In the $P^{\text{th}}$ second,a particle describes an angular displacement of $\beta \text{ rad}$. If it starts from rest,the angular acceleration is:

$A$ car is moving at a speed of $72 \, km/h$. The diameter of its wheels is $0.25 \, m$. If the wheels stop after completing $20$ revolutions upon applying brakes,the angular retardation produced by the brakes is ....... $rad/s^2$. (in $.5$)

$A$ wheel initially at rest,begins to rotate about its axis with constant angular acceleration. If it rotates through an angle $\theta_1$ in first $2 \ s$ and a further angle $\theta_2$ in the next $2 \ s$,the ratio $\theta_1 : \theta_2$ is

$A$ fan rotating with an angular speed of $600 \, rev/min$ comes to rest after $60$ revolutions when switched off. The time taken to come to rest is ........ $(sec)$.

$A$ racing car is travelling along a track at a constant speed of $40\ m/s$. $A$ $T.V.$ cameraman is recording the event from a distance of $30\ m$ directly away from the track as shown in the figure. In order to keep the car under view in the position shown,the angular speed with which the camera should be rotated is ........ $rad/s$.

If the position vector of a particle is $\vec{r} = (3\hat{i} + 4\hat{j}) \text{ m}$ and its angular velocity is $\vec{\omega} = (\hat{j} + 2\hat{k}) \text{ rad/s}$,then its linear velocity is (in $\text{m/s}$):