question_answer

A mass M is broken into two parts of masses m and M - m. How are m and (M - m) related, so that force of gravitational attraction between two parts is maximum. (a) \[\frac{M}{4},\frac{3M}{4}\]                (b) \[\frac{M}{3},\frac{2M}{3}\] (c) \[\frac{M}{2},\frac{M}{2}\]                   (d) \[\frac{M}{10},\frac{9M}{10}\]

Answer:

Suppose that the two parts of the mass M i.e., m and M - m are placed at a distance r apart. Then, gravitational force between the two parts will be maximum, if \[\frac{dF}{dm}=0\]or \[\frac{dF}{dm}\left[ \frac{gm(M-m)}{{{r}^{2}}} \right]=0.\]Since G and\[{{r}^{2}}\]are constants, we have \[\frac{dF}{dm}[m(M-m)]=0\]or \[1\times (M-m)+m(0-1)=0\]or \[m=\frac{M}{2}\]