In a satellite if the time of revolution is $T$, then $K.E.$ is proportional to

In order to shift a body of mass $m$ from a circular orbit of radius $3R$ to a higher radius $5R$ around the earth, the work done is

If $g$ is the acceleration due to gravity on the earth's surface, the gain in the potential energy of an object of mass $m$ raised from the surface of the earth to a height equal to the radius $R$ of the earth, is

The escape velocity from a planet is $V_e.$  A tunnel is dug along the diameter of the planet and a small body dropped into it. The speed of the body at the centre of the planet will be

A particle of mass $M$ is situated at the centre of a spherical shell of same mass and radius $a$. The gravitational potential at a point situated at $\frac {a}{2}$ distance from the centre, will be

The masses and radii of the earth and the moon are $M_1, R_1$ and $M_2, R_2$ respectively. Their centres are distance $d$ apart. The minimum speed with which particle of mass $m$ should be projected from a point midway between the two centres so as to escape to infinity is