Archimedes' Principle

Hydrostatic buoyant force on submerged objects and physical conditions for equilibrium and flotation.

Conceptual context

Archimedes' Principle is one of the fundamental laws of fluid statics and is the cornerstone of naval engineering and aerostatics. Originally formulated by the Greek sage Archimedes of Syracuse in the 3rd century BC, this principle states that any body submerged, totally or partially, in a fluid at rest experiences an upward vertical force called buoyancy (or buoyant force), whose magnitude is exactly equal to the weight of the volume of fluid that the body has displaced to occupy its place.

This concept is revolutionary because it shows that flotation is a relation of parity between the density of the body and that of the medium.

🟢 Essential level

Intuitively, we can think of Archimedes' Principle as the "resistance" that a fluid offers to being displaced from its original position. When we submerge an object in water, the fluid is forced to move aside to make room for the body; however, because of its internal pressure, the fluid tries to recover that space by pushing the object outwards.

• The buoyant force $E$: This is the force we feel as a "help" when we try to lift someone inside a swimming pool. Things feel lighter because the water is doing part of the work for us, always pushing upwards, towards the surface.
• Displaced volume $V_{des}$: If we fill a glass of water to the brim and drop in a marble, the water that spills out is the displaced volume. Archimedes discovered that the weight of that spilled water is identical to the force with which the water pushes the marble upwards.
• The contest of forces: To know whether something floats or sinks, we only need to see who wins the contest: the weight $W$ (pulling downwards, towards the centre of the Earth) or the buoyant force $E$ (pulling upwards, towards the freedom of the surface). If the buoyant force is greater than the weight, the object rises; if the weight is greater, the object sinks.

This principle is the reason why a small iron nail sinks without remedy, while an enormous transatlantic liner made of thousands of tonnes of the same iron manages to stay afloat in a stable way. The difference lies in the fact that the ship, being hollow, displaces an amount of water whose weight far exceeds the weight of the metal itself.

🔵 Formal level

Basic formulas

Before applying the buoyant force, identify the volume that is actually submerged, because the force depends on the displaced fluid and not on the weight of the body. A partially submerged body displaces only the part that lies below the surface. From a mathematical and operational point of view, the buoyant force $E$ is quantified through the properties of the fluid and the geometry of the body. The fundamental equation defining this magnitude is:

Archimedes' Law

Real student questions

• Does buoyancy depend on depth? No, as long as the liquid is incompressible and the object is totally submerged. The difference of pressure between the upper and lower faces remains constant.
• Why does iron sink but ships do not? Because of the mean density. A ship contains air inside it, so that its total density (mass/volume) is lower than that of water.
• Is there buoyancy in space? In microgravity there is no hydrostatic pressure gradient, so buoyancy disappears and objects simply drift.

Final synthesis

Archimedes' Principle is the law of lightness in the fluid world. It teaches us that buoyancy is a consequence of geometry and density. Understanding that the buoyant force is the weight of the displaced fluid is the key to mastering any device that sails the seas or crosses the skies.