The gravitational force on a body located a distance R from the center of a unif

Question

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, 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, 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 a low orbit, r = 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.

Explanation / Answer

Refer to http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/earthole.html Write an equation of forces

m dv/dt = -G Mm/r^2
or
dv/dt = -G M /r^2
since
dr/dt=v and the mass M is a function of radius we have
M(r)=(4/3)?r3
where ? - is the density of the Earth where ? - is the density of the Earth d2r/dt2 = -G (4/3)?r3 / r2    d2r/dt2 = -G (4/3)? r   
we have a form of a differential equation D2r + kr = 0    k= G (4/3)? Solving this equation we get the period of oscilation T= 2pv(Rmax/g) I ran out of time If you have question please send me a message

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