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Direct flights to Beijing from the East Coast (eg, New York City) typically take -15 hours. A hypothetically faster way to travel to the opposite side of the Earth would be to drop an object through:a tunnel drilled through the center o shown to the right. f the Earth along its diameter, as Rp (a) One could prove, using three-dimensional integral calculus techniques, that the gravitational force on an object located on the surface of a (hypothetical) solid, uniform sphere of radius r results only from the gravitational interaction between the mass and the mass contained within this (hypothetical) sphere of radius r. Using this fact: Show that the gravitational force on an object of mass m located a distance r from the center of the Earth within such a tunnel would obey a Hookes Law-type force, F-kr. Derive an expression for k in terms of the constants m, G, the mass of the Earth ME, and the radius of the Earth Rg. (b) Calculate the time (in hours) it would take for an object released from rest to fall through such a hypothetical tunnel to th e opposite side of the Earth.

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