such a list of natural laws, even so encapsulated, may seem dreadfully
complex at first; yet when we consider it, we see that the list is really not all
that long. Approached logically, against a comprehensible background or
frame of reference, it will appear less and less complex, and more and more
comprehensible as we go along. But at this point we must recognize clearly
that all these laws are descriptions of what and how, not discussions of
why. When we ask why, even today, we are in trouble. And the physicist
more than anyone else grows weary of the age-old questions which he still
cannot even begin to answer:
Why can't matter either be created or destroyed? We don't know but
nobody has ever been able to do it yet.
Why can't energy either be created or destroyed? We don't know but
nobody has ever been able to do it yet.
But who says it can't be done? Nobody says so except the thousands
upon thousands of scientists who have tried and failed with monotonous
regularity over the centuries.
Well, what is matter? We don't know. We're still trying to find out.
What is energy? We don't know. We're still trying to find out.
More than any others, those last two questions are the real challenge to
modern physicists. It is because these two questions have not yet fully
been answered that physicists remain at work today, after thousands of
years of study and in spite of their inability to explain why. And it seems
likely that they will remain in business for some time to come.
The Methods of Discovery 53
What is matter? What is energy? There is no better way for us to start
considering the riddles of the "normal" universe of human experience than
with these questions in mind. In succeeding chapters we will see what
answers the classical physicists came up with as they tried to describe the
universe they saw and the way things happened in it. In the process we
will see how some of these great laws we have listed above were first de
fined, and see more clearly exactly what they mean.
Part II
The Universe of Classical Physics
CHAPTER 5
Assumptions, Observations, and Measurements
By the last quarter of the nineteenth century the first three groups of the
great natural laws outlined in the last chapter had been defined, proved,
entrenched, and fully accepted throughout the world of science. These are
the laws we think of today as the "fundamental laws of classical physics"
the law of universal gravitation, the conservation laws, the laws of in
ertia, and the laws of entropy and thermodynamics. As little as seventy
years ago it was firmly believed that these laws alone described virtually
every phenomenon and occurrence that existed in all the universe. Arising
as they did from a thousand years of labor, they were regarded by the
physicists of the 1880s and 1890s with great satisfaction; through these
great "unshakable" laws, it appeared that all the workings of the universe
had really been pretty well defined and described.
Today, of course, we know that those who held this pleasantly self-
satisfied attitude were in for a rude awakening. They were sitting on a
powder keg, with Michelson and Morley all set to touch off the fuse. But
from one point of view, physicists of that day were perfectly right. They
could hardly be blamed for assuming that the universe they knew the
universe of everyday human experience was all the universe there was.
And so far as that "normal" universe was concerned, these classical laws
of physics were indeed perfectly valid. Except for a few loose ends, such
as the question of the "ether wind," and the puzzle of the nature of light,
these laws did indeed describe virtually everything that had ever been ob
served in nature.
What is more, those same great laws are just as valid today in describ
ing and predicting things in the "normal" universe as they were then. They
still work. They still apply. We still need to understand these laws in order
to understand the things we see happening around us. We still use them
as powerful tools in solving our everyday problems of mechanics, optics,
acoustics, or electrical engineering.
More to the point here, we need to understand these laws in order to
understand what was found wrong with them and why. It was a painful
reappraisal of these classical laws, beginning at the end of the nineteenth
century, which led directly to the explosive revolution in physics which
57
58 The Universe of Classical Physics
took place later. Our everyday world in turn has been revolutionized by
that great scientific explosion. It will be worthwhile reviewing those
classical laws to see better how and why that revolution in physics came
about.
NOTHING STARTS FROM NOTHING
As we have seen, these classical laws of physics evolved from centuries
of observation and experiment. They did not, however, arise from scientific
investigation alone. Earlier we saw that Euclid could never have developed
his elegant and complex system of plane geometry out of thin air. He
started with a few basic assumptions which he considered self-evident, so
obviously true that they required no proof to support them. Then, on the
basis of these assumptions (or axioms, as we call them) he created his
system of geometry. Beyond these axioms, every single proposition re
quired rigorous logical proof before it could be accepted as a valid basis
for further propositions.
Euclid's axioms were simple. They included such statements as "The
shortest distance between any two points is a straight line"; or: "Any two
lines which, extended indefinitely, never meet at a single point are parallel."
Those axioms have stood the test of time splendidly well. Within the limits
of plane geometry, they have never been disproved. They are taught today
in precisely the same terms as Euclid taught them. True, we know now
that there are other useful systems of geometry in which the shortest
distance between two points is not a straight line, and in which parallel
lines may indeed intersect one another, but Euclid's axioms still apply per
fectly well within certain practical limits. Within those limits they are still
enormously useful in solving everyday geometric problems, as any engineer,
architect, or surveyor can testify.
By the same token, the great classical laws of physics outlined in the
last chapter were also based upon certain assumptions that were accepted
without question as self-evident. The first of these assumptions, so obvious
as to seem ridiculous to mention, was the assumption of reality. In order to
explore anything about the universe, the scientist first had to assume that
the universe really did exist that he existed, that other people existed
apart from him, and that the earth, the planets, the stars, and all the rest
of the universe also existed in fact.
Second, the scientist had to assume that he could learn something about
the universe by means of his senses by seeing it, feeling it, smelling it,
and measuring it and that this was the only way that he could learn any
thing about it. This assumption, of course, conflicted with the ancient
Greek notion that something could be learned about the universe by means
of logic and reason alone. Modern physicists disagree on another basis;
they argue, for example, that pure theoretical mathematics has, in fact,
Assumptions, Observations ; and Measurements 59
suggested or predicted any number of previously unsuspected phenomena
which simply could never have been suspected or detected any other way.
Modern physicists also realize that observation by means of the senses has
some built-in problems: Often the very act of measuring something alters
the thing that is being measured, But such considerations were quite un
known to the early physicists. Bath Galileo and Newton assumed without
question that their sensory observations were the only avenue to learning
anything, and that their measurements yielded valid information. And just
as well, too; without those "self-evident" assumptions, the scientific study
of nature could never have gotten off the ground!
Third, early scientists assumed that there was regularity in the universe
or to put it another way that the universe was orderly. This meant,
simply, that everything that happened in the universe happened in accord
ance with certain natural laws, and that there were no exceptions to those
laws ever, anywhere. The implications of this assumption are a bit more
profound than first meet the eye. Essentially, this was an assumption
that cause and effect ordered the universe. It implied that if scientists
could discover all the natural laws there were, understand them perfectly,
and then apply them to things that were currently happening in the universe,
they should then be able to predict precisely what would be happening any
time in the future, anywhere in all creation.
To take a simple example, suppose a cannonball is fired at a given
instant. Suppose a good scientist then takes precise measurements of the
mass of the ball, its muzzle velocity as it leaves the cannon, the air resistance
it encounters in flight, the direction and angle the cannon is tilted, the
windage affecting the ball's trajectory through the air, and the force of
gravity acting to pull it toward the ground. If cause and effect prevail, the
scientist should then be able to predict, in advance, precisely where and
when the ball will strike the earth, and with exactly what force.
Or suppose a scientist has a box a foot square containing twenty Ping-
pong balls, and gives the box a vigorous shake so that the balls go bouncing
from the walls and colliding with each other in an apparently chaotic
scramble. But is it really chaotic? Not so, if cause and effect prevail. By
applying the natural laws describing the movements of material bodies and
the forces acting upon them, the scientist should be able to predict in ad
vance precisely where any given Ping-pong ball would be found ten seconds
after the box was shaken^ provided that its location before the shake was
known and all other factors (including the force and direction of the shake)
were measured exactly. It is granted that accurate measurements in this
case might be fantastically difficult, and that the calculations involved would
be a tall order even for a high-speed computer. But, at least in theory, a
correct and accurate prediction could be made.
The assumption that all things occurred as a result of cause-and-effect
relationships in an orderly, regular universe was very comforting to the
60 The Universe of Classical Physics
early physicists. It was also very useful. Without this basic assumption it is
unlikely that any of the classical laws of physics could have been derived at
all. There was, of course, no way for those early scientists to guess that a
young man named Werner Heisenberg, living centuries later, might suc
cessfully challenge the assumption's validity. For the world of physics those
early workers were investigating, it was a perfectly valid assumption and
it remains just as valid today, within certain practical limits, in our under
standing of things happening in the "normal" universe of our everyday
experience.
A fourth basic assumption was that all the laws of nature apply with
equal validity throughout all regions of the universe, throughout all time.
In other words, it was assumed that the universe was uniform from one part
to another, and from one time to another. Thus, an object was assumed
to have the same mass in some far corner of a distant galaxy as it had
here on earth. It would respond to a given force in exactly the same way,
no matter where in the universe the force might act upon it. Once again,
this was an utterly necessary assumption if any kind of "rules of the road"
were to be outlined at all. And again, this assumption proved perfectly
valid within certain limits, but was challenged at the beginning of the
twentieth century. We know today that things which occur in our "normal"
universe of experience do not necessarily occur under all conditions in all
parts of the universe. Objects or particles accelerated to velocities approach
ing the velocity of light, for example, behave quite differently than the
same objects or particles traveling at low velocities.
Finally, early physicists assumed that time was uniform, flowing steadily
from past to present to future. It was assumed that natural phenomena
always occurred in chronological sequence. Natural laws would be the
same today as they were yesterday, and still the same tomorrow, or next
year, or next century. Just as things were assumed to happen through
cause and effect, the sequence of cause and effect was always assumed to
be forward in time. Of course, there was no logical proof of this assump
tion. It was taken for granted on the basis of experience. It was a very
necessary assumption, if scientists were to have any hope of finding any
orderly patterns in the apparent chaos of natural occurrences. Even today
this assumption is considered valid for almost all practical purposes. Of
course, we might argue that we have not really been observing nature very
long, and have no way of being certain that natural laws do not gradually
change in some way with the passage of great stretches of time. What is
more, certain very peculiar phenomena observed recently in the micro-
cosmic world of elementary particles actually do appear to be taking
place backward in time! Thus modern physicists may ultimately find that
even this assumption is only valid within certain limits.
But however vigorously or successfully these basic assumptions may have
been challenged in modern times, they were unchallenged and unchal-
Assumptions, Observations, and Measurements 61
lengeable in the days when the classical laws of physics were being out
lined. Together they formed a solid foundation on which scientists could
begin, with the use of the scientific method, to observe and describe things
they found happening in the universe around them, and to begin to
formulate, however imperfectly, the first orderly system of natural law.
OBSERVATIONS AND MEASUREMENTS
Armed with these assumptions, the first task of the early physicists was
the observation, description, and measurement of natural phenomena. But
for this work not to be wasted, the things being measured first had to be
defined, useful methods of description found, and units for measurement
agreed upon.
If we look closely at the classical laws of physics we see that all of
them were concerned with matter (the material substance of the universe)
and with various kinds of motion of material objects resulting from the ac
tion of certain forces. Indeed, the whole work of classical physics has been
summed up as the study of matter and motion. But how can we describe
a material object to distinguish it from some other similar but different
object? How can motion of an object be described, or a force acting upon it
be defined?
First of all, a material object to qualify as such must occupy a certain
amount of space; we can observe and measure the space that it occupies.
It has linear dimensions length, width, and height which we can meas
ure and record if we can agree on what units to use. It has other charac
teristics of shape which can be described roundness, for example, or
flatness, or angularity. It has consistency hardness, softness, solidity,
fluidity, firmness, doughiness, etc.
Other physical characteristics can be noted: If we bend it it may change
shape or not; if it does change, it may spring back to its original shape
when we stop bending it, or it may remain bent. If we drop it, it may
shatter into pieces, it may bounce, it may splash, or it may just go whonk
and sit there. Even an "object" which lacks any or all of these physical
characteristics (a certain volume of gas, for example, may have no set
limits of linear dimension, nor any of the other characteristics we have
mentioned) will still have weight and occupy space; we might hesitate to
call it an "object" but it certainly qualifies as "matter" and can be de
scribed in some unique and distinguishable fashion.
Nor need we stop here. Other measurable characteristics can be used
to describe a given object. Its temperature can be measured and recorded.
So can its position with reference to other things around it the particular
point in space that it occupies can be identified. Even two objects that are
virtually identical in every way can be distinguished one from the other by
pinning down their respective locations in space through linear coordinates;
62 The Universe of Classical Physics
two objects cannot occupy exactly the same space at exactly the same time.
(This is true even of a mixture of two gases in a closed container. We may
have trouble telling one from the other and even more trouble separating
them, but no single molecule of gas occupies the same space as any other
molecule.) We can further describe some objects in terms of color, others in
terms of roughness or smoothness, scratchiness or slipperiness.
Finally, we can describe and identify a given object by stating what, if
anything, it is doing at a given time. It will either be at rest, or in some kind
of motion. If in motion, it may be moving in a straight line in a given
direction, or in a circle around a fixed pivot point, or some combination of
the two. It may be moving at a constant velocity, covering the same dis
tance during each unit of time, or with positive acceleration (moving faster
and faster with each unit of time) or with negative acceleration (moving
more and more slowly decelerating with each unit of time). If it is
accelerating positively or negatively it is either doing so regularly (with
a constant increase in acceleration or deceleration for each unit of time)
or irregularly (as if the force causing the motion were varying in strength
from moment to moment).
Certain of these things were soon found to be easy to describe and
measure while others proved suprisingly difficult. Physicists soon dis
covered, for example, that while it was relatively easy to describe regularly
accelerating motion, describing irregular or varying acceleration merely
caused confusion and did not contribute any more knowledge. It was easier
to describe an object moving with a constant velocity than with a velocity
which kept changing all the time. Angular motion in a circle was more
convenient to study than in a stretched-out ellipse. Furthermore, the
simpler forms of motion seemed to be the more basic ones that occurred
in nature anyway, while more complex motion could almost always be
broken down into combinations of two or more simpler forms. For ex
ample, the most commonplace form of accelerated motion occurring in
nature was the downward acceleration of an object under the influence of
gravity. This was soon found to be a perfectly regular acceleration if the
object was allowed to fall freely. Thus scientists sought out "idealized
conditions" in which to study the behavior of objects at rest or in motion,
and would always try to set up their experiments so as to rule out all
useless and irritating irregularities right from the very beginning.
The early physicists also soon realized that certain characteristics of
matter and motion were very slippery to describe and thus could cause all
sorts of confusion, while others were far less complicated to use. The weight
of an object, for example, could vary a great deal depending on how far
from the center of the earth it was when its weight was measured. A steel
ball would weight a tiny bit less atop a 10,000-foot mountain than at sea
level, and would weigh less by far if weighed on the surface of the moon,
since its weight would be directly related to the gravitational force acting on
Assumptions, Observations, and Measurements 63
it. To use its weight to describe it accurately, you therefore always had to
specify where it was weighed which was a bore. Far better to measure a
characteristic which did not change from place to place for example, the
amount or quantity of matter in an object, the characteristic which came to
be called the object's mass. While an object's weight might vary from place
to place (thanks to varying gravitational forces acting on it) the quantity
of matter that it contained its mass would be the same wherever and
whenever it was measured.* Thus, early scientists preferred to describe an
object according to its mass rather than its weight, just because it was
simpler and less confusing.
Once we have settled on some reasonably reliable characteristics to use in
describing an object and how it is behaving, we must next agree on some
units of measurement so that someone else measuring the same character
istics independently will be able to understand what exactly we have
measured and how. Suppose scientist A measured the mass of an iron ball
in kilograms, while scientist B chose to measure it in cocos (one coco being
equivalent to the mass of an averaged-sized coconut). One says the ball
has a mass of 10 kilograms, while the other insists that its mass is 19.6
cocos. Obviously the two are going to have trouble communicating their
findings or even comparing them. If they both knew that they were meas
uring the mass of the same iron ball, they might well deduce that 1 kilogram
is equivalent to 1.96 cocos but the two scientists, being equally lazy,
might wrangle forever about who was going to have to do the work of con
verting the other's units. Don't laugh the world of science has had pre
cisely this sort of problem on its hands since time immemorial, and is still
struggling with it today. In the United States weights are measured in
pounds; in England the unit is the stone; in France, the kilogram. In
measuring linear distances you can take your choice among inches, feet,
miles, rods, furlongs, centimeters, angstroms, astronomical units, or light-
years. That is just in the United States alone; other countries offer further
possibilities.
Fortunately, most scientists got tired of having to convert continuously
from one unit to another, and settled among themselves upon the metric
system commonly used in Europe and much of the rest of the world for all
kinds of measurements. In this system linear distances are measured in
meters, centimeters, millimeters, or kilometers; weights and masses are
measured in grams and kilograms; areas are measured in square centi-
* Or so it seemed. Today, of course, physicists know that the mass of an object
can also vary quite noticeably, if the object is moving at a high enough velocity.
But nobody knew that in the days when the basic laws of motion were being studied,
and even today no tangible object in our "normal" universe ever moves fast enough
for us to worry about measurable change in its mass. For all intents and purposes,
in the universe of human experience, we are still quite safe in saying that the mass
of an object is constant wherever it may be.
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Assumptions, Observations, and Measurements 65
meters, square meters, etc.; and fluid volumes in cubic centimeters, etc.
Table 1 sets forth the units, metric and otherwise, which are most com
monly used in the world of science today, together with their equivalents.
Throughout this book we will proceed to use these units even though
some of them may be quite unfamiliar (see Table 1 ) .
In this chapter we have discussed a number of factors which entered
into the first scientific exploration of the laws of physics, both from the
viewpoint of the early scientists and from the retrospective view of present-
day knowledge. Of course those early physicists did not have retrospect
to help them. But in spite of this, and in spite of all the flaws in their
knowledge and impediments to their work that we recognize today, some
eight hundred years ago a real and fundamental exploration of nature
began. It proved to be incredibly fruitful. Armed with a few "self-evident"
assumptions, a few ideas of how objects and forces might be described, and
a few basic units for measurement, a handful of scientific pioneers with the
first glimmerings of a scientific method of study to work with began to
observe and measure certain physical characteristics of matter, and to
study the manner in which certain forces seemed to affect and alter the
ordinary motion of material objects.
Their "self-evident" assumptions, as it happened, were mostly wrong.
They were not entirely sure how to describe what they were observing, nor
even exactly what they were trying to describe. Their measurements were
sorely limited by the built-in boundaries of five human senses. Yet in a
few hundred years they built up an amazing groundwork of valid observa
tion, and developed a truly awesome structure of natural law based on that
observation. And in addition, all unknowing, they laid the foundation for a
vast scientific revolution which is continuing on to this very day.
CHAPTER 6
The Riddle of Falling Objects
Matter and motion were the first order of the day. Galileo sought to ex
plain the behavior of material objects in motion. So did Copernicus. So
did Newton and a dozen others. To describe the enduring and predictable
characteristics of matter in the universe, and to describe the various forces
that might act on material objects and what happened when they did
that was the goal. The earliest scientific explorations of physics were aimed
at finding universally valid descriptions of what would always happen to
any kind of material object, anywhere in the universe, when it was acted
upon by one or another kind of force,
We know today that there are a variety of forces that can act upon
material objects. Sometimes the result of such action is a change in the
physical state of the object in question: A cube of solid ice under pressure
turns into liquid water. Sometimes a change in the shape of the object can
occur: A rubber band stretches; an impacting bullet flattens. Sometimes
the object itself may be unchanged, but a change occurs in the direction
or manner in which it is moving (or not moving, as the case may be).
Many of the forces we are speaking of are part of our everyday observa
tion. The action of human muscles to move an object from one place to
another, perhaps with the aid of a lever or a pulley, is one such force.
Friction is another force that influences the behavior of objects. Centrif
ugal and centripetal forces affect objects moving with angular speed, that
is, rotating on an axis or Mowing a curving path of motion. Of all the
forces affecting all material objects in the universe, perhaps the most com
monly recognized and still the most mysterious is the force of gravity.
We could pick no better place than this to begin our exploration of the
nature of matter and motion.
Gravity must certainly have been among the first natural forces that
mankind became aware of at the dawn of history. Surely it was the first
that man tried to explain, define, and put to practical use. Throughout the
centuries gravity has remained the most constant and omnipresent of all
forces men have had to grapple with in their everyday world. Gradually its
characteristics were explored and defined and it was indeed put to practical
66
The Riddle of Falling Objects 67
use; yet even today physicists are as much at a loss as ever to explain
what exactly it is or why it exists in the first place.
We can readily see why gravitational force would have been the most
persistently observed and studied of all natural forces in human experience.
Even Ug the caveman (Chapter 2) knew that if he let go of an object
held in his hand, it would fall to the ground, and that it would hit his toe
a good solid whack unless he got his toe out of the way. He soon learned
that it didn't matter what the object was or how much it weighed (if he
disregarded leaves sailing upward in the breeze); any object that he let
go of would immediately begin moving, and always in the same direction:
down. Indeed, Ug the caveman came to learn that any object that was not
already on the ground and was not supported off the ground in some way
would fall down in the same general fashion. He might even have been
clever enough to notice that it was the wind a gust of moving air which
he could feel that pushed the leaf upward. On a still day even the leaf
would fall down.
Thousands of years probably passed before anyone seriously tried to
measure how fast a released object fell under the influence of this mysterious
force that pulled unsupported objects downward. But even Ug the caveman
must have been puzzled by this force and wondered about it, if he was at
all perceptive. He knew that he could force objects to move in one direction
or another by pushing them, striking them, pulling them, or throwing
them. He could usually even predict roughly how far a given object would
move in what direction when he did the pushing. But this strange, invisible
force that acted on objects that nobody had pushed was something else
again. It was as if all the objects on earth were somehow attached to the
ground by invisible cords constantly straining to pull them down. Ug found
that he could "defy" this mysterious force, at least temporarily, by hurling
a rock, for example, straight up into the air. But he also found that that
rock would then inevitably rise more and more slowly, come to a stop in
the air, and then turn around and come back down again with just as much
force as he had thrown it upward. In the long run, gravity always won!
We can also see how early men might have made some perfectly com
mon-sense assumptions about this invisible force "obvious" assumptions
that were considered self-evident, things which "everybody knew" were
true. Efforts to explain why objects fell to the ground didn't get very far,
but even to Ug it was obvious that objects must naturally belong on the
ground; otherwise why would they keep falling down there? Millenniums
later even the great Aristotle couldn't improve much on this "self-evident
fact" that things fell to the ground because that was their "natural place."
Similarly, common sense said that heavy objects fell to the ground faster
than lighter ones. After all, they were clearly heavier; therefore they had
to fall faster. What was more, since no one had ever produced a vacuum in
those days, casual observation actually "proved" that light objects fell more
68 The Universe of Classical Physics
slowly than heavy ones: A leaf floated gently down, while a rock fell thud!
And it stood to reason that the harder a rock was thrown in a horizontal
direction, the longer time it would stay in the air before falling to the
ground.
Nobody really knows if Galileo ever personally stood atop the tower of
Pisa and dropped a ball of iron and a ball of wood simultaneously, as the
legend goes, to see which struck the ground first. There are spoil-sports in
modern physics who claim that he probably never did. But it seems certain
that various early ordnance experts and artillery men had already begun to
Strong
Bowman
Fig. 3 Fast arrow travels twice as far as slow arrow in the same time, but both
arrows, shot simultaneously, will strike ground at the same instant.
poke holes in some of the "self-evident truths" about the force of gravity
and its action on moving objects, long before Galileo ever appeared on the
scene. Surely some early bowmen or catapultists must have realized that an
arrow shot from a bow, or a stone from a catapult, began to be pulled
earthward the very instant it started on its path through the air, and that
two such objects thrown horizontally at the same instant would ultimately
come to earth at precisely the same time, no matter how hard one might
have been thrown compared to the other. The object thrown harder would
travel farther horizontally before it hit ground than the one thrown more
gently, but it would remain in the air no longer (see Figure 3).
Those early artillerymen must also have discovered that the only way to
The Riddle of Falling Objects 69
extend the ultimate range of a projectile, such as an arrow fired from a
bow, was to tilt the arrow up away from the earth before it was released.
Even then, there was a point of maximum gain: the range of the arrow
could be extended by tilting its trajectory up to an angle of about 45 de
grees; tilt it any higher and it began to lose range rather than gain it.
Essentially, however, they must have come to realize that any object
thrown, hurled, or tossed at or near the surface of the earth ultimately
became a jailing object, sooner or later, thanks to the inexorable pull of
gravity.
Whether Galileo did or didn't stand atop the tower of Pisa is unim
portant. What was important was that Galileo was the first to demonstrate
two key facts about the behavior of falling objects: First, that except for
the influence of air resistance, the mass of an object had nothing to do with
the rate of speed at which it fell; in other words, that the downward ac
celeration of gravity was precisely the same for heavy bodies as it was
for light bodies; and second, that the gravitational acceleration of falling
bodies was always uniform and constant at any given place that it was
measured.
As I. Bernard Cohen once pointed out: "In studying the science of the
past, students very easily make the mistake of thinking that people who
lived in earlier times were rather more stupid than they are now." It is
easy, for instance, for us to consider Aristotle "stupid" for declaring that
objects fell down because their "natural place" was on the ground, or to
regard his contemporaries as "stupid" for accepting such an "explanation"
of the force of gravity. Likewise, we may think Aristotle rather dull for
teaching that objects of different weights would fall at different speeds
without making any effort to prove such a contention. But in all fairness,
we must remember that Aristotle inherited a philosophical point of view
of the universe and a philosophical attitude toward exploring it, not a
scientific attitude. It was not because of stupidity that Aristotle scorned
experimentation; it was simply that no one had yet begun to realize how
crucial experimentation might be to the support and proof of physical
theories. Indeed, we might well have expected him to champion human
reason and logic as far superior to the brute labor of experiment as a
method of investigating nature. We might also have expected him, as a
philosopher, to be far more concerned with why things happened than with
trying to observe exactly what happened or how.
Doubtless there were many people before Galileo who had private
reservations about certain of Aristotle's pronouncements. But Galileo in
the seventeenth century was Aristotle's challenger. Galileo was an experi
menter. From the results of his own and others' experiments, he set out
to describe the what and how of gravity. But in trying to observe how fast
objects fell and with what sort of acceleration, Galileo faced two problems
that made any experimentation difficult. First, there was the problem of
- o The Universe of Classical Physics
air resistance, which he recognized clearly as a factor that had confused
the earlier Greeks. The second problem merely compounded the first:
Objects falling freely in air fell so fast that it was all but impossible to make
accurate measurements of how they fell over short distances; and when
the distance of the fall was increased so as to provide longer time periods
for measurement, the effect of air resistance became more and more ex
aggerated.
What about air resistance? We can see for ourselves how confusing this
must have been to early observers of the forces of gravity simply by hold
ing a glass marble and a Ping-pong ball side by side at the same height
and then releasing them at the same time. Although the fall to the floor may
be only four feet, there is no question but that the marble strikes the floor
sooner than the Ping-pong ball. We can repeat this experiment innumer
able times and the same thing will happen every time. Furthermore, if we
stand on a table and increase the distance of the fall from four to eight
feet, we will find the difference in the times the two objects reach the floor
to be even more exaggerated.
From this simple experiment we could draw either of two different
conclusions, one correct, the other incorrect. We might conclude that
Aristotle was right: The heavier of the two objects (the marble) fell faster
and therefore struck the floor sooner. We could argue that we had ex
perimented and measured carefully and observed this behavior with our
own eyes. On the other hand, we might conclude that both objects would
have fallen at the same speed except that something or other impeded the
fall of one object and not the other.
On the face of it, the first conclusion seems to make more sense even
though it is the wrong one. Indeed, it might seem to be the inevitable con
clusion, if we did not know that the space between the point of release
and the floor was not empty, but was actually filled with a very real
substance air which could conceivably impede the downward fall of
light objects. But suppose we didn't realize this, and accepted the first
conclusion. Where would it lead us? It would lead us straight into trouble
the moment we looked farther than the special circumstances of our
original experiment.
Suppose, for example, that we repeated the same experiment with a
strong draft blowing horizontally through the room. Would we observe the
same results? Of course not; we would then see the Ping-pong ball fall to
the floor at an angle instead of straight down. It might reach the floor
halfway across the room if the breeze was strong enough! Again, suppose
we used identical-sized marbles, but with one made of glass and one of
lead. Now, oddly enough, we might not detect any difference in time of fall,
either from four feet or from eight, even though lead is just as much
heavier than glass as the glass marble was heavier than the Ping-pong
ball. Well, could it be the fact that the Ping-pong ball was hollow that
The Riddle of Falling Objects 71
made the difference? Perhaps we substitute a tennis ball, which is clearly
heavier than a marble, but hollow like a Ping-pong ball, and try again.
And again find that the marble and the tennis ball strike the ground at
virtually the same time.
Thus, if we had accepted the first (and seemingly inevitable) conclusion
that heavier objects fall faster than lighter ones further experiments
would soon force us to conclude that gravity made no consistent sense
whatever. Those experiments would seem to show that sometimes heavier
objects fall to the ground faster than lighter objects, but that sometimes
they don't in other words, that gravity is a completely quixotic force
that behaves one way one time and another way another. We would even
have to conclude that it sometimes pulled objects straight down and some
times pulled them sideways at an angle. If that first conclusion were correct,
it would have to mean that gravity was a totally whimsical and inconsistent
force, subject to physical rules that kept changing with the conditions.
One of Galileo's greatest contributions to science was that he could
not and would not believe that the laws of nature were whimsical, incon
sistent, and ever-changing. Nature, he insisted, was consistent if nothing
else. The same laws that governed the behavior of one free-falling object
had to govern the behavior of any free-falling object anywhere in the uni
verse. The laws of nature, he was convinced, were simple, logical, and
reasonable; when something was observed to occur that made a natural
law seem whimsical or inconsistent, it was only as a result of some un
suspected factor which the observer was missing, and thus was failing to
take into account.
Galileo, for example, realized that something other than a consistent,
unchanging, downward gravitational force was influencing the behavior
of various falling objects but what was the missing factor? Obviously,
objects falling through the air met with a certain amount of resistance
just from the air itself; they literally had to push aside the air through
which they were falling. Heavy, compact objects managed this with
little difficulty, and thus showed little effect from air resistance; but light,
feathery objects had more trouble shouldering their way through the air
and were noticeably retarded in falling.
So air resistance had to be ruled out somehow if the effect of gravity,
free of any other influence, was to be tested and measured. But how could
this be done? Galileo realized that the only way to really test and measure
the behavior of falling objects would be to measure their fall in some way
in which air resistance could have no effect whatever, because they were
falling not through air, but through a perfect vacuum. Unfortunately, though
Galileo could imagine a perfect vacuum (i.e., a container in which all air
had been pumped out, so that it contained no air at all), he had no way
actually to create such unheard-of conditions.
So what could Galileo, or any scientist, have done? It would have been
7 2 The Universe of Classical Physics
easy to quit and go home. Instead, Galileo found an ingenious way to dodge
the problem and come up with the correct answer in spite of it. The device
he used, now a time-honored method in the study of all sorts of physical
phenomena, was the device we know today as the "thought experiment."
Galileo couldn't create a vacuum in which objects of various weights
could be observed falling, but he could imagine such conditions. By doing
so, and then by imagining how objects would fall under such "ideal"
(though unachievable) conditions, Galileo came up with several shrewd
guesses:
First, he guessed that all falling bodies would fall in the same way, and if
released together, would fall together and reach the ground together, re
gardless of their respective weights.
Second, he guessed that these objects would fall with constant accelera
tion that is, that they would increase in downward speed steadily, with
the same increase in speed each successive second until they struck the
ground.
Now admittedly these were guesses about what would happen under
certain ideal circumstances. But armed with these guesses, Galileo set about
diligently testing them in real experiments in which he made allowances
for the complications that would have to arise because of air resistance. He
found that the results of these experiments, once the necessary allowances
were made, coincided exactly with the results he had imagined in his
thought-experiment.
In other experiments he tried in other ways to minimize the effect of
air resistance by using large, dense objects, for example. He tried to im
prove the accuracy of his time measurements by using counterweights to
slow down the velocity with which objects fell. Again and again, his ob
servations coincided with the results he expected from his thought experi
ment.
Now this all sounds very neat. But was this real experimental proof that
Galileo's guesses were right? Of course not. Galileo didn't have actual
experimental proof. Suggestive supporting evidence, yes. Proof, no. And
precisely because he didn't have actual proof at a time of growing skepti
cism toward taking things for granted, other scientists set about to obtain
actual experimental proof. Galileo had no vacuum with which to test out
the "ideal case" of his thought experiment. He very probably doubted that
such a vacuum could ever be achieved. But some years later another
Italian scientist named Torricelli discovered that a vacuum could be
created. Torricelli took a long glass tube sealed at one end and filled with
mercury, and then inverted it open end down into a pan of mercury. He
discovered that the mercury in the glass tube fell to a certain level and then
stabilized, leaving an empty space at the sealed top end of the tube. Torri
celli reasoned that since that "empty" space up there in the end of the
tube had originally been filled with mercury, and since no air could have
The Riddle of Falling Objects 73
leaked in when the tube was inverted, that space really was empty it had
to be a vacuum!
