Question

Find the mass of Uranus (in kg) based on the fact that its moon Ariel has an average orbital radius of 199,900 km and a period of 2.52 days.

Answer 1

Gravitational constant = G = 6.67 x 10^-11 N.m²/kg²

Mass of Uranus = M

Mass of the moon Ariel = m

Average orbital radius of the moon Ariel = R = 199900 km = 199900 x 10³ m = 1.999 x 10⁸ m

Orbital period of the moon Ariel = T = 2.52 days = 2.52 x (24x60x60) sec = 217728 sec

Orbital sped of the moon Ariel = V

VT = 2R

The gravitational force of Uranus on the moon Ariel provides the moon with the required centripetal force for the circular motion.

M = 9.97 x 10²⁵ kg

Mass of Uranus = 9.97 x 10²⁵ kg