The moment of inertia of a solid cylinder about its own axis is equal to its moment of inertia about an axis passing through its center of gravity and perpendicular to its length. What is the relationship between its length $L$ and radius $R$?

If $I$ is the moment of inertia of a thin circular disc about an axis passing through the tangent of the disc and in the plane of the disc,what is the moment of inertia of the same circular disc about an axis perpendicular to the plane and passing through its centre?

Each spherical shell has a radius $R$ and mass $M$. They are connected by a light massless rod. Calculate the moment of inertia about the axis $xx'$.

$A$ rigid body of mass $m$ is rotating with angular velocity $\omega$ about an axis at a distance '$a$' from the center of mass $G$. The radius of gyration about an axis passing through $G$ is $K$. The rotational kinetic energy of this body about the new parallel axis will be:

The moment of inertia of a thin square plate $ABCD$ of uniform thickness about an axis passing through the centre $O$ and perpendicular to the plane of the plate is given by which of the following expressions? Here,$I_1, I_2, I_3,$ and $I_4$ are the moments of inertia about axes $1, 2, 3,$ and $4$ respectively,which lie in the plane of the plate as shown in the figure.

$A$ uniform cylinder has a radius $R$ and length $L$. If the moment of inertia of this cylinder about an axis passing through its centre and normal to its circular face is equal to the moment of inertia of the same cylinder about an axis passing through its centre and perpendicular to its length,then