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