$A$ thin uniform wire of mass $m$ and linear density $\rho$ is bent in the form of a circular ring. The moment of inertia of the ring about a tangent parallel to its diameter is

$A$ thin disc of mass $M$ and radius $R$ has mass per unit area $\sigma (r) = kr^2$,where $r$ is the distance from its centre. Its moment of inertia about an axis passing through its centre of mass and perpendicular to its plane is

Two discs having moment of inertia $I_1$ and $I_2$ are made from the same material and have the same mass. Their thicknesses and radii are $t_1, t_2$ and $R_1, R_2$ respectively. The relation between the moment of inertia of each disc about an axis passing through its centre and perpendicular to its plane and its thickness is:

The radius of gyration of a cylindrical rod about an axis of rotation perpendicular to its length and passing through its center is $k$. Given that the length of the rod is $10 \sqrt{3} \ m$,find the value of $k$ in meters.

Four particles,each of mass $m$,are placed at the corners of a square of side $l$. The radius of gyration of the system about an axis passing through the center of the square and perpendicular to its plane is:

Two solid spheres ($A$ and $B$) are made of metals of different densities $\rho_A$ and $\rho_B$ respectively. If their masses are equal, the ratio of their moments of inertia $(I_B / I_A)$ about their respective diameter is