## The Compound Pendulum

The simple pendulum models a mass swinging on a rigid rod of zero mass. In practice of course the mass of the rod is not zero. The compound pendulum takes the non – zero mass of the rod into account.

A compound pendulum consists of a mass ( here I will assume it to be a circular disc) attached to the end of a rod of length and mass The rod is hung and the rod plus mass is allowed to swing in a vertical plane. The moment of inertia of the rod about its centre is so the moment of inertia about the pivot at the top is The moment of inertia of the disc about its centre is so the moment of inertia of the disc about the pivot from the centre of the disc is The moment of inertia of the pendulum is then Now the pendulum can be modelled as a particle with moment of inertia free to rotate in the vertical plane. If it is displaced an angle to one side it will experience a restoring torque towards the vertical due to the mass of and due to the rod of so the total restoring torque is and this will be equal to the the neagive of the product of and angular acceleration The negative indicates and are in the opposite sense. If is small, and writing  Which becomes where The compound pendulum will execute simple harmonic motion.

We can check the formula by putting then just as for a simple pendulum. 