Question

A mass M is split into two parts m and (M - m) which are separated by a certain distance. What ratio m/M maximises the gravitational force between the parts?

Answer 1

$\text{If }r\text{ is the distance between }m\text{ and }(M-m)\text{ the gravitational force will be}\\\\F=\dfrac{Gm(M-m)}{r^2} = \dfrac{G}{r^2}(mM-m^2)\\\\\text{For }F\text{ to be maximum }\dfrac{dF}{dm}=0\text{ and }r\text{ is constant}\\\\\dfrac{G}{r^2}\times\dfrac{d}{dm}(Mm-m^2)=0\therefore\:\dfrac{d}{dm}(mM-m^2)=0,\\\\M-2m=0\:\:\:\:\:\:\:\:\:\:\dfrac{m}{M}=\dfrac{1}{2}\\\\\text{The force will be maximum when the parts have equal mass.}$