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Do heavier objects actually fall a TINY bit faster?
Heavier things will accelerate the earth more than light things. The acceleration of earth is equal to G*m2/r2. For a 1000 kg mass on the earth’s surface (~6 x 106 m), this translates into an acceleration of about 2 x 10-21 meters per second2 .
The very best accelerometers can measure things on the order of 10-9 m/s2 , so you’re also right in that we cannot measure this.