Irodov Problem 1.217

May 19, 2026

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Problem Statement

A particle moves in a field U = −α/r (gravitational or Coulomb). Find the relationship between total energy E and period T of circular orbit.

Given

U = −α/r, circular orbit. m.

Concepts & Formulas

Circular orbit: mv²/r = α/r² → v² = α/(mr). E = KE + U = α/(2r) − α/r = −α/(2r). Kepler’s 3rd: T² ∝ r³. Also T = 2πr/v = 2πr/√(α/(mr)) = 2π√(mr³/α).

Step-by-Step Solution

Step 1: v² = α/(mr) → KE = α/(2r).
Step 2: E = α/(2r) − α/r = −α/(2r) → r = −α/(2E).
Step 3: T = 2π√(mr³/α) = 2π√(m/α)·r^{3/2}.
Step 4: T = 2π√(m/α)·(−α/(2E))^{3/2} = π α/(√(2/m))·(−E)^{−3/2}.

Worked Calculation

T = πα√(m/2)·|E|^{−3/2}.

Boxed Answer

T = πα√(m/2) · |E|^{−3/2}

Physical Interpretation

This is the quantum analogue of Kepler’s 3rd law expressed in energy — higher (less negative) energy means larger orbit and longer period.