Irodov Problem 1.164 – Sphere Rolling Down an Incline

Problem Statement

A solid sphere of mass m and radius R rolls without slipping down an incline of angle θ. Find the acceleration of the center of mass.

Given Information

• Mass m, radius R
• Angle θ
• No slipping

Physical Concepts & Formulas

$$ma=mg\sin\theta-f,\quad I_{cm}\alpha=fR,\quad a=R\alpha$$

Step-by-Step Solution

Step 1: Newton’s 2nd law: ma = mg sinθ − f (friction up slope).

Step 2: Rotational: (2/5)mR²·(a/R) = fR → f = (2/5)ma.

Step 3: Substitute: ma = mg sinθ − (2/5)ma → a(1+2/5) = g sinθ → a = 5g sinθ/7.

Worked Calculation

a = 5g sinθ/7

Answer

$$\boxed{a=\frac{5g\sin\theta}{7}}$$

Physical Interpretation

Rolling friction uses some force to create rotation, reducing acceleration vs sliding (g sinθ). Hollow spheres (smaller I fraction) roll even slower; disks accelerate at 2g sinθ/3.