$A$ uniform solid sphere of mass $M$ and radius $a$ is surrounded by a concentric uniform thin spherical shell of mass $0.5 M$ and radius $1.5 a$. The gravitational potential energy of a unit mass kept at a distance of $2.5 a$ from the center is

Potential energy of a satellite having mass $m$ and rotating at a height of $6.4 \times 10^6 \ m$ from the Earth's surface is

$A$ body of mass $m$ rises to a height $h = R/5$ from the earth's surface,where $R$ is the earth's radius. If $g$ is the acceleration due to gravity at the earth's surface,the increase in potential energy is:

Gravitational potential at the centre of curvature of a hemispherical bowl of radius $R$ and mass $M$ is $V.$

If one wants to remove all the mass of the earth to infinity in order to break it up completely,the amount of energy that needs to be supplied will be $\frac{x}{5} \frac{GM^2}{R}$,where $x$ is .......... (Round off to the Nearest Integer). ($M$ is the mass of earth,$R$ is the radius of earth,$G$ is the gravitational constant)

Two masses $m_1$ and $m_2$ are initially at rest and are separated by a very large distance. If the masses approach each other subsequently due to gravitational attraction between them,their relative velocity of approach at a separation distance of $d$ is: