A space station moves on a circular orbit of radius about the Earth

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Rotation, rigid bodies and accelerated frames · Gravitation and central-force motion

Action and Symmetry, Ch Non-inertial frames from one Lagrangian

Problem

A space station moves on a circular orbit of radius r=6.771×106 mr = 6.771\times10^{6}\ \mathrm m about the Earth, with GNM⊕=3.986×1014 m3 s−2G_{\rm N}M_\oplus = 3.986\times10^{14}\ \mathrm{m^3\,s^{-2}}, and it turns once per orbit so that the same face always points at the Earth. Use axes fixed to the station, with origin at its centre of mass: xx along the outward vertical, yy along the direction of motion and zz perpendicular to the orbital plane. Treat the Earth as a point mass and keep only terms linear in the distance from the station's centre of mass. ((a)) Explain why the station's angular velocity Ω\mathbf\Omega is constant and points along zz, with Ω2=GNM⊕/r3\Omega^2 = G_{\rm N}M_\oplus/r^3. ((b)) Show that an object at rest relative to the station at position x\mathbf x has acceleration x¨i=(−Tij+Cij)xj\ddot x_i = (-T_{ij} + C_{ij})x_j relative to it, with Cij=Ω2δij−ΩiΩjC_{ij} = \Omega^2\delta_{ij} - \Omega_i\Omega_j, and that in these axes −Tij+Cij-T_{ij} + C_{ij} is diagonal with entries 3Ω23\Omega^2, 00 and −Ω2-\Omega^2. ((c)) Evaluate the three entries. One object is released at rest relative to the station 1.00 m1.00\ \mathrm m above its centre of mass, and another 1.00 m1.00\ \mathrm m ahead of it along the track. Find the initial acceleration of each relative to the station, and compare the first with the 2.6×10−6 m s−22.6\times10^{-6}\ \mathrm{m\,s^{-2}} found for axes that do not turn. ((d)) Check the trace of your array against the chapter's statement about a turning, falling frame.

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