Today, of course, we know that Torricelli's vacuum was not 100 per
cent perfect. That "empty" space actually did contain a few atoms of
vaporized mercury, perhaps a few molecules of oxygen, nitrogen, or
carbon dioxide from air that had been entrapped in the mercury and was
drawn out by the pressure of the vacuum. But Torricelli's vacuum was so
very close to the perfect vacuum needed for Galileo's "ideal case" that
the imperfection was totally negligible and Galileo's guesses were sub
stantiated. Later, air pumps provided more convenient vacuums in tall
glass pipes. The crucial experimental proof of Galileo's conclusions finally
came a century later when Sir Isaac Newton released a bit of goose down
and a gold coin simultaneously at the top of an evacuated glass pipe and
found that even this pair of objects fell side by side, with constant down
ward acceleration, and struck the bottom of the tube at the same time.
We have gone into this seemingly simple and insignificant matter at
such length for two very important reasons. First, it is an excellent example
of how exceedingly difficult it was for those early physicists, groping in the
dark, really to pin down anything so that they could say, "This is true and
we can prove it." Galileo saw the problem, guessed the answer, paved the
wa y yet brilliant men searched another hundred years to find actual ex
perimental proof.
But more important, we can see in this example how answers had to be
torn away from nature by hook or crook, guile, cleverness, and ingenuity.
Many nonscientists find themselves very uneasy with such scientific pro
cedures as Galileo's "thought experiments" and "experimental proofs"
based upon "ideal conditions" that don't actually exist. How can anyone
really prove anything this way? It is all very nice for a physicist to guess
that in an ideal case two objects will fall together in a vacuum, regardless
of their weight, and with constant acceleration. It's even nicer to be able
to prove this by experimental observation later, once you can create a
vacuum in a tube to test it out. But where does this really get us? The fact
is that our world is not a vacuum. It is a world covered with a blanket of
air, and in this real world, objects of different weights (or of different sizes
or shapes) do not fall together. Leaves flutter down in the breeze, while
apples fall thud!
From the nonscientist's point of view, this objection is perfectly valid.
We do not live in a world of ideal conditions, and it is in our real world
that we have to cope with the behavior of things. What, in fact, we come up
with is not one but two sets of rules: the rules that govern the ideal case,
and the rules that govern the real case. In the ideal case, two objects do
fall together. They do increase in speed with constant acceleration so that
the longer they fall the faster they are moving and the harder they hit
bottom. In the real case, however, not only do objects of different weights,
74 The Universe of Classical Physics
sizes, or shapes fall at different speeds because of air resistance, but any
object that falls freely meets with air resistance. The more it accelerates
and the faster it falls, the greater the air resistance it meets and the greater
the resultant "friction lag."
Indeed, if it were possible to drop a marble out of an airplane at 30,000
feet and then to measure its pattern of acceleration, we would find that it
accelerated downward at a constant rate per second only for the first part
of the fall, up where the air is thin. As it fell into denser layers of at
mosphere, its acceleration would increase more and more slowly until at a
certain point it wouldn't increase its acceleration any more at all; it would
just continue to fall at precisely the same velocity, without further accelera
tion, until it struck the ground. Furthermore, in this real world, something
else would happen that Galileo never dreamed of in his "ideal" case: The
falling marble would heat up as it shouldered its way down against air
resistance. Much-faster-moving objects, such as meteorites entering the
earth's atmosphere, actually heat to incandescence and vaporize before
striking the ground. We can calculate that the marble falling from 30,000
feet would not become that hot before it reached ground, but we might
have quite a job actually documenting experimentally what did happen to
it under these "real- world" conditions!
Perhaps we could study these problems more conveniently by dropping
the marble down through a medium that offers even more resistance than
air: measuring its fall down through a long tube of water. This, at least,
would be a manageable experiment. So suppose we release an iron marble
at the top of a thirty-foot tube of water and measure its acceleration as it
falls. What would we see? We would see precisely the same thing happen
as we had guessed would happen to the marble in our thought-experiment.
The marble's downward acceleration would not be constant; it would de
crease second by second until the marble was falling down the water column
with a uniform speed, and it would then maintain that uniform speed until
it reached bottom. Furthermore, if we had delicate enough instruments for
measuring temperature, we would discover that the temperature of the
whole system of marble-falling-through-water would have increased a meas
urable amount. In this case, however, it would not be the ball alone that
increased in temperature, but ball, water, tube, and all!
We know today that each of these curious observations can be ex
plained completely and accurately in terms of our knowledge of modern
physics. We know that air resistance (or water resistance, as the case may
be) are in fact forms of friction one of the natural forces we have yet to
discuss. We know that any falling object at the beginning of its fall pos
sesses a certain form of energy "potential energy" which is steadily
converted into another form of energy "kinetic energy" throughout the
period of its fall. We also know that as the falling object encounters the
opposing resistance of friction during its fall, part of its accumulating
The Riddle of Falling Objects 75
kinetic energy is converted by that opposing frictional force into still an
other form of energy heat. We further know that what happens to the
heat energy depends on the heat-conducting or heat-insulating qualities
of all materials concerned: iron ball, water, tube, etc.
If it seems that we are getting farther and farther away from our dis
cussion of the force of gravity and its characteristics as we examine this
"real" case, indeed we are. We are getting into a whole complex snarl of
considerations that are only peripherally related to gravity and its effects.
And it doesn't matter that all these other physical considerations are be
side the point; we can't escape them in the "real" world. The only possible
way we can extricate gravitational force alone from this mess and examine
it by itself is by imagining "ideal" situations which we know are flatly im
possible, in hopes of isolating the thing we are trying to study from a
wildly confusing array of other things we really don't want to have to deal
with right then.
Of course, such isolation of phenomena in "ideal" cases or "thought
experiments" may not always be practical in terms of the "real" world, but
it is a perfectly legitimate procedure if we know what we are doing, and
above all, it allows us to learn things we couldn't learn otherwise. What is
more, sometimes those "ideal" and "imaginary" conditions are not so im
practical as they seem. Maybe Galileo couldn't imagine a vacuum existing;
but today we are sending real human beings into a part of the universe
where relative vacuum does indeed exist: the space beyond earth's atmos
phere that we have been so busily exploring lately! Try to tell an astronaut
that there are no practical applications of the laws of gravitation under
"ideal" conditions of free fall, and see what he says as he frantically tries
to retrieve a pencil that has slipped from his fingers and is merrily floating
around the inside of his space capsule. Try to tell the first man who breaks
his anchor cable during a "space walk" outside his ship that the laws of
gravity, the laws of motion, and the laws of inertia have no "real" meaning!
You might find him eagerly offering to trade places with you if he and
his teammates weren't quite so busy trying to apply those very laws to
grapple him back aboard ship before he embarks on a long, cold, and
endless space walk to nowhere.
In fact, we will see throughout this book more and more areas in which
the "ideal case" of the scientist and the "real world" of the man in the
street impinge, and we will find more and more reasons why each one of
us, scientist or not, needs a clear understanding of basic laws of physics,
however "impractical" or "useless" they may seem. The time when these
laws can be ignored with impunity is passing fast. Tomorrow this will be a
body of knowledge each one of us will need perhaps even for survival.
Galileo and the men who followed after him clearly recognized the part
that air resistance played in obscuring what the real characteristics of
gravitational force were. They got around this first by setting up imaginary
76 The Universe of Classical Physics
thought experiments involving ideal conditions, then by actually creating
those ideal conditions as best they could, and then making the necessary
corrections in their observations and calculations; and they found that
it worked. They ended up with answers they couldn't have obtained any
other way. Gradually the use of thought experiments involving ideal con
ditions became one of the most widely used and most fruitful of all tech
niques of scientific investigation.
We will use this technique ourselves again and again throughout this
book to make various points. We need not feel the least nervous about it.
If such thought experiments are used with care and integrity they do not
in any way invalidate actual experimental data that may be obtained, nor
the conclusions drawn from them. On the contrary, they may well show the
direction that actual experiment must take to produce any usable results
whatever. Virtually all natural physical phenomena are not simple occur
rences, but are complex combinations of many forces acting at the same
time to produce a given result. The scientist is constantly laboring to try to
separate one such force from another so that he can study each force
separately instead of trying (usually in vain) to analyze everything that
is happening all at once.
Of course, once various laws of nature are known, then it may be pos
sible to examine two or more forces acting in different ways at the same
time, and successfully predict the net result on the basis of what is already
known about each of these forces. This is particularly true today when
modern computers can handle the mathematical leg-work so simply and
quickly. But in searching out a natural law that is not yet known or under
stood the scientist always seeks to study just one factor or force at a time.
Thus even though Galileo recognized that air resistance was a force acting
to modify the behavior of objects falling under the force of gravity, he had
to sidestep this "extraneous" force temporarily in order to make sense of
the phenomenon he was trying to examine.
The other major problem in studying gravitational force was the simple
fact that objects fell so fast that it was all but impossible to measure time
intervals accurately, or actually to observe or measure the state of things
at various points in the course of an object's fall. Galileo and later
physicists got around this problem by dreaming up a number of ingenious
devices for slowing down the movement of objects acting under the force
of gravity by "diluting" gravitational force, so to speak. One way of doing
this was to study the movement of objects sliding or rolling down a ramp,
rather than trying to measure their behavior in free fall. Galileo had al
ready guessed that Aristotle was wrong, and that any two objects released
simultaneously would fall together (under ideal conditions) at the same
velocity and with the same acceleration, regardless of their comparative
weights. He further guessed that the acceleration of falling objects (that
The Riddle of Falling Objects 77
is, the increase in their downward speed per unit of time) would remain
constant second after second, or minute after minute for as long as they
continued falling. Galileo found times and speeds hard to measure when
objects were allowed to fall freely; but by rolling polished metal balls down
a parchment-coated trough held at an angle of 20 degrees from the hori
zontal, he found that the time required in the downward movement of the
ball from the top of the trough to the ground was greatly lengthened (and
thus more easily measured) because the ball also had to roll horizontally
down the length of the trough. Indeed, if the trough were held at a very
slight incline to the ground, and if it were long enough, the steel ball might
take as much as 15 or 20 seconds to move the vertical distance of a single
inch from the top of the trough to the ground (see Fig. 4).
Fig. 4 Use of an inclined trough to "dilute gravity" by slowing down the
motion of a ball falling under the force of gravity. The shallower the incline,
the more the ball's rate of motion is slowed down.
But were the natural laws governing the movement of a ball rolling
down an inclined plane the same as those ruling its behavior falling straight
down? Galileo didn't know; but he guessed correctly that exactly the
same rules would apply. He then proceeded with experiments, rolling many
different-sized balls of widely differing weights down identical troughs
parallel to each other, and found ( 1 ) that they always rolled side by side,
regardless of their weights; and (2) that they always accelerated at a con
stant rate and reached the ground at the same time, with the same final
velocity. By raising the ends of the trough higher and higher, he found that
78 The Universe of Classical Physics
the ratio of time required per unit of distance traveled always remained
the same: The same rules seemed to apply no matter to what angle the
troughs were tilted. Tilt the troughs almost vertical and the balls would
roll faster and reach the ground sooner but they would still roll together
and their final velocities would still be in the same proportion to the vertical
distance they had traveled.
Obviously, it didn't matter a whit whether the motion being studied was
that of an object falling freely, or rolling down an inclined plane: The rules
were the same, and the results were the same. In fact, the tilted troughs
were nothing more than a device to "dilute" or slow down the effect of
gravity on a falling object, much as a motion picture can be slowed down
by varying the speed the machine is running. Nothing basic is changed; the
man seems to leap over the barrel much more slowly than before, but the
distance and pattern of his leap, from beginning to end, remain in exact
proportion to the time that the leap seems to take!
Other means were eventually devised to dilute the effect of gravity. The
movement of a pendulum, for example, was one way of studying the be
havior of a falling body in "slower motion" than if it were falling freely
to the ground. Again, an ingenious device called "the Atwood machine"
was invented essentially a cord going over a pulley with weights on either
end, so that the weight on one end counterbalanced the weight on the other.
When a weight was released from a platform at the top of a marked scale
on the Atwood machine and allowed to fall, its fall was not "free," but was
impeded by the counterforce of a slightly smaller weight that had to be
drawn up the other side of the scale. Even so, the same gravitational rules
seemed to apply (in slowed-down proportion) to weights on the Atwood
machine as to objects in free fall,
GRAVITATIONAL PULL VS. THE "OBJECTNESS" OF THINGS
Thus, by finding ways of diluting, or modifying the effect of, gravitational
force on freely falling objects, early observers began to learn a great deal
about the effects of gravity. As a fringe benefit, they also discovered that
the effects of gravity on objects were essentially the same whether the object
was suddenly released to fall freely, straight down to the ground, or
whether the fall was impeded by counterweights, inclined planes, or what
not.
But these early observations of gravity revealed something else of very
great importance: that there was a fundamental difference between the
measurable downward gravitational pull on a given object what we call
its weight; and the basic, irreducible amount of matter that the object con
tained its mass. We raised this question of the weight versus the mass of
an object briefly back in Chapter 5 when we spoke of the trouble that arose
in trying to use the weight of an object as a reliable identifying character-
The Riddle of Falling Objects 79
istic. Now we can see more clearly what the problem was when we con
sider some actual observations recorded by those early scientists.
They discovered, as we have seen, that the gravitational pull on a given
object could vary, depending upon where measurements were being taken.
For example, an object falls to the ground a tiny bit more slowly at the
top of a high mountain than if its fall is measured at sea level. The same
object even falls more slowly at sea level somewhere on the equator than
it would at sea level in northerly latitudes. Now, this observation was quite
unexpected and perplexing. Scientists had always assumed that a given ob
ject would have the same weight (i.e., be pulled downward by gravity by
the same amount) no matter where the object was located but now simple
observation proved this wrong! An object suspended, say, from the same
spring balance in two different places might register two different weights.
Simple observation also showed that another characteristic of the object,
similar to its weight but not quite the same, remained unchanged wherever
the object might be. If the object were placed on a rough horizontal surface,
exactly the same amount of force would be required to drag it horizontally
across the board for a given distance, no matter where the experiment might
be done. In short, early physicists discovered that an object's weight, always
assumed to be a constant, was in fact variable, while another quality of the
object related to its weight (but not the same) did remain constant and did
not change from place to place.
Obviously this second quality would be more useful in describing the
object than its weight but what was it? Early physicists were hard put to
say. An object's weight was nothing more than a measurement of the
amount of downward tug exerted upon it by the invisible but apparently
omnipresent force of gravity. Why this amount of downward tug should
vary from place to place for the same object was a mystery, but vary it
did. The quality of the object which did not seem to change, even when
its weight did, was its resistance to being dragged horizontally and that
quality could be counted upon to remain constant anywhere certainly a
more useful quality to measure than one that kept shifting around from
place to place!
That constant quality of a material object, its resistance to being dragged
horizontally which was so similar to its weight, yet not quite the same, had
no name. Even today it is a difficult concept for most of us (including
fledgling physicists!) to grasp. Actually, we might call that quality anything
we choose. We could speak of it, for instance, as an object's "basic object-
ness" in other words, the basic amount of the object that was there. We
could demonstrate that the "objectness" of a lump of iron the actual
amount of iron that was there in the lump would remain the same any
where we might measure it, be it at sea level, on a mountain top, or on the
surface of the moon, even though the weight of that lump of iron might be
10 per cent less on a mountain top than at sea level, and 85 per cent less on
80 The Universe of Classical Physics
the moon! As long as none of the iron was allowed to rust away, or be
dissolved by acids, or something of that sort, the "objectness" of the lump
would always remain constant any place.
Early physicists grappling with this concept chose a different term than
"objectness" to describe it. They spoke of this constant, unchanging quality
of an object as its "mass" and defined mass as "the amount of matter
present in an object." That amount of matter might weigh more or less in
different places; where gravitational force was strong, it would weigh
more, and where gravitational force was weak it would weigh less but
the amount of matter would remain the same. Later they even speculated
that the mass of one object could be compared to the mass of another
object made of the same material (at least theoretically) by counting the
number of atoms in the two objects and comparing the results. Thus, they
speculated, two lumps of pure iron with exactly the same number of iron
atoms in them ought to have exactly equivalent mass.
Today the modern physicist would agree that this speculation would
probably be absolutely true and universally applicable anywhere in the
universe, so long as both lumps of iron happened to be at a temperature
of absolute zero, and were completely at rest. We now know that the mass
of an object does vary to an infinitesimal degree with changes in
temperature, although we would have quite a job measuring the change.
We also know that an object's mass increases, if only infinitesimally, the
moment it is set in motion. Thus if one of the lumps of iron is moving, even
very slowly, while the other remains at rest, the mass of the moving lump
will be slightly greater, even though the two lumps contain exactly the same
number of iron atoms. And if one of the objects is moving very fast in
deed close to the speed of light its mass will increase enormously,
although it still has the same number of atoms as the object at rest.
As we will see later, these become exceedingly important considerations
under certain circumstances, but from our standpoint here, we can ignore
them as quibbles. In our everyday world, the universe the early physicists
were exploring, we can regard the "objectness" of an object its mass as
opposed to its weight as a constant unchanging measurement of the
amount of matter in that object wherever it may be on this planet or any
other. But we recognize that the tug of gravity upon the mass of an object
(i.e., the object's weight) may be quite different at different places on earth,
and still more different on other planets or in orbit in space, where gravita
tional forces may be stronger or weaker than on earth. Modern space
science has fortunately helped make this concept clear to most nonsci-
entists. We can readily grasp that there is obviously just as much of an
astronaut present when he is walking about on the surface of the moon as
there was when he was entering his space capsule at Cape Kennedy. Yet
the moon's gravitational pull upon him is only one-sixth that of the earth's,
so that his 200-pound weight on earth becomes a mere 34 pounds.
The Riddle of Falling Objects 81
Thus, Galileo and his contemporaries made many observations of the
force of gravity acting upon falling objects, and evolved the very useful
concept of an object's mass as distinct from its weight. From his observa
tions, Galileo came up with a number of general statements describing how
gravity affected objects, whether they were falling freely, swinging like a
pendulum from a pivot point, or rolling down an inclined plane. These
"general laws of falling objects" were a far cry from the great law of uni
versal gravitation that Isaac Newton would later derive but at least they
were a respectable start. What these laws of Galileo really were, in effect,
were the laws describing local gravitational effects on objects allowed to
fall under special circumstances: objects, for example, suspended close to
the earth's surface and then allowed to fall.
We will see later how Newton took these "local" laws and extended them
to the behavior of all objects anywhere in the universe. But first, we must
consider certain other forces affecting the motion of objects that drew the
attention of Galileo and the other early physicists. Here even more per
plexing riddles were encountered in particular, the riddles of friction and
of inertia.
CHAPTER 7
The Riddles of Friction and Inertia
It was a great step forward when early physicists finally pinned down a
few general rules that reliably described the behavior of material objects
in certain familiar patterns of motion. These rules could be used, for ex
ample, to predict how a solid object such as a rock would move if it were
allowed to fall freely from the hand or to roll down an inclined plane
(assuming, of course, that these things were occurring somewhere near the
surface of the earth). With some minor modifications these same rules
helped to describe other, related forms of motion the movement of a
swinging pendulum, for instance, or the behavior of a rock tossed into
the air. When they were applied at various geographical locations on the
earth's surface (on a mountain top, at sea level, etc.) these rules of motion
furthermore demonstrated why a distinction had to be made between an
object's weight (a measure of the gravitational force acting on it at a given
moment, which could vary greatly depending on where it was weighed) and
the object's mass (a measure of the quantity of matter it contained, always
the same under ordinary circumstances of human experience).
We must remember that these rules made no attempt to explain -what
gravitational force was, nor why it existed. They merely described how
gravitational force had been observed to act. They showed gravity's effect
on the motion of material objects at or near the earth's surface, and some
findings were a little surprising. For example, it was found that the earth's
gravity exerts a continuous force upon objects whether they are in motion
or not. An iron ball held in your outstretched hand will not fall to the
ground as long as your hand continues to support it. It remains "at rest"
yet gravity is constantly acting upon it just the same, tugging it ground-
ward, so to speak. You can even feel the downward pull. But what, then,
keeps the ball from falling? Obviously, your hand must actually be pushing
the ball upward, against gravity, with a forde exactly equal to the gravita
tional force pulling it downward. Similarly, when a man stands "at rest" on
the ground, the surface of the earth must be pushing upward against his
feet with a force equal to the gravitational force seeking to pull him down
82
The Riddles of Friction and Inertia 83
toward the center of the earth.* Of course, in this context the net result
in either case is a state of equilibrium neither ball nor man is set into
motion by either force acting on it and this whole discussion may seem
rather pointless. But as we will very soon see, this notion of force and
counterforce always existing together was later recognized as an extremely
important concept in physics, and we will be encountering it repeatedly as
we go along.
Gravity, however, was not the only force observed by early physicists
to act upon objects and influence their behavior, nor was the falling motion
of an object as a result of gravity the only form of motion that was studied.
Indeed, a number of forces quite unrelated to gravity were observed (or
at least postulated) and investigated, even as early as the Greeks. Among
these were two very puzzling kinds of force which seemed to exist side by
side quite universally, apparently closely related to each other, but acting
upon objects in directly opposite ways. One of these forces was a resistant
force that was observed to make moving objects slow down or to ob
struct their movement. The other seemed to be a propelling force that kept
moving objects moving in their paths in direct opposition to the resistant
force. Between the two, the behavior of these opposing forces seemed ex
tremely confusing and contradictory to early investigators.
Today we know that only one of these two puzzling "forces" actually
exists; the resistant force, which we know as "friction." But we can easily
see why early physicists were trapped into assuming that a "propelling
force" also had to exist to keep a moving object moving. Consider the
problem as they saw it: A light object, such as a feather or a light piece
of wood, falls to the ground much more slowly when released than a dense
object such as a lead ball. Why? Even the Greeks recognized that some
resistant force in the air, some "air resistance," impeded the fall of lighter
objects, while more dense objects were affected by it less, if at all. They
could also see that another force acted to pull the feather or piece of wood
down toward the ground in spite of the air resistance: the force we know
as gravity. As long as the gravitational force pulling the object downward
* By the physicist's logic, the weight of an object (as distinguished from its mass)
is actually defined as the precise "equal-but-opposite" upward force necessary per
fectly to counterbalance the downward gravitational force acting on the object, thus
preventing it from accelerating downward. To confuse things further, the physicist
will argue that neither the ball nor the man under discussion is really "at rest" at
all, but in each case is "in motion" as a result of the two external forces acting
upon them the downward gravitational force and the upward force acting in the
opposite direction. However, since these forces are exactly equal in magnitude and
exactly opposite in direction, the physicist would speak of the resultant "motion" of
the ball or the man as "static motion." Unfortunately, such technical distinctions in
terminology (i.e., "motionless motion") sound like pure double talk to most laymen;
in this book we will settle for more comprehensible (if less sophisticated) terms.
84 The Universe of Classical Physics
was greater than the air resistance impeding its fall, the object would
eventually reach the ground.
But suppose the same light piece of wood were thrown horizontally
through the air. Here again air resistance would impede its flight but now
no force of gravity is acting to pull it or push it along in a horizontal
direction. What then keeps it moving long after it has left the hand? What
keeps an arrow moving against air resistance long after it leaves the bow?
To the Greeks it seemed self-evident that some sort of propelling force
had to be pushing or pulling an object through the air as long as it con
tinued moving. They even devised some colorful (if spurious) explanations
of how this "propelling force" might be generated. They speculated, for
example, that the air that was pushed out of the way by the tip of a flying
arrow flowed back along the shaft and filled a vacuum that the arrow left
behind it, thus shoving the arrow onward from behind. We know now
that such a thing would be tantamount to lifting ourselves by our own
bootstraps quite impossible but it seemed a plausible idea to the Greeks.
And it seemed even more plausible if one compared the "air resistance"
impeding an arrow in its flight with the "ground resistance" that was en
countered when an object was pushed along the ground. If a sculptor
wanted to move a block of marble from one side of a hall to the other he
would have to push it or drag it forcibly every inch of the way; the
moment he stopped pushing, the block would stop moving. So why shouldn't
an arrow flying through the air require a continuing force to keep it moving
against "air resistance" just as much as the marble required a continuing
force to keep it moving against "ground resistance"?
The Greeks made many casual observations of the way such resistant
forces acted to slow down moving objects, whether they were moving in air,
on land, or in the water. They also speculated about the nature of the
"propelling forces" they thought necessary to keep such objects moving.
It remained for Galileo and later physicists of the seventeenth and eight
eenth centuries to define just what these "resistant forces" were and how
they behaved and to show that the mysterious "propelling force" of the
Greeks simply didn't exist at all.
THE FORCES OF RESISTANCE
Today we know that the "resistant force" that slows a feather's fall or
makes a chunk of marble difficult to push across the floor is a familiar
and universal force that arises any time that an object moves in physical
contact with another object. We call this resistance frictional force, or
more simply, just friction. The "air resistance" which impedes the fall of a
feather is nothing more than the frictional force created when the air
molecules surrounding the feather literally rub against it as it falls
the force of gravity has to pull the feather down through the air. A much
The Riddles of Friction and Inertia 85
greater frictional force has to be overcome when the sculptor (by sheer
muscle power! ) pushes his heavy block of marble across the rough stone
floor. In fact, an obstructing force is actually created the moment he even
attempts to slide the block, even if he can't make it budge; that force
reaches its maximum at the point that the block begins to move. But the
block would not even start to move unless the sculptor could apply enough
force to exceed the maximum frictional force obstructing its movement, at
least by a little bit!
This obstructing force, called friction, tends to impede the motion of any
object rubbing against another. But the amount of friction arising in a
given case depends upon many different factors. The rougher the surfaces
of the two objects that are in contact, for example, the greater the friction
when one of the objects tries to slide past the other. Similarly, the more
tightly the two surfaces are pressed together, the greater the frictional
forces created when one moves relative to the other whether the surfaces
are pressed together by gravitational force (as when a brick is "pressed"
against the sidewalk by gravity) or by some other force (such as a clamp
pressing two pieces of wood together, for instance). Oddly enough, fric
tional force does not depend upon the surface areas in contact; just as much
force would be required to overcome frictional resistance and slide a brick
along the sidewalk if the brick were lying on its narrow edge as if it were
lying on its broad side. On the other hand, if the narrow edge of the brick
were ground smooth from sliding along the sidewalk, then less frictional
resistance would arise when it was moved than if it were lying on a
rougher edge.
Equally important, the amount of friction present in a given case will
vary greatly depending on whether the objects in contact are in the solid,
the liquid, or the gaseous state. In a later chapter we will discuss a number
of commonly occurring "states" or physical forms of matter in more detail,
and examine more closely just how one state differs from another. For now
it is sufficient to point out that in solid matter the component atoms and
molecules are bound together in a more or less rigid crystalline structure,
so that when two solid surfaces grate against each other each holds its
shape stubbornly, and the friction created is comparatively great. In a
liquid the molecules are less rigidly bound and can usually "give" a little.
Thus, a boat can move through water with far less friction than if it were
dragged across the ground. As for a gas, each molecule is quite free of all
the others and can easily be "pushed aside," so that the frictional force
arising when a solid object "slides past" matter in a gaseous state is gen
erally very small indeed. Thus an arrow flying through the air is "impeded"
very little by the friction of the air molecules rushing past it but some
friction still arises. Meteorites plunging into our atmosphere from outer
space, for instance, encounter enough frictional resistance to heat them to
incandescence, and we see them at night as brilliant "shooting stars."
86 The Universe of Classical Physics
Friction, in short, is a resisting force, surprisingly ubiquitous, which
tends to obstruct or impede the motion of objects in relation to one another,
with the amount of frictional force present in any given case depending
heavily on certain specific physical characteristics of the objects involved.
But what about the "propelling force" necessary to keep objects moving
in spite of frictional resistance? In order to see more clearly exactly what
effect friction has upon the motion of objects, and to separate fact from
fallacy in regard to the mysterious "propelling force" of the Greeks, a
simple imaginary experiment will be helpful.
Suppose we had a device, much like the spring-operated shooting
mechanism on a pinball machine, with which we could give a carefully
calibrated push to any moving object we chose (so long as it was of man
ageable size). Suppose we satisfied ourselves, after repeated tests, that this
device would always move our test object in a certain specified direction
with a certain fixed force for a uniform period of time, say for one-tenth
of a second. In other words, our "shooter" would deliver an impulse to
the test object that would be exactly the same every time it was applied.
Then imagine that we use this "shooter" to deliver an impulse to a
small block of wood, shoving it horizontally across a rough table top. What
would be likely to happen? If the impulse were forceful enough, we would
expect the block to continue moving across the table for a short distance
after the impulse was over before it came to a stop.
Next, suppose we used the same shooter to move the same block of
wood across a highly polished counter, and then across the slippery sur
face of a frozen pond. We would find the block sliding farther on the
polished counter than on the rough table top, and farther still on the frozen
pond before it came to rest. But why? In each case the shooter delivers
precisely the same impulse to the same block of wood; why then does the
block move perhaps ten inches in the first case, ten feet in the second, and
ten yards in the third?
Obviously what is happening has nothing to do with the wooden block
alone, nor with the shooting mechanism, nor with the impulse it delivers.
The behavior of the block in each case depends on the interaction between
the block and the particular surface upon which it is sliding at the time.
In each case the moving block is met by a resisting force friction which
tends to slow it down. Indeed, with such an imaginary setup as this, we
have attempted to eliminate all factors that might affect the movement
of the block except frictional resistance. We see that when the block is
sliding and scraping across a rough wooden surface, friction is great and
the block slows down quickly. Much less frictional force impedes the
block's movement when it slides across the polished counter top, even
though solid is still rubbing on solid. As for the icy surface of the pond, if
we had keen microscopic vision and looked very closely we would see
that the block was not really sliding on ice at all, but on a thin film of water
The Riddles of Friction and Inertia 87
melted from the ice under the block. In such a case frictional resistance is
far less than in a case in which a solid was sliding on a solid. (We can also
see why a lubricant such as oil or grease can often reduce friction between
solid surfaces so effectively. Such a lubricant not only fills in the cracks and
crevices on both surfaces and thus smooths out the roughness, but also
forms a thin slippery film of liquid between the surfaces in contact. )
This kind of experiment tells us a good deal about the nature of friction
the resistant force that slows down the movement of a moving object.
We would expect friction to arise any time we moved an object relative to
another (or even attempted to move it). We might even be able to guess
quite accurately how much friction would arise in a given case. If we had
actually measured the velocity of our wooden block from moment to
moment while it was still moving in each of the three test shots of our
experiment, we would have found a constant decrease in its velocity every
second after the impulse was over, but the rate of decrease would vary ac
cording to the amount of frictional resistance. Thus the wooden block
sliding on rough wood would slow down and come to a stop very abruptly.
When sliding on the polished counter top, the velocity of the block would
decrease at a more leisurely constant rate. When the block was sliding on
ice its velocity would decrease comparatively little with each passing
second.
But what is the "propelling force" that keeps pulling the moving block
along against the resistance of friction? We have seen that when friction
is reduced (as in the case of the block sliding on ice) the block continues
moving for a remarkably long distance before it stops. What, then, keeps
it moving? Or to ask a more pointed question: What would the block do
if there were no frictional resistance impeding it at alll Common sense tells
us that if any kind of "propelling force" continued to push the moving
block and no resistant force of any sort acted to impede it, the block
would continue to accelerate indefinitely after the impulse had ended.
And yet no one has ever seen any such thing happen.
THE FORCE THAT WASN'T THERE
Of course, in Galileo's time physicists could only imagine conditions in
which a moving object encountered no friction at all. The best they could
do was to set up experiments under relatively friction-free conditions, and
then attempt to extend their findings to conditions which to them were
totally imaginary and unobtainable: conditions of no friction whatever. We
approached friction-free conditions in our own imaginary experiment when
we had our shooter push a block of wood across the glassy surface of a
frozen pond, and found that it continued to move for a comparatively long
distance perhaps ten yards after the impulse before frictional resistance
had brought it to a stop. Suppose in the same situation that we substituted
88 The Universe of Classical Physics
a block of dry ice for the wooden block, taking care that the two test-
objects had exactly the same weight. What would happen if we pushed
the block of dry ice across the frozen surface of the pond with the same
impulse we had applied to the wooden block? We might well not believe
our eyes, for the block of dry ice would continue to move and move and
move, fifty yards, a hundred yards, perhaps as far as half a mile, before it
finally came to rest! Again, if we had very keen microscopic vision, we
would discover the reason very easily. Dry ice is nothing other than frozen
and solidified carbon dioxide gas, and where the dry ice touched the surface
of the pond, a thin layer of frozen carbon dioxide melted from the block
into its gaseous state, so that the block was really sliding across the ice
on a thin layer of carbon dioxide gas.
