A bead is threaded on a straight rigid rod that rotates in a horizontal plane about a…

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 bead is threaded on a straight rigid rod that rotates in a horizontal plane about a fixed pivot at one end, at a constant angular speed ω=3.0 rad s−1\omega = 3.0\,\mathrm{rad\,s^{-1}}. The rod has length 0.80 m0.80\,\mathrm{m} and the bead can slide freely along it; its position is given by the distance rr from the pivot, 0≤r≤0.80 m0 \le r \le 0.80\,\mathrm{m}. At t=0t=0 the bead is at r=0.30 mr=0.30\,\mathrm{m} with r˙=0.20 m s−1\dot r = 0.20\,\mathrm{m\,s^{-1}}, and the rod's angle is θ=ωt\theta = \omega t. ((a)) Write the bead's real displacement dr\mathrm d\mathbf r, in the polar unit vectors r^\hat{\mathbf r} and θ^\hat{\boldsymbol\theta}, over a short time dt\mathrm dt at this instant, keeping both the sliding and the rotation. ((b)) Write the bead's virtual displacement δr\delta\mathbf r at the same instant. State in words which term of part (a) it drops, and why. ((c)) State the dimension and the shape, as a geometric object, of this system's configuration space, and name its generalised coordinate.

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