Two astronauts each have a mass of $75.0\text{ kg}$ are connected by a $10\text{ m}$ long rope of negligible mass. They are isolated in space and orbit around common centre of mass with a speed of$10 {\text{ms}}^{-1}$ relative to each other, as shown in diagram.


              

Two particles $\mathrm{A},\mathrm{ }\mathrm{B}$ are moving on two concentric circles of radii $R_1$ and $R_2$ with equal angular speed $\omega .$ At $t=0,$ their positions and direction of motion are shown in the figure:

The relative velocity $\overrightarrow{V_A}-\overrightarrow{V_B}$ at $t=\frac{\pi }{2\omega }$ is given by:

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