Here we have reduced friction to about as bare a minimum as we can
attain here on earth. The frictional resistance between the carbon dioxide
block and the icy surface of the pond is very slight indeed, but still enough
to decrease the velocity of the moving block of dry ice slowly, bit by bit,
at a constant rate. Indeed, if we were to compare the effect of friction
on the block of dry ice on the pond with its effect on the block of wood
on the pond, we would find that each of these objects slowed down at a
rate exactly proportional to the amount of friction present. The less friction,
the less rapidly an object would slow down. And in the case of the dry
ice, it appeared at first that the block might just go on moving at the
same velocity and in the same direction forever, the frictional resistance
was so very slight.
But one thing the block of dry ice did not do: Even under these prac
tically friction-free conditions, it did not accelerate in the slightest. And
indeed, by this time we suspect very strongly that there is no propelling or
accelerating force pushing a moving object against friction. But to be
certain, suppose we find a place where there is virtually no frictional re
sistance of any sort acting to slow down our test object. Suppose, for
example, that we could transport our shooting device and the wooden
block into outer space far from any sun or planet, and there use the
shooter to launch the block still farther into space, free from contact with
any other surface. Under such circumstances, with no frictional resistance
possible, we would certainly expect any "propelling force" acting on the
block to make it accelerate (that is, to move continually faster and faster)
after it left the shooter. But if we actually measured the velocity of the
block under such circumstances after launching it with the shooter, we
would find no such thing happening. True, the block would not slow down
in the absence of frictional resistance or any other impeding force. But
neither would it speed up. Once the impulse was applied, the block would
continue to move in the same direction at exactly the velocity it had at
tained by the end of the impulse, and would keep on moving at this same
The Riddles of Friction and Inertia 89
constant velocity literally forever neither accelerating nor decelerating
unless or until it encountered some outside force acting upon it either to
speed it up, slow it down, or to change its direction in some way.
Of course, in Galileo's time physicists could only imagine conditions in
which a moving object encountered no friction at all. The best they could
do was to set up experiments in which friction was reduced to the barest
possible minimum, and then attempt to extend their findings to ideal
conditions of no friction whatever which, to them, were totally imaginary
and unattainable. But even so, Galileo very soon shrewdly guessed that
the mysterious "propelling force" the Greeks had imagined had to exist to
keep moving bodies moving simply did not exist at all. He saw clearly that
a moving object, once it had been started moving by some force or another,
tended to continue moving without any need for any additional propelling
force, just as long as no outside force acted upon it in any way. He also
saw that the opposite was equally true: An object sitting undisturbed at
rest would not suddenly start moving of its own accord, but would remain
at rest until some outside force some impulse acted upon it to start it
moving. That impulse might be the powerful force of a bowstring against the
end of an arrow, the short, sharp blow of a hammer on a peg, or the steady,
continuing pull of gravity on an object released from the fingers but some
outside force had to act to get an object moving.
Galileo performed many experiments testing the behavior of objects at
rest and objects in motion, and found that invariably his results seemed to
support the truth of his shrewd guesses. Other scientists also experimented
along the same lines; Leonardo da Vinci, for example, described frictional
resistance simply but accurately in his notebooks. Finally, a hundred years
after Galileo, Sir Isaac Newton summarized all these observations of the
behavior of bodies in motion and at rest in the form of two simple but
sweeping laws of motion which he believed applied to all objects, whether
they were moving or at rest, anywhere in the universe:
1. Any object in motion will continue in motion in the same direction
and at a constant velocity unless acted upon by some (resultant) outside
force.
2. Any object at rest will remain at rest unless acted upon by some
(resultant) outside force.
Taken together, these laws have come to be known as the "laws of
inertia." First investigated by Galileo and finally formalized by Sir Isaac
Newton, they were among the first of a very few sweeping laws of nature
which became the foundation for all the modern work in physics which
came later. The great importance of these early natural laws lay in the very
fact that they were believed to be accurate, universal descriptions of rela
tionships that occurred in nature. They were believed to be general (i.e.,
90 The Universe of Classical Physics
to apply to any object, no matter how large or small, with no exceptions)
and universal (i.e., applying to objects at any location anywhere in the
universe no matter how near or far away).
Furthermore, these laws were believed to mean exactly what they said.
When they spoke of an object "at rest" they meant an object at complete
rest or inaction; when they spoke of an object "in motion in the same
direction and at a constant velocity" they meant an object in perfectly uni
form motion. The laws recognized that any number of outside forces might
be acting upon an object at any time, but that a change in that object's
state of rest or motion would only come about if there were a net resultant
force at work. Two exactly equal forces acting on the object in precisely
opposite directions, for example, would in effect cancel each other out and
have no effect whatever on the object's motion; they would exert no
resultant force. Finally, these laws stated explicitly that only forces ex
ternal to an object could affect its motion one way or the other, a very sig
nificant qualification, as we will soon see.
In short, the laws of inertia simply stated that all objects would always
continue indefinitely in the same state that they were already in, whether
that be a state of rest or a state of uniform motion, until some external force
came along to change things and that no change in an object's velocity
or direction could possibly come about unless it was caused or brought
about by some external force.
We know, of course, that a variety of forces can and do act upon objects
to bring about changes in their velocity or direction of motion. We have seen
that frictional forces can slow a moving object down to a stop, or even
act upon an object at rest to keep it at rest if another force attempting
to make it move is not great enough to overcome the frictional resistance.
We know that gravitational force here on earth will act on a flying arrow
to pull it down to the ground eventually, no matter how swiftly it leaves the
bow, and we can imagine that any gravitational force anywhere in the uni
verse would act as an "outside force" capable of changing or modifying the
velocity or direction of any object within its reach. Still another force
capable of acting on an object is the kind of force that the shooter in our
imaginary experiment applied to the wooden block: essentially a collision
of one object (the shooter) with another object (the block) producing
what we spoke of as an "impulse" a force acting on an object in a
given direction for a given time.
But why are the laws of inertia so extremely important and basic to an
understanding of how things work? If we look to see how these laws affect
the behavior of moving objects in the face of action of each of these three
kinds of "external force" in a variety of situations we will discover some
rather surprising things things which have a direct application to our
everyday life here on earth.
We have already seen the effect that friction has on a moving object;
The Riddles of Friction and Inertia 9!
acting as an outside force, frictional resistance will slow a moving object
down, and may alter the direction of its motion as well. In either case, the
object's velocity is changed.
But how does a moving object behave when the outside force acting upon
it is a gravitational force alone? Here on earth if we set our wooden block
moving horizontally through the air, gravitational force would pull it down
to the ground so that its path in flight or trajectory would be a curving
line. In this case, gravity as the "outside force" acts on the moving block
to change its direction of motion, pulling it out of the straight-line path it
would follow if no such force were acting on it.
Low Velocity Block
Captured!
Fig. 5a Trajectory of an object moving through space and "captured" by
earth's gravitational field.
Suppose, however, that we knew that the earth's gravitational field ex
tended far out into space beyond its atmosphere (as, of course, it does) and
imagine that our wooden block were moving through outer space in a
straight line with a constant velocity and were to approach within the
earth's gravitational field. Here again, gravity as an external force would
act upon the moving block to change its velocity and direction, and again
(if the block's original velocity were not very great) it would follow a
curving path down to the surface of the earth, "captured" in effect by the
earth's gravitational force (see Fig. 5a).
This is clear enough; but we know that the earth has a moon. Suppose
that the moon also has a gravitational field, however great or small it may
be in comparison to that of the earth, and suppose that our wandering block
approaches and passes between the earth and the moon at just exactly such
a point and at just such an angle that at all times the gravitational pull of
92 The Universe of Classical Physics
Earth on one side would be exactly equal and exactly opposite to the
gravitational pull of the moon on the other side (see Fig. 5b). How would
the block behave under these circumstances? We can see that it would not
change in direction to curve either way, with both forces equally balanced
but what about its velocity? Surely, we might think, these opposing forces
acting on the block would at least tend to slow it down some in its flight.
Fig. 5b Object passing between earth and moon. The gravitational tug of the
earth on the object always equals that of moon, so net effect of gravity is zero.
But nothing of the sort would happen. Assuming no other forces were
acting upon the block, the resultant outside force of the two opposing
gravitational fields would be zero. According to Newton's laws of inertia,
therefore, there could be no resultant force acting on the block at all and
it would continue placidly on its way without any change whatever in either
its direction or velocity!
So far we have been talking about objects moving at a uniform, relatively
low velocity. What would happen if higher velocities were involved? We
have seen that our wooden block approaching and entering earth's gravita-
The Riddles of Friction and Inertia 93
tional field from outer space and traveling at a low velocity was forced to
curve sharply from its straight-line path and nose-dive down to the earth's
surface. What about the other extreme? Suppose the block approached
from outer space and entered earth's gravitational field at an extremely high
velocity. Earth's gravity would still be an outside force acting upon the
block, so that its path would have to curve toward the earth. At the same
time, the block's velocity might be high enough to carry it on beyond the
pull of the earth's gravity and ultimately free again. In such a case the
block's direction and velocity would be modified by its encounter with this
outside force but it would "escape capture" and continue hurtling into
space beyond the earth, traveling in a new direction and with a new
velocity (see Fig. 5c).
High Velocity Block
Escaped
Fig. 5c Object moving at high velocity changed in velocity and direction by
earth's gravity, but not captured.
But suppose the block were traveling at a velocity somewhere between
these two extremes. Suppose its velocity as it approached the earth's gravita
tional field was just enough to balance the gravitational pull of earth upon
it. Then the block's path would be curved toward the earth each instant,
but its velocity would still remain high enough that at each instant it sought
to continue in a straight line or tangent away from the earth on its own
path. If the gravitational pull of the earth at each instant was exactly
balanced by the velocity of the block seeking to travel in its own inertial
straight line, the block could neither quite fly off into space again free of
earth's gravitational pull, nor could the earth quite pull it down to the
surface. Rather, the block would continue to curve toward the earth, and
curve toward the earth, and curve toward the earth, and continue in
definitely revolving around and around the earth at the same velocity and
the same distance away from the surface in effect, a new satellite in a
permanent orbit (see Fig. 5d).
At first glance this might seem like a highly fanciful idea, but if we
think about it we see that there is nothing fanciful about it. After all, earth
94 The Universe of Classical Physics
has a somewhat larger satellite, the moon, which travels in a stable orbit
around the earth at the same average distance and with the same average
velocity that it did thousands of years ago. We can begin to see why Galileo,
and later Johannes Kepler, and ultimately Isaac Newton, after studying
the behavior of gravitational forces on earth, began looking to the sky and
wondering if the moon's wandering path around the earth was not a result
of a cosmic tug-of-war between earth's gravitational pull upon the moon
and the moon's continuing struggle to pull away from earth and hurtle
out to space in a straight line according to the dictates of the laws of inertia.
We will return to this line of thought again presently, because it will lead
us to a better understanding of how Newton ultimately arrived at the laws
of universal gravitation (as opposed to the laws of local gravitational effects
Medium Velocity Block
Trapped in Orbit
Fig. 5d The capture of an object in orbit, a balance between earth's gravita
tional attraction and the object's own inertial (centrifugal) force due to its
original motion.
which Galileo worked out). However, before we abandon our block of
wood traveling through space, obeying the laws of inertia, we should turn
our attention for a moment to the question of what constitutes an "out
side force" capable of acting upon an object at rest or an object in motion
and what is not, in fact, an "outside force." Again, the answers may not
be quite what we would at first imagine.
Suppose, for example, that instead of launching our wooden block into
space we transport the block of dry ice instead, and send it traveling in a
straight line at a constant velocity in a place where no gravitational force
could reach and no frictional force existed. Suppose, however, that the block
of dry ice (merely a chunk of frozen carbon dioxide gas) should happen
to move into an area of space which happened to be unusually warm so
that the dry ice evaporated completely into an expanding cloud of carbon
dioxide gas. The original block, indeed, would cease to exist altogether,
transformed into something quite different! Would this not be a violation of
the laws of inertia? Surely such a transformation from within the "object"
The Riddles of Friction and Inertia 95
which was not an outside -force would nevertheless act to alter its direction
or velocity, would it not?
The answer is that it would not. True, the original block of dry ice is
transformed into a cloud of gas, and the molecules of gas in that cloud,
stimulated by the increased temperature, might diffuse and move every
which way, creating a thinner and thinner cloud of carbon dioxide gas of
greater and greater diameter. But in the absence of any outside resultant
force acting upon it, that cloud of gas as a whole would continue to move in
exactly the same direction and at exactly the same velocity as if it were
still an untransformed block of dry ice. In other words, the encounter with
this area of heated space might bring about an internal change or trans
formation within the system, but it would cause no change whatever in
the behavior of the system as a whole.
In fact nothing that might occur within a system, no matter how forceful,
could alter the over-all behavior of the system as a whole, no matter liow
much that system's shape or appearance might be changed. When Newton
spoke of "some outside resultant force," he meant some force arising
strictly from outside the moving system, not some force that arose in any
way from within the moving body itself.
We can see this more clearly, perhaps, if we imagine our moving object
not as a block of dry ice traveling through space but as an old-fashioned
Bolshevik bomb with the fuse lit. After the bomb travels along for a period
of time at a constant velocity in a perfectly straight line, the fuse burns
down and the bomb explodes, throwing fragments off in all directions.
Certainly this would appear to be a case in which an "inside force" had a
grave effect upon the direction and velocity of the bomb!
But once again, appearances deceive us. Of course, the fragments of
the bomb would go flying in all directions, some backward in the direction
the bomb was coming from, some forward, some off to the sides, up or
down. But the total net result of the explosion would add up to nothing
more than a change in the over-all shape of the bomb. Just as we had a
solid block of dry ice suddenly transformed into an expanding cloud of gas,
here we would have a bomb that suddenly "expanded" violently in all
directions, with pieces thrown out at a hundred different velocities. But if
we were to add up all the velocities, directions, and masses of every single
one of the resulting fragments we would come up with a net result of zero:
no change in the over-all velocity of the bomb fragments taken as a whole.
Of course, each of those bomb fragments individually would continue to
move in its own direction at its own velocity forever unless individually
acted upon by some outside force, but the system (which previously was a
nice solid Bolshevik bomb but now has been transformed into an expanding
cloud of shrapnel) would continue to move as a whole in the same direction
and with the same velocity as the original bomb was moving.
96 The Universe of Classical Physics
We could look at the explosion another way. It would be as though the
bomb, before its explosion, had been enclosed in an infinitely elastic,
stretchable plastic envelope. After the bomb exploded, the envelope con
taining all the fragments would be stretched out into a completely dif
ferent shape than the original bomb, but in spite of this, the entire envelope
and its contents would continue moving at its original velocity and in its
original direction until an outside force affected it.
Now suppose that if instead of a round Bolshevik bomb, our floating ob
ject had been a bomb in the form of a tube open at both ends with one end
pointing in the direction of its motion and the other end pointing in the
opposite direction. Suppose that instead of breaking the bomb up into
shrapnel, under these circumstances the explosion simply blew a cloud of
hot gas out of both ends. There would still be no change in the velocity
or direction of the system as a whole even though an expanding cloud of
gas was blossoming from either end of the tube. Then, finally, suppose that
the front end of this tube-shaped bomb were plugged up so that the only
escape for the explosive gases would be from the rear end. In this case,
when the explosion occurred, the shell of the bomb (which now would
essentially be a rocket) would certainly show a change in velocity: It would
suddenly leap forward as a result of the gust of hot gases escaping out the
open rear end. The shell of the bomb would end up with a much higher
velocity going one way while the escaped gases would move with an
equally high velocity in the opposite direction. But even in this extreme
case the velocity and direction of the system as a whole (bomb shell plus
released gases) would remain completely unchanged; if the shell and the
gases were all enclosed in an infinitely elastic stretchable bag, we would see
a greater and greater distention of the bag in the fore-and-aft direction,
but except for this distortion in shape the entire system would continue
moving just exactly as it was moving before the explosion.
If we think about this for a moment, we see that here is the reason that
we cannot lift ourselves up by our own bootstraps no matter how hard we
may try. We may exert all sorts of effort, straining our muscles and tugging
for all we are worth, but all of the force that we are applying in an attempt
to pull our own feet off the ground comes from within the system. There is
no force from outside the system (i.e., from outside our bodies and our
boots) which is acting to lift us, and consequently we can bring about no net
change in our position. Of course, if a block and tackle were attached to
the ceiling and we ran a rope through our bootstraps and then through the
pulleys we could hoist ourselves up off the ground with no trouble, but
the force that is lifting us is coming from outside of the body-and-bootstrap
system; in fact, we are utilizing the upward pull of the ceiling on the block
and tackle (the ceiling is obviously outside the system) in order to provide
ourselves with "some place to stand" or, more accurately, "something to
hang on to." (This is assuming that we consider the "system" in this case
The Riddles of Friction and Inertia 97
to include only ourselves and our bootstraps; if we included the building
and its ceiling in the "system," then hoisting ourselves up with a block and
tackle would constitute nothing more than moving ourselves around within
the system, and we would not bring about any net change in the position
of this new and larger "system" unless we could find something for the
building to hang on to. )
As we have seen there are a number of clearly distinguishable "outside
forces" that can affect the behavior of objects in motion and objects at
rest. Gravity is one such force. Friction is another. Both are forces that can
alter the direction of a moving object or alter its velocity once it has been
set in motion. Other outside forces can have similar effects: A magnetic
field, for example, can exert a powerful outside force upon certain kinds of
moving objects objects composed of iron, for example, or objects which
happen to possess a certain peculiar characteristic known as "electrical
charge" (a characteristic we will discuss in detail later). But perhaps the
most common of all outside forces that we observe in our everyday world
is a force that is easy to identify, easy to understand, and highly instructive
to consider when we are attempting to figure out the peculiar behavior of
objects in motion and at rest. This is the force of one object striking another
the simple phenomenon of a collision. We know from long experience
that the velocity and direction of a moving object can be altered very
sharply and radically if it happens to run into some other object. What is
more, in our everyday world Newton's laws of motion should allow us to
predict in advance precisely what the end result would be any time one
object collided with another, providing only that we know the masses of
the objects, their velocities, and the directions in which they are moving.
CHAPTER 8
Push and Push Back: The Riddle of Collisions
What happens when an irresistible force meets an immovable object?
Surely every one of us must have puzzled over this moldy old grade school
conundrum at one time or another. The unwary ones may even have
accepted the far-from-satisfactory answer that "heat is produced" and
dismissed the matter without further thought. On closer inspection, how
ever, we can see that the question itself is absurd: Its very terms are self-
contradictory. A truly "irresistible" force would have to be a force so com
pletely overwhelming that it would be capable of moving any object it acted
upon, even if that "object" contained all the existing matter in the universe.
A truly "immovable" object, on the other hand, would have to be an ob
ject so incredibly massive that no imaginable force could possibly make it
budge, no matter how enormous that force might be. It is hard enough
for us to imagine either such a force or such an object existing in the ab
sence of the other; obviously the two could not possibly exist simultane
ously in the same universe!
But absurd or not, this conundrum is still intriguing. For one thing, it
forces us to pause and reconsider exactly what we mean when we speak ot
a "force" acting upon something. For another, it implies certain things
about the behavior of objects in motion that such great early experimenters
as Galileo and Sir Isaac Newton came to believe were universally true,
after years of thoughtful experiment and observation of things in the world
around them.
First, this conundrum implies that any change in an object's state of
rest or motion can occur only if some outside force acts upon the object.
In the last chapter we saw how the great classical physicists were led
inexorably to this conclusion. Indeed, they went a step further: They began
to realize that the only way a "force" could be identified at all was by
observing and measuring the effect it had on the motion of some object. A
cannonball flying through the air exerts no force upon anything (except a
few air molecules) until it strikes a target somewhere. Neither does a rough
warehouse floor exert any frictional force upon a packing case stored upon
it until someone comes along and tries to push the packing case. Odd as
it may seem, a force cannot really be said to exist at all until it somehow
98
Push and Push Back: The Riddle of Collisions 99
acts upon some object, and then its magnitude can only be deduced
indirectly, as it were from the change that we observe in the object's
motion as a result of the force acting upon it. And if we are tempted to
challenge this slightly sneaky idea, we need only consult our own everyday
experience for proof. We see leaves fluttering and treetops swaying and
deduce from this that a wind is blowing. We discover that a gravitational
force is present by its action upon a vase that we drop on the floor. We
detect frictional force only when we attempt to slide one object across
another. In each case, both the force and its magnitude are identified solely
by the change in an object's motion resulting from the action of the force.
But our conundrum has an even more subtle implication. It suggests to
us that the amount of change in an object's state of rest or motion must be
directly related not only to the magnitude and direction of the force acting
upon it, but also to the mass of the object that is, to the object's inertia,
its built-in resistance to any change in its motion. A truly "irresistible"
force acting upon any object would cause the maximum possible change
in that object's motion, regardless of the object's mass. Such a force would
be able to accelerate the entire mass of the universe up to the speed of
light in the direction the force was acting. It would just as readily be able
to slow a beam of light (which has some mass, as we will see later) from
light-speed down to a standstill, if the direction of the force happened to
oppose the direction of the light beam. At the other extreme, a truly
"immovable" object would offer so much resistance to any change in its
state of rest or motion that no force could budge it even an inch, no matter
how great that force might be.
Happily, we are unlikely ever to witness either of these outside extremes,
but we can see that any interaction between an object and a force acting
upon it must fall somewhere between these extremes of total immovability
and total irresistibility. Time and again the early experimenters found that
any change occurring in the motion of any given object was always directly
proportional to the magnitude of the force acting upon it the greater the
force, the greater the change in the object's motion. On the other hand,
they also found that the more massive the object, the more it resisted
change in its state of rest or motion, and the less change any given force
would be able to bring about when acting upon it.
Today this proportional relationship between an object's mass, the force
acting upon it, and the change that occurs in its state of rest or motion
seems self-evident. Common sense tells us on one hand that a given force
will influence the motion of a comparatively light object more than it will
a more massive one, but that on the other hand a given object's motion will
be influenced more by a powerful force than by a weaker one. If we tried
to play Ping-pong with a golf ball, we would have trouble getting the
massive ball across the net with a standard lightweight paddle. If instead we
used a paddle made of lead, we might be able to play the game all right
Ioo The Universe of Classical Physics
using a golf ball, but we would be likely to knock an ordinary standard
Ping-pong ball into the next county on the first serve.
Finally, our irresistible force-immovable object conundrum suggests that
whenever any force is brought to bear upon an object to push it around, the
object must counter with a resisting force of equal magnitude acting in
the opposite direction. An object being pushed literally pushes back. In the
extreme case, our imaginary "immovable" object would be just as un
willing to be moved at all as our "irresistible" force would be unwilling to be
resisted, so at best we would end up with a Mexican standoff. To imagine
a more familiar example, we might substitute a twenty-pound sledge
hammer for our "irresistible" force and a brick wall for our "immovable"
object. If we then swung the hammer against the wall with all our strength,
we might expect the hammer's force at the moment of impact to "move" the
wall, at least to some degree: A few bricks would be chipped or cracked,
perhaps even crushed. But the sledge hammer would not continue to plow
its way through the wall, unaffected by the encounter; we would see it
bounce back from the impact, perhaps with so much force that it is torn
out of our hands!
But in such a case, what force could possibly be acting upon the hammer
to bring it to a halt against the wall and then thrust it violently back in the
opposite direction? This could only happen if the wall were to exert some
counterforce against the hammer at the same time the hammer hits the wall.
If we could somehow measure the forces acting at the moment of interaction
between wall and hammer, we would find the wall's impact on the hammer
to be exactly equal to the hammer's impact on the wall, but acting in the
opposite direction. Furthermore, if we were to experiment with other such
interactions or collisions, we would soon find that an equal but opposite
counterforce is always present any time any force acts upon any object.
In somewhat simpler terms, we can say that for every action (of a force
upon an object, for example) there is an equal but opposite reaction (of
the object resisting the action of the force, for example).
If this idea seems confusing and obscure, take heart: Even physics
majors find it difficult to grasp. Part of our trouble is that we tend to over
look or ignore the "reaction" part of the equation that is always present
when forces and objects interact in the world around us. We just don't
ordinarily think in terms of the wall hitting the hammer back, even though
it obviously does so, any more than we think of the ground pushing upward
against our feet, or of the rough floor pushing back against the packing case
we are trying to move. Yet when we hold a steel ball at rest in an out
stretched hand for a while, our tired arm muscles soon tell us that we have
been pushing the ball upward at the same time and with the same force as
gravity has been tugging it downward. Even if we fail in our efforts to move
the packing case so much as an inch across the floor, we nevertheless
know that we have been pushing with might and main against some real
Push and Push Back: The Riddle of Collisions IOI
force that is opposing our efforts and preventing the box from moving.
When we fire a rifle, we expect the bullet to be driven forward by the
force of the explosion and imbed itself in a tree but we also feel the
rifle butt slam against our shoulder in recoil. The only reason that the rifle
does not fly backward as far and as fast as the bullet flies forward is
simply that the rifle itself has much more mass to be moved backward than
the bullet has to be driven forward, and even at that we are likely to end
up our target practice with a black-and-blue shoulder if we aren't careful.
The fact is that all of these characteristics we have been discussing of
the interactions of forces and objects can easily be observed and confirmed
in the world around us every day, providing we know what to look for. For
the most part, people rarely look. They simply take these things for granted
without even trying to describe what is actually happening. And if we
find these characteristics of moving objects hard to pin down when we try
to describe them, we can take comfort that the early physicists found them
just as hard to comprehend, if not harder. Galileo spent decades trying
to figure out how to describe the behavior of moving objects with accuracy.
From his experiments he came to recognize all the characteristics that we
have been discussing but he never did find a way to express them as con
cise rules which could then be applied to the motion of all objects, large or
small, anywhere in the universe. It remained for Isaac Newton, starting
where Galileo and others had left off, to work out three simple, general
statements which he believed accurately described all possible forms of
motion throughout the universe. Today these statements are known as
Newton's laws of motion, and can be briefly summarized here:
Law I: Any object in a state of rest (or of uniform motion in a straight
line) will remain at rest (or in uniform motion in a straight line) unless
acted upon by some external resultant force. (We have already seen some
of the implications of this statement, and we will soon see more.)
Law II: When an external force acts upon an object, the change in the
object's motion is proportional to the force and occurs in the direction
that the force is acting.
Law III: When any force is brought to bear upon an object, an equal
force is brought to bear acting in the opposite direction;
or:
For every action there is an equal but opposite reaction.
(We will consider the implications of laws II and III more closely later in
this chapter.)
At this point we must remind ourselves once again that Isaac Newton
did not come up with these three principles by means of any divine
revelation, nor did he regard them as irrevocable "laws of nature" at the
102 The Universe of Classical Physics
time he formulated them. He merely considered them as useful working
rules, tentative conclusions based upon hundreds of years of observation
and experiment. In effect, he was saying: 'These rules seem to describe
what happens any time a force acts upon an object, or any time one object
interacts with another. As far as we know now, these rules always apply,
with no exceptions. Let's consider them to be true until some new evidence
shows up to prove them false."
This was surely a reasonable stand to take, and scientists of the day
accepted it. But as time passed, no such new evidence showed up; re
peated crafty attempts to find exceptions to Newton's "working rules" in
variably failed. By the beginning of this century, most physicists had come
to accept these rules as broad, universal laws of nature, essential to any
understanding of how things work in the universe.
This is not to say that Newton's laws of motion became any easier to
comprehend as time went by; to this day physicists themselves cannot fully
agree upon just how the laws of motion should be interpreted. As casual
bystanders we might be tempted to ignore them as obscure and meaning
less, if it were not for the fact that these rules are profoundly important
to us in the conduct of our daily lives. Any time we sip a cup of coffee,
throw a baseball, walk to the grocery, or slam the garage door, we
are in fact utilizing the laws of motion to fulfill our needs, whether con
sciously or not. Those laws guide and limit virtually every move we make.
What is more, they enable us to make useful and accurate predictions about
things that have not yet happened. Every time we drive a car around the
block we embark upon a multitude of half-conscious computations, judg
ments, and predictions of what is going to happen next, all based upon the
laws of motion. And when we see a sand-lot baseball flying through the air
toward our biggest plate-glass window, we do not need Isaac Newton to
tell us that that window is going to be smashed to shards unless we can
stop the ball before it reaches its target.
To understand more clearly just what the laws of motion actually mean,
and to see how they enable us to predict how things are going to work in
the world around us, it will be helpful to examine more closely one of the
most familiar and commonplace of all the interactions we observe every
day: the collision of one moving object with another.
THE COSMIC POOL TABLE
What actually happens when one object crashes into another? In our
everyday experience it often is difficult to say, precisely, because of the red
herrings that lead us astray. In some collisions the colliding objects are
shattered. In others one object or another may be bent out of shape, altered
beyond recognition, heated to incandescence, even vaporized! We have
already seen how such extraneous factors as air resistance or friction
Push and Push Back: The Riddle of Collisions 103
interfered with early observations of the influence of gravity on falling ob
jects, leading to many confusing or downright misleading experimental re
sults. If we wish to concentrate solely on the effects of collision forces upon
moving objects, we must rule out all other forces and effects, as far as
possible. In short, we must try to imagine a "perfect collision" of two
moving objects occurring under ideal conditions. In actuality we could never
hope to find the "ideal test objects" necessary to fulfill such rigorous speci
fications, but we can find a very close approximation from our everyday
experience: the collision of two ordinary billiard balls on the smooth green
felt of a large pool table.
There are several reasons that billiard balls lend themselves so splendidly
to our purpose. For one thing, most of us are already familiar with their
collision behavior from personal experience. We know that when two
billiard balls collide, they inevitably bounce away from each other, with
some alteration in the velocity and direction of motion of each ball. We
even recognize from experience that what we see happening after a collision
of two billiard balls depends a great deal upon the velocity and direction of
motion of each ball before the collision and upon the angle at which the
collision takes place. In short, we recognize a cause-and-effect relationship
between the conditions before the collision and the new conditions after
it has occurred, and we already have some idea of what to expect when
billiard balls collide. All we really need to do is fill in the details of what
we actually observe under a variety of collision circumstances.
Furthermore, the physical properties of billiard balls lend themselves well
to our needs. For one thing, billiard balls are substantially massive objects,
unlikely to be affected much by such minor forces as air resistance, cross-
wind drafts across the pool table, or whatnot (whereas Ping-pong balls, in
contrast, would be). For another thing, we can safely assume that any two
billiard balls we might choose would be very nearly identical in mass
certainly nearly enough that we could ignore the effect of any minor
differences.
Again, because of their respective masses (and their attendant qualities
of inertia i.e., resistance to any change in their state of rest or motion)
we can expect billiard balls to behave very much the same as objects in
free fall, relatively unaffected by frictional or gravitational forces. We ex
pect a billiard ball at rest on the table to remain at rest unless some out
side force starts it moving. Once it is set in motion, however, we expect
it to continue rolling (at least for a while) at a relatively constant velocity
in the direction the force acted upon it, unless it is again acted upon by
another external force.
Of course, we acknowledge from the beginning that these things are only
approximately true. Billiard balls are influenced by earth's gravity, as we
would soon learn if we dropped one on our toes. But if our pool table is
perfectly level, gravity would influence any one ball exactly as much as
IO4 The Universe of Classical Physics
any other, so the effects of gravity would be canceled out, as far as our
experiment was concerned. As for friction, we know that it will indeed
slow down a rolling billiard ball a bit at a time, but not enough to cloud
our experiment. As long as the balls are moving at a fair velocity and we
observe their behavior over relatively short distances, we can imagine their
movement to be virtually frictionless.
Finally, billiard balls provide an ideal sort of collision to observe. When
they strike each other, only a tiny surface area of one actually comes in
contact with the other and then only briefly: The impact is nearly in
stantaneous, and the balls bounce freely away from each other almost
immediately after colliding. What is more, billiard balls will not be sig
nificantly deformed at the instant of collision, as two soft-rubber balls would
be, nor do they have any tendency to stick together when they strike, as
two balls of well-chewed bubble gum might. A physicist would speak of a
collision between billiard balls as comparatively elastic that is, a collision
in which virtually all of the force of the collision is transmitted directly to
the colliding objects almost instantaneously upon impact, so that very little
energy is dissipated into heat or exhausted in the physical deformation of
one or both of the objects.
In short, billiard balls offer a reasonable approximation of the "ideal"
collision conditions we are seeking. So what happens when one billiard
ball collides with another? First, let's imagine that we have two shooting
devices, such as the one we used in the last chapter, installed at opposite
ends of a pool table and use them to start two billiard balls rolling toward
each other in a straight line, each ball moving with exactly the same
constant velocity as the other but in opposite directions. When the two
balls reach the exact center of the table we are not surprised to see them
collide with each other smackl and then bounce smartly apart again.
