Archimedes' Principle - Explanation

Explanation

Archimedes' two-part treatise on hydrostatics, called On Floating Bodies, states that:

Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.

— Archimedes of Syracuse

with the clarifications that for a sunken object the volume of displaced fluid is the volume of the object. Thus, in short, buoyancy = weight of displaced fluid. Archimedes' principle is true of liquids and gases, both of which are fluids. If an immersed object displaces 1 kilogram of fluid, the buoyant force acting on it is equal to the weight of 1 kilogram (as a kilogram is unit of mass and not of force, the buoyant force is the weight of 1 kg, which is approximately 9.8 Newtons.) It is important to note that the term immersed refers to an object that is either completely or partially submerged. If a sealed 1-liter container is immersed halfway into the water, it will displace a half-liter of water and be buoyed up by a force equal to the weight of a half-liter of water, no matter what is in the container.

If such an object is completely submerged, it will be buoyed up by a force equivalent to the weight of a full liter of water (1 kilogram of force). If the container is completely submerged and does not compress, the buoyant force will equal the weight of 1 kilogram of water at any depth, since the volume of the container does not change, resulting in a constant displacement regardless of depth. The weight of the displaced water, and not the weight of the submerged object, is equal to the buoyant force.

Objects weigh more in air than they do in water. If a 30-kilogram object displaces 20 kilograms of fluid when it is immersed, its apparent weight will be equal to the weight of 10 kilograms (98 newtons). Similarly, when submerged, a 3-kilogram block may have an apparent weight of 1 kilogram. The "missing weight" is equal to the weight of the water displaced, the weight of 2 kilograms (19.6 newtons), which is the buoyant force. The apparent weight of a submerged object is its weight under standard conditions minus the buoyant force.

Consider a completely submerged cube. The fluid exerts pressure on all six faces, but as long as the cube is not tilted, the forces on the four vertical faces balance each other out. The pressure difference between the bottom and the top face is directly proportional to the height (difference in depth). Multiplying the pressure difference with the area of a face give the net force on the cube - the buoyancy, or the weight of the fluid displaced. It makes no difference how deep the cube is placed because, although the pressures are greater with increasing depths, the difference between the pressure up against the bottom of the cube and the pressure down against the top of the cube is the same at any depth. Whatever the shape of the submerged body, the buoyant force is equal to the weight of the fluid displaced.