12.64

A dumbbell has a mass m on either end of a rodof length 2a. The center of the dumbbell is a distancer from the center of the Earth, and the dumbbell isaligned radially. If r≫a, the difference in thegravitational force exerted on the two masses by the Earth isapproximately4GmMEa/r3.(Note: The difference in force causes a tension in the rodconnecting the masses. We refer to this as a tidalforce.)
Suppose the rod connecting the two masses m is removed. Inthis case, the only force between the two masses is their mutualgravitational attraction. In addition, suppose the masses arespheres of radius a and massm=43πa3ρ that touch each other.(The Greek letter ρ stands for the density of themasses.)

Part A

Write an expression for the gravitational force between themasses m.

Express your answer in terms of the variables a,ρ, and appropriate constants.

F=?

Part B

Find the distance from the center of the Earth, r, forwhich the gravitational force found in part A is equal to the tidalforce (4GmMEa/r3).This distance is known as the Roche limit.

Express your answer in terms of the variables ME,ρ, and appropriate constants.

r=?

Part C

Calculate the Roche limit for Saturn, assumingρ=3330kg/m3. (The famous rings of Saturn are within theRoche limit for that planet. Thus, the innumerable small objects,composed mostly of ice, that make up the rings will never coalesceto form a moon.)

Express your answer using three significant figures.

r_S=?

Please explain the answers. Thanks