Irodov Problem 1.165 – Hollow Cylinder Rolling Down Incline

Problem Statement

A hollow cylinder (thin shell) of mass m and radius R rolls without slipping down an incline of angle θ. Find the acceleration.

Given Information

• Thin cylindrical shell: I_cm = mR²
• Angle θ

Physical Concepts & Formulas

$$a=\frac{g\sin\theta}{1+I_{cm}/(mR^2)}$$

Step-by-Step Solution

Step 1: For thin shell: I_cm = mR².

Step 2: a = g sinθ/(1 + mR²/mR²) = g sinθ/2.

Step 3: Compare: solid cylinder gets 2g sinθ/3, sphere gets 5g sinθ/7.

Worked Calculation

a = g sinθ/2

Answer

$$\boxed{a=\frac{g\sin\theta}{2}}$$

Physical Interpretation

Objects with more mass at the rim (larger I) roll slower. Racing objects: sphere wins, then solid cylinder, then hollow cylinder, then ring (slowest at g sinθ/2).