Question

Explain why the faster Earth spins, the less a person weighs, whereas the faster a space station spins, the more a person weighs.

Answer 1

EARTH SPINS FASTER:
When the Earth spins faster your angular velocity increases which will lessen the effect that the centripetal force (due to gravity) has on keeping you from flying off into space...so you weigh less (think about Newton's First Law). In essence gravity has not changed at all...it's just not affecting you as much.

A decent analogy is a merry-go-round. The faster it spins when you're on it, the more you feel pulled outward from the axis of rotation and the "lighter" you feel.

SPINNING SPACE STATION:
A space station that's spinning and is designed to produce artificial gravity uses the exact same concept but in a different way. The personnel spaces are positioned to where they are forced outward against the inner hull of the rotating portion of the ship. The faster it spins, the more you are forced against this inner hull.

You can imagine the same merry-go-round analogy but this time, picture it with a wall around the circumference of it. The faster it spins when you're on it, the more you feel pulled outward from the axis of rotation just like before. However, now you have the inner wall pushing back on you and keeping you from flying off. The end result is that you're pressed against the inner wall which opposes your outward motion and you end up feeling "heavier".

I found this image which may illustrate the spinning space station concept better: