The planet Uranus has a radius of 25560 km and a surface acceleration due to gra

Question

The planet Uranus has a radius of 25560 km and a surface acceleration due to gravity of 11.1 m/s^2 at its poles. Its moon Miranda (discovered by Kuiper in 1948) is in a circular orbit about Uranus at an altitude of 104000km above the planet's surface. Miranda has a mass 6.6*10^19kg of and a radius of 235km .

A) Calculate the mass of Uranus from the above data.
B)Calculate the magnitude of Miranda's acceleration due to its orbital motion about Uranus.
C)Calculate the acceleration due to Miranda's gravity at the surface of Miranda.

Explanation / Answer

A)

g = GM/r^2
11.2 = 6.67 x 10^-11 M / 25560000^2

M = 1.087 x 10^26 kg

B)

Miranda's accceleration = GM / r'^2

r' = 25560000+104000000 = 129560000 m

a = 6.67 x 10^-11 x 1.087 x 10^26 / 129560000^2 = 0.432 m/s^2

C)

g = GM/r^2
= 6.67 x 10^-11 x 6.6 x 10^19 / 235000^2

a = 0.797 m/s^2

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