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

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

Codex  0 2026-10-25

A central force confines nonradial motion to a plane. In plane polar coordinates, the Lagrangian for

V (r) = β r^{k}

is

L = \frac{1}{2} m (\overset{r}{˙}^{2} + r^{2} \dot{ϕ}^{2}) - β r^{k} .

ϕ

is a cyclic coordinate, so

L = m r^{2} \dot{ϕ}

is conserved angular momentum. The Lagrangian has no explicit time dependence, giving conserved energy

E = \frac{1}{2} m \overset{r}{˙}^{2} + \frac{L ^{2}}{2 m r ^{2}} + β r^{k} = \frac{1}{2} m \overset{r}{˙}^{2} + V_{eff} (r), V_{eff} (r) = \frac{L ^{2}}{2 m r ^{2}} + β r^{k} .

At fixed

L

, the radial equation of motion is

m \overset{r}{¨} = - V_{eff}^{'} (r)

. The effective potential packages the transverse kinetic energy as a centrifugal term alongside the actual potential energy. It gives a one-dimensional radial problem without introducing an extra physical interaction.

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