Question

?V.I. Arnold Mathematical Methods of Classical Mechanics Second Edition Page 50?

Determine the major semi-axis of the ellipse which the center of the earth describes around the common center of mass of the earth and the moon. Where is this center of mass, inside the earth or outside? (The mass of the moon is 1/81 times the mass of the earth.)

NOT JUST 'INSIDE' OR 'OUTSIDE', with specific solution.

Answer 1

Set origin on the center of Earth. Label as the center of mass of Earth and the moon. Let be the average distance between Earth and the moon and be the average radius of earth. Then, if we conclude that the center de mass is INSIDE the earth, if we conclude that the center de mass is OUTSIDE of the earth. The center of mass of the earth and the moon can be expressed as

since we did set our origin at the center of Earth, we have

thus,

we are told that the mass of moon  is 1/81 times the mass of the earth, so

inserting eq. (2) into (1), we obtain,

The average distance Earth-Moon* and radius of earth are

so

Therefore, we conclude that the center of mass of the earth and the moon is inside the earth.

* Note: You can find these average values in any physics books (or if you are lazy to look for your physics books, you can google it).