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One of the biggest issues that I've heard of in building a space elevator is the high tensile strength of the cord used to tether the the elevator to the ground.

If there's a structure that's at geostationary orbit, shouldn't the tension of the cord be theoretically zero (seeing as from the point of view of the tether point on the ground, the satellite would remain at the same distance above it)? If so, could other, weaker materials be used?

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At the geostationary orbit, the growth (gradient) of tension will be practically zero, but the value will be far, far from zero. Actually, it will be the highest in the whole tether.

After all, the whole length of the tether, that goes all the way down to Earth surface, is moving progressively slower than what orbital speed at corresponding altitude is - and as result, is pulled down by Earth's gravity. Any part of the tether, therefore, must carry the weight of the tether below it, reduced only as much as gravity drag is reduced by the velocity of the tether so far. This all cumulates to the maximum at GEO, and then drops off again, with the pull in the opposite direction (away from Earth) as the tether extends farther, to a counter-balance at its end, as its velocity exceeds local orbital velocity, creating an outwards pull - that keeps the tether taut and prevents it from falling all the way down to Earth.

That also means there is no limit (well, within reason) to the size of an orbital station placed at GEO on the tether, as it doesn't contribute to the strain. Any other station creates an extra strain either way - although in case of one near the far end of the tether this is actually welcome as it contributes to the pull that keeps the tether taut.

Actually, the least strained point of the tether is right at Earth surface, where there is no more mass hanging down from the tether - only as much strain as needed by surplus counter-balance necessary to allow the extra mass of traveling payload, improve rigidity of the tether and combat winds and rotation of Earth that could otherwise wrap it around Earth surface.

Also, one practical way of building the space tower is to deliver the (rolled) tether to GEO and unroll it there, both directions simultaneously, up and down, so that the "downwards" part's pull evens up with the "upwards" one. The net pull remains zero, but as the two ends travel apart, the strain grows immensely.

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    $\begingroup$ This is easiest to see by example - a rope thrown down from a building is also geostationary. As the height increases, so does the strain. The only "unintuitive" part happens when it reaches past GEO, as you've noted in your answer. $\endgroup$ – Ordous Apr 1 '16 at 15:14

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