Question

Two masses $M$ and $4M$ are at a distance $r$ apart on the line joining them, $P$ is point where the resultant gravitational intensity is zero (such a point called null point). The distance of $P$ from the mass $M$ is:

• A. $\frac{r}{5}$
• B. $\frac{r}{3}$
• C. $\frac{2r}{3}$
• D. $\frac{4r}{5}$

Answer 1

The correct option is B $\frac{r}{3}$

Let the null point be at a distance of $x$ from the mass $M$.
$I_M-I_{4M}=\frac{GM}{x^2}-\frac{G\left(4M\right)}{(r-x{)}^2}$

Since this is equal to zero,
$\frac{GM}{x^2}=\frac{G\left(4M\right)}{(r-x{)}^2}$
$\Rightarrow {\left(\frac{r-x}{x}\right)}^2=4$
$\Rightarrow \frac{r-x}{x}=2$
$\Rightarrow \frac{r}{x}-1=2$
$\Rightarrow \frac{r}{x}=3$
$\Rightarrow x=\frac{r}{3}$