A radar tracks a weather balloon whose position relative to the radar dish is…

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

A radar tracks a weather balloon whose position relative to the radar dish is described by spherical polar coordinates, currently r=5.0 kmr = 5.0\,\mathrm{km}, θ=π/4 rad\theta = \pi/4\,\mathrm{rad}. ((a)) Write the scale factors hrh_r, hθh_\theta, hϕh_\phi at this point. ((b)) State the direction and magnitude of ∂r^/∂θ\partial\hat{\mathbf{r}}/\partial\theta at this point. ((c)) Using the line element ds2=hr2 dr2+hθ2 dθ2+hϕ2 dϕ2\mathrm{d}s^2 = h_r^2\,\mathrm{d}r^2 + h_\theta^2\,\mathrm{d}\theta^2 + h_\phi^2\,\mathrm{d}\phi^2, find the length ds\mathrm{d}s of a small displacement with dr=20 m\mathrm{d}r = 20\,\mathrm{m}, dθ=0.010 rad\mathrm{d}\theta = 0.010\,\mathrm{rad}, dϕ=0.0050 rad\mathrm{d}\phi = 0.0050\,\mathrm{rad} from this point.

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