A non-rotating rocky planet is a uniform sphere of radius and density

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Relativity & Gravitation · The equivalence principle and gravity as geometry

Relativity and Gravitation, Ch Gravity as curved spacetime: redshift and local inertial frames

Problem

A non-rotating rocky planet is a uniform sphere of radius

R = 3.20\times10^{6}\ \mathrm{m}

and density

\rho = 4.50\times10^{3}\ \mathrm{kg\,m^{-3}}

. On its surface stands a vertical tower of height

h = 40.0\ \mathrm{m}

. A laser fixed at the base sends light straight up to a detector fixed at the top; both are at rest on the planet, and there is no atmosphere. Take

G = 6.674\times10^{-11}\ \mathrm{N\,m^{2}\,kg^{-2}}

and

c = 2.998\times10^{8}\ \mathrm{m\,s^{-1}}

. ((a)) Find the gravitational field strength

g

at the planet's surface. ((b)) Find the fractional frequency shift

\Delta\nu/\nu = (\nu_{\rm rec} - \nu_{\rm em})/\nu_{\rm em}

of the light as measured by the detector at the top, working to first order in

gh/c^{2}

and treating

g

as uniform over the tower.

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