Quirks and Quarks is a science show on CBC radio.
This segment is about space elevators, and fun, too.
Although the gravity is significantly less, and the atmosphere insignificant, I doubt that the rotational velocity of the moon is significant enough. I’m not positive, but I think you would only be able to put it in one of two spots: either straight out of the earth side, using earths gravity to keep it up (very long), or on the opposite side, using the rotation of the moon around the earth to keep it up (even longer).
Of course, if you were content with a normal elevator, rather than the flying cable thing, that would probably be possble from any point. I don’t know if you could make it more than a few miles tall, though.
Doing it on mars would be between the difficulty of doing it on earth and doing it on the moon, since mars has less atmosphere and less gravity than earth, but more of both than our moon. The day is only 38 minutes longer than an earth day, which is one of those weird coincidences that seem to crop up in scince a bit more often than they should.
Although perhaps more possible, I don’t see why you’d do it. Other than perhaps mining mars for materials for long range expeditions, I’d think Mars would have a lot more imports than exports.
Ben
Short answer: for the same reason ants the size of people would collapse under their own weight and foot long bees can’t fly. Scaling up changes the physics involved. And we’re talking about mega-scaling up here.
Long answer: 
The mass of the cable itself becomes the limiting factor.
BTW, AnderwL’s point. Actually you can have stuff in lower orbits. E.g., a 12 hour orbit can be easily arranged to avoid the cable when crossing the equator. But you’ll need to expend some fuel on occasion to avoid drifting. Ditto other nice fractions of 24 hours.
When I first saw the reference to Clarke in this thread I was ROTFL. And then others continue to cite him.
Didn’t any of you guy see Mysterious Worlds? So much for his credibility.
Quite true, which is why you need miraculously high strength-to-weight materials to make the idea work. If you had the right materials, it would be workable, in theory.
Yeah, there are orbital resonances you can set up which will keep missing the tether. The real problem is going to come from junk - spent boosters, dead sattalites, bits of debris and dropped tools. The kind of stuff that doesn’t make orbital correnctions or stay where you want it to reliably. It only takes one good hit to take out your tether.
Clarke did mention this point in Fountains of Paradise, but pretty much glossed it over.
The other problem is that in order for the tether to stay stable, your upward-going traffic has to be balanced by your downward-going traffic. Everytime you lift something on your tether, it saps a bit of momentum from the anchor at the end. You have to repay that momentum by dropping equal weight from the end to the groundbringing in lots of asteroid-mined materials doing a lot of asterois mining at the same time you’re shipping supplies to off-planet colonies, I suppose.
AndrewL has nicely summed up the potential problems there, so I’ll just go from his post. First of all, while you can’t get the orbital debris to dodge the tether, you can get the tether to dodge the debris, if the base is mobile. Picture it, for instance, on an oil rig-type floating platform in the ocean. It would, of course, take time for the effects of moving the base to propogate to orbital height, but that’s relatively easily calculated, and you can move it in advance.
Now, as to the up-down balance. If you’re shipping materials (such as water) to other planets, how are you getting it down on the other end? Why, with another elevator, of course. And that Mars elevator, say, will also have to balance its books. Just send up one ton of rocks on Mars for every ton of water you take down, and send it back to Earth to be taken down as ballast on our elevator. Or, if there’s something on Mars more valuable than rock, send that instead. It doesn’t matter, as long as the mass is right.
The only problem with that is that the stuff you put up the elevator is still in orbit around the earth, and would require more energy to send it to mars. Essentailly, what I’m saying is that it’d be simpler to just have propulsion on the top of the cable, using an ion drive or something like that. You could ship whatever it is you plan to eject from the ion drive up the elevator, and you could either use solar panels or have power cables from an earth-based power station to keep the cable top at speed.
That’s one area where the orbital ring with dangling cables has an advantage: the orbital ring could be accelerated or decelrated against earth magnetic field using electricity generated by solar panels. Then, when you want to send a shipment up, it would just slow the ring down a bit. If you needed emergency power, or wanted to deorbit the ring and kill a whole bunch of people, it’d be easy as flipping a switch.
Of course, if you did have enough ion drive propulsion, you wouldn’t need to even put the cable head past geosynchronous orbit. It’d need a lot of propulsion, though.
After all this discussion, I’m thinking that by the time we need this sort of capability bad enough to be willing to spend the money on it, we’ll have perfected fusion to the degree that we can just harvest solar wind and make it into whatever element we want.
Dang, you could make about fifty scifi books on this thread alone.