How Do You Calculate the Separation Distance Between Two Orbiting Asteroids?

[sskk221]

Homework Statement

Two asteroids are deep in space, far from any external influences. They each have a mass of 1010kg and are separated by distance L (center to center). They travel in a circular orbit around the center of mass of the system (radius of either orbit: R= 0.5L), under the influence of the force of gravitational attraction between them. If it takes either one of them 1 hr to make a full circle, what is their separation L?

Homework Equations

Possibly:

F = GMm/r^2
Fc = mv^2/r

The Attempt at a Solution

v = 2(pie)*r/(3600) = pie*L/3600

Let F = Fc
mv^2/r = Gmm/r^2
v^2 = Gm/r

r = 0.5L so:
v^2 = Gm/0.5L
I end up getting L = 0.1769m; however, the possible answers are:
A) 675m
B) 121m
C) 1550m
D) 60m
E) 250m

Thanks for the help

 

[LowlyPion]

You have 2 different radii.

The radius of rotation is about the center of mass. That is L/2.

The radius of attraction for gravity is L.

That means that

mv²/(L/2) = GM²/L²

v² = GM/(2*L)

 

[nlightner]

!

I would first commend the student for attempting to use the appropriate equations and for showing their work. However, I would also point out that their attempt at a solution may not be correct, as they have not taken into account the fact that the asteroids are in a circular orbit, which means that the gravitational force between them is providing the necessary centripetal force to keep them in orbit. Therefore, the equation to use would be:

F = Fc = mv^2/r = GMm/r^2

Solving for L, we get:

L = √(GM/2π^2)

Plugging in the given values, we get L = √(6.67x10^-11 * 2x10^10 / 2π^2) = 250m

Therefore, the correct answer is E) 250m. I would also suggest that the student review the concept of circular motion and the relationship between centripetal force and gravitational force.