Buoyancy Calculator
Enter an object's density (or its mass) and volume, pick a fluid, and the calculator tells you whether it floats or sinks, how much of it sits below the surface, and what it weighs underwater. A second mode applies Archimedes' principle directly, F = ρVg, for any unknown. The working is shown, and the fluid and planet menus are one click. Free, no sign-up.
Results
Floats
The object
50.09%
Share of volume below the surface
500.0 kg/m³
Object density ρ
98.07 N
Weight in air W = mg
98.07 N
Buoyant force acting
97.71 N
Hold-down force to keep it fully underwater
97.71 N
Extra load it can carry before sinking
0.01002 m³
Submerged volume
The working
ρ_object / ρ_fluid = 500 / 998.2 = 0.5009
submerged fraction = ρ_object / ρ_fluid = 50.09 %
F_b = W = m g = 10 × 9.80665 = 98.07 N
extra load before sinking = ρ_fluid V g − W = 97.71 N
Assumes a still fluid of uniform density and an object whose density is uniform (a hollow boat floats because its average density is below the fluid's — enter its total mass and the volume it would displace when submerged). Surface tension, air trapped on the object and the buoyancy of the air around it are ignored.
How to Use the Buoyancy Calculator
- 1. Choose the mode. Will it float? takes an object and a fluid and gives the verdict, the submerged fraction and the apparent weight. Buoyant force solves F = ρVg for the force, the displaced volume or the fluid density.
- 2. Describe the object. Pick a material (pine, ice, steel, gold and others) to fill in its density, or type your own. You can describe the object by its mass and volume instead, and the calculator works out the density. Every box has a unit menu, and scientific notation such as 5e-3 works.
- 3. Choose the fluid and the gravity. Fresh water, seawater, mercury, ethanol, glycerol and air are built in. The gravity menu loads the surface gravity of the Moon, Mars and every planet; the verdict does not change with gravity, but the forces do.
- 4. Read the result. You get the verdict, the share of the object below the surface, its weight, the buoyant force that acts, and either its apparent weight when held underwater or the force needed to hold it down.
Archimedes' Principle
Buoyant force
F_b = ρ_fluid × V_displaced × g
Floating object
submerged fraction = ρ_object / ρ_fluid
Apparent weight, fully submerged
W_apparent = m g − ρ_fluid V g
A body in a fluid is pushed upward by a force equal to the weight of the fluid it displaces. The reason is pressure: fluid pressure grows with depth, so the push on the bottom of the object is larger than the push on its top, and the difference is exactly ρ_fluid × V × g. The result does not depend on what the object is made of, only on how much fluid it pushes aside.
Compare the two forces. If the object's density is below the fluid's, the buoyant force from full submersion would exceed its weight, so it rises until it displaces just enough fluid to match its weight: it floats with a fraction ρ_object/ρ_fluid of its volume underwater. If it is denser, the buoyant force falls short, and it sinks; underwater it still weighs less than in air, by exactly the buoyant force. See OpenStax University Physics, section 14.4.
Worked Examples
- A pine block in fresh water (the default). A 20 L block of pine at about 500 kg/m³ has a mass of 10 kg and weighs 98.07 N. It floats with 50.09% of its volume below the surface, and it could carry another 97.71 N, about 10 kg, before sinking.
- An iceberg. Ice at 917 kg/m³ floats in seawater with 89.46% of its volume underwater, and in fresh water with 91.87%. That is the origin of the saying that only about a tenth of an iceberg shows.
- A 1 L block of steel. It weighs 76.98 N in air and displaces 1 L of water, so the buoyant force is 9.789 N. Held underwater it feels as if it weighed 67.19 N: it sinks, but not by as much as you might expect.
- Lead on mercury. Lead (11.34 g/cm³) is denser than water, but mercury (13.546 g/cm³) is denser still, so lead floats on mercury with 83.71% submerged.
- An object of known mass. A 0.5 kg object with a volume of 0.6 L has a density of 833.3 kg/m³ and floats in water; the same object in ethanol (789 kg/m³) would sink.
- A cubic metre of air. Displacing 1 m³ of air (1.225 kg/m³) gives 12.01 N of buoyant force, which is the starting point for how balloons and airships lift.
Why steel ships float
Buoyancy depends on the object's average density, not its material. A ship's hull encloses a great deal of air, so its total mass divided by the volume it would displace when submerged is well below the density of water. To model a ship or a hollow object, enter its whole mass and the full volume it takes up; the calculator then shows how deep it sits.
Assumptions and Limits
- Still fluid, uniform density. Real water gets denser with depth and cooler or saltier water is denser; the presets use representative values near 20 °C.
- Uniform object. The submerged fraction applies to the object's total volume. A lopsided object may tip until its centre of mass and centre of buoyancy line up, which this tool does not model.
- Tiny forces ignored. The buoyancy of the air around the object and surface tension are left out, so a very light object weighed in air differs slightly from its true mass.
- Typical densities. Wood, concrete and steel vary from sample to sample; use a measured density when the answer matters.
Frequently Asked Questions
- What is the formula for buoyant force?
- F_b = ρ_fluid × V × g, where ρ_fluid is the density of the fluid, V is the volume of fluid displaced (the submerged part of the object) and g is the gravitational acceleration. A fully submerged cubic metre in fresh water feels about 9.79 kN of upward force.
- How do you tell whether an object will float or sink?
- Compare densities. If the object's density is less than the fluid's, it floats; if it is greater, it sinks; if they match, it hovers. Divide the mass by the volume to get an object's density, then compare it with 998 kg/m³ for fresh water or about 1025 kg/m³ for seawater.
- How much of a floating object is underwater?
- The fraction ρ_object / ρ_fluid. A 500 kg/m³ pine block floats about half submerged in fresh water. The same block floats slightly higher in seawater, with about 48.8% below the surface, because seawater is denser.
- What is apparent weight?
- It is what a scale reads when the object is supported underwater: the true weight minus the buoyant force, W − ρ_fluid V g. A 7.85 kg block of steel has a weight of 76.98 N but an apparent weight of about 67.19 N in water.
- Does buoyancy depend on gravity?
- The forces do, but the verdict does not. Both weight and buoyant force are proportional to g, so a block that floats on Earth floats on the Moon, with the same fraction submerged, just with smaller forces on each side.
- How did Archimedes use this to test a crown?
- According to the story told by the Roman writer Vitruvius, Archimedes compared the water displaced by the king's crown with that of an equal weight of gold. A crown mixed with a lighter metal would take up more volume and displace more water. Modern accounts doubt the details, but the principle is real.