At the Earth's surface

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Relativity & Gravitation · Newtonian gravitation as a field theory

The Geometry of Gravity, Ch The tidal field: what free fall cannot remove

Problem

At the Earth's surface, take the

x

-axis along the upward radius and the

y

-axis horizontal, so that

[\mathcal{E}_{ij}] = k\,\mathrm{diag}(-2,1,1)

with

k = GM_\oplus/R_\oplus^3

. Use

GM_\oplus = 3.986\times10^{14}\ \mathrm{m^3\,s^{-2}}

and

R_\oplus = 6.371\times10^{6}\ \mathrm{m}

. A second set of axes is turned about the

z

-axis by

\theta = 30\,{}^\circ

, with

x' = x\cos\theta + y\sin\theta

and

y' = -x\sin\theta + y\cos\theta

. ((a)) Using

\mathcal{E}_{i'j'} = O_{i'}{}^{i}O_{j'}{}^{j}\mathcal{E}_{ij}

, find

\mathcal{E}_{x'x'}

\mathcal{E}_{x'y'}

and

\mathcal{E}_{y'y'}

. ((b)) Two free particles are released at rest

1.00\ \mathrm{m}

apart along the

x'

-axis. Find the

x'

and

y'

components of their relative acceleration. ((c)) Find the size of that relative acceleration again, working in the radial axes instead, and compare.

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