Use kg, kg, mean Sun distance km, and m /s…

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

Use

M_{\rm moon}=7.35\times10^{22}

kg,

M_\odot = 1.989\times10^{30}

kg, mean Sun distance

d_\odot = 1.496\times10^{8}

km, and

G_{\rm N}M_\oplus = 3.986\times10^{14}

^3

^2

. ((a)) Estimate the ratio of the lunar to the solar tidal acceleration at the Earth's surface at the moment the Moon is at its farthest point (apogee),

d_{\rm moon}=4.055\times10^{5}

km, treating both bodies as point masses. ((b)) A free-floating experiment sits inside a satellite in geostationary orbit, radius

r=42\,164

km from the Earth's centre. Using the point-mass tidal-tensor magnitude

|T|\approx 2G_{\rm N}M_\oplus/r^3

, estimate the greatest length the experiment may have if the tidal acceleration between its two ends is to stay below a tighter tolerance,

a_{\rm tol}=1.0\times10^{-8}

m/s

^2

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