Two discs of same moment of inertia $I$ are rotating about their central axes perpendicular to their planes with angular velocities $\omega_1$ and $\omega_2$. They are brought into contact face to face,such that their axes of rotation coincide. The expression for the loss of energy during this process is:

$A$ child with mass $m$ is standing at the edge of a disc with moment of inertia $I$,radius $R$,and initial angular velocity $\omega$. See the figure given below. The child jumps off the edge of the disc with tangential velocity $v$ with respect to the ground. The new angular velocity of the disc is

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

An equilateral triangle $ABC$ is made of a uniform wire. Two identical beads are initially at $A$. The triangle is set to rotate about a vertical axis $AO$. The beads are then released simultaneously from rest to slide down along $AB$ and $AC$. Neglecting friction,which quantities are conserved as the beads slide down?

$A$ 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 $N$?

$A$ disc of mass $1 \, kg$ and radius $0.1 \, m$ is rotating with angular velocity $20 \, rad/s$. What is the angular velocity (in $rad/s$) if a mass of $0.5 \, kg$ is placed on the circumference of the disc?