PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Relativity & Gravitation · The equivalence principle and gravity as geometry
Relativity and Gravitation, Ch Experimental foundations: free fall, Lorentz invariance and clock tests
Problem
A torsion balance hangs from a vertical fibre. A rigid beam of negligible mass carries two test bodies of different composition, each of inertial mass m=10.0g, at the two ends, each a distance ℓ=3.00cm from the fibre. The fibre gives a restoring torque κφ for a twist φ, with κ=2.50×10−8Nmrad−1. The two bodies fall towards the Sun with laboratory-frame accelerations g1 and g2 whose Eötvös parameter is η; the horizontal component of the Sun's field at the laboratory has magnitude gˉh=5.00×10−3ms−2 (the same for both bodies to leading order) and its direction turns through 2π once per mean solar day, about 86400s, as Earth turns once relative to the Sun. The beam is horizontal, the fibre's own drift and all other torques on the twist coordinate are ignored, and the balance's free torsional period is much shorter than a day, so the beam follows the slowly varying torque quasi-statically.
((a)) Show that the torque about the vertical fibre axis is Γ=mℓηgˉhsinθ, where θ is the angle between the beam and the horizontal direction to the Sun, and say why the part of the solar field common to the two bodies contributes nothing. Explain in what sense the instrument makes a null measurement.
((b)) Show that the twist is periodic with the solar day and give its amplitude φmax in terms of the given quantities.
((c)) Evaluate φmax for η=1.0×10−13.
((d)) If the smallest twist amplitude the instrument can distinguish at the diurnal frequency is 5.0×10−13rad, what upper bound on η does a null result give?