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 maximises the gravitational force between the parts ?

Explanation

If r is the distance between m and (M-m) the gravitational force between them will be
`F=G(m(M-m))/(r^(2))=G/(r^(2))(mM-m^(2))`
For F to be maximum dF/dm=0 as M and r are constants.
`impliesd/(dm)[G/(r^(2))(mM-m^(2))]=0, implies M-2m=0`
`impliesm/M=1/2`.So the force will be maximum when the parts are equal.