$A$ thin circular ring of mass $M$ and radius $R$ is rotating with a constant angular velocity $2 \; rad \; s^{-1}$ in a horizontal plane about an axis vertical to its plane and passing through the center of the ring. If two objects,each of mass $m$,are attached gently to the opposite ends of a diameter of the ring,the ring will then rotate with an angular velocity (in $rad \; s^{-1}$) of:

$A$ 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 $N$?

$A$ smooth uniform rod of length $L$ and mass $M$ has two identical beads of negligible size,each of mass $m$,which can slide freely along the rod. Initially,the two beads are at the center of the rod and the system is rotating with angular velocity ${\omega _0}$ about an axis perpendicular to the rod and passing through the midpoint of the rod (see figure). There are no external torques. When the beads reach the ends of the rod,the angular velocity of the system is

$A$ particle of mass $m$ moves with a velocity $v_0$ in a circle of radius $R_0$ on a smooth horizontal plane. The mass is tied to a string passing through a hole in the smooth plane as shown in the figure. The tension in the string is gradually increased,and finally,the mass $m$ moves in a circle of radius $\frac{R_0}{2}$. What is the ratio of the final kinetic energy to the initial kinetic energy?

State and explain the law of conservation of angular momentum.

$A$ hot solid sphere is rotating about a diameter at an angular velocity $\omega_0$. If it cools so that its radius reduces to $\frac{1}{\eta}$ of its original value,its angular velocity becomes .............