Buoyant force:
the upward push of a fluid on a submerged object

What Archimedes' principle says

Lower a bucket into a pool and you can feel it getting lighter. The water is pushing up on it. Archimedes' principle states that the push equals the weight of the fluid that the object has shoved out of the way.

This sounds odd at first, because the force has nothing to do with the object's own weight or material. It depends on how much fluid is displaced and on how heavy that fluid is. A steel ship floats because its hull displaces a huge volume of water, and a steel nail sinks because it displaces very little.

The origin of the force is pressure. Fluid pressure is greater on the lower surface of a submerged body than on the upper one, and that difference over the body's size adds up to the upthrust.

The upthrust formula

The object's density does not appear. It matters only when you compare upthrust with the object's weight, which is the sink-or-float test further down.

Volume must be in cubic metres. A litre is 0.001 m³, so a 20-litre container is 0.02 m³. Forgetting that conversion produces answers that are wrong by a factor of a thousand.

Worked example: a 20-litre object under water

A sealed 20-litre drum is held completely under fresh water. How hard does the water push it upward?

The drum displaces 20 kg of fresh water, and 20 kg weighs 196.2 N, so the numbers agree with the principle. Whenever the fluid is water, the upthrust in newtons is about 9.81 times the displaced volume in litres.

To know whether the drum rises, compare 196.2 N with its own weight. If it has a mass of 5 kg, its weight is 5 × 9.81 = 49.05 N, so the net force is 196.2 − 49.05 = 147.15 N upward and it shoots to the surface.

Float, sink or hover: the comparison

A shortcut for solid objects: if the average density is less than the fluid's density, the object floats. Ice at about 917 kg/m³ floats in water at 1000 kg/m³, and roughly 92 percent of it sits below the surface.

For a floating body the upthrust equals its weight, not the full-submersion value. That balance sets how deep it sits: the displaced volume simply adjusts until the weight of displaced fluid matches the object's weight.

Where buoyancy sums go wrong

• Using the object's density instead of the fluid's in the formula. The fluid's density is the one that counts.
• Using total volume for a floating object. Only the part below the surface counts.
• Working in litres or cm³ without converting to m³.
• Believing that heavier objects always sink. A 100,000-tonne tanker floats because its average density, hull and air included, is lower than water's.
• Ignoring the fluid. A dense object may float in mercury and sink in water.

How deep a floating body sits

A floating object settles until the water it displaces weighs exactly as much as the object. That gives a neat shortcut: the fraction of volume below the surface equals the ratio of the object's density to the fluid's.

Take a wooden block 0.5 m × 0.2 m × 0.1 m, so its volume is 0.01 m³, with a density of 600 kg/m³. Its mass is 6 kg and its weight 58.86 N. The submerged fraction is 600 ÷ 1000 = 0.6, so it displaces 0.006 m³, and 1000 × 9.81 × 0.006 = 58.86 N, matching the weight as it should.

Add load and the block sinks further until the extra weight is balanced by the extra displaced water. Ship load lines are painted on the hull for exactly this reason, with different marks for fresh water, tropical water and winter seas.

Finding upthrust with a spring balance

You can check all of this at a kitchen table. Hang a small object from a spring balance and note the reading in air. Lower it fully into a container of water, without touching the sides, and read again. The drop in the reading is the upthrust.

Suppose the balance shows 2.50 N in air and 2.10 N when submerged. The loss of 0.40 N means the displaced water weighs 0.40 N, so its volume is 0.40 ÷ (1000 × 9.81) ≈ 4.1 × 10⁻⁵ m³, or about 41 mL. The object's density then follows from its mass, 2.50 ÷ 9.81 = 0.255 kg: 0.255 ÷ 0.0000408 ≈ 6,250 kg/m³.

That is the method Archimedes is said to have used to test a crown, and it still works for finding the density of irregular objects without any geometry.

Where the upthrust figure is used

Ship designers use the displaced volume at the load line to work out how much cargo a vessel can carry. Divers add or remove lead weights so that their weight is nearly equal to buoyancy, and then small changes in lung volume let them rise or sink gently.

Hot-air balloons and airships rely on the same law in air. Air has a density of about 1.2 kg/m³, so a 1000 m³ envelope displaces about 1,200 kg of air, which is roughly 11.8 kN of upthrust before the weight of the structure and gas is subtracted.

A force calculator that returns weight from mass and acceleration is useful in this process: it turns a mass in kilograms into the weight in newtons that you set against the buoyant force.