A thin spherical shell of radius carries a uniform surface mass density over the polar cap

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

Conditions of Validity, Ch A coordinate system is a map: the metric, the scale factors and where a chart fails

Problem

A thin spherical shell of radius R=0.50 mR = 0.50\ \mathrm{m} carries a uniform surface mass density σ0=1200 kg m−2\sigma_0 = 1200\ \mathrm{kg\,m^{-2}} over the polar cap 0≤θ≤α0\le\theta\le\alpha, with α=π/6\alpha=\pi/6, where θ\theta is the colatitude measured from the pole θ=0\theta=0. The spherical chart fails at θ=0\theta=0: det⁡g=R4sin⁡2θ=0\det g = R^4\sin^2\theta = 0 there and the longitude ϕ\phi is undefined at that single point. ((a)) Write the surface area element dA\mathrm{d}A of the spherical chart in terms of det⁡g\sqrt{\det g} for fixed RR. ((b)) As in an improper integral treated by excising the bad point and taking a limit, exclude a small cap of angular radius ε\varepsilon (0<ε<α0<\varepsilon<\alpha) about the pole and compute the mass M(ε)M(\varepsilon) of the remaining region ε≤θ≤α\varepsilon\le\theta\le\alpha. ((c)) Let ε→0+\varepsilon\to0^+ and give the total mass of the cap 0≤θ≤α0\le\theta\le\alpha. Explain, in terms of det⁡g\sqrt{\det g}, why excising the pole and taking the limit reproduces the naive calculation that ignores the coordinate failure.

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