Rolling without slipping: connect the two motions

Why is the bottom of a rolling wheel instantaneously at rest?

A wheel rolls right along the floor. Its center moves forward, while rotation moves the bottom point backward relative to the center. Without slipping, these two velocities exactly cancel at the floor.

A useful starting point: Two wheels lock together: conserve L, track the energy →

Words and symbols before equations

No-slip rolling

The contact point has zero velocity relative to the surface at that instant.

R

Wheel radius, in m.

v_CM and a_CM

Center-of-mass velocity and acceleration along the surface.

ω and α

Angular velocity and angular acceleration; below, clockwise is positive to match rightward rolling.

Connect the picture to the physics

For rightward rolling with clockwise taken positive, v_CM=Rω. Over a rolling interval, Δx_CM=RΔθ, and a_CM=Rα for constant R. If using counterclockwise positive instead, the corresponding signed relation has a minus sign. State the convention before substituting.

Relative to the center, a rim point has tangential speed R|ω|. Add its vector velocity to the center’s velocity. At the bottom the two cancel; at the top they add to 2v_CM. The center itself moves at v_CM. The material point touching the ground changes as the wheel turns.

Zero instantaneous contact velocity does not mean the whole wheel is at rest or that the contact point has zero acceleration. In the ideal rigid-body model, static friction at a stationary contact does not dissipate energy. Friction may be zero during steady rolling on a level surface with no other horizontal forces.

A worked example, step by step

A wheel of radius 0.5 m rolls right without slipping at v_CM=2 m/s. Find ω and the ground-frame velocities of its top and bottom points.

Predict. Change one thing. Explain.

Change center speed and radius. Predict which changes ω. Compare the arrows for center, top and bottom velocities in the ground frame; they share a common velocity scale.

On narrow screens, swipe or scroll diagrams sideways to read all labels.

Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant physical relationship or contact condition to justify your prediction.

Show what you understand.

Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.

Original written challenge

A wheel of radius 0.2 m rolls right without slipping at 3 m/s. Its center accelerates right at 1 m/s². (a) Find angular speed. (b) Find clockwise angular acceleration. (c) Find top speed. (d) Explain the contact-point velocity.

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Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.