The angular speed of a body changes from $\omega_1$ to $\omega_2$ without applying a torque but due to changes in moment of inertia. The ratio of radii of gyration in two cases is

$A$ circular disc of mass $M$ and radius $R$ is rotating about its axis with angular speed $\omega_{1}$. If another stationary disc having radius $\frac{R}{2}$ and same mass $M$ is dropped co-axially onto the rotating disc,gradually both discs attain a constant angular speed $\omega_{2}$. The energy lost in the process is $p \%$ of the initial energy. The value of $p$ is:

When torque applied on a system is zero,which of the following will be constant?

Suppose all the people in the world line up at the equator and all start running at speed $v_{rel}$ relative to the surface of the Earth along the equatorial circle. The initial angular velocity of the Earth is $\omega_0$. Let the moment of inertia of the Earth be $I_E$,the moment of inertia of all people be $I_P$,and the radius of the Earth be $R$. Which of the following statements is correct?

$A$ disc with moment of inertia $I_1$ is rotating with an angular velocity $\omega$ about an axis passing through its center and perpendicular to its plane. Another disc with moment of inertia $I_2$ is placed on it,rotating about the same axis. What will be the combined angular velocity of both discs?

$A$ thin circular ring of mass $M$ and radius $r$ is rotating about its axis with an angular speed $\omega$. Two particles each of mass $m$ are now attached at diametrically opposite points. The angular speed of the ring will become