# Buoyancy

#### Prerequisites When a solid is added to a liquid, it may dissolve, forming a liquid with a new composition. However, if the solid is not soluble, it stays intact and either floats or sinks, depending on its relative mass compared to that of the liquid. If the solid weighs less than the volume of water it displaces, its buoyant force will be greater than its weight and it will experience a net upward force. If it weighs more than its buoyant force, it will sink.

To see this explicitly, look for the net force on the solid by doing a free-body diagram for the solid inside the liquid. There are two forces acting on the solid: Its weight pulling it down, and a buoyant force of the liquid pushing it up (since the pressure that generates an upward force on the bottom surface of the solid is greater than the pressure that generates a downward force on the top surface of the solid pushing it down). The buoyant force is equal to the weight of the displaced water by Archimedes' Principle:

$$F^{BF}_{fluid \rightarrow solid} = \rho_{fluid} \; g V_{solid}$$

Since the buoyant force is the weight of the displaced water, it is the mass of the water, determined from the density and the volume of the water displaced by the solid, times the gravitational field (in N/kg).

The weight of the object — the force of the earth's gravity pulling it down — can be expressed in a similar form using the density of the solid to get its mass, $m_{solid} = ρ_{solid} \; V_{solid}$ and therefore its weight is:

$$F^{W}_{earth \rightarrow solid} = \rho_{solid} \; g V_{solid}$$

The result is that when the solid body is completely imbedded in the liquid, the net force on the solid, or its buoyancy is dependent on the relative density of the solid and the liquid. In this case the net force is given by:

$$F^{net}_{solid} = (\rho_{liquid} - \rho_{solid})V_{solid}g$$

where $\rho_{solid}$ and $\rho_{liquid}$ are the densities of the solid and liquid, and $V_{solid}$  is the volume of solid.  Here we have chosen the positive direction as up (an arbitrary choice), so if the liquid is denser than the solid the net force pushes the solid up and the solid floats, while if the solid is more dense than the liquid, the net force is down and the solid sinks.

Once the solid reaches the surface of the liquid, part of it may partially rise above the surface. As a result, it displaces less liquid and the buoyant force goes down. (But its weight will stay the same!) It will come to rest when the volume displaced is just enough that the forces balance:

$$F^{BF}_{fluid \rightarrow solid} = \rho_{fluid} \; g V_{submerged}$$

$$F^{W}_{earth \rightarrow solid} = \rho_{solid} \; g V_{solid}$$

At equilibrium, the two forces balance so

$$F^{BF}_{fluid \rightarrow solid} = F^{W}_{earth \rightarrow solid}$$

which implies

$$\rho_{fluid} \; g V_{submerged} = \rho_{solid} \; g V_{solid}$$

The $g$ cancels. Solving for the fraction of the solid submerged, we get

$$\frac{V_{submerged}}{V_{solid}} = \frac{\rho_{solid}}{\rho_{fluid}}$$

This gives the interesting result that the fraction of the object that is submerged is equal to its specific gravity — its density relative to the fluid in which it is floating.

Biological materials are often close in density to that of water since water is such a large component of tissues.  The fraction of the human body made of water varies from person to person, with an average of about 55% in women and 60% in men.  Therefore, the average density of the human body will be somewhat close to that of water (1000 kg/m3). However, density varies for different biological materials.  Vertebrates will have some fraction of their bodies made of bone. Bone is composed of calcium phosphate which has a density around 2000 kg/m3, twice the density of water. However, muscle has a density of 1050 kg/m3, much closer to that of water.  There are some parts of the body with densities less than water including fat (915-945 kg/m3) and air in the lungs (1.14 kg/m3). The proportion of the body made of bone, muscle, fat, and air will set the overall or average body density.

For aquatic organisms, body density is particularly important as it enables them to control their depth. To stay at a constant depth, an aquatic organism needs to be neutrally buoyant, such that the mass of its body is equal to the mass of the volume of water being displaced.  This means that the buoyant force of the water exactly balances the body's weight. While many of an animal’s tissues are essentially water, animals must compensate for the heavier components like bones, by having a bag of air which has a much lower density. Fish hold this air in a swim bladder, while mammals hold the air in their lungs.

To control buoyancy, aquatic organisms usually adjust the volume of their internal air chamber. Increasing the volume occupied by air will decrease their average density, giving them a lower density than water and allowing them to rise. Decreasing the volume occupied by air will increase their average density making them more dense than water, and causing them to sink. The ideal gas law indicates that are two ways to increase the gas volume:  either increase the number of air molecules (i.e. breathing in) or decrease the pressure.

The change in pressure with water depth leads to an interesting challenge. As an organism rises, the overall pressure decreases and the volume of the internal gas chamber will increase. The resulting decrease in average density then makes the animal rise faster, further decreasing the pressure, increasing the gas chamber volume and decreasing average density yet again. The opposite happens as an animal descends. The pressure increases, decreasing the internal air volume, making the animal more dense and causing them to descend faster.

Anyone who has learned to scuba dive has experienced the problems of trying to control buoyancy to be able to get down to depth to see an interesting fish and avoid bobbing up to the surface in mid dive. To avoid these feedbacks and rapid depth changes, it is necessary to carefully control the internal air volume. Mammals can control their ascent by releasing air from their lungs. Some fish can also expel air from their swim bladders, though others do not have any external connection and must rely on gas exchange from the swim bladder into or out of the blood. Fish without an external connection typically remain at relatively constant depths, changing only slowly over a period of days or weeks. Workout: Buoyancy

Karen Carleton, Joe Redish 10/25/11, & Wolfgang Losert 10/31/2012

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