Not how gravity works. Smaller objects do not fall faster. The moon having less mass would mean it affects Earth less. Overall there is less attraction from the combined gravity of both objects.
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This is correct. Think of the moon as "falling" around earth.
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I prefer to think of the Earth as "falling" around the moon. (and the Sun...)
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The moon still contains a great mass. Also fun fact: Every 100+ years the moon get one centimeter closer.
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The moon drifts away form Earth every year, not closer. About 4 cm each year, to be exact.
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I could be wrong on this but i swear i read that it is getting farther not closer
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You are absolutely right.
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Farther away actually!
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Oh I know it does. But it would still have less, assuming the Hive took the excavated moon dirt somewhere else. Probably slightly increasing in the amount it gets closer all the time.