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Question

Two bodies of masses $M_{1}$ and $M_{2}$ are placed at a distance $R$ apart. Then at the position where the gravitational field due to them is zero, the gravitational potential is

- G \frac{\sqrt{M_{1}}}{R}

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- G \frac{\sqrt{M_{2}}}{R}

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- {(\sqrt{M_{1}} + \sqrt{M_{2}})}^{2} \frac{G}{R}

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- {(\sqrt{M_{1}} - \sqrt{M_{2}})}^{2} \frac{G}{R}

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Solution

The correct option is C $- {(\sqrt{M_{1}} + \sqrt{M_{2}})}^{2} \frac{G}{R}$
Let O be the point where gravitational field due to the masses is Zero.
Thus, $\frac{G M_{1} m}{x^{2}} - \frac{G M_{2} m}{(R - x)^{2}} = 0 ⟹ x = \frac{\sqrt{M_{1}} R}{\sqrt{M_{1}} + \sqrt{M_{2}}}$
Now gravitational potential at point O, $V_{o} = [- \frac{G M_{1}}{x^{2}}] + [- \frac{G M_{2}}{(R - x)^{2}}]$
Put the value of $x$ , $⟹ V_{o} = - (\sqrt{M_{1}} + \sqrt{M_{2}})^{2} \frac{G}{R}$

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