Irodov Problem 1.168 – Rolling Cylinder on Accelerating Surface

Problem Statement

A cylinder of mass m rolls without slipping on a plank that accelerates at a₀. Find the acceleration of the cylinder’s center.

Given Information

• Plank acceleration a₀
• Cylinder: mass m, radius R, I = ½mR²

Physical Concepts & Formulas

Rolling constraint on moving surface: a_cyl − a_plank = −Rα (no slip condition)

Step-by-Step Solution

Step 1: Friction f on cylinder: ma_cyl = f.

Step 2: Rotation: (½mR²)α = fR → α = f/(½mR) = 2f/(mR).

Step 3: No-slip: a_plank − a_cyl = Rα = 2f/m = 2a_cyl → a₀ − a_cyl = 2a_cyl → a_cyl = a₀/3.

Worked Calculation

a_cylinder = a₀/3

Answer

$$\boxed{a_{cylinder}=\frac{a_0}{3}}$$

Physical Interpretation

The cylinder lags behind the plank. In the plank’s frame, it appears to roll backward at 2a₀/3. Static friction provides the coupling between translational and rotational motion.