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

Answer: (A)

$\frac { 1 } { 2 }$

Answer 2

Gravitational force

$F = \frac { G m _ { 1 } m _ { 2 } } { r ^ { 2 } } = \frac { G x M ( 1 - x ) M } { r ^ { 2 } } = \frac { G M ^ { 2 } } { r ^ { 2 } } x ( 1 - x )$ For

maximum value of force $\frac { d F } { d x } = 0$ $\therefore$

$\frac { d } { d x } \left[ \frac { G M ^ { 2 } x } { r ^ { 2 } } ( 1 - x ) \right] = 0$

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