First slide

Gravitational field intensity

Question

Two bodies of masses M₁, and M₂ are kept separated by a distance d. The potential at the point where the gravitational field produced by them is zero, is :

Easy

A

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

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

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

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

Solution

$\begin{array}{l} \frac{{GM}_{1}}{x^{2}} = \frac{{GM}_{2}}{(d - x)^{2}} \Rightarrow \frac{d - x}{x} = \sqrt{\frac{M_{2}}{M_{1}}} \\ \Rightarrow \frac{d}{x} = 1 + \sqrt{\frac{M_{2}}{M_{1}}} \Rightarrow x = \frac{\sqrt{M_{1}} d}{\sqrt{M_{1}} + \sqrt{M_{2}}} \end{array}$
Now potential $V = \frac{{GM}_{1}}{x} - \frac{{GM}_{2}}{d - x} = - \frac{G}{d} (M_{1} + M_{2} + 2 \sqrt{M_{1} M_{2}})$

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