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 m = 4/3πa3ρ that touch each other. (The Greek letter r 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 the previous problem. This distance is known as the Roche limit.

(c) Calculate the Roche limit for Saturn, assuming ρ = 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.)