Let us now move on to consider the dynamics of rotation. Here a new concept,force, must be introduced. Let us inquire whether we can invent something which we shall call the torque(L torquere, to twist)which bears the same relation ship to rotation as force does to linear movement. A force is the thing that is needed to make linear motion, and the thing that makes something rotate is a rotary force”'ora“¨ twisting force,”le, a torque. Qualitatively, a torque is a "twist" what is a torque quantitatively? We shall get to the theory of torques quantitatively by studying the work done in turning an object, for one very nice way of defining a force is to say how much work it does when it acts through a given displacement. We are going to try to maintain the analogy between linear and angular quantities by equating the work that we do when we turn something a little bit when there are forces acting on it, to the torque times the angle it turns through. In other words, the definition of the torque is going to be so arranged that the theorem of work has an absolute analog force times distance is work, and torque times angle is going to be work. That tells us what torque is. Consider, for instance, a rigid body of some kind with various forces acting on it, and an axis about which the body rotates. Let us at first concentrate on one force and suppose that this force is applied at a certain point(x, y). How much work would be done if we were to turn the object through a very small angle? That is easy. The work done is △W=Fx△x+F"△y (18.10) We need only to substitute Eqs. (18.6)and (18.7)for Ax and Ay to obtain △W=(xFv-yF2)△. (1811) That is, the amount of work that we have done is, in fact, equal to the angle through which we have turned the object, multiplied by a strange- looking combination of the force and the distance. This"strange combination""is what we call the torque. So, defining the change in work as the torque times the angle, we now have the formula for torque in terms of the forces.(Obviously, torque is not a completely ew idea independent of Newtonian mechanics-torque must have a definite definition in terms of the force When there are several forces acting, the work that is done is, of course, the sum of the works done by all the forces, so that Aw will be a whole lot of terms, all added together, for all the forces, each of which is proportional, however, to 40 We can take the A0 outside and therefore can say that the change in the work is qual to the sum of all the torques due to all the different forces that are acting, times A0. This sum we might call the total torque, T. Thus torques add by the ordinary laws of algebra, but we shall later see that this is only because we are working in a plane. It is like one-dimensional kinematics, where the forces simply add algebraically, but only because they are all in the same direction. It is more complicated in three dimensions. Thus, for two-dimensional rotation, yi Fri (18.12) ∑ (18.13) It must be emphasized that the torque is about a given axis. If a different axis is hosen,so that all the xi and yi are changed, the value of the torque is (usually) Now we pause briefly to note that our foregoing introduction of torque, through the idea of work, gives us a most important result for an object in equilib rium: if all the forces on an object are in balance both for translation and rotation, not only is the net force zero, but the total of all the torques is also zero, because if an object is in equilibrium, no work is done by the forces for a small displacement Therefore, since AW= T 48=0, the sum of all the torques must be zero. So here are two conditions for equilibrium: that the sum of the forces is zero, and nat the sum of the torques is zero. Prove that it suffices to be sure that the sum of torques about any one axis(in two dimensions)is zero