Buoyancy Calculator

Calculate buoyant force, submerged volume, or fluid density using F_b = ρVg (Archimedes' Principle).

🚢 Fluid Mechanics📐 F = ρVg⚓ Archimedes
Fluid density (ρ) kg/m³
Submerged volume (V) m³
Common densities: Water = 1000, Seawater = 1025, Air = 1.225 kg/m³
⚠️ Please enter valid positive numbers.

What Is Buoyancy?

Buoyancy is the upward force a fluid exerts on a submerged or floating object. Archimedes' Principle (discovered ~250 BCE) states: the buoyant force on an object equals the weight of the fluid it displaces. The formula is F_b = ρ·V·g, where ρ is the fluid density (kg/m³), V is the volume of fluid displaced (m³), and g is gravitational acceleration (9.8 m/s²). The result is in newtons.

An object floats when its average density is less than the fluid density (F_buoyancy > Weight), and sinks when its average density is greater. For a floating object, F_b = weight exactly, meaning ρ_fluid × V_submerged = ρ_object × V_total. A ship floats by displacing water equal to its total weight — the hollow steel hull gives a low average density despite steel being 8× denser than water.

The buoyant force depends only on the volume of fluid displaced and the fluid density — not on the shape, material, or depth of the object. A 1 m³ block submerged in water experiences 9,800 N upward regardless of whether it's steel, wood, or hollow. This independence from depth is sometimes counterintuitive — pressure increases with depth, but so does the pressure below the object, keeping the net upward force constant.

Buoyancy has applications far beyond ships. Submarines control depth by adjusting the amount of water in ballast tanks (changing average density). Hot air balloons rise because heated air is less dense than surrounding cool air. Fish use swim bladders to achieve neutral buoyancy. Even objects in the atmosphere experience buoyancy from air (why helium balloons rise and objects weigh slightly less in air than in vacuum).

Formula Reference Table

QuantityFormulaNotes
Buoyant force (F_b)F_b = ρ · V · gρ = fluid density, V = displaced volume
Displaced volume (V)V = F_b / (ρ · g)Volume of fluid pushed aside
Fluid density (ρ)ρ = F_b / (V · g)kg/m³
Floating conditionF_b = Wρ_fluid × V_sub = ρ_obj × V_total
Apparent weightW_app = W − F_bWeight felt while submerged
Water densityρ_water ≈ 1000 kg/m³Seawater ≈ 1025; Air ≈ 1.225

3 Worked Examples

Example 1
Wood Block in Water

A wood block (ρ = 600 kg/m³) has volume 0.01 m³. What fraction floats above water?

  • F_b when fully submerged: F_b = 1000 × 0.01 × 9.8 = 98 N
  • Weight: W = 600 × 0.01 × 9.8 = 58.8 N
  • Since W < F_b, wood floats. Fraction submerged = ρ_wood/ρ_water = 600/1000 = 60%
  • 40% of the block floats above the waterline
✓ 60% submerged, 40% above water
Example 2
Ship Displacement

A 5,000 tonne (5×10⁶ kg) cargo ship floats in seawater (ρ = 1025 kg/m³). Find displaced volume.

  • Floating: F_b = Weight → ρ_sw × V × g = mg
  • V = m/ρ_sw = 5×10⁶ / 1025
  • V = 4,878 m³ of seawater displaced
✓ Displaced volume = 4,878 m³
Example 3
Crown Density (Archimedes Problem)

A 0.5 kg crown weighs 4.5 N in water. Find its density. (Apparent weight = 4.5 N, actual = 4.9 N)

  • Actual weight: W = 0.5 × 9.8 = 4.9 N
  • Buoyant force: F_b = 4.9 − 4.5 = 0.4 N
  • Volume: V = F_b/(ρ_water × g) = 0.4/(1000×9.8) = 4.08×10⁻⁵ m³
  • Density: ρ = m/V = 0.5/4.08×10⁻⁵ = 12,255 kg/m³ (pure gold = 19,300; this crown is not pure gold!)
✓ Crown density ≈ 12,255 kg/m³ — not pure gold

