A Foucault pendulum swings at London

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

Invariance, Ch Curvature

Problem

A Foucault pendulum swings at London, latitude λ=51.5∘N\lambda = 51.5^\circ\mathrm{N}. Treat the Earth as a sphere of radius R⊕=6371 kmR_\oplus = 6371\,\mathrm{km} rotating once per sidereal day (23.93 h23.93\,\mathrm{h}). ((a)) Using the pendulum's precession rate Ωsin⁡λ\Omega\sin\lambda from mechanics (with Ω\Omega the Earth's angular speed, one full turn per sidereal day), find the angle through which the plane of swing precesses, relative to the ground, in one sidereal day. ((b)) Find the holonomy angle, the turn of the transported vector against its own starting direction, for a vector carried parallel round the latitude circle at λ=51.5∘N\lambda = 51.5^\circ\mathrm{N} (equivalently, round the boundary of the polar cap lying north of that latitude). ((c)) Show that the two angles found in (a) and (b) add to 360∘360^\circ, and explain in one or two sentences why this must happen.

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