How much does the mass of the earth effect moon gravity?

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Phenomniverse
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As I understand it, the effect of gravity is greatly diminished over long distances. Even so, the mass of the moon is sufficient to exert an influence on earth, for example on tides. My question is, to what extent does the mass of the Earth influence gravity on the moon. Gravity on the moon is quoted as being 1.622m/sec squared. But how much is that effected if you're standing on the moon with the Earth directly above your head, compared to standing on the opposite side of the moon with the Earth directly underfoot?
 
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Try applying Newtons law to a nominal 1kg test mass on the surface of the moon. There will be two forces one due to the moon and the other due to the earth. The general equation is...

F = Gm1m2/r2
 
If you were in a cave in the centre of the moon you would not notice any gravitational force.