Problem 4 Harmonic EarthFor something spherically symmetric like the Earth, the amount ofmass that contributes to pulling you towards the center is onlythe mass within the radius you are at. Thus, if you are atthe surface of the Earth this would just be the total mass ofthe Earth, but as you move into the Earth less mass is 'under'you. (a) Assuming the Earth has uniform mass density, showthat the total mass within a radius r is: Mr = M()³, whereR and M are the radius and total mass of the Earth respec-tively.=(b) Now suppose a tunnel was drilled through the entire Earth as shown and you jump in. Make afree-body diagram for when you are a distance r from the center of the Earth. Write the gravitationalforce in terms of the gravitational constant G, your mass m, the total mass of the Earth M, the radius ofthe Earth R, and the distance you are from the center r. (c) Compare your expression to Hooke's Law.Argue/show that the frequency is w = √g/R where R is the radius of the Earth and g is the (typical)acceleration near the surface of the Earth. (d) How long in minutes (plug in numbers) would it take foryou to return to the location you jumped in?

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