$A$ tube of length $L$ is filled completely with an incompressible liquid of mass $M$ and closed at both the ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity $\omega$. The force exerted by the liquid at the other end is

Write the change in magnitude and direction of angular velocity with respect to time for a body rotating about a fixed axis.

$A$ solid sphere of radius $R$ is placed on a smooth horizontal surface. $A$ horizontal force $F$ is applied at a height $h$ from the lowest point. For the maximum acceleration of the centre of mass,which is correct?

The left end of a massless stick with length $l$ is placed on the corner of a table,as shown in the figure. $A$ point mass $m$ is attached to the center of the stick,which is initially held horizontal. It is then released. Immediately afterward,what normal force does the table exert on the stick?

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?
$(A)$ $v = \sqrt{\frac{k}{2m}} R$
$(B)$ $v = \sqrt{\frac{k}{m}} R$
$(C)$ $L = \sqrt{mk} R^2$
$(D)$ $L = \sqrt{\frac{mk}{2}} R^2$

$A$ uniform disc of radius $R$ and mass $M$ is free to rotate only about its axis. $A$ string is wrapped over its rim and a body of mass $m$ is tied to the free end of the string as shown in the figure. The body is released from rest. Then the acceleration of the body is