## The Parallel Axis Theorem

The Parallel Axis Theorem

If the moment of inertia of a body of mass M about an axis through its centre of mass is then the moment of inertia of the body about a parallel axis at a distancefrom the first axis is

Proof

The moment of inertia about axis Az is given by :

|Separate into distinct integrals

The coordinates of the centre of mass are

andand

as the axes are chosen so that the centre of gravity lies on the z axis so

From the diagramandand substituting in the equation for I, we have: