Irodov Problem 1.166 – Cylinder Rotating on Rough Surface

May 17, 2026

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Problem Statement

A cylinder of mass m and radius R is given an initial angular velocity ω₀ while its center is at rest. Friction coefficient μ. Find the time before rolling begins and final velocity.

Given Information

Initial: ω₀, v_cm = 0
Friction μ, mass m, radius R

Physical Concepts & Formulas

$$f=\mu mg,\quad ma=f,\quad I_{cm}\alpha=fR$$

Step-by-Step Solution

Step 1: a = μg (forward), α = −μmgR/(½mR²) = −2μg/R (slowing rotation).

Step 2: v = μgt, ω = ω₀ − 2μgt/R. Rolling when v = ωR: μgt = (ω₀−2μgt/R)R → t = ω₀R/(3μg).

Step 3: v_f = μg·ω₀R/(3μg) = ω₀R/3.

Worked Calculation

t = ω₀R/(3μg), v_f = ω₀R/3

Answer

$$\boxed{t=\frac{\omega_0 R}{3\mu g},\quad v_f=\frac{\omega_0 R}{3}}$$

Physical Interpretation

Friction simultaneously accelerates translation and decelerates rotation until rolling condition is met. Final speed is 1/3 of what it would be if all rotation converted to translation.