Irodov Problem 1.160 – Bicycle Wheel Gyroscope

Problem Statement

A bicycle wheel (moment of inertia I, radius R) spins at ω and is held at the end of a horizontal rod of length l. Find the precession rate.

Given Information

• Spin angular momentum L = Iω
• Rod length l
• Mass of wheel m

Physical Concepts & Formulas

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

Step-by-Step Solution

Step 1: Torque about pivot: N = mgl.

Step 2: Precession: Ω = N/L = mgl/(Iω).

Step 3: For I = mR²: Ω = gl/(R²ω).

Worked Calculation

Ω = mgl/(Iω)

Answer

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

Physical Interpretation

A fast-spinning wheel precesses slowly when held at the end of a rod, appearing to defy gravity. The wheel ‘falls’ by precessing horizontally rather than vertically.