Physics
How To Calculate Buoyancy
Buoyancy is the upward push a fluid gives anything placed in it. It depends on how much fluid is pushed aside, not on how heavy the object is, which is why a steel ship floats and a steel nail does not.
Quick Answer
buoyant force = fluid density x displaced volume x gravity
- F
- Buoyant force pushing upwards, in newtons
- rho
- Density of the fluid, about 1000 kilograms per cubic metre for water
- V
- Volume of fluid displaced, in cubic metres
- g
- Acceleration due to gravity, about 9.81 metres per second squared
The upward force equals the weight of the fluid the object pushes out of the way. Multiply the fluid's density by the submerged volume and by gravity. Fifty litres of water displaced gives about four hundred and ninety newtons of lift, enough to hold up thirty kilograms.
What Is Buoyancy?
Buoyancy is the upward force a fluid exerts on any object placed in it. It exists because pressure increases with depth, so the bottom of a submerged object feels a greater push than the top, and the difference between those two pushes is a net force directed upwards.
Archimedes stated the result over two thousand years ago: the buoyant force equals the weight of the fluid displaced by the object. That single sentence contains the whole calculation. You do not need to know what the object is made of, only how much fluid it pushes aside.
That is why a steel ship floats while a steel nail sinks. Both are made of the same material, but the ship's hull is a shape that displaces an enormous volume of water for its weight, while the nail displaces almost nothing. Density of the object as a whole, not of the material, is what decides the outcome.
The formula is fluid density times displaced volume times gravity. Water has a density of about one thousand kilograms per cubic metre, so displacing fifty litres, which is 0.05 cubic metres, gives a buoyant force of about four hundred and ninety newtons.
Comparing the buoyant force with the object's weight tells you everything. If the buoyant force exceeds the weight, there is a net upward force and the object rises. If they are equal, the object hovers in equilibrium. If the weight is larger, it sinks, and the net force downwards is the difference.
A floating object is a special case of that comparison. It settles at exactly the depth where the submerged volume displaces water weighing the same as the whole object. This is why you can read a ship's cargo from the depth of its Plimsoll line, and why a hydrometer floats higher in a denser liquid.
The ratio of the object's weight to the maximum buoyant force is the fraction that ends up submerged. An object whose weight is sixty per cent of the buoyant force available at full submersion will float with sixty per cent of its volume under the surface. Ice, at about ninety two per cent of the density of water, floats with almost all of itself submerged.
Density of the fluid matters as much as volume. Sea water, at roughly 1025 kilograms per cubic metre, is denser than fresh water and provides about two and a half per cent more lift. That small difference is enough to change how deeply a ship sits in the water, which is why load lines are marked differently for salt and fresh water.
Temperature and salinity both change fluid density, and therefore buoyancy. Warm water is slightly less dense than cold, which is part of why a swimmer feels more supported in the sea than in a cool lake, and why a hot air balloon rises through the cooler air around it.
Buoyancy applies to gases as well as liquids. The same formula works with the density of air, which is why a helium balloon rises and why the buoyant force on a person is about a newton, far too small to notice. The principle is identical; only the density changes.
It is worth being precise about which volume to use. The formula needs the submerged volume, not the total volume. For an object floating on the surface, only the part below the waterline counts, which is exactly why the submerged fraction and the density ratio are the same number.
Formula
F = fluid density x volume x g
The buoyant force equals the weight of the fluid pushed aside, so only the displaced volume and the fluid's density matter.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| F | Buoyant force | N | Upward force from the fluid. |
| rho | Fluid density | kg/m3 | Mass per cubic metre of the fluid. |
| V | Displaced volume | m3 | Volume of fluid pushed out of the way. |
| g | Gravity | m/s2 | Acceleration due to gravity, about 9.81. |
W = mass x g
Compare this with the buoyant force to decide whether the object rises, floats or sinks.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| W | Weight | N | Downward force due to gravity. |
| m | Mass | kg | Mass of the object in kilograms. |
| g | Gravity | m/s2 | Acceleration due to gravity, about 9.81. |
submerged fraction = object weight / buoyant force
When the object floats, this is the proportion of its volume sitting below the surface.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| f | Submerged fraction | fraction | Proportion of the volume below the waterline. |
| W | Weight | N | Downward force due to gravity. |
| F | Buoyant force | N | Upward force available at full submersion. |
How To Calculate Buoyancy
- 1
Find the submerged volume
In cubic metres. Fifty litres is 0.05 cubic metres, and for a floating object only the part below the surface counts.
- 2
Look up the fluid density
Fresh water is about 1000 kilograms per cubic metre, sea water about 1025, and air about 1.2. The fluid, not the object, sets this number.
- 3
Multiply density by volume by gravity
One thousand times 0.05 times 9.81 gives about four hundred and ninety newtons of upward force.
- 4
Work out the object's weight
Mass times gravity. A thirty kilogram object weighs about two hundred and ninety four newtons, which is less than the lift available, so it floats.
- 5
Compare the two to get the net force
Four hundred and ninety minus two hundred and ninety four leaves about one hundred and ninety six newtons upwards, and the object floats with sixty per cent of its volume submerged.
