20 Rotation in space 20-1 Torques in three dimensions In this chapter we shall discuss one of the most remarkable and amusing 20-1 Torques in three dimensions consequences of mechanics, the behavior of a rotating wheel. In order to do this 20-2 The rotation equations using we must first extend the mathematical formulation of rotational motion the principles of angular momentum, torque, and so on, to three-dimensional space cross We shall not use these equations in all their generality and study all their conse- 20-3 The gyroscope quences, 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 20-4 Angular momentum of a solid apply them to a very few situations of special interest First, we notice that if we have a rotation in three dimensions whether of a igid body or any other system, what we deduced for two dimensions is still right. That is, it is still true that xFy- yFr is the torque"in the xy-plane, "or the torque "around the z-axis. "It also turns out that this torque is still equal to the rate of change of xpv -ypa, for if we go back over the derivation of Eq (18. 15)from Newtons laws we see that we did not have to assume that the motion was in a plane;when we differentiate xpy-ypr, we get xFy-yFr, so this the oren still right. The quantity xP, -DPz, then, we call the angular momentum belonging to the xy-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 yz-plane, and it is clear from symmetry that if we just substitute y for x and z for y, we would find yF, - zF, for the torque and yp, -zp, would be the angular momentum associated with the yz-plane. Of course we could have another plane, the zx-plane, and for this we would find zFx -xF:= d/dt(zp2 -xp) That these three equations can be deduced for the motion of a single particle quite clear. Furthermore, if we added such things as xp,- yp, together for many particles and called it the total angular momentum, we would have three kinds for the three planes xy, yz, and zx, and if we did the same with the forces, we would talk about the torque in the planes xy, yz, and zx 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 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 equations! "Interestingly enough, it turns out that if we were to work out the combination x'Fy-y'F2 for another plane, measuring the x, Fy, etc, in that plane, the result can be written as some combination of the three expressions for the xy, yz-and zx-planes. There is nothing new. In other words, if we know what the three torques in the xy, yz-, and zx-planes are, then he 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 xyz-axes, Joe has worked out all his torques and his angu- lar momenta in his planes. But Moe has axes x', y, z in some other direction. To make it a little easier, we shall suppose that only the x and y-axes have been turne Moe's xand y are new, but his z happens to be the same. That is, he has new planes, let us say, for yz and zx. He therefore has new torques and angular momenta which he would work out. For example, his torque in the x'y' plane would be equal to x'Fy-y' Fr and so forth. What we must now do is to find the relation ship between the new torques and the old torques, so we will be able to make a