Earth-Moon Center of Mass?

Question

The Earth has a mass of 5.98*10^24 kg, the Moon has a mass of 7.35*10^22 kg, and their center-to-center distance is 3.85*10^8 m. How far from the cneter of the Earth is the Earth-Moon center of mass? Is the Earth-Moon center of mass above or below the surface of the Earth? By what distance? (As the Earth and Moon orbit one another, their centers orbit about their common center of mass.)

Answer 1

Solve this problem as if you had a balance beam scale with the Earth on one side and the Moon on the other side. The point at which they balance is the center of mass. It is just the point at which m1*d1 = m2*d2 where m1 = Earth mass , d1 = Distance from center of Earth to center of Earth-Moon system, etc. Also use the equation d1 + d2 = 3.85*10^8. Now since you know the masses solve the two equations for d1 and d2.

Answer 2

I remember from physics that the center of mass is slightly below the earth's surface. I believe it's something about 200 miles down, but don't remember for certain. Also, the moon doesn't revolve around the Earth, they both revolve around this common center of mass. I know that doesn't answer the entire question, but at least it's a start.

Answer 3

The numbers are sufficient to figure this out. The barycenter is about a thousand miles below the earth's surface. The arithmetic is exactly the same as finding the fulcrum point of a teeter-totter with different weights on each end.

Answer 4

It is below the surface of the earth, but I don't know the distance.

Answer 5

The center-of-mass is at the distance C from the center of the Earth.

Moon Mass * (3.85E8 - C) = Earth mass * (C)
M * 3.85e8 - M*C = E * C

Solve for C:
M * 3.85e8 = E*C + M*C = C * (E + M)
M * 3.85e8 / (E + M) = C