Irodov Problem 1.161 – Free Rotation of Asymmetric Body

May 17, 2026

—

Problem Statement

A rigid body has principal moments of inertia I₁ < I₂ < I₃. It rotates freely (no torque). Describe the stability of rotation about each principal axis.

Given Information

Principal moments: I₁ < I₂ < I₃
Free rotation (N = 0)

Physical Concepts & Formulas

Euler’s equations: I₁ω̇₁ = (I₂−I₃)ω₂ω₃, cyclic.

Step-by-Step Solution

Step 1: Rotation about I₁ (smallest): stable. Small perturbation → bounded oscillation.

Step 2: Rotation about I₃ (largest): stable. Same reason.

Step 3: Rotation about I₂ (intermediate): UNSTABLE. Known as the ‘tennis racket theorem’ or intermediate axis theorem.

Worked Calculation

Stable: axes 1 and 3. Unstable: axis 2.

Answer

$$\boxed{\text{Rotation about axes of max and min }I\text{ is stable; intermediate }I\text{ is unstable}}$$

Physical Interpretation

This is the ‘Dzhanibekov effect’ — a book thrown spinning about its spine will flip periodically. Space stations must avoid rotation about the intermediate axis.