$A$ thin circular ring of mass $M$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Four objects,each of mass $m$,are kept gently on the opposite ends of two perpendicular diameters of the ring. The new angular velocity of the ring will be:

$A$ circular platform rotates in a horizontal plane about a vertical axis passing through its center. $A$ tortoise sits on the edge of the platform. The platform is given an initial angular velocity $\omega_0$. The tortoise now moves along a chord of the platform with a constant velocity (relative to the platform). How does the angular velocity $\omega(t)$ of the platform change with time $t$?

Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are $0.1 \; kg \cdot m^{2}$ and $10 \; rad \cdot s^{-1}$ respectively,while those for the second one are $0.2 \; kg \cdot m^{2}$ and $5 \; rad \cdot s^{-1}$ respectively. At some instant,they get stuck together and start rotating as a single system about their common axis with some angular speed. The kinetic energy of the combined system is ........... $J$.

Two coaxial discs,having moments of inertia $I_1$ and $\frac{I_1}{2}$,are rotating with angular velocities $\omega_1$ and $\frac{\omega_1}{2}$ respectively,about their common axis. They are brought into contact with each other and thereafter they rotate with a common angular velocity. If $E_f$ and $E_i$ are the final and initial total energies,then $(E_f - E_i)$ is

State the law of conservation of angular momentum and explain it with the example of a girl sitting on a swivel chair.

Explain with illustration the pure translation and combination of translation and rotation motion of a rigid body.