$A$ mass is whirled in a circular path with constant angular velocity and its linear velocity is $v$. If the string is now halved keeping the angular momentum same,the linear velocity is

$A$ thin circular ring of mass $M$ and radius $R$ is rotating in a horizontal plane with angular velocity $\omega$ about an axis passing through its center and perpendicular to the plane. If another ring of the same size but mass $\frac{M}{4}$ is placed coaxially on the first ring,then the new angular velocity of the system is:

The angular momentum of a system of particles is not conserved:

$A$ 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?

$A$ thin uniform rod of mass $M$ and length $L$ is rotating about a perpendicular axis passing through its centre with a constant angular velocity $\omega$. Two objects,each of mass $\frac{M}{3}$,are attached gently to the two ends of the rod. The rod will now rotate with an angular velocity of

$A$ smooth uniform rod of length $L$ and mass $M$ has two identical beads of negligible size,each of mass $m$,which can slide freely along the rod. Initially,the two beads are at the center of the rod and the system is rotating with angular velocity $\omega_0$ about its axis perpendicular to the rod and passing through its mid-point (see figure). There are no external torques. When the beads reach the ends of the rod,the angular velocity of the system is: