If the earth were to suddenly contract to $1/n^{th}$ of its present radius without any change in its mass, the duration (in $hrs.$ ) of the new day will be nearly

A ball is thrown on a lawn in such a way that it initially slides with a speed $v_0$ without  rolling. It gradually picks up rotation motion. Find the speed of the ball at which there will  be rolling without slipping-

The mass per unit length of a rod of length $l$ is given by : $\lambda = \frac{M_0x}{l}$ ,where $M_0$ is a constant and $x$ is the distance from one end of the rod. The position of  centre of mass of the rod is

If $\vec F$ is the force acting on a particle having position vector $\vec r$ and $\vec \tau $ be the torque of this force about the origin, then

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

A circular stage is free to rotate about vertical axis passing through centre. $A$ tortoise is  sitting at corner of stage. Stage is provided angular velocity $\omega_0$. If tortoise start  moving along one chord at constant speed with respect to stage then how the angular  velocity of stage $\omega(t)$ vary with time $t$ :-