Why isn't it possible to build a space elevator at the north pole? Why does it have to be built on the equator?
A "space tower" could be built at the north pole, but only if materials capable of supporting its weight were available. The "space tower" should be supported by the Earth's crust below it, but the crust will be flexible under the enormous load and over a long time.
A space elevator with cables makes use of the centrifugal forces caused by the rotation of the earth. These forces' vertical component is highest at the equator, while it is absent at the poles.
If you did build a tower at the north pole, a payload would only gain height but no speed. If released, it would fall down instantly.
From a space elevator at the equator, a payload could be lifted to the height of a geostationary orbit. If released, the payload would stay in orbit because the necessary height and speed for an orbit are met.
Because a space elevator would connect to a geostationary satellite.
Geostationary satellites can only exist above the equator.
As an assist to the current answers.
Try imagining an analogy:
First you hold a rope with your arms held out horizontally and you spin. The result would be the rope spinning with you, tending towards the horizontal the faster you go.
Now imagine spinning around but with the rope above your head. It will only fall down onto your head. You would need a rigid body to stand upright without spinning.
What is a space elevator, anyway?
The usual form of a 'space elevator' is a thing in geostationary orbit to which a cable is attached between the thing and the Earth surface, along with some adjustment or counterbalance to keep the centre of mass of the structure at the geostationary altitude (or it will no longer be geostationary).
The geostationary orbit is the only one in which an orbiting body (like the thing at the top of the elevator) stays a roughly constant distance from a point on the Earth's surface, where the cable would terminate. If the anchor point has to move to account for a non-geostationary orbit, that makes things either a bit harder (a floating base station that drifts around the ocean) or impossibly hard (an aircraft that must be kept continually fuelled and in motion to follow the orbit's groundtrack and counteract the drag on tens of kilometres of cable), and that's for subsonic relative motion on the Earth.
But what about the poles! Well, it would be different.
This would be a very different kind of elevator. Instead of a lump in orbit and a cable hanging down (mostly), this would have to be an entirely ground-supported structure - there is no geostationary orbit over the poles, unless you count plummeting or have a plan for a huge hole through the Earth's core. So this would be a great space tower. On the plus side, it would only need to reach some height in outer atmosphere (maybe 160 km), not geostationary distances (36,000 km). On the down side, you couldn't just release a thing at the top and have it orbit, it would need to be fired off to reach the high LEO orbital speeds (~7.8 km/s).
Having started the tower discussion at the pole, it needn't remain there; an edge-of-space tower could be positioned anywhere, albeit the latitude would determine the initially accessible orbits. Thus something closer to the equator would be likely more useful.
So, is it harder to build a 160 km tower or a 36,000 km cable?
In practise the longest continuous cable we've ever made is only 5 km long (http://www.kingpin-manufacturing.co.uk/blog/the-worlds-longest-cables/), and the tallest structure is 0.8 km. Apart from the Burj Khalifa, the next tallest structure was a guyed tower, so it seems more feasible for that type of structure to be extended several times over than to invent cables that can support their own weight over 36,000 km.
We'll need the tower anyway
Even the cable version would need a tower on the ground to provide a counter to the effects of wind, and in fact to avoid dragging the orbital mass off-course (which would be hard to correct and put strain on the whole cable), the sensible (!) thing would be to build a tower quite high in the atmosphere so that the cable doesn't suffer these lateral forces.
Which leads us to the conclusion that whichever kind of elevator we want, we have to build a tower most of the way to space.
Just how tall would the tower need to be? About 50 km.
At 160 km, orbits are just about stable. But we could build one to 120 km and only add a relatively small amount of additional drag to overcome with a rocket. For a short elevator like this there is inevitably a tradeoff between the height of the tower and the size of the rocket needed to reach orbit from the top. Given that you need a sizeable rocket anyway, there's a reasonable argument for just building one to 50 km (the top of the ozone layer, to avoid environmental catastrophe).
Step back, look at the balloons.
But wait, all we're doing now is hoiking (technical term) a reasonably large rocket to an altitude of 50 km so we can fire it. So why build a tower? According to Wikipedia, we have sent balloons to 53 km before. And expensive though balloons are, they're a lot less expensive than a 50 km steel tower.
