Ever since people set out to sea in ships, the issues of buoyancy and stability have been of importance. In spite of this, the treatment it receives in textbooks is often lacking. Following is an overview of the subject; basic understanding of the principles is essential in performing the experiment and interpreting the results.
Buoyancy
Buoyancy is ultimately what makes things float, such as the buoy in Figure 1. This is true whether the material the boat is made of is lighter than water (like the balsa wood rafts Thor Heyerdahl and his crew crossed the Pacific with in 1947) or heavier than water. The latter would include objects from the buoy shown to the ships of the U.S. Navy.
The basic concept is very simple: for anything placed in a fluid medium, the upward force the medium exerts on the body is equal to the weight of the fluid the body displaces. This is not only true of bodies placed in water; it is also true of those in air. The difference is that, for those in air, the weight of the air displaced is usually not enough to “float” the aircraft. A notable exception are dirigibles such as the “Goodyear blimp,” which is filled with helium, a gas lighter than air. Another lighterthanair gas used is hydrogen. This is very combustible, as everyone was reminded of when the Hindenburg caught fire in New Jersey in 1938.
Most buoyancy applications are marine ones, and it is those we will concentrate on in this experiment. We will also concentrate on rectangular forms and flatbottomed vessels, which simplifies the math somewhat. However, these principles can be extended to just about any floating craft.
Using a flatbottomed craft also makes it easier to understand why displacing a fluid works. Consider first the following: how the force of the fluid on the flat hull of a craft varies with depth^{1}:
For a fluid at rest, the hydrostatic pressure increases linearly with depth, thus
(1)
where p is the hydrostatic pressure, γ is the unit weight of the water, and D is the depth from the water’s surface to the bottommost point of the vessel, usually called the draught. This distance from the water line to the top of the rectangle (the gunwale) is called the freeboard; the results of inadequate freeboard can be seen in Figure 3.
In any case, for a vessel of beam (width) W and a length L the volume it displaces is given by the equation
(2)
Combining and rearranging these two equations,
(3)
For the boat to float, it has to be in static equilibrium, and so the downward force of the weight of the boat W_{boat} must equal the upward force F_{buoyant}. Therefore,
(4)
So we’ve established a relationship between the weight of the boat and the volume of water it displaces. The “far right” hand side only applies to boats with a flat bottom and straight sides.
What this means is that there are three ways we can weigh an existing boat:

We can simply weigh it on a scale. For small boats this isn’t too difficult; larger ones can be tricky. We can then estimate how far it will sink into the water.

We can measure the freeboard, then obtain D and, knowing L, W and the unit weight of water, we can compute the weight of the boat. This works easily for rectangular boats; for real boats, you have to determine the relationship between the actual waterline and the displacement, then see where the actual waterline ends up.
 We can use an overflow method, which is okay for small experiments (like Archimedes used) but not so hot on a larger scale. But this illustrates our concept.
Procedure for determining volume of water displacement^{2}: