Question

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

• A. $1:4$
• B. $1:3$
• C. $1:2$
• D. $1:1$

Answer 1

The correct option is C $1:2$

The two parts of masses are $m$ and $\left(M-m\right)$.

The gravitational force between the parts is given as,

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

The condition for maximum force is given as,

$\frac{dF}{dm}=0$

$\frac{d}{dm}\left(\frac{Gm\left(M-m\right)}{r^2}\right)=0$

$\frac{G}{r^2}\left(M-2m\right)=0$

$\left(M-2m\right)=0$

$\frac{m}{M}=\frac{1}{2}$

Thus, the ratio $\frac{m}{M}$ is $\frac{1}{2}$.