So, it's cables or balloons. And the cable version is iffy.
So if you want to build an elevator at the pole, you might as well just use a balloon, and you can use it anywhere. I would add the caveat that the geostationary elevator is no panacea; it only makes geostationary orbits easy, and once you've put up all the cameras and telecoms kit you're going to want different orbits anyway.
What would we do with a space elevator, apart from satellite launches?
One question we can ask is what else we might do with cheaper lifting to space. Some imagine space holidays to the sky anchor in geostationary orbit. But that would turn out to be a slightly underwhelming holiday: If you actually use a vehicle that climbs the cable, you're going to be limited to maybe 150 kph; to permit the length of cable it will be fairly thin and not very robust, and the downside of tearing it is fairly huge. So speeding up it at speeds you wouldn't do on a highway in a car is unrealistic. But at 150 kph, 30,000 km is 200 hours, or over 8 days. The first hour of the holiday would be zipping up through the atmosphere. The next week would be like being on an increasingly remote cruise; the Earth gets smaller and smaller and space gets increasingly vast.
Can you go faster up the cable?
To put people up to the sky anchor really you'd need to send them up in a rocket, or perhaps zip them up near the cable on a rail gun. Either way it's either much more expensive (which was the whole point of the elevator) or requires a more complex (i.e. impossible) cable.
Well what would be a fun space holiday?
Let's go back to the short 50 km tower version of the elevator (at the pole, if you like). A trip could happily involve being sent at less ridiculous speeds up the tower, and then being gleefully shot into space on an orbital or suborbital path; a few hours to either encircle the globe or be flung to the other side, ending by gliding down to the ground on a winged lander.
Wait, isn't that global space travel?
Basically, yes. A suborbital flight could pitch you from one side of the world to the other in a couple of hours from the top of the tower. You'd probably want another tower (or perhaps just a balloon) at the other end or that's a particularly boring return journey.
On the plus side, we've managed to get above the ozone layer before we play with rockets, which is a great improvement on surface-launched orbital and suborbital vehicles. And with technology that is real now, not imagined for the future.
The idea is not as stupid or absurd as it seems at first glance (to me, at least). The elevator's cable would not extend vertically though as in a space elevator on the equator, but instead at an angle, probably almost horizontally.
Perhaps it helps to imagine moving the base of an existing equatorial space elevator. Moving the base north would pull the counterweight above (not at) the geostationary orbit north, off the equatorial plane. (Let's suppose a move which is slow enough to neglect Coriolis forces.) Earth's gravity would increasingly pull at a (small) angle to the tether, creating a displacement force parallel to the earth axis. The counter-weight's latitudinal position of equilibrium would therefore be south of the base's position.
When moving closer to the pole, the base would also move closer to the earth's rotational axis, pulling the counter-weight in. The radial distance traveled eventually is the earth's radius.
Because the rotational speed and the centripetal force of the counter-weight are linked to the orbit's radius, the tether would have to be lengthened as a compensation. (As others have mentioned, this probably increases the maximum forces on the tether, which could be a problem given that the load in the optimal equator case is already close to the specs of the best proposed materials.)
As a result, an ideal, weightless tether would leave the north pole almost, but not quite horizontally, because the counter-weight would be a bit south of the pole.
Interestingly, in this scenario the space elevator could rotate at any speed (and, coupled to the speed, orbit) the tether can sustain; it is not limited to geo-syncronicity any longer. Other than geostationary orbits would mean that the tether circles slowly around its anchor, which is possible only close to the poles. But this is only a thought-experiment with a weightless tether.
All realistic tethers would be far from weightless (in fact, the tether's own weight is the ultimate constructive problem); therefore it would bend substantially under earth's gravity and form some kind of catenary, as anybody who has ever flown a kite will easily understand. Consequently a base at the very pole would be quite pointless; the tether would lie on the ground anyway for hundreds or thousands of kilometers. But a base somewhere on the northern or southern hemisphere is not immediately unreasonable or absurd; it will just not be vertical.
