How this instrument works
Archimedes' principle names a mass before it names a force. Trace round whatever you submerge, imagine that outline filled with surrounding fluid instead, and weigh it — that is displaced mass, and your object gets lifted by precisely its weight. Both numbers appear on this sheet because they answer separate questions: kilograms say how much fluid had to move, newtons say how hard a rope, crane or forearm must pull to hold everything still.
That bath story reaches us through Vitruvius, the Roman architect, writing around 25 BCE, some two centuries after Archimedes died at Syracuse — and as told, it barely works. One kilogram of gold occupies about 52 millilitres, an identical mass of silver about 95, so adulterating a crown shifts displaced volume by mere tens of millilitres. Spread across the tub, that is a rise well under one millimetre, lost among ripples and whatever drips off a lifted arm. Galileo said so in La Bilancetta in 1586 and set out what Archimedes had more likely used: a balance with one pan hanging in water, weighing apparent weights against each other instead of squinting at a waterline.
Two assumptions hold this formula up — still fluid of one density reaching right around your object, plus submerged volume that stays constant. Divers break that second one routinely, since neoprene compresses under pressure: gear trimmed to hover at ten metres displaces measurably less at thirty, and the descent can begin running away. Very small objects break it from below, where surface tension carries loads Archimedes never accounted for and holds up things this arithmetic condemns to sink. Note too that displaced mass is geometry multiplied by density and travels anywhere, whereas upthrust in newtons inherits local gravity — those same ten litres lift with a sixth as much force on our Moon.
- Set Fluid density to whichever medium does your lifting. Its default, 1000 kg/m³, is fresh water; that menu also reads g/cm³ and lb/ft³ if your handbook prints either.
- Enter Submerged volume: on any floating hull, only what sits below its waterline counts. Type litres, gallons or cubic feet and conversion happens for you.
- Read Mass of fluid displaced in kilograms — naval architects call this figure displacement, and for anything afloat it matches an object's own mass.
- Read Upthrust in newtons, or switch that output over to kgf, lbf or kN to suit whichever rigging or load tables you work from.
Worked example — ten litres of lift in a quarry lake
A diver in a flooded quarry vents ten litres of air into her lift bag to raise a dropped tool. Fluid density keeps its fresh-water default of 1000 kg/m³, and Submerged volume is 10 litres, which this instrument converts to 0.01 m³. Mass of fluid displaced comes out at 1000 × 0.01 = 10 kg, so Upthrust reads 10 × 9.80665 = 98.0665 N.
Ten kilograms of lift from ten litres of gas — the coincidence that lets buoyancy wings be quoted in either unit, which is why divers size kit by volume and never bother converting. Take that bag out to sea, where Fluid density becomes 1025 kg/m³, and displacement climbs to 10.25 kg with upthrust at 100.52 N, a 2.5% bonus. Try mercury at 13534 kg/m³ and an identical ten litres would shift 135.34 kg.
Questions
Why does this calculator give a mass as well as a force?
Because Archimedes' rule is naturally stated as the weight of fluid, and mass is what you actually weigh. Mass of fluid displaced is ρ times V and needs no gravity at all; Upthrust multiplies that by 9.80665 m/s² to reach newtons. Naval and diving work quotes kilograms, rigging and structural work wants newtons, so both live on one sheet. Switch Upthrust over to kgf and that pair reads identically, since one kilogram-force is defined as exactly 9.80665 N.
Does upthrust grow stronger the deeper something goes?
No, not in a liquid you can treat as incompressible. Pressure climbs steadily with depth, but pressure on an upper face climbs by just as much, and only their difference lifts. That difference depends on how tall your object is, never on how far down it sits, so any sealed steel drum feels identical push at 3 metres and at 300. Gases behave otherwise: air compresses, weather balloons shrink as they descend, and their Submerged volume falls away along with their lift.
Is a ship's displacement tonnage the same quantity as this?
Exactly the same. Any vessel rated at 100,000 tonnes displacement moves 100,000 tonnes of seawater out of its way, which at 1025 kg/m³ means roughly 97,600 cubic metres of hull below its waterline. Because anything afloat displaces its own mass, cargo can be weighed by arithmetic alone: surveyors read draught marks before and after loading, convert through the ship's hydrostatic tables to submerged volume, and whatever displaced mass has been gained is cargo — one weighbridge that never needs calibrating.
Did Archimedes really work this out in the bath?
Probably not that way. Our only source is Vitruvius, writing roughly two hundred years afterwards, and his water-level trick is far too coarse to catch any diluted crown, where levels shift by fractions of a millimetre. Archimedes' own surviving treatise on floating bodies derives everything from a postulate about fluid pressure, with no tub anywhere in it. Galileo proposed the hydrostatic balance in 1586 as one method that would genuinely have worked, and it does: comparing an object's weight in air against its weight in water separates alloys easily.
Why do divers sink faster the deeper they descend?
Because Submerged volume shrinks on them. Neoprene is foamed with gas bubbles, and at 30 metres ambient pressure is four times atmospheric, squeezing those bubbles toward one quarter of their surface size. Displaced volume drops, displaced mass drops with it, upthrust follows, and somebody trimmed neutral near the surface turns distinctly negative down deep. Compensating means feeding gas into a wing or drysuit going down, then venting it coming back up.
Can something float even though it is denser than the fluid?
Yes, whenever a force other than buoyancy holds it up. A steel sewing needle laid gently on still water floats at about 7850 kg/m³ because surface tension supports it along the dimple it makes; break that film with a drop of detergent and down it goes. Water striders exploit an identical effect. Archimedes' rule has nothing to say here, since it accounts only for displaced fluