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

In circular motion, the centripetal acceleration is given by

When an object is shot from the bottom of a long smooth inclined plane kept at an angle $60^{\circ}$ with horizontal. it can travel a distance $\mathrm{x}_{1}$ along the plane. But when the inclination is decreased to $30^{\circ}$ and the same object the shot with the same velocity, it can travel $x_{2}$ distance. Then $x_{1}: x_{2}$ will be

An object moves at a constant speed along a circular path in a horizontal plane with centre at the origin. When the object is at $x =+2\,m$, its velocity is $-4 \hat{ j }\, m / s$. The object's velocity $(v)$ and acceleration $(a)$ at $x =-2\,m$ will be

A stone tied to $180 cm$ long string at its end is making 28 revolutions in horizontal circle in every minute. The magnitude of acceleration of stone is $\frac{1936}{ x }\,ms ^{-2}$. The value of $x.........\left(\text { Take } \pi=\frac{22}{7}\right)$

If $\theta$ is angle between the velocity and acceleration of a particle moving on a circular path with decreasing speed, then .........