What is the formula for lift?

The buoyancy is given by the formula: F_B = ρ*V*g. Here, ρ is the density of the fluid, V is the volume of displaced fluid and g is the acceleration due to gravity (on Earth about 9.81 m/s²). Since ρ * V represents the mass of the displaced fluid, the buoyancy force also corresponds to the weight of this fluid.

How do you determine whether an object will float or sink?

Density is compared. If the average density of an object is less than that of the liquid it will float; if greater, it will sink; and if equal, it will remain suspended at any depth. A floating object sinks until the weight of displaced fluid equals its own weight.

Why do objects appear lighter in water?

Since the buoyancy force is an upward force on the object, the scale will read a value that is equal to the actual weight minus the buoyancy. This apparent weight is what you feel when lifting a rock in a swimming pool. If the buoyancy is greater than the actual weight, then the apparent weight becomes negative and the object rises up.

How can you use Archimedes' principle to determine an object's density?

First weigh the object in air and then fully immersed in a liquid of known density while suspended. The reduction in weight corresponds to the buoyancy, from which you can calculate the displaced volume. If you divide the mass of the object by this volume, you get the density: ρ_Object = ρ_Liquid * m_Air / (m_Air - m_Liquid).

Archimedes' Principle Calculator

Buoyancy inputs

Fluid density

Displaced volume

Gravitational acceleration

Buoyant force

Enter any three of the four boxes above and the fourth is solved in place, along with the results panel.

Object

Analyse a specific object

Add the object's mass to check whether it floats or sinks, and how much it weighs when submerged.

Please input values into the fields

Charts

Show the buoyancy charts

Plot the buoyant force against displaced volume, and weigh the buoyant force against the object's weight.

Archimedes' principle states that a body immersed in a fluid experiences an upward buoyant force equal to the weight of the displaced fluid. This simple concept explains why a steel ship floats while a steel nail sinks, and allows us to calculate the buoyancy, density or volume of displacement given two known values.

If you enter three of the four values - liquid density, displaced volume, gravity or buoyancy - this tool will calculate the missing value. By opening the object panel and adding a mass, you can check whether an object floats or sinks, how heavy it is when fully submerged, and how much of a floating object is above water.

The importance of Archimedes' principle.

When an object is immersed in a liquid it displaces some of the liquid. The liquid exerts a counter force which acts upwards and is called buoyancy. The magnitude of this force exactly equals to weight of the displaced liquid. It does not matter what the object is made of, only how much liquid it displaces and how heavy that liquid is.

This principle is named after Archimedes because of the following story: It is said that when he stepped into a full bath tub, water spilled over the sides onto the floor. As he watched the water overflow, he discovered that the volume of irregularly shaped objects can be measured by the amount of water they displace and from that determined the object's density.

Formula

The buoyancy acting on a submerged object is the product of the density of the fluid and the displaced volume, multiplied by the acceleration due to gravity.

F_B = \rho \cdot V \cdot g

where F_B is the buoyancy force, rho (Greek letter) is the density of the fluid, V is the volume displaced by the object and g is the acceleration due to gravity which on Earth's surface is approximately 9.81 meters per second squared. Since the product of density and volume gives mass, it can be deduced from the same equation that buoyancy is equal to the weight of the displaced fluid.

F_B = m_{fluid} \cdot g \qquad m_{fluid} = \rho \cdot V

By rearranging the equation missing values can be calculated. If an input field is left blank this calculator will perform that calculation.

\rho = \frac{F_B}{V \cdot g} \qquad V = \frac{F_B}{\rho \cdot g}

Example calculation:

Assume a rock displaces one liter of water. The density of water is 1000 kilograms per cubic meter and one liter equals 0.001 cubic meters so:

F_B = 1000 \cdot 0.001 \cdot 9.81 = 9.81 \ \text{N}

The mass of the displaced water is 1 kilogram, which weighs 9.81 Newtons. This is the buoyancy itself. If the rock weighs 25 Newtons in air, then it will weigh about 15.2 Newtons in water, calculated as the difference between 25 and 9.81. Although the weight of the rock is less in water, it still sinks because its weight is greater than the buoyancy.

Will an object float or sink?

One compares the average density of the object to that of the liquid. If the density of the object is less than that of the liquid it will float; if it's greater, it will sink; and if they are equal, it will hover without rising or sinking. This explains why a beach ball floats on top of water while a marble sinks.

A floating object attains its stable position when the weight of the displaced liquid is equal to the weight of the object itself. The proportion of the object below the surface is exactly equal to the ratio of the two densities.

