$A$ straight rod of length $L$ is released on a frictionless horizontal floor in a vertical position. As it falls and slips,the distance of a point on the rod from the lower end,which follows a quarter circular locus,is

Write the formula expressing the relationship between linear velocity and angular velocity for a particle in rotational motion.

The potential energy of a particle of mass $m$ at a distance $r$ from a fixed point $O$ is given by $V(r) = kr^2 / 2$,where $k$ is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius $R$ about the point $O$. If $v$ is the speed of the particle and $L$ is the magnitude of its angular momentum about $O$,which of the following statements is (are) true?
$(A)$ $v = \sqrt{\frac{k}{2m}} R$
$(B)$ $v = \sqrt{\frac{k}{m}} R$
$(C)$ $L = \sqrt{mk} R^2$
$(D)$ $L = \sqrt{\frac{mk}{2}} R^2$

The spool shown in the figure is placed on a rough horizontal surface and has an inner radius $r$ and an outer radius $R$. The angle $\theta$ between the applied force and the horizontal can be varied. The critical angle $\theta$ for which the spool does not roll and remains stationary is given by:

$A$ rigid lamina is rotating about an axis passing perpendicular to its plane through point $O$ as shown in the figure. The angular velocity of point $B$ with respect to $A$ is ....... $rad/s$.

$A$ mass $m$ hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass $m$ and radius $R$. Assuming the pulley to be a perfect uniform circular disc,the acceleration of the mass $m$,if the string does not slip on the pulley,is