A geosynchronous satellite is a satellite whose orbital period matches the rotation of the Earth. Calculate the height above the surface of the Earth which a satellite must have in order to be in a geosynchronous orbit. (Please enter your answer in units of kilometers)
Answers
Answer:
A satellite in such an orbit is at an altitude of approximately 35,786 km (22,236 mi) above mean sea level. It maintains the same position relative to the Earth's surface.
Explanation:
Answer:
Geosynchronous means that the satellite has same period as the earth, back to the same place in 24 hours.
T =24hrs = 86400 s
And let
h = height of the satellite from the surface of the earth.
r = radius of the satellite from the center of the Earth
R
E
= earth radius
M
E
= mass of the earth
The gravitational pull from the earth causes the satellite to go in orbit (otherwise it flies away. Hence the gravity is the cause of the centripetal force
F
G
=
m
s
v
2
r
⇒
G
m
E
m
s
r
2
=
m
s
v
2
r
r
v
2
=
G
M
E
Because the orbital speed
v
=
2
π
r
T
⇒
r
(
2
π
r
T
)
2
=
G
M
E
⇒
r
3
=
G
M
E
4
π
2
T
2
r
=
(
G
M
E
4
π
2
T
2
)
⅓
r
=
R
E
+
h
h
=
r
−
R
E
=
(
G
M
E
4
π
2
T
2
)
⅓
−
R
E
Substitute the proper values to compute h. The key is to recognize T is 24 hours.