Referring to the previous problem, suppose the rod connecting the two masses m is removed....

Referring to the previous problem, 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   that touch each other. (The Greek letter p stands for the density of the masses.) (a) Write an expression for the gravitational force between the masses m. (b) 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 found in Problem. This distance is known as the Roche limit. (c) Calculate the Roche limit for Saturn, assuming p = 3330 kg/m3. (The famous rings of Saturn are within the Roche limit for that planet. Thus, the innumerable small objects, composed mostly of ice, that make up the rings will never coalesce to form a moon.)

Problem

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 4GmME a/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 ]