\text{fraction submerged} = \frac{\rho_{object}}{\rho_{fluid}}

The density of ice is about 917 kilograms per cubic meter, while seawater has a density of about 1025 kg/m3. So for an iceberg that floats, about 89% of its total volume is hidden below the surface of the water, which is equivalent to dividing 917 by 1025. The expression "only the tip of the iceberg" derives from this fact.

Useful density values:

Density is a value that you will often need to check. Common density values are given in kilograms per cubic meter and it's useful to have them handy.

Fluid	Density (kg/m3)	Solid	Density (kg/m3)
Fresh water	1000	Cork	240
Seawater	1025	Ice	917
Vegetable oil	920	Oak wood	750
Ethanol	789	Aluminium	2700
Petrol	750	Iron	7870
Mercury	13534	Lead	11340

Density determination by weight measurement

Archimedes' principle can also be used as a measuring instrument. First, an object is weighed in air and then completely immersed in a liquid of known density. The apparent weight loss corresponds to the buoyancy force, from which the displaced volume can be calculated. By dividing the mass of the object by this volume, one obtains the density of the object.

\rho_{object} = \rho_{fluid} \cdot \frac{m_{air}}{m_{air} - m_{fluid}}

Suppose a sample of metal has a mass in air of 200 grams and an apparent mass in water of 175 grams. The mass has been reduced by 25 grams, which means that 25 grams of water have been displaced, or 25 cubic centimeters. The density is about 8 grams per cubic centimeter, calculated by dividing 200 by 25, and this is close to the value for brass. This was the method used by jewellers to test whether a crown was made from pure gold.

Fields of application

The construction of ships and submarines is based on this principle. The shape of the hull ensures that enough water is displaced to carry the load. Hydrometers use this principle to determine the density of a liquid by measuring the depth of immersion of a float. Hot air balloons and airships also work on this principle, rising because they displace the surrounding denser air with hot or light gases.

This calculator is for learning and quick estimation purposes only. The buoyancy calculation assumes the object is fully submerged and treats the object as homogeneous. In actual measurements temperature, surface tension and entrapped air must be considered.

Frequently asked questions

What is Archimedes' principle?: The Archimedes' principle states that a body immersed in a fluid experiences an upward buoyant force equal to the weight of the displaced fluid. This principle applies to any object in any liquid, whether it floats or sinks.
What is the formula for lift?: The buoyancy is given by the formula: F_B = ρ*V*g. Here, ρ is the density of the fluid, V is the volume of displaced fluid and g is the acceleration due to gravity (on Earth about 9.81 m/s²). Since ρ * V represents the mass of the displaced fluid, the buoyancy force also corresponds to the weight of this fluid.
How do you determine whether an object will float or sink?: Density is compared. If the average density of an object is less than that of the liquid it will float; if greater, it will sink; and if equal, it will remain suspended at any depth. A floating object sinks until the weight of displaced fluid equals its own weight.
Why do objects appear lighter in water?: Since the buoyancy force is an upward force on the object, the scale will read a value that is equal to the actual weight minus the buoyancy. This apparent weight is what you feel when lifting a rock in a swimming pool. If the buoyancy is greater than the actual weight, then the apparent weight becomes negative and the object rises up.
How can you use Archimedes' principle to determine an object's density?: First weigh the object in air and then fully immersed in a liquid of known density while suspended. The reduction in weight corresponds to the buoyancy, from which you can calculate the displaced volume. If you divide the mass of the object by this volume, you get the density: ρ_Object = ρ_Liquid * m_Air / (m_Air - m_Liquid).

Related calculators

Buoyant Force CalculatorCalculate the buoyant force on a submerged or floating object with Archimedes' principle (F = rho V g). Solve for density, volume, or gravity, and check whether an object floats or sinks.

Density CalculatorFind density, mass, or volume from the other two with rho = m / V. Enter any two values and the calculator solves for the third, with specific gravity, a float or sink read, and unit conversion built in.

Buoyancy CalculatorWork out whether something floats and by how much. Enter an object's volume and weight to get its net buoyancy, average density, submerged fraction and spare load capacity, size a helium, hydrogen or hot air balloon from the lift it has to produce, or turn hull dimensions into displacement and deadweight.

Bump Force CalculatorCalculate the bump force on a vehicle's suspension from spring rate, displacement, damping coefficient, and vertical velocity. Solve for any variable and see the spring-vs-damper split.

Axial Loading CalculatorCalculate axial stress, strain, and deformation of a member under tension or compression, with a material preset, a factor-of-safety check, and reverse solving.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Archimedes' principle (Encyclopaedia Britannica)
Definition, history, and statement of the buoyancy law.
Buoyancy (Wikipedia)
Buoyant force, the displaced-fluid derivation, and floating equilibrium.
NIST: standard acceleration of gravity
The standard value of g used to convert mass to weight (9.80665 m/s2).