Two small bolts of negligible mass drift freely near a coasting spacecraft at a…
PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Newtonian gravitation as a field theory · Newtonian gravitation as a field theory
Relativity and Gravitation, Ch Tidal forces and the Newtonian tidal field
Problem
Two small bolts of negligible mass drift freely near a coasting spacecraft at a distance r=7.00×106m from the centre of the Earth (M=5.972×1024kg, treated as a point mass). One bolt is a distance ξ further from the Earth than the other, both lying on the same radial line; no forces other than gravity act. Take the x axis along the outward radius.
((a)) Starting from the field of a point mass, gx(x)=−GM/x2, and the first-order expansion of a function, derive the relative acceleration of the outer bolt with respect to the inner one and show that it is Δax=2GMξ/r3, a stretch.
((b)) Deduce Exx from your result, quote Eyy and Ezz for a point mass in these axes, and show that the three diagonal entries sum to zero.
((c)) Evaluate the stretch for ξ=1.20m.
((d)) The bolts are released at rest relative to one another. Estimate their extra separation after 120s, treating the relative acceleration as constant over that time, and comment on whether that is justified.