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Physics · 术语表

什么是 Rolling without slipping?

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

Two solids S1S_1 and S2S_2 touch at a point II. The slip velocity of S1S_1 on S2S_2 is v→slip=v→I∈S1−v→I∈S2\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 II at this instant. S1S_1 rolls without slipping on S2S_2 when v→slip=0→\vect v_{\text{slip}} = \vect 0. For a wheel of radius RR rolling without slipping on a fixed ground, with centre velocity vGv_G and angular velocity ω\omega, the condition reads vG=Rωv_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 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 II, vG=Rωv_G = R\omega at the centre, 2vG2v_G at the top. Right: every velocity is perpendicular to the segment joining the point to II and proportional to its length: at each instant the wheel turns about its contact point.

示例

Example 1.5 (Velocity field of a rolling wheel)

For the rolling wheel, v→M=Ω→∧IM→\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=Rωv = R\omega, the top at 2v2v, and a point of the rim at height hh moves at ω2Rh\omega\sqrt{2Rh}, perpendicular to IM→\vect{IM}. A spoke valve traces a cycloid; a stone flung from the tread leaves at the speed of the rim point, up to 2v2v — which is why a lorry’s mudflaps matter.

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