Buoy

Classical Mechanics Level 3

A buoy is attached to the bottom of a river by an anchor chain. Due to a strong flood, the water level rises higher than the length of the anchor chain, so that the buoy gradually sinks into the flood. What happens when the waterline keeps rising?

Details and Assumptions: The buoy is open at the bottom, so that water can penetrate into the interior. However, the water is held back by an air bubble, which also causes the buoyancy of the buoy.

The buoyancy force increases, so that the anchor chain tears at some point and the buoy drifts away The buoyancy decreases and the buoy sinks to the bottom of the river The buoyancy remains constant, leaving the buoy in place below the waterline

2 solutions

Markus Michelmann
Jan 8, 2018

The buoyancy force according to the Archimedean principle corresponds to the volume $V$ of the displaced water: $F = \rho g V$ with the density of the water, $\rho$ , and the gravitational acceleration $g$ . In our case, the volume $V$ is approximately equal to the volume of the enclosed air. According to the ideal gas law, this volume is given by $V = \frac{n R T}{p}$ with the molar amount of substance, $n$ , the general gas constant $R$ , the temperature, $T$ , and the pressure $p$ . Since both the air quantity $n$ and the temperature $T$ are constant, the volume depends only on the pressure $p$ . The pressure is due to the atmospheric pressure and the gravitational pressure of the water: $p = p_0 + \rho g h$ with the height difference $h$ between the river level and the water surface in the buoy. As the water level rises, so does the pressure in the buoy, compressing the air inside and decreasing the volume of displaced water. Finally, the buoyancy force is no longer sufficient for the buoy to float, causing the buoy to sink to the ground. It is crucial that the air in the buoy is compressible, while the surrounding water is virtually incompressible and therefore has a constant density.

I do not agree. The buoyancy increase but does not means that anchor chain it will get broken because that depends among other factors, like the links strength for which the chain was designed. The fact is the level of the free surface rises and so the buoyancy, since you keep tied the buoy, air inside increase its pressure by the same reason, and this pressure act on upper inner surface of buoy. Just take glass of water and do the experiment in a bucket or in the kitchen sink, pour water on it till a half full for instance, now you put glass upside down and open the faucet slowly to keep filling the bucket or sink you will feel need to impress force down in order to balance the increase of pressure of the air inside, once the water rises over the top of buoy the force remains constant. The reduction of volume of air inside of the buoy it will compensate by the increase of pressure by the the Boyle's Law P1V1 = P2V2.

So if we assume the volume of due to the hull of buoy small compared to full volume, the problem is just a gas problem, no room for Archimedes

I also dissent in the plausible final course of the event at least until the pressure due to water level be such that air liquefied at normal temperature, and collapse similarly to cavitation phenomena.

Mara Jares - 3 years, 4 months ago

I agree with you, that the buoyancy of the bouy alone would be not sufficient to break the anchor chain. But this answer is already wrong, so it does not matter.

The fact, that the buoy is floating, is because of the buoyancy force and Archimedes princple. A body swims, if the mass of the displaced water is bigger than the mass of the body. Here, the mass of the buoy (including the enclosed air) is constant, but the volume of the displaced water changes, since the enclosed gas is compressible. Since the gas volume is reduced by increasing water level, the volume of the displaced water is also reduced by the same amount. At same point, the mass of the displaced water is not enough to take the weight of the bouy, so that the bouy sinks to the ground.

The fact, that a buoy sinks to the ground at high floods is an useful feature, because an bouy below the water line serves no purpose but is potential danger for ships.

Although, the physics of this problem is not that complicated, the answer is kinda contra-intuitive. This problem was shown as a experiment in a german TV-show called "Knoff-hoff-Show" in the early 1990's. A selection of experiments later appeared as a book, so that I have taken this problem together with the explanation essentially from this book. I would put a video link for the original experiment to my solution, but unfortunately I can not find it on the internet.

Markus Michelmann - 3 years, 4 months ago

I do not see the reduction in volume of water will imply less buoyancy, because air pressure acting on ceiling of buoy will increase precisely in the amount equivalent to the difference of water level, acting like a pneumatic cylinder. In my opinion, assuming the floating surface being constant, buoyancy will increase linearly till the buoy is fully covered with water. From this status on, the buoyancy of buoy remains constant since the weight of water on top of buoy will balance the increase of pressure in the confined chamber of air inside the buoy. Probably in the experiment you have cited in the top of buoy there is a relief valve, so when pressure rises to a prescribed level the relief valve open just enough to free an amount of compressed air to sink the buoy. Or buoy can have a small compartment on top that allows water inlet when start to submerge. In my opinion someting is missing.

In river Ebro in Spain we have a problem alike and it is a headache to regulate the buoys, because density of water change due to amount of sediment the river carry

Mara Jares - 3 years, 4 months ago

Gediminas Sadzius
Jan 11, 2018

The crossection of the buoy resembles a crossection of a boat. If we turn the buoy upside down, it will resemble a boat with water leaking in. From experience - too much water in the boat will sink it! The air in the boat will not be compressed if it is completely open on top, which is different from the situation with the buoy. However the density of the air in the buoy will not change much due to water leaking in, as the pressure increase will only be tens (x10) of meters of water column above the atmospheric pressure, depending on the vertical dimension of the boy.

Buoy

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