$A$ 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 plate is rotating in a horizontal plane about an axis passing through its center and perpendicular to the plate with an angular velocity $\omega$. $A$ person sits at the center holding two dumbbells. When he stretches out his hands,the moment of inertia of the system becomes triple. If $E$ is the initial kinetic energy of the system,then the final kinetic energy will be $\frac{E}{x}$. The value of $x$ is $....$

$A$ circular platform of moment of inertia $100 \ kg\ m^2$ with a boy of mass $20 \ kg$ standing at the centre completes one rotation in $10 \ s$. If he walks $2 \ m$ away from the centre along the radius,then one rotation (with respect to the ground) will be completed in $.... \ s$.

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 $B$?

$A$ horizontal circular platform of radius $R = 0.5 \ m$ and mass $M = 0.45 \ kg$ is free to rotate about its axis. Two massless spring toy-guns,each carrying a steel ball of mass $m = 0.05 \ kg$,are attached to the platform at a distance $r = 0.25 \ m$ from the centre on its either sides along its diameter (see figure). Each gun simultaneously fires the balls horizontally and perpendicular to the diameter in opposite directions. After leaving the platform,the balls have a horizontal speed of $v = 9 \ m/s$ with respect to the ground. The rotational speed of the platform in $rad/s$ after the balls leave the platform is:

An ant is sitting at the edge of a rotating disc. If the ant reaches the other end, after moving along the diameter, the angular velocity of the disc will