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):-

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?

$A$ 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 $t = 2.0 \ s$?

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

$A$ plank with a uniform sphere placed on it rests on a smooth horizontal plane. The plank is pulled to the right by a constant force $F$. If the sphere does not slip over the plank,which of the following is true?

For every particle of a rigid body in rotational motion,which of the following is true?