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Find speed of a satellite placed at geostationary orbit

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A 270 kg communication satellite is placed in a geostationary orbit 35,780 km above a relay Earth ground level. What is the speed of the satellite in orbit?
 
Classical Mechanics - Find speed of a satellite placed at geostationary orbit Classical Mechanics - Find speed of a satellite placed at geostationary orbit - Physics Problems
A 270 kg communication satellite is placed in a geostationary orbit 35,780 km above a relay Earth ground level. What is the speed of the satellite in orbit?
 
Geostationary Orbit

A geostationary orbit is a circular orbit directly above the Earth's equator approximately 35,786 km above ground. Any point on the equator plane revolves about the Earth in the same direction and with the same period as the Earth's rotation.
The period of the satellite is one day or approximately 24 hours. To find the speed of the satellite in orbit we use Newton’s law of gravity and his second law of motion along with that we know about centripetal acceleration. The inward and outward forces on the satellite must equal each another (by Newton's first law of motion).
Fcentripetal = Fcentrifugal
By Newton's second law of motion:
F = ma
Newton's second law
where:
ms – Mass of satellite
ag – Gravitational acceleration
ac – Centrifugal acceleration


The centripetal acceleration provided by Earth's gravity:
centripetal acceleration
where:
Me - Mass of Earth in kilograms (5.9742 x1024 kg)
G - Gravitational constant (6.6742 x 10-11 N m2 kg-2 = 6.6742 x 10-11 m3 s-2 kg-1)


Magnitudes of the centrifugal acceleration derived from orbital motion:
centrifugal acceleration
where;
ω - Angular velocity in radians per second.
r - Orbital radius in meters as measured from the Earth's centre of mass.
centrifugal acceleration
centrifugal acceleration

From the relationship
Fcentripetal = Fcentrifugal centripetal force = centrifugal force
We note that the mass of the satellite, ms, appears on both sides, geostationary orbit is independent of the mass of the satellite.
Image
Image
r (Orbital radius) = Earth's equatorial radius + Height of the satellite above the Earth surface
r = 6,378 km + 35,780 km
r = 42,158 km
r = 4.2158 x 107 m

Image
Speed of the satellite
Speed of the satellite is 3.0754 x 103 m/s

Note:
Geostationary orbits are useful because they cause a satellite to appear stationary with respect to a fixed point on the rotating Earth. The satellite orbits in the direction of the Earth's rotation, at an altitude of approximately 35,786 km above ground.
Geostationary orbits can only be achieved very close to the ring 35,786 km directly above the equator.
 
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