Real-World Applications

🚢
Ship Design
Naval architects design hull shapes and ballast systems so ships displace water equal to their total weight. Cargo ships float lower as they load — the Plimsoll mark indicates maximum safe loading depth.
🤿
Scuba Diving
Divers achieve neutral buoyancy by adjusting their buoyancy compensator device (BCD) — an inflatable vest. Adding air increases volume (more F_b); releasing air allows sinking. Perfect neutral buoyancy allows effortless depth maintenance.
🎈
Balloons & Airships
Hot air balloons work because heated air (ρ ≈ 0.9 kg/m³) is less dense than surrounding cool air (1.225 kg/m³). F_b = ρ_cool × V × g; W = ρ_hot × V × g + basket weight. Net lift = (ρ_cool − ρ_hot) × V × g.
🐟
Fish Swim Bladders
Bony fish regulate buoyancy using swim bladders — gas-filled internal organs that change volume. Inflating the bladder increases V without changing mass, reducing average density and allowing the fish to rise.
⛏️
Density Measurement
Unknown materials can be weighed in air and in water to find density without measuring volume directly: ρ_obj = m × g/(m×g − W_apparent) × ρ_water. This is the classic Archimedes method.

Common Mistakes to Avoid

⚠️
Using object density instead of fluid density in F_b = ρVg

The ρ in F_b = ρVg is the density of the FLUID (water, air, etc.), not the object. The object's density determines whether it sinks or floats, but the buoyant force depends on the displaced fluid.

⚠️
Using total object volume instead of submerged volume

For partially submerged floating objects, V in F_b = ρVg is only the submerged portion. A floating object displaces fluid equal to only the volume below the waterline.

⚠️
Ignoring buoyancy of air

For precise weight measurements, objects experience a small buoyant force from air (F_b = 1.225 × V × 9.8). For light objects or large volumes, this can be significant. This is why balances in vacuums give slightly different readings.

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Confusing weight and buoyant force

An object sinks if W > F_b; floats if W ≤ F_b. The actual buoyant force when floating equals exactly the object's weight — not the maximum buoyant force if fully submerged.

⚠️
Not accounting for temperature effects on density

Water density peaks at 4°C (1000 kg/m³) and decreases with temperature. At 20°C, ρ_water ≈ 998.2 kg/m³. For high-precision calculations, use the actual fluid density at operating temperature.

Frequently Asked Questions

Why does a steel ship float if steel is denser than water? ▾
A ship is not solid steel — it's mostly hollow space (cargo holds, living quarters, engine rooms). The average density of the entire ship (steel + enclosed air spaces) is less than water. The hull shape displaces enough water so that the upward buoyant force equals the ship's total weight.
What is Archimedes' Principle? ▾
Archimedes (c. 250 BCE) discovered that the buoyant force on an object equals the weight of the fluid displaced. Legend says he realized this while noticing water overflow when he entered a bath — and the insight allowed him to determine if the king's crown was pure gold without damaging it.
Does depth affect buoyant force? ▾
In a uniform fluid (constant density), no — the buoyant force depends only on volume displaced and fluid density, not depth. At greater depths, pressure increases both above and below the object, but the net upward force stays the same. However, for compressible objects (foam, gas-filled cavities), compression at depth reduces volume, reducing F_b.
What is neutral buoyancy? ▾
Neutral buoyancy means the object neither sinks nor floats — it remains at rest at any depth. This requires the object's average density to exactly equal the fluid density. Submarines achieve this by adjusting ballast water. Astronauts train for zero-gravity in neutral buoyancy pools.
How does temperature affect buoyancy? ▾
Fluids become less dense when heated (generally), so buoyant force decreases slightly at higher temperatures (F_b = ρ(T)·V·g). This is why hot air rises — hotter air has lower density, producing less buoyant force per unit volume from surrounding cooler air, creating a net upward lift. Wait — actually warmer air produces a net upward buoyant force when surrounded by denser cooler air.
Why do dead fish float? ▾
When a fish dies, bacteria produce gases inside the body, increasing volume and thus buoyancy. The increased volume with approximately the same weight gives a density less than water, causing the carcass to float to the surface.
What is the metacenter and why does it matter for ships? ▾
The metacenter is the point about which a tilted ship rotates. For stability, the metacenter must be above the center of gravity. When a wave tilts the ship, the buoyant force shifts to the low side, creating a righting moment only if M is above G. This is why ships have ballast and low centers of gravity.
How do submarines control depth? ▾
Submarines have ballast tanks that can be filled with water (increasing weight, causing descent) or blown out with compressed air (decreasing weight, causing ascent). By precisely controlling the amount of water in these tanks, submarines achieve any desired density from less than seawater (floating) to more (diving) to equal (neutral buoyancy at any depth).

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