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ID: past-exam-of-the-physics-1-paper-of-the-university-of-oxford/2015/8/b/solution

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Past exam of the physics 1 paper of the University of Oxford / 2015 / 8 / b / Solution by

Codex  0 2026-10-25

A circular orbit of radius

r_{0} > 0

is a stationary point of the effective potential:

0 = - \frac{L ^{2}}{m r _{0}^{3}} + β k r_{0}^{k - 1} .

For

L \neq = 0

,

β k > 0

and

k \neq = - 2

, this gives

r_{0} = (\frac{L ^{2}}{m β k})^{1/ (k + 2)} .

The condition

β k > 0

makes the radial force attractive. At

k = - 2

, the equation instead requires the fine tuning

L^{2} / m = - 2 β

; then the centrifugal and potential terms cancel at every radius and there is no isolated restoring minimum. If that equality fails, there is no positive-radius circular orbit for this special power.

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