Someone argues: "An upright coin rolling on a table without slipping needs four…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Lagrangian mechanics · Lagrangian mechanics

Action and Symmetry, Ch Multipliers, non-holonomic constraints and forces outside the potential

Problem

Someone argues: "An upright coin rolling on a table without slipping needs four numbers to place it, (x,y,θ,ψ)(x,y,\theta,\psi), but the two no-slip conditions x˙=aψ˙cos⁡θ\dot x=a\dot\psi\cos\theta, y˙=aψ˙sin⁡θ\dot y=a\dot\psi\sin\theta are two equations relating the four coordinates' rates of change. Two equations must eliminate two of the unknowns, so the coin's configuration is really described by only two independent numbers, not four." A coin of radius a=15 mma=15\,\mathrm{mm} starts at the origin with θ=0\theta=0, ψ=0\psi=0. It rolls once, without slipping, right round a circle of radius rc=0.20 mr_{\rm c}=0.20\,\mathrm m painted on the table, returning to the same point (x,y)(x,y) with the same heading θ\theta. ((a)) Find the coin's spin angle ψ\psi (in radians) on returning, and state whether it equals the value it started with. ((b)) Use this to explain what is wrong with the argument quoted above, and say in one sentence what a non-holonomic constraint actually restricts.

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