Fine, but what actually happens during the collision! How do the velocities
and directions of the balls after collision compare with their velocities and
directions before the collision?
We could make some shrewd guesses without even measuring, just on
the basis of common sense and experience. First, we would expect the balls
to collide head-on, since they were approaching each other dead-ahead
on the same line. Further, since each ball has the same mass as the other
and is moving toward the other with the same velocity we can imagine that
each will strike the other with identical force at the moment of impact, so
that whatever happens to one ball as a result of the collision will also
happen to the other in mirror-image fashion.
Similarly, we might guess that certain other things might be observed and
measured:
1. Since the balls are moving toward each other head-on, each will
obstruct the movement of the other at the moment of impact, so that for
Push and Push Back: The Riddle of Collisions 105
a split second during collision the balls will be standing motionless side by
side at the center of the table. Each ball will have brought the other to a
complete halt.
2. Since neither ball can pass through the other, and since each exerts
an equal force on the other, the two balls will be pushed away from each
other in opposite directions as a result of the collision. Thus the direction of
motion of each ball will be exactly reversed.
3. Since the force that each ball can exert on the other depends upon the
mass of the ball and its velocity at the moment of impact, and since the
balls approaching collision have identical masses and equal velocities in
opposite directions, we would expect the balls to bounce away from each
other after collision at exactly equal velocities in exactly opposite directions
and to continue moving away from each other with equal constant veloci
ties until some other outside force (such as the end of the pool table)
forces another change.
4. Finally, since each ball approaching the collision is carrying a certain
amount of energy with it (let's call it "energy of motion" for the moment)
and since practically none of that energy of motion is lost in heat or expended
in deforming the balls during the collision, we would expect each ball to
throw virtually all of its energy of motion into pushing the other ball away
at moment of impact. As a result, the velocity imparted to each ball as a
result of the collision will be exactly equal to the velocity of the other ball
before the collision, but in reversed direction; and since the velocity of
each ball before collision is equal to that of the other, each will bounce
away after collision with exactly the same velocity it carried into the col
lision.
In short, the net result of our imaginary "ideal" collision should be a
complete reversal of the direction of each ball, in mirror-image fashion,
with the after-collision velocities of the balls precisely equal to their before-
collision velocities but directed in opposite directions. Indeed, we might
see the same result (at least as an illusion) if we rolled one billiard ball into
a head-on collision with its own image in a mirror!
Of course we must bear in mind that this "ideal" collision would never
actually come about on a real pool table. Because of the frictional resistance
of the table top, the balls would not be moving toward each other at
perfectly constant velocities, but rather with constantly decreasing velocity,
however slight the decrease. Their collision would not be perfectly elastic,
because some of the energy carried by each ball would be consumed in
correcting a slight distortion of the surface of each ball caused by their
impacts. Each ball would, in fact, flatten slightly at the point of impact and
then spring back to normal shape immediately after. Some energy (not
much, but some) would be converted into heat at the same time each
ball would become slightly warmer at collision point. Thus, in reality, we
io6 The Universe of Classical Physics
know the balls will move away from each other with a slightly lower
velocity than they had the instant before collision and then (thanks again
to friction) will continue to slow down as they move away from each other
in opposite directions. We could eliminate friction completely only if we
could stage the collision somewhere in outer space; and even there we
would have to conjure up completely undentable billiard balls and arrange
a collision in which no energy whatever is converted to heat before we
could achieve a perfectly elastic collision in which the balls really would
retire from each other at precisely the same velocity they had had before
the collision.
Indeed, the more closely we compare a real-life pool table collision with
our imaginary one, the more red herrings we find. But the truly amazing
thing is not how far away from ideal results we would come in an actual
billiard ball collision, but rather, how close we would come! If we had
actually measured velocities and energies throughout our considerably-less-
than-ideal pool table collision, we would have found that what actually
happened there approached our ideal predictions surprisingly closely
so closely, in fact, as to suggest strongly that // ideal conditions had been
possible, our experimental results would have been precisely as we had
predicted. Thus such imaginary experiments, although impossible to actu
ally perform, can still be a perfectly valid way sometimes the only way
to learn what is true and what is not.
But in imagining impossibly "ideal" conditions for our billiard ball col
lision, we have made one quite unwarranted assumption. We have assumed,
because the balls bounced away from each other in mirror-image fashion,
that each ball must have transmitted all of its energy of motion to the
other, and vice versa, during the moment of impact. But how can we be
so sure there was an exchange of energy? Suppose we had interposed a
thick plate of steel in the middle of the table, so that each of the balls
struck the steel plate instead of the other ball (see Fig. 6). In such a case
both balls would have rebounded exactly as if they had collided with each
other, even though nothing could possibly have been transferred from one
ball to the other. How do we explain that? And if there was an exchange
of some sort between the balls when they actually collided, what exactly
was exchanged?
Obviously, as long as both balls are of identical mass, approaching col
lision with equal velocities but moving in precisely opposite directions, we
will have trouble answering these questions and determining exactly what
does happen. We can clarify the question by repeating the experiment under
slightly different conditions. Suppose this time we insert a lead weight into
the center of one of the balls so that ball A has twice the mass of ball B.
Everything else we keep the same. We beef up the shooter pushing the
heavy ball A so that both balls are again set moving toward each other
just as before, with equal constant velocities in precisely opposite directions,
Push and Push Back: The Riddle of Collisions IO7
on a head-on collision course. What effect will the doubled mass of ball A
have on the results of the collision?
Once again we will see the two balls smack together at the center of the
table. Once again they will bounce away from each other in opposite direc
tions but this time we would find that the more massive ball A would
bounce away at only half its original velocity, while the less massive ball
B would spring away at twice its original velocity! And we would observe
the same thing every time we repeated the experiment.
Ball A
->l
Before Collision
V/ = V 2
Ball B
Fig. 6
Well, what is happening here? At the instant of collision, massive ball
A exerts a force call it force A on less massive ball B, while ball B
simultaneously is exerting a force, force B, on ball A (see Fig. 7). As a
result of these forces acting, each ball is stopped and its direction of motion
reversed the after-collision course of each ball is in the direction of the
opposing force of the other ball acting on it. All this is the same as before,
except that in this collision the after-collision velocity of each ball has been
changed. The after-collision velocity of the more massive ball A has been
reduced by half, while that of the less massive ball B has been doubled, as
though ball A gave up some of its velocity to ball B at the instant of collision.
What could account for this change? Clearly it must be the difference in
mass of the two balls, since nothing else had been altered in the second
experiment. And indeed, accurate measurements would reveal that the
change in velocity of each ball was inversely proportional to the mass of
the ball: Massive ball A ended up with half its former velocity after col
lision, lighter ball B with twice its former velocity. If we did the same
io8
The Universe of Classical Physics
experiment with ball A bearing four times the mass of ball B, we would
find the four-times-as-massive ball A bouncing away from the collision
with only one-fourth of its former velocity, while ball B would have jour
times its former velocity after collision, and so on for any difference in
mass of the balls that we might arrange.
From this, it might seem that "might makes right" in the case of col
lisions: The big guy pushes the little guy harder and farther than the little
guy can push the big guy when the two run into each other. But is the
mass of the billiard balls the only factor that can alter the results of their
collision? Another variation in our experiment will help us find an answer.
Ball A
V,
-TY
^^
Before Collision
o--
After Collision
V 3 only 1/2 of V
Ball B
V v twice V,
Fig. 7
To find out, let us make a different kind of change in the ground rules.
This time imagine that ball A and ball B are identical in mass again but
that this time we set ball B rolling into the collision with twice the velocity
of ball A (see Fig. 8). What will happen in this case? Once again the
balls will collide and bounce away in opposite directions, but this time the
slower-moving ball A will bounce away with twice the velocity than it
had coming into the collision, while the faster ball B will bounce away with
only half its former velocity.
Here again it would appear that the faster ball B has given up or trans
ferred some of its higher velocity to the slower ball A at the instant of
collision, but this time the exchange could not be blamed on any difference
Push and Push Back: The Riddle of Collisions IO ^
in the mass of the balls. The only difference that could possibly account
for this exchange is the difference in the before-collision velocities of the
balls. If we ran a multitude of tests in which ball B entered the col
lision at higher and higher velocities compared to ball A, we would find
that the after-collision velocity of ball A would increase in direct proportion
to the velocity of ball B before collision.
Thus we begin to see that what happens in an ideal instantaneous head-on
collision between two billiard balls (or any other freely moving objects)
must depend not only on the masses of the colliding objects, but also upon
their respective velocities at the instant of collision. Furthermore, we see
Ball A
o-oo
Before Collision
V l0 nly 1/2 of V,
Ball B
-O-OHXJ
After Collision
V, twice V
Fig. 8
that in every collision, each ball applies a force to the other which is pro
portional to both its mass and its velocity, and that each ball then reacts
in accordance with the force applied to it by the other.
So far we have limited ourselves to a very special kind of collision in
which the billiard balls approach each other from exactly opposite direc
tions, with no spin or "English," and collide squarely with each other
head-on. Thus the only change in direction of the balls that we have seen
after collision has been ISO-degree reversal of their directions along the
same straight line. But what would happen if we set the balls rolling toward
each other at an angle? Would the direction each ball was moving at the
moment of impact have any influence on the results? Any billiard en
thusiast can tell us the answer: The angle of impact of two billiard balls
has a very important effect upon what the balls will do after collision.
But how much effect, or what sort?
no The Universe of Classical Physics
Suppose we take ball A and ball B, identical in mass, and start them
rolling toward each other from two adjacent corners of the pool table, so
that they will collide at a 90 degree angle, taking care that both balls
approach collision at precisely equivalent velocities (see Fig. 9a). What
will happen when they collide? Once again, each ball will exert a force
upon the other at the instant of collision: Ball A smacks ball B and ball
B smacks ball A. Once again, these collision forces result in a change of
direction for each ball. Ball A, approaching from the left, bounces away
from the collision to the left as though it had turned a square corner of
90 degrees. Simultaneously, ball B approaching from the right would bounce
away to the right in mirror-image fashion. But the velocity of each ball
after the collision would be exactly the same as it was before. The only
result of the collision would be a change in direction of motion of each ball
exactly what we saw in our first experiment when balls of equal mass
collided head-on while approaching each other with equal velocities.
With further experiment, we would obtain similar results no matter at
what angle the balls approached collision. If the angle is very narrow (as
in Fig. 9b) the balls will diverge after collision at an equivalently narrow
angle; if the angle is very wide (Fig. 9c) they will diverge at an equivalently
wide angle after collision.
Obviously, direction of motion of the colliding balls is important to what
happens in such "glancing blow" collisions. Does the mass or the before-
collision velocity of the respective balls also play a part? To find out, we
can vary the circumstances of the collision as we did before, first making
ball A twice as massive as ball B, then making ball B approach collision
at twice the velocity of ball A. In such instances we would discover
perhaps to our surprise that alteration of the masses of the balls, or of
their initial velocities, or both, has no effect whatever on the angle they
bounce away from each other. This seems to depend solely on the angle
at which they approached each other. But we would see, once again, the
same apparent "exchange" of velocities of the balls in relation to their
masses and bef ore-collision velocities: the higher- velocity (or more mas
sive) ball would appear to transmit some of its velocity to the lower- velocity
(or less massive) ball.
Finally, consider one other situation. Suppose ball A, with mass identical
to ball B, is not moving at all. We simply place it at rest in the center of
the table and then start ball B rolling toward it along a straight line at a
constant velocity. Here again a head-on collision would occur, but we
would see a curious thing happen. At the instant of collision, the moving
ball B would stop dead, while ball A, formerly at rest, would bounce away
from the collision with precisely the same velocity as ball B had had before
the collision.
What has happened here? At first it might seem that only the stationary
ball, ball A, had any force acting upon it. It was just sitting there minding
o
o
v. -V
90 C
y*
90
V,
Ball A
o
Boll B
r~v i
30 C
Ball A
v^-
vr
o 30
<-,
Ball B
Q
v ~ 7
120 C
120 C
yi\
r\ i >
f
Ball A
BaliB
Pig. 9
112 The Universe of Classical Physics
its own business when ball B ran into it, so to speak, and started moving
only when the collision force of ball B was applied to it. But if we think
carefully, we see that ball B must also have had a force applied to it. After
all, before the collision it was moving, whereas after collision it was not.
Some force must have acted upon it in a direction opposite the direction
it was moving to bring it to a halt and that force could only have been
applied by the stationary ball A at the instant of collision.
Thus once again we see that a force and an equal but opposite counter-
force must both have been present at the moment of collision. Ball B
pushed stationary ball A with sufficient force to set ball A moving, while
ball A pushed moving ball B in the opposite direction with sufficient counter-
force to bring it to a stop. What is more, if the velocity that ball A "ac
quired" in the collision was the same as the velocity that ball B "lost,"
then the force and counterforce must have been equal in magnitude but
acting in opposite directions.
What would happen if the stationary ball A were twice as massive as
the moving ball B? We would expect the lighter ball B to have much less
effect on the twice-as-massive ball A than it would have if the balls had
the same masses, as long as ball B approached at the same velocity as in
the previous test. And our expectation would prove correct. As before,
moving ball B would come to a halt at the instant of collision, while sta
tionary ball A would bounce away, but this time ball A would have only
half the velocity after collision that ball B had before it. Does this mean that
ball B exerted only half as much force on ball A this time? Not at all;
it exerted exactly the same force as before, but that force had to act upon
twice as much mass in ball A as before, so that the change in ball A's
velocity could only be half as great.
In all of these imaginary situations, we have simply been watching New
ton's law of motion at work. Ruling out such confusing influences as fric-
tional and gravitational forces and rotation, or spin, which might have
affected the motion of our billiard balls, we have seen that the changes
in motion and direction of the balls occurred in a predictable and uniform
manner according to some sort of specific rules or principles. Indeed, in
each case the collision results seemed to depend directly upon the relation
ship between at least three easily distinguished factors: the respective
masses of the billiard balls, their respective velocities as they approached
collision point, and the respective directions in which the balls were moving.
Galileo and other early physicists came to realize that these three factors
mass, velocity, and direction of motion each played a vital role in
determining the behavior of all material objects known to them, whether
at rest or in motion. They realized that some outside force had to act on
an object if its state of rest or of motion was to be altered. They also
realized that the period of time during which a force acted upon an object
also played a part in determining how much change was to come about
Push and Push Back: The Riddle of Collisions 1 13
in the object's state of rest or motion, whether the time was a split second
(as in our billiard-ball collisions) or prolonged over many minutes (as
when we exert a force to push a piano across the room an inch at a time).
These early scientists groped for ways to express the relationship between
these factors in the form of broad universal principles which might serve
as good rules of thumb to help them understand all forms of motion
they might encounter in other words, as tentative laws of motion.
Newton succeeded better than any before him in piecing together the
observations that had been made, making shrewd guesses as to what these
working rules had to be, and then experimenting further to test the rules
as he worked them out. The laws of motion that he finally set forth seemed
to describe the motion behavior of all known objects, whether at rest or
in motion, and whether moving in isolation or interacting with other
objects.
It was a breathtaking scientific breakthrough to have discovered such
simple yet apparently universal natural laws as these. Newton's laws of
motion must have seemed like a haven in the storm for scientists of the
day a patch of solid, reliable ground appearing at last in the vast quag
mire of confusing observations, superstitious dogmas, and conflicting
theories that beset the early physicists. But in working out his laws of
motion, Newton made yet another discovery that was perhaps even more
staggering: He discovered that there were certain properties of matter or
energy that seemed never to change, ever, anywhere in the universe, no
matter what forces were brought to bear and no matter how objects might
move about or interact with each other. For reasons unknown, certain
baseline characteristics of the universe seemed always and invariably to
be conserved, unaltered since creation and perpetually inalterable.
THE VITAL CONCEPT OF CONSERVATION
It seemed that matter, for example, could neither be created nor de
stroyed by any force or interaction in the universe. Matter could be moved
about, altered in shape, even forced into chemical reaction with other
matter, but the total quantity of matter in the universe always remained
totally unchanged. Similarly, energy seemed always to be conserved: It
could be changed from one form to another, even transmitted from one
part of the universe to another, but none could ever be destroyed and
none freshly created. Even the discovery, later, that matter and energy
were really two manifestations of the same thing did not alter this rule of
conservation; the two separate conservation rules were simply combined
into one that was more all-inclusive.
The idea that the universe has certain unchanging and inalterable prop
erties may seem commonplace to us today, but in Newton's time it was
by no means self-evident. The concept evolved bit by bit over the centuries
H4 The Universe of Classical Physics
as suggestive experimental evidence began to accumulate. As it was, the
most familiar conservation laws were not the first to be affirmed. Long
before physicists were convinced that neither matter nor energy could ever
be created or destroyed, Newton's study of objects in motion began turning
up evidence that yet another very important universal property was always
conserved the property of moving objects which physicists call momentum.
We ordinarily use the word "momentum" very loosely as a vague refer
ence to the apparent ability of something to continue moving of its own
accord against some kind of resistance. We say, "The halfback plowed
through to the goal line on his own momentum," or "The train picked up
momentum as it raced down the grade," or even "Let's get the job done
before we lose our momentum."
To the physicist, however, "momentum" has a much more precise and
specific meaning. To understand what the physicist means when he says,
"The momentum of any closed system is always conserved," and to see
why such an odd-sounding and unfamiliar natural law should be so
important to our understanding of how things work in the universe, we
must return to our imaginary billiard ball experiments once again and
consider some things which we deliberately ignored before.
First, suppose we place two billiard balls with identical masses at one
end of a long pool table, and start them rolling simultaneously toward the
far end of the table by means of two shooters. Instead of using identical
impulses to push the balls, however, let us imagine that ball A is pushed
twice as hard as ball B, and thus rolls toward the end of the table with
twice the velocity of ball B. Which ball then has the greater force as they
roll toward the end of the table?
Instinctively we might feel that ball A must obviously have twice the
force of ball B but our instinct would be wrong. In point of fact, neither
ball would have any force at all as it rolls along (assuming "ideal" condi
tions of no friction and no gravity), until it collides with something. Then
and only then would either ball exert a force upon whatever it happened
to collide with.
Even so, we see ball A hustling down the table, increasing its lead over
ball B with every passing second. Surely ball A must have twice as much
of something. But what? If we could review our experiment in slow motion,
we would discover the answer. To begin with, each ball was at rest at one
end of the table. The velocity of each was zero. Then a shooter exerted a
force on each ball, and in each case that force was applied to each ball
for a brief but measurable period of time. In each case, this force-applied-
for-a-period-of-time had a similar effect: The instant the force was applied
to either ball, the ball's state of motion began to change in the direction
the force was applied. The shooter's force caused each ball to begin to
accelerate from a state of rest to a state of progressively swifter motion,
and the acceleration of each ball continued until the shooter stopped
Push and Push Back: The Riddle of Collisions
pushing it. The instant the accelerating force of the shooter stopped acting
on either ball, that ball stopped accelerating and thereafter continued to
roll down the table at precisely the velocity it had attained when the
accelerating force stopped acting on it f In the case of ball A, that velocity
was exactly twice the velocity ball B had attained, so that during each
subsequent second ball A continued to roll twice as far down the table as
ball B rolled during the same second.
At this point we must be wary or we will fall into a trap. Because ball
A attained twice the velocity of ball B, thanks to the action of its shooter,
we might be tempted to conclude that ball A's shooter exerted twice the
force that ball B's shooter exerted. But this does not necessarily follow.
As we have seen, the velocity ball A attained depended not only upon the
size or magnitude of the force its shooter exerted upon it, but also upon
the length of time that force continued to act. A gentle force acting on ball
A for two seconds would impart precisely the same velocity to the ball as
a force twice that magnitude acting for only one second. Thus we see that
the outstanding performance of ball A in our experiment compared to
ball B depended not on the force of either shooter alone, nor on the time
either shooter was acting alone, but on the force of the shooter multiplied
by the time it was acting. We have already spoken of this combination of
force times time as the impulse of the shooter on the ball, and we can
clearly see that the impulse that started ball A moving had to be twice
the magnitude of the impulse that was applied to ball B.
We can begin to glimpse what a mare's nest the early physicists found
themselves in when they tried to define what a "force" was, or specify what
effect a "force" might have on an object. They could see that a "force"
always had to be associated with a specific direction it would obviously
be impossible for a single force acting on an object to move it in opposite
directions at the same time, for example, or to move it in all directions
at once. They could also see that one "force" might be of greater size
or magnitude than another, but how could this be demonstrated? How
could they ever pin down what a "force" was when the effect of its action
on a given object invariably got all scrambled up with the length of time
that it acted?
Newton was one of the first to recognize that a "force" was really an
intangible and abstract concept somehow related to observed changes in
the velocity or direction of material objects, rather than a tangible "thing"
that one could isolate, measure, and describe. He also recognized that the
only possible way a given force could be measured or even be identified
as existing at all was in terms of some observable and measurable change
in the state of rest or motion of some object. Thus, the only way we could
describe or measure the force of the shooter that moved ball A (over a
period of time) from a state of rest to a state of motion (i.e., rolling at a
constant velocity down the pool table) would be to describe it in terms
n6 The Universe of Classical Physics
of what it managed to move the mass of ball A and the final velocity
that mass had achieved by the time the force stopped acting on it.
In other words, like the pessimistic fisherman who assumes that there
are no trout in the stream until one tugs on his line, we must assume that no
force is present at all until we observe some change occurring in some
object's state of rest or motion. When we do see such a change occurring,
we know some force of some magnitude is at work bringing about the
change. We can measure the length of time that force is acting on the
object by starting a stopwatch the instant we see some change in its motion
begin to occur (whether the change be acceleration, deceleration, or merely
a change in the direction of its motion) and then stopping the watch the
instant we see the change in the object's motion cease. Then since we
know that a force was acting and for how long, and since we know the
object's mass and can compare its velocity before and after the force acted
upon it, we can calculate the magnitude of the force that brought about
the change. We can also calculate the total resultant change in the object's
motion, as well as determining how fast the force caused the change to
take place the rate of change in the object's motion.
We have already seen these ideas in action on our imaginary pool table.
The impulse (i.e., the force times the length of time it acted) starting ball
A down the table was clearly twice as great as the impulse brought to bear
on ball B. Due to the double-strength impulse propelling it, ball A ended
up with twice the resultant velocity that ball B acquired. In our earlier ex
periments with colliding billiard balls we ignored the time factor involved
when two balls smacked together and bounced apart because the time
involved was so short that the collisions seemed to take but an instant.
Now we realize that the time involved, however short, is nevertheless an
important part of the picture if we are talking about measuring the forces
acting upon the balls during their collisions, or if we are trying to measure
comparative changes in the velocities of those billiard balls.
You will also recall from those billiard ball collisions that we recognized
that something was transmitted or exchanged from one ball to the other
and vice versa when they collided and bounced away, but we couldn't
quite pin down what, exactly, that something was. The reason for our
confusion was simply that we were trying to juggle too many interrelated
factors at once. Early physicists had the same problem, and tried to simplify
their concepts of what happened when forces acted upon objects to bring
about changes in their motion by combining certain closely related factors
and using special blanket terms to describe these combinations. We have
already seen that the concept of a force acting upon an object cannot be
separated from the length of time during which it acts, so the term impulse
was used to indicate "a given force multiplied by the length of time it
acts on an object" or more simple "force X time."
Similarly, physicists realized that an object with a given mass moving
Push and Push Back: The Riddle of Collisions 1 17
at a constant velocity possessed a certain formidable capacity as it moved
along its way. Although it had no "force" connected with it, it possessed
the capacity to bring a force to bear upon any other object it happened to
encounter or collide with. Just how much of such threatening .capacity
was possessed by any given object moving at any constant velocity de
pended on both the mass of the object and what its velocity happened
to be. An object of little mass moving at high constant velocity might
have the same "capacity to exert a force on something else" as an object
with much greater mass moving at a much lower constant velocity.
In either case, this "capacity to exert a force" would be altered the
instant the object in question actually collided with another object and
began, so to speak, to "put its capacity to work" by exerting a force on
the other object. It would, in fact, lose its "capacity to exert a force" by
actually exerting that force, while the other object that was struck would
gain the same amount of "capacity to exert a force" in the course of the
collision, and carry it off with it as it was pushed away by the collision
force, bouncing away in a changed direction with a changed constant
velocity. But, of course, that other object also had its own "capacity to
exert a force" which it lost to the first object by exerting a counterforce
during the collision at the same time it picked up the first object's "capacity
to exert a force." In short, in the course of the "push and push back"
of a collision between these two objects, each gave up its "capacity to exert
a force" to the other, and each moved away from collision with a new
direction and velocity dictated by the "capacity to exert a force" the other
ball had had prior to collision, and had brought to bear during the collision.
If we consider this closely, we notice four interesting things: First, in
the course of the collision, there was obviously an exchange of capacity
to exert a force between the colliding objects; what one object lost, the
other gained, and vice versa.
Second, the final or resultant velocity and direction of each object after
the collision was determined by the capacity to exert a force possessed
by the other object before the collision, and vice versa. If the capacity to
exert 'a force of the first object that is, its combined mass times velocity
was greater than that of the second object, the force actually exerted by
the first object upon the second in the collision would cause a greater
change in the second object's velocity and direction than the force exerted
by the second object could bring about in the resultant velocity and direc
tion of the first object. But if, as it appears, the little guy gets shoved around
by the big guy in this encounter, we must recognize that the little guy takes
on the big guy's "capacity" to shove the next guy around in the course of
some similar encounter later. If "might makes right," then there are some
equalizing compensations involved!
Third, in such a collision the end results depend not on the comparative
masses of the colliding objects alone, nor upon the velocities or directions
i !g The Universe of Classical Physics
of the colliding objects alone, but upon the combined mass times velocity
of each object. In any interaction it is this combined property of mass
times velocity of each interacting object in other words, the capacity
to exert a force of each object that makes the difference and determines
the results.
Finally, and more significant than anything else, the total combined
capacity to exert a force of both objects is precisely exchanged from one
object to the other and vice versa in the collision, but none of it is lost, nor
is any new capacity acquired. If we think of the two colliding objects as
a "closed system" if we imagine, for example, that they are the only
two objects existing in the whole universe, with nothing else whatever
capable of influencing their motion behavior in any way then we could
say that the total capacity to exert a force of any closed system is always
conserved or, if we preferred, in any closed system the capacity to exert
a force always remains unchanged: It can neither be diminished nor in
creased (i.e., created or destroyed).
If we find something strangely familiar about these statements, it is no
wonder. What we have reasoned through and expressed here is nothing
more nor less than a conservation law in this case, the law of conserva
tion of capacity to exert a force. We defined capacity to exert a force as
a property of any object, big or small, moving at a constant velocity any
where in the universe, and saw that that property was equal to the mass of
the object multiplied by its velocity. We also saw that that property was
not a tangible "thing" but rather a quality or capacity possessed by any
moving object which could only be "used" or "spent" if and when the
object came into interaction or collision with another object somewhere,
sometime. All the same, that shadowy capacity had very real meaning in
terms of what might happen in the universe. The baseball flying toward
the picture window has a very real and threatening quality about it, even
though it is perfectly harmless flying through the air: It has the capacity
to exert a force sufficiently great to smash the window if some other object
(such as our hand) is not interposed before the ball and window reach
collision point.
For simplicity physicists use a different term to describe the threatening
mass-times-velocity capacity of a moving object. What we have called
"capacity to exert a force" physicists call momentum-, and they define the
momentum of any moving object, just as we have, as the mass of the
object multiplied by its velocity. Thus, obviously, any object sitting at
rest (and thus possessing no velocity) has no momentum or more accu
rately, its momentum equals its mass multiplied by zero velocity, which
like anything else multiplied by zero equals zero. The moment a force
(such as the collision force of another billiard ball striking the first) begins
to act on the object, its "state of motion" (in this case its state of rest)
is changed. The object begins moving, its velocity increasing steadily from
Push and Push Back: The Riddle of Collisions ! I9
a state of rest (zero velocity) through a period of constantly increasing
motion (acceleration) as long as the collision force continues to act on it.
And as the object's velocity increases from velocity zero to velocity one
to velocity two to velocity three and so forth, instant by instant it acquires
momentum from the object colliding with it, and that acquired momentum
(the object's mass times its velocity at any given instant) steadily increases
from initial momentum zero to momentum one, to momentum two to
momentum three and so forth, instant by instant, as the collision force
continues to act upon it. This acquired momentum, of course, does not
appear like manna from heaven; it is acquired from the colliding object,
and that object's momentum is steadily given up throughout the collision
in precise balance with the momentum acquired by the other object. And
again, if the colliding objects together were considered a "closed system"
isolated from any outside force, there would be an exchange of momentum
between them, but no net gain or loss of momentum in the course of the
interaction. The total momentum of the system would be conserved.
From the above, we can see that the period of time the force acts is im
portant in a collision of objects, because it is during that time that the
momentum of each object is changing. The impulse of a collision (force
times the time the force acts) is the factor that determines the change in
momentum of either object in a collision, so that the magnitude of the
force acting on an object can at last be pinned down and identified as equal
to the change in momentum of the object per unit of time the force acts.
In short, the force acting on an object in any collision is equal to the rate
of change in the object's momentum.
Granted that this seems a long and tricky way around to identify the
magnitude of a given force acting on something but it is the only way
a force in action can be separated from the length of time it acts. Similarly,
the concept of momentum, its exchange between interacting objects and
its conservation within any closed system, is the only way we can really
describe what happens when one object pushes another. We can see now
that some of the ordinary, everyday uses of the word "momentum" are
more accurate than we realize. The line-rushing halfback does indeed have
"momentum" (mass times velocity) as he plunges through to the goal,
and the momentum he loses in the plunge is gained by opponents he
scatters in his wake. A train does indeed "pick up momentum" as it
rolls down the grade: With the same mass, its velocity increases, so
momentum is being acquired, in this case from the gravitational field which
is pulling the train down the grade. Later the train will lose momentum
back to the earth's gravitational field as it rolls up the grade on the other
side of the valley. But in the "closed system" composed of the train and
earth's gravitational field, momentum is merely exchanged between train
and gravitational field; it is never increased or decreased within the system.
Finally, we see how the concept of momentum explains our earlier bil-
I2O
The Universe of Classical Physics
liard ball experiments. Before, we spoke of the collisions as "almost instan
taneous." But in each case the collision forces lasted long enough for
momentum to be exchanged between the colliding balls. Before, we recog
nized that something seemed to be exchanged between the colliding billiard
balls, at least in some cases (as, for example, when one ball had twice the
mass of the other, or twice the velocity of the other) but we could not say
quite what it was that was exchanged. Now we see that it was momentum
that was exchanged between the colliding billiard balls, and that exchange
took place in every collision, not just certain kinds. In fact, the outcome
of the collision was determined in every case by the exchange of momentum
between the balls during the collision. But we only saw outward evidence
of an exchange in those cases where one ball had greater momentum (due
either to greater mass or greater velocity) than the other; when momentum
was equal on either side of the collision each ball acquired exactly the
same momentum it lost and no outward evidence of the exchange seemed
apparent except, of course, that the balls changed direction.
It would be foolish to pretend that physicists created anything new
when they began using the word "momentum" to mean "mass times veloc
ity of a given object." They merely gave a new name to a combination of
two already known measurable quantities. But new or not, the concept
of momentum proved extremely useful in trying to sort out how objects
in motion behave and what really happens when they collide or interact.
Since so much of early physics had to do with studying the characteristics
of matter and motion, the concept of momentum was a powerful tool.
But the discovery that momentum was always conserved in any "closed
system" interaction between objects that is, in any interaction unaffected
by any outside force was an enormously important gain for science. And
of all the natural laws worked out by Newton and other classical physicists,
the law of conservation of momentum remains one of the strongest even
today.
Most of us have heard of various "laws of conservation" off and on
since early grade school, but what exactly are "conservation laws"? Why
are they considered so important and so powerful? Actually, there is no
magic connected with them; the conservation laws are simple statements
that there are a few distinctive properties of matter and energy that never
change, no matter what happens. It is this simple fact and this alone that
makes them so important to anyone trying to unravel and explain all the
myriad peculiar things happening all around us in the universe. If nothing
else was clear to early scientists, it was clear that we lived in a universe of
constant, bewildering movement and change. The sun, moon, and stars
moved; earth's surface changed from moment to moment; forces acted
upon objects. Physicists attempting to describe in some sort of orderly,
sensible fashion just how things worked in this constantly changing universe
desperately needed something solid to hang onto, some firm ground that
Push and Push Back: The Riddle of Collisions I2 i
never moved. They needed a few things that were invariably stable and
unchanging to use as a baseline against which to measure other things.
The conservation laws provided such a baseline.
Understand again that these laws were not blindly accepted as gospel.
