Use Gauss\'s Law to prove this theorem:The gravitational force on a body located

Question

Use Gauss's Law to prove this theorem:The gravitational force on a body located at distanceRfrom the center of a uniform spherical mass is due solely to the massm(R) lying at distancer<Rfrom the center of the sphere. The strength of the force is the same as that of a point mass atr=0.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 equals the time needed for a satellite to orbit the Earth in a low orbit withr?REarth.In deriving this result, you need to treat the Earth as a uniformly dense sphere, and neglect all friction and any effects due to the Earth's rotation.Give the time in minutes.

Explanation / Answer

Gauss' Law for gravitation will say that the gravitational flux is proportional to the mass enclosed in a Gaussian Surface: ? ? M I'm going to need to work out the proportionality constant, because I usually deal with Gauss' Law for electric charge. The gravitational field (force per mass) at a distance r from a point mass is g = F/m = GM/r

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