Rolling without slipping

Intermediate

Contact point is momentarily at rest, so v = ωR.

Key relationv_cm = ωR

The point of contact is instantaneously at rest, so v_cm = ωR. Static friction makes it possible and does no work.

Worked example

A 0.3 m radius wheel at 6 m/s has ω = 20 rad/s.

Common mistake

Top of the wheel moves at 2v.

In more detail

Rolling without slipping means the contact point on the surface has zero velocity. For a wheel of radius R, this requires v = Rω at the centre.

Car tyres roll without slipping during normal driving, giving grip.
Bowling balls roll without slipping down the lane to reach the pins.
A wheel rolling along a road without slipping has its contact point momentarily at rest, which is what keeps it from skidding.

What is the velocity of the contact point?

Zero, because the rotation and the translation cancel there.

Does rolling without slipping need friction?

Yes, static friction at the contact point maintains the condition.

Why is the contact point at rest?

Its velocity from translation is equal and opposite to its velocity from spinning, so the net velocity at the contact is zero.