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

The rectangular block shown in the figure is rotated about the $x-x$,$y-y$,and $z-z$ axes passing through its center of mass $O$. Its moment of inertia is:

$A$ thin wire of length $L$ and uniform linear mass density $\rho$ is bent into a circular loop with centre at $O$ as shown. The moment of inertia of the loop about the axis $XX'$ is

State and prove the theorem of perpendicular axes and the theorem of parallel axes.

The moment of inertia of a disc about an axis passing through its centre and perpendicular to its plane is $I$. What is the ratio of the moment of inertia about a parallel axis tangential to its rim to the moment of inertia about a parallel axis passing through a point midway between the centre and the rim (in $:1$)?

Two spheres each of mass $M$ and radius $R$ are connected with a massless rod of length $4R$. The moment of inertia of the system about an axis passing through the centre of one of the spheres and perpendicular to the rod will be