municates the force to the rim through the spokes. ""Wait, " someone says, "what about the particles that are going back on the other side? It does not take long to decide that there must be a force in the opposite direction on that side. The net force that we have to apply is therefore zero. The forces 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 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 is that this is a solution of an equation. That is, if the torque is given, and if we get e 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 most general motion a spinning body can undergo as the result of a given torque. The general motion involves also a " wobbling"about the mean precession. This "wobbling "is called nutation Some people like to say that when one exerts a torque on a gyroscope, it turns and it precesses, and that the torque produces the precession. It is very strange that hen one suddenly lets go of a gyroscope, it does not fall under the action of gravity, but moves sidewise instead! Why is it that the downward force of the gravity, which we know and feel, makes it go sidewise? 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 is precessing nicely. What really happens, in detail, is the following If we were to hold the axis absolutely fixed, so that it cannot precess in any manner out the top is spinning) then there is no torque acting, not even a torque from gravity, because it is balanced by our fingers. But if we suddenly let go, then there will instantaneously be a torque from gravity. Anyone in his right mind would k that the top would fall, and that is what it starts to do as can be seen if the Is not spi The gyro actually does fall, as we would expect. But as soon as it falls, it 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. This 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 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 preces- Fig. 20-5. Actual motion of tip of sional drift(Fig. 20-5). The slower the wheel spins, the more obvious the nu- axis of gyroscope under gravity just after releasing axis previously held fixed tation Is When the motion settles down, the axis of the gyro is a little bit 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 is an abso lute miracle. It is a wonderful thing, but it is not a miracle. If we were holdin the axis absolutely horizontally, and suddenly let go, then the simple precession equation would tell us that it precesses, that it goes around in a horizontal plane But that is impossible! Although we neglected it before, it is true that the wheel has some moment of inertia about the precession axis, and if it is moving about that axis, even slowly, it has a weak angular momentum about the axis. Where did it come from? If 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? The answer is that the cycloidal motion of the end of the axis damps down to the average, steady inotion of the center of the equivalent rolling circle. That is, it set les down a little bit low. Because it is low, the spin angular momentum now h mall 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 bit to the gravity; by turning its axis down a little bit, it maintains the rotation about the vertical axis. That, then, is the way a gyroscope works