A mechanical model would say that it would just get to the other side and then fall back the other way, get back to the exact point where it started and repeat. A better one that factors in resistance to motoin would say that it stops short of the other side and oscillates from side to side, stopping closer to the centre each time, until it finally stops perfectly still at the centre. And is probably crushed. Of course, what would actually happen is that there would be a huge eruption of lava that incinerates the ball, the person dropping the ball and everything else anywhere nearby.

From what I believe this is how it would go.

The drill bits and anything, would melt once it broke through the crust and in to the mantle. So it's impossible.

If it was possible, the drill bits would get stuck at the centre of the earth as it's made of iron and is magnetic (if the drill bits were made out of magnetic material). So, still impossible.

If they weren't magnetic, the drill bits would get crushed by the enormous gravitational forces. Basically it would be like a polythene cup being crushed in your hands. Still impossible.

If we could make an anti-gravity field or the equivalent, it would go all the way through the earth.

Now for the ball. If it was made of any magnetic field it would fall down to the centre and get stuck in the middle. It might start spinning as it's in the middle of a magnetic field. (That's counting that the ball didn't melt).

Or if it was non-magnetic, it fall to the centre and then go through (depending on speed, mass, shape, etc.) for a certain distance. Then it would back. Like Mathsyperson's post.

This is all assuming the hole was drilled straight, etc.

So, I declare it impossible for the above mentioned matter to ever happen.

True. Nothing is impossible, only improbable.

We are constrained by physical laws which make drilling the hole impossible unless it turns out we were wrong about those laws. You are limited by materials and energy. It's like trying to build a fusion reactor. We have been able to produce fusion for 50 years or so, but only in bombs--we can't contain it. The properties of matter say it's impossible to produce a material that can withstand the temperatures of fusion without ionizing and turning into plasma themselves. So you need to generate a magnetic field strong enough to keep it from touching anything, and we haven't been able to do that yet.

Or space travel. We pretty much reached our limits on space travel technology in the 60's and 70's, because there does not exist a fuel with sufficient energy density to propel a passenger ship at a reasonable rate. It takes our unmanned probes years to reach Jupiter. The only candidate is antimatter, which we cannot produce or store with near the efficiency required for practical application.

Talking about the hole, there doesn't exist a material with sufficient tensile strength to extend from the surface of the earth to the center--it would break under its own weight. That is, unless we discover one (buckminsterfullerine looks promising).

If we could make an antigravity field to balance the weight, there's still the problem of having a motor big enough to turn it. Even with a nuclear-powered gas turbine, I'm not sure you could produce enough power.

And if you had an antigravity field, how much power would it require? Could you project it along the length of the tunnel?

All of this is ignoring the presence of magma.

A thought just occurred to me. We could perform this experiment on a smaller scale in space. Not that we'd want to...