##### Parallel Axis Theorem:

If the moment of inertia of a rigid body about an axis(n) passing through its centre of mass is I_{cm} , then the moment of inertia of the body about an axis parallel to 1 , at a distance d from the first one, is given by

$\displaystyle I = I_{c.m} + m d^2 $

This theorem is known as the parallel axis theorem.

Illustration : Using the parallel axis theorem, find the M.I. of a sphere of mass m about an axis that touches it. Given that I_{c.m.} = (2/5) mr^{2}

Solution :

I_{p} = I_{cm} + m (OP)^{2}

⇒ I_{p} = I_{o} + m (OP)^{2}

= (2/5)m r^{2} + mr^{2}

= (7/5) mr^{2}

### Perpendicular Axis Theorem

If the moment of inertia of a plane lamina about two mutually perpendicular axes in its plane are I_{x} and I_{y} , then its moment of inertia about a third axis (z) perpendicular to both the axes and passing through the point of intersection is

$ \displaystyle I_Z = I_X + I_Y $

This theorem is known as the perpendicular axis theorem.

Illustration : Using perpendicular axes theorem, find the M.I. of a disc about an axis passing through its diameter.

Solution : According to perpendicular axis theorem,

$ \displaystyle I_Z = I_X + I_Y $

we know that I_{x} = I_{y} due to the geometrical symmetry of the disc.

⇒ I_{x} = I_{y} = I_{Z}/2

where I_{Z} = M.I of the disc about z axis passing through its center perpendicular to its plane = mr^{2}/2

⇒ M.I = I_{x} = I_{y} = mr^{2}/4