Equation of Motion of a Mass on a Smooth Pulley

Pulleys have mass, and this is often ignored when analysing the motion of a mass on a pulley. Because the mass of the pulley is not zero, the pulley has a moment of inertia and a force is required to make it turn. This force is supplied by the tension in the string produced by a masshanging from a string wound around the pulley.

If the mass of the pulley isthen the moment of inertia of the pulley about its centre is

Applying the equationto the pulley gives

Applyingto the mass gives

We have the simultaneous equations

Adding these gives

Substituting this into (1) above gives