Assume a hole is drilled through the center of the Earth. It can be shown that a

Question

Assume a hole is drilled through the center of the Earth. It can be shown that an object of mass m at distance r from the center of the Earth is pulled toward the center only by the material in the shaded portion (the picture of earth looks like a donut, the inside is the shaded area). Assume Earth has a uniform density p. Write down Newton's law of gravitation for an object at a distance r from the center of the Earth and show that the force on it is of the form of Hooke's Law, F = -kr, with an effective force constant of k = (4/3)(pi)pGm, where G is the gravitational constant.

Explanation / Answer

If the body is at r, the gravitational force on it is only due to the volume of earth (4pir^3/3 ) It has a mass (4pir^3/3 imes ho = 4pir^3/3 imes M/(4piR^3/3)) Where R is the radius of the earth and M is the mass of earth. It then becomes (F = -(GMm/R^3)r), negative because it attracts and the value in the bracket is always constant - G, M, m, R This is now of the type (F = -kx), where (k = GMm/R^3) Hence proved!

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