$A$ 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.

$A$ round disc of moment of inertia $I_{2}$ about its axis perpendicular to its plane and passing through its centre is placed over another disc of moment of inertia $I_{1}$ rotating with an angular velocity $\omega$ about the same axis. The final angular velocity of the combination of discs is

If the mass of the Earth is kept constant and its radius is reduced to half, then the duration of the day will become ........ $hr$.

$A$ 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

$A$ 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 an axis perpendicular to the rod and passing through the midpoint of the rod (see figure). There are no external torques. When the beads reach the ends of the rod,the angular velocity of the system is

$A$ 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 $O$.