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 will be `F=G(m(M-m))/(r^(2))=(G)/(r^(2))(mM-m^(2))`
For F to be maximum `(dF)/(dm)=0` and `(d^(2)F)/(dm^(2))lt 0` as M
and r are constants, i.e. `(d)/(m)[(G)/(r^(2))(mM-m^(2)]=0`
i.e. M-2m=0
or `(m)/(M)=(1)/(2)`, i.e., the force will be maximum when the two parts are identical.