Now let us consider a single force, and try to figure out, geometrically, what this strange thing xFy -yFr amounts to. In Fig. 18-2 we see a force F acting at a point r. When the object has rotated through a small angle Ae, the work done, of course, is the component of force in the direction of the displacement times the displacement. In other words, it is only the tangential component of the force that counts, and this must be multiplied by the distance r4e. Therefore we see that the torque is also equal to the tangential component of force(perpendicular to the radius) times the radius. That makes sense in terms of our ordinary idea of the torque, because if the force were completely radial, it would not put any twist""on the body it is evident that the twisting effect should involve only the part of the force which is not pulling out from the center, and that means the tangential component. Furthermore, it is clear that a given force is more effective on a long arm than near the axis. In fact, if we take the case where we push right on the axis, we are not twisting at all! So it makes sense that the amount of twist or torque, is proportional both to the radial distance and to the tangential com Fig. 18-2. The torque produced by ponent of the force a force There is still a third formula for the torque which is very interesting e hav just seen that the torque is the force times the radius times the sine of the angle a in Fig. 18-2. But if we extend the line of action of the force and draw the line os the perpendicular distance to the line of action of the force(the lever arm of the force) we notice that this lever arm is shorter than r in just the same proportion as the tangential part of the force is less than the total force. Therefore the formula for the torque can also be written as the magnitude of the force times the length of the lever arm The torque is also often called the moment of the force. the origin of this term is obscure, but it may be related to the fact that"moment"is derived from the Latin movimentum, and that the capability of a force to move an object(using the force on a lever or crowbar) increases with the length of the lever arm. I mathematics 'moment"means weighted by how far away it is from an axis 18-3 Angular momentum Although we have so far considered only the special case of a rigid body, the properties of torques and their mathematical relationships are interesting also even when an object is not rigid. In fact, we can prove a very remarkable theorem ust as external force is the rate of change of a quantity p, which we call the total momentum of a collection of particles, so the external torque is the rate of change of a quantity L which we call the angular momentum of the group of particles To prove this, we shall suppose that there is a system of particles on which there are some forces acting and find out what happens to the system as a result of he torques due to these forces. First, of course, we should consider just one particle. In Fig. 18-3 is one particle of mass m, and an axis O; the particle is not going around the sun, or in some other curve llt ing in an ellipse, like a planet necessarily rotating in a circle about O, it may be mot loving somehow, and there are forces on it, and it accelerates according to the usual formula that the x-com- ponent of force is the mass times the x-component of acceleration, etc. But let us see what the torque does. The torque equals xF, -yFx, and the force in the Fig. 18-3. A particle moves ab x-or y-direction is the mass times the acceleration in the x- or y-direction (1814) Now, although this does not appear to be the derivative of any simple quantity, it is in fact the derivative of the quantity xm(dy/dr-ym(dx/do (2)-m(2)-m()+(a)m( (1815 (a)-()m()=m(2m)-m(