They were constantly being tested and challenged, and the more challenges
they survived, the more important they became. The law of conservation
of momentum very early became the most powerful and important of them
all, and remains so even today. For one reason, the law covers all sizes,
sorts, and varieties of moving objects known to Man, and all sorts and
varieties of forces acting on them, no matter what the force might be.
The law says that when two objects interact, whether they be galaxies or
subatomic elementary particles, their total combined momentum before
interaction will still be present after interaction. No matter what forces,
changes, upheavals, or holocausts the interaction itself may involve, nothing
occurring within the interacting system can ever alter the total momentum
present. Thus the law is especially powerful because it applies equally to
all the different worlds of physics; it crosses the boundaries between the
cosmic universe, our everyday world of experience, and the microworld
of nuclear physics, as valid in one world as another.
Of course physicists have tried to challenge the law of conservation of
momentum repeatedly and tirelessly throughout the years. They are still
trying today as new knowledge is gained, as new and puzzling phenomena
are observed and recorded. So far the law has survived every challenge;
no one has ever, even once, found a single exception to it. That is not to
say that no one ever will, but such a flawless record makes us wonder.
For now, at least, it remains one of the strongest, most fundamental, and
universal laws of nature ever discovered. In the next chapter we will see
more clearly just how important it really is, and why,
CHAPTER 9
Motion, Momentum, and Universal Gravitation
From the very beginning in this book it has been our contention that the
great laws of physics, however difficult they may be for nonscientists to
comprehend, nevertheless have very real and practical significance in our
everyday lives. If knowledge of these natural laws is not of practical use
to us, if they do not help us understand things that are happening all
around us in the world of our personal experience, then we would do well
to leave them to the scientists to understand. The fact that a revolutionary
concept proposed by Isaac Newton three centuries ago opened up sweep
ing vistas to the scientists of the day is not enough; as laymen and non-
scientists trying to grasp such a concept, it is perfectly reasonable and
proper that we should ask: So what? How does this affect my life today?
What use is it for me to understand this obscure and confusing idea?
We should not feel embarrassed, therefore, to ask just such questions
about the whole perplexing concept of conservation of momentum that
we have been discussing. So physicists recognize this natural law even
today as powerful and well established, unshaken so far by any challenge
so what? How does this law touch our lives? What use can we make
of it? What does it allow us to do that we couldn't do without it?
That the concept of momentum and its conservation is hard to grasp we
cannot argue. It is one of the most difficult ideas we will encounter any
where in this book so difficult, in fact, that many elementary textbooks
of physics side-step it entirely. Yet the fact remains that the law of con
servation of momentum (or certain of its consequences) touches our lives
continuously. Whether we are aware of it or not, we are using this natural
law constantly in our everyday encounters with the universe around us.
Specifically, we use it to help us predict what is going to happen next,
on the basis of what is happening now, and our predictions are so accurate
that we rarely indeed come up with the wrong answers. When we do, for
the most part, it is only because conditions are somehow unfamiliar or
unusual.
We live in a world of motion in which forces of all kinds are continually
acting upon material objects. We pick up a pen, kick a football, watch
leaves fluttering in the wind, see the moon rise and set; very few things in
122
Motion, Momentum, and Universal Gravitation 123
our lives remain motionless and unchanged for very long. The concept
of momentum tells us how to predict the future behavior of objects, or
groups of objects, at rest or in motion as a result of forces acting upon
them. The billiard player wins or loses according to his skill in predicting
precisely what will happen next to all the balls on the table if he applies
a certain force on a certain ball in a certain manner in a certain direc
tion. An automobile driver may live or die according to his ability to
predict with split-second accuracy just what will happen on a fast freeway
if he accelerates his car a certain amount in a certain .direction at a certain
time. Momentum concerns us all.
What is more, the concept of momentum simplifies our everyday calcula
tions immensely in a variety of ways. To take a single crucial example, it
shows us that we can accurately predict the over-all behavior of a whole
group of dissimilar objects as a result of the action of some outside force
upon that group, even if that group or "system" of objects is very large
and spread out over huge areas of space, with individual objects within
the group all moving in different directions at different speeds in the most
complicated fashion imaginable.
We might never be able to calculate what would happen individually to
each object in such a group as a result of the action of the outside force,
and we wouldn't need to. The concept of momentum shows us that we can
find the correct answer for the group as a whole by imagining that the
combined masses of all the objects in the group are concentrated in a single
imaginary point in space the "center of mass" of the group and then
calculating what would happen to that single imaginary massive point as a
result of the outside force acting on it. But how would we know our answer
was correct if we couldn't actually add up the varying behaviors of the
individual parts? We would know simply because it is a long-established
law of nature that the total momentum of any isolated group or "system"
that is, the combined mass times velocity of all individual "members" of
the system is always conserved, so that the system as a whole will always
behave as if all its mass were concentrated in a single point moving at a
single velocity and direction, and thus possesses a single unchanging re
sultant "group momentum."
CENTER OF MASS AND "AVERAGE BEHAVIOR"
The idea that any object (or any group or "system" of objects we choose
to name, no matter how large and diverse) might have all its mass con
centrated in some single imaginary point in space which then behaves as if
it were the whole object (or the whole group of objects) is important enough
to bear closer inspection. There is some faint aura of double talk about this
notion, some slippery quality that makes us draw back and say, "Now,
wait a minute. Is this really true?"
124 The Universe of Classical Physics
A couple of examples may convince us that it is.
First, suppose we have some awkward, lopsided object like a hickory
baseball bat to experiment with and want to determine its mass so that we
can measure what happens to it when it is thrown or dropped. Since we
know that an object's mass will equal its weight if the measuring is done
at sea level, we can easily discover the baseball bat's mass the exact
quantity of matter it contains simply by weighing it on a good spring
balance.
Now consider that it doesn't matter just how we go about weighing the
bat. Its mass would be precisely the same whether we laid it horizontally
across the pan of the balance, suspended it from the balance by a thread,
or somehow balanced it on end in order to weigh it. If we wanted to do
things the hard way we might balance it horizontally or vertically on the
point of a thumb tack, or even on the ultrafine point of a needle same
value for its mass in any case. Balanced horizontally on a needle point it
would look "off center" because one end of the bat is thicker and heavier
than the other, but we would still come up with a constant value for its
mass.
With the bat balanced that way on a needle point, however, we have a
singular situation. The only contact between the delicately balanced ball
bat and the weighing device is at a single tiny spot that point of the needle;
yet the spring balance registers the same weight as if the bat were lying
flat on the pan. Obviously the force of gravity is pulling the bat down on the
needle point precisely as if all the mass of the bat were concentrated at that
single point of contact and all the rest of the bat had no mass at all. At
the same time, the needle is pushing upward against the bat as if all the
bat's mass were concentrated on that single contact point. Clearly there is
some mysterious, dimensionless "point in space" somewhere within that
ball bat which acts as if it contains the bat's entire mass!
To the physicist, this imaginary point somewhere within the baseball
bat is known as its center of mass. Laymen are more familiar with the term
"center of gravity" since we are accustomed to measuring objects here on
earth within earth's gravitational field where the mass of an object is roughly
equal to its weight. But we can see that an object floating somewhere out
in space might be "weightless" in the absence of gravitational forces but
would still have the same mass there as anywhere else in the universe.
In other words, our baseball bat would always have the same center of
mass wherever it might be, whereas it might well have no center of gravity.
Furthermore, in any experiment involving the baseball bat, any place in
the universe, we could always treat the bat as if all of its mass were
contained within a single point.
Now suppose we are interested not in one object alone but in a closed
system of two or more objects taken together. For our purposes a "closed
system" might consist of any wild combination of two or more related ob-
Motion, Momentum, and Universal Gravitation 125
jects that we care to pick, as long as we agree to think of those objects taken
together as behaving as an independent and isolated group, cut off in
some way from the rest of the universe so that all other objects or forces
are by definition "outside" the system. We can imagine a "closed system"
as something akin to an ordinary artistic mobile hanging from the ceiling
by a thread; its sundry parts may twist and turn individually, but anything
that happens to it as a whole must be a result of some force acting from
outside it. Just as we cannot lift ourselves by our own shoelaces, a mobile
cannot jump loose from its mooring because of any chance concerted
action of the individual parts making it up. Yet the very fact that a mobile,
however large and complex it may be, can be suspended from a thread
attached at a single point suggests that such a closed system of objects
has a center of mass just as the baseball bat had.
The fact is that virtually any group of objects, no matter whether in
motion or at rest, no matter whether close together or scattered all over
the universe, can be considered a "closed" or "isolated" system so long
as we are willing to exclude everything else in the universe other than the
group's constituents as being "outside." And for any such closed system
of objects a center of mass exists for the system an imaginary point
in space which, if we could find it, would behave exactly as if it contained
all the mass of the system moving with a velocity equal to the resultant
combined velocities of each of its constituent parts.
Consider our solar system, for example. Here we have a confusing
collection of planets moving around the sun in their various orbits at
various distances and with varying speeds, many of them equipped with
their own satellites whirling around them in even more perplexing fashion.
At first glance this unruly collection of celestial rubble would seem to
show little evidence of cohesion as a closed system of objects. Yet sure
enough, the entire solar system has a center of mass located somewhere
near the surface of the sun, and no internal movement of the sundry
component planets, individually or in concert, can cause that center of
mass to move an inch. The fact that the solar system's center of mass is
moving through space, just as the solar system as a -whole is moving through
space, is a result of forces from outside the solar system acting on the
system as a whole (or on its center of mass) to change its velocity and
direction.
The same can be said for any other closed system, whether it be
an exploding bomb sending fragments out in all directions, a pair of
billiard balls colliding and bouncing away from each other, a collection
of oxygen and nitrogen molecules moving about at random inside a closed
container, or any other group of objects you care to mention. In any such
case, since we know that the momentum of the system remains unchanged
by anything going on within it, we can predict how the system as a whole
will behave by regarding the entire system as a single pinpoint-sized object
126 The Universe of Classical Physics
located at the center of mass of the system which behaves in accordance
with the average behavior of all of the system's component parts.
One important implication of the law of conservation of momentum is
that if no outside force is acting upon a closed system of objects, then there
can be no change in the velocity of the center of mass of that system. If
no outside force of any kind were acting upon our solar system, for
example, then all the motion of all the planets and their satellites, each
with its own individual momentum, could have no effect whatever on the
velocity of the solar system as a whole. Its center of gravity would remain
at rest if it were at rest, or would move at whatever constant velocity it
always had had. The net effect of all that planetary motion would be zero
no change. If the sun had four thousand massive planets moving about it
instead of nine or so, the net effect of all that motion would still be zero.
Nothing that happened -within the solar system could budge its center of
mass in the slightest. But if an outside force, however small, began to act
on the system, things would be different. The velocity of the system as a
whole (as represented by its center of mass) would immediately be affected,
changing its motion in proportion to the magnitude of the force and in the
direction the force was acting.
Of course we know that this is precisely what is happening to our solar
system. It is not in a state of rest, but is constantly moving as a whole
in an orbit around the center of the galaxy. And as we will soon see, the
outside force that moves our solar system is the resultant of two conflicting
forces: a gravitational force tending to drag the solar system in toward
the center of the galaxy, and an opposing centrifugal force tending to
drive the solar system away from the galactic center on a straight-line
trajectory.
If we look at this idea more closely, we will see that this implication of
the law of conservation of momentum is nothing more than a restatement,
in slightly different terms, of Newton's first law of motion: Any object
(or the center of mass of any closed system of objects) which is at rest
will remain at rest unless acted upon by some resultant outside force; and
any object (or the center of mass of any closed system of objects) which is
in motion will remain in motion in a straight line at a constant (i.e.,
unchanging) velocity unless acted upon by some resultant outside force.
The fact that we substitute the center of mass of a closed system of objects
for a single object, or substitute the average behavior of the constituents
of a closed system for the constant velocity of a single object doesn't alter
the law in the least. If anything, it strengthens and reaffirms the law,
showing us that it extends to the behavior (as a whole) of closed systems
of objects as well as to the behavior of individual objects at rest or in
motion.
Thus we can see that the law of conservation of momentum is very
closely related to the laws of motion, and together with them helps us
Motion, Momentum, and Universal Gravitation 127
understand what to expect when forces act upon objects or groups of
objects around us. It provides a powerful tool for predicting whether or not
a given object will remain at rest under given circumstances, or in what
manner its velocity and direction will change in response to a given force.
It buries forever the common-sense idea that a continuing force is needed
to keep a moving object moving, yet at the same time it forces us to
recognize that any change in an object's velocity or direction must be the
result of the action of some outside force, whether we are aware of the
force or not.
It was such a simple realization as this, so hard come by after years of
experiment and observation by the classical physicists, that enabled Newton
to recognize at last that the force of gravity which pulled objects to the
ground when they were released from his hand might be the same force
that kept the planets moving in their orbits, and to extend Galileo's limited
concept of gravity as a purely local phenomenon into the great law of
universal gravitation which Newton finally defined.
To understand how the one group of concepts led to the other in those
wonderful days in history, however, we must first fill in a few important
bits of background that are still missing. Up to now we have gotten away
with using certain key terms without defining them too fastidiously. Now
we must consider certain fine shades of meaning which we previously
ignored. In particular we need to understand precisely what we are saying
when we use such terms as speed, velocity, direction, and acceleration.
THE SIGNIFICANCE OF DIRECTION
We have already seen how a word like "momentum" can have a broad,
general meaning in common usage, yet mean something far more precise
and specific to the physicist. Oddly enough, there are also certain pairs
of closely related terms which the layman may use interchangeably, as
if they were synonyms, but which have very distinct individual meanings
to the physicist, and still other terms which do not necessarily mean what
we assume they mean at all.
Take "speed" and "velocity," for example. Ordinarily we use these
words interchangeably to describe how fast something is moving. We
assume that "a speed of sixty miles per hour" means precisely the same
as "a velocity of sixty miles per hour" and meanwhile, the physicist
cringes. To him the words have distinctly different meanings.
But when should we use "speed" and when "velocity"? The difference
is all a matter of direction.
"Speed" is properly used as an abstract description of a state of motion,
whether swift or dawdling, high or low. Speed is measured in terms of
distance traveled per unit of time: A car may move at a speed of sixty
miles per hour, a snail at a speed of two millimeters per second. A star's
I2 g The Universe of Classical Physics
light rushes away from its source at a speed of 186,000 miles per second,
which we must agree is a pretty high speed. But even though it is concerned
with the description of an object's motion, the term speed tells us nothing
whatever about the direction in which the motion is occurring.
The term "velocity" does, and therein lies the difference. Velocity is
properly used as a specific and complete description of an object's motion,
telling us not only its speed but also the direction it is moving. In fact, the
term velocity is defined as speed in a specified direction. When the physicist
speaks of an object's speed he is describing the size or magnitude of its
rate of motion in the abstract, without reference to anything else in the
universe. When he speaks of the object's velocity, he is coming down to
earth, so to speak: he is describing the magnitude of its rate of motion with
specific directional reference to something else, whether it be to the ground,
to the azimuth, to himself, to another observer, or whatnot. And this
directional reference to something else always implies speed in a stated
direction.
Thus velocity is measured in terms of distance traveled per unit of time
in a given (or understood) direction with reference to something else. A car
moves with a velocity of sixty miles per hour north (with reference to the
ground); part of the sun's light moves with a velocity of 186,000 miles
per second toward the earth (with -reference to the sun) ; a Saturn V rocket
must achieve a velocity of seven miles per second away from earth's center
in order to "escape" from earth's gravitational field and carry astronauts
to the moon.
This seemingly quibbling distinction between certain quantities which
include a directional element and others which do not is actually so
important in exact, descriptive sciences such as physics and mathematics
that scientists use special terms to distinguish them. Quantities which have
no directional element are called "scalar quantities" because they can be
fully described by a number indicating magnitude alone, or represented
by a point on a scale. Speed is one such nondirectional scalar quantity; so is
time, which has no direction (at least not in terms of our three familiar
linear dimensions). The mass of an object is likewise a scalar quantity,
for its magnitude does not depend on the direction the object is moving;
in fact, an object's mass remains the same even if the object isn't moving.
On the other hand, quantities which do include an inseparable directional
component are called "vector quantities," and are always described both
by a number indicating magnitude and by a specific direction, like a
signpost saying "10,000 miles to Nowhere Much," with an arrow attached.
Velocity is one such vector quantity; so is momentum since it is defined
as the mass of an object times its velocity. So also is acceleration, defined
as the rate of change in an object's velocity (i.e., the total change in an
object's velocity divided by the interval of time in which the change
occurred). After all, it would be impossible to describe an object's acceler-
Motion, Momentum, and Universal Gravitation 129
ation without specifying in which direction the acceleration took place.
For clarity we might emphasize the distinction between scalar quantities
and vector quantities in a simple table (Table 2) :
TABLE 2
Scalar Quantities Vector Quantities
Speed: Velocity:
60 miles per hour (no direction) 60 miles per hour, thataway
Mass: Momentum:
3,000 kilograms (no direction) 3,000 kilograms x 60 miles per hour,
thataway
Time: Acceleration:
10 seconds (no direction) 6 miles per hour, per second, thataway
Clearly there is a distinction, then, between scalar quantities and vector
quantities. But why all the fuss about it? To the nonscientist the difference
may well seem pointless; after all, it is perfectly true that the need to
distinguish between speed and velocity rarely occurs in our daily life.
Normally we can see the direction most objects are moving with reference
to ourselves or to the ground and see little need to specify. Even though we
almost always mean "velocity" when we speak of "speed/' nobody gets
confused. But the distinction becomes very important in physics when
we recall that the most fundamental natural laws describing the motion
of objects the laws of motion all include very careful reference to the
direction an object is moving. They tell us that any object in motion will
remain in motion at a constant velocity in a straight line (that is, in the
direction it is already going) unless acted upon by some outside force.
When an outside force does act on an object, its velocity will change in the
direction the force is acting. And for every action of a force on an object
there is always an equal reaction in the opposite direction.
We have already seen examples of these principles in action. We have
also seen that any time a force acts on an object steadily over a period of
time and thus produces a steady or uniform change in the object's velocity,
that change in velocity per unit of time that is, the rate of change in
velocity is spoken of as "acceleration."
Here we have a case of a word which means something more than we
may think. Ordinarily we think of acceleration only in terms of an increase
in an object's velocity per unit time. To the physicist, however, acceleration
means any kind of change in an object's velocity per unit of time. If a
force acts on a billiard ball to increase the "speed" aspect of the ball's
velocity, the resulting acceleration is called "positive acceleration." But
if some force acts to slow down the "speed" part of the ball's velocity,
the rate of change is still called acceleration in this case, "negative
acceleration" or more colloquially, "deceleration."
130 The Universe of Classical Physics
The notion that a force might cause an object to "accelerate to a stop"
seems a little ridiculous at first, but this is only because we normally
ignore the full meaning of the term "accelerate." The fact is that objects are
"accelerating to a stop" all the time. A billiard ball does this when it
collides with another and then bounces away in the opposite direction.
An automobile does the same thing when we step on the brake; so also
does a rock which we toss into the air. But if the idea of negative accel
eration seems a bit odd, there is still another form of acceleration that
is even more peculiar.
Remember that an object's velocity is its speed in a given direction.
Remember also that when an object's velocity is changed by the action of
a force, its acceleration is a measure of its change in velocity per unit of
time that the force is acting. But since velocity is a vector quantity with
a "speed" part inseparable from a "direction" part, a force can cause a
change in an object's velocity merely by altering its direction of motion
slightly by pushing the object off course, so to speak without either
increasing or decreasing its speed in the slightest. A strong crosswind, for
example, acting on a sailboat could change the boat's velocity from 10
knots due north to 10 knots northeast. The "speed" part of the boat's
velocity would remain the same; only the "direction" part is changed
but this action of the crosswind would still result in a true change in the
boat's velocity! And by the same token, the amount of change in the
boat's velocity per unit of time that the crosswind is acting is a true accel
eration of the boat even though the speed of the boat is unchanged.
But what can we call this kind of acceleration? Obviously it cannot be
either positive or negative acceleration since the boat is neither speeding
up nor slowing down. A new word is needed to describe such "sideways
acceleration" or acceleration arising solely from change in a moving object's
direction. The term generally used in physics is "angular acceleration":
the acceleration of an object along a curving path as a result of a uniform
and continuing change in its direction, describable and measurable in terms
of an angle of a circle.
Once again we see that direction of motion plays a critical role any
time we attempt to describe the behavior of moving objects as a result of
forces acting upon them. And the concept of angular acceleration imme
diately draws our attention to a form of motion we have barely considered
so far. The laws of motion and the law of conservation of momentum
very nicely enable us to describe and predict the behavior of objects as long
as they are moving in straight lines, but what about the multitudes of
objects in the universe which normally move in curves, parabolas, ellipses,
or circles? Must we find a whole new set of natural laws to describe such
motion? Fortunately not, for we shall see that the old laws apply perfectly
well with certain minor but significant modifications.
Motion, Momentum, and Universal Gravitation 131
ANGULAR VELOCITY AND CENTRIFUGAL FORCE
Earlier we employed "ideal" billiard balls on an imaginary "ideal"
pool table to see how objects moving in straight lines behave in accordance
with the laws of motion when acted upon by various forces. We also saw
that when our ideal billiard balls collided, momentum might be exchanged
between the colliding pair but the total momentum of the "closed system"
of two colliding balls was always conserved, the momentum lost by one
was gained by the other, and vice versa. Indeed, the laws of motion seemed
to suggest that the motion of such objects had to be in a straight line.
But suppose now that we alter our billiard ball experiment a bit. Suppose
once again that we have one target ball resting motionless in the center
of the table call it ball A and then use our shooting device to start
ball B, equal in mass to ball A, rolling toward it on a collision course. But
suppose that this time we have anchored ball A to the table by attaching a
six-inch bit of thread to it and tacking the other end of the thread firmly
to the table top. Now what happens when ball B smacks into ball A?
Certainly not the same thing that happened the first time we tried this
experiment! Earlier, when neither ball was attached to the table in any
way, you will recall that moving ball B came to a halt when it collided
with stationary ball A, while ball A was sent rolling away from ball B by
the force of the collision. In other words, the two balls exchanged momen
tum in the collision; the originally stationary and momentumless ball A
picked up all of ball B's momentum while ball B became stationary and
momentumless. But this time something rather different happens. Once
again moving ball B collides with stationary and momentumless ball A.
Once again momentum seems to be exchanged in the course of the
collision, for Ball B comes to a halt, while ball A starts moving but this
time, not in a straight line. This time, like a dog on a leash, ball A moves
away from collision along a circular path with the tacked-down end of the
thread as a pivot point and the six-inch length of the thread as the radius
of the circle (see Fig. 10).
But what has happened to Newton's first law of motion? It claims that
an object in motion will remain in motion in a straight line unless acted
upon by some outside force, doesn't it? Indeed it does. But here, ball A,
once set in motion, is continuing in motion along a circular path. How can
this be? The collision seemed identical with the one before. The billiard
balls are the same ones we used before. Why are the results of the col
lision different this time?
The answer, of course, is right before our eyes in Newton's first law.
Ball A set in motion by the collision would remain in motion in a straight
line unless acted upon by some outside force. The fact that we see ball A
moving in a circle rather than a straight line indicates that there must be
1^2 The Universe of Classical Physics
some outside force making it do so, whether we recognize the farce when
we see it or not. But what could the force be? There is only one way such
a force could be acting: through the thread attaching the ball to a fixed
point on the table six inches away!
If we look at this collision through our slow-motion lens and examine
the forces acting on ball A instant by instant from the time of collision
on, we see clearly what is happening. The instant before collision ball B
is approaching collision point with momentum equal to the ball's mass
times its velocity. Ball A at that instant has no force acting upon it; it is
at rest, with a velocity of zero and hence a momentum equal to zero.
Tack ,
BallA
!
6m
Ball!
Before Collision
After Collision
Fig. 10 When Ball B, attached to table by a thread and tack, is struck by Ball
A, straight-line velocity VI is transformed into angular velocity V2. When
Ball B completes its circular "orbit" and strikes Ball A, angular velocity V2 is
re-transformed into straight-line velocity VI.
Then in the course of the collision ball B exerts a force on ball A setting
it in motion, while ball A exerts a counterforce on ball B bringing it to rest.
Ball B has lost its velocity (and hence its momentum) while ball A has
gained that lost velocity in a straight line in the direction ball B had been
moving; it has picked up ball B's lost momentum. So far everything is the
same as in the previous experiment. But this time the moment ball A starts
to move off in a straight line with its newly acquired momentum, a new
force begins acting through the thread to pull it off course. That new force
in fact begins tugging ball A in toward the pivot point at the same time that
Motion, Momentum, and Universal Gravitation
133
ball A's inertia keeps tugging it away from the pivot point along the
straight-line tangent that the ball would follow if the in-pulling force
were not acting upon it.
The result? A compromise. Ball A at each instant moves in the only di
rection permitted by the two opposing forces acting upon it. This means
that at each instant it is pulled off course exactly as much as it is tending
to move back on course. Neither the off-course force nor the on-course
force can overcome the other, so the ball follows a resultant circular path,
constantly pulled in by the thread and pulled out by its own momentum
(see Fig. 11 a). Assuming, as we did before, conditions of no friction and no
Fig. 11 Fl represents the inertial
force acting on Ball A; F2 is the
centripetal (in-pulling) force ex
erted through the thread; Rl is the
resultant path of motion of the ball.
If thread breaks, centripetal force
can no longer act, so Ball A then
follows course determined by iner
tial force Fl.
gravity, we could say that the circling ball is "caught in orbit" around the
pivot point, and would continue circling the pivot point forever unless
some other outside force acted upon it, or unless one or the other of the
two opposing forces acting upon it suddenly quits acting.
Of course the latter case is not impossible. Suppose that the thread
tugging the ball in toward the pivot point were a slightly frayed thread,
and suddenly broke under the continuing strain of tugging at this stubborn
ball. The instant that the thread breaks, the in-pulling force can no longer
act on the ball. So what will it do then? Obviously, it will then have only
one force acting upon it the collision force which gave it straight-line
velocity and momentum. Thus, if the thread snaps, the ball will continue
moving in a straight line out at a tangent to the pivot point from the instant
that the in-pulling force ceases (see Fig. lib).
134 The Universe of Classical Physics
But barring such an "accidental" occurrence, we have ball A trapped in
a circular path of motion by the effect of two forces acting upon it from
the time of collision onward. What can we say about its velocity (which
must include an element of direction), or about its momentum gained from
ball B at time of collision? What happens to our principle of conserva
tion of momentum, which insists that the momentum lost by ball B must
equal the momentum gained by ball A? Earlier we saw both velocity and
momentum as straight-line qualities, always associated with direction of
an object's motion in a straight line. What can we say about these things in
the present case in which ball A's motion follows a circular path?
First, if we could measure carefully, we would find that the speed part
of ball A's velocity after collision is constant, just as the former speed
part of ball B's velocity was constant before collision. Ball A's speed in its
circular orbit is neither increasing nor decreasing. The direction part of its
velocity, however, is changing constantly, instant by instant, as it is forced
to move off course. Thus we must say that ball A is accelerating constantly
around a pivot point at a distance of six inches even though its speed
neither increases nor decreases. It undergoes a constant angular accelera
tion as it moves. But what can we call the odd kind of velocity ball A has
acquired a velocity in which the direction keeps uniformly changing
from instant to instant? Because this is a special kind of velocity, physicists
give it a special name and call it "angular velocity."
We can see that the angular velocity of ball A is different from straight-
line velocity because the direction part of its velocity is uniformly changing.
But what controls the speed part of its angular velocity? The collision force
of ball B originally set ball A in motion in the first place, and contributed
a straight-line element to its resulting angular velocity "in orbit," but the
inward tugging of the thread must also contribute something. And with
further experiment, we would find that it does. If the thread holding ball A
in orbit were shortened to three inches half its original length we would
find the speed part of the ball's angular velocity to be twice what it was
before; but if we increased the thread to twelve inches twice its original
length the speed part of ball A's angular velocity would be only one-half
what it was originally.
In other words, wherever angular velocity is concerned, the speed ele
ment is inversely proportional to the distance between the moving object
and the pivot point, i.e., to the radius of the circle. When the radius is
made smaller, the object speeds up proportionally, and when it is made
large, the object slows down proportionally. But distance from object to
pivot point is not the only factor influencing the speed part of the object's
angular velocity; the object's mass also plays a part. If we substituted a ball
with twice the mass for ball A, its speed in orbit would be only half that
of ball A, whereas if we used a ball only half as massive as ball A, it would
travel in orbit at double the speed achieved by ball A.
Motion, Momentum, and Universal Gravitation 135
If this all seems to sound vaguely familiar we ought not to be surprised,
for we are talking about a relationship between an object's mass and its
velocity (in this case, its angular velocity) which sounds a great deal like
our old friend momentum. In discussing straight-line motion, we saw that
the momentum of a moving object was equivalent to its mass times its
(straight-line) velocity. We also saw that in our present billiard ball experi
ment, moving ball B had straight-line momentum at the instant before col
liding with the motionless but "captive" ball A, and gave up that momentum
during the collision. Ball A at the same time took on something very much
akin to the momentum lost by ball B but not quite the same because the
newly acquired velocity of the captive ball A was angular velocity. Thus
ball A acquired a "different kind" of momentum equivalent to its mass times
its angular velocity which was greater or smaller according to the distance
of the ball from the pivot point.
Just as the "captive" ball A acquired a "different kind" of velocity from
ball B in the collision, a velocity described as "angular velocity," it also
acquired ball B's momentum transformed into a different kind. This dif
ferent kind of momentum is also given a special name, "angular mo
mentum," since its velocity factor is angular velocity. In either case the
difference arises from the motion of the object in a circular path rather than
along a straight line due to the continuing action of an inward-pulling force
upon the object. Thus, if we define an object's angular momentum as its
mass times its angular velocity, we would then find that all of Newton's
laws of motion apply to an object moving in a circular path around a
pivot point exactly as if the object were moving freely in a straight line.
Furthermore, we would find that angular momentum is conserved in any
closed system of objects moving with angular velocity, just as straight-line
momentum is conserved and that straight-line momentum can be trans
lated into angular momentum and vice versa any time an object moving in
a straight line interacts with an object capable only of angular or rotational
motion. In fact, if we followed out the results of our imaginary collision
between straight-moving ball B and fettered ball A on our friction-free and
gravity-free pool table, we would see an amazing thing: Ball B colliding
with ball A would set it into angular motion in a circle around its pivot
point while the impact would bring ball B to a halt. But with all ball B's
momentum translated into angular momentum of ball A, ball A would
swing completely around its circle to smack in turn into the back side of
ball B. In this impact ball A's angular momentum would be completely
given up to ball B, translated back into straight-line momentum, so that
ball A would halt and ball B would continue on its interrupted straight-
line course in the same direction it had been traveling before its first col
lision, and at the same velocity it had originally!
Clearly, t momentum and angular momentum are completely equivalent
except that the latter includes an additional "rotational movement" factor
136 The Universe of Classical Physics
related to the distance between object and pivot point the radius of the
circle. In a case in which an object has angular velocity, and thus angular
momentum, the "mass" part of the momentum constitutes an inertial
force urging the object out on a straight-line course; and indeed, the physi
cist speaks of the mass of an object in rotational motion as its "moment of
inertia." The force applied to the rotationally moving object through the
thread that holds it to the pivot point is an off-center force acting on the
object, not changing its speed but pulling it constantly off its straight-line
course; such a force is called a "torque" (i.e., a twisting force pushing the
object off balance and thus into rotational motion). And the laws of mo
tion and conservation of momentum apply equally to straight-line motion
and rotational motion, as long as all the factors in rotational motion include
the radius-of-a-circle factor and an angle factor in place of a distance
factor.
ANGULAR MOTION AND NATURAL LAW
The whole concept of rotational movement of objects, and the notion
that angular momentum had to be analogous, in some way, to straight-line
momentum, had kept the early physicists puzzled and confused. To New
ton, however, the problem was not "How are the two different?" but rather
"How are they the same? What are the common denominators?" Above all,
Newton was convinced that the universe was orderly and that such natural
laws as existed were few, simple, and broadly applicable. Motion, he must
have reasoned, was motion, and he could not have believed for a moment
that one set of rules would apply to one kind of motion and another differ
ent set of rules to another kind. When he saw the close analogy that
seemed to exist between rotational or angular motion on the one hand and
straight-line motion on the other, and when he saw that rotational motion
seemed to be no more than a variation of straight-line motion which oc
curred when a second force in addition to straight-line inertial force was
acting on a body, he recognized that his laws of motion must apply equally
well to either form of motion so long as a "rotational motion factor," so to
speak, was taken into consideration in the case of rotational motion.
