$A$ thin circular disc of mass $M$ and radius $R$ is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity $\omega$. If another disc of same dimensions but of mass $M/2$ is placed gently on the first disc co-axially,then the new angular velocity of the system is:

$A$ solid sphere is rotating freely about its symmetry axis in free space. The radius of the sphere is increased keeping its mass same. Which of the following physical quantities would remain constant for the sphere $?$

If the earth were to suddenly contract to half of its present radius, what would be the duration of the day (in $text{ h}$)?

$A$ disc of mass $M$ and radius $R$ is rolling with angular speed $\omega$ on a horizontal plane as shown. The magnitude of angular momentum of the disc about the origin $O$ is

$A$ horizontal platform is rotating with uniform angular velocity around the vertical axis passing through its centre. At some instant of time, a viscous fluid of mass $m$ is dropped at the centre and is allowed to spread out and finally fall off. The angular velocity during this period

$A$ 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 $N$?