$A$ circular ring of mass $m$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Two particles of mass $M$ are gently attached to the ring at the ends of a diameter. The ring will now rotate with an angular velocity $\omega'$ equal to:

$A$ stone of mass $m$,tied to the end of a string,is whirled around in a circle on a horizontal frictionless table. The length of the string is reduced gradually,keeping the angular momentum of the stone about the centre of the circle constant. Then,the tension in the string is given by $T = Ar^n$,where $A$ is a constant and $r$ is the instantaneous radius of the circle. The value of $n$ is equal to

$A$ rigid spherical body is spinning around an axis without any external torque. Due to a change in temperature,the volume increases by $1\%$. Its angular speed:

$A$ man is standing on the edge of a circular platform. The platform is rotating with a constant angular velocity about an axis passing through its center and perpendicular to its plane. If the man walks along the radius towards the axis, the angular velocity will ........ .

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

$A$ 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?