A small toy car is moving with a constant speed of $10 \mathrm{m} {\mathrm{s}}^{-1}$ in a circle of radius $\frac{16}{\mathrm{\pi }} \mathrm{m}$. The circle lies in the horizontal plane. At time $t=0$, the car is at point $P$ and moving in an anticlockwise direction. At this instant, a stone is projected from the car with a velocity $\left(-6\hat{\mathrm{i}}+8\hat{\mathrm{k}}\right) \mathrm{m} {\mathrm{s}}^{-1}$ relative to the car. The angular displacement ($\text{in} \mathrm{radian}$) of the car when the stone is at the highest point of its trajectory is,

If the kinetic energy of a particle of mass $m,$ performing uniform circular motion in a circle of radius $r,$ is $E,$ find the acceleration of the particle.

A mass $\text{m}$ moves in a circle on a smooth horizontal plane with velocity ${\mathrm{v}}_0$ at a radius ${\mathrm{R}}_0$ . The mass is attached to a string which passes through a smooth hole in plane as shown.

The tension in the string is increased gradually and finally $\text{m}$ moves in a circle of radius $\frac{{\mathrm{R}}_0}{2}$. The final value of the kinetic energy is:

In the given figure, $a=15 \mathrm{m} {\mathrm{s}}^{-2}$ represents the total acceleration of a particle moving in the clockwise direction in a circle of the radius $R=2.5 \mathrm{m}$at a given instant of time. The speed of the particle is