Central Force Motion · Radial Oscillation

Displace the circular orbit and watch the radius oscillate

The particle starts in a circular orbit under an attractive inverse-square force. When its angular momentum is kept fixed and the radius is changed slightly, the effective radial force acts like a linear restoring force about r0.

2026 Paper 2 Q4 · radial oscillation about r₀
Small radial displacement · angular momentum fixed · radius varies periodically
Observation stage
Displacement size 0.12 r₀
Circular-orbit balance
At r₀, inward inverse-square attraction is exactly balanced by the angular-momentum barrier.
Expressionℓ²/(mr₀³) = k/r₀²
Meaningr₀ = ℓ²/(mk)
Circular orbitequilibrium radius r₀
Radial displacementsmall x = r − r₀
Restoring forceFeff ≈ −(k/r₀³)x

Derivation path

01 / CIRCULAR ORBIT

Balance the radial forces at r₀

For the unperturbed circular orbit, the angular-momentum term balances the inverse-square attraction.

ℓ²/(mr₀³) = k/r₀² ⇒ r₀ = ℓ²/(mk)
02 / LINEARISE

Expand the effective radial force

For r = r₀ + x with x ≪ r₀, the first-order force is proportional to −x.

mẍ = −(k/r₀³)x
03 / PERIOD

Read off the angular frequency

The radial motion is simple harmonic with ω² = k/(mr₀³). Substitute r₀ = ℓ²/(mk).

ω = mk²/ℓ³
Time period of radial oscillation
T = 2π/ω = 2πℓ3/(mk2)
Correct option: (A)