Question #89592
The Moon has a mass of M = 7.3 · 1022 kg, a radius of R = 1.7 · 106 m and a rotation period of
T = 27.3 days. Scientists are planning to place a satellite around the Moon that always remains
above the same position (geostationary).
(a) Calculate the distance from the Moon’s surface to this satellite.
(b) Explain if such a Moon satellite is possible in reality.
1
Expert's answer
2019-05-13T10:49:19-0400

In the case of a stationary circular rotation of a satellite of mass m, the Moon’s gravitational force (gravity) Fg and the centrifugal force Fc act on it, they balance each other (Fg=Fc).


Fc=mv2/(R+h)Fc=mv^2/(R+h)

where v - speed of the spacecraft, m - mass of the spacecraft, R - radius of the moon, h - height of the spacecraft above the surface of the moon.


Fg=GMm/(R+h)2Fg=GMm/(R+h)^2

G – gravitational coVnstant, M - the mass of the Moon

Equating forces and using the formula for speed


v=2π(R+h)/Tv=2\pi(R+h)/T

(T- rotation period):



h=cubicroot(T2GM/(4π2))Rh=cubic root(T^2GM/(4\pi^2))-R

h=8.7107mh=8.7*10^7 m

h>>Rh>>R

Too far from the moon, this is impossible


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