What is Rolling without slipping? — Physics

Definition 1.4 University Physics — Year 2 · Chapter 1 — Rigid-Body Mechanics

Two solids $S_1$ and $S_2$ touch at a point $I$ . The slip velocity of $S_1$ on $S_2$ is $\vect v_{\text{slip}} = \vect v_{I \in S_1} - \vect v_{I \in S_2}$ , the difference of the velocities of the two material points that coincide at $I$ at this instant. $S_1$ rolls without slipping on $S_2$ when $\vect v_{\text{slip}} = \vect 0$ . For a wheel of radius $R$ rolling without slipping on a fixed ground, with centre velocity $v_G$ and angular velocity $\omega$ , the condition reads $v_G = R\omega$ (signs chosen so that a wheel rolling to the right turns clockwise).

A wheel caught by a slow shutter: the spokes near the road, almost at rest, stay sharp; those at the top, moving at twice the bicycle’s speed, blur.

A wheel rolling without slipping. Left: the velocities of a few points — zero at the contact point I, v_G = R at the centre, 2v_G at the top. Right: every velocity is perpendicular to the segment joining the point to I and proportional to its length: at each instant the wheel turns about its contact point. — A wheel rolling without slipping. Left: the velocities of a few points — zero at the contact point $I$ , $v_G = R\omega$ at the centre, $2v_G$ at the top. Right: every velocity is perpendicular to the segment joining the point to $I$ and proportional to its length: at each instant the wheel turns about its contact point.

Examples

Example 1.5 (Velocity field of a rolling wheel)

For the rolling wheel, $\vect v_M = \vect\Omega\wedge\vect{IM}$ : at each instant the wheel rotates about its contact point. The contact point is at rest, the centre moves at $v = R\omega$ , the top at $2v$ , and a point of the rim at height $h$ moves at $\omega\sqrt{2Rh}$ , perpendicular to $\vect{IM}$ . A spoke valve traces a cycloid; a stone flung from the tread leaves at the speed of the rim point, up to $2v$ — which is why a lorry’s mudflaps matter.

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