$A$ mass $M$ is divided into two parts $xM$ and $(1 - x)M$. For a given separation,the value of $x$ for which the gravitational attraction between the two pieces becomes maximum is

Two stars of masses $m_1$ and $m_2$ are parts of a binary star system. The radii of their orbits are $r_1$ and $r_2$ respectively,measured from the centre of mass of the system. The magnitude of the gravitational force that $m_1$ exerts on $m_2$ is

The mass of the Earth has been determined through:

$A$ uniform sphere of mass $M$ and radius $R$ is surrounded by a concentric spherical shell of the same mass but radius $2R$. $A$ point mass $m$ is kept at a distance $x$ $(R < x < 2R)$ in the region bounded by the spheres as shown in the figure. The net gravitational force on the particle is:

$A$ point mass '$m$' is located at a distance '$r$' from the end of a uniform thin rod of mass '$M$' and length '$L$' as shown in the figure. The magnitude of the gravitational force of attraction is

The $SI$ unit of $\frac{G}{g}$ is $(g = \text{acceleration due to gravity}, G = \text{gravitational constant})$