Buoyancy Calculator
Calculate the buoyant force on an object submerged in a fluid using Archimedes' principle. The buoyant force equals the weight of displaced fluid: F_b = ρVg.
Buoyancy is the upward force that fluids exert on objects placed in them. It is the reason ships float, hot-air balloons rise, fish hover at depth, and helium balloons drift toward the ceiling. The principle was discovered by Archimedes in the 3rd century BC during his famous "Eureka!" bathtub moment: an object immersed in fluid experiences an upward force equal to the weight of fluid it displaces.
The formula is elegantly simple: F_b = ρVg, where ρ is the fluid density, V is the volume of fluid displaced, and g is gravitational acceleration. Whether the object sinks, floats, or hovers depends on how the buoyant force compares to the object's weight. If buoyancy exceeds weight, the object rises; if weight wins, it sinks; if they balance, the object floats at a specific depth.
For floating objects, only enough volume sinks to displace fluid equal to the object's weight — a 100-ton steel ship floats by displacing 100 tons of water with its hull. For fully submerged objects (like submarines or rocks), the volume of displaced fluid equals the object's full volume, and net force depends on density comparison.
Common applications: naval architecture (ship displacement and stability), hot-air ballooning, submarine ballast systems, hydrometer measurements, density determination via water displacement, and any introductory fluid mechanics problem.
Inputs
Water = 1000, Seawater = 1025
Results
Buoyant Force
98.1 N
Net Force
49.05 N
Floats?
Yes
Buoyancy Results
| Parameter | Value |
|---|---|
| Fluid Density | 1000 kg/m³ |
| Displaced Volume | 0.01 m³ (10.0000 L) |
| Object Mass | 5 kg |
| Object Density | 500.00 kg/m³ |
| Buoyant Force | 98.1 N |
| Object Weight | 49.05 N |
| Net Force | 49.05 N |
| Displaced Fluid Mass | 10.0000 kg |
| Will Float? | Yes (buoyancy ≥ weight) |
| Formula | F_b = ρVg |
Formula
How to use this calculator
- Enter fluid density (water = 1000 kg/m³, seawater = 1025, air = 1.225).
- Enter displaced volume (for fully submerged, this is object volume).
- Enter object mass to compare buoyant force to weight.
- Standard gravity is 9.81 m/s² (Earth surface).
- Calculator returns buoyant force and tells if object floats or sinks.
- For floating objects, use displaced volume = object_mass / fluid_density.
Worked examples
Floating wood block
**Scenario:** A 0.5 m × 0.3 m × 0.1 m wood block (volume 0.015 m³) with mass 7.5 kg in water. Floats or sinks? **Calculation:** Density of block = 7.5 / 0.015 = 500 kg/m³ (less than water at 1000). Block floats. Fraction submerged: 500/1000 = 50%. So 50% of the block (0.0075 m³) sits below water; buoyant force = 1000 × 0.0075 × 9.81 = 73.6 N, equal to the block's weight (7.5 × 9.81). **Result:** Block floats with half its volume submerged. Adding weight on top sinks it further until enough volume is displaced to balance the new total weight.
Steel ship hull
**Scenario:** A 100,000-ton (10⁸ kg) cargo ship in seawater. How much water must it displace? **Calculation:** F_b must equal weight: ρVg = mg → V = m / ρ = 10⁸ / 1025 ≈ 97,560 m³. **Result:** Ship hull must displace nearly 100,000 m³ of seawater — about 25 Olympic pools. Steel is 7,850 kg/m³ (denser than water), but the hull's shape encloses enough air that average density is below water's. This is why ship design focuses on hull shape: a steel cube would sink.
