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tỉme A¿, the axis has turned to a new position, tilted at an angle AØ with the |
--- Trang 371 --- |
/ =1 F AL |
x ớy . lọ y |
Fig. 20-2. A gyroscope. |
horizontal. Since the major part of the angular momentum is due to the spin on |
the axis (very little is contributed by the slow turning), we see that the angular |
momentum vector has changed. What is the change in angular momentum? The |
angular momentum does not change in rmagn#tude, but it does change in đứecclion |
by an amount A0. The magnitude of the vector AE is thus AÙ, = bọ A0, so |
that the torque, which is the time rate of change of the angular momentum, is |
7= AL/At = Lạ A0/At = LạO. Taking the directions of the various quantities |
into account, we see that |
T =f) x Lạ. (20.15) |
'Thus, if €) and ọ are both horizontal, as shown in the fgure, 7 is 0ertzcøl. To |
produce such a torque, horizontal forces #" and —.F' must be applied at the ends |
of the axle. How are these forces applied? By our hands, as we try to rotate the |
axis of the wheel into the vertical direction. But NÑewton's Phird Law demands |
that equal and opposite forces (and equal and opposite forqgues) act on 0s. This |
causes us to rotate in the opposite sense about the vertical axis z. |
This result can be generalized for a rapidly spinning top. In the familiar case |
of a spinning top, gravity acting on its center of mass furnishes a torque about |
the point of contact with the floor (see Fig. 20-3). 'This torque is in the horizontal |
direction, and causes the top to precess with its axis moving in a circular cone |
about the vertical. If ©J ¡is the (vertical) angular velocity of precession, we again |
fnd that |
T = dL/dt = © x Lạ. |
Thus, when we apply a torque to a rapidly spinning top, the direction of the |
precessional motion is in the direction of the torque, or at right angles to the |
forces producing the torque. |
We may now claim to understand the precession of gyroscopes, and indeed |
we do, mathematically. However, this is a mathematical thing which, in a sense, |
--- Trang 372 --- |
Fig. 20-3. A rapidly spinning top. Note that the direction of the |
torque vector ¡is the direction of the precession. |
appears as a “miracle.” It will turn out, as we go to more and more advanced |
physics, that many simple things can be deduced mathematically more rapidly |
than they can be really understood in a fundamental or simple sense. This is a |
strange characteristic, and as we get into more and more advanced work there are |
circumstances in which mathematics will produce results which mo one has really |
been able to understand in any direct fashion. An example is the Dirac equation, |
which appears in a very simple and beautiful form, but whose consequences are |
hard to understand. In our particular case, the precession of a top looks like some |
kind of a miracle involving right angles and circles, and twists and right-hand |
serews. What we should try to do is to understand it in a more physical way. |
How can we explain the torque in terms of the real forces and the accelerations? |
W© note that when the wheel is precessing, the particles that are going around |
the wheel are not really moving in a plane because the wheel is precessing |
(see Fig. 20-4). As we explained previously (Fig. 19-4), the particles which are |
crossing through the precession axis are moving in curued paths, and this requires |
application of a lateral force. This is supplied by our pushing on the axle, which |
`v— /2[ ]⁄*, x/LATER |
: hi si NOW |
_ấ 4 = |
m. VÀNG |
-⁄ ` R⁄” `*EARLIER |
Fig. 20-4. The motion of particles in the spinning wheel of Fig. 20-2, |
whose axIs Is turning, ¡is in curved lines. |
--- Trang 373 --- |
then communicates the force to the rim through the spokes. “Wait,” someone |
says, “what about the particles that are goïng back on the other side?” It does |
not take long to decide that there must be a force in the opposite direclion on |
that side. The net force that we have to apply is therefore zero. The ƒorces |
balance out, but one of them must be applied at one side of the wheel, and the |
other must be applied at the other side of the wheel. We could apply these forces |
directly, but because the wheel is solid we are allowed to do it by pushing on the |
axle, since forces can be carried up through the spokes. |
'What we have so far proved is that if the wheel is precessing, it can balance |
the torque due to gravity or some other applied torque. But all we have shown 1s |
that this is ø solution of an equation. 'Phat is, 1f the torque is given, and Zƒ ue |
get the spinning started right, then the wheel will precess smoothly and uniformly. |
But we have not proved (and it is not true) that a uniform precession is the |
tmos‡ general motion a spinning body can undergo as the result oŸ a given torque. |
The general motion involves also a “wobbling” about the mean precession. 'This |
“wobbling” is called nu‡ation. |
Some people like to say that when one exers a ÿorque on a øyroscope, i% ©urns |
and it precesses, and that the torque øroduces the precession. Ït is very sirange |
that when one suddenly lets go of a gyroscope, it does not ƒœl! under the action |
of gravity, but moves sidewise insteadl Why ¡is it that the dourmwuard force of the |
gravity, which we knou and ƒeel, makes it go sideu#se? All the formulas in the |
world like (20.15) are not going to tell us, because (20.15) is a special equation, |
valid only after the gyroscope 1s precessing nicely. What really happens, in detail, |
1s the following. lf we were to hold the axis absolutely fñxed, so that it cannot |
precess in any manner (but the top is spinning) then there is no torque acting, |
not even a torque from gravity, because it is balanced by our fñngers. But iŸ we |
suddenly let go, then there will instantaneously be a torque from gravity. Anyone |
in his right mind would think that the top would fall, and that is what it starts |
to do, as can be seen If the top is not spinning too fast. |
'The gyro actually does fall, as we would expect. But as soon as it falls, 1t is |
then turning, and If this turning were to continue, a torque would be required. In |
the absence of a torque in this direction, the gyro begins to “fall” in the direction |
opposite that of the missing force. 'Phis gives the gyro a component of motion |
around the vertical axis, as it would have in steady precession. But the actual |
motion “overshoots” the steady precessional velocity, and the axis actually rises |
again to the level from which it started. The path followed by the end of the |
axle is a cycloid (the path followed by a pebble that is stuck in the tread of an |
--- Trang 374 --- |
automobile tire). Ordinarily, this motion is too quick for the eye to follow, and it |
damps out quickly because of the friction in the gimbal bearings, leaving only |
the steady precessional drift (Eig. 20-5). The slower the wheel spins, the more |
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