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Simple Harmonic Motion Of The Center Of The Earth

The gravitational force on a body located a distance R from the center of a uniform spherical mass is due solely to the mass lying at distance r<R or r=R, measured from the center of the sphere. This mass exerts a force as if it were a point mass at the origin.
Use the above result to show that if you drill a hole through the earth and then fall in, you will execute simple harmonic motion about the earth's center. Find the time it takes you to return to your point of departure and show that this is the time needed for a satellite to circle the earth in low orbit with r approximately equaling R(earth). In deriving this result, you need to treat the earth as a uniformly dense sphere, and you must neglect all friction and any effects due to the earth's rotation.

Solution Preview

First we have to find the formula for the force due to gravity on a mass m at distance r from the center of the Earth of mass M and radius R. Since the Earth is a spherical body the gravitational field that it produces will have spherical symmetry and we can always consider it to be coming from a point source at ...

Solution Summary

The solution is comprised of an explanation for the simple harmonic motion of the center of the earth. Time to return to point of departure and time needed for a satellite to circle the earth is also considered in the solution.