Question

Two masses M and 4M are separated by a distance d. What will be the distance of mass 1 kg from M and 4M respectively, so that net gravitational force on it is zero?

• A.
  $\frac{d}{2},\frac{d}{2}$
• B.
  $\frac{d}{3},\frac{2d}{3}$
• C.
  $\frac{2d}{3},\frac{d}{3}$
• D.
  $\frac{d}{4},\frac{3d}{4}$

Answer 1

The correct option is B

$\frac{d}{3},\frac{2d}{3}$
Let mass 1 kg be placed at a distance x from mass M as shown in the figure.




Gravitational force on mass 1 kg due to mass 4M:

$F=\frac{G\left(4M\right).\left(1\right)}{(d-x{)}^2}=\frac{4GM}{(d-x{)}^2}...\left(i\right)$
Gravitational force on mass 1 kg due to mass M:

$F^{\prime }=\frac{GM\left(1\right)}{x^2}=\frac{GM}{x^2}...\left(ii\right)$
For net gravitational force to be equal to zero, $F-F^{\prime }=0\Rightarrow \frac{4GM}{(d-x{)}^2}-\frac{GM}{x^2}=0\Rightarrow \frac{4GM}{(d-x{)}^2}=\frac{GM}{x^2}\Rightarrow (d-x{)}^2=4x^2\Rightarrow d-x=2x\Rightarrow x=\frac{d}{3}$

Distance of mass 1 kg from mass $M=\frac{d}{3}$

Distance of mass 1 kg from mass $4M=d–x=d-\frac{d}{3}=\frac{2d}{3}$

Hence, the correct answers is option (b).