A particle of mass $M$ is placed at the centre of a uniform spherical shell of mass $2M$ and radius $R$. The gravitational potential on  the surface of the shell is

The height at which the weight of a body becomes $1/16^{th}$, its weight on the surface of earth (radius $R$), is

A satellite is launched into a circular orbit of radius $R$ around earth, while a second satellite is launched into a circular orbit of radius $1.02\, {R}$. The percentage difference in the time periods of the two satellites is -

A body of mass $m$ is situated at distance $4R_e$ above the Earth's surface, where $R_e$ is the radius of Earth how much minimum energy be given to the body so that it may escape

The two planets have radii $r_1$ and $r_2$ and their densities $p_1$ and $p_2$ respectively. The ratio of acceleration due to gravity on them will be

A satellite moving with velocity $v$ in a force free space collects stationary interplanetary dust at a rate of $\frac{{dM}}{{dt}} = \alpha v$ where $M$ is the mass (of satellite + dust) at that instant . The instantaneous acceleration of the satellite is