Angular speed of Uranus around the sun?

Question

A physics homework question :)
Uranus has a radius of 2.33e7m and is 2.87e12m from the sun. The sun has a radius of 6.96e8.

Answer 1

f = GmM/r^2 = mW^2r = c and F = GnM/R^2 = nw^2R = C; where f and F are the forces of gravity on the Earth and Uranus respectively. M is the mass of the Sun, m is Earth mass, n is Uranus mass, r is distance of Earth from the Sun (= 93 million miles), and R = 2.87 X 10^12 meters from the Sun to Uranus. W = 2 pi/year the angular velocity of Earth and w = the angular velocity of Uranus, which is what you are looking for.

F/f = GnM/R^2//GmM/r^2 = nw^2R/mW^2r; so that (n/m)(r/R)^2 = (n/m)(w/W)^2(R/r) Then (r/R)^3 = (w/W)^2 and we can solve for w = W(r/R)^3/2; where W = 2 pi/year, r = 93 million miles, R = 2.87 X 10^12 meters and with converting everything to the same units, you can do the math.

The physics is this. The gravitational forces f and F are offset by the respective centrifugal forces c and C, which is why the two planets stay in orbit around the Sun. By taking the ratio f/F = c/C, we can solve for the angular velocity of one of the planets if we know their respective distances from the Sun and the angular velocity of one of the planets. The equation, which we derived, is (r/R)^3 = (w/W)^2. This is a variant of Kepler's equation.