Answer:
The radius R of the orbit of the geosynchronous satellite is
Explanation:
By means of the equation of the orbital speed, the orbital period can be known:
(1)
Where r is the orbital radius and T is the orbital radius
r can be isolated from equation 1:
(2)
Notice that, from equation 2 is necessary to find the velocity before the orbital radius can be determined. That can be done through the Law of Universal gravity.
(3)
Then, replacing Newton's second law in equation 3 it is gotten:
(4)
However, a is the centripetal acceleration since it is a circular motion:
(5)
Replacing equation 5 in equation 4 it is gotten:
(6)
Where v is the orbital speed, G is the gravitational constant, M is the Earth mass, and r is the Earth radius.
Finally, equation 6 can be replaced in equation 2:
(7)
Where r is the orbital radius, T is the orbital period, G is gravitational constant and M is the Earth mass.
Since it is a geosynchronous satellite, it will have the same orbital period of the Earth (24 hours).
It is necessary to express the period in seconds:
⇒
Hence, all the values can be replaced in equation 7:
So the radius R of the orbit of the geosynchronous satellite is