A mass M is broken into two parts of masses `m_1 and m_2` . How are `m_1 and m_2` related so that force of gravitational attraction between the two parts is maximum ?

Explanation

Let `m_1 =m` then `m_2=M-m`.
Gravitational force between the two parts when they are placed at a distance r apart is,
`F=G(m(M-m))/(r^2)=(GmM)/(r^2)-(Gm^2)/(r^2)`
Differentiating with respect to m, we have,
`(dF)/(dm)=(GM)/(r^2)-(2Gm)/(r^2)`
For F to be maximum ,`(dF)/(dm)=0`
`therefore (GM)/(r^2)=(2GM)/(r^2) or, M=2m or, m=M/2`
`therefore m_1=m_2 =M/2`