Irodov Problem 1.151 – Conservation of Angular Momentum

May 17, 2026

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Problem Statement

A particle moves under a central force. Show that angular momentum is conserved and find the rate of change of angular momentum for a particle in a uniform gravitational field.

Given Information

Central force: F directed toward fixed center
Gravity: F = −mgĵ

Physical Concepts & Formulas

$$\frac{dL}{dt}=\mathbf{r}\times\mathbf{F}=\boldsymbol{\tau}$$

Step-by-Step Solution

Step 1: For central force: τ = r × F = r × (F·r̂) = 0. Thus L = constant.

Step 2: For gravity: τ = r × (−mgĵ) = −mg(r × ĵ).

Step 3: |τ| = mgx (where x is horizontal distance from reference point), direction along ẑ.

Worked Calculation

τ = mgx (for uniform gravity about a fixed origin)

Answer

$$\boxed{\frac{dL}{dt}=mgx\text{ (torque by gravity about origin)}}$$

Physical Interpretation

Central forces conserve angular momentum — the basis for Kepler’s second law. Gravity is central only about Earth’s center.