Calculating Centripetal Force on Moon & Earth: Mass, Distance, and Acceleration

[kapitanma]

Homework Statement

I)The mass of the moon is 7.35*10^22 kg. In inertial coordinates, the moon orbits the Earth in 27.3 days at an average distance of 3.84*10^5 kg. Calculate the centripetal force on the moon.
II)The mass of the moon is .0123 times that of earth. Since the Earth is experiencing the same magnitude of force, it too is being accelerated. In these inertial coordinates what is the radius of the circular path the Earth follows?

I calculated the first part easily, using F = M(v^2)/r and T = 2pr/v.

So the force acting on the Earth is the same magnitude as the one I calculated in part I. However, I have no idea how to relate the radius of the Earth's path in part II. I have two unknowns for the Earth's velocity, and radius, and so I am not sure how to approach this problem with my current knowledge. I'm not sure if I'm missing something conceptual, or missing a formula here. Any help would be appreciated.

 

[rl.bhat]

When the moon moves around earth, the center of mass of Earth and moon remains at rest.
The Earth rotates around this center of mass. If its distance from the center of the Earth is r, then
Me*ω^2*r = Mm*ω^2*(d-r), where ω is the angular velocity which is the same for moon and earth.
Now substitute the values and find r.