Rolling constraint (${\mathbf{v}}_P=0$)

For a rigid body rolling on a stationary support, the instantaneous contact point $P$ has zero velocity: ${\mathbf{v}}_P={\mathbf{v}}_G+\mathbf{\omega }\times \overrightarrow{GP}=0$. For a wheel of radius $R$ moving horizontally and a counterclockwise angle $\varphi$, this gives $\dot{x}+R\dot{\varphi }=0$, hence $x+R\varphi$ is constant. Maintaining this mechanical constraint generally requires sufficient static friction and a nonnegative normal force.