Question

State and prove perpendicular axis theorem.

Answer 1

Perpendicular axis theorem:

This perpendicular axis theorem holds good only for plane laminar objects. The theorem states that the moment of inertia of a plane laminar body about an axis perpendicular to its plane is equal to the sum of moments of inertia about two perpendicular axes lying in the plane of the body such that all the three axes are mutually perpendicular and have a common point.

Let the X and Y – axes lie in the plane and Z-axis perpendicular to the plane of the laminar object. If the moments of inertia of the body about X and Y-axes are I_X and I_Y respectively – and I_Z is the moment of inertia about Z-axis, then the perpendicular axis theorem could be expressed as,
I_Z = I_X + I_Y

To prove this theorem, let us consider a plane laminar object of negligible thickness on which lies the origin (O). The X and Y – axes lie on the plane and Z-axis is perpendicular to it as shown in the figure. The lamina is considered to be made up of a large number of particles of mass m. Let us choose one such particle at a point P which has coordinates (x, y) at a distance r from O. 

Perpendicular axis theorem

The moment of inertia of the particle about the Z-axis is, mr².

The summation of the above expression gives the moment of inertia of the entire lamina about the Z-axis as, I_Z = ∑ mr²

Here, r² = x² + y²

Then, I_Z = ∑ m (x² + y²)

I_Z = ∑ m x² + ∑ m y²

In the above expression, the term ∑ m x² is the moment of inertia of the body about the Y-axis, and similarly the term ∑ m y²is the moment of inertia about the X-axis. Thus,

I_X = ∑ m y² and I_Y = ∑ m x²

Substituting in the equation for Iz gives,

I_Z = I_X + I_Y

Thus, the perpendicular axis theorem is proved.