A particle of mass m, and angular momentum ℓ is moving in a circular orbit of radius r₀ under the influence of an attractive force F(r) directed radially inward with magnitude k/r². Keeping its angular momentum unchanged, the particle is displaced radially by a small distance δr ≪ r₀, due to which its radial distance varies periodically. The corresponding time period is:
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Question

A particle of mass m, and angular momentum ℓ is moving in a circular orbit of radius r₀ under the influence of an attractive force F(r) directed radially inward with magnitude k/r². Keeping its angular momentum unchanged, the particle is displaced radially by a small distance δr ≪ r₀, due to which its radial distance varies periodically. The corresponding time period is:

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G1 Physics · Accepted Answer

The effective potential is U(eff) = ℓ²/(2mr²) − k/r. For the circular orbit, dU(eff)/dr = 0 gives r₀ = ℓ²/(mk). For a small radial displacement, the effective spring constant is U''(r₀) = k/r₀³. Thus ω² = U''/m = k/(mr₀³). Substituting r₀ = ℓ²/(mk) gives ω = mk²/ℓ³. Hence T = 2π/ω = 2πℓ³/(mk²).