Two bodies of masses M₁ and M₂ are placed at a distance 'r'. The gravitational potential at the point where the gravitational field is zero.

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$- \frac{G}{r} (M_{1} + 2 \sqrt{M_{1} M_{2}} + M_{2})$

$- \frac{G}{r} {(M_{1} + M_{2})}^{2}$

$- \frac{G}{r} (\sqrt{M_{1}} + \sqrt{M_{2}})$

$- 2 \frac{G}{r} (M_{1} + M_{2})$

answer is A.

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Detailed Solution

At a point P, resulting gravitational field is

$\vec{E_{P}} = \vec{E_{1}} + \vec{E_{2}} = 0$

$\Rightarrow \frac{{GM}_{1}}{x^{2}} = \frac{{GM}_{2}}{{(r - x)}^{2}}$

$\Rightarrow x = \frac{r}{\sqrt{\frac{M_{2}}{M_{1}}} + 1}$

$V_{P} = V_{1} + V_{2} = - \frac{{GM}_{1}}{x} - \frac{{GM}_{2}}{r - x}$

substituting the value of x,

$V_{P} = - \frac{G}{r} [M_{1} + M_{2} + 2 \sqrt{M_{1} M_{2}}]$

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