Why doesn't water fall out of a bucket when it is rotated in a vertical circle?
When a tethered object is rotated from the point where it's tethered, there are several different vector forces acting on it. The key element here is that the force vector causing the bucket to move in a circle is separate from the force vector keeping the water in the bucket.
If a person simply held the bucket, the water would remain in it. Likewise, if they flipped the bucket upside down without swinging it, the water should fall out. Therefore, when the bucket is at the top of the arc, it must be experiencing a force equal to, or greater than, the force of gravity trying to pull it down. It can't experience this force by itself; it must do so as a component of the rotational motion, and that rotational motion is the result of three force vectors: gravity, the tangential force on the bucket, and the centripetal force on the bucket.
When a person swings a bucket, he or she must exert an increasing amount of force on its handle just to hold on to it, because rotation in a circle is basically the same thing as lots of very small and rapid adjustments to one plane of a vector, which all add up to the appearance of a circle. This is the centripetal force, and without it the bucket would simply fly off in whatever direction its tangential force was directing it at the moment of release. The byproduct of enough centripetal force is that the bucket will move in a circle and keep the water inside it. This force will also cause the bucket to feel like it has different weights as its interaction with the gravity vector changes.