Rolling Without Slipping
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
A wheel rolls without slipping when its contact point doesn’t slide: the center moves at v = Rω, and the point touching the ground is momentarily at rest. A rolling object has both kinetic energies: ½mv² + ½Iω².
1. What rolling looks like
2. The ramp race
Roll objects down the same ramp and they all turn the same mgh into kinetic energy. The ones that put less of it into spinning have more left for moving forward, so they’re faster.
Mass and size don’t matter, only the shape: how much of the mass is near the rim.
3. Example, then try it
A 2.0 kg solid cylinder (I = ½MR²) rolls at 4.0 m/s. Find its translational and rotational kinetic energy.
Translational: ½(2.0)(4.0)² = 16 J.
Rotational: ½(½MR²)(v/R)² = ¼Mv² = ¼(2.0)(4.0)² = 8.0 J. One-third of the 24 J total is rotational.
A solid sphere (I = ⅖MR²) rolls from rest down a ramp, dropping 1.4 m. How fast is it moving at the bottom? Compare a hoop rolling down the same ramp. (g = 10 m/s²)
Sphere energy: mgh = ½mv² + ½(⅖mR²)(v/R)² = ½mv² + ⅕mv² = 0.7mv².
Solve: v² = gh ÷ 0.7 = (10)(1.4) ÷ 0.7 = 20 → v ≈ 4.5 m/s. The mass cancels.
Hoop: mgh = ½mv² + ½(mR²)(v/R)² = mv² → v² = 14 → v ≈ 3.7 m/s.
Compare: the sphere wins, because the hoop puts half its energy into spinning.
Answer: sphere ≈ 4.5 m/s, hoop ≈ 3.7 m/s. (A frictionless sliding block would reach about 5.3 m/s.)
4. Common mistakes
A solid disk and a hoop with the same mass and radius are released from rest at the top of the same ramp and roll without slipping. Which is moving faster at the bottom?
The disk has the smaller rotational inertia (½MR² vs. MR²), so less of its energy goes into spinning and more into moving forward.
Ready to practice?
5 questions on Rolling Without Slipping, each with a hint and an explanation for every wrong answer.