What is the perpendicular axis theorem?

Perpendicular Axis Theorem

This is applicable only to a plane lamina. The moment of inertia of a plane lamina about an axis perpendicular to its plane is equal to the sum of the moments of inertia of the lamina about any two mutually perpendicular axes, passing in its own plane, intersecting each other at the point through which the perpendicular axis passes.

 

Let us consider a plane lamina lying in the XOY plane. The lamina is made up of a large number of particles. Consider a particle of mass 'm' at P. From P, PN and PNl are drawn perpendicular to X-axis and Y-axis, respectively.

Now PNl= x, PN = y


Moment of inertia about X-axis = my2

Moment of inertia of the whole of lamina about X-axis

 

Moment of inertia of the whole of lamina about Y-axis

 
Moment of inertia of the whole of lamina about Z-axis