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obvious the nutation is.
` 7x7 =Z
Fig. 20-5. Actual motion of tip of axIs of gyroscope under gravity Just
after releasing axis previously held fixed.
'When the motion settles down, the axis of the gyro is a little bít lower than
it was at the start. Why? (These are the more complicated details, but we bring
them in because we do not want the reader to get the idea that the gyroscope 1s
an absolute miracle. It 7s a wonderful thing, but it is not a miracle.) IÝ we were
holding the axis absolutely horizontally, and suddenly let go, then the simple
precession equation would t$ell us that it precesses, that it goes around in a
horizontal plane. But that is impossiblel Although we neglected it before, it is
true that the wheel has sornme moment of inertia about the precession axis, and
1f it is moving about that axis, even slowly, it has a weak angular momentum
about the axis. Where did it come from? Tf the pivots are perfect, there is no
torque about the vertical axis. How then does it get to precess if there is no
change in the angular momentum? 'Phe answer is that the cycloidal motion of
the end of the axis damps down to the average, steady motion of the center of
the equivalent rolling circle. 'Phat is, it settles down a little bit low. Because it is
low, the spin angular momentum now has a small vertical component, which is
exactly what ¡is needed for the precession. So you see it has to go down a little,
in order to go around. It has to yield a little bít to the gravity; by turning is
axis down a little bit, it maintains the rotation about the vertical axis. That,
then, is the way a gyroscope WwOorks.
--- Trang 375 ---
20-4 Angular momentum of a solid body
Before we leave the subject of rotations in three dimensions, we shall discuss,
at least qualitatively, a few effects that occur in three-dimensional rotations that
are not self-evident. The main efect is that, in general, the angular momentum
of a rigid body is no necessari in the same direction as the angular velocity.
Consider a wheel that is fastened onto a shaft ïn a lopsided fashion, but with
the axis through the center of gravity, to be sure (Fig. 20-6). When we spin
the wheel around the axis, anybody knows that there will be shaking at the
bearings because of the lopsided way we have it mounted. Qualitatively, we
know that in the rotating system there is centrifugal force acting on the wheel,
trying to throw its mass as far as possible from the axis. 'This tends to line
up the plane of the wheel so that it is perpendicular to the axis. To resist this
tendenecy, a torque is exerted by the bearings. lf there is a torque exerted by the
bearings, there must be a rate of change of angular momentum. How can there
be a rate of change of angular momentum when we are simply turning the wheel
about the axis? Suppose we break the angular velocity œ into components œ1
and œs perpendicular and parallel to the plane of the wheel. What is the angular
mmomentum? “The moments of inertia about these two axes are đjƒƒferent, so the
angular momenbum components, which (in these particular, special axes only)
are equal to the moments of inertia times the corresponding angular velocity
components, are in a đjfƒferent ratio than are the angular velocity components.
'Therefore the angular momentum vector is in a direction in space øø‡ along the
axis. When we turn the object, we have to turn the angular momentum vector
in space, so we must exert torques on the shaft.
Lìị = hư
N Ầ L
L U ^
La = laua «
Fig. 20-6. The angular momentum of a rotating body Is not necessarily
parallel to the angular velocity.
Although it is much too complicated to prove here, there is a very important
and interesting property of the moment of inertia which is easy to describe and to
--- Trang 376 ---
use, and which is the basis of our above analysis. This property is the following:
Any rigid body, even an irregular one like a potato, possesses three mutually
perpendicular axes through the ƠM, such that the moment of inertia about one
of these axes has the greatest possible value for any axis through the ƠM, the
moment of inertia about another of the axes has the minwửnwm possible value,
and the moment of inertia about the third axis is intermediate between these two
(or equal to one of them). These axes are called the pr/ncipal azes of the body,
and they have the important property that If the body is rotating about one
of them, its angular momentum is in the same direction as the angular velocity.
For a body having axes of symmetry, the principal axes are along the symmetry
a%Xes.
z4@:~———_
lổ JÄÌ t |
| J⁄ ⁄ s
F. dã ⁄
Fig. 20-7. The angular velocity and angular momentum of a rigid
body (4> B> C).
TÝ we take the z-, -, and z-axes along the principal axes, and call the
corresponding principal moments of inertia A, Ö, and Œ, we may easily evaluate
the angular momentum and the kinetic energy of rotation of the body for any
angular velocity œ0. IÝ we resolve œ into componenfs œ„, (œ„, and œ; along the
Z-, -, z-axes, and use unit vectors ?, 7, k, also along zø, , z, we may write the
angular momentum as
TL Au„¿ + Buy 2 + Cu¿k. (20.16)
--- Trang 377 ---
The kinetic energy of rotation is
KE = š(Au2 + Bưu + C2) (20.17)
--- Trang 378 ---
Tho lÍtrrreorerc Ê)setÏletéor-
21-1 Linear diferential equations
In the study of physics, usually the course is divided into a series of subjects,
such as mechanics, electricity, optics, etbc., and one studies one subJect after the
other. Eor example, this course has so far dealt mostly with mechanics. But a
strange thing occurs again and again: the equations which appear in diferent
fields of physics, and even in other sciences, are often almost exactly the same,
so that many phenomena have analogs in these different fñelds. To take the
simplest example, the propagation oŸ sound waves is in many ways analogous to
the propagation of light waves. IÝ we study acoustics in great detail we discover
that much of the work is the same as it would be iŸ we were studying opties in
great detail. 5o the study of a phenomenon in one field may permit an extension
of our knowledge in another field. It is best to realize from the first that such
extensions are possible, for otherwise one might not understand the reason for
spending a great deal of time and energy on what appears to be only a small