Some of the deepest mines in the world are in South Africa and are roughly 3.5 k

Question

Some of the deepest mines in the world are in South Africa and are roughly 3.5 km deep. Consider the Earth to be a uniform sphere of radius 6370 km. What is the percentage difference in the gravitational acceleration at the bottom of the 3.5-km-deep shaft relative to that at the Earth's mean radius? That is, what is the value of What is the percentage difference in the gravitational acceleration at the bottom of the 3.5-km-deep shaft relative to that at the Earth's mean radius? That is, what is the value of (asurf a3.5km)/asurf? i keep getting -0.016% or 6.52% but these answers are correct, can someone help me out here

Explanation / Answer

M = 4/3 pi R^3 p where p = density of earth and R = radius of mass enclosed F1 = G M m / R1^2 = G 4/3 pi R1^3 m / R1^2 = G 4/3 pi m R1 F2 = G 4/3 pi m R2 where m is an arbitrary mass F1 / F2 = R1 / R2 = 7 So R2 = R1/7 = 6370/7 = 910 km The shaft would have to be (6370 - 910) = 5460 km deep

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