Physics Classical Mechanics

A hole is drilled with smooth sides straight through the center of the earth to the other side of the earth. The air is removed from this tube (and the tube doesn't fill up with water, liquid rock or iron from the core). An object is dropped into one end of the tube and just reaches the opposite end. You can assume the earth is of uniform mass density. You can neglect the amount of mass drilled out and the rotation of the earth.

(a) The gravitational force on an object of mass m located inside the earth a distance r<re from the center (re is the radius of the earth) is due only to the mass of the earth that lies within a solid sphere of radius r . What is the magnitude of the gravitational force as a function of the distance r from the center of the earth? Express your answer in terms of the gravitational of the r, m, g, and re (enter r_e for re).

Note: you do not need the mass of the earth me or the universal gravitation constant G to answer this question but you will need to find an expression relating me and G to g and re .


(b) How long would it take for this object to reach the other side of the earth? Express your answer in terms of the gravitational constant at the surface of earth g, m, and re as needed (enter r_e for re).

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