Imagine two astronauts.
The first is in a satellite in geostationary orbit around the earth.
The second is on top of a 35,786 km tall tower that has been built from the equator to space. This means that that the top of the tower is at the same height as the geostationary satellite.
Naive logic would seem to indicate that the astronaut in the satellite would experience free-fall (as do all satellites in orbit), whereas the astronaut on top of the tower would feel the effects of gravity (albeit at a reduced level due to being 35,786 km above the surface of the earth.
However, I can't believe that this would be the case. What's the flaw in my logic? Is the acceleration due to the earth's gravity zero at 35,786 km above the earth?