Question

Find the moment of inertia of a uniform square plate of mass m and edge a, about one of its diagonals?

Answer 1

Assume that the MI in this uniform sq. plate on one of the axis = I.

Therefore the MI of it's diagonal = I

MI on centre of mass to will be perpendicular to the plane, means both diagonals are perpendicular Since this is sq. plate.

Assume, MI perp. to the plate & passing by it's center = I'

Now they become mutually perpendicular to each other and further passing by the center

Therefore according to moment of inertia, perpendicular axis theorem.

$I+I =I'$

$2I =\frac{ma^2}{6}$

$I = \frac{ma^2}{12}$

This is the moment of inertia about the diagonal plate

Hope it Helps :-)