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, 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.)
Suppose the rod connecting the two masses m is removed. In this case, the only force between the two masses is their mutual gravitational attraction. In addition, suppose the masses are spheres of radius a and mass m=43πa3ρ that touch each other. (The Greek letter ρ stands for the density of the masses.)
Find the distance from the center of the Earth, r , for which the gravitational force found in part A is equal to the tidal force ( 4GmMEa/r3 ). This distance is known as the Roche limit.
Express your answer in terms of the variables ME , ρ , and appropriate constants.