Irodov Problem 1.163 – Thin Disk Rolling Without Slipping

Problem Statement

A disk of mass m and radius R rolls without slipping on a horizontal surface with velocity v of its center. Find the total kinetic energy.

Given Information

• Mass m, radius R
• Center velocity v
• Rolling without slipping

Physical Concepts & Formulas

$$T=\frac{1}{2}mv^2+\frac{1}{2}I_{cm}\omega^2,\quad \omega=v/R,\quad I_{cm}=\frac{1}{2}mR^2$$

Step-by-Step Solution

Step 1: Translational KE: ½mv².

Step 2: Rotational KE: ½·(½mR²)·(v/R)² = ¼mv².

Step 3: Total: T = ½mv² + ¼mv² = ¾mv².

Worked Calculation

T = ¾mv²

Answer

$$\boxed{T=\frac{3}{4}mv^2}$$

Physical Interpretation

For a rolling disk, rotational KE is half the translational KE. Total is 3/2 times translational alone. Compare: rolling sphere = 7/10 mv², rolling ring = mv².