Question

If mass M is split into two parts, m and $(M-m)$ which are then separated by a certain distance. What ratio of m/M maximizes the gravitational force between the two parts.

• A. $1/3$
• B. $1/2$
• C. $1/4$
• D. $1/5$

Answer 1

The correct option is B $1/2$

$F=\frac{Gm(M-m)}{r^2}=\frac{G}{r^2}(mM-m^2)$

Force will be maximum if

$\frac{dF}{m}=0\Rightarrow \frac{d}{dm}=\left[\frac{G}{r^2}(mM-m^2)\right]=0\Rightarrow \frac{G}{r^2}(M-2m)=0\therefore M-2m=0\Rightarrow M=2m\Rightarrow \frac{M}{m}=2or,\frac{m}{M}=\frac{1}{2}$

$F$ will be maximum when$\frac{m}{M}=\frac{1}{2}$