Then if m; is the mass of that point, the total kinetic energy of the whole thing just the sum of the kinetic energies of all of the little pieces T=∑mn2=∑m)2 Now w- is a constant, the same for all points.Thus T=如32∑mn2= At the end of Chapter 18 we pointed out that there are some interesting phenomena associated with an object which is not rigid, but which changes from one rigid condition with a definite moment of inertia, to another rigid condition Namely, in our example of the turntable, we had a certain moment of inertia I1 with our arms stretched out, and a certain angular velocity w When we pulled our arms in, we had a different moment of inertia, I2, and a different angular veloc ty, w2, but again we were"rigid. The angular momentum remained constant, since there was no torque about the vertical axis of the turntable. This means that I 1G1=I202. Now what about the energy? That is an interesting question With our arms pulled in, we turn faster, but our moment of inertia is less, and it looks as though the energies might be equal. But they are not, because what does balance is Iw, not Iw. So if we compare the kinetic energy before and after, the kinetic energy before is HI1wf=2Lwl, where L=I1@1= 12w2 is the angular rward, by the have T=是Lo w2> w1 the kinetic energy of rotation is greater than it was before. So we had a certain energy when our arms were out, and when we pulled them in, we were turn ing faster and had more kinetic energy. What happened to the theorem of the conservation of energy? Somebody must have done some work. We did work When did we do any work? When we move a weight horizontally, we do not do any work. If we hold a thing out and pull it in, we do not do any work. But that is when we are not rotating! When we are rotating, there is centrifugal force on the weights. They are trying to fly out, so when we are going around we have to pull the weights in against the centrifugal force. So, the work we do against the centrifugal force ought to agree with the difference in rotational energy, and of course it does. That is where the extra kinetic energy comes from There is still another interesting feature which we can treat only descriptively, as a matter of general interest. This feature is a little more advanced but is worth pointing out because it is quite curious and produces many interesting effects Consider that turntable experiment again. Consider the body and the arms separately, from the point of view of the man who is rotating. After the weights are pulled in, the whole object is spinning faster, but observe, the central part of the body is not changed, yet it is turning faster after the event than before. So. if we were to draw a circle around the inner body, and consider only objects inside the circle, their angular momentum would change; they are going faster. Therefore there must be a torque exerted on the body while we pull in our arms. No torque exerted by the centrifugal for that among the forces that are developed in a rotating system, centrifugal force is not the entire story, there is another force. This other force is called Coriolis force, and it has the very strange property that when we move something in a rotating system, it seems to be pushed sidewise. Like the centrifugal force, it is an apparent force. But if we live in a system that is rotating, and move something radially, we find that we must also push it sidewise to move it radially. This sidewise push which we have to exert is what turned our body around Now let us develop a formula to show how this Coriolis force really works Suppose Moe is sitting on a carousel that appears to him to be stationary. But from the point of view of Joe, who is standing on the ground and who knows the right laws of mechanics, the carousel is going around. Suppose that we have drawn a radial line on the carousel, and that Moe is moving some mass radially along this line. We would like to demonstrate that a sidewise force is required to do that do this by paying attentio