Furthermore, this idea seemed to be confirmed from observations in
nature, for it was obvious to all physicists in those days that angular mo
tion was in fact present in nature at least as commonly as straight-line mo
tion. Indeed, it had become clear, in the centuries before Newton, that
angular motion was the rule rather than the exception, at least so far as the
relative motion of the sun, the moon, and the planets was concerned. For
almost a thousand years men had believed firmly that the sun, the moon,
and all the planets moved in some kind of complex circular fashion with
the earth at the center of them all. In 1543, after a lifetime of study, Nico-
laus Copernicus had finally overthrown this deeply entrenched and hallowed
Motion, Momentum, and Universal Gravitation 137
idea: He had demonstrated that the observed facts simply did not fit the
theory. He contended, rather, that although the moon did indeed revolve
in circles around the earth, the earth and the other known planets turned in
even more huge circular orbits with the sun as their "pivot point."
Copernicus was so shaken by the enormity of this idea, and so aware
that the full prestige of Aristotle and the full earthly power of the Church
supported the old earth-centered theories of Ptolemy that he waited until
the year of his death to publish his own contradictory findings. Even then
they were by no means universally accepted by contemporary scientists.
But the real strength of his observations lay in the simple fact that anyone
with a good telescope could repeat them and come up with the same
answers he had come up with, and his theories set such great astronomers
as Tycho Brahe in Denmark to work studying the heavens again.
After Brahe's death in 1601, his student Johannes Kepler pieced to
gether the huge volume of observations Brahe had made, and confirmed
once and for all that the planets followed curving paths of motion around
the sun. Their rotation was not in a perfect circle, however, as was the case
with our "captive" billiard ball; instead, Kepler showed that the earth and
the other planets moved in an oval or elliptical path around the sun, with
the sun not at the center but occupying the position of one of two foci of
each such ellipse.
These observations had all become known during Newton's lifetime
or even earlier; he was perfectly aware of them when he was working out
his universal laws of motion, first published in 1687. He realized that such
massive objects as the earth or the other planets ought to be moving in
straight lines and indeed would have to be unless some hitherto unrecog
nized force were acting to "pull them off course." The observed fact that
they were moving not in straight lines but in great curving orbits was not a
matter of divine whim, but the result of a force that was continuously tug
ging planets away from the straight-line courses they would otherwise be fol
lowing. We saw the same thing, on a small scale, with our captive billiard
ball. It did not follow a circular path because nature had suddenly changed
her mind about straight-line motion, but because straight-line motion was
rendered impossible for that billiard ball by a force acting on it through
the thread to pull it in toward the thumbtack pivot point. Similarly, New
ton saw that by all rights earth's moon ought to be flying out into space
away from the earth along a straight line, and the fact that it obviously
was not doing so could mean only one thing: that some force was at work
continuously tugging the moon in toward the earth, against its own better
judgment, so to speak.
At the same time, Newton thought about other examples of angular
motion which were observable closer to home. It was known that a projectile
such as a rock hurled in a horizontal direction began falling toward the
earth the moment it was released from the hand. Its straight-line momentum
138 The Universe of Classical Physics
kept it moving horizontally, all right, but some other force simultaneously
pulled it off course toward the ground. Thus the rock's trajectory was never
a straight-line path, but rather a curving line representing the resultant
instant-by-instant compromise between two opposing forces. Obviously, the
"other force" that opposed the rock's straight-line inertial force by tugging
it down toward the ground was nothing more than Galileo's old, familiar
force of gravity. If the rock were hurled a second time with greater force
than before (and thus with greater straight-line momentum and greater
straight-line velocity) it would be able to travel considerably farther before
gravity finally brought it to earth, but it would still begin to fall off course
toward the ground the instant it was released. Its trajectory would be a
longer, flatter curve than before, but gravity again would win.
But now suppose we took that rock to a high mountain top, and launched
it horizontally with such an exceedingly high velocity that the curving
surface of the earth fell away from it just as fast as gravitational force
pulled it downward. Imagine at the same time that there was no atmos
phere, so that the rock was unopposed by any air resistance. What would
happen then? When would the rock finally reach the ground?
The answer was clear to Newton: It wouldn't. In such a case, that rock
would continue to travel around and around the earth, always seeking to
follow the straight-line path away from earth's surface dictated by its
inertial force, yet always tugged earthward by a second force, gravity. Be
cause of its very high velocity, that rock would never fall downward far
enough nor fast enough to strike the earth's surface, even though it would
continually be falling toward the center of the earth just as any tossed
rock falls toward the center of the earth. Thus the rock's direction of motion
would continually be changing so that it would continually be accelerating
downward without any change of speed perpetually accelerating toward
the center of the earth.
Newton realized that the continued acceleration of a rock under these
imaginary circumstances could be nothing other than acceleration due to
gravity. If the rock were slower moving, with less straight-line momentum,
gravitational force would presently tug it down to earth. If it were faster
moving, with greater momentum, it would pull farther and farther away
from earth's center and presently "escape" on a straight-line course out into
space. But if the rock's outward inertial force exactly counterbalanced
gravity's tugging, that rock would continue to circle the earth in orbit, over
and over again, indefinitely.
Very much, indeed, the way the moon circles the earth in orbit, over
and over again, indefinitely . . . !
It is easy now to see where Newton's reasoning was taking him. It was
obvious that some unrecognized force was acting across empty space to
hold the moon on a leash, so to speak, forcing it to travel in perpetual orbit
around the earth instead of flying off on a straight-line tangent into space.
Motion, Momentum, and Universal Gravitation 139
Could it be possible that that "unrecognized force" was the force of gravity
pulling the moon continually earthward even as it struggled continually to
move out and away from earth? Could it be possible that earth's gravita
tional force was not just a local phenomenon! occurring on the surface of
the earth, as Galileo had imagined, but a force that could act upon objects
across such enormous expanses of space as the distance from earth to
the moon?
Newton saw that it had to be possible, no matter how incredible it
seemed. The moon's motion in orbit followed the laws of rotational motion
perfectly, so long as a factor for earth's gravitational pull was always
taken into account. But Newton also realized that far more was afoot than
just this, if his own laws of motion were really valid. Gravitational force
could not be an exclusive property of the earth alone, for if the earth were
exerting a gravitational force on the moon, influencing its path of motion,
the moon must simultaneously be exerting an equal but opposite gravita
tional force on the earth!
The idea must have seemed preposterous, but Newton was too immacu
late a scientist to accept a law of nature when it proved convenient and
disregard it when it became clumsy. He may even have seen proof that
the moon had gravity tugging away at the earth in the waxing and waning
of the tides: whenever an ocean was facing the moon, the tides were low,
as though the whole mass of the ocean's water were drawn up into a
bulge by some powerful moonward force; a few hours later, when the
ocean was turned away from the moon that force was relaxed, allowing the
tides to rise at the ocean's edges. But if the moon also had gravitational
forces at work, why didn't the earth revolve around the moon instead of
vice versa? Obviously because the mass of the earth was so much greater
than that of the moon that the moon's gravity could not overcome earth's
momentum as readily as earth's gravity could overcome the moon's.
The idea was awkward, but it fitted in with other observations. The
earth perhaps held the moon on a controlling gravitational leash but in
the case of earth's motion around the sun, it was earth that seemed to be
on a leash, moving around the sun in a perpetual elliptical orbit so huge
that it took the earth a full year to make one circuit. This could only mean
that the sun also possessed a gravitational force a force so powerful it
could reach out to affect the motion of planets millions of miles away from
it. But like our captive billiard ball moving in a faster circle when the
thread was short and more slowly when the thread was long, the sun's
effect on the planets seemed related to the distance they were away. The
inner planets Mercury and Venus circle the sun at higher angular velocity
than the earth does, while Mars's angular velocity is lower, and far Jupiter's
velocity in orbit is ponderously slow.
Bit by bit, Newton fitted the pieces together. Gravity was not an ex
clusive property of earth, nor was it a one-way street, but a mutual force
140 The Universe of Classical Physics
of attraction existing between any two celestial bodies, a force precisely the
same in nature as the familiar "local" gravitational force existing between
the proverbial apple over Newton's head and the earth below him. But as a
final stroke of genius, Newton went one step further: He realized that
gravitational attraction existed not just between the sun, the moon, and the
planets, but that it was a universal property of all objects a force of at
traction which exists, quite literally, between any two objects anywhere in
the universe, no matter how large or small they may be, no matter how
close together or far apart, no matter where they are located nor how fast
or slow they are moving.
But the magnitude of the gravitational force between any two objects,
Newton realized, is not necessarily the same as between two other objects;
it varies from one situation to the next. By applying his laws of motion to
what was known of the orbital behavior of the sun, the moon, and the
planets, he searched for some consistent principle or relationship by means
of which anyone could calculate the gravitational force between any two
objects anywhere. He found, for one thing, that the magnitude of the force
depends upon the respective masses of the objects: Massive objects at
tract each other more strongly than less massive objects. Again, the dis
tance between the objects makes a difference, regardless of how massive
they are: The farther apart they are, the less they attract each other. In
any case, the force of gravity is always exerted in a straight line between
the objects in question or between their centers of mass and in any case
a fixed, unchanging number, a "universal gravitational constant" is needed
in order to compare the magnitude of the force between one pair of objects
with the magnitude of the force between any other pair of objects, no
matter how dissimilar they may be. It was this gravitational constant that
Newton calculated which permitted him to set the masses of any two ob
jects, the distance between them, and the gravitational force between them
in the proper proportion in effect, to provide a "yardstick of perspective"
for comparing the gravitational force between two grains of sand located
200,000 miles apart, for example, and two earth-sized planets located
200,000 miles apart. Without such a constant the magnitudes of these two
gravitational forces could not be compared in terms of actual values. It
might be possible to say that the force present in one case would be larger
or smaller than in the other case, but not how much larger or smaller.
These calculations were laborious, but as in so many cases it was the
basic idea, not the mathematical details, that threw open the door. The
idea that every object in the universe attracted every other object in the
universe was breathtaking in its sweeping implications; the task of determin
ing how much attractive force existed between any two given objects on
the basis of their respective masses and the distance between them was more
of a mopping-up exercise. Newton finally worked out a famous formula
Motion, Momentum, and Universal Gravitation 141
relating the masses of any two given objects, the distance between their
centers of mass and the magnitude of gravitational force that existed be
tween them in simple mathematical terms that could be applied universally
to any two objects one might choose. In its simplest form, this formula
states that the gravitational force F that exists between any two objects
anywhere (call them Mass 1 and Mass 2) always increases in direct pro
portion to the product of their two masses (Mass 1 times Mass 2) and
always decreases in direct proportion to the distance between their two
centers multiplied by itself. This unvarying proportional relationship can
be expressed for any two objects, using the symbol oc to indicate propor
tionality, as follows:
Mass 1 X Mass 2 ^ (Ml x M2)
F oc or
R X R R*
but to obtain specific values for the gravitational force F existing between
any two specific objects, it is necessary to use Newton's universal gravita
tional constant G as a proportionality constant in an algebraic equation:
Most of us have encountered this familiar "inverse square" law of
universal gravitational attraction between objects at one time or another.
Once the gravitational constant G, always the same in all cases, had been
measured, this equation made it possible to calculate the amount or magni
tude of gravitational force that existed between any two objects in the
universe. Essentially, the law of universal gravitation as expressed above
means simply this:
1 . That an attractive force exists between any two objects in the universe;
2. That in any given case the magnitude of the force depends directly
upon the product of the two objects' masses according to a specific rule,
so that the larger the product of the two masses, the greater the gravita
tional attraction between them;
3. That in any given case the magnitude of the gravitational force be
tween two objects also depends directly upon the distance between the two
masses according to a specific rule, so that the force decreases by a factor
of the square of the distance. In short, the force between two objects is
greatest when they are the closest together, and falls off in geometric pro
gression as the objects move apart;
4. That in any given case the force exerted upon one object by the other
is equal in magnitude to the force exerted by the other object upon the
first, and this mutual force of attraction is always exerted along a straight
line between the centers of mass of the two objects.
142 The Universe of Classical Physics
This law of universal gravitation was a revolutionary concept when
Newton first expressed it, not merely because it enabled physicists to
measure gravitational forces existing between the planets and thus predict
the effect those forces would have on planetary motion, but even more be
cause it demonstrated that the force of gravity, once thought to be a strictly
local phenomenon affecting only the motion of objects on or near the
earth's surface, was in fact a universal force that existed between any two
objects in the universe to one degree or another. What was more, Newton's
law of gravitation explained clearly why gravitational forces seemed to
be active in some cases and absent in others. If the gravitational force of the
sun were great enough and could extend far enough into space to pull the
earth away from its normal straight-line course and force it to move in a
curving "captive" orbit around the sun, why weren't all sorts of smaller
objects on the earth's surface pulled together by their mutual gravitational
attraction? The answer was that the masses of such small objects were
so small that the gravitational attraction between them was unmeasurable,
far too small to produce enough force to overcome the inertia of the ob
jects and make them move toward each other.
But was it possible, then, to prove that any force really existed between
two objects of comparatively small mass? Indeed it was; in his laboratory in
England in the year 1797-98, almost a century after Newton published his
theory of universal gravitation, Henry Cavendish set up an experiment to
demonstrate whether gravitational attraction between two objects could ac
tually be measured in the laboratory or not. For his test objects he used two
heavy iron dumbbells each balanced and suspended from a thin wire in a
glass container which excluded any air currents which might cause the dumb
bells to turn. After the dumbbells had become perfectly still, Cavendish gently
moved them closer together, and found that at a certain distance apart the
metal bells could be seen to attract each other, and the amount of attraction
could actually be measured by the movement of a needle-pointer attached
to each wire. Not a force of great magnitude, this attraction, but a force
just the same a measurable force that could be nothing but the attraction
of each dumbbell for the other. And to top it off, Cavendish found that
the magnitude of the attractive force he measured was almost precisely the
magnitude predicted for dumbbells of this mass held at this distance apart,
according to Newton's equation!
Gravitational attraction between objects was real, beyond doubt, and
Newton's equation for calculating the magnitude of the attractive force in
any given case was valid, but the force of attraction between small objects
is so very tiny that it has no effect on the motion of the objects. If one of
the objects is very massive as massive as the earth, for example and
one very small only as massive as an apple, say each will attract the
other with equal gravitational force, but the effect of the earth's gravitational
Motion, Momentum, and Universal Gravitation !43
force upon the apple's motion is far greater than the effect of the apple's
gravitational force on the earth's motion, so that the apple is pulled toward
the earth far more readily than the earth is pulled toward the apple.
Similarly, the motion of the earth is affected more readily by the gravita
tional attraction of the far more massive sun than the sun's motion is
affected by earth's gravitational pull. But here distance too plays its part.
Earth's gravitational effect on the motion of its nearby moon is greater
than the more distant sun's effect on the moon's motion, so that the
moon is held in orbit around the earth. The distance between earth
and moon is so much less than the distance between moon and sun that
the earth-moon gravitational force is much greater than the moon-sun
gravitational force and thus the moon orbits the earth and not the sun.
If by some disaster, however, the earth were suddenly to vanish com
pletely from space and thus suddenly cease to exert its gravitational at
traction upon the moon, the moon would then shift its allegiance to the
sun, responding to the sun's gravitational pull. Indeed, the orbits of all the
planets and their various satellites are as they are solely as a result of
the interplay of conflicting gravitational forces acting upon them. Not only
does the sun exert a powerful gravitational attraction on the earth; each of
the planets, large or small, exerts its own gravitational attraction on each of
the other planets (and vice versa) in a continuing cosmic tug-of-war.
In fact, sometimes the gravitational effect of one planet upon another
can actually be observed to shift or alter a planet's "normal" orbit around
the sun, at least temporarily. We know, for example, that the planet
Neptune is "perturbed" in its usual orbit around the sun from time to time
because of the conflicting gravitational attraction of its neighboring planet
Uranus at such times as both planets happen to be on the same side of the
sun at the same time and approach comparatively near to each other in
their orbits. We also know that the orbit of Uranus is similarly "per
turbed" by Jupiter's gravitational attraction under similar circumstances.
As a matter of fact, it was the observation of a mysterious "perturba
tion" or temporary alteration in the normal orbit of Uranus around
the sun that first convinced astronomers of the 1700s that another still-
undiscovered planet must be present somewhere in space out beyond
Uranus. From time to time the nice, orderly orbit of Uranus seemed to go
temporarily out of whack for no apparent reason; astronomers reasoned that
only a strong gravitational force between Uranus and some other massive
planetary body passing comparatively nearby could possibly throw a planet
the size of Uranus even temporarily off course. By measuring the amount
that Uranus was bounced around by this shadowy stranger in space, as
tronomers were able to predict not only that the planet Neptune had to be
out there, but could even make surprisingly accurate predictions of the
shape of its orbit, how far beyond Uranus it lay, how long it took to make
144 The Universe of Classical Physics
one turn around the sun, and how massive it was. And when the planet
was at least spotted in the telescope and studied, virtually all these earlier
predictions proved to be correct!
The final cataloguing of Newton's laws of motion, the twin concepts of
conservation of momentum and conservation of angular momentum, and
the discovery that gravity was a universal force that kept the planets in their
orbits and existed between any two objects anywhere in the universe were
among the greatest achievements of a Golden Age of classical physics. In
the foregoing chapters we have discussed at some length how these discov
eries were made and how these concepts grew, because taken together they
formed a coherent picture of how things work in the world of our every
day experience, the only world of physics those early scientists could ex
amine. It may have seemed that Sir Isaac Newton received the lion's share
of the credit for expressing these great, basic concepts, but Newton himself
once wrote, "If I have seen further than other men, it is by standing upon
the shoulders of giants." And indeed, these concepts arose not from the
work of any one human mind but from a great intellectual ferment that
had been evolving for centuries, beginning with Galileo, embracing the
lifework of such giants as Copernicus, Tycho Brahe, Paracelsus, and
Johannes Kepler, and was reaching full flower in the time of Newton.
Out of this ferment came a few concepts, a few simple but sweeping
working rules that seemed to describe a great deal that was known to be
happening in the universe, and that seemed always to be verified whenever
they were put to test. These laws of classical physics strongly confirmed
the ancient assumption that everything happened in the universe in an
orderly fashion. They predicted the end results of the motion and interac
tion of objects and the effects of forces acting upon objects, no matter
whether here on earth or in the far reaches of space, no matter whether the
objects or the forces were large or small. Because they seemed so universal
in application, they were exceptionally useful laws, for they explained what
was happening and allowed men to calculate what would happen next.
There was nothing about these classical laws of nature to suggest why
they existed. In working out the law of universal gravitation, Newton made
no attempt to explain why massive objects attracted each other, nor what
caused the force of gravity to exist in the first place. The law merely de
scribed what happened. It stated under what circumstances gravitational
forces existed, and what effects these forces had upon the motion of objects
whether on earth or in the farthest reaches of the heavens. And when it
came to helping physicists understand the world they saw around them,
these laws seemed powerful indeed. Indeed, they were so apparently uni
versal, and seemed to describe so many natural phenomena that physicists
of a hundred years ago began to feel that they were approaching the end
of the road to discovery. The ancient dream of discovering a few simple,
Motion, Momentum, and Universal Gravitation 145
interrelated principles which together would describe everything that hap
pened in the universe seemed at least to be coming within reach. Only a
few knots still remained to be unraveled.
Or so it seemed. As we will see, it was not to be quite that simple; those
complacent physicists were heading for a rude awakening, their dream soon
to be shattered. But before we consider the strange discoveries that shattered
that dream, we must take time to look more closely at the nature of the
physical matter that composes the universe as we see it, and discuss a
strange and puzzling manifestation of matter which has only been clearly
recognized as such in the last hundred years or so a form of matter that we
know more familiarly as "energy."
CHAPTER 10
The Forms and Shapes of Matter
In the previous few chapters we have been discussing in some detail quite
a wide variety of abstract ideas and esoteric concepts. We have been con
cerning ourselves with frictional forces, air resistance, and other forces
difficult to pin down precisely. We have been considering the behavior
of imaginary objects existing under imaginary conditions in the course of
imaginary experiments. We have also been dealing with that most mysteri
ous and peculiar force of them all, the force of gravitation, capable of reach
ing its shadowy fingers across millions of miles of space in order to juggle
planets around in their orbits, yet so weak that its presence between two
small objects in an earthly laboratory can be measured only with difficulty,
if at all* We have encountered a curious quantity known as "momentum"
which can, it seems, be transferred froin one object to another under cer
tain circumstances, and even be translated from one form to another, but
can never be created or destroyed. We have explored the historical devel
opment of certain deceptively simple natural laws, and have seen some
surprising implications of these laws as they were understood and inter
preted by physicists in the seventeenth, eighteenth, and nineteenth cen
turies.
Many of these concepts, admittedly, have been difficult to grasp, and
even more difficult for us to relate to the "real" world of earth, sky, and
sea in which we live the world of our own experience. And in pursuing
these concepts we have, again admittedly, by-passed some very practical
and fundamental aspects of the universe that would help us greatly in mak
ing sense out of the more esoteric concepts. It may be with a sense of relief,
therefore, that we now come down to earth for a while and consider one
of the most basic and commonplace of all facts that we must cope with
every day: the fact that the physical universe we live in is made of
material objects composed of a very solid and tangible stuff known as
matter, and that it is material objects of this sort which we constantly see
acted upon, altered, or influenced by the wide variety of forces we have
been discussing.
Earlier we had a great deal to say about various kinds of objects at rest
or objects in motion. For the most part we selected objects that were
146
The Forms and Shapes of Matter 147
familiar to us all in one form or another. But we neglected to pause and
consider the "matter" of which those objects were composed. It is im
portant now that we do so before we move on, for the matter making up
the physical universe exists in a bewildering variety of shapes and forms,
and possesses some surprising and extraordinary characteristics which we
need to know about.
Of course in our everyday lives we rarely give much thought to the nature
of material objects all around us. Most of us feel relatively confident deal
ing with chairs, books, drinking water, or sledge hammers. We are natively
aware that matter exists in various forms and have learned to use the vary
ing characteristics of different forms of matter to our own best advantage.
When we sit down on a chair, we expect it to support us, whether comfort
ably or uncomfortably, and when we spill a cup of coffee on the rug we
expect it to make quite a mess and be difficult to recover intact in its
original form. Practically speaking, we know pretty well how far we can go
with any given object on the basis of the form of matter that composes it:
we use steel girders rather than wooden beams to support our bridges,
because we know that steel has a degree of strength that wood lacks; we
would not normally try to wash with a dry bar of soap any more than we
would try to dry our hands with sandpaper; and we can predict with some
accuracy that the man who tries breathing water instead of air is likely to
find himself in trouble.
All such "common-sense" considerations are a natural and normal
part of our everyday working knowledge of the universe around us. We
don't have to be told these things. But in addition, we are also aware that
much that happens in the universe around us depends very heavily upon
certain particular characteristics of matter in one form or another, and that
the effect that a given force may have upon an object may vary widely
according to what form the matter making up that object happens to be in.
The thirsty man on the desert will draw small comfort from a canteen full
of water vapor, although the month before he was bemoaning the flash
flood that swept away his house. The wings of a small airplane are specially
built to provide the lift necessary to get the plane off the ground in regions
where earth's atmosphere is the most dense; the wings on the new super
sonic transports must be quite remarkably different if those planes are to
fly at great speeds and high altitudes, while wings of any sort on the lunar
excursion modules designed for soft landings on the moon would be to
tally useless. All three of these craft are designed to move men from one
place in the universe to another in response to various kinds of driving
forces, but it is the form of matter through which they must move that
determines the details of their structure.
But if we feel reasonably confident of the predictable nature of the mun
dane material objects around us, we must bear in mind that not everyone
has been as confident as we are either in past ages or even in modem times.
148 The Universe of Classical Physics
The ancient Greek philosophers quite seriously questioned whether matter
existed at all, except as the creation of their own imaginations. With the
evolution of experimental science (and particularly of experimental physics)
this particular notion seemed pointless to pursue and was generally forgot
ten. Scientists came to assume as a basic working axiom that matter was
indeed real and did indeed exist quite independently of the influence of the
human mind. Yet, strangely enough, in modern times physicists have come
full circle and once again are questioning seriously whether matter as such
has any real physical existence that can ever be specifically defined, and if
so, they are wondering precisely what in the universe actually is "matter"
as such and what isn't. For practical purposes, of course, we must go along
with the assumption of the classical scientist that matter is real and does
exist, but we will be able to understand the uncertainty of the modern
physicist, and the truly remarkable discoveries he is making because of
such concerns, far better if we pause briefly to review the shapes and forms
of matter as it was regarded by the physicists of Newton's time and later.
THE ANCIENT ELEMENTS
Since time immemorial men must have been worrying about the shape
or form in which substances in the natural world around them were to be
found, and about the elementary substances they felt must in one com
bination or another go to make up all those substances. Even the prehuman
ancestors of man were apparently aware that some substances were more
practically useful than others. When those ancient creatures first took up
hand weapons to help them in the hunt, they seem to have found the long
bone from the foreleg of certain antelope to be a more satisfactory club
than objects made either of wood or of stone. And even the later cavemen
must have been aware on a very practical level that various natural sub
stances were more useful in one form than in another, and that some
substances, under certain circumstances, could be altered or changed into
quite different substances while others could not. A fine dry log thrown into
a fire would flare up brightly, then shrink, crack, snap, turn glowing red,
and ultimately change first to a charred black lump and then to a powdery
ash; yet the stones on which the fire was built remained unchanged. They
learned that water poured on a fire would vanish into a cloud of hot white
vapor, but that in the midst of this process the fire would go out. Then there
was the fire itself, a mysterious and unpredictable thing, sometimes flaring
up in long orange flames, sometimes glowing red, sometimes merely giving
off heat. Certainly fire was a strange and fearsome thing to those ancient
men, capable of warming them and frightening off predators on cold nights,
yet equally capable of crippling or killing them if they got too close to it
and who was to say what fire was?
Of course no one knows how much those early men actually thought
The Forms and Shapes of Matter 149
about such things, except that various primitive societies worshiped fire as
a deity. Later societies in Egypt and Babylon merely used it as a controllable
tool and were clearly aware of the practical utility of a wide variety of sub
stances which could be shaped and changed in multitudes of ways to make
them more useful. Metals such as copper, tin, lead, or gold could be smelted
out of various kinds of rocks. Weapons and utensils could be fashioned,
first out of rock or clay and then out of metal. Wood could be carved and
shaped to man's will, and animal hides could be converted into durable
leather.
By the time of the ascendancy of the Greeks in the ancient world, men
had come a long way from the caveman's vague musings about the nature
of matter. Nor were all of the ancient Greeks hung up in sterile debate
about whether matter was real or not. By Aristotle's time three basic ideas
had become well established about the nature of matter. All three were to
guide men's thoughts for millenniums, and all three retain a certain degree
of validity even to this day. First, there was the conviction that all the
material substance in the universe was composed of various quantities and
combinations of a relatively few basic "elementary substances." Second,
there was the notion that these elementary substances were discontinuous
by their very nature; that is, that although they might be divided and sub
divided into extremely tiny particles, eventually one would reach a final
"indivisible unit" of each basic substance which could no longer be sub
divided. And finally, there was the fundamental idea that these elementary
substances existed in a certain fixed quantity in the universe which could
neither be increased by any means nor destroyed in any way.
Of course the elementary substances as described by the ancient Greeks
were far different from the "elements" as we understand them today. The
Greeks believed that all material objects in existence could be accounted
for in terms of combinations of four basic "elements": earth, air, water,
and fire. They regarded mud or clay as a combination of earth and water,
for example, while the substance issuing forth from a sulphurous hot spring
was composed mainly of water and fire with a little earth thrown in. Fur
thermore, they argued that various special circumstances could either
separate the component "elements" of a substance fairly readily or else
change one "element" into another.
For example, the Greeks believed that when a log was burned it gave
up its "fire" component (you could see the flames rising up) as well as its
"air" component (you could see the smoke) until nothing remained but the
log's "earth" component (the pile of charred ashes) which could no longer
be changed by further burning.
Later, as metallurgy developed and as alchemists began heating, dis
solving, or mixing substances in hopes of transforming them into precious
metals, it became harder and harder to explain observed changes in matter
purely on the basis of the four basic elements of the Greeks. Even so,
150 The Universe of Classical Physics
the idea persisted for centuries, as did so many other philosophical ideas
of the Greeks. The idea of the four basic elements became related and
entangled with the ancient medical notion that health or illness was de
termined by a balance or imbalance of four "humors" within the body, and
it was not until well into the Renaissance that these ideas were finally
reluctantly discarded. Yet even today we see a curious and prophetic
symbolism in the Greek's idea of four basic elements. Certainly three of
the four could be seen to represent the three forms of matter most familiar
to us today solid, liquid, and gas and in modern times we might even
regard the fourth Greek element, fire, as symbolic of energy, now con
sidered by physicists as merely another manifestation of matter.
The second idea of the Greeks, that matter could not be endlessly sub
divided into infinitely smaller and smaller portions, but was basically com
posed of tiny but finite indivisible particles, proved even more significant.
This was a purely intuitive idea, unsupported by a single scrap of experi
mental evidence; yet it appeared repeatedly throughout the centuries as
scientists again and again tackled the problem of defining the exact nature
of these tiniest of elemental particles. On repeated occasions in history
physicists and chemists thought they had at last found the answer, but as
we shall see, the mystery of the exact nature of the elementary particles
composing matter is even more perplexing to modern physicists than it was
to the alchemists of a thousand years ago.
Finally, the Greeks became convinced that while matter might be changed
from one form to another, there was no way that any part of it could ever
be destroyed, nor could even the tiniest fragment of matter be created which
did not already exist. Once again this was an intuitive idea; the Greeks
simply assumed that all the matter the universe ever had contained or ever
would contain was already there in one form or another, and that the total
quantity could never be altered. Of all these ancient ideas, this one above
all seemed vulnerable to experimental challenge. It was by no means a
self-evident truth; indeed, it often seemed to be contradicted by casual
observation of things appearing in nature. Over the centuries, as the sci
entific method of investigation began to evolve, scientists made repeated
attempts to prove that matter could be created from nothing, or that it
could be destroyed under certain circumstances but these attempts in
variably came to grief. Even today continuing attempts in modern laborato
ries continue to fail, although modern-day understanding of the inseparable
interrelationship between matter and energy has forced certain modifications
or extensions of the ancient concept.
In the days of Isaac Newton, however, and even up to the middle of the
last century, no one had even begun to suspect any intimate relationship
between matter as such and energy as such. Up to that time physicists had
largely been concerned with the universe as they could observe and study it
directly the universe of sensory experience and nothing in that world had
The Forms and Shapes of Matter I ^ I
yet appeared to suggest that any such relationship between matter and
energy existed. As far as classical physicists were concerned, all the matter
in the universe was believed to exist either in a solid or a fluid state, with
the fluid state further subdivided either into liquid form or gaseous form.
Each of these forms or states of matter was familiar and observable. Each
had certain specific characteristics all its own. In terms of our everyday
world the individual characteristics of these three major states of matter are
as critically important to us today as they were to the early scientist. It
will be worthwhile to see what common characteristics these states or forms
of matter share, and to see in what ways they are distinctly different from
each other and why.
FORM AND SUBSTANCE: THE SOLID STATE
We need only a glance around us to confirm that the most substantial
part of the world around us is composed of matter in the solid state. Solid
objects are so commonplace to our experience that we might even be
tempted to assume that most of the universe is made up of matter in the
solid state. In this assumption we would be wrong, but there can be no
doubt that from the standpoint of functional utility matter in the solid state
is by far the most useful to man and provides him with the most durable
and permanent of the artifacts he needs for his everyday life.
This is a direct result of certain common characteristics of matter in
the solid state. Solids, for example, tend to be rigid and hold their shape
under considerable stress and strain. Whatever the components of a solid bit
of matter may be, those components seem to be locked into a durable forma
tion and resist being shoved out of that formation. Although certain kinds
of solids which we know as metals can be hammered or bent or pulled
from one shape to another, solids in general cannot be compressed very
much nor stretched very much. They tend to retain their shape without
external support and resist (to a greater or lesser degree) the action of forces
to break them into fragments or to twist them out of shape.
Furthermore, solids characteristically possess a quality known as elas
ticity, a tendency, when acted upon by some distorting force, to spring
back to their original shape as soon as the force is removed. We ordinarily
think of elasticity in terms of some object such as a rubber band, which
quite obviously can be greatly distorted or compressed, and which has an
equally obvious powerful tendency to pull itself back into shape when the
force is removed. But the behavior of a rubber band is actually an extreme
and dramatic example of the quality of elasticity possessed to a much lesser
degree by innumerable other solid objects which we would not ordinarily
think possessed it at all. A solid ivory billiard ball, for example, would
hardly impress us with its elasticity; yet it demonstrates surprising elastic
qualities. When it is struck by another ivory billiard ball, both balls
152 The Universe of Classical Physics
momentarily "give a little" or flatten at their point of contact, then im
mediately spring back to their original shape as they bounce apart. Even
such a brittle solid object as a sheet of plate glass will bend and buckle
slightly in a gust of high wind and then spring back to its original shape
as soon as the wind declines.
