Two spheres of masses (m) and (M-m) are prepared from a sphere of mass M. They are kept at a distance x. For what ratio of `m/M` the gravitational attraction between them will be maximum

Explanation

The gravitational force of attraction between the two spheres is `F=(Gm(M-m))/x^2=((GMm)/x^2)-(Gm^2)/x^2`
The force to be maximum, `Df/dm=GM/x^2=0=2Gm/x^2 therefore Gm/x^2=2Gm/(x^2)
thereforeM=2m or m/M=1/2`