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I have a question about space elevators. With the rotation of a planet, what would prevent the scenario where the orbital space station is the fishing hook or weight, the elevator is the fishing line, and the planet is the fishing reel; and the planet is rotating wrapping the elevator around it and the space station down toward it?

The way space elevators work is basically a function of centripetal force. Not unlike a child's skipping ball:

Image courtesy ShopTV

We have the same deal. The Earth is rotating (representing the ring around the child's ankle). The elevator is the string attached to the ring. The space station is the ball at the end of the string. Why doesn't the string reel in like a fishing reel? For the same reason it doesn't reel in on the child's leg. Because the "ring" (the planet Earth), the string, and the space station are all moving at the same rotational speed.

But... heh heh.... what could make it reel in?

Let's introduce drag to the space station. In normal space, there isn't any drag.¹ So the elevtator and space station move at the same rotational speed as the Earth and happily stay in place.

But let's introduce something, like the the ancient concept of an aether that the space station must be pushed through. In this case, the Earth would reel in the space station like a fishing reel because the station is now moving slower than the Earth's rotational speed — and since everything's attached, rather than acting like a child's skipping ball, now we have a yo-yo in its upward motion.

So, the reel² question is: what would cause drag on your space station? Is there a flaw in the planet's magnetosphere that allows solar wind to affect the station? Is the hand of god holding it back? Did something hit it? The possibilities are endless — but without an infinitely³ powerful thruster system to reposition and balance the rotational speed, there will always be something that can cause the elevator to reel in like a fishing reel.

_{¹â€ƒWell... there is. It's complicated. But for the purposes of this question it's easier to say there isn't. Unless you get into the finer points of orbital mechanics and argue over the answer to "what's drag?" there isn't any drag in normal space. So say we all.}

_{²â€ƒI'm sorry, I couldn't help myself. :-)}

_{³â€ƒThe is the philisophical concept of infinite, not the mathematical concept of infinite. You only need enough thrusting power to correct after something hit you that wasn't strong enough to rip you off the tether. Still, it's an awful lot of thrusting power and unlikely to be built due to expense. In other words, it's a great idea for a story.}

The answer is: careful engineering of such space elevator.

"Fishing reel" scenario would may realize if space elevator's counterweight doesn't have enough speed, and instead of being geo- (or planet-) stationary is beginning to rotate with respect to the surface. So, its altitude will be lowering, and the space elevator's line would start accumulating on the ground.

Unless deceleration of the counterweight is very controlled so that its stationary position is more or less maintained, the strain on the line would likely became so strong that it will snap before it's lowered to the ground.

Fishing hook and line are affected by gravity, and the friction of wherever they are (even air has friction).

Space has no friction. Space station is in orbit, meaning it will stay there even if there is no elevator. The cable is not in orbit, it is hanging down from the station, and pulling it down. That's why the station will need to have a counterweight above it, to balance out the weight of the cable.

As a crude and imperfect analogy, take a small weight on a string (e.g. a yo-yo, or a phone charger), and spin it around. The weight is the station, the wire is the cable, and you are the planet. Is it wrapping around you?

Most stuff you wanted to know about space elevators can be found on the Wikipedia page.

The upshot of all current designs that employ a cable held in place by tension alone is that you need a cable and counterweight with center of mass at or slightly above geostationary orbit. In geostationary orbit, a space station located there will be in free fall at always the same location above the surface because the speed of the Earth's rotation and the speed at which the space station at GEO travel match up exactly. On the tether below GEO you experience a net force towards the ground, on the tether above GEO you experience a net force upward towards the counterweight.

The counterweight, on the other hand, moves faster than a free fall orbit would. You can house a space station there too which has the benefit of slinging stuff near or beyond Earth's escape velocity into interplanetary space when released from the space station, depending on the distance between counterweight and geosynchronous orbit.