If the radius of the earth becomes $x$ times its present value,the new period of rotation in hours is (in $x^2$)

The angular velocity of a body changes from $\omega_1$ to $\omega_2$ without the application of any external torque,but this change occurs due to a change in its moment of inertia. The ratio of the radii of gyration in the two cases will be:

$A$ disc of mass $M$ and radius $R$ is moving with an angular velocity $\omega$ as shown in the figure. Find the angular momentum of the disc about the reference point $O$.

$A$ thin circular ring of mass $M$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Four objects,each of mass $m$,are kept gently on the opposite ends of two perpendicular diameters of the ring. The new angular velocity of the ring will be:

$A$ cockroach of mass $\frac{M}{2}$ starts moving with velocity $V$ on the circumference of a disc of mass $M$ and radius $R$. What will be the angular velocity of the disc?

$A$ swimmer can easily perform somersaults in the air while jumping from a height if he: