A bead of mass g is threaded on a rigid wire of fixed shape (heights in metres for in…

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

Action and Symmetry, Ch Symmetry, Noether's theorem and what is conserved

Problem

A bead of mass m=10.0m = 10.0 g is threaded on a rigid wire of fixed shape y=w(x)=x2/(1.00 m)y=w(x)=x^2/(1.00\ \mathrm{m}) (heights in metres for xx in metres), carried sideways at a steady speed u=2.00 m s−1u = 2.00\ \mathrm{m\,s^{-1}}, with xx measured along the wire's own horizontal axis. In addition to gravity (g=9.8 m s−2g=9.8\ \mathrm{m\,s^{-2}}) and the wire's normal constraint force, a linear drag force acts on the bead, opposing its velocity relative to the wire, with Rayleigh dissipation function F=12bx˙2[1+w′(x)2]\mathcal{F} = \tfrac12 b\dot x^2\left[1+w'(x)^2\right], b=0.0500 kg s−1b = 0.0500\ \mathrm{kg\,s^{-1}}. ((a)) Show that the generalised force from F\mathcal{F} gives h˙=−2F\dot h = -2\mathcal{F} for this system, where hh is the Jacobi integral of the coordinate xx. ((b)) Evaluate h˙\dot h at the instant the bead is at x=0.100x=0.100 m moving with x˙=1.20 m s−1\dot x = 1.20\ \mathrm{m\,s^{-1}} along the wire.

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