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Unformatted text preview: I2 Characteristics of Force 12—1 What is a force? Although it is interesting and worth while to study the physical laws simply
because they help us to understand and to use nature, one ought to stop every once
in a while and think, “What do they really mean?” The meaning of any statement
is a subject that has interested and troubled philosophers from time immemorial,
and the meaning of physical laws is even more interesting, because it is generally
believed that these laws represent some kind of real knowledge. The meaning of
knowledge is a deep problem in philosophy, and it is always important to ask,
“What does it mean?" Let us ask, “What is the meaning of the physical laws of Newton, which we
write as F = ma? What is the meaning of force, mass, and acceleration?” Well,
we can intuitively sense the meaning of mass, and we can deﬁne acceleration if we
know the meaning of position and time. We shall not discuss those meanings,
but shall concentrate on the new concept of force. The answer is equally simple:
“If a body is accelerating, then there is a force on it.” That is what Newton’s laws
say, so the most precise and beautiful deﬁnition of force imaginable might simply
be to say that force is the mass of an object times the acceleration. Suppose we
have a law which says that the conservation of momentum is valid if the sum
of all the external forces is zero; then the question arises, “What does it mean,
that the sum of all the external forces is zero?” A pleasant way to deﬁne that
statement would be: “When the total momentum is a constant, then the sum of the
external forces is zero.” There must be something wrong with that, because it is
just not saying anything new. If we have discovered a fundamental law, which
asserts that the force is equal to the mass times the acceleration, and then deﬁne the
force to be the mass times the acceleration, we have found out nothing. We could
also deﬁne force to mean that a moving object with no force acting on it continues
to move with constant velocity in a straight line. If we then observe an object
not moving in a straight line with a constant velocity, we might say that
there is a force on it. Now such things certainly cannot be the content of physics,
because they are deﬁnitions going in a circle. The Newtonian statement above,
however, seems to be a most precise definition of force, and one that appeals to
the mathematician; nevertheless, it is completely useless, because no prediction
whatsoever can be made from a deﬁnition. One might sit in an armchair all day
long and deﬁne words at will, but to ﬁnd out what happens when two balls push
against each other, or when a weight is hung on a spring, is another matter al
together, because the way the bodies behave is something completely outside any
choice of deﬁnitions. For example, if we were to choose to say that an object left to itself keeps its
position and does not move, then when we see something drifting, we could say
that must be due to a “gorce”——a gorce is the rate of change of position. Now we
have a wonderful new law, everything stands still except when a gorce is acting. You
see, that would be analogous to the above deﬁnition of force, and itwould contain no
information. The real content of Newton’s laws is this: that the force is supposed
to have some independent properties, in addition to the law F = ma; but the
speciﬁc independent properties that the force has were not completely described
by Newton or by anybody else, and therefore the physical law F = ma is an
incomplete law. It implies that if we study the mass times the acceleration and i‘ call the product the force, i.e., if we study the characteristics of force as a program 12—] 12—1 What is a force? 12—2 Friction 12—3 Molecular forces 12—4 Fundamental forces. Fields
12—5 Pseudo forces 12—6 Nuclear forces of interest, then we shall ﬁnd that forces have some simplicity; the law is a good
program for analyzing nature, it is a suggestion that the forces will be simple. Now the ﬁrst example of such forces was the complete law of gravitation,
which was given by Newton, and in stating the law he answered the question,
“What is the force?” If there were nothing but gravitation, then the combination
of this law and the force law (second law of motion) Would be a complete theory,
but there is much more than gravitation, and we want to use Newton’s laws in
many different situations. Therefore in order to proceed we have to tell something
about the properties of force. For example, in dealing with force the tacit assumption is always made that
the force is equal to zero unless some physical body is present, that if we ﬁnd a
force that is not equal to zero we also ﬁnd something in the neighborhood that
is a source of the force. This assumption is entirely different from the case of the
“gorce” that we introduced above. One of the most important characteristics of
force is that it has a material origin, and this is not just a deﬁnition. Newton also gave one rule about the force: that the forces between interacting
bodies are equal and opposite—action equals reaction; that rule, it turns out, is
not exactly true. In fact, the law F = ma is not exactly true; if it were a deﬁnition
we should have to say that it is always exactly true; but it is not. The student may object, “I do not like this imprecision, I should like to have
everything deﬁned exactly; in fact, it says in some books that any science is an exact
subject, in which everything is deﬁned.” If you insist upon a precise deﬁnition of
force, you will never get it‘ First, because Newton’s Second Law is not exact, and
second, because in order to understand physical laws you must understand that
they are all some kind of approximation. Any simple idea is approximate; as an illustration, consider an object, . . .
