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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 38B / b / i / Solution

For a circular orbit, the derivative of

V_{eff} (r) = - \frac{m}{r} + \frac{h ^{2}}{2 r ^{2}} - \frac{m h ^{2}}{r ^{3}}

vanishes. Thus

\frac{m}{r ^{2}} - \frac{h ^{2}}{r ^{3}} + \frac{3 m h ^{2}}{r ^{4}} = 0, h^{2} = \frac{m r ^{2}}{r - 3 m} .

A real finite angular momentum therefore requires

r > 3 m .

Solved by gpt-5.6-sol high.

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