Whether stationary or in circular motion, the ball displaces the same amount of water. Yet, the upthrust acting on the ball varies continuously during the circular motion to provide the required centripetal force.
Consider when the bucket is at the bottom most position.
If the bucket is stationary, then
- A column of water of height h would have weight hAρg. This weight must be supported by the upward pressure force exerted by the water below. Hence the water pressure at depth h must be hρg.
- The pressure gradient results in the ball experiencing an upthrust U = ρVg, where V is the volume of displaced water.
- ρVg –mg = 0. Hence the ball is stationary.
If the bucket is in circular motion, then
- A column of water of height h would have weight hAρg. In addition to supporting this weight, the upward pressure forces exerted by the water below must also provide for the required centripetal force of hAρ(rω2). Hence the water pressure at depth h must be hρ(rω2+g).
- The larger pressure gradient results in the ball experiencing a larger upthrust U = ρV((rω2+g).
- The net force experienced by the ball is thus ρV(rω2+g) –mg = ρVrω2 = mrω2. Hence the ball does the same circular motion as the bucket of water.
When the bucket is in circular motion, the water pressure becomes larger because the water molecules would be pressing harder into one another. This is equivalent to the water experiencing an artificial g that is directed in the centrifugal direction, and an “upthrust” that is always in the centripetal direction. The magnitude of this artificial g would be varying continuously (in the exact manner as the tension in the rope, and normal contact force in the bucket) during the circular motion, so as to keep the water and the ball in circular motion.