what is an object? Philosophers are always saying, “Well, just take a chair for
example.” The moment they say that, you know that they do not know what
they are talking about any more. What is a chair? Well, a chair is a certain thing
over there . . . certain?, how certain? The atoms are evaporating from it from time
to time—not many atoms, but a few—dirt falls on it and gets dissolved in the paint;
so to deﬁne a chair precrsely, to say exactly which atoms are chair, and which
atoms are air, or which atoms are dirt, or which atoms are paint that belongs to
the char is impossible. So the mass of a chair can be deﬁned only approximately.
In the same way, to deﬁne the mass of a single object is impossible, because there
are not any single, leftalone objects in the world—every object is a mixture of a
lot of things, so we can deal with it only as a series of approximations and idealiza
trons. The trick is the idealizations. To an excellent approximation of perhaps one
part in 101°, the number of atoms in the chair does not change in a minute, and if
we are not too precise we may idealize the chair as a deﬁnite thing; in the same way
we shall learn about the characteristics of force, in an ideal fashion, if we are not
too precise. One may be dissatisﬁed with the approximate view of nature that
physics tries to obtain (the attempt is always to increase the accuracy of the
approximation), and may prefer a mathematical deﬁnition; but mathematical
deﬁnitions can never work In the real world. A mathematical deﬁnition Will be
good for mathematics, in which all the logic can be followed out completely, but
the physical world is complex, as we have indicated in a number of examples, such
as those of the ocean waves and a glass of wine. When we try to isolate pieces of it,
to talk about one mass, the wine and the glass, how can we know which is which,
when one dissolves in the other? The forces on a single thing already involve
approximation, and if we have a system of discourse about the real world, then
that system, at least for the present day, must involve approximations of some
kind. This system is quite unlike the case of mathematics, in which everything can
be deﬁned, and then we do not know what we are talking about. In fact, the glory
of mathematics is that we do not have to say what we are talking about. The glory
is that the laws, the arguments, and the logic are independent of what “it" is. If
we have any other set of objects that obey the same system of axioms as Euclid’s 1 2—2 geometry, then if we make new deﬁnitions and follow them out with correct logic,
all the consequences will be correct, and it makes no difference what the subject
was. In nature, however, when we draw a line or establish a line by using a light
beam and a theodolite, as we do in surveying, are we measuring a line in the sense
of Euclid? No, we are making an approximation; the cross hair has some width,
but a geometrical line has no width, and so, whether Euclidean geometry can be
used for surveying or not is a physical question, not a mathematical questlon.