A space elevator consists of a tether between a point on the Earth's surface and an object in space. That object must remain at a fixed position relative to the ground station - directly overhead (on a straight line connecting the Earth's center of mass and the ground station, and extended out into space) and at a fixed distance.
Above the pole, an object will respond to Earth's gravity by falling straight down because it has no orbital velocity around Earth.
To remain a fixed distance from Earth without some form of support (thrust, a tower, etc), an object must be in orbit around it. To remain at a fixed position over a point on the ground, the orbit must be of the geostationary kind. To support the weight of a tether and payload, the object to be used as a "space anchor" must actually be in an equatorial orbit but at a greater than geostationary altitude.
The whole point of a space elevator is to get a payload out of Earth's gravity well. Lifting it to what is essentially a geostationary orbit does this because the payload can be released at this point and it will not fall back to Earth because it is in orbit.
Two main reasons.
centrifugal force This can be best described with an example, if you take a bucket of water and spin it around, the water will not pour out of the bucket (The bucket is inline with the pane it is moving on, perpendicular to it). This is important because an orbiting body will experience this force and it can act against gravity as a way of counter balancing the weight of the tether/elevator. Its a force that acts at a right angle to the direction of travel along a curved path, that points away from the central point of the of that path, if it were a complete circle. https://en.wikipedia.org/wiki/Centrifugal_force
Centerpidial force is basically the opposite of this and points toward the central point. https://en.wikipedia.org/wiki/Centripetal_force Gravity often provides the Centerpidial force of orbiting bodies. Its the force that holds a moving body to a circular path.
Geosynchronous orbit This is an orbit where the orbiting body stays stationary over the surface of the body it orbits. It never sets. This is important because you need to anchor your tether to the ground. Otherwise it will be whipping around the surface which will make it a navigation hazard and very difficult to grab a hold of.
You can also think of this (a space elevator) as a ball tied to a pole by a long string. The pole is the earth, the ball is an orbiting satellite and the string is the space elevator.
If you send the ball flying it will "orbit" the pole because of the tension of the rope doesn't allow it to fly off. Now imagine the pole rotates at a rate that matches the ball where the rope doesn't wind up around the pole. This is basically what you need to have a space elevator. Of course this is a simplification and doesn't take into account of gravity, but you get the idea.
The only place that really satisfies this is near the equator and in a geosynchronous orbit.
If you did a polar orbit, your satellite does not stay directly over the pole. Therefor your elevator will transverse across the surface of the earth, probably whipping around.
If you just positioned it statically over the pole, there would be no centrifugal force and it would just fall from orbit. Most people don't realize you are not weightless when you are in low earth orbit because of lack of gravity. You are weightless because the centrifugal force (from your momentum) is exactly opposite and balanced with the centerpidial force of your orbit (from gravity). Now because it's balanced there is basically only one speed you can go at any given altitude and maintain your orbit. If you speed up your orbit/altitude increases. If you slow down your orbit decreases.
Ok so if you follow so far, the trick with a space elevator is to actually orbit a tiny bit faster at a given altitude. Without the tether your altitude would increase because your centrifugal force is greater then your centerpidial force, but because you are tethered this will put tension on tether and help releave some of the weight that it feels. Basically you will be pulling up on the tether.
Hopefully I remembered that right and it made sense.
A Space Fountain can be built at the North Pole, and could in principle reach any altitude. If you step off the top of it, you are still not in orbit, but if it's tall enough the delta-V you need to get into orbit, or escape the Earth altogether could quite small.
Keep in mind that the earth does not spin on an exact axis but rather has a 'wobbling' effect. You'd need to account for that when setting this up.
Since 2000, the Earth's axis has jumped eastward by about 7 inches a year, a "massive swing," Adhikari said. Another pattern of wobbling that occurs every six to 14 years has been vexing scientists for more than a century. The researchers examined both in their study.
The movements of the Earth's axis are crucial to understand, since they affect the performance of satellites and global positioning systems. The polar axis also could become a powerful indicator for scientists studying climate change, Adhikari said.
If you have two or more tethers for one space-based counter weight you could have one tether at the North Pole and the others connected to locations in the southern hemisphere. Then tweak the tether lengths and you could get the counter weight to fly in a stable orbit (around the equator).