$A$ thin circular ring of mass $M$ and radius $R$ is rotating about a transverse axis passing through its centre with constant angular velocity $\omega$. Two objects each of mass $m$ are attached gently to the opposite ends of a diameter of the ring. What is the new angular velocity?

$A$ rotating table completes one revolution in $10 \ s$. Its moment of inertia is $100 \ kg \cdot m^2$. $A$ man of mass $50 \ kg$ stands at the center of the rotating table. If the man moves $2 \ m$ away from the center,what will be the angular velocity of the table in $rad/s$?

$A$ space station consists of two living modules attached to a central hub on opposite sides by long corridors of equal length $L$. Each living module contains $N$ astronauts of equal mass $m$. The mass of the space station structure is negligible compared to the mass of the astronauts,and the size of the central hub and living modules is negligible compared to the length of the corridors. Initially,the space station is rotating with angular velocity $\omega$ such that the astronauts feel an artificial gravitational field of strength $g = \omega^2 L$. Two astronauts,one from each module,move into the central hub. The remaining astronauts now feel an artificial gravitational field of strength $g'$. What is the ratio $g'/g$ in terms of $N$?

$A$ 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 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

$A$ ring of mass $M$ and radius $R$ is rotating with angular speed $\omega$ about a fixed vertical axis passing through its centre $O$ with two point masses each of mass $\frac{M}{8}$ at rest at $O$. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant,the angular speed of the system is $\frac{8}{9} \omega$ and one of the masses is at a distance of $\frac{3}{5} R$ from $O$. At this instant,the distance of the other mass from $O$ is