Answer to Question #157283 in Field Theory for Hiran Fatah

Question

The Earth and the Moon may be considered to be uniform spheres that are isolated in space. The Earth has radius R and mean density ρ. The Moon, mass m, is in a circular orbit about the Earth with radius nR,

 The Moon makes one complete orbit of the Earth in time T.

Show that the mean density ρ of the Earth is given by the expression

ρ = 3πn^3/GT^2

Answer 1

$$\frac{GmM}{(nR)^2}=m\omega^2(nR)\\ \frac{4\pi G \rho R^3 }{3(nR)^2}=\omega^2(nR)=\frac{4\pi^2}{T^2}(nR)\\ \rho=\frac{3\pi n^3}{GT^2}$$