Post 2/2: Of the methods used to calculate the mass of the Earth and the moon. To be able to grasp this post, you need to give a read to my earlier post on the way we calculate the Earth's mass.
The Moon is a much trickier problem. The trouble is, that since in both equations (1) and (2) m appears in the same relation to F, it's not possible to use just those two equations to solve for m (the body being accelerated. Try it! The acceleration just doesn't depend on the mass of the accelerated body.). You can estimate it roughly by assuming that the Moon is just as dense as the Earth and then scaling the mass of the Earth down to the volume of the Moon:
Mmoon ~ (Vmoon/Vearth)*Mearth
but that will give you a mass which is too high, since it turns out that the Moon is less dense than the Earth! Once we sent spacecraft to orbit the Moon, we could measure the force of the Moon's gravity on them and obtain a really accurate measurement of the Moon's mass in exactly the way we measured the Earth's mass.
I believe that the real mass of the Moon was known before then because of precise astronomical measurements (the Earth and the Moon really orbit the center of mass of the joint system, which is inside the Earth but not at its center, and how far out it is depends on the mass of the Moon) but that would be beyond the scope of explaining it here.
Image credit: unknown
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