Examples
Example 1: A thirty kilogram block displacing fifty litres
- Displaced volume
- 0.05 m3
- Fluid density
- 1000 kg/m3
- Object mass
- 30 kg
- Gravity
- 9.81 m/s2
| Step | Calculation | Result |
|---|---|---|
| Buoyant force | 1000 x 0.05 x 9.81 | 490.5 |
| Object weight | 30 x 9.81 | 294.3 |
| Net upward force | 490.5 - 294.3 | 196.2 |
| Submerged fraction | 294.3 / 490.5 x 100 | 60% |
Result: Fifty litres of water gives 490.5 newtons of lift against a weight of 294.3, so the block floats with 60 per cent of its volume submerged and 196.2 newtons of net upward force.
Example 2: A lighter object in sea water
- Displaced volume
- 0.02 m3
- Fluid density
- 1025 kg/m3
- Object mass
- 15 kg
- Gravity
- 9.81 m/s2
| Step | Calculation | Result |
|---|---|---|
| Buoyant force | 1025 x 0.02 x 9.81 | 201.105 |
| Object weight | 15 x 9.81 | 147.15 |
| Net upward force | 201.105 - 147.15 | 53.955 |
| Submerged fraction | 147.15 / 201.105 x 100 | 73.17% |
Result: In denser sea water the same twenty litres gives 201.105 newtons of lift, so a fifteen kilogram object floats with 73.17 per cent submerged and only 53.955 newtons of net lift left.
Example 3: An object that barely floats
- Displaced volume
- 0.03 m3
- Fluid density
- 1000 kg/m3
- Object mass
- 29 kg
- Gravity
- 9.81 m/s2
| Step | Calculation | Result |
|---|---|---|
| Buoyant force | 1000 x 0.03 x 9.81 | 294.3 |
| Object weight | 29 x 9.81 | 284.49 |
| Net upward force | 294.3 - 284.49 | 9.81 |
| Submerged fraction | 284.49 / 294.3 x 100 | 96.67% |
Result: A twenty nine kilogram object displacing thirty litres is only just lighter than the water it moves, so it floats with 96.67 per cent of its volume under the surface and just 9.81 newtons of spare lift.
Calculator
Buoyant force
490.5
- Object weight
- 294.3
- Net upward force
- 196.2
- Volume submerged when floating
- 60.00%
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the Buoyancy calculator page.
Common Mistakes
Using the object's volume instead of the displaced volume
For a floating object only the submerged part displaces fluid. Using the whole volume overstates the buoyant force and predicts that objects float far higher than they actually do.
Assuming the object's material density decides whether it floats
It is the average density of the whole object that matters. A steel ship floats because its hull encloses a large volume of air, bringing the average density below that of water.
Forgetting that gravity appears in both terms
Both the buoyant force and the weight are multiplied by g, so it cancels when you compare them. It does not cancel when you want the forces in newtons, which is where leaving it out causes trouble.
Treating fresh and sea water as identical
Sea water is about two and a half per cent denser, which is enough to change how deeply a vessel sits and to alter the load a ship may legally carry. Using a single density for both is a small but real error.
Ignoring the container in a laboratory measurement
A submerged object also raises the water level and pushes down on the balance. Careful measurements account for the apparent weight loss rather than weighing the object in air and in water separately without correction.
Assuming buoyancy only applies to liquids
Gases exert buoyant force too, using the same formula with the gas density. Hot air balloons and helium balloons work on exactly this principle, and the effect on a person in air is real but tiny.
FAQ
What is Archimedes principle in simple terms?
The upward force on an object in a fluid equals the weight of the fluid it pushes out of the way. A body that displaces ten kilograms of water feels an upward force equal to the weight of those ten kilograms.
Why do some objects float and others sink?
It depends on whether the buoyant force can match the object's weight. If the object's average density is less than the fluid's, the water it displaces at full submersion weighs more than the object, and it floats.
Does the shape of an object affect buoyancy?
Not the force itself, which depends only on displaced volume, but shape decides how much volume is displaced for a given weight. That is exactly why a hull shape can make a steel vessel float.
What is the buoyant force on a person in water?
About nine hundred and eighty newtons for someone displacing a hundred litres, which is close to a hundred kilograms of lift. Most people float because their average density is only slightly above that of water, and lungs full of air tip the balance.
How does sea water change buoyancy?
It is denser, roughly 1025 kilograms per cubic metre against 1000 for fresh water, so it provides about two and a half per cent more lift for the same displaced volume. Ships ride slightly higher in the sea than in a river.
Does buoyancy depend on depth?
Not on the buoyant force, which depends only on displaced volume and fluid density. Pressure rises with depth, but it rises on all sides equally, so the net upward force stays the same as long as the fluid's density does.
References
- [1]Wikipedia, Buoyancy — https://en.wikipedia.org/wiki/Buoyancy
- [2]Wikipedia, Archimedes principle — https://en.wikipedia.org/wiki/Archimedes%27_principle
- [3]Wikipedia, Density — https://en.wikipedia.org/wiki/Density
- [4]Wikipedia, Displacement — https://en.wikipedia.org/wiki/Displacement_(fluid)