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change. When we do this, all kinds of wonderful things happen, because of the
law of the conservation of angular momentum: lf the external torque is zero, then
the angular momentum, the moment of inertia times omega, remains constant.
Initially, we were rotating with a large moment of inertia 1 at a low angular
velocity œ, and the angular momentum was ;ư. Then we changed our moment
oŸ inertia by pulling our arms in, say to a smaller value f¿. 'Then the product Tœ,
which has to stay the same because the total angular momentum has to stay the
same, was Ïaœs2. So hư = la. That 1s, IÍ we reduce the moment of inertia, we
have to #nwcrease the angular velocity.
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I9
(orefor' of IWerss: /Wort©ortÉ @Ÿ Írt©rfier
19-1 Properties of the center of mass
In the previous chapter we found that iŸ a great many Íorces are acting on a
complicated mass of particles, whether the particles comprise a rigid or a nonrigid
body, or a cloud oŸ stars, or anything else, and we ñnd the sum of all the forces
(that is, oŸ course, the external forces, because the internal forces balance out),
then if we consider the body as a whole, and say it has a total mass jM, there is
a certain point “inside” the body, called the cenmter oƒ mass, suụch that the net
resulting external force produces an acceleration of this point, jus as though the
whole mass were concentrated there. Let us now discuss the center of mass in a
little more detail.
The location of the center of mass (abbreviated CM) is given by the equation
» THẠT¡
Tcw Sâm (19.1)
This is, of course, a vector equation which is really three equations, one for
cach of the three directions. We shall consider only the z-direction, because
1ƒ we can understand that one, we can understand the other two. What does
Xe = È})m;+¡/S `m¿ mean? Suppose for a moment that the object is divided
into little pieces, all of which have the same mass ?n; then the total mass 1s
simply the number / of pieces times the mass oŸ one piece, say one gram, or any
unit. Then this equation simply says that we add all the z's, and then divide
by the number of things that we have added: Xe =?n})z;/mN =3 `z/;/N.
In other words, Xem is the average of all the +ˆs, If the masses are equal. But
Suppose one of them were twice as heavy as the others. 'Phen in the sum, that +
would come in twice. 'This is easy to understand, for we can think of this double
mass as being split into two equal ones, just like the others; then in taking the
average, of course, we have to count that + twice because there are two Immasses
--- Trang 347 ---
there. Thus X is the average position, in the z-direction, of all the masses, every
mass being counted a number of times proportional to the mass, as though 1$
were divided into “little grams.” From this it is easy to prove that X must be
somewhere between the largest and the smallest z, and, therefore lies inside the
envelope including the entire body. It does not have to be in the mater?al of the
body, for the body could be a cirele, like a hoop, and the center of mass is in the
center of the hoop, not in the hoop itself.
Of course, if an object is symmetrical in some way, for instance, a rectangle,
so that it has a plane of symmetry, the center of mass lies somewhere on the
plane of symmetry. In the case of a rectangle there are two planes, and that
locates it uniquely. But ïŸ it is just any symmetrical object, then the center of
gravity lies somewhere on the axis of symmetry, because in those circumstances
there are as many positive as negative 4s.
: ` QCM
Fig. 19-1. The CM of a compound body lies on the line Joining the
CM's of the two composite parts.
Another interesting proposition is the following very curious one. Suppose
that we imagine an object to be made oŸ two pieces, 4 and Ö (Eig. 19-1). Then
the center oŸ mass of the whole object can be calculated as follows. First, ñnd the
center of mass of piece 4, and then of piece Ø. Also, fnd the total mass of each
pilece, ƒ4 and Míp. hen consider a new problem, in which a pø¿n‡ mass Ma 1s
at the center of mass of object 4, and another øø¿n‡ mass Míp is at the center
of mass of object #. 'The center of mass of these two point masses is then the
center of mass of the whole object. In other words, if the centers of mass of
various parts of an object have been worked out, we do not have to start all over
again to find the center of mass of the whole object; we Just have to put the
pleces together, treating each one as a point mass situated at the center oŸ mass
of that piece. Let us see why that is. Suppose that we wanted to calculate the
center of mass of a complete object, some of whose particles are considered to
be members of object A and some mermbers of object . The total sum À `?nm;#;
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can then be split into two pieces—the sum 3) ¿?m¿#¿ for the A object only, and
the sum 3 }„ m¿z; for object Ö only. Now if we were computing the center of
mass of object 4 alone, we would have exactly the first of these sums, and we
know that this by itselfis Ma Xa, the total mass of all the particles in A tỉmes
the position of the center of mass of 4, because that is the theorem of the center
of mass, applied to object A. In the same mamner, jus6 by looking at object Ö,
we get Míp Xp, and of course, adding the two yields ÄMƒ XeM:
= MAXaA+ MpXn. (19.2)
NÑow since Ä⁄ is evidently the sum of Mu and ăpg, we see that Eq. (19.2) can
be interpreted as a special example of the center of mass formula for two point
objects, one of mass f4 located at Xa and the other of mass Mfp located at Xg.
The theorem concerning the motion of the center of mass is very interesting,
and has played an Important part in the development of our understanding of
physics. Suppose we assume that Newton”s law is right for the small component
parts of a much larger object. Then this theorem shows that Newton's law is also
correct for the larger object, even iŸ we do not study the details of the object, but
only the total force acting on it and its mass. In other words, Newton's law has
the peculiar property that If it is right on a certain small scale, then 1% will be
right on a larger scale. Ifwe do not consider a baseball as a tremendously complex
thing, made of myriads of interacting particles, but study only the motion of the
center of mass and the external forces on the ball, we fnd #' = ma, where F'
1s the external force on the baseball, m 1s Its mass, and ø is the acceleration of
1ts center of mass. 5o = rmœ is a law which reproduces itself on a larger scale.
(There ought to be a good word, out of the Greek, perhaps, to describe a law
which reproduces the same law on a larger scale.)
Of course, one might suspect that the first laws that would be discovered
by human beings would be those that would reproduce themselves on a larger
scale. Why? Because the actual scale of the fundamental gears and wheels of the
universe are of atomie dimensions, which are so much finer than our observations
that we are nowhere near that scale in our ordinary observations. So the first