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 thhe 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` as M and r are constant.
i.e., `(d)/(dm)[(G)/(r^(2))(mM-m^(2))]=0` i.e., `M-2m=0[because(G)/(r^(2))ne0]`
or `(m)/(M)=(1)/(2)` i.e., the force will be maximum when the two parts are equal.