# Parallel Axis Theorem , Perpendicular Axis Theorem

##### Parallel Axis Theorem:

If the moment of inertia of a rigid body about an axis(n) passing through its centre of mass is Icm , 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 Ic.m. = (2/5) mr2

Solution :

Ip = Icm + m (OP)2

⇒ Ip = Io + m (OP)2

= (2/5)m r2 + mr2

= (7/5) mr2

### Perpendicular Axis Theorem

If the moment of inertia of a plane lamina about two mutually perpendicular axes in its plane are Ix and Iy , 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 Ix = Iy due to the geometrical symmetry of the disc.

⇒ Ix = Iy = IZ/2

where IZ = M.I of the disc about z axis passing through its center perpendicular to its plane = mr2/2

⇒ M.I = Ix = Iy = mr2/4