$A$ circular plate of mass $M$ and radius $R$ has its density varying as $\rho(r) = \rho_0 r$,where $\rho_0$ is a constant and $r$ is the distance from its center. The moment of inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is $I = aMR^2$. The value of the coefficient $a$ is:

Two spheres each of mass $M$ and radius $R/2$ are connected by a massless rod of length $2R$ as shown in the figure. The moment of inertia of the system about an axis passing through the center of one of the spheres and perpendicular to the rod is:

$A$ square plate of mass $M$ and edge $L$ is shown in the figure. The moment of inertia of the plate about the axis in the plane of the plate passing through one of its vertices making an angle $15^{\circ}$ with the horizontal is:

$A$ thin square plate $ABCD$ has a uniform thickness. What is the moment of inertia about an axis passing through its center and perpendicular to its plane?

The moment of inertia of a rod of mass $M$ and length $L$ about an axis passing through its center and normal to its length is $\alpha$. Now,the rod is cut into two equal parts,and these parts are joined symmetrically to form a cross shape. The moment of inertia of the cross about an axis passing through its center and normal to the plane containing the cross is:

The radius of gyration $K$ of a hollow sphere of mass $M$ and radius $R$ about an axis $XY$ is equal to $R$. The distance of that axis from the center of the sphere is $h$. The value of $h$ is