The plank in the figure moves a distance $100 \ mm$ to the right while the centre of mass of the sphere of radius $150 \ mm$ moves a distance $75 \ mm$ to the left. The angular displacement of the sphere (in radian) is (there is no slipping anywhere):-

$A$ solid body rotates about a fixed axis such that its angular velocity depends on $\theta$ as $\omega = k \theta^{-1}$,where $k$ is a positive constant. At $t = 0$,$\theta = 0$. Then,the time dependence of $\theta$ is given as ..........

$A$ 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)

When a ceiling fan is switched off,its angular velocity falls to $\frac{1}{3}$ of its initial value while it makes $24$ rotations. How many more rotations will it make before coming to rest?

$A$ bob of mass $m$ is tied by a massless string whose other end is wound on a flywheel (disc) of radius $R$ and mass $m$. When released from rest,the bob starts falling vertically downwards. If the bob has covered a vertical distance $h$,then the angular speed of the wheel will be (There is no slipping between the string and the wheel,$g$ is the acceleration due to gravity).

$A$ thin but rigid semicircular wire frame of radius $r$ is hinged at $O$ and can rotate in its own vertical plane. $A$ smooth peg $P$ starts from $O$ and moves horizontally with constant speed $v_0$,lifting the frame upward as shown in the figure. Find the angular velocity $\omega$ of the frame when its diameter makes an angle of $60^{\circ}$ with the vertical.