Irodov Problem 1.158 – Gyroscope Precession Rate

Problem Statement

A gyroscope with angular momentum L spins about its axis. A torque N acts perpendicular to L. Find the precession angular velocity.

Given Information

• Angular momentum L (spin)
• Torque N perpendicular to L

Physical Concepts & Formulas

$$\boldsymbol{\Omega}_{prec}=\frac{\mathbf{N}}{L},\quad\Omega_{prec}=\frac{N}{L}$$

Step-by-Step Solution

Step 1: dL/dt = N (torque equation).

Step 2: Since N ⊥ L, |L| stays constant and L precesses.

Step 3: dL = L dφ in time dt, so L(dφ/dt) = N → Ω = N/L.

Worked Calculation

Ω = N/L

Answer

$$\boxed{\Omega_{prec}=\frac{N}{L}}$$

Physical Interpretation

Larger spin angular momentum → slower precession. This is why a fast-spinning top precesses slowly and maintains its orientation — a key principle in gyroscopic navigation.