$A$ string of length $L$ is fixed at one end and carries a mass of $M$ at the other end. The mass makes $(\frac{3}{\pi})$ rotations per second about the vertical axis passing through the fixed end of the string as shown. The tension in the string is . . . . . . $ML$.

$A$ proton of mass $1.6 \times 10^{-27} \, kg$ moves in a circular orbit of radius $0.10 \, m$ under a centripetal force of $4 \times 10^{-13} \, N$. The frequency of revolution of the proton is about:

$A$ cage revolves around a vertical circle of radius $R$ with constant linear speed $\sqrt{gR}$. The cage is connected to the revolving arm in such a manner that a boy of mass $m$ remains always vertical while standing on a weighing machine kept inside the cage. It is found that

$A$ ball of mass $m = 0.5 \ kg$ is attached to the end of a string having length $L = 0.5 \ m$. The ball is rotated on a horizontal circular path about a vertical axis. The maximum tension that the string can bear is $324 \ N$. The maximum possible value of angular velocity of the ball (in $rad/s$) is

An aircraft executes a horizontal loop of radius $9 \ km$ at a constant speed of $540 \ kmh^{-1}$. The wings of the aircraft are banked at an angle of (Acceleration due to gravity $g = 10 \ ms^{-2}$)

$A$ circular overbridge has a radius of $20\,m$. What is the maximum speed with which a car can cross the bridge without leaving contact with the overbridge at the highest point (in $,m/s$)? $(g = 9.8\,m/s^2)$