However, from an experimental standpoint, not a mathematical standpoint, we
need to know whether the laws of Euclid apply to the kind of geometry that we
use in measuring land; so we make a hypothesis that it does, and it works pretty
well; but it is not precise, because our surveying lines are not really geometrical
lines. Whether or not those lines of Euclid, which are really abstract, apply to the
lines of experience is a question for experience; it is not a question that can be
answered by sheer reason. In the same way, we cannot just call F = ma a deﬁnition, deduce everything
purely mathematically, and make mechanics a mathematical theory, when me
chanics is a description of nature. By establishing suitable postulates it is always
possible to make a system of mathematics, just as Euclid did, but we cannot make
a mathematics of the world, because sooner or later we have to ﬁnd out whether
the axioms are valid for the objects of nature. Thus we immediately get involved
with these complicated and “dirty“ objects of nature, but with approximations
ever increasing in accuracy. 12—2 Friction The foregoing considerations show that a true understanding of Newton’s
laws requires a discussion of forces, and it is the purpose of this chapter to introduce
such a discussion, as a kind of completion of Newton’s laws. We have already
studied the deﬁnitions of acceleration and related ideas, but now we have to study
the properties of force, and this chapter, unlike the previous chapters, will not be
very precise, because forces are quite complicated. To begin with a particular force, let us consider the drag on an airplane
ﬂying through the air. What is the law for that force? (Surely there is a law for
every force, we must have a law!) One can hardly think that the law for that force
will be simple. Try to imagine what makes a drag on an airplane ﬂying through
the air—the air rushing over the wings, the swirling in the back, the changes going
on around the fuselage, and many other complications, and you see that there is
not going to be a simple law. On the other hand, it is a remarkable fact that the
drag force on an airplane is approximately a constant times the square of the
velocity, or F ~ cv2. Now what is the status of such a law, is it analogous to F = ma? Not at all,
because in the ﬁrst place this law is an empirical thing that is obtained roughly by
tests in a wind tunnel. You say, “Well F = ma might be empirical too.” That is
not the reason that there is a diﬂerence. The difference is not that it is empirical,
but that, as we understand nature, this law is the result of an enormous complexity
of events and is not, fundamentally, a simple thing. If we continue to study it more
and more, measuring more and more accurately, the law will continue to become
more complicated, not less. In other words, as we study this law of the drag on an
airplane more and more closely, we ﬁnd out that it is “falser” and “falser,” and
the more deeply we study it, and the more accurately we measure, the more compli
cated the truth becomes; so in that sense we consider it not to result from a simple,
fundamental process, which agrees with our original surmise. For example, if the
velocity is extremely low, so low that an ordinary airplane is not ﬂying, as when
the airplane is dragged slowly through the air, then the law changes, and the drag
friction depends more nearly linearly on the velocity. To take another example,
the frictional drag on a ball or a bubble or anything that is moving slowly through
a viscous liquid like honey, is proportional to the velocity, but for motion so fast
that the ﬂuid swirls around (honey does not but water and air do) then the drag
becomes more nearly proportional to the square of the velocity (F = c222), and 12—3 , s~w .__~c a» > DIRECTION OF MOTION Fig. 12—1 . The relation between fric
tional force and the normal force for
sliding contact. if the velocity continues to increase, then even this law begins to fail. People who
say, “Well the coefficient changes slightly,” are dodging the issue. Second, there
are other great complications: can this force on the airplane be divided or analyzed
as a force on the wings, a force on the front, and so on? Indeed, this can be done,
if we are concerned about the torques here and there, but then we have to get
special laws for the force on the wings, and so on. It is an amazing fact that the
force on a wing depends upon the other wing: in other words, if we take the airplane
apart and put just one wing in the air, then the force is not the same as if the rest
of the plane were there. The reason, of course, is that some of the wind that hits
the front goes around to the wings and changes the force on the wings. It seems a
miracle that there is such a simple, rough, empirical law that can be used in the
design of airplanes, but this law is not in the same class as the basic laws of physics,
and further study of it will only make it more and more complicated. A study of
how the coeﬁicient c depends on the shape of the front of the airplane is, to put
it mildly, frustrating. There Just is no simple law for determining the coefﬁcient
in terms of the shape of the airplane. In contrast, the law of gravitation is simple,
and further study only indicates its greater simplicity. We have just discussed two cases of friction, resulting from fast movement in
air and slow movement in honey. There is another kind of friction, called dry
friction or sliding friction, which occurs when one solid body slides on another.
