Buoyancy Formula — F_b = ρ × g × V

This buoyancy calculator computes buoyant force using the buoyant force formula F_b = ρ_fluid × g × V_submerged — Archimedes' principle — and determines whether objects float or sink by comparing densities, with full step-by-step working and a real-time float/sink diagram.

Archimedes' principle states: any object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced. The buoyancy formula is:

The buoyant force formula F_b=ρgV reveals a critical insight: buoyant force depends on the FLUID density and the SUBMERGED VOLUME — not on what the object is made of. Two objects of identical size immersed in the same fluid experience the same buoyant force regardless of their material, density, or mass.

Alternative forms of the buoyancy equation: F_b = ρ_fluid × g × V_displaced (volume of fluid displaced), F_b = m_displaced × g (weight of displaced fluid), F_b = W_fluid_displaced (weight of fluid pushed aside). All are equivalent statements of Archimedes' principle.

How to Calculate Buoyancy Force — Step-by-Step

Follow this five-step method to calculate buoyancy force reliably using the buoyant force formula F_b=ρgV:

1. Identify the fluid and find its density in kg/m³ (water = 1000, salt water = 1025, mercury = 13534)
2. Determine the submerged volume — use the full object volume if completely submerged, or the partial submerged volume if floating
3. Apply F_b=ρgV: multiply fluid density × g × submerged volume
4. Convert units as needed: 1 N = 0.2248 lbf = 0.1020 kgf
5. Verify using Archimedes' principle: compute mass displaced (ρ×V) and its weight — should equal F_b

Float or Sink — Comparing Densities

The simplest rule in buoyancy: compare the object's average density to the fluid density. If ρ_object < ρ_fluid → floats. If ρ_object > ρ_fluid → sinks. If equal → neutral buoyancy. This is a complete determination — no other properties matter.

When floating, the fraction submerged at equilibrium is exactly:

Iceberg example: ice density = 917 kg/m³, fresh water = 1000 kg/m³. Fraction submerged = 917/1000 = 91.7%. Only 8.3% remains visible above water — the origin of "tip of the iceberg." In salt water (1025 kg/m³): fraction = 917/1025 = 89.5% submerged.

Human body example: average human density ≈ 985 kg/m³. In fresh water (1000 kg/m³): 985 < 1000 → technically floats, but barely (98.5% submerged — very little freeboard). In salt water (1025 kg/m³): fraction = 985/1025 = 96.1% submerged — floats more comfortably. This explains why ocean swimming is easier than pool swimming.

How Is Density Related to Buoyancy?

Density and buoyancy are directly and intimately related at two levels:

• Fluid density determines the buoyant force magnitude: F_b = ρ_fluid × g × V. Higher fluid density → greater buoyant force for the same submerged volume. The Dead Sea (1240 kg/m³) exerts 24% more buoyant force than fresh water (1000 kg/m³) on the same object.
• Object density (relative to fluid density) determines whether the object floats or sinks, and if it floats, how much is submerged.

The relationship between density and buoyancy is captured entirely in one comparison: ρ_object vs ρ_fluid. This is why the Dead Sea makes everything float — its density (1240 kg/m³) exceeds almost all biological materials (human body 985 kg/m³, wood 120-800 kg/m³). It is physically impossible to sink in the Dead Sea without deliberate effort.

Similarly, mercury (13,534 kg/m³) is so dense that many metals float in it — lead (11,340 kg/m³), aluminum (2,700 kg/m³), iron (7,874 kg/m³) all float in mercury. Only gold (19,300 kg/m³) and platinum (21,450 kg/m³) are dense enough to sink in mercury.

Apparent Weight — How Much Do You Weigh in Water?

When an object is submerged, it feels lighter because the buoyant force partially counteracts gravity. The apparent weight in fluid is:

This principle is used to measure density experimentally (Archimedes' original discovery — he used it to test whether the king's crown was pure gold). From apparent weight measurements:

ρ_object = ρ_fluid × W_true / (W_true − W_apparent) = 1000 × 49.03/6.24 = 7,858 kg/m³ ≈ 7,850 kg/m³ ✓

How much do you weigh in water? A 70 kg person with density 985 kg/m³ has volume 70/985 = 0.0711 m³. Buoyant force in fresh water: F_b = 1000 × 9.807 × 0.0711 = 697 N = 71.1 kgf. Apparent weight = 70 × 9.807 − 697 = 686 − 697 = −11 N — slightly negative, meaning the person is positively buoyant!

Does Buoyant Force Change with Depth?

No — for incompressible fluids and rigid objects, the buoyant force F_b=ρgV does NOT change with depth. It depends only on fluid density and submerged volume, not on how deep the object is. This is one of the most common misconceptions in introductory physics.

Why the confusion? Pressure does increase with depth (P = ρgh). So pressure on the bottom face of a submerged object is greater at depth. But pressure on the top face also increases by the same amount — the pressure difference (which creates buoyancy) remains ρg×height_of_object = constant regardless of depth.

Buoyancy in Air — Hot Air Balloons and Helium

The buoyancy formula F_b=ρgV applies identically to air as a fluid. Air density at sea level ≈ 1.204 kg/m³ (20°C). Buoyancy in air works the same way as buoyancy in water — lighter-than-air objects float upward.

Scuba Diving Weight Calculator — How Much Weight Do You Need?

Scuba divers need weight belts because the human body and wetsuits are positively buoyant in salt water — without ballast weights, divers float up and cannot descend. The scuba weight buoyancy calculator determines the correct ballast for neutral buoyancy.

The calculation sums all buoyancy contributions: body (+), wetsuit (+), tank (−/+), and weights (−) to find the total, then determines how much weight brings the sum to zero (neutral buoyancy). Rule of thumb: 5-10% of body weight in salt water with a wetsuit. Always validate in water — start conservative and add weight gradually.

Common Mistakes in Buoyancy Calculations

Buoyancy Worked Examples — 8 Problems

Frequently Asked Questions

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