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If you stand on the north pole and dig a hole so deep that you emerge at the south pole, do you emerge head first or feet first?? (and yes I know you couldnt actually do this but in theory..)

I know that if you stand on the North Pole with a compass the compass needle will point to "S" no matter which way you turn.

Does that help ?

Were it possible to go right through, by the time you reached the centre of the earth, gravity would have meant that rather than digging below your feet to go down you would have to dig above you head to go up which is of course impossible.

Gravity is actually a very weak force, a baby can counteract it by lifting something, but by the time you got to the centre of the earth its effect though weak would be to tell you you were the wrong way up, (or down) and what had been definately 'down' was now certainly 'up'

[quote]If you stand on the north pole and dig a hole so deep that you emerge at the south pole, do you emerge head first or feet first?? (and yes I know you couldnt actually do this but in theory..)[/quote]

You mean like if you dive into the hole in freefall, or climb?

If you climb, you can emerge any way you want. If you jump/dive, you'll fall until you reach the centre, your momentum will keep you going past the centre for a while, and then you'll slow and gravity will pull you back, finally settling at the centre.

Next week, if a tree falls in a forest and no one is there to hear it, does it make a sound.

No.