An apple falls from a tree, Does the moon move as a result?

[Tom Hammer]

I am interested in the following thought experiment: an apple falls from a tree on Earth. Does the moon move (although slightly)? I can see an argument for it doing so—the center of mass of the Earth has moved slightly away from the moon, so the moon would feel a slightly smaller force and would be less attracted, thus moving away. Is this reasoning correct?

 

[tnich]

I think you should first ask if the Earth moves as a result. If so, what is the effect on the center of mass of the Earth/apple system?

 

[Anachronist]

...the center of mass of the Earth has moved slightly...

Um, no. The apple falls toward Earth and the Earth falls toward the apple. The center of mass does not move at all.

You have three masses here. Let's assume a dumbbell-shaped object consisting of 2 spherical masses of 1 metric kiloton each, connected by a 1 meter rod of negligible mass. This represents the apple separated from the Earth by the tree trunk. A third spherical mass of 2 kilotons is some distance away, say 1 km. This represents the moon. These two 2-kt masses (the dumbbell and the bigger sphere) are in a stable orbit around a common center exactly midway between the center of mass of each.

Now, remove the rod. The two 1 kt masses will fall toward one another but their center of mass does not change, so the orbit does not change and the other 2 kt mass remains unperturbed.

Such a system would impose a tidal force on the dumbbell so that it would want to orient itself with the rod pointing at the common orbital center, but even taking this into account, breaking the rod would still not affect the "moon" mass. The two masses falling toward one another will simply not fall directly toward one another, ending up as a single object with a slight spin, but even then their center of mass remains unchanged in its orbit relative to the large mass 1km away.