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.


(a) How deep would a mine shaft have to be for the gravitational acceleration at the bottom to be reduced by a factor of 3 from its value on the Earth's surface?


(b) 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?

Explanation / Answer

so only mass inside the radius you are looking at influences a a = G Minside/r^2 Minside = M r^3/R^3 a = G M r/R^3 but GM/R^2 = g a = g r/R a) g/9 = g r/R r = R/9 but that is radius not depth, depth = R-r = 8 R/9 = 8*6370/9=5662 km b) g - a/g = (g - gr/R)/g = 1 - r/R but we want in terms of depth, d d = R-r so r = R-d g - a/g = 1 -(R-d)/R = d/R = 3.5/6370=5.49E-4

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.