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

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

Two racing cars of masses $m_1$ and $m_2$ are moving in circles of radii $r_1$ and $r_2$ respectively. Their speeds are such that each makes a complete circle in the same time $t$. The ratio of the angular speeds of the first to the second car is

A plank is moving in a horizontal direction with a constant acceleration $\alpha \hat{ i }$. A uniform rough cubical block of side $l$ rests on the plank and is at rest relative to the plank. Let the centre of mass of the block be at $(0, l / 2)$ at a given instant. If $\alpha =g / 10$, then the normal reaction exerted by the plank on the block at that instant acts at

The instantaneous angular position of a point on a rotating wheel is given by the equation $\theta (t) = 2t^3 -6t^2$. The torque on the wheel becomes zero at $t$ $=$ ........ $\sec.$

If the equation for the displacement of a particle moving on a circular path is given by:

                               $\theta  = 2t^3 + 0.5$

Where $\theta $ is in radian and $t$ in second, then the angular velocity of the particle at $t = 2\,sec$ is $t=$ ....... $rad/sec$