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would be a centripetal force if Ï were to run around a circle with velocity 0x.”
'This is simply the centripetal force that Moe would expect, having nothing to do
with rotation. In addition, Moe is quite aware that there is another centripetal
force that would act even on objects which are standing still on his carousel. “This
1s the third term. But there is another term in addition to these, namely the
second term, which is again 2n. 'Phe Coriolis force ¿ was tangential when
the velocity was radial, and now it is radial when the velocity is tangenmtial. In
fact, one expression has a minus sign relative to the other. The force is always
in the same direction, relative to the velocity, no matter in which direction the
velocity is. The force is at right angles to the velocity, and of magnitude 2m0.
--- Trang 362 ---
}ŸOof(tífOIt ít SJ06EC©
20-1 Torques ỉn three dimensions
In this chapter we shall discuss one of the most remarkable and amusing
consequences of mechanics, the behavior of a rotating wheel. In order to do
this we must first extend the mathematical formulation of rotational motion,
the principles of angular momentum, torque, and so on, to three-dimensional
space. We shall not se these equations in all their generality and study all theïr
consequences, because this would take many years, and we must soon turn to
other subjects. In an introductory course we can present only the fundamental
laws and apply them to a very few situations oŸ special interest.
Pirst, we notice that If we have a rotation in three dimensions, whether of a
rigid body or any other system, what we deduced for two dimensions is still right.
That is, it is still true that z#„ — 9F „ is the torque “in the #-plane,” or the
torque “around the z-axis.” It also turns out that this torque ¡s still equal to the
rate of change oÝ #Ðp„ — 1p„, for iÝ we go back over the derivation of Eq. (18.15)
from Newton's laws we see that we did not have to assume that the motion was In
a plane; when we diferentiate #py — px, we get øF — 9F+„, so thìs theorem ¡is still
right. The quantity #ø„ — px, then, we call the angular momentum belonging
to the z#-plane, or the angular momentum about the z-axis. This being true, we
can use any other pair of axes and get another equation. For instance, we can use
the z-plane, and it is clear from symmetry that iŸƒ we just substitute for z and z
for , we would find ¿/È; — zÈ;, for the torque and ø„ — zø„ would be the angular
tmmomentum associated with the z-plane. Of course we could have another plane,
the zz-plane, and for this we would ñnd zF„ — #EF¿ = d/dt (zp„ — #p;).
'That these three equations can be deduced for the motion of a single particle
is quite clear. Furthermore, if we added such things as #p„ — #p„ together for
many particles and called it the total angular momentum, we would have three
kinds for the three planes ø, z, and zz, and if we did the same with the forces,
--- Trang 363 ---
we would talk about the torque in the planes z, z, and zz also. Thus we would
have laws that the external torque associated with any plane is equal to the rate
of change of the angular momentum associated with that plane. 'This is Just a
generalization of what we wrote in two dimensions.
But now one may say, “Ah, but there are more planes; after all, can we not take
some other plane at some angle, and calculate the torque on that plane from the
forces? Since we would have to write another set of equations for every such plane,
we would have a lot of equationsl” Interestingly enouph, it turns out that iÝ we
were to work out the combination #2 — “F„; for another plane, measuring the
+", Fụ:, ec., in that plane, the result can be written as some cørmbination oŸ the
three expressions for the z-, z- and zz-planes. 'Phere is nothing new. In other
words, if we know what the three torques in the z-, z-, and zz-planes are, then
the torque in any other plane, and correspondingly the angular momentum also,
can be written as some combination of these: six percent of one and ninety-two
percent of another, and so on. 'This property we shall now analyze.
Suppose that in the zz-axes, Joe has worked out all his torques and his
angular momenta in his planes. But Moe has axes #, z/, z” in some other direction.
To make it a little easier, we shall suppose that only the z- and „-axes have
been turned. Moe's zø and + are new, but his z” happens to be the same. That
1s, he has new planes, let us say, for z and zz. He therefore has new torques
and angular momenta which he would work out. For example, his torque in the
z/-plane would be equal to #2 — F>/ and so forth. What we must now do
1s to find the relationship between the new torques and the old torques, so we
will be able to make a connection from one set oŸ axes to the other. 5omeone
may say, “Phat looks just like what we did with vectors.” And indeed, that is
exactly what we are intending to do. Then he may say, “Well, isnˆt torque jusE a
vector?” It does turn out to be a vector, but we do not know that right away
without making an analysis. 5o in the following steps we shall make the analysis.
'W© shall not discuss every step in detail, since we only want to illustrate how it
works. The torques calculated by Joe are
Tự„ụ = #Eụ — Uy,
Ty =UF; — zÈy, (20.1)
Tyy —= Zl„ — œF;.
W© digress at this point to note that in such cases as this one may get the wrong
sien for some quantity if the coordinates are not handled in the right way. Why not
--- Trang 364 ---
write 7z = 2 — F;? 'The problem arises from the fact that a coordinate system may
be either “right-handed” or “left-handed” Having chosen (arbitrarily) a sign for, say
Tx„, then the correct expressions for the other two quantities may always be found by
interchanging the letters zz in either order
# OT #
Z ~— Z —>
Moe now calculates the torques in his system:
Tag! — #' Fụ — V Fy,
Tụ!z! = '.Eà, — Z Fụ, (20.2)
Ty? — z' Fụ„ — x'EQ, *
Now we suppose that one coordinate system is rotated by a ñxed angle Ø, such
that the z- and z/-axes are the same. (This angle Ø has nothing to do with
rotating objects or what is goïng on inside the coordinate system. Ít is merely
the relationship between the axes used by one man and the axes used by the
other, and is supposedly constant.) Thus the coordinates of the two systems are
related by
+ = #øcos 0 + 1 sin 0,
ˆ = cos0 — #sin 0, (20.3)
z' =z.
Likewise, because force is a vector it transforms into the new system in the
same way as do zø, , and z, since a thing is a vector I1 and only if the various
components transform in the same way as z, , and z:
tạ = Fạ cos Ø + Fý, sin Ø,
Tàu = Fy cosØ — F„ sìn Ú, (20.4)