$A$ thin circular ring of mass $M$ and radius $r$ is rotating about its axis with a constant angular velocity $\omega$. Two objects,each of mass $m$,are attached gently to the opposite ends of a diameter of the ring. The ring will now rotate with an angular velocity:

$A$ uniform rod of length $8 a$ and mass $6 m$ lies on a smooth horizontal surface. Two point masses $m$ and $2 m$ moving in the same plane with speed $2 v$ and $v$ respectively strike the rod perpendicularly at distances $a$ and $2 a$ from the mid point of the rod in the opposite directions and stick to the rod. The angular velocity of the system immediately after the collision is:

If the earth were to suddenly contract to half of its present radius, what would be the duration of the day (in $text{ h}$)?

Two wheels $A$ and $B$ are mounted on the same axis. The moment of inertia of wheel $A$ is $I_A$ and its angular velocity is $\omega_A$,while wheel $B$ is initially at rest. When they are coupled together by a clutch,they rotate with a common angular velocity $\omega$. What is the moment of inertia of wheel $B$?

If the radius of the earth is reduced to three-fourth of its present value without any change in its mass,then the duration of the day on earth will be . . . . . . hours $30$ minutes.

$A$ 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