$A$ mass $m$ is supported by a massless string wound around a uniform hollow cylinder of mass $m$ and radius $R$. If the string does not slip on the cylinder, then with what acceleration will the mass be released? (Assume $g =$ acceleration due to gravity)

As shown in the figure,a bob of mass $m$ is tied by a massless string whose other end portion is wound on a flywheel (disc) of radius $r$ and mass $m$. When released from rest,the bob starts falling vertically. When it has covered a distance of $h$,the angular speed of the wheel will be

$A$ $3.0\ kg$ bobbin consists of a central cylinder of radius $5.0\ cm$ and two end plates each of radius $6.0\ cm$. It is placed on a slotted incline,where friction is sufficient to prevent sliding. $A$ block of mass $4.5\ kg$ is suspended from a cord wound around the bobbin and passing through the slot under the incline. If the bobbin is in static equilibrium,what is the angle of tilt $\theta$ of the incline?

$A$ rigid body is rotating with variable angular velocity $\omega = (a - bt)$ at any instant of time $t$. The total angle subtended by it before coming to rest will be ($a$ and $b$ are constants).

$A$ hemisphere of radius $R$ and mass $4m$ is free to slide with its base on a smooth horizontal table. $A$ particle of mass $m$ is placed on the top of the hemisphere. The angular velocity of the particle relative to the hemisphere at an angular displacement $\theta$ when the velocity of the hemisphere is $v$ is:

$A$ binary star system consists of two stars,one of which has double the mass of the other. The stars rotate about their common centre of mass: