$A$ solid sphere of radius $r = 5 \text{ cm}$ is rotating about an axis at a distance $d = 10 \text{ cm}$ from its center, as shown in the figure. If the radius of gyration about that axis is $\sqrt{x} \text{ cm}$, then the value of $x$ is:

Four hollow spheres,each with a mass of $1\, kg$ and a radius $R = 10\, cm$,are connected with massless rods to form a square with a side of length $L = 50\, cm$. In case-$1$,the masses rotate about an axis that bisects two sides of the square. In case-$2$,the masses rotate about an axis that passes through the diagonal of the square,as shown in the figure. Compute the ratio of the moments of inertia $I_1/I_2$ for the two cases.

Radius of gyration of a thin uniform circular disc about the axis passing through its centre and perpendicular to its plane is $K_{c}$. Radius of gyration of the same disc about a diameter of the disc is $K_d$. The ratio $K_c: K_d$ is

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

Find the moment of inertia of the system of particles about the axis $OA$ passing through point $O$.

The moment of inertia of a uniform semicircular disc of mass $M$ and radius $r$ about an axis perpendicular to the plane of the disc passing through its center is: