A particle moves from points $1$ to $2$ during a time $\frac{1}{2} \mathrm{s}$. Find the average angular speed of the particle over $\frac{1}{2} \mathrm{s}$,

$\left(\mathrm{i}\right)$ in the anticlockwise sense

$\left(\mathrm{ii}\right)$ in the clockwise sense

A particle starts from the point $1$ and reaches at the same point by revolving once in anti-clockwise sense as shown in the figure $\left(\mathrm{i}\right)$ and once in anti-clockwise sense and again in clockwise sense as shown in the figure $\left(\mathrm{ii}\right)$. If the time taken in both cases is equal to $2 \mathrm{s},$ find the average angular velocity of the particle.

A particle is moving with constant speed in a circular path. Find the ratio of average velocity to its instantaneous velocity when the particle describes an angle $\mathrm{\theta }=\frac{\mathrm{\pi }}{2}$.

A particle is moving in a circle with constant speed $v.$ If the radius of the circle is $R,$ find the angular velocity of the particle relative to any point $A$ on the perimeter of the circle.

An aeroplane moves with constant velocity $v$ parallel to $x$ -axis at a height $y=h.$ Find the

(a) angular velocity
(b) angular acceleration of the aeroplane relative to $O$ as the function of time $t.$ Assume that at $t=0, x=0.$

Find the angular velocity of $A$ with respect to $B$ in the figure given below:

Two satellites $1$ and $2$ are orbiting with angular velocities $\omega$ and $2\omega$ relative to the centre of earth respectively, in the same plane. If the radii of their orbits are $r$ and $3r$ respectively, find the angular velocity of $1$ relative to $2\left({\overrightarrow{\omega }}_{1,2}\right)$ and angular velocity of $2$ relative to $1\left({\overrightarrow{\omega }}_{2,1}\right).$

Find the time period of the meeting of the minute hand and second hand of a clock.