$A$ star of mass $M$ and radius $R$ is made up of gases. The average gravitational pressure compressing the star due to the gravitational pull of the gases making up the star depends on $R$ as

Two identical particles of mass $1 \, kg$ each go round a circle of radius $R$,under the action of their mutual gravitational attraction. The angular speed of each particle is:

$A$ geostationary satellite is at a height $h$ above the surface of the Earth. If the Earth's radius is $R$,which of the following statements is correct?

Let $\omega$ be the angular velocity of the earth's rotation about its axis. Assume that the acceleration due to gravity on the earth's surface has the same value at the equator and the poles. An object weighed at the equator gives the same reading as a reading taken at a depth $d$ below the earth's surface at a pole $(d << R)$. The value of $d$ is

$A$ rocket is fired vertically from the surface of Mars with a speed of $2 \; km/s$. If $20 \%$ of its initial kinetic energy is lost due to Martian atmospheric resistance,how far will the rocket go from the surface of Mars before returning to it? (Mass of Mars $= 6.4 \times 10^{23} \; kg$,radius of Mars $= 3395 \; km$,$G = 6.67 \times 10^{-11} \; N m^2 kg^{-2}$)

$A$ body situated on the surface of the earth becomes weightless at the equator when the rotational kinetic energy of the earth reaches a critical value '$K$'. The value of '$K$' is given by [$g$ = gravitational acceleration on earth's surface,$M$ = mass of the earth,and $R$ = radius of the earth].