6.5 Rolling Without Slipping Dynamics

What rolling without slipping means

Rolling without slipping occurs when the contact point of a round object has zero velocity relative to the surface. This gives the constraint v_cm = rω and a_cm = rα. Static friction (not kinetic) enables this motion — it does no work and doesn't dissipate energy, but it provides the torque needed to change rotational speed.

Newton's second law for rolling objects

For an object rolling down an incline, apply ΣF = ma_cm along the incline (gravity component minus friction) and Στ = Iα about the center of mass (friction provides torque). Combine with a_cm = rα to solve for acceleration. Objects with larger rotational inertia (I) relative to mass accelerate more slowly down the incline because more energy/torque goes into spinning.

Energy approach to rolling

Since static friction does no work, total mechanical energy is conserved: KE_total = ½mv_cm² + ½Iω². Using energy conservation (e.g., ΔPE = KE_trans + KE_rot) is often faster than forces for finding final speed at the bottom of an incline, and gives the same result as the force/torque method.

Key terms

Rolling without slipping condition

The contact point has zero velocity relative to the surface; v_cm = rω and a_cm = rα.

Role of friction in rolling

Static friction supplies the torque needed for angular acceleration but does no work since the contact point doesn't slide.

Newton's 2nd law setup for rolling on incline

ΣF = ma_cm (along incline) and Στ = Iα (about center of mass), linked by a_cm = rα.

Effect of larger moment of inertia on rolling speed

Larger I means more energy goes to rotational KE, so the object has a slower translational speed and acceleration down an incline.

Energy conservation for rolling objects

ΔPE = ½mv_cm² + ½Iω², valid because static friction does no work during rolling without slipping.

Try an AP-style question

For an object rolling without slipping, what provides the torque that makes it spin?

1. Gravity at the center of mass
2. Kinetic friction
3. The normal force
4. Static friction at the contact point