But just as solids possess varying degrees of elasticity, they also possess
varying limits to their elasticity. A brittle object like a plate-glass window
has a very narrow limit of elasticity; let a gust of wind distort it just a whis
ker beyond its limit and it will shatter. Even a highly elastic object such as
a rubber band will break if stretched too far, and once broken cannot then
be restored merely by pushing the broken ends together.
We know today that the relative brittleness, malleability, rigidity, or elas
ticity of a solid object is directly related to the manner in which its com
ponent molecules and atoms are locked together. In many solids these
components are bound tightly into a rigid crystalline structure which per
mits very little distortion. The internal atomic structure of a quartz crystal,
for example, is such that if it is struck with a hammer it will shatter
into smaller and smaller fragments of quartz, even down to a very fine
powder; yet each fragment will retain the same basic crystalline structure
that it possessed as a part of the original chunk. Metals too possess char
acteristic crystalline structures, but in many metals the interatomic bonds
holding the crystals together are comparatively weak, so that metals such as
gold or silver can be hammered and shaped quite readily without shattering.
In the case of a highly elastic substance such as rubber, the component
molecules are made up of extremely long serpentine chains of atoms only
loosely bonded one to another so that the atoms and molecules can be
stretched apart, yet bound firmly enough that they tend to pull each other
back together once the distorting force relaxes. Still other solid substances
such as phosphorus, certain forms of sulphur, caramelized sugar, beeswax
and glass have no distinguishing internal crystalline structure at all and are
called amorphous substances from the Latin term meaning quite literally
"without shape." Some such substances, in fact, come so close to an in
distinct no-man's land between matter in the solid state and matter in the
liquid state that we would be hard put to specify whether they should be
classified as very soft and formless solids or very dense and viscous liquids.
Solids as a group, however, do generally exhibit some degree of crystalline
rigidity; their component atoms and molecules tend to remain locked to
gether in fixed formations relative to each other, capable of some very
limited localized motion or vibration but incapable of moving freely out
of the vicinity of their immediate neighbors. This characteristic is a result
of powerful binding forces between the atoms which hold them relatively
close one to another with insufficient atomic or molecular motion to over
come these bonds. As a consequence, matter in the solid state is generally
The Forms and Shapes of Matter 153
more dense than matter in the liquid state (with a few notable exceptions
such as mercury).
But in addition to these more or less familiar characteristics, solids have
certain other characteristics which we might not ordinarily consider. Al
though they retain their shape, they also tend to expand in volume when
heated, and to contract in volume when cooled. This expansion or contrac
tion of solids as a result of temperature variations is ordinarily so slight
that we fail to notice it, but it is a very real occurrence just the same. It
would be perfectly possible, for example, to use a narrow iron rod as the
temperature indicator hi a thermometer if the instrument were equipped
with a microscopic temperature scale to register changes in length of the
rod with changes of temperature, and if we had a microscope with which
to take readings. And, in fact, the property of metals to expand and con
tract with temperature variations is actually utilized in the temperature-
regulating mechanism of many delicate thermostats in which an expanding
strip of metal bends sufficiently to break contact and shut off a furnace
when room temperature reaches a given level, and then bends back and
reestablishes contact to turn the furnace on again when temperature of the
strip has dropped to a certain point. For the same reason, railroad tracks
are laid with end-to-end gaps between strips of rail at regular intervals
so that the tracks will not buckle in the heat of the summer sun.
Furthermore, solids characteristically melt into liquid form if heated
sufficiently, providing that some other chemical or physical change does not
take place first at a temperature lower than the melting point of the solid.
As we know, it is easy to melt ice or paraffin in a saucepan on the stove
with the use of comparatively little heat. Glass or even iron will melt at
considerably higher temperatures, but if we try to melt a stick of wood,
for example, the cellulose of which it is composed will break down and
separate into water vapor, carbon dioxide gas, and carbon before it will
melt, and the carbon itself will combine with oxygen from the air and be
dissipated before its melting point can be reached.
Among solids which can be melted into liquid form by heating, we notice
a curious thing: With the singular exception of one solid frozen water
or ice the solid form of a substance tends to be more compact, more dense,
than the same substance in its liquid form. The reason again is related to
the internal molecular structure of the substance; when a solid has been
heated sufficiently for its molecules to break free of the intermolecular
binding forces that held them rigid, the distance between the individual
molecules is increased so that fewer molecules are present in a given volume
of the liquid than in the same volume of the solid.
The very close relationship between the internal atomic structure of
various solid materials and such physical properties as hardness or soft
ness, elasticity or brittleness, is especially well illustrated in certain solid
!54 The Universe of Classical Physics
substances which can exist in two or more solid states with quite different
physical characteristics. There would seem to be little relationship between
the crystalline hardness of diamonds, the powderiness of charcoal, and the
slippery "greasiness" of graphite; yet all three are not only examples of
matter in the solid state, but are actually different forms of the same pure
element. In diamonds, the carbon atoms are locked together in an excep
tionally rigid crystalline formation; in charcoal powder the pattern of the
carbon atoms is far less dense, while in graphite the carbon atoms are held
together loosely in flat sheets which slide readily over each other. The
element phosphorus may occur as a soft yellowish-white material with a
waxy consistency which can burst into flame spontaneously when exposed to
air, or in the form of a dry, dull-red powder which is stable in air and
ignites only at a much higher temperature. Metallic sodium, one of the most
violently chemically active of all the elements, is as soft as butter and also
burns spontaneously when exposed to air; yet in chemical combination
with other substances forms exceptionally stable compounds such as sodium
chloride., sodium carbonate, or sodium sulphate, most of which exhibit very
distinctive crystalline characteristics.
Finally, solids in general are found to lack certain identifying character
istics of liquids or gases. In general, solids do not tend to evaporate as
liquids do, nor to diffuse from one area of space to another, as gases do.
Furthermore, solids in general are not good solvents. It is possible to mix
a quantity of one solid such as flour with a quantity of another such as sand,
but we do not end up with a solution of flour in sand or a solution of sand
in flour. Two solids mixed together in such a way may interreact chemically
with each other to form a new substance which is different from either
ingredient, but unless a chemical interreaction occurs the particles of two
solids mixed together remain intact, with particles of one lying side by side
and intermixed with particles of the other. In the case of the flour-and-sand
mixture, it would be perfectly possible physically to separate the mixture
again with the aid of a good enough microscope and delicate enough tools,
picking out the sand grains and putting them in one pile and the particles
of flour in another, to end up with precisely the same quantity of flour
and sand unmixed which had originally gone into the mixture.
It may seem that we have been hedging a bit in this discussion of the
characteristics of solid materials by using such terms as "in general," "as a
group," and "usually." And, in fact, we are hedging a bit, but for a good
reason. For one thing, the myriad solid substances with which we are
familiar exhibit such an enormous variety of different identifying char
acteristics that it is difficult if not impossible to find any single identifying
characteristic common to all solids. The best we can do is to enumerate
a few general characteristics commonly shared by most solids, while freely
admitting that some exceptions exist in the case of each particular char
acteristic. Second, as we have seen, there are many substances which seem
The Forms and Shapes of Matter 1 55
to fall in a gray area which lies somewhere in between matter in a solid state
and matter in a liquid state. Graphite, for example, has certain liquidlike
characteristics of slipperiness or greasiness, while taffy appears for all the
world like a solid substance until you start chewing it.
Finally, and perhaps most baffling of all, there are a number of solid
substances, with more and more appearing in recent years, which are not
quite solids, or at least do not behave quite the way other solids behave.
Among these substances are those which, while "solid" enough, possess a
characteristic known as plasticity a capacity to be molded, tugged, or
pulled into a permanently altered shape by the action of comparatively
gentle forces acting steadily over a period of time. A stick of sealing wax
which would shatter into fragments under the sudden force of a hammer
blow will bend into a 90-degree angle if one clamps one end hi a vise and
leaves a weight hanging on the other end overnight. A rod of glass can be
molded with gentle pressure when heated red-hot in a flame, without ever
actually melting. A modern example of this class of solid is the vinyl plastic
phonograph record which will bend and curve out of shape if left out in the
direct sunlight or stored too close to a hot-air register.
Perhaps the strangest of all examples of a solid which is not exactly solid
is the silicone putty recently marketed as a children's toy. A ball of this
stuff is solid but pliable, easily molded with the fingers. A ball of it left on
the table will gradually flatten out over a period of time into a sort of a
puddle under the effect of gravity. If we drop it on the floor it will bounce
like a rubber ball, but if we whack the same piece with a hammer it will
shatter into a dozen pieces. If we try to tug it into two pieces gently it will
stretch and stretch almost without limit, yet if we wrench it with a sudden
shearing force it will snap. If we press it gently with a thumb it will flatten
out to reveal a perfect fingerprint impression, and if two portions are
snapped apart they can be rejoined again simply by placing the broken
ends in contact.
Substances such as these certainly have some "solid" characteristics, but
in some respects behave more like extremely thick and viscous fluids. The
chemists and physicists of Newton's time who were busy classifying sub
stances into three main categories of solid, liquid, and gas were not
acquainted with many such peculiar substances and largely tended to ignore
those few they did know. But in our modern age of shirts that melt on the
ironing board and shoes made of impermeable plastic that still allows air to
circulate through it, we encounter a great many "peculiar" solids and other
in-betweens. To complain that these substances are impossible to classify
is to blame the shoe for the faults of the foot.
156 The Universe of Classical Physics
THE CONVENIENT QUALITIES OF LIQUIDS
If much of the commonplace usefulness of matter in the solid state
derives from its strength, rigidity, and durability of structure and from its
built-in resistance to the effects of stresses and strains, the equally con
venient qualities of matter in the liquid state derive in large part from the
very absence of these same peculiar characteristics. A liquid may be so thin
as to have practically no substance at all (as with ethyl chloride, an aromatic
fluid which evaporates so fast that a finger dipped into it will be dry by the
time it is pulled out) or thick, viscous, and sticky (as with honey, for ex
ample, or axle grease); but whether thick or thin, the one common char
acteristic of all liquids is the absence of any rigid internal crystalline struc
ture. The molecules of a liquid are free to move around, over, above, or
below their neighboring molecules with far more freedom than the molecules
in a solid.
Consequently, liquids have no fixed and rigid shape of their own but
can be poured freely from one container to another, spread out in a puddle
on a flat surface, or be physically separated into portions which subsequently
can freely rejoin and commingle, but ultimately take the shape of whatever
container they happen to be deposited in. It would be impossible to describe
the "shape" of a glass of water except by describing the shape of the vessel
formed by the containing walls and the bottom of the glass. You may pour
sufficient water into the glass to have a column of water several inches
high and the width of the glass but if you then smash the glass you end
up not with a column of water but with a puddle on the table.
Liquids and solids have certain distinctly similar characteristics, and
others which are sharply differing. Like solids, liquids cannot readily be com
pressed by squeezing or crushing forces, nor stretched when subjected to
pulling forces. Indeed, they have no quality of elasticity whatever. They do
tend to expand in volume when heated, or to contract in volume when
cooled, much as solids do, but usually to a greater degree. Like solids,
liquids generally have clearly distinguishable borders and to remain within
coherent boundaries of volume unless physically separated into smaller
portions by outside forces; but unlike solids, many liquids tend to evaporate
quite readily, converting bit by bit into a gaseous state even at temperatures
far below their boiling points.
But perhaps the single most unique characteristic of liquids not to say
one of the most fortunate, from the point of view of earth's living creatures
is their ability not only to mix freely with other substances, but actually
to dissolve other substances so that the substance being dissolved (the
"solute") and the liquid which has dissolved it (the "solvent") can no
longer be separated without a great deal of bother. This ability to dissolve
other substances is something quite apart from the ability to form chemical
combinations with other substances, although many liquids are capable of
The Forms and Shapes of Matter 157
that also. In a true solution, both the dissolving liquid and the substance
dissolved in it remain distinct and autonomous chemical entities, each
essentially unchanged except for being so thoroughly mixed and interspersed
that physical separation has become difficult if not impossible. And indeed,
the dissolving of one or another substance in a liquid brings about such a
thorough intermixture that the dissolved substance or solute is broken down
into molecule-sized particles, and often even those molecules are further
separated into electrically charged fragments of molecules called ions. And
because such a thorough intermixing of substances in solution takes place,
the solution of some substance in a liquid often brings about a change in
the physical characteristics of either the dissolved substance, the dissolving
liquid, or both, even when no chemical alteration in these substances
occurs.
Thus, for example, when water (a colorless, tasteless liquid) and salt (a
crystalline solid) are intermixed in solution, the salt loses its "solid matter"
property to become part of the liquid solution, while the water loses its
tasteless property and assumes a salty flavor. Furthermore, the dissolv
ing of salt into water produces a solution that freezes at a temperature
several degrees below the freezing point of pure water. But even in spite
of these changes, if the proper steps are taken at least one or the other
of the intermixed substances can be separated and recovered in its origi
nal form. If a solution of salt and water is allowed to sit in a flat dish,
exposed to air, the water will gradually evaporate leaving a residue of solid
crystalline salt on the dish, and careful measurement would show that the
residue contained all of the salt that had been mixed in the water, and that
it was chemically indistinguishable from the salt before it was placed in
solution. Recovering the water free of the salt is quite a different matter,
but even this can be accomplished in a clumsy fashion by boiling the solu
tion dry, collecting the water vapor in a separate container, and then
distilling it back into liquid form by passing it through the coils of a con
denser.
Just because the ability to dissolve other substances is a commonplace
characteristic of liquids does not necessarily mean that any given liquid will
dissolve any other substance we mix into it in any quantity that we desire.
Liquids in general are highly individual and selective with regard to which
substances they will dissolve and which they will not, and in regard to the
quantity of a given substance they will admit to solution. A given liquid
may dissolve one substance quite readily, yet not dissolve another at all.
Even when a substance can be dissolved by a given liquid, the liquid may
dissolve only a tiny amount of it before becoming "saturated," whereas
it will dissolve enormous quantities of some other substance without be
coming saturated. Thus the capacity of a given liquid to dissolve other
substances is comparable to a country's immigration quotas: A country may
admit a large number of people of one nationality, only a few people of
158 The Universe of Classical Physics
another nationality, and none at all of a third, according to fixed quota
laws. If a given quota of immigrants is exceeded, the country simply refuses
to admit any more and sends them back home just as a liquid that is
saturated by a substance it has been dissolving simply refuses to dissolve
any more. You can dissolve surprising quantities of salt in a tumbler of
water but, if you continue adding salt, a point will be reached at which
any additional salt crystals will simply sit surrounded by the solution in
the bottom of the glass and will not dissolve no matter how much you stir
nor how long you wait.
Usually when we think of liquids dissolving substances, we think of a
solid substance being dissolved by a liquid. But in truth, a liquid may dis
solve surprising quantities of various gases, or may even dissolve another
liquid. Fish and other water-breathing fauna depend for their lives on the
oxygen gas dissolved in water, either from the contact of the water with the
oxygen and air, or from oxygen released by water plants as a by-product
of photosynthesis. In the case of two liquids in solution water and ethyl
alcohol, for example, or gasoline and motor oil it is difficult to say which
liquid dissolves which, so we speak of such liquids as "mutually soluble."
Even in such cases, however, each liquid will have an upper limit, a satura
tion point, above which it will no longer dissolve more of the other liquid.
In many mutually soluble liquids (again, such as ethyl alcohol and water)
the saturation points may be infinitely high, so that either will dissolve an
infinite quantity of the other; but in the case of two liquids which are
only very slightly soluble in each other water and ethyl ether, for ex
ample only a very tiny amount of one liquid (the ether) will dissolve
in the water, while virtually no water dissolves in the ether. In such a case,
when excess ether is added to a container of water, an "interface" between
the ether and water will form, with the lighter liquid (in this case the
ether) rising to the top and the water sinking to the bottom.
There are, of course, instances in which one liquid will not dissolve even
a tiny amount of another, or vice versa; in such a case we speak of the two
liquids as "mutually insoluble." Any two such liquids placed in the same
container will simply separate one from the other with the more dense
liquid beneath and the less dense liquid forming a layer on the top with a
clear interface at the line of demarcation. Water and mineral oil, for ex
ample, are two such mutually insoluble liquids and literally cannot be forced
to dissolve one another unless an emulsifier is added.
So far we have been talking about a true solution of one liquid in another
(or lack of it). But there are other ways by which two liquids can be
intermixed that do not involve a true solution of one in another at all.
If we mix olive oil and vinegar together, one liquid may seem to dissolve
in the other at first, but if the container holding this "solution" is allowed
to stand, the two liquids will gradually separate, with the vinegar gradually
The Forms and Shapes of Matter ^
sinking to the bottom and the oil rising to the top. Obviously such a com
bination is not a true solution at all. In such cases we speak of the two
liquids as being "miscible" capable of being mixed together quite
thoroughly by stirring or shaking, but never forming a solution. Finally,
further to confuse things, there is yet another kind of "phony solution" in
which certain substances which are insoluble in a given liquid nevertheless
divide into such tiny particles when they are mixed into the liquid that the
particles remain permanently suspended in the liquid and evenly distributed
throughout its volume, never settling out or separating even though they are
not really dissolved. Such a mixture is called a "colloidal suspension." A
familiar example is homogenized milk, in which the butterfat has been
mechanically broken down into such tiny particles in the watery liquid of
the milk that it remains suspended. Even skim milk is essentially a perma
nent suspension of butterfat droplets in water; although the butterfat is
not dissolved in the water, the only way to separate it out from the water
is by addition of some such agent as lemon juice which causes the butterfat
particles to cling together in a sticky curd which can be filtered or centri-
fuged out of the water or from which the water can be decanted.
Finally, liquids are quite as capable of dissolving gases, sometimes in
great quantity, as they are capable of dissolving solids or other liquids.
Cola drinks and other carbonated beverages depend upon the ability of
water to dissolve quantities of carbon dioxide gas under pressure and re
duced temperatures for their distinctive tanginess. Here we see a splendid
demonstration of the fact that the amount of gas dissolved in a given volume
of liquid depends upon increased pressure and decreased temperature.
Try drinking a bottle of Coke sometime after it has been left standing open
in the hot sun for a few hours, and see whether things go better or not.
The capacity to dissolve other substances is a distinctive identifying
characteristic of any matter in liquid form; it is also a highly convenient
characteristic which permits liquids to lend themselves to all sorts of
practical uses. Literally hundreds of products that we use each day, from the
coffee we drink in the morning to the soap we wash the dishes in to the
fuel we burn to the perfumes and colognes we use, all are solutions of one
substance dissolved in another. And of all known liquids, water is perhaps
the most ubiquitous solvent, willing and eager to dissolve an endless variety
of other substances. We depend on this vast solution-forming capability of
water for our very lives; not only are all the cells in our body constantly
bathed in nutrients dissolved in water in the body and giving up waste pro
ducts to be dissolved and carried away, but it was the presence on the face
of the earth millions of years ago of a warm sea of water containing salt
and other dissolved substances that provided a medium in which life on our
planet first came about or was even possible.
But if water and other solution-forming liquids are important to man
160 The Universe of Classical Physics
(and perhaps innumerable other forms of life throughout the universe)
there is a third commonplace state of matter, more quixotic than either
solid or liquid, which is equally deserving of our attention.
THE EFFERVESCENT GASES
Just as solids and liquids have certain characteristics in common and
other characteristics sharply in contrast, liquids and gases are similar in
some ways and dissimilar in others. In fact liquids and gases are so similar
in so many ways that scientific classification frequently lumps them together
as "fluids," but their differences are such that we rarely have difficulty
distinguishing matter in one state from matter in the other.
Like a liquid, matter in the form of gas has no fixed shape of its own,
but tends to fill and take the shape of any closed container into which it is
placed. Also like a liquid, a gas can flow freely from one place to another,
and demonstrates the same sort of swirls and eddies as a liquid when it is
flowing from one place to another. Similarly, quantities of two different
gases enclosed together in a container will diffuse and intermix uniformly
one with another, much like two mutually soluble liquids, but here we en
counter a major dissimilarity. Unlike liquids, gases do not dissolve other
substances, nor are gases fussy about which other gases they will mix
with. While mutually insoluble liquids simply will not mix at all, any gas
will intermix with any other gas placed in the same container, each behaving
precisely as if it were the only gas around (unless, of course, the two gases
enter into chemical reaction and form quite different substances which may
not be gaseous in nature at all. Both hydrogen and oxygen are gases, for
example, but when mixed together in the same container may combine
explosively to form a liquid, water).
The most singular characteristic of matter in the form of a gas is, in fact,
that it has no physical coherence whatever. A quantity of a liquid released
from a closed container will flow and change its shape just as a gas will, but
it will also maintain a coherent delimited physical form of some kind even
if that form happens to be one or more puddles on the floor, each of which
has physical coherence. A gas, on the other hand, will show no physical
coherence at all if released from a closed container; it will diffuse freely
out of the container without limit, and freely intermix with any other gas
that happens to be around. In the absence of any confining forces, it will
continue to diffuse and expand indefinitely. If it were not for the confining
force of gravity acting to limit the diffusion and expansion of the gases
in earth's atmosphere, those gases would long since have been dissipated,
just as our atmosphere's hydrogen and helium were, by diffusion through a
vast expanse of empty space in the universe around us, and would still be
diffusing. This is precisely why the planet Mars is believed to have as sparse
and tenuous an atmosphere as it has, completely devoid of very light gases
The Forms and Shapes of Matter 161
such as hydrogen and containing only a tiny amount of oxygen: Mars's
gravitational force, far weaker than earth's, has been able to "contain"
only the heavier gases in its atmosphere while the lighter gases have leaked
away a bit at a time over the ages.
Indeed, the only time that a gas could be said to have a coherent physical
shape upon release from an enclosing container would be in the event that
it was released in an environment in which it is surrounded by a liquid
in which it is not soluble. In such a case, the gas would rise to the surface
of the liquid (having less density than the liquid) and in the course of rising
would be contained in spherical bubbles. But even this is not a case of a
volume of gas assuming a "natural" physical shape; when released into the
container of liquid, it is effectively "enclosed" by the pressure of the liquid
on all sides of it boxing it in exchanging one container for another, so
to speak. Conceivably the same things might hold true in the case of a very
light gas such as hydrogen that is released from a closed container into
into an environment of a very heavy, dense gas under pressure an environ
ment such as might be found in the heavily compressed and bitterly cold
atmosphere of methane and ammonia gas near the surface of the planet
Uranus. Under such conditions the hydrogen might conceivably be con
fined in the form of "bubbles" within the dense gaseous atmosphere of such
a heavy planet; but more likely even under such extreme circumstances
there would be plenty of room between the molecules of the "confining"
heavy gas to permit the hydrogen plenty of room simply to diffuse and mix
into the surrounding atmosphere.
Ordinarily we think of a gas as an effervescent stuff without form or
substance, but like any other form of matter any gas has mass and
occupies space. The weight of our earthly atmosphere pressing against the
ground at sea level is equal to almost 15 pounds for every square inch of
the earth's surface; and if we have sneaky doubts that the gases in our
atmosphere indeed occupy space, we need only watch a sky diver float to
earth under his parachute. Obviously something that occupies space is mak
ing that parachute balloon out, while the parachute equally obviously is
moving down through something which has to be pushed aside in order for
it to descend.
Furthermore, matter in gaseous form tends to expand in volume when
heated or contract in volume when cooled, just as solids or liquids do,
except that the expansion or contraction of a gas under these influences
is far more marked and dramatic. The air in a blown-up air mattress, for
example, can expand enough to split open the seams if the mattress is left
out unprotected in the hot sun; but if the outside air cools enough at
night, the air inside the mattress may contract so much in volume that
more air must be added for comfortable sleeping. Finally, gases possess one
unique characteristic that both solids and liquids lack: gases are "compres
sible." Just as a gas can and will diffuse and expand without limit unless
1 62 The Universe of Classical Physics
contained or confined by some outside force, so a gas can be squeezed or
compressed by an outside force, its volume diminishing in direct proportion
to the pressure exerted by the force. But there is a bottom limit to a gas's
compressibility. From Newton's third law we know that any gas that is
being compressed by an outside force is at the same time pressing back
against the force that confines it with an equal force. Thus if a gas's pressure
outward against the confining walls of a container into which it is being
pressed exceeds the containing strength of those confining walls, the con
taining vessel will burst. Alternatively, if the confining walls of the container
are strong enough and the compressing outside force great enough,
molecules of the gas will be forced so close together that interatomic forces
of attraction can take over and the gas may "condense" into a liquid which
will then no longer be significantly compressible.
The physicists of Newton's day were convinced that all the matter in the
universe existed in one of these three major states either in the solid
state, as a liquid, or as a gas. Today, of course, we know that matter can
exist in certain other more exotic states under special circumstances. In a
number of physics laboratories, for example, modern physicists work with
hydrogen atoms in an extremely rarefied gaseous state in which the nuclei
of the atoms are -stripped of their electron components. These particles,
nothing more than naked hydrogen nuclei or "protons," are confined within
powerful magnetic fields and are artificially accelerated to great speeds.
Matter in such a state as this, which we might think of as a superrarefied,
superheated gas is known as a "plasma" and can be considered as a quite
separate and unique state of matter characterized by its own peculiar
properties. Of course, here on earth it may require a $35 million collection
of machinery in order to convert a tiny amount of ordinary gaseous hydro
gen into a plasma state in which it is maintained for only 1/1,000 second,
but even here matter can be converted into such a state for long enough
at least to demonstrate that it exists. Elsewhere, most of the visible matter in
our universe (stars, etc.) exists in the plasma state, and it is entirely pos
sible that the universe contains more matter in the plasma state in the
unimaginable reaches of space between the stars and galaxies than exists
in all the other states put together in more familiar corners of the universe.
Similarly, modern astronomers are convinced that the universe also
contains uncounted multitudes of dark, cold aggregates of burnt-out star
ash, formed from the densely packed nuclei of atoms that once fueled stars
that are now long dead, all their available energy expended. Such compacted
nuclei with their electrons stripped away would have to form incredibly
dense matter unlike any solid ever encountered in our earthly experience.
It has been calculated that a cubic inch of such hypercondensed star ash
would have a mass of tons, and such material would certainly have to be
considered a separate and distinct state of matter. So would the strange
substances both liquid and solid that have recently been studied in modern
The Forms and Shapes of Matter 163
low-temperature physics laboratories. As we will see later, when the
temperature of certain substances is reduced to within a few degrees of
absolute zero the point at which all molecular motion is believed to stop
these substances suddenly take on physical and electromagnetic proper
ties totally unlike any other substances known. Here again we might con
sider such exotic substances as existing in a separate and distinct state of
matter.
For the sake of completeness we need to acknowledge that matter can
exist in such bizarre and exotic states under certain special conditions or
extreme environments. For practical purposes, however, there is no need
right now for us to concern ourselves with matter in these peculiar states;
for the moment we will concentrate on the conclusions reached by the
classical physicists about the nature of matter based on their knowledge of
the three major and commonplace states of matter they knew, dealing with
the others at a more appropriate place.
CONVERSION AND CONSERVATION
Today we know that the differing characteristics of matter in the solid,
liquid, or gaseous state are directly related to the internal atomic structure
of the substance. The elementary units of a substance in the solid state are
indeed "locked together" in more or less rigid geometrical patterns, and
even when the geometrical structure of a solid is temporarily distorted by
one kind of stress or another those elementary units- can pull back into their
original pattern when the stress is relieved. The atoms of a liquid are not
locked together in quite this unyielding fashion; yet they are still tightly
enough assembled, with sufficient binding force between them to give the
liquid a coherent volume even if the liquid flows freely and takes on the
shape of the container holding it. We will have much more to say about
these interatomic binding forces in a later chapter, for the precise nature
of these forces remains one of the major problems modern physicists are
still wrestling with.
In a gas, however, the elementary units making it up are far more widely
separated than in either a liquid or a solid and are free of the effect of
internal binding forces which are present only when elementary particles
are in comparatively close proximity to each other. Thus the atoms or
molecules of a gas are free to move quite independently of one another
and can move at random within the confines of any container holding them,
capable of being pushed closer together by external pressures but also capa
ble of diffusing without limit as long as no containing force acts upon them.
Of course the physicists and chemists of the seventeenth and eighteenth
centuries did not know anything to speak of about the submicroscopic
structure of matter in any state, although the basic idea that all matter was
composed of small indivisible particles had been kicking around for over a
164 The Universe of Classical Physics
thousand years and was soon to be revised and expanded by such men as
John Dalton. But if they did not know precisely why matter could occur
in three quite different states, those early scientists gathered together a
remarkable amount of information about how matter behaved in each of
the three major states, and about the conditions necessary to change or
convert matter in one state into another state.
Most familiar substances seemed to exist in nature in one or another
state by preference under normal conditions, but a great many substances
could be converted from one state into another more or less readily and
consistently under certain specified conditions. Gold normally was found
in the solid state in nature, but if heated to a certain unnaturally high
temperature it would change into a liquid. Another heavy metal, mercury,
was already a liquid in its natural state, but if heated to a sufficiently high
temperature could be converted into a noxious violet-colored gas, while
sulphur dioxide which occurred in nature in the vicinity of sulphur de
posits as an evil-smelling, pungent gas could easily be condensed into a
colorless oily liquid when it was cooled and compressed. A few substances,
such as the resinous sap of pine trees or the tallow used in making candles,
might be found in either solid or liquid state in nature, depending upon the
prevailing temperature. But of all known substances, there was one and
only one that could normally be found in nature in any of the three major
states depending upon the particular circumstances that prevailed.
This substance, of course, was water. It could be found in its solid state
in the Arctic or Antarctic ice packs, or in rigid sheets that formed on rivers
and canals even in the Temperate Zone during the winter. As a liquid, it fell
as rain, emerged as underground springs or flowed down mountainsides
in cascading torrents, while in its gaseous state it could be observed as
water vapor in hot springs and geysers, or saturating the air in regions with
moist climates.
Even the earliest scientific observers recognized that the particular state
of matter of any given substance seemed to be a function of its temperature.
A solid, heated sufficiently, would melt and become a liquid, providing that
it did not undergo some chemical change or breakdown in the process,
and the resulting liquid when heated still further would presently begin to
boil and become a gas. It was also observed that each substance that could
be converted from one state to another had its own characteristic temper
atures at which the change would take place, and while these "change-of-
state" temperatures might vary widely from one substance to another, a
given substance could be relied upon to change from solid to liquid, or
liquid to gas, or vice versa, quite consistently and reliably when heated or
cooled to the appropriate temperature. Thus it became customary in de
scribing various substances to list as physical properties of a given substance
its freezing point and its boiling point, in the case of a liquid, its melting
point and vaporization point in the case of a solid, or its condensation point
The Forms and Shapes of Matter i$$
and freezing point in the case of a gas whenever those temperatures could
be measured under standardized pressure conditions (i.e., at normal atmos
pheric pressure at sea level). Of course, certain of the transition-point tem
peratures could not be measured at all for certain substances because they
underwent various kinds of chemical alterations before melting point or
boiling point could be reached; coal would burn before it would melt,
for example. With other substances transitions from one state to another
required such extremely high or extremely low temperatures that it was
difficult to achieve them. Many metals simply could not be heated to a high
enough temperature to measure their boiling points except in a vacuum,
while gases such as hydrogen or oxygen had such extremely low conden
sation points that twentieth-century technology was required to cool them
down sufficiently to liquefy them, and helium gas would not condense into a
liquid until its temperature was reduced to within four degrees of absolute
zero, to 265 degrees Centigrade!
Among the considerable variety of substances that could be studied in
two or more different states, certain other interesting general characteristics
were observed. For one thing, most substances would change in an orderly
manner from solid state to liquid state, and then from liquid state to a gas,
and vice versa, providing necessary temperatures could be reached before
chemical changes in the substances occurred. Some few substances, how
ever, seemed to ignore this orderly rule: Solid crystals of iodine, for ex
ample, would "sublimate" directly into a gas when heated without ever
passing through a liquid state, and iodine vapor when cooled sufficiently
would sublimate directly back into solid crystals.
Similarly, a chunk of dry ice (frozen carbon dioxide) would evaporate
directly into carbon dioxide gas without passing through a liquid stage, at
least under ordinary conditions. Yet under quite extraordinary laboratory
conditions, and with a great deal of effort, carbon dioxide gas can be
cooled down under sufficient pressure to force it into a liquid state before it
freezes. Just why these particular substances happen to deviate from the
general rule nobody knows nor cares, for that matter, since we really
have no practical use either for elemental iodine in the liquid state or for
liquefied carbon dioxide.
