Question

Mass M is divided into two parts xM and (1 − x)M. For a given separation, the value of x for which the gravitational attraction between the two pieces becomes maximum is

• A. $\frac{1}{2}$
• B. $\frac{3}{5}$
• C. $1$
• D. $2$

Answer 1

The correct option is A

$\frac{1}{2}$

Gravitational force $F=\frac{G{m}_1{m}_2}{r^2}=\frac{GxM(1-x)M}{r^2}=\frac{G{M}^2}{r^2}x(1-x)$

For maximum value of force $\frac{dF}{dx}=0\therefore \frac{d}{dx}\left[\frac{G{M}^{2}x}{r^2}\right(1-x\left)\right]=0$

$\Rightarrow \frac{d}{dx}(x-x^2)=0\Rightarrow 1-2x=0\Rightarrow x=\frac{1}{2}$