Irodov Problem 1.157 – Conical Pendulum Angular Momentum

Problem Statement

A conical pendulum (mass m, string length l, half-angle θ) rotates uniformly. Find the angular momentum about the suspension point.

Given Information

• Mass m, string l, half-angle θ
• Uniform circular motion

Physical Concepts & Formulas

$$T\cos\theta=mg,\quad T\sin\theta=m\omega^2 r,\quad r=l\sin\theta$$

Step-by-Step Solution

Step 1: ω² = g/(l cosθ), r = l sinθ.

Step 2: v = ωr = sinθ√(gl/cosθ).

Step 3: L about suspension = mvl = ml sinθ·sinθ√(gl/cosθ) = ml²sinθ·sinθ√(g/(l cosθ)).

Worked Calculation

L = m l² sin²θ · √(g/(l cosθ))

Answer

$$\boxed{L=ml^2\sin^2\theta\sqrt{\frac{g}{l\cos\theta}}}$$

Physical Interpretation

The angular momentum vector is not vertical — it precesses around the vertical with the same frequency as the rotation. The torque by gravity causes this precession.