Q: Mass M is split into two parts m and (M-m), which are then separated by a certain distance. What is the ratio of (m/M) which maximise the gravitational force between the parts ?

Sol: If r is the distance between m and (M-m), the gravitational force between them will be

$\large F = G \frac{m(M-m)}{r^2}$

$\large F = G \frac{(M m-m^2)}{r^2}$

For F to be maximum dF/dm = 0 as M & r are constants.

$\large \frac{d}{dm}[\frac{G}{r^2}(M m – m^2)] = 0 $

M – 2 m = 0

$\large \frac{m}{M} = \frac{1}{2}$

So,the force will be maximum when the parts are equal .