Physics Gravitational Force and Circular Motion

Question

The distance from the Sun to Uranus is 19.1 times the distance from the Sun to the Earth. How many Earth years does it take Uranus to orbit the Sun?

Answer 1

Use Kepler's 3rd law.

R_e = radius of earth orbit

T_e = period of earth orbit = 1 year

R_u = radius of Uranus orbit = 19.1R_e

T_u = period of Uranus orbit

R_e³ / T_e² = R_u³ / T_u²

T_u² = (R_u / R_e)³ T_e² = (19.1R_e / R_e)³ T_e² = 19.1³ T_e²

T_u = [√(19.1³)] T_e = 83.47 years