The tip of a robot arm sweeps in a plane at a distance from the arm's pivot

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

Mathematical Methods for Physics, Ch Curvilinear and orthogonal coordinates

Problem

The tip of a robot arm sweeps in a plane at a distance r=3 mr = 3\,\mathrm{m} from the arm's pivot. The controller does not report the angle; it reports only the direction of increasing θ\theta at the tip, θ^=(−0.500, 0.866)\hat{\boldsymbol{\theta}} = (-0.500,\ 0.866), with θ\theta measured from the +x+x axis. ((a)) Find θ\theta at the tip, and write r^\hat{\mathbf{r}} there in Cartesian components. ((b)) Give the Cartesian components of ∂θ^/∂θ\partial\hat{\boldsymbol{\theta}}/\partial\theta at the tip. ((c)) Write the scale factors hrh_r, hθh_\theta and, using them, find the area dA\mathrm{d}A of the small curvilinear cell swept out by changing the coordinates by dr=0.10 m\mathrm{d}r = 0.10\,\mathrm{m}, dθ=0.050 rad\mathrm{d}\theta = 0.050\,\mathrm{rad} from the tip.

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