Gravitational potential where field is zero – Rankers Physics
Topic: Gravitation
Subtopic: Gravitational Potential

Gravitational potential where field is zero

Two bodies of masses \(m\) and \(M\) are placed at distance \(d\) apart. What is the gravitational potential (\(V\)) at the position where the gravitational field due to them is zero is \(V\) :
\(V = -\frac{G}{d}(m+M)\)
\(V = -\frac{G}{d} m\)
\(V = -\frac{GM}{d}\)
\(V = -\frac{G}{d}(\sqrt{m}+\sqrt{M})^2\)

Solution:

Let the point of zero field be at distance \(r_1\) from \(m\) and \(r_2\) from \(M\). Then \(\frac{\sqrt{m}}{r_1} = \frac{\sqrt{M}}{r_2}\), with \(r_1 + r_2 = d\). Solving gives \(r_1 = \frac{\sqrt{m}d}{\sqrt{m}+\sqrt{M}}\) and \(r_2 = \frac{\sqrt{M}d}{\sqrt{m}+\sqrt{M}}\). Thus, \(V = -\frac{Gm}{r_1} - \frac{GM}{r_2} = -\frac{G}{d}(\sqrt{m}+\sqrt{M})^2\).

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