Question

Mass M is divided into two parts x M 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 = $\frac{G{m}_1{m}_2}{r^2}$ = $\frac{GxM(1-x)M}{r^2}$ = $\frac{{GM}^2}{r^2}$ x (1 - x)

For maximum value of force $\frac{dF}{dx}$ = 0

$\therefore$ $\frac{d}{dx}$ [ $\frac{{GM}^2}{r^2}$ x (1 - x) ] = 0

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

$\Rightarrow$ 1 - 2x = 0 $\Rightarrow$ x = $\frac{1}{2}$