Motion, Forces and Energy Torque, Angular Momentum and Its Conservation Fsinf The tendency of a force to rotate an object about some axis is called TORQUE. Torque is defined as t = F sin f r r f Fcosf z F1 Line of action d d1 O d2 In the figure to the right, there are two forces Acting in opposite senses. One (F1) causes Anti-clockwise rotation, the other clockwise. Net torque about O-axis, t = F1d1 – F2d2 F2

Torque and angular acceleration y dFt For any element dm, we know from Newton’s 2nd Law that dFt = (dm) at where at is the tang- ential acceleration. dm r q x O The torque dt = r dFt = (r dm) at From earlier we saw that at = r a So we can write: dt = (r2dm)a The net torque is given by: This is the rotational analogue of Newton’s 2nd Law, F = ma

Angular Momentum and Its Conservation x y z vi mi r We can find the angular momentum of this disk by considering each mass element, mi. Differentiating gives: The net external torque acting on a rigid object rotating about a fixed axis is equal to the moment of inertia about the rotation axis multiplied by the object’s angular acceleration.

Conservation of Angular Momentum The total angular momentum of a system is Constant if the resultant external torque acting On the system is zero: Stext = dL / dt = 0 M OR/ Angular momentum before a redistribution of mass is equal to the angular momentum after such a redistribution. R1 R2 Therefore if we know M, m, R1, R2 and w1, then we can find w2. Man walks inwards from R1 to R2.