In this case a force is needed to maintain motion. This is called a frictional force,
and its origin, also, is a very complicated matter. Both surfaces of contact are
irregular, on an atomic level. There are many points of contact where the atoms
seem to cling together, and then, as the shdmg body is pulled along, the atoms
snap apart and vibration ensues; something like that has to happen. Formerly
the mechanism of this friction was thought to be very simple, that the surfaces
were merely full of irregularities and the friction originated in lifting the slider
over the bumps; but this cannot be, for there is no loss of energy in that process,
whereas power is in fact consumed. The mechanism of power loss is that as the
slider snaps over the bumps, the bumps deform and then generate waves and
atomic motions and, after a while, heat. in the two bodies. Now it is very remark
able that again, empirically, this friction can be described approximately by a
simple law. This law is that the force needed to overcome friction and to drag one
object over another depends upon the normal force (i.e., perpendicular to the
surface) between the two surfaces that are in contact. Actually, to a fairly good
approximation, the frictional force is proportional to this normal force, and has
a more or less constant coefﬁcient; that is, F = MN, (12.1) where y is called the coeﬂfcient of friction (Fig. 12—1). Although this coefﬁcient is
not exactly constant, the formula is a good empirical rule for judging approxi
mately the amount of force that will be needed in certain practical or engineering
circumstances. If the normal force or the speed of motion gets too big, the law fails
because of the excessive heat generated. It is important to realize that each of these
empirical laws has its limitations, beyond which it does not really work. That the formula F = ,uN is approximately correct can be demonstrated by
a simple experiment. We set up a plane, inclined at a small angle 0, and place a
block of weight W on the plane. We then tilt the plane at a steeper angle, until
the block just begins to slide from its own weight. The component of the weight
downward along the plane is W sin 0, and this must equal the frictional force F
when the block is sliding uniformly. The component of the weight normal to the
plane is Wcos 0, and this is the normal force N. With these values, the formula
becomes Wsin 9 = MW cos 9, from which we get it = sin 6/cos 0 = tan 0. If
this law were exactly true, an object would start to slide at some deﬁnite inclination.
If the same block is loaded by putting extra weight on it, then, although W is
increased, all the forces in the formula are increased in the same proportion, and
W cancels out. If it stays constant, the loaded block will slide again at the same
slope. When the angle 6 is determined by trial with the original weight, it is found 12—4 that with the greater weight the block will slide at about the same angle. This will
be true even when one weight is many times as great as the other, and so we con
clude that the coefﬁcient of friction is independent of the weight. In performing this experiment it is noticeable that when the plane is tilted
at about the correct angle 6, the block does not slide steadily but in a halting fashion.
At one place it may stop, at another it may move with acceleration. This behavior
indicates that the coefﬁcient of friction is only roughly a constant, and varies from
place to place along the plane. The same erratic behavior is observed whether the
block is loaded or not. Such variations are caused by different degrees of smooth
ness or hardness of the plane, and perhaps dirt, oxides, or other foreign matter.
The tables that list purported values of ,u for “steel on steel,” “copper on copper,”
and the like, are all false, because they ignore the factors mentioned above, which
really determine ,u. The friction is never due to “copper on copper," etc., but to
the impurities clinging to the copper. In experiments of the type described above, the friction is nearly independent
of the velocity. Many people believe that the friction to be overcome to get
something started (static friction) exceeds the force required to keep it sliding
(sliding friction), but with dry metals it is very hard to show any difference. The
opinion probably arises from experiences where small bits of oil or lubricant are
present, or where blocks, for example, are supported by springs or other ﬂexible
supports so that they appear to bind. It is quite difﬁcult to do accurate quantitative experiments in friction, and the
laws of friction are still not analyzed very well, in spite of the enormous engineering
value of an accurate analysis. Although the law F = nN is fairly accurate once the
surfaces are standardized, the reason for this form of the law is not really under
understood. To show that the coefﬁcient p is nearly independent of velocity
requires some delicate experimentation, because the apparent friction is much
reduced if the lower surface vibrates very fast. When the experiment is done at
very high speed, care must be taken that the objects do not vibrate relative to one
another, since apparent decreases of the friction at high speed are often due to
vibrations. At any rate, this friction law is another of those semiempirical laws
that are not thoroughly understood, and in view of all the work that has been
done it is surprising that more understanding of this phenomenon has not come
about...
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 Spring '09
 LeeKinohara
 Physics, Force

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