Here's one I was asked at an interview:
Consider a hypothetical situation where a hole is drilled, from the North Pole, right through the centre of the Earth to the South Pole, along the Earth's axis of rotation.
A small sphere is then dropped into the hole, describe it's motion in terms of conservation of energy and simple harmonic motion and say what happens when it's kinetic energy has fully dissipated.
Now, consider if we moved the hole, so instead of it being along the axis of rotation we put it say... on the equator, going through the centre of the Earth again and to another point on the equator. How will the Earth's rotation effect the sphere on this occasion?
Now, say we had a step ladder and decided to drop the sphere from the top of the step ladder, how would this effect the motion?
And finally, what if we dropped the ball from thousands of feet in the air, what would happen if we dropped it from directly above the hole? How could we be more shrewd with our choice of drop point in order to achieve a better result?
The worst hour of my life...
Horrifying for a (then) 16 year old.
But as there seems to be some interest in dynamical systems on here I thought I'd give the bored people something to do!