A block of mass is attached to a spring of stiffness and slides on a horizontal table

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

Action and Symmetry, Ch Constraints, generalised coordinates and Lagrange's equations twice over

Problem

A block of mass m=0.40 kgm=0.40\,\mathrm{kg} is attached to a spring of stiffness k=6.4 N m−1k=6.4\,\mathrm{N\,m^{-1}} and slides on a horizontal table. In addition to the spring force it feels a quadratic air-drag force −cx˙∣x˙∣-c\dot x|\dot x|, with c=0.25 kg m−1c=0.25\,\mathrm{kg\,m^{-1}}, opposing its motion (no other constraint force does work). ((a)) Using the derivation of the Euler-Lagrange equation from the first variation of the action (Chapter 1), show that changing to the rescaled coordinate Q=2xQ=2x leaves the spring-only equation of motion in the form Q¨+ω02Q=0\ddot Q + \omega_0^2 Q = 0, and give ω0\omega_0. ((b)) State, with a reason, which route to Lagrange's equations, the variational route with L=T−VL=T-V or d'Alembert's route with a generalised force, must be used to include the quadratic drag. Write the complete equation of motion for xx, then convert it into an equation for QQ, giving the coefficient of Q˙∣Q˙∣\dot Q|\dot Q|.

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