Question

Uranus completes one revolution about its own axis every 17.24 hours. What is the radius of the orbit required for a satellite to revolve about Uranus with same period?

Answer 1

Start with force supplying centripetal acceleration of the satellite, F =m v^2/r, at distance r from the center of Uranus is equal to the force of gravity of Uranus at that distance,
F = m M G / r^2, where M is the mass of Uranus and G the universal gravitational constant. Equating these forces,

v^2 = M G / r

An orbit will be the distance 2 pi r, so the velocity is simply that distance divided by the time, T, needed to travel it, the period.

v = 2 pi r / T.

By squaring this, we get v^2 and we can set it equal to our other expression for v^2, so that

v^2 = (2 pi)^2 r^2 / T^2 = M G / r

and we can solve this to show that

r^3 = M G T^2 / (2 pi)^2

For the orbit of the satellite around Uranus, as true for the planets around the sun, the square of the radius is proportional to the cube of the period.

All you need now is the mass of Uranus, M.