A rolling rigid body combines translation of its center with rotation about its center. Rolling without slipping means that the point of the body touching a stationary surface is instantaneously at rest relative to that surface.

For a wheel or cylinder of radius $R$, if the center advances a distance $x$ while the body rotates through an angle $\theta$, the no-slip condition requires

$$x=R\theta$$

in magnitude. Differentiating gives the corresponding speed relation

$$v_{\rm CM}=R|\omega|,$$

and, when the signs are chosen consistently,

$$a_{\rm CM}=R|\alpha|$$

for the tangential acceleration relation.

These equations are constraints imposed by contact without slipping. They are not additional force laws.

Velocities of points on a rolling wheel

Suppose a wheel rolls to the right with center-of-mass speed $v_{\rm CM}$. Relative to the center, points on the rim move with tangential speed $R|\omega|$. Under the no-slip condition this equals $v_{\rm CM}$.

At the instant of contact with the ground, the rotational velocity of the contact point is opposite the translational velocity of the center, so the two cancel:

$$v_{\rm contact}=0.$$

At the top of the wheel they point in the same direction, so

$$v_{\rm top}=2v_{\rm CM}.$$

The contact point is therefore an instantaneous point of zero velocity. It does not remain fixed to the ground: a different material point comes into contact as the wheel rolls.

Worked example

A wheel of radius $0.30,\mathrm m$ rolls without slipping at

$$v_{\rm CM}=6.0,\mathrm{m/s}.$$

Its angular speed is

$$|\omega|=\frac{v_{\rm CM}}{R} =\frac{6.0}{0.30} =20,\mathrm{rad/s}.$$

At that instant, the point touching the ground has zero velocity relative to the ground, while the top point moves at

$$v_{\rm top}=12,\mathrm{m/s}.$$

If $v_{\rm CM}$ and $R|\omega|$ do not satisfy the no-slip relation, the contact point moves relative to the surface and the body is slipping or skidding.

Demonstration: pure rolling

Walter Lewin's rolling-cylinder demonstration makes the no-slip constraint and the dependence on mass distribution visible experimentally.