Irodov Problem 1.159 – Spinning Top Precession

May 17, 2026

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Problem Statement

A spinning top of mass m has its center of mass at distance l from the pivot. It spins with angular velocity ω and is inclined at angle θ. Find the precession angular velocity.

Given Information

Mass m, CM distance l
Spin angular velocity ω
Inclination angle θ

Physical Concepts & Formulas

$$N=mgl\sin\theta,\quad L=I\omega,\quad \Omega=\frac{N}{L\sin\theta}=\frac{mgl}{I\omega}$$

Step-by-Step Solution

Step 1: Torque by gravity: N = mgl sinθ.

Step 2: Component of L perpendicular to spin axis (in precession plane): L_⊥ = L sinθ = Iω sinθ.

Step 3: Ω = N/L_⊥ = mgl sinθ/(Iω sinθ) = mgl/(Iω).

Worked Calculation

Ω = mgl/(Iω)

Answer

$$\boxed{\Omega=\frac{mgl}{I\omega}}$$

Physical Interpretation

Precession rate is independent of inclination angle θ! Faster spin → slower precession. This counterintuitive result is fundamental to gyroscopic stabilization systems.