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The gravitational pull of the Moon cannot be measured on the Earth using abalance; precision of the balance has nothing to do with it.
The Earth is in free fall toward the common center of mass in the Earth-Moonsystem.
That might be true if the earth were pointlike. But it's not.laboratory is not.
The "center" of the earth is in free-fall toward the center of mass, but your
Tides are easily detectible. http://www.monmouth.com/~jsd/physics/tides.htm