Rolling without Slipping (Dynamics)

Rolling without slipping links translational and rotational motion through a physical constraint: the point of the object in contact with the surface must be momentarily at rest relative to the surface.

The Rolling Constraint

For a wheel of radius $r$ rolling on a flat surface, the arc length $s$ that unrolls must equal the linear distance $x$ traveled by the center of mass.

Because this geometric constraint holds throughout the motion, linear and angular quantities are related by:

Velocity Vectors on the Rim

To determine the velocity of any point on the rim, we apply vector addition. The total velocity $\vec{v}$ is the sum of the translational velocity of the center of mass $\vec{v}_{\mathrm{CM}}$ and the tangential velocity $\vec{v}_{\mathrm{T}}$ relative to the center.

Substituting the magnitude $v_{\mathrm{T}} = \omega r = v_{\mathrm{CM}}$, we can analyze key points:

1. Contact Point: The tangential velocity points backward, directly opposing the forward translational velocity. The net velocity is zero.

2. Top of Wheel: The tangential velocity points forward, parallel to the translational velocity. The velocities adds up.