Chapter 12, Problem 61PCE

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 4GmM_Ea/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.]