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Period of small radial oscillations of a nudged circular orbit under an inverse-square force

A particle on a circular orbit under F = -k/r^2 is nudged outward with its angular momentum unchanged and its distance oscillates. U_eff'' = k/r0^3 with r0 = l^2/(mk) gives T = 2 pi l^3/(mk^2) - option A - the same as the orbital period, so the orbit closes.

Sub-topics covered

• Central force motion with conserved angular momentum
• Effective potential U_eff = l^2/(2mr^2) - k/r and the circular orbit at its minimum
• Small radial oscillations: omega^2 = U_eff''(r0)/m
• Inverse-square force: radial and orbital periods equal, closed (elliptical) orbits

What it teaches

• With l fixed the radial motion is that of a particle in U_eff = l^2/(2mr^2) - k/r; the circular orbit is its minimum, r0 = l^2/(mk)
• U_eff''(r0) = 3l^2/(m r0^4) - 2k/r0^3 = k/r0^3, so omega_r^2 = k/(m r0^3)
• T = 2 pi sqrt(m r0^3/k) = 2 pi l^3/(m k^2) - option A - equal to the orbital period, so the nudged orbit closes (an ellipse)
• Option B 2 pi sqrt(m/k) has the wrong units for k in N m^2 (the spring formula); C and D are a third and a fifth of the right value
• For other power laws k/r^n the radial period is T_orbit/sqrt(3 - n) and the orbit traces a rosette

Simulation ID

• ADV-2026-P2-PHY-Q04 · revision 2

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