Parallel and perpendicular axes theorems

These two theorems may be useful in considering problems on moments of inertia.

(a) Parallel axes theorem

The moment of inertia I of a body about any axis is equal to the moment of inertia I_G about a parallel axis through the centre of gravity of the body plus Mb², where M is the mass of the body and b is the distance between the two axes. (See Figure 1)

(b) Perpendicular axes theorem

For any plane body (e.g. a rectangular sheet of metal) the moment of inertia about any axis perpendicular to the plane is equal to the sum of the moments of inertia about any two perpendicular axes in the plane of the body which intersect the first axis in the plane.

This theorem is most useful when considering a body which is of regular form (symmetrical) about two out of the three axes. If the moment of inertia about these axes is known then that about the third axis may be calculated.