Irodov Problem 1.216

Problem Statement

A particle of mass m moves in a potential field U(r) = αr² (isotropic harmonic oscillator). Find the period of revolution in a circular orbit of radius R.

Given

U = αr², circular orbit radius R. m.

Concepts & Formulas

For circular orbit: centripetal force = restoring force. F = −dU/dr = −2αr. So m·v²/R = 2αR → v² = 2αR²/m. Period T = 2πR/v.

Step-by-Step Solution

Step 1: F = 2αR (inward).
Step 2: m v²/R = 2αR → v = R√(2α/m).
Step 3: T = 2πR/v = 2πR/(R√(2α/m)) = 2π√(m/(2α)) = π√(2m/α).

Worked Calculation

T = π√(2m/α).

Boxed Answer

T = π√(2m/α)

Physical Interpretation

The period is independent of orbit radius R — all circular orbits in a harmonic potential have the same period, analogous to a spring-mass system.