Answer
The water is held inside the bucket with centripetal force when spun at a sufficient speed.
Work Step by Step
As the water comes over the top of the arc, it is moving in a curved path of radius r, and is therefore accelerating. The net force must be directed downward/inward, toward the center of the circle.
The 2 forces acting on the water at that instant are gravity (pulling downward) and the normal force from the bucket (also pushing downward). The net force is downward.
At a critical minimum speed $v = \sqrt{gr}$, the normal force is exactly zero, and gravity provides all of the centripetal force to keep the water moving in a circle. If the bucket is moving more slowly than that, water will spill out.
