Buoyancy Calculator | Archimedes
Calculate buoyant force using Archimedes' principle (F = ρVg). Determines float or sink status, fraction submerged, apparent weight, net force, and operating leverage across Earth, Moon, Mars, and Jupiter gravity. Includes fluid and material density presets.
Fluid
SG = 1.000
Displaced Volume *
Volume of fluid pushed aside by the object
Gravitational Acceleration
m/s², choose a planet preset or enter a custom value
Object Properties (optional, enables apparent weight & float/sink analysis)
Enables apparent weight calculation
Enables float/sink analysis & fraction submerged
Material density presets:
Press Enter to calculate · Esc to reset
What Is the Buoyancy Calculator | Archimedes?
Archimedes' principle states that any object wholly or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. This deceptively simple principle, formulated around 250 BCE, underpins everything from ship design to submarine navigation to the float valves in your toilet cistern.
The buoyant force depends on three things: the density of the surrounding fluid, the volume of fluid displaced, and the local gravitational acceleration. Notably, the object's own material, weight, or shape does not affect the buoyant force directly, only the volume it displaces matters.
This calculator computes Fb, shows the force diagram, and, when you provide object density or mass, tells you whether an object floats or sinks, what fraction of it sits below the surface, and what it would weigh on a scale while submerged (apparent weight).
Formula
Archimedes' Principle, Core Formula
Derived Quantities
Float / Sink Rule
Common Fluid Densities
| Fluid | Density (kg/m³) | Specific Gravity |
|---|---|---|
| Fresh water | 1,000 | 1.000 |
| Salt water | 1,025 | 1.025 |
| Ethanol | 789 | 0.789 |
| Gasoline | 700 | 0.700 |
| Vegetable oil | 920 | 0.920 |
| Honey | 1,420 | 1.420 |
| Mercury | 13,534 | 13.534 |
| Air (sea level) | 1.225 | 0.001225 |
Gravity on Other Bodies
| Body | Gravity (m/s²) | Buoyancy effect |
|---|---|---|
| Earth | 9.81 | 100%, baseline |
| Moon | 1.62 | 16.5% of Earth, 6× lighter apparent weight |
| Mars | 3.72 | 38% of Earth, affects subsurface brine exploration |
| Jupiter | 24.79 | 253% of Earth, extreme buoyancy conditions |
How to Use
- 1Select a fluid: Click a preset (Fresh Water, Salt Water, Mercury, etc.) or enter a custom density in kg/m³.
- 2Enter displaced volume: Type the volume of fluid the object pushes aside and choose a unit (m³, cm³, L, or mL). For a fully submerged object, this equals its total volume.
- 3Set gravity: Choose a planet preset (Earth, Moon, Mars, Jupiter) or enter a custom value in m/s². Defaults to Earth gravity (9.81 m/s²).
- 4Add object properties (optional): Enter object mass (kg) to calculate apparent weight and net force. Enter object density (or select a material preset) to see float/sink analysis and fraction submerged.
- 5Calculate: Press Enter or click "Calculate Buoyancy". Read the buoyant force, force diagram, and all derived metrics.
Example Calculation
A solid steel cube with side length 20 cm is fully submerged in fresh water. Does it sink or float, and what does it weigh on an underwater scale?
Why ships don't sink, Archimedes applied
Understanding Buoyancy | Archimedes
Archimedes' Principle, Two Millennia of Engineering
According to legend, Archimedes of Syracuse leapt from his bath crying "Eureka!" after realising that he could measure the crown's volume by the water it displaced, and therefore detect whether the goldsmith had fraudulently substituted silver. Whether or not the bath story is literally true, the insight it represents is real and profound: a submerged object displaces its own volume of fluid, and the fluid pushes back with a force equal to the displaced fluid's weight.
This principle, formalised as Fb = ρVg, has not changed in 2,200 years. It explains why aircraft carriers weighing 100,000 tonnes float, why submarines can hover at any depth, why a helium balloon rises, and why measuring body fat by water displacement is more accurate than skinfold calipers.
What Determines the Buoyant Force?
