Two large spherical objects of mass M each (uniformly distributed) are fixed as shown in figure. A small point mass m is projected from point A heading towards center C 2 of second sphere. The minimum

Question

Two large spherical objects of mass M each (uniformly distributed) are fixed as shown in figure. A small point mass m is projected from point A heading towards center C 2 of second sphere. The minimum velocity of point mass so that it can reach upto second object at point B is n 3 G M 5 R . Then calculate n . [Neglect other gravitational forces]

TestHub · Accepted Answer

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,
$\frac{1}{2} m V_{0}^{2} - \frac{G M m}{R} - \frac{G M m}{9 R} = - 2 \frac{G M m}{5 R} + 0$
Solving, $V_{0} = \frac{8}{3} \sqrt{\frac{G M}{5 R}}$
$\Rightarrow n = 8$

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