Angular momentum around a vertical axis cannot change because of the(friction- less)pivot of the chair, so if we turn the axis of the wheel into the vertical. then the wheel would have angular momentum about the vertical axis, because it is now spinning about this axis. But the system(wheel, ourself, and chair)cannot have a vertical component, so we and the chair have to turn in the direction opposite to the spin of the wheel, to balance it First let us analyze in more detail the thing we have just described. What is surprising,and what we must understand, is the origin of the forces which turn us and the chair around as we turn the axis of the gyroscope toward the vertical Figure 20-2 shows the wheel spinning rapidly about the y-axis. Therefore its angular velocity is about that axis and, it turns out, its angular momentum is like wise in that direction. Now ose that we wish to rotate the wheel about the y x-axis at a small angular velocity n; what forces are required? After a short ti At, the axis has turned to a new position, tilted at an angle 46 with the horizontal Fig. 20-2. A gyroscope 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 as changed. What is the change in angular momentum? The angular momentum does not change in magnitude, but it does change in direction by an amount 4e The magnitude of the vector AL is thus AL= Lo 40, so that the torque, which he time rate of change of the angular momentum,isr=△L/△t=Lo△/△t= Loa. Taking the directions of the various quantities into account, we see that g×Lo Thus, if Q2 and Lo are both horizontal, as shown in the figure, r is vertical. To produce such a torque, horizontal forces F and-F must be applied at the ends of 2, 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 Newton's third Law demands that equal and opposite forces(and equal and opposite torques)act on us. 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 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 a is the(vertical)angular velocity of precession, we again find that Fig. 20-3. A rapidly spit a/dn=a×Lo lote that the direction of the torque vector is the direction of the precession. 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 appears as a"miracle. It will turn out, as we go to more and more advanced ∠ LATER physics, that many simple things can be deduced mathematically more rapidl than they can be really understood in a fundamental or simple sense. This is NOw strange characteristic, and as we get into more and more advanced work there are circumstances in which mathematics will produce results which no one has really EARLIER 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 g.20-4.The motion of particles in hard to understand. In our particular case, the precession of a top looks like some the sp wheel of Fig. 20-2, whose kind of a miracle involving right angles and circles, and twists and right-hand hould try to do is to understand in a more physical way. How can we explain the torque in terms of the real forces and the accelerations? We 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 through the precession axis are moving in curved paths, and this requires application of a lateral force. This is supplied by our pushing on the axle, which then com- 20-6