Helium balloon lift
**Scenario:** A 1 m³ helium balloon at sea level. Lift force? **Calculation:** Lift = (ρ_air − ρ_He) × V × g = (1.225 − 0.179) × 1 × 9.81 = 10.26 N (≈ 1.05 kg lifting force). **Result:** Balloon lifts about 1 kg per cubic meter. A standard party balloon (~0.005 m³) lifts only ~5 g — slightly more than the balloon's own mass, so it just barely floats. Lifting a small toy needs many balloons or a much larger one.
When to use this calculator
**Use buoyancy calculations for:**
- **Naval architecture**: ship displacement, draft, freeboard. - **Hot-air ballooning**: lift force from temperature difference. - **Helium/hydrogen balloons**: party balloons through weather balloons. - **Submarines**: ballast tank calculations. - **Hydrometers**: floating depth measures liquid density. - **Iceberg melt physics**: only 9% above water for fresh ice in seawater. - **Density determination**: weigh object in air vs water (Archimedes method). - **Engineering**: offshore platforms, oil rigs.
**Floating vs submerged:**
- **Fully submerged**: V = full object volume; net force depends on density comparison. - **Partially submerged (floating)**: only enough volume sinks to displace equal weight.
**Stability (not just buoyancy):**
A floating object can be balanced or unstable. Ship stability depends on: - **Center of gravity**: lower is more stable. - **Metacenter**: must be above center of gravity for stable righting. - **Hull shape**: wide hulls more stable than narrow.
A boat with high center of gravity (poorly loaded cargo) can capsize even though it's positively buoyant.
**Submarine ballast:**
Submarines control buoyancy by flooding/emptying ballast tanks. Adding water increases mass → net negative buoyancy → dive. Blowing tanks with air → positive buoyancy → surface. Hovering requires exact balance ("neutral buoyancy").
**Hot air balloon physics:**
Heating air reduces its density: ρ_hot = ρ_cold × T_cold / T_hot (ideal gas, constant pressure). Example: 300 K to 380 K air → density ratio 300/380 = 0.79. Each m³ of hot air "weighs" 0.79 × 1.225 = 0.97 kg, vs 1.225 kg cold. Net lift: 0.26 N/m³.
**Iceberg "tip of the iceberg":**
Pure ice: ρ = 917 kg/m³. Fresh water: 1,000. Submerged fraction = 0.917. In seawater (1,025): submerged fraction = 0.895 — about 89.5% under water. The famous "10% above" figure is approximately correct.
**Apparent weight in fluid:**
Diver in seawater: weighs less than on land. A 70 kg person volume ~70 L = 0.07 m³: F_b = 1,025 × 0.07 × 9.81 = 704 N (apparent weight = 70 × 9.81 − 704 = −17 N → slightly buoyant). Most people float in seawater, slightly sink in fresh water.
**Common applications:**
- **Civil engineering**: pile design with groundwater uplift. - **Aerospace**: lighter-than-air vehicles (airships). - **Oceanography**: instrument design (drifters, sondes). - **Recreation**: scuba diving (buoyancy compensator BCDs). - **Industrial**: oil-water separators, floatation cells in mining.
**Pitfalls:**
- **Confusing volume vs mass**: F_b depends on displaced volume × fluid density, not object mass directly. - **Forgetting fluid density variation**: warm water less dense than cold; salt water denser than fresh. - **Ignoring partial submersion**: a floating object's displaced volume ≠ total volume. - **Stability vs buoyancy**: positive buoyancy doesn't guarantee stability (boats can capsize). - **Air buoyancy in weighing**: precision scales need air buoyancy correction (~0.1% effect).
Common mistakes to avoid
- Confusing object volume with displaced volume for floating objects.
- Forgetting that fluid density varies with temperature and salinity.
- Mixing units (grams + m³ instead of kg + m³).
- Ignoring buoyancy of air for precise weight measurements.
- Assuming a positively buoyant object is stable (it might not be).
- Forgetting to account for compressibility of gas at depth.
- Calculating buoyancy with object density instead of fluid density.
- Treating buoyant force as a property of the object rather than the displaced fluid.