$A$ uniform disc of radius $R$ and mass $M$ can rotate on a smooth axis passing through its centre and perpendicular to its plane. $A$ force $F$ is applied on its rim as shown in the figure. What is the tangential acceleration of the rim?

$A$ 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:

$A$ wheel of radius $1.5 \ m$ rotates with a constant angular acceleration of $10 \ rad/s^2$. Its initial angular speed is $(\frac{60}{\pi}) \ rpm$. What will be its angular speed and angular displacement at $t = 2.0 \ s$?

$A$ block of mass $2\,kg$ hangs from the rim of a wheel of radius $0.5\,m$. On releasing from rest,the block falls through $5\,m$ height in $2\,s$. The moment of inertia of the wheel will be ...... $kg \cdot m^2$.

Point masses $m_1$ and $m_2$ are placed at the opposite ends of a rigid rod of length $L$ and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point $P$ on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity $\omega_0$ is minimum,is given by

$A$ 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.