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`. Gravitationa force of attraction between the two masses when placed distance `r` apart will be
`F=(Gm(M-m))/(r^(2))`
`(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`
`implies G/(r^(2))(M-2m)=0`
or `M=2m` or `m=M/2`