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A dumbbell has a mass m on either end of a rod of length 2a. The center of the dumbbell is a distance r from the center of the Earth, and the dumbbell is aligned radially. If r ≫ a, show that the difference in the gravitational force exerted on the two masses by the earth is approximately 4GmMEa/r3. (Note: The difference in force causes a tension in the rod connecting the masses. We refer to this as a tidal force.) [Hint: Use the fact that 1/(r − a)2 − 1/(r + a)2 ~ 4a/r3 for r ≫ a.]
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