AP Physics C: Mechanics · Unit 6 · Topic 4 of 5 · 6 min lesson

Rolling

A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.

Quiz this topic · 5 questions

The big idea

Rolling without slipping links spin and motion: v_cm = Rω and a_cm = Rα. The contact point is momentarily at rest. A rolling object’s kinetic energy has two parts: ½Mv² + ½I_cmω².

1. Forces on a rolling object

Static friction at the contact point provides the torque that makes the ball spin faster as it speeds up.

v = Rω, a = Rαno slipping

K = ½Mv² + ½I_cmω²= ½(M + I/R²)v²

a = g sin θ / (1 + I/MR²)down an incline

v = √(2gh / (1 + I/MR²))after dropping h

Top of the wheelmoves at 2v

2. Shape decides the race

Solid sphereI/MR² = 2/5: fastest roller

Solid disk or cylinderI/MR² = 1/2

HoopI/MR² = 1: slowest

Frictionless slideno spin: faster than any roller

Mass and radius cancel out; only the shape factor I/MR² matters. Objects that store less energy in spin have more left for moving forward.

Static friction does no work on a rolling object, because the contact point doesn’t slide. Mechanical energy is conserved.

3. Example, then try it

Example

A 2.0 kg hoop rolls at 3.0 m/s. Find its kinetic energy.

For a hoop I = MR², so ½Iω² = ½MR²(v/R)² = ½Mv². K = ½Mv² + ½Mv² = Mv² = (2.0)(3.0)² = 18 J, half of it rotational.

Try it · work it on paper first

A 2.0 kg solid sphere rolls without slipping down a 30° incline. Find its acceleration, the friction force on it, and the minimum μ_s needed. How fast is it going after dropping 1.4 m? (g = 10 m/s²)

Acceleration: a = g sin θ ÷ (1 + 2/5) = 5.0 ÷ 1.4 ≈ 3.6 m/s².

Friction: torque fR = I(a/R), so f = (2/5)Ma = (0.40)(2.0)(3.57) ≈ 2.9 N, up the slope.

Minimum μ_s: f/N = 2.9 ÷ (2.0 × 10 × cos 30°) = 2.9 ÷ 17.3 ≈ 0.17.

Speed: v² = 2gh ÷ 1.4 = 28 ÷ 1.4 = 20, so v ≈ 4.5 m/s.

Answer: 3.6 m/s², 2.9 N of friction, μs ≥ 0.17, 4.5 m/s after 1.4 m.

4. Common mistakes

Forgetting rotational energy.Using only ½Mv² gives a speed that’s too high.

Using kinetic friction.Rolling without slipping involves static friction, which does no work.

“Heavier objects roll faster.”Only the shape factor matters.

Check yourself

Two solid cylinders of different masses and radii roll without slipping down the same ramp from rest. Which reaches the bottom first?

a = g sin θ / (1 + I/MR²), and I/MR² = ½ for every solid cylinder. Mass and radius cancel, so they tie.

Ready to practice?

5 questions on Rolling, each with a hint and an explanation for every wrong answer.

Quiz this topic All of Unit 6