- ›Fluid density (ρ): denser fluids produce larger upthrust. It is much easier to float in the Dead Sea (salinity ≈ 34%) than in a fresh-water lake, the buoyant force is roughly 24% greater.
- ›Displaced volume (V): the more fluid an object pushes aside, the greater the upward force. A hollow ball displaces more water than a solid ball of the same mass, which is why shape matters for floating.
- ›Gravitational acceleration (g): on the Moon (g = 1.62 m/s²), the buoyant force is only 16.5% of its Earth value, but so is the object's weight, so the float/sink outcome is identical. What changes is the actual force magnitudes.
What does NOT affect buoyant force
Why Objects Float or Sink, The Density Rule
Whether an object floats or sinks depends purely on how its average density compares to the fluid's density:
- ›ρ_obj < ρ_fluid → Floats: the maximum possible buoyant force (fully submerged) exceeds the object's weight, so the object bobs up until equilibrium is reached at a partial submersion where Fb = W.
- ›ρ_obj = ρ_fluid → Neutral buoyancy: the object hovers at any depth without rising or sinking. This is what divers call "being neutrally buoyant" and what submarines achieve by flooding ballast tanks to a precise level.
- ›ρ_obj > ρ_fluid → Sinks: even fully submerged, Fb cannot overcome gravity. The net downward force continues to accelerate the object until it hits the bottom.
Typical material densities and their fate in water:
| Material | Density (kg/m³) | Buoyancy behaviour in common fluids |
|---|---|---|
| Styrofoam | 50 | Floats in almost any liquid |
| Pine wood | 500 | Floats in water (SG = 0.5) |
| Ice | 917 | Floats in water (SG = 0.917) |
| Aluminum | 2,700 | Sinks in water, floats in mercury |
| Concrete | 2,300 | Sinks in water |
| Steel | 7,850 | Sinks in water, floats in mercury |
| Gold | 19,300 | Sinks in water and mercury |
Fraction Submerged, How Deep Does a Floating Object Sit?
When an object floats, it sinks just deep enough so that the buoyant force on the submerged portion equals the object's full weight. The equilibrium condition gives:
This simple ratio is all you need. Ice has a density of 917 kg/m³ in fresh water (density 1000 kg/m³), so the fraction submerged = 917/1000 = 91.7%, meaning only 8.3% of an iceberg is visible above the surface. This is where the phrase "tip of the iceberg" originates.
Apparent Weight, What Objects Weigh Underwater
When you weigh an object while it is submerged (e.g. on an underwater scale, or by hanging it on a spring balance into a water tank), the scale reads the apparent weight:
- ›The apparent weight is always less than the true weight in any fluid denser than a vacuum.
- ›Hydrostatic weighing, measuring apparent weight in water, is used to calculate body fat percentage with high accuracy (error ±1–3%).
- ›Dentists and metallurgists use underwater weighing to detect counterfeit materials, since each material has a unique apparent weight signature.
- ›If Fb > W, the "apparent weight" is negative, meaning the object would need to be held down rather than held up.
Real-World Applications of Archimedes' Principle
| Application | How buoyancy is used | Key principle |
|---|---|---|
| Ship design | Hull volume displaces enough water to equal ship weight | Archimedes' principle |
| Submarine control | Variable ballast tanks flood or empty to control depth | F_b = ρVg adjustment |
| Hot air balloon | Hot air (lower density) creates buoyancy in cooler air | ρ_gas × V_balloon × g |
| Fish swim bladder | Gas-filled organ adjusts fish depth without swimming | Neutral buoyancy |
| Density measurement | Object's weight loss in fluid reveals its density | Hydrostatic weighing |
| Life jacket design | Low-density foam provides excess buoyancy over body weight | Safety margin factor |
| Oil/gas exploration | Drilling mud density tuned to balance formation pressure | Fluid column pressure |
Frequently Asked Questions
What is Archimedes' principle?
- ›Archimedes' principle states that any object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces.
- ›This force is often called upthrust.
- ›The principle applies to any fluid, liquids and gases alike.
- ›It does not matter whether the object floats or sinks, the buoyant force is always present.
