Four spheres of diameter $2a$ and mass $M$ are placed with their centres on the four corners of a square of side $b$. The moment of inertia of the system about an axis passing through one of the sides of the square is ..........

Consider a thin uniform square sheet made of a rigid material. If its side is $a$,mass is $m$,and the moment of inertia about one of its diagonals is $I$,then:

The moment of inertia of a disc of mass $M$ and radius $R$ about a tangent to its rim in its plane is

For the given uniform square lamina $ABCD$,whose centre is $O$,

The rectangular plate shown in the figure is rotated in turn about $x-x'$,$y-y'$ and $z-z'$ axes passing through its centre of mass $O$. Its moment of inertia is

$(a)$ Prove the theorem of perpendicular axes. (Hint: Square of the distance of a point $(x, y)$ in the $x-y$ plane from an axis through the origin and perpendicular to the plane is $x^{2}+y^{2}$)
$(b)$ Prove the theorem of parallel axes. (Hint: If the centre of mass of a system of $n$ particles is chosen to be the origin,$\sum m_{i} r_{i}=0$)