Would two astronauts one in a satellite, one on top of a tower have the same experience?

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?

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A (hypothetical) construction up to (or in most schemes, down from) geo stationary orbit is sometimes called a "beanstalk" or a "orbital tether". You'll find a few questions and comments concerning them here and there on the site. You'll also find some reasonable treatments in the science fiction literature. –  dmckee Oct 20 '12 at 19:37
@dmckee - I read "Fountains Of Paradise" years ago :) We were discussing this over lunch! –  ChrisF Oct 20 '12 at 19:49
Even if you're standing on the ground at sea level at the equator, your weight is slightly reduced by the rotation of the Earth. On a stationary tower at geosynchronous height, that effect would completely cancel out the force of gravity. –  Keith Thompson Oct 20 '12 at 20:25
@dmckee: I heard it called a "space elevator" most commonly. –  Ron Maimon Oct 21 '12 at 1:17
They would both be free-falling. YOur mountain dweller could stand on top of the mountain, and would feel no upward force from the ground. The magic of geostationary orbit is that the gravitational force $G\frac{mM}{r^{2}}$ on an object is exactly equal to the centripetal force $\frac{mv^{2}}{r}$ required to keep something moving at the speed $v$ equivalent to one lap around the Earth per day. Since gravity provides no less or more force than exactly the centripetal force required to keep your person above the same spot on the ground, your person would not need the ground to hold herself up. She would feel weightless while standing on the mountaintop.