Moment of Inertia


1. Introduction
The angular acceleration of a body subjected to a torque depends upon it's moment of inertia. This is a measure of the distribution of the mass within the body. Where a body is pivoted about a fixed point it is the moment of inertia about this fixed point that is the appropriate value. Where a body is free to move in an unconstrained manner, then the linear motion (translation) of the centre of mass is considered and the rotation about the centre of mass - so in this case it is the moment of inertia about the centre of mass that is required.
2. Derivation
The section below shows in general terms how the moment of inertia is derived.

3. Two Theorems

a) The parallel axis theorem. This states that if the moment of inertia of a body of mass m about an axis through it's centre of mass is Ig then it's moment of inertia about another axis a parallel to the first and a distance d from it, is given by
Ia = Ig + m d2
The perpendicular axis theorem states that the moment of inertia of a thin lamellar body about it's polar axis, is equal to the sum of the moments of inertia about any 2 axes in the plane of the lamellar that are perpendicular. So assuming the body lies in the 'x - y' plane, the polar axis is the 'z' axis:
Ip = Izz = Ixx + Iyy

David J Grieve. Modified: 10th March 2011, original: 15th November 2001.