Two bodies of mass $m$ and $9m$ are placed at a distance $R$. The gravitational potential on the line joining

Question

Two bodies of mass $m$ and $9m$ are placed at a distance $R$. The gravitational potential on the line joining the bodies where the gravitational field equals zero, will be: ($G=$ gravitational constant)

Accepted Answer

1. **Identify the Neutral Point:** The gravitational field is zero at a point where the forces due to the two masses cancel each other out. Let this point be at a distance $x$ from mass $m$. Consequently, it is at a distance $(R-x)$ from mass $9m$.
   $$ \frac{Gm}{x^2} = \frac{G(9m)}{(R-x)^2} $$
   Taking the square root of both sides:
   $$ \frac{1}{x} = \frac{3}{R-x} \Rightarrow R - x = 3x \Rightarrow 4x = R \Rightarrow x = \frac{R}{4} $$
   The distance from $9m$ is $R - \frac{R}{4} = \frac{3R}{4}$.

2. **Calculate Gravitational Potential:** The total gravitational potential $V$ is the scalar sum of the potentials due to each mass at that point ($V = -\frac{GM}{r}$).
   $$ V = -\frac{Gm}{x} - \frac{G(9m)}{R-x} $$
   Substitute the distances:
   $$ V = -\frac{Gm}{R/4} - \frac{9Gm}{3R/4} $$
   $$ V = -\frac{4Gm}{R} - \frac{36Gm}{3R} $$
   $$ V = -\frac{4Gm}{R} - \frac{12Gm}{R} = -\frac{16Gm}{R} $$

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