Irodov Problem 1.213

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Problem Statement

A particle of mass m moves under the influence of a central force F(r). Show that its angular momentum about the force center is conserved.

Central force: F always along r (toward or away from center). m.

For a central force, torque τ = r × F = 0 (since r ∥ F). Therefore dL/dt = τ = 0 → L = const.

Step 1: τ = r × F.
Step 2: Central force: F = f(r)·r̂, so F is parallel to r.
Step 3: r × r̂ = 0 → τ = 0.
Step 4: dL/dt = τ = 0 → L = constant.

τ = r × F = 0 since F ∥ r. Therefore L = const.

L = const (angular momentum conserved under central force)

This is the foundation of Kepler’s second law: planets sweep equal areas in equal times because L = const under the Sun’s central gravity.