Irodov Problem 1.162 – Euler Angles and Body-Fixed Frame

May 17, 2026

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Problem Statement

A disk of mass m, radius R spins at ω about its symmetry axis, which makes angle θ with vertical. Find the angular momentum vector and torque needed to maintain this motion.

Given Information

Disk: mass m, radius R
Spin ω at angle θ to vertical

Physical Concepts & Formulas

$$I_{axis}=\frac{1}{2}mR^2,\quad L=I\omega$$

Step-by-Step Solution

Step 1: L = Iω = ½mR²ω along symmetry axis.

Step 2: L_vertical = L cosθ, L_horizontal = L sinθ.

Step 3: If precessing at Ω: torque needed N = ΩL sinθ = Ω·½mR²ω sinθ.

Worked Calculation

N = ½mR²ωΩ sinθ

Answer

$$\boxed{L=\frac{1}{2}mR^2\omega,\quad N=\frac{1}{2}mR^2\omega\Omega\sin\theta}$$

Physical Interpretation

The required torque maintains the tilted precessing spin. Without this torque, the axis would change orientation. Gyroscopes maintain direction precisely because large torques are needed to change L.