AkCalculators

Buoyancy — Archimedes' principle and floatation

Buoyancy describes the upward force that a fluid exerts on an object immersed in it. Archimedes' principle states that this buoyant force equals the weight of the fluid displaced by the object. Expressed mathematically:

F_b = ρ_fluid × V_displaced × g

Here ρ_fluid is the fluid density, V_displaced is the volume of fluid displaced (equal to the submerged volume of the object), and g is gravitational acceleration. This simple relation explains why ships float, why balloons rise (with buoyancy due to displaced air), and why some objects partially submerge to reach equilibrium.

Floating condition and submerged fraction

A freely floating object reaches equilibrium when buoyant force equals its weight: ρ_fluid V_displaced g = m_object g. Canceling g gives:

V_displaced = m_object / ρ_fluid

For a uniform object with total volume V_total and density ρ_object, the fraction submerged f = V_displaced / V_total = ρ_object / ρ_fluid (assuming no trapped air and uniform distribution). If ρ_object < ρ_fluid the object floats with a submerged fraction less than 1; if ρ_object > ρ_fluid it sinks.

Shape, stability and center of buoyancy

While Archimedes' principle determines the magnitude of the buoyant force, stability depends on the relative positions of the center of gravity and the center of buoyancy. The center of buoyancy is the centroid of the displaced volume and can shift as the object heels or tilts, which can create righting or capsizing moments. Designers of ships and floating platforms manage these moments by controlling weight distribution and hull geometry.

Applications

Buoyancy is central to naval architecture, offshore engineering, submersible design, and everyday tasks like determining whether objects float in water. It also applies to aerostatic lift in gases—hot-air balloons and helium-filled balloons rise because the air displaced weighs more than the balloon system.

Worked examples

Example 1 — Buoyant force: A submerged volume of 0.02 m³ in freshwater (ρ = 1000 kg/m³) experiences F_b = 1000 × 0.02 × 9.80665 ≈ 196.13 N upward.

Example 2 — Floating fraction: A wooden block with density 600 kg/m³ and total volume 0.05 m³ floats in water (ρ = 1000 kg/m³) with submerged fraction f = 600 / 1000 = 0.6 — so 60% submerged.

Example 3 — Required displaced volume: An object of mass 200 kg in seawater (ρ ≈ 1025 kg/m³) requires V_displaced = m / ρ ≈ 0.1951 m³ to float.

Temperature, salinity and compressibility effects

Fluid density varies with temperature and salinity for liquids and with pressure and temperature for gases. In ocean engineering, salinity and temperature profiles (thermohaline structure) affect buoyancy and stability of submerged bodies. Compressibility affects deep submersibles where pressure changes significantly with depth.

Practical measurement tips

Measure displaced volume experimentally by submerging an object in a container and measuring overflow, or calculate it from geometry for regular shapes. When working with hulls or vessels, use hydrostatic tables and consider payload distribution to ensure stability and reserve buoyancy.

Using this calculator effectively

Enter any two independent values among fluid density, displaced volume, object mass and buoyant force to compute the others. Override g if performing calculations for other planets. Enable step-by-step derivations for teaching or lab documentation, and export CSV for records. For stability analysis and complex hull forms use specialized naval architecture tools.

Archimedes' principle is elegant and remarkably practical. It links mass, volume and fluid properties to produce predictions that guide shipbuilding, submersible design, and many other engineering tasks.