Theorems related to Moment of Inertia

Theorem of Parallel Axes

We have to obtain the moment of inertia of a body with mass M and with the centre of mass at C about a given line AB. Visualise a line CZ parallel to AB through C.

Let I₀ be the moment of inertia of the body about CZ respectively.
If the perpendicular distance between AB and CZ is d
Then I, the moment of inertia of the body about AB is going to be

I = I₀ + Md²



Theorem of Perpendicular axes

This theorem is applicable only to the plane bodies. Let X and Y axes be chosen in the plane of the body and Z=axis perpendicular to this plane, and the three axes are mutually perpendicular.

Ix, Iy, Iz are moment of inertia of the body about x,y, and z-axes respectively.

According to the theorem
Iz = Ix + Iy

(Chapter: Rotational Mechanics)