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/(dm) [G/r^(2) (mM-m^(2))]=0`
`implies G/r^(2) (M-2m)=0" i.e." M-2m=0" "[ :' G/r^(2) ne 0]`
or `m/M=1/2`, i.e., the force will be maximum when the two parts are identical.