A body performing unifom circular motion completed $140$ revolution in a second. Its angular speed is .......... $rad / s$

A student skates up a ramp that makes an angle $30^{\circ}$ with the horizontal. $He /$ she starts (as shown in the figure) at the bottom of the ramp with speed $v_0$ and wants to turn around over a semicircular path xyz of radius $R$ during which he/she reaches a maximum height $h$ (at point y) from the ground as shown in the figure. Assume that the energy loss is negligible and the force required for this turn at the highest point is provided by his/her weight only. Then ( $g$ is the acceleration due to gravity)

$(A)$ $v_0^2-2 g h=\frac{1}{2} g R$

$(B)$ $v_0^2-2 g h=\frac{\sqrt{3}}{2} g R$

$(C)$ the centripetal force required at points $x$ and $z$ is zero

$(D)$ the centripetal force required is maximum at points $x$ and $z$

A stone of mass $0.3\,kg$ attached to a $1.5\,m$ long string is whirled around in a horizontal circle at a speed of $6\,m s ^{-1}$. The tension in the string is $............\,N$

A particle of mass ${m}$ is suspended from a ceiling through a string of length $L$. The particle moves in a horizontal circle of radius $r$ such that ${r}=\frac{{L}}{\sqrt{2}}$. The speed of particle will be:

A particle starting from rest, moves in a circle of radius $r$. It attains a velocity of $\mathrm{V}_{0} \;\mathrm{m} / \mathrm{s}$ in the $\mathrm{n}^{\text {th }}$ round. Its angular acceleration will be

Two particles having mass $M$ and $m$ are moving in a circular path having radius $R$ and $r$. If their time period are same then the ratio of angular velocity will be