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.)

Question

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.)

Write an expression for the gravitational force between 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.

Answer 1

To find the expression for the gravitational force between the masses, we can use Newton's law of universal gravitation. According to this law, the force between two masses is given by:

F = G * (m1 * m2) / r^2

Where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.

In this case, since the masses are spheres of radius a and mass m, the mass m1 is equal to m2 and is given by m = 4/3 * π * a^3 * ρ, where ρ is the density of the masses.

So, the expression for the gravitational force between the masses is:

F = G * (m * m) / r^2
= G * (4/3 * π * a^3 * ρ) * (4/3 * π * a^3 * ρ) / r^2
= (16/9) * (G * π^2 * a^6 * ρ^2) / r^2

Now, to find the distance from the center of the Earth, r, for which the gravitational force is equal to the tidal force (4GmMEa/r^3), we can set the two forces equal to each other:

(16/9) * (G * π^2 * a^6 * ρ^2) / r^2 = 4GmMEa/r^3

Simplifying and rearranging the equation, we get:

(16/9) * π^2 * a^6 * ρ^2 = 4 * m * ME * a

Canceling out common terms and rearranging further, we get:

r = (4/3) * a * (M/ρ)^(1/3)

where M is the mass of the Earth and ρ is the density. This distance r is known as the Roche limit.