$A$ body changes its angular speed from $\omega_1$ to $\omega_2$ without any external torque applied. This change is due to a change in its moment of inertia. What is the ratio of the radii of gyration in the two cases?

$A$ small disk is attached to the end of a light inextensible string,which passes through a hole in a frictionless horizontal tabletop. Initially,the disk moves on a circle of radius $R$ with kinetic energy $K_0$. The string is then slowly pulled so that the disk finally rotates on a circle of radius $\frac{R}{\eta}$. What is the work $W$ done in pulling the string?

$A$ force $\vec F = \alpha \hat i + 3\hat j + 6\hat k$ is acting at a point $\vec r = 2\hat i - 6\hat j - 12\hat k$. The value of $\alpha$ for which angular momentum about the origin is conserved is:

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:

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

$A$ thin circular ring of mass $M$ and radius $R$ is rotating about a transverse axis passing through its centre with constant angular velocity $\omega$. Two objects each of mass $m$ are attached gently to the opposite ends of a diameter of the ring. What is the new angular velocity?