## Acceleration of a Pulley by a Weight

A simplifying assumption often made is that a pulley has no mass or moment of inertia. This is of course false. A uniform disc of radiushas moment of inertiaabout an axis through the centre of the disc perpendicular to the plane of the disc.

If a massis tied to a string which is then wound around the pulley, then let loose, and the pulley is free to turn, then the massapplies a torqueto the system.

The moment of inertia of the whole system is

The angular accelerationis given by

Another way to show this is using energy.

The kinetic energy of the system is

This will be equal to the gravitational potential energy lost by the massas it falls a distanceEquating these – gives

Now useSince the system starts from rest