A stone of mass $1\,kg$ is tied to end of a massless string of length $1\,m$. If the breaking tension of the string is $400\,N$, then maximum linear velocity, the stone can have without breaking the string, while rotating in horizontal plane, is $.......\,ms^{-1}$

Two particles $A$ and $B$ are moving in uniform circular motion in concentric cirdes of radius $r_{A}$ and $r_{B}$ with speed $v_A$ and $v_B$ respectively. The time period of rotation is the same. The ratio of angular speed of $A$ to that of $B$ will be

A particle $P$ is sliding down a frictionless hemispherical bowl. It passes the point $A$ at $t = 0$. At this instant of time, the horizontal component of its velocity is $v$. A bead $Q$ of the same mass as $P$ is ejected from $A$ at $t = 0$ along the horizontal string $AB$ (see figure) with the speed $v$. Friction between the bead and the string may be neglected. Let ${t_P}$ and ${t_Q}$ be the respective time taken by $P$ and $Q$ to reach the point $B$. Then

A particle moving in a circle of radius $R$ with uniform speed takes time $\mathrm{T}$ to complete one revolution. If this particle is projected with the same speed at an angle $\theta$ to the horizontal, the maximum height attained by it is equal to $4 R$. The angle of projection $\theta$ is then given by :

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

A car is moving with speed $30$ $m/\sec $ on a circular path of radius $500\, m$. Its speed is increasing at the rate of $2m/{\sec ^2},$ What is the acceleration of the car ........ $m/sec^2$