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

Question

The planet Uranus has a radius of 25560km and a surface acceleration due to gravity of 11.1m/s^2 at its poles. Its moon Miranda (discovered by Kuiper in 1948) is in a circular orbit about Uranus at an altitude of 104000 km above the planet's surface. Miranda has a mass of 6.6x10^19 kg and a radius of 235 km

(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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