$A$ 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 now rotates with an angular velocity $\omega '$ equal to:

$A$ small bead of mass $m$ moving with velocity $v$ gets threaded on a stationary semicircular ring of mass $m$ and radius $R$ kept on a horizontal table. The ring can freely rotate about its centre. The bead comes to rest relative to the ring. What will be the final angular velocity of the system?

If the radius of the Earth becomes $1/n^{th}$ of its present value,without any change in its mass,then the duration of the day becomes:

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 ........... $J$.

$A$ solid sphere is rotating freely about its symmetry axis in free space. The radius of the sphere is increased keeping its mass same. Which of the following physical quantities would remain constant for the sphere $?$

$A$ 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 gently placed at opposite ends of a diameter of the ring. The new angular velocity of the ring will be