Flotation

Static equilibrium of submerged bodies in fluids. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Flotation is the static equilibrium of a submerged body in a fluid, representing the practical application of Archimedes' Principle. A body floats when the upward thrust exactly compensates for its gravitational weight. This principle allows determining the draft and stability of any object, from icebergs to cargo ships, based on the geometric and mechanical interaction between the solid and the liquid medium.

🟢 Essential level

Flotation is essentially a competition between two opposing and constant forces: weight (gravitational pull pulling the object toward the center of the Earth) and thrust (hydrostatic upward force that the fluid exerts). For an object to float on the surface, the maximum thrust that the fluid is capable of generating must be at least equal to the weight of the object. If the weight is greater than this maximum thrust (which occurs when the object is fully submerged and gravity still wins the competition), the object will inevitably sink until it finds a solid support at the bottom of the container.

The fundamental key to flotation is not simply the total mass of the object, but its average density. This concept explains why a solid iron nail sinks quickly, while a steel ship of thousands of tons floats elegantly: the ship is not a solid block of steel, but a shell enclosing a large volume of air.

By averaging the density of the steel with that of the air contained inside, the total density of the ship turns out to be less than that of water. The hull design allows the ship to "trick" the fluid, displacing an enormous amount of water whose weight far exceeds the weight of the metal structure itself.

Furthermore, flotation is a process of automatic self-adjustment of nature. When you place an object in the water, it begins to descend due to the effect of gravity, displacing fluid in its path. As more volume enters the water, the upward thrust increases proportionally.

The object stops descending at the precise moment when the thrust $E$ equals the weight $P$. If the object is very light, it will barely need to submerge a small fraction of its body to reach this equilibrium; if it is very heavy but less dense than the fluid, it will submerge almost completely before stopping.

🔵 Formal level

Basic formulas

Before predicting whether a body floats, compare its mean density with that of the fluid and calculate the submerged fraction at equilibrium, because the result depends on shape and not only on material. A hollow body can float even if the material is denser than water.

The physical-mathematical analysis of flotation is based on the application of Newton's first law to hydrostatic systems. For a body in flotation equilibrium, the sum of vertical forces must be zero. If we define the vertical axis as positive upwards, the force balance is expressed as equality between buoyancy and weight.

By substituting the fundamental expressions for hydrostatic thrust and gravitational weight, we obtain a relationship that links the densities and volumes involved in the phenomenon: The physical equality can be read as: fluid density times gravitational acceleration times submerged volume equals object density times gravitational acceleration times total volume.

Where $\rho_{f}$ represents fluid density, gravitational acceleration is a common factor, $V_{\sum}$ is the submerged part volume (carena volume), $\rho_{obj}$ the average body density, and $V_{tot}$ the total volume. By simplifying gravity on both sides, we arrive at the operational relationship for the submerged fraction $f_s$:

Submerged Volume Fraction

Real student questions

• Why do I float better in the sea than in a pool? Because saltwater is denser than freshwater. Since $\rho_{f}$ is greater, the submerged fraction $f_s$ needed to balance your weight is smaller, so you protrude more from the surface.
• What happens if the object is hollow but has a leak? Its average density increases as water replaces air. When the average density exceeds that of the external fluid, the maximum thrust can no longer compensate for the weight and the object sinks.
• Can something float in the air? Yes, helium or hot air balloons float following exactly the same laws. As long as the density of the whole (balloon + gas + load) is less than the surrounding air density, there will be a positive flotation fraction.

Final synthesis

Flotation is the balance between weight and hydrostatic thrust. Governed by the density ratio, it defines the behavior of any body interacting with a fluid. Mastering its laws allows predicting draft and stability in naval design and understanding natural phenomena.