- ›Mathematically: F_b = ρ_fluid × V_displaced × g
What is the buoyant force formula?
- ›F_b = ρ_fluid × V_displaced × g
- ›ρ_fluid is the fluid density in kg/m³ (e.g. fresh water = 1000, salt water ≈ 1025)
- ›V_displaced is the volume of fluid pushed aside, in m³
- ›g is gravitational acceleration in m/s² (Earth = 9.81)
- ›The result F_b is in Newtons (N)
- ›Equivalently: buoyant force = weight of displaced fluid (mass = ρ × V, weight = mass × g)
Why do ships float even though they are made of steel?
- ›Steel has a density of about 7,850 kg/m³, nearly 8 times denser than water. A solid steel block sinks instantly.
- ›A ship, however, is not a solid steel block, it is a hollow shell containing a huge volume of air.
- ›The ship's average density (steel hull + interior air + cargo spaces) is far less than water.
- ›The large hull volume displaces an enormous weight of water, generating enough buoyant force to support the entire ship.
- ›This is why a ship can carry thousands of tonnes of cargo, the volume of displaced water increases as the ship sits lower, providing more buoyant force.
What is the difference between buoyant force and apparent weight?
- ›Buoyant force is the upward force exerted by the fluid on the submerged object.
- ›Apparent weight is what you would read if you weighed the object while it was submerged.
- ›Apparent weight = True weight − Buoyant force
- ›For example: a 10 kg rock in air weighs 98.1 N. If the buoyant force is 20 N, the apparent weight is 78.1 N.
- ›If the buoyant force exceeds the true weight (e.g. a balloon), the apparent weight becomes negative, meaning the object must be held down.
Does the depth of submersion affect the buoyant force?
- ›For incompressible fluids and rigid objects, depth does not change buoyant force.
- ›The buoyant force depends only on the volume of fluid displaced and the fluid density, not the depth.
- ›A 1 L object displaces 1 L at 1 m depth and still displaces 1 L at 100 m depth, giving the same F_b.
- ›However, in very deep water, compressible objects (like a human body or foam) can be compressed, reducing their volume and thus their buoyancy.
- ›Also, very deep water is very slightly denser due to pressure, minutely increasing buoyancy, but for practical calculations this is negligible.
How does a submarine control its depth?
- ›Submarines have ballast tanks, large chambers that can be filled with seawater or emptied with compressed air.
- ›To dive: seawater floods the ballast tanks, increasing the submarine's average density above that of seawater → net downward force → sinks.
- ›To surface: compressed air blows water out of the tanks, decreasing average density below seawater → net upward force → rises.
- ›To hover at a specific depth: the tanks are adjusted until average density equals seawater density → neutral buoyancy.
- ›Trim tanks (smaller tanks) provide fine depth control.
What is specific gravity and how does it relate to buoyancy?
- ›Specific gravity (SG) = density of a substance ÷ density of fresh water (1000 kg/m³).
- ›SG is dimensionless, it tells you how many times denser a material is than water.
- ›If SG < 1: the material floats in water (e.g. ice at SG 0.917, wood at SG 0.5–0.9).
- ›If SG = 1: the material is neutrally buoyant in water.
- ›If SG > 1: the material sinks in water (e.g. steel at SG 7.85, gold at SG 19.3).
- ›Salt water has SG ≈ 1.025, so materials with SG between 1.000 and 1.025 float in salt water but sink in fresh water.
How does buoyancy work in gases (balloons, blimps)?
- ›Buoyancy in gases follows exactly the same formula: F_b = ρ_air × V_balloon × g.
- ›Air at sea level has a density of about 1.225 kg/m³.
- ›A helium balloon rises because helium is much less dense than air (0.164 kg/m³), so the balloon's average density is below air's.
- ›A hot air balloon works because heating air reduces its density, the hot air inside is lighter than the surrounding cool air.
- ›A blimp uses a very large gas volume to displace enough air to lift its structure and payload.
- ›The principle is identical to a ship in water, just with a much less dense "fluid" (air) and correspondingly much larger volume requirements.