But another even more curious and quite unique exception to the general
rule has far greater importance the very existence of life as we know it
on earth depends upon it. Most substances in the solid state are more dense
than the same substances in the liquid state, and even less dense in the
gaseous state than in the liquid state. A striking exception to this rule is
water, which through some fortuitous happenstance is significantly less
dense a degree or two below its freezing point than it is in liquid form. The
extreme good fortune of this curious variation from the general rule is easy
to see: If ice were more dense than water at the freezing point, and were
thus heavier than water, all of the lakes, rivers, and even oceans on the face
1 66 The Universe of Classical Physics
of the earth would have frozen solid from the bottom up during the cold
seasons, and once frozen would never again have melted completely in most
regions, so that any form of life which required warm salty seas in which
to develop could never have survived long enough to propagate.
Over centuries of observation of matter in its various states scientists
came to recognize many curious variations from what seemed to be the
normal rule, but at the same time they began to recognize one character
istic of any kind of matter regardless of its state to which no exceptions of
any sort were ever found. A given quantity of matter could be changed
from solid state to liquid state or from liquid state to gaseous state, or vice
versa; it could be ground up or evaporated or compressed by any number
of physical forces, and might undergo any number of chemical combina
tions or break down into a variety of chemical compounds; but regardless
of what was done to it, no matter what kind of interaction in which it
might be involved, the total quantity of the matter in question always
remained the same. None was ever destroyed and no new matter was ever
created.
Often in the course of chemical reactions between substances, totally
new and different substances would be formed with strikingly different
appearances and properties. Sometimes new and insoluble substances would
be formed and precipitate out of solutions as a result of chemical reactions,
and not infrequently gaseous by-products of chemical reactions might be
inadvertently released to diffuse into the atmosphere and be lost to the
four winds. But whenever truly meticulous measurements were made and
great care was exercised to collect all the end products of some physical
change or chemical interaction of matter, the total quantity of matter that
resulted, in whatever state it might be found, was invariably found to be
precisely equal to the total quantity of matter that existed before the
physical change or chemical reaction.
This idea that the total quantity of mass of matter was always conserved
in any kind of interaction was by no means self-evident to the casual
observer of nature. It was, in fact, vigorously disputed by a great many
very excellent physicists and chemists as late as the eighteenth and nine
teenth centuries. In many cases it was extremely difficult to take accurate
enough measurements to tell whether the principle was valid or not,
and even as more and more evidence of its validity accumulated this
principle, which was known first as the theory and later the law of the
conservation of matter, became one of the most widely challenged and
tested of all the classical laws of physics precisely because there were
so many kinds of interactions in which it seemed that a certain amount
of matter had been destroyed or had appeared out of nowhere. But by
the beginning of the twentieth century the law had been so thoroughly and
repeatedly tested and proved that scientists regarded it as a very rock of
The Forms and Shapes of Matter 167
stability, fully as reliable as Newton's laws of motion or the law of con
servation of momentum.
Later, as we will see, the law of conservation of matter had to be mod
ified or, more accurately, expanded to include certain manifestations
of matter that had never even been dreamed of previously; but with those
necessary expansions the law today remains as valid and unshakable as
ever. New challenges to its validity even now continue to arise with
tiresome regularity, but by now the law has withstood so many such
challenges that it seems unlikely ever to be shaken unless some totally
unsuspected and unpredictable item of knowledge is suddenly unearthed.
Even the most vigorous of the recent challengers, a group of astronomers
and cosmologists led by Dr. Fred Hoyle of England, have recently begun
to hedge their bets and question the validity of their own challenges. Con
ceivably one day some key item of new knowledge mil appear and a suc
cessful challenge will be mounted, a challenge the law cannot answer; but
so far as is known today, it remains as one of the very few physical in-
variables in the universe. And as we shall see as we learn more of the
changing and bewildering world of modern physics, any invariable truth
at all is a pearl of great price.
Without question, the history of scientific investigation in the last four
centuries has been in great measure the history of multitudes of observers
gathering a huge quantity of knowledge of the many different character
istics of matter in the solid, liquid, or gaseous state. But so far we have
ignored one characteristic of matter of which scientists in the last four
centuries have become increasingly aware. Many substances could be
converted from one state of matter to another; substances could be dis
solved in one another; substances could be encouraged to form chemical
combinations with one another, mixed with one another, forced to expand
and diffuse freely or to be massively compressed.
But any time that any such physical or chemical change was brought
about, it was first necessary that something be done, that certain require
ments be fulfilled, before such changes or interactions would take place.
In order for any change or interaction to occur, energy had to be applied
in one form or another.
Indeed, even in the case of objects interacting with each other in keep
ing with Newton's laws of motion and the law of conservation of momentum,
energy inevitably seemed to be involved one way or another in any change
whatsoever. So it was not surprising that the same scientists who were
observing and studying the various states of matter and confirming the
law of conservation of matter again and again found themselves simul
taneously observing and studying the characteristics of another far less
tangible entity in the physical universe an entity known as energy
1 68 The Universe of Classical Physics
defining it, discovering the various forms in which it manifested itself,
unearthing the relationship existing between energy in one form and energy
in another form, and ultimately discovering that just as the universe
seemed to contain a fixed and inalterable quantity of matter which could
be changed in form but which could neither be created nor destroyed, so
also the universe seemed to contain a fixed and inalterable quantity of
energy which could be converted from one form to another but could
neither be created nor destroyed.
But whereas matter could be pinned down, pinched, squeezed, measured,
and manipulated, the study of the nature of energy proved to be a far
more elusive and frustrating game. The search still continues today in
the laboratory of the modern physicist, but he could not even have begun
without the groundwork that was laid by generations of classical physicists
before him.
CHAPTER 11
The Manifestations of Energy
Of all of the concepts that have evolved from the experiments and observa
tions of physicists since the time of Newton, perhaps one of the most
crucially important, yet most confusing and obscure to the average non-
scientist, is the concept of energy in its various manifestations. And once
again we find that the major barrier to understanding is semantic. We
are in trouble from the first with our use of language.
We have already encountered more than once the gulf that exists be
tween the common usage of terms and the more precise scientific use of
the same terms. Remember, for example, the trouble we had when we
tried to define precisely what a "force" was. We found it a rather vague
entity variously described as a "push," a "pull," and "impulse" (that is,
a force acting over a period of time), or even existing in a fuzzy and in
definable "field," as in the case of gravitational force. When it came right
down to fundamentals, we found that the closest we could approach
defining what a force might be was in terms of the effect it had (what
ever "it" was) when it acted upon some object to cause some change
in its motion. To many of us this seemed suspiciously similar to defining
the "haves" and the "have-nots" as "those people who have" and "those
people who don't have," respectively. Unsatisfactory as this may be, it
is fairly typical of what happens any time we attempt to discuss an abstract
concept in concrete terms: We can do only as well as our language
permits us to do.
Now we encounter the same difficulty when we attempt to describe
precisely what energy is. To the nonscientist the term inevitably brings a
variety of vague and nonspecific images to mind. The dictionary defines
the word "energy" as "vitality of expression"; "the capacity of acting";
"power forcefully exerted"; or "the capacity for doing work." But then,
what exactly does "vitality of expression" mean? What is a "capacity"
for doing anything, or even an "ability"? Ordinarily we tend to equate
energy in our minds with somehow stirring around and getting things done;
but we also hear of "suppressed energy"; we speak of "mental energy" or
read of a modern painting "radiating energy," and so on into the night
Of course all of these various uses of the word have one thing in com-
169
170 The Universe of Classical Physics
mon: they all suggest some sort of capacity for doing something. But what
is the "something" that energy implies the capacity for doing?
Rather than get ourselves thoroughly snarled up in words, it might
be better for us to recognize here and now that energy as a physicist uses
the term has a more specific and well-defined meaning than any of the
commonplace connotations assigned to the word. To the physicist, energy
is a natural phenomenon, not a thing but a concept. He relates the concept
of energy specifically to a capacity for doing work: a capability for chang
ing the motion of an object, for forcing a substance to change from one
state of matter into another, or for bringing about an interaction between
substance A and substance B. Within this limited meaning of the word,
the physicist further regards energy as a capacity that exists in a number
of different forms, and a capacity which can, in any given form, be nailed
down precisely and measured in some kind of comprehensible unit.
THE CASE OF THE BROKEN TOE
One of the most common and familiar forms of energy that we encounter
in our everyday lives is simple physical or mechanical energy the form
of energy that is constantly being acquired or released by physical objects
in mechanical motion. But how can we define mechanical energy? Rather
than try to define it, first let us see an example of how it can be acquired,
how it can be released, and how it can be measured and described in
meaningful units.
Consider the following commonplace situation: A man finds a good-
sized rock lying in his driveway. Picking it up with one hand, he raises
it four feet into the air, intending to toss it aside. Unfortunately, before
he can throw it he loses his grip, dropping it on his foot and breaking a toe.
Now what has happened here? No matter what words we use to
describe this particular sequence of events, certain basic things are clear.
First, it is obvious that at the beginning the rock represented no direct
immediate threat to the man (although it might possibly have damaged his
car if he had driven over it) . It was merely sitting there in the driveway
minding its own business with no capability to roll off the driveway, leap
up and fly, or anything else. We could say that at the beginning the rock
possessed no mechanical energy at all, and would not have acquired any
if the man had simply let it alone,
But when he picked the rock up and lifted it to four feet above the
ground, the whole picture changed as a result of this action. In doing this
the man instilled in the rock a capacity that it did not have while it was
resting on the driveway: the capacity to strike his toe with a certain
measurable force. Of course, in the instant before it slipped from his
fingers while it was suspended four feet above the ground, the rock had
not yet done any damage; it had merely acquired a "capacity'' or "potential"
The Manifestations of Energy 171
for doing something. We could then say that the rock had acquired energy.
We could even measure the energy it had acquired in any arbitrary units
we wished to use, perhaps choosing units that were clearly related to some
one of many possible things the rock had acquired the capacity or potential
to do. We could say, for example, that at the instant before it slipped from
the man's fingers four feet above the ground the rock had been "charged"
(like a storage battery) with one broken toe's worth of energy. In saying
this we would imply that the rock possessed enough energy in the event
that it was dropped to break one of the man's toes, but not enough to
break two or four or six, nor so little that it could not break at least one.
Now granted, physicists would not ordinarily find "one broken-toe's-
worth" a particularly useful or versatile unit for measuring or describing
a quantity of energy, although it is a perfectly valid unit for us to use
under these circumstances. Instead, they have found certain generally
useful words to describe the form or forms of energy involved in this
sequence of events, and have selected units for measuring it that are
somewhat more universally relevant and practical. The rock sitting on
the driveway clearly had no capacity to do anything on its own; it had
no energy. In lifting it four feet above the ground against the -pull of
gravity the man, by virtue of the effort expended by his muscles, instilled
in the rock a "potential capacity" to break a toe.
This as-yet-unexercised capacity might be spoken of as "potential
energy." As long as the man held the rock suspended and motionless, that
potential capacity remained entirely potential; the rock was quite as in
capable of doing anything there under those circumstances, suspended by
the man's grip on it, as it was when it was resting on the driveway
provided the man didn't let go of it. The instant that he did let go of it,
things changed abruptly. The rock's "potential energy" was immediately
transformed into a different, more active mechanical energy which did
indeed have the capability of breaking a toe when it struck it. Physicists
would speak of this "energy-in-action" or "released potential energy" as
"kinetic energy." The inert, immobile, and utterly harmless "one broken
toe's worth" of potential energy possessed by the rock when it was held
suspended four feet above the ground was very rapidly transformed into
"one broken-toe's-worth" of kinetic energy by the time it reached the
man's toe and it is quite obviously the rock's kinetic energy, not its
potential energy, that the man had to thank for the broken toe he received.
Was this transformation from potential energy to kinetic energy some
thing which occurred instantaneously when the rock was dropped? Not
quite, as we can easily demonstrate. Suppose, for example, that the man
happened to have his foot on a two-foot-high apple box at the time the
rock slipped from his fingers. In such a case the rock would have fallen
only two feet instead of four when it struck his toe, and would not strike
it with enough force actually to cause a fracture. It might sting a little,
172 The Universe of Classical Physics
but at the collision point only half the potential energy the rock had
acquired in being lifted had been transformed into kinetic energy. If the
rock were stopped at that point two feet above the ground and again held
suspended it would still possess some of its potential energy an amount
we might describe as "one broken-toe's-worth minus 24 inches" of potential
energy which could yet be transformed into kinetic energy in the event
that the rock were allowed to fall the remaining two feet to the ground.
Indeed, if we think this through carefully, we see that the potential
energy the rock acquired when it was lifted four feet above the ground
would not be transformed instantaneously into kinetic energy the moment
it was dropped, but that the conversion of the potential energy into kinetic
energy would take place gradually and steadily throughout the length and
time of the rock's fall, so that if it were stopped at any given place between
its release point and the end point of its fall it would there have a ratio of
potential energy to kinetic energy directly proportional to the distance it
had fallen at that point, and to the time it had taken to fall.
In virtually all examples of mechanical motion of objects or interaction
of moving objects we see precisely the same interchangeability of potential
energy (unreleased capability to do something) and kinetic energy (energy
of action in which the capability to do something is released) in operation.
We can see even more clearly the relationship between kinetic "energy
in action" and potential or "stored" energy if we return to our imaginary
"ideal" billiard table on which frictional forces, gravitational forces, and
other red herring forces have been conveniently eliminated for our benefit.
Imagine, then, rolling a billiard ball on our table in a straight head-on
collision course with the perfectly elastic springy cushion at the far end of
the table. As the ball is moving toward the cushion it has a certain amount
of kinetic energy or energy in action. It also has a measurable momentum
equal to its mass multiplied by its velocity. After it strikes the cushion,
we see it rebound in the opposite direction with precisely the same
momentum it had before (except that the direction of its velocity has been
reversed). Furthermore, if we could measure its kinetic energy on the
rebound, we would find that it possessed precisely the same kinetic energy
moving in the opposite direction as it had before striking the cushion.
But what happened during the collision? Obviously, in order for its
direction to have been changed, the billiard ball striking the cushion must
have been slowed down and ultimately brought to a complete stop, then
speeded up and thrust away again in the opposite direction. But what
happened to the ball's kinetic energy during this process of slowing down,
stopping, and speeding up again? At the moment the ball was at a dead
stop a split-second photograph would have revealed that the cushion
touching the ball had been compressed and distorted out of shape. Se
quential split-second photographs taken subsequently would show the
cushion expanding again to resume its normal shape and thus pushing
The Manifestations of Energy 173
the ball away in the opposite direction. But where did the cushion get the
energy to push the ball away? We said that the rock sitting dead still on
the driveway possessed no energy at all; would it not also be true that
at the split second when the ball has come to rest against the cushion, at
a dead stop, and the compressed cushion is in completely motionless con
tact with it, that this whole ball-and-cushion system is at that split second
of time completely without any energy at all?
The answer, of course, is no. What actually happens is that as the ball
strikes the cushion it begins to lose its kinetic energy steadily, and has lost
it completely by the time it has come to an absolute stop. But that kinetic
energy has not been destroyed. Rather, it has been transferred to something
else, namely the cushion, and converted into a form which we might call
"energy of compression." But since the cushion is elastic and seeks to
return to its normal shape, it must have acquired a potential capacity to
push the billiard ball away while it is in the process of recovering its normal
shape. Thus the energy of compression that the cushion has acquired is
just another name for potential energy which could be converted again
into kinetic energy when the compressed cushion begins to push the ball
away.
We can see this transfer of energy from one object to another and from
one form to another very clearly if we regard the ball-cushion collision
in detail as a sequence of events much the same as the case of the man
picking up the rock. The instant before the ball strikes the cushion, the
cushion possesses no energy at all, and 100 per cent of the energy in the
closed system of ball-and-cushion is in the form of the ball's kinetic energy.
The instant that the ball encounters the cushion and begins pressing it in
and deforming it, some part of the ball's kinetic energy is being transferred
to the cushion and stored there in the form of energy of compression.
The more the ball presses in the cushion, the more its kinetic energy
is so transferred to the cushion and converted into energy of compression
or potential energy until the moment that the ball has finally come to
rest.
At that point the ball has no energy of any kind left, while the cushion
has 100 per cent of the ball's previous kinetic energy invested in it, so
to speak, but totally converted from the form of energy in action (kinetic
energy) into the form of energy of compression (potential energy).
But at this instant the ball-and-cushion system is clearly not stable. The
ball may be at a dead stop and thus possess no energy at all, but the
cushion it is compressing has been deformed by the ball's pressure. Because
of the cushion's elasticity, it seeks to snap back to its normal configuration
again. Although it possesses all of the ball's former kinetic energy hi the
form of potential energy, there is nothing to prevent this potential energy
from immediately being triggered and allowed to begin changing back into
kinetic energy again. Thus the compressed cushion with its potential
174 The Universe of Classical Physics
energy is in precisely the same unstable condition as the rock was in during
the instant after the man lost his grip on it: The potential energy is there
and nothing restrains it from being released.
So how is this unstable state of affairs resolved? In the ball-and-cushion
system, a complete reversal of the conversion and transference of energy
takes place. The cushion begins to convert its potential energy into kinetic
energy while simultaneously transferring it back to the billiard ball once
again. And the fact that careful measurement would show that the billiard
ball rebounding from the cushion has the same kinetic energy (under these
ideal, frictionless conditions) as it had before its collision with the cushion
must mean that at the moment it had come to rest all its kinetic energy
had been totally converted and stored in the cushion as potential energy,
and then all recovered once again so that there was no energy lost or de
stroyed, and none created, at any time during the interaction.
From this ball-and-cushion example we can see a very interesting char
acteristic about energy and a universal characteristic. We saw that the
energy of compression was nothing other than potential energy, "energy
stored and available for use," so to speak, as opposed to kinetic energy
or "energy already in action." We can see that in this interacting system
of ball-and-cushion (or in any other interacting system in the universe)
energy within the system can be converted from one form to another and
back again, and can be transferred from one object or part of the inter
acting system to another and back again but the total amount of energy
in the system remains constant despite these conversions and transferences.
This means, in effect, that kinetic energy and potential energy in an inter
acting system are completely interchangeable and completely equivalent
in quantity or magnitude. They could be interchanged in part or in toto,
but whatever part is interchanged in either direction must always be
exactly equivalent in quantity on one side of the interchange as it was
on the other side, regardless of what form it is in.
ENERGY, FORCE, AND THE SEMANTIC BARRIER
Physicists of the 1700s and 1800s clearly recognized the differences
between kinetic energy and potential energy, and were fully aware that
one could be converted into the other or transferred from one object to
another. But they were not by any means certain that some energy was
not lost or gained in such transfers, nor for that matter were they con
vinced that all the energy in the universe was a constant quantity that
could never be changed. If anything, a great many of their laboratory
experiments seemed to demonstrate that there was in fact some destruction
of either potential or kinetic energy within such interacting systems of
objects in motion as billiard balls striking cushions, bullets striking wooden
barriers, or railroad trains colliding in the night.
The Manifestations of Energy 175
One reason for confusion in this area was that scientists in those days
had not yet learned how important it was to be fastidious in their use
of words and definitions. They commonly got the everyday meanings of
various terms confused with their technical meanings. In everyday con
versation we often use such words as "energy," "force," "work," and
"power" more or less interchangeably. For the most part, of course, this
works out splendidly; we succeed in getting our ideas across and save
having to bother with absolute precision. When we are struggling trying
to wrench a nail out of a plank with a hammer, it makes little difference
whether we say, "I'm trying to work this nail out," or "I'm trying to force
this nail out." No one cares whether we say, "I haven't the power to budge
it" or "I haven't the energy to budge it." But when it came down to
describing physical phenomena in precise terms, science was in trouble
with such vagueness and had to agree to use one particular, specific word
to describe a particular, specific thing or concept. Thus in present-day
physics the word "force" is an abstract term referring to something that
causes an object to change its velocity in some way. Even physicists fight
their own intuitive inclination to think of a "force" as a "push," a "pull,"
or a "tug" and instead make themselves think of a "force" as something
acting upon an object for a certain period of time, measurable only in
terms of the resulting change in the object's momentum.
Similarly, we ordinarily think of "work" as practically anything that
causes us to exert ourselves and make ourselves tired, but to the physicist
"work" is done only when a force moves an object through a certain dis
tance, with the amount of "work" always calculated by multiplying the
distance the object has been moved by the force that moved it. In some
ways such a definition seems contrary to common sense: A man might
completely exhaust himself trying unsuccessfully to push the Washington
Monument one foot north, but the physicist would say that he had done
no "work" at all unless he actually succeeded in moving it. "Power" is
an even fuzzier word in common usage, implying some potential for doing
something but really carrying a wide variety of different common meanings.
To the physicist, "power" has only one meaning: the rate at which work
is done. The man who succeeds in pushing the Washington Monument one
foot north in half the time it takes another man would be said to have
twice the power of the other man. In physics power is always represented
as some form of work divided by the time it takes to accomplish the work.
Finally, whenever work is done (that is, when some force has moved
a body through a certain distance) we usually find that energy has been
changed from one form to another in the process. Thus, when the man
let the rock slip out of his hands, the force of gravity performed work on
the falling object and its potential energy was transformed into kinetic
energy. Similarly, when the billiard ball struck the cushion, the force of
the rolling ball performed a certain amount of work in compressing the
176 The Universe of Classical Physics
cushion (as its kinetic energy was transformed into potential energy), and
then in reverse fashion the cushion performed an equal amount of work
upon the billiard ball in shoving it away as its potential energy was
converted back to kinetic energy again.
Vagueness about the use of such terms made it difficult for early physi
cists to assess really clearly what was happening in their experiments. The
idea that the entire universe might contain a certain finite total amount of
energy in some form or another which could never be destroyed nor in
creased had an ancient and respectable history; it seemed that men had
wanted rather badly to believe in the law of conservation of energy for
centuries! Yet repeated attempts to prove that energy was always con
served frequently led to failure, whether because of crude or inaccurate
measurements, through confusion about just how energy should be defined,
or through overlooking extraneous forces that were at work in the experi
ments. As a consequence, the principle of conservation of energy re
mained a theoretical dream rather than a well-established law of nature
for a long, long time.
Yet one type of evidence, of a rather negative kind, seemed consistently
to substantiate that energy was always conserved. Continuing attempts
to create energy to get energy for nothing, so to speak invariably failed.
Throughout the history of science we can find a long, sad succession of
men with all sorts of ingenious ideas for building perpetual motion
machines machines which, once started running, would keep on running
and continue doing work without any additional input of energy. Some of
these hopeful inventors were no doubt charlatans, but many others were
perfectly earnest in their conviction that there really -was some way to get
mechanical energy for nothing, if only they could just figure out how.
But the fact that such attempts always and invariably failed must presently
have convinced more and more scientists that in fact there was just no way
that this could possibly be done, and that some immutable law of nature
would be violated if there were.
THE OTHER FORMS OF ENERGY
In spite of the multitudes of skeptics, Newton and many other physicists
of his day had soon become convinced that mechanical energy potential
energy and kinetic energy was always conserved, and had the courage
to base their lives' work on this conviction. But sure as they were that
mechanical energy was conserved, even these men were not so clearly cer
tain that all possible kinds of energy were likewise conserved.
Of course, in those days there was no knowledge of the internal structure
of atoms, of the kinds of energy that served to bind the particles in atomic
nuclei together, nor of the forces of attraction and repulsion that exist
within atoms and between atoms. But there were certain kinds of energy
The Manifestations of Energy !77
recognized other than kinetic and potential energy. Heat was one such
form of energy. Electromagnetic energy came to be recognized, although it
seemed to have no connection whatever with potential or kinetic energy.
There was also clear evidence that chemical energy also was present in
the interaction of various substances to form new chemical compounds.
Just what these forms of energy were was not understood, but they were
recognized to exist. Other forces such as friction, cohesion, and surface
tension were observed and studied, and recognized to involve energy in
one way or another, but again no one seemed able to build the vital idea-
bridge between the potential and kinetic energy present in interacting
systems of objects and these other forms of energy.
In a way this is surprising, for some of these other forms of energy were
regularly observed to appear as red herrings in orderly scientific study of
conservation of mechanical energy under laboratory conditions. Very often,
for instance, the total energy of an interacting system seemed to be
measurably less after the interaction than before, apparently indicating
that a portion of the mechanical energy in the system had been destroyed;
yet heat would simultaneously appear during the interaction. A wheel
spinning on an axle, for example, could be given a carefully measured
amount of rotational (kinetic) energy by applying a carefully measured
force to it, but no matter how well the axle might be lubricated, the kinetic
energy of the spinning wheel would gradually seem to be dissipated and
the wheel would slowly stop spinning, while at the same time the hub and
the axle would invariably become hotter and hotter.
To us today the connection seems obvious, but it was not in those
early days. Nevertheless, by the end of the eighteenth century a good deal
of interesting data about these other forms of energy began to turn up.
In the year 1 800 Alessandro Volta invented the battery and proved beyond
doubt that certain chemical reactions could result in the production of an
electric current in other words, that chemical energy could be converted
into electrical energy. But the electric current produced in this way could
in turn produce heat if it was passed through a high-resistance wire; it
could even produce light, if the wire became hot enough to glow. Further
more, if a wire carrying an electric current were wound in a coil around
a steel rod, the flow of electrical energy through the wire induced magnetism
in the steel rod, which could then produce mechanical motion of iron filings
or nails. Conversely, mechanical energy through friction in a generator
could produce electricity, and a full circle of conversion of energy from
one form to another could be demonstrated: electricity to magnetism to
mechanical motion to friction to electricity!
At roughly the same time other observations were made. In 1882 a
European physicist named Thomas Seebeck demonstrated that heat could
produce an electric current directly when applied to the junction between
two different metals. Just at the turn of the century the American Benjamin
178 The Universe of Classical Physics
Thompson (who became notorious in Europe as "Count Rumford") ob
served that when a cannon barrel was being bored with a drill, so much
heat was produced that the process had to be cooled repeatedly in order
to avoid melting either the drill bit or the cannon barrel or both. Finally,
during the 1840's, James Prescott Joule of England actually measured the
amount of heat that could be obtained from a given amount of mechanical
energy and showed that the conversion of mechanical energy into heat
could take place without any loss.
Thus by about 1850 it appeared that the true implications of the law
of conservation of energy were finally becoming clear. It was not just the
potential energy and kinetic energy in a mechanical interacting system
that were interchangeable and always conserved; all forms of energy
were interchangeable, and the total energy in any form in a given system
was always conserved. The reason that an accurate balance of mechanical
energy before and after a collision could not be achieved was simply that
part of the kinetic energy of the colliding objects was converted into heat
and dissipated into the air, thus warming up the room but becoming
difficult to measure, rather than being converted 100 per cent into potential
energy. Friction was recognized as a force that "stole" useful mechanical
energy from a mechanical system or a machine by converting it into useless
heat energy. In fact, it was finally demonstrated that energy in any form
could by the employment of the proper means be converted or transformed
into energy of any other kind, and that this could be done without any loss
of energy whatever. And so another conservation law became a solid
cornerstone of modern physics, one of the very few invariables that
physicists came to count upon.
But the last chapter in the development of the law of conservation of
energy was not yet quite closed for, as we have seen, any law of nature
can stand only as long as it answers all challenges, and in the early 1900s
this natural law came up against a challenge that seemed to shake it to
its very roots. In a way, this seemed a catastrophic blow, for by then the
idea of conservation of energy was well entrenched as one of the most
elegant, satisfying, and useful of all laws of nature and perhaps least vul
nerable of any. But at about this time a number of chemical substances
were discovered which we now know contained "radioactive" elements,
and which seemed to have some strange characteristics that would have
been quite impossible if the law of conservation of energy were indeed true.
For one thing, some of these radioactive substances seemed constantly
to give off heat, even though they were not apparently interacting with
any other substances in any way. Even worse, some of these substances
were found to hurl chunks of themselves away quite spontaneously, thus
reducing the mass of the substance that was left. This clearly did not jibe
with classical ideas of how matter ought to behave; physicists were well
enough acquainted with such things as dynamic explosions, but this was
The Manifestations of Energy 179
something else again. The chunks that were thrown off by the radioactive
substances were in many cases hurled away with incredibly high energy,
with no indication of just where all that energy came from.
Then, further to confound the experts, when techniques were devised
to make careful measurements it was found that the total mass of such a
substance before a chunk of itself was hurled away was significantly
greater than the mass of the substance remaining after an emission plus
the mass of the particle it had emitted. This seemed to physicists very much
like subtracting two from five and ending up with only IVz as a result.
Here was a case in which the total mass of a system seemed to decrease
by a tiny amount in the course of an interaction, while at the same time
a rather enormous amount of energy seemed to appear unbidden out of
nowhere! This kind of natural occurrence simply did not add up in terms
of conservation of mass or of conservation of energy, yet the fact that it
indeed occurred forced physicists much against their will to begin to-
wonder if these conservation laws were really as valid as they had been
thought to be after all.
Most of us know the end of the story. Albert Einstein came to the rescue
with an idea that is now familiar to us all: that energy itself has mass, and
conversely that solid matter is in reality nothing more than another form
of energy. Granted that energy does not have very much mass, and granted
that a very tiny quantity of matter is equivalent to (or represents) a per
fectly staggering amount of energy. But even so, the concept explained
how mass could be "lost" and energy "created" in cases when radioactive
substances spontaneously hurled pieces of themselves into space. The
"missing" mass after such an occurrence was later found experimentally
to be exactly the amount of mass necessary to produce or be converted
into the amount of energy the emitted particle flew off. with.
Later we will discuss in more detail the significance of Einstein's familiar
equation for the equivalence of mass and energy E = mc 2 . For now it is
enough to point out that modern physicists regard mass and energy as
essentially two manifestations of the same thing, "convertible" from one
to the other and back again under various special circumstances; and to
note that the amount of mass "represented" by a given amount of energy
is very tiny, whereas the amount of energy "represented" by a given mass
of matter is extremely large.
In modern times physicists do not speak of the law of conservation of
matter or the law of conservation of energy separately, simply because they
can't. For all practical purposes the two laws have been combined into
a single all-inclusive law which can be stated as follows:
In any closed system the matter and/or energy that it contains can
neither be created nor destroyed; all the matter and/or energy of such a
system invariably remain constant.
180 The Universe of Classical Physics
And we must remember, whenever we think of that law of nature, that the
entire universe may someday be found to be a closed system.
MAXWELL'S DEMON AND THE PUZZLE OF ENTROPY
Today we know that the law of conservation of mass and/or energy still
remains one of the strongest of all natural laws. We also know of the wide
variety of forms in which energy can be found, and that it is possible to
convert energy in one form into energy in another form without loss. A list
of the major forms in which energy is encountered would include the fol
lowing:
Mechanical energy
Potential energy
Kinetic energy
Chemical energy
Electrical energy or, more accurately, electromagnetic energy
Heat energy (including heat produced by friction)
Atomic and nuclear energy (including the energy produced when
matter is converted into its equivalent amount of energy)
We have seen that it is possible to convert energy from any one of these
forms into energy of any other form without ever either creating any new
energy (thus increasing the total amount present in the universe) or destroy
ing any energy (thus decreasing the total amount of energy in the universe).
In many cases, little or no difficulty is encountered in converting energy from
one form to another, and at least theoretically it should be possible to con
vert 100 per cent of a given amount of energy in one form into any other
form desired if the proper technique for accomplishing this is known.
Yet when we get down to the practicalities of actually doing this, we soon
discover somewhat surprisingly that here is a case in which nature seems to
play favorites. While energy in one form can be converted into energy in any
other form, it is much more difficult to convert it into certain forms than
into certain other forms, while energy in any form seems to be converted so
very readily into certain other forms that we have difficulty avoiding those
forms and converting it instead into the specific form that we want.
It seems very much as though the various major forms of energy occupy
comparatively higher or lower rungs on a ladder, so that it is relatively easy
to convert higher forms of energy into lower forms, but exceedingly difficult
to convert lower forms of energy into higher forms. And of all forms of
energy known, the form that occupies the lowest rung on the ladder the
form of energy that all the other forms can be most readily converted into
The Manifestations of Energy jgj
and the form that is most difficult to convert into any other form is in many
ways the least useful form of energy that we know: heat energy.
In fact, energy in the form of heat persistently keeps turning up at times
and in places where it is simply not wanted at all. We discussed at some
length the conversion of kinetic energy to potential energy and back again,
and found that under imaginary ideal conditions 100 per cent of the kinetic
energy of a system could be converted to potential energy and then 100
per cent of the potential energy could be converted back to kinetic energy.
But any time such a conversion is attempted under real conditions, some of
the energy involved in each conversion is invariably converted into heat and
dissipated into the atmosphere, whether we like it or not. Thus when we have
a wheel spinning around an axle, its kinetic energy will gradually be con
verted to heat and the axle and hub will warm up. Then eventually the axle
will cool and the heat energy will be dissipated into the air.
This does not mean, of course, that the
