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:

$A$ particle of mass $m$ moves with a velocity $v_0$ in a circle of radius $R_0$ on a smooth horizontal plane. The mass is tied to a string passing through a hole in the smooth plane as shown in the figure. The tension in the string is gradually increased,and finally,the mass $m$ moves in a circle of radius $\frac{R_0}{2}$. What is the ratio of the final kinetic energy to the initial kinetic energy?

$A$ 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 $O$.

$A$ circular disk of moment of inertia $I_t$ is rotating in a horizontal plane,about its symmetry axis,with a constant angular speed $\omega_i$. Another disk of moment of inertia $I_b$ is dropped coaxially onto the rotating disk. Initially,the second disk has zero angular speed. Eventually,both the disks rotate with a constant angular speed $\omega_f$. The energy lost by the initially rotating disk to friction is

$A$ disc of moment of inertia $I_1$ is rotating in a horizontal plane about an axis passing through its center and perpendicular to its plane with a constant angular speed $\omega_1$. Another disc of moment of inertia $I_2$ having zero angular speed is placed coaxially on the rotating disc. Now,both the discs are rotating with a constant angular speed $\omega_2$. The energy lost by the initial rotating disc is

$A$ uniform rod $AB$ of length $L$ and mass $M$ is pivoted at its center $O$ such that it can rotate freely in a vertical plane. The rod is initially in a horizontal position. $A$ particle $S$ of the same mass $M$ falls vertically with velocity $v$ and strikes the rod at point $C$. $C$ is the midpoint between $O$ and $B$. Find the angular velocity of the rod immediately after the particle strikes it.