Two bodies of masses $\dpi{100} m\: and \: 4m$ are placed at a distance $\dpi{100} r$. The gravitational potential at a point on the line joining them where the gravitational field is zero is Option 1) $zero$ Option 2) $-\frac{4Gm}{r}$ Option 3) $-\frac{6Gm}{r}$ Option 4) $-\frac{9Gm}{r}$

S Sabhrant Ambastha

As we discussed in

Gravitational field due to Point mass -

$\dpi{100} F=\frac{GmM}{r^{2}}$

$\dpi{100} I= \frac{F}{m}= \frac{GMm}{r^{2}m}$

$\dpi{100} \therefore I= \frac{GM}{r^{2}}$

$\dpi{100} G \rightarrow Gravitational\: constant$

$\dpi{100} M\rightarrow mass\: of\: earth$

- wherein

As the distance (r) of test mass from point (M) Increases I decreases.

$\dpi{100} I\propto \frac{1}{r^{2}}$

$\dpi{100} I= 0\: at\: \left ( r= \infty \right )$

$\frac{Gm}{r^{2}}=\frac{G(4m)}{(r-x)^{2}}$

$\left [ \frac{x}{\left ( r-x \right )} \right ]^2=\frac{1}{4}$

$\Rightarrow x=\frac{r}{3}$

$U=\frac{-Gm}{x}-\frac{G\left ( 4m \right )}{\left ( r-x \right )}$

$=\frac{-Gm}{\left ( \frac{r}{3} \right )}-\frac{G\left ( 4m \right )}{\left ( r-\frac{r}{3} \right )}$

$= -\frac{3Gm}{r}-\frac{3G\left ( 4m \right )}{2r}$

$= -\frac{9Gm}{2r}$

Option 1)

$zero$

Incorrect option

Option 2)

$-\frac{4Gm}{r}$

Incorrect option

Option 3)

$-\frac{6Gm}{r}$

Incorrect option

Option 4)

$-\frac{9Gm}{r}$

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