fluids question about buoyant force & submerged volume

Question

There is a man standing in a pool. Besides him, there is a raft floating, at rest, in the pool. How does the water level change when the man lays down on the float?

Answer 1

Hi John D.,

The water level goes up. Initially, the man is displacing a volume of water equal to the volume of the portion of his body which is submerged. Since he is not completely submerged, said volume is less than his total volume and since he is not floating, said displaced mass of water is less than his total mass. And the raft is displacing water mass equal to its mass. When he flops onto the raft (note: he does NOT LAY, he LIES!) he will then be displacing water mass equal to his entire mass. That is a greater mass of water than previously displaced, so the level goes up.

Another way of conceptualizing: the weight of the tank, water, and him is constant, no matter what he is statically doing. So in the initial state, some of his weight is being directly carried by his feet on the tank bottom, and the pressure at the tank bottom carries the rest of everything else. When he reclines, the water pressure at the tank bottom now carries ALL of everything. In order for it to do that, the depth (surface to tank bottom) must be greater, hence, the level has risen.

Answer 2

When the man lays down on the raft, the water level in the pool will not change.

Here's the reasoning:

1. Initial State:
2. The man is standing in the pool. He displaces a volume of water equal to his own volume. This causes the water level to rise by an amount proportional to the volume of his body submerged in the water.
3. Final State:
4. The man lays down on the raft. The raft, along with the man on it, now displaces a volume of water equal to the combined weight of the man and the raft.

According to Archimedes' principle, the buoyant force (which keeps the raft and the man floating) is equal to the weight of the displaced water. Therefore, whether the man is standing in the water or lying on the raft, the weight of the water displaced must equal the man's weight (plus the raft's weight in the final scenario).

Since the weight of the displaced water does not change, the overall water level in the pool remains the same. The transition of the man's weight from being directly supported by the water to being supported by the raft (which in turn is supported by the water) doesn't alter the total displacement.

Therefore, the water level in the pool remains unchanged when the man lays down on the raft.