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these phenomena involve nothing but combinations of NÑewton”s laws, but it is
sometimes hard to believe that only #' = ma is at work.
The more complicated objects we deal with can be of several kinds: water
fowing, galaxies whirling, and so on. “The simplest “eomplicated” object to
analyze, at the start, is what we call a r/g/đ body, a solid object that is turning
as it moves about. However, even such a simple object may have a most complex
motion, and we shall therefore first consider the simplest aspects of such motion,
in which an extended body rotates about a fized azs. A given point on such a
body then moves in a plane perpendicular to this axis. Such rotation of a body
about a fñxed axis is called pÏiøne rotatfion or rotation in two dimensions. We
shall later generalize the results to three dimensions, but in doïng so we shall
fnd that, unlike the case of ordinary particle mechanics, rotations are subtle and
hard to understand unless we first get a solid grounding in two dimensions.
The first interesting theorem concerning the motion of complicated objects
can be observed at work if we throw an object made of a lot of blocks and spokes,
held together by strings, into the aïr. Of course we know it goes in a parabola,
because we studied that for a particle. But now our object is no‡ a particle; 1%
wobbles and it jiggles, and so on. I$ does go in a parabola though; one can see
that. Wha# goes in a parabola? Certainly not the point on the corner of the
block, because that is jiggling about; neither is i9 the end of the wooden stick,
--- Trang 334 ---
or the middle of the wooden stick, or the middle of the block. But something
goes In a parabola, there is an efective “center” which moves In a parabola. So
our first theorem about complicated objects is to demonstrate that there 7s a
mean position which is mathematically defñnable, but not necessarily a point of
the material itself, which goes in a parabola. That ¡is called the theorem of the
center of the mass, and the proof of it is as follows.
W©e may consider any object as beïing made of lots of little particles, the atoms,
with various forces among them. Let ¿ represent an index which defines one of
the particles. (There are millions of them, so ¿ goes to 102, or something.) Then
the force on the ?th particle is, of course, the mass times the acceleration of that
particle:
E¡ = m;(dŠr;/d12). (18.1)
In the next few chapters our moving objects will be ones in which all the
parts are moving at speeds very much slower than the speed of light, and we shall
use the nonrelativistic approximation for all quantities. In these circumstances
the mass is constant, so that
E¡ = dẦ(m¿r;)/dfẺ. (18.2)
T we now add the force on all the particles, that is, IÝ we take the sum of all
the #;'s for all the diferent indexes, we get the total force, #'. Ôn the other
side of the equation, we get the same thing as though we added before the
diferentiation: Z(S )
¡ Th¿T;
» E;=P=—=“— (18.3)
Therefore the total force is the second derivative of the masses times their
positions, added together.
Now the total force on all the particles is the same as the ezternal force. Why?
Although there are all kinds of forces on the particles because of the strings, the
wigplings, the pullings and pushings, and the atomie forces, and who knows what,
and we have to add all these together, we are rescued by Newton's Third Law.
Between any two particles the action and reaction are equal, so that when we
add all the equations together, if any two particles have forces between them I1
caneels out in the sum; therefore the net result is only those forces which arise
from other particles which are not included in whatever object we decide to sun
over. So if Eq. (18.3) is the sum over a certain number of the particles, which
--- Trang 335 ---
together are called “the object,” then the ez#ernal force on the total object 1s
equal to the sum of ai the forces on all its constituent particles.
NÑow it would be nice if we could write Bq. (18.3) as the total mass times
some acceleration. We can. Let us say /Mƒ is the sum of all the masses, i.e., the
total mass. Then if we deffne a certain vector i? to be
R=À mr//(M, (18.4)
Eq. (18.3) will be simply
FP= d (MR)/dt? = M(d°R/dt2), (18.5)
since # is a constant. Thus we fñnd that the external force is the tota]l mass
times the acceleration of an imaginary point whose location is l. This poïnt is
called the cen#er oƒ mmass of the body. It is a point somewhere in the “middle7
of the object, a kind of average r in which the diferent ?;'s have weights or
Importances proportional to the masses.
W© shall discuss this important theorem in more detail in a later chapter, and
we shall therefore limit our remarks to two points: First, if the external Íorces are
zero, if the object were floating in empty space, it might whirl, and jiggle, and
twist, and do all kinds of things. But the center oƒ mass, thìs artiflcially invented,
calculated position, somewhere in the middle, +uiÏl moue u#th a constant 0elocitg.
In particular, if it is initially at rest, ít will stay at rest. So If we have some kind
of a box, perhaps a space ship, with people ín it, and we calculate the location
of the center of mass and fñnd it is standing still, then the center of mass will
continue to stand still if no external forces are acting on the box. Of course, the
space ship may move a little in space, but that is because the people are walking
back and forth inside; when one walks toward the front, the ship goes toward
the back so as to keep the average position of all the masses in exactly the same
place.
ls rocket propulsion therefore absolutely impossible because one cannot move
the center of mass? No; but of course we fñnd that to propel an interesting part
of the rocket, an uninteresting part must be thrown away. In other words, if we
start with a rocket at zero velocity and we spit some gas out the back end, then
this little blob of gas goes one way as the rocket ship goes the other, but the
center of mass is still exactly where it was before. So we simply move the part
that we are interested in against the part we are not interested in.
--- Trang 336 ---
The second point concerning the center of mass, which is the reason we
introduced ït into our discussion at this time, is that it may be treated separately
from the “internal” motions of an object, and may therefore be ignored in our
discussion oŸ rotation.
18-2 Rotation of a rigid body
Now let us discuss rotations. Of course an ordinary object does not simply
rotate, it wobbles, shakes, and bends, so to simplify matters we shall diseuss the
motion of a nonexistent ideal object which we call a rigid body. 'Phis means an
object in which the forces bebween the atoms are so strong, and of such character,
that the little forces that are needed to move it do not bend it. Its shape stays