Question

Two large spherical objects of mass $M$ each (uniformly distributed) are fixed as shown in the figure. A small point mass $m$ is projected from point $A$ heading towards centre $C_2$ of the second sphere. The minimum velocity of point mass so that it can reach up to the second object at point $B$ is $\frac{n}{3}\sqrt{\frac{GM}{5R}}$ . Then calculate $n$ . [Neglect other gravitational forces]

Answer 1

Answer: 8

If the ball can reach the mid point of line connecting spheres, then it will get attracted towards B. (At midpoint gravitational intensity is zero)
Using energy conservation,
$\frac{1}{2}m{V}_0^2-\frac{GMm}{R}-\frac{GMm}{9R}=-2\frac{GMm}{5R}+0$
Solving, $V_0=\frac{8}{3}\sqrt{\frac{GM}{5R}}$
$\Rightarrow n=8$