Parallel transport of a vector once round the latitude circle at latitude (colatitude…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Rotation, rigid bodies and accelerated frames

Invariance, Ch The metric, the connection and geodesics

Problem

Parallel transport of a vector once round the latitude circle at latitude λ\lambda (colatitude θ0=90∘−λ\theta_0 = 90^\circ-\lambda) rotates it by 2πcos⁡θ02\pi\cos\theta_0 relative to the local north-east frame, and the Foucault pendulum's swing plane precesses through a full turn in the period T(λ)=Tsid/sin⁡λT(\lambda) = T_{\rm sid}/\sin\lambda, where Tsid=23.93 hT_{\rm sid} = 23.93\,\mathrm{h} is the sidereal day. ((a)) Tabulate T(λ)T(\lambda) at λ=90∘,60∘,45∘,30∘,10∘\lambda = 90^\circ, 60^\circ, 45^\circ, 30^\circ, 10^\circ. ((b)) Find the latitude at which the Foucault period equals exactly twice the sidereal day. ((c)) Describe, with a formula, how T(λ)T(\lambda) behaves as λ→0∘\lambda\to0^\circ (the equator), and explain physically why the transported vector's rotation rate vanishes there.

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