A particle is moving with constant speed in a circular path. When the particle turns by an angle $90^{\circ}$, the ratio of instantaneous velocity to its average velocity is $\pi: x \sqrt{2}$. The value of $x$ will be $.........$

A particle $P$ is moving in a circle of radius $'a'$ with a uniform speed $v$. $C$ is the centre of the circle and $AB$ is a diameter. When passing through $B$ the angular velocity of $P$ about $A$ and $C$ are in the ratio

If a particle covers half the circle of radius R with constant speed then

$A \,10\, kg$ ball attached to the end of a rigid massless rod of length $1\, m$ rotates at constant speed in a horizontal circle of radius $0.5\, m$ and period $1.57 \, sec$ as in fig. The force exerted by rod on the ball is ........ $N$.

A point $P$ moves in counter-clockwise direction on a circular path as shown in the figure. The movement of '$P$' is such that it sweeps out a length  $s = t^3+5$, where s is in metres and $t$ is in seconds. The radius of the path is $20\ m$. The acceleration of '$P$' when $t = 2\ s$ is nearly ..........  $m/s^2$

Two 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 they make complete circles in the same time $t$. The ratio of their centripetal acceleration is