State and prove theorem of parallel axis.

Explanation

The moment of inertia of a body about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of the distance between the two parallel axes.

As shown in figure Z and Z. are two parallel axes separated by a distance a. `therefore OO.=a`
The Z-axis passes through the centre of mass O of the rigid body.
From theorem of parallel axes
`I_(Z)=I_(Z)+Ma^(2)orI_(Z)=I_(C)+M(OO.)^(2)`
where `I_(Z)andI_(Z)` are the moments of inertia of the body about the Z and Z. axes respectively.
M = total mass of the body
a = perpendicular distance between the two parallel axes.
If moment of inertia of object of any shape about the axis passing through its centre is given, then moment of inertia about the axis parallel to the axis passing through its centre can be determined.