Universal Gravitation?

Question

Uranus requires 84 years to circle the sun. Find Uranus' orbital radius as a multiple of Earth's orbital radius.

What formula do I need to use and what does this question actually mean? Thanks!

Answer 1

Force of gravity on earth mass e by Sun M is F = GeM/r^2 = e(Wr)^2/r = eW^2r = C the centripetal force with W = 1 cycle/year.

Force of gravity on Uranus mass u by Sun M is f = GuM/R^2 = u(wR)^2/R = uw^2R = c the centripetal force with w = 1 cycle/84 years

GM/r^2 = W^2r; and GM/R^2 = w^2R

GM/r^3 = W^2; and GM/R^3 = w^2

W^2/w^2 = R^3/r^3; so that R^3 = r^3*(W/w)^2

W = 1 cycle/year, w = 1 cycle/84 years, r = 1 Earth orbital radius

Solve for R in terms of Earth radii.