What would happen if you built a gigantic pole from the Earth's equator?

The strongest material physically possible would still bend. It would have to be made of some exotic material with special atomic properties. If the tip was moving at 1/2 light speed, the bar would bend completely over, you couldn't get it to extend out any more. That would happen when it was about 1 billion kilometers long.

The strongest, stiffest material that might realistically be possible using ordinary matter, could only make it out to 70,000 km or so. That's about 1/6th of the way to the Moon.

There are people who want to do just that (to a point...I don't think they plan on extending all the way out to the Moon's orbit).

The concept is known as a Space Elevator. Here's the pitch: you use some kind of high-strength (but light-weight) material to anchor a cable to Earth's equator. The other end extends thousands of miles into space, well past the geostationary position. The geostationary position is the altitude where an object would take exactly one day to orbit Earth, and would therefore remain at the exact same point in Earth's sky. The far end of the cable is tethered to a counterweight. If the counterweight were detached, it would take more than one day to orbit Earth; the attachment of the cable therefore causes it to move faster and requires extra force to hold it in place. The extra force keeps the cable pulled tight.

The advantage of a Space Elevator is that it would be much, much, much cheaper to lift payloads into space...once you got it built. The disadvantage is that it would be incredibly expensive and difficult to build the Elevator using current materials. Carbon nanotubes may be used in the future, but we'd have to be able to make a LOT of carbon nanotubes at a low price, and we're just not there yet.

But you asked about going past the Moon. I suspect that would be impossible. For one, the Moon crosses the plane of Earth's equator twice each orbit, which means that about every two weeks there would be an opportunity for the Moon to collide with the cable. Even if the Moon didn't actually hit the cable, its gravity would cause such strain that the cable would most likely snap. Even if you took the Moon out of the equation, you need more and more tensile strength as the length of the cable increases, which means that engineering a trans-lunar Space Elevator would make a more modest Space Elevator look easy by comparison.

I hope that helps. Good luck!

Well this would be pointless unless you wanna use it as a space elevator and make a port on the tip of the pole. why would you want it to extend past the moon? the earth travels at 18.5 miles a second, not to mention how fast its spinning. anything this "tether" would collide with is gonna make it a bad day. it might cause a shift in the axis? this is just a bad idea period.

Well, nothing much, really. Nothing interesting could happen, except:

The pole breaking from the huge centrifugal forces

The pole hitting some satellite/asteroid

The pole going so fast it starts experiencing relativistic effects.

I'm not sure what exactly would happen in the last scenario, but it definitely wouldn't go faster than light, if that's what you were hoping for ;)

I don't understand why this would be done? It would most likely hit a lot of space junk and other things we have put into the earths orbit.