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 radius
has moment of inertia
about an axis through the centre of the disc perpendicular to the plane of the disc.
If a mass
is 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 torque
to the system.
The moment of inertia of the whole system is
The angular acceleration
is 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 distance
Equating these – gives
Now use
Since the system starts from rest
