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 of attraction between them when placed
distance `r` apart will be `F = (Gm(M - m))/(r^(2))`.
Differentiating it w.r.t.`m`, we get
`(dF)/(dm) = (G)/(r^(2)) [m(d)/(dm) (m - m) + (M - m) (dm)/(dm)]`
`= (G)/(r^(2))[m (-1) + M - m] = (G)/r^(2)(M - 2m)`
If `F` is maximum, then `(dF)/(dm) = 0`,
`:. (G)/(r^(2)) (m - 2m) = 0` or `M = 2m`
or `m = (M)/(2)`