Two bodies of mass and are placed at a distance . The gravitational potential on the line joining the bodies where the gravitational field equals zero, will be (G = gravitational constant) [2023]

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Solution

Q.
Two bodies of mass $m$ and $9 m$ 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) [2023]

1 $- \frac{16 G m}{R}$

2 $- \frac{20 G m}{R}$

3 $- \frac{8 G m}{R}$

4 $- \frac{12 G m}{R}$

Ans.
(1)

Let gravitational field at point $P$ is zero which is situated at a distance $x$ from mass $m$ .

$∴$    $\frac{G m}{x^{2}} = \frac{G (9 m)}{{(R - x)}^{2}}$

or,   ${(R - x)}^{2} = 9 x^{2}$

or,   $\frac{R - x}{x} = 3 or x = \frac{R}{4}$ ....(i)

Gravitational potential at point $P,$ $V_{P} = - \frac{G m}{x} - \frac{G (9 m)}{R - x}$

or   $V_{P} = - \frac{G m}{R / 4} - \frac{G (9 m)}{3 R / 4} (using (i))$

or $V_{P} = - \frac{G m}{R} (4 + 12) = - 16 \frac{G m}{R}$

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