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 ..........  $m/s^2$

A clock has $75 \mathrm{~cm}, 60 \mathrm{~cm}$ long second hand and minute hand respectively. In $30$ minutes duration the tip of second hand will travel $x$ distance more than the tip of minute hand. The value of $x$ in meter is nearly (Take $\pi=3.14$ ) :

A smooth wire of length $2\pi r$ is bent into a circle and kept in a vertical plane. A bead can slide smoothly on the wire. When the circle is rotating with angular speed $\omega$ about the vertical diameter $AB$, as shown in figure, the bead is at rest with respect to the circular ring at position $P$ as shown. Then the value of $\omega^2$ is equal to

A particle is revolving in a circular path of radius $25 \,m$ with constant angular speed $12 \,rev/min$. Then the angular acceleration of particle is .......... $rad / s ^2$

Two bodies $A$ & $B$ rotate about an axis, such that angle $\theta_A$ (in radians) covered by first body is proportional to square of time, & $\theta_B$ (in radians) covered by second body varies linearly. At $t = 0, \theta \,A = \theta \,B = 0$. If $A$ completes its first revolution in $\sqrt \pi$ sec. & $B$ needs $4\pi \,sec$. to complete half revolution then; angular velocity $\omega_A : \omega_B$ at $t = 5\, sec$. are in the ratio

In uniform circular motion, the velocity vector and acceleration vector are