State and explain the theorem of perpendicular axis with an example.

Explanation

Statement : The moment of inertia about an axis perpendicular to two other axes acting in the same plane with their point of intersection being a point on it and the (third) axis passing through the common point, is equal to the sum of moments of inertia about the two axes
e.g: `I_z=I_x+I_y`
Let M be the mass of the disck of radius R.
M.I. about a point passing through the centre and perpendicular to the plane containing X and Y is `I_z=(MR^2)/(2)`
Since X and Y are in the same plane `I_x=I_y`
`:." " I_z=I_x+I_y` becomes `I_z=2I_x`
hence `I_x=(I_z)/(2)=(MR^2)/4`
i.e. moment of inertia of a circular disc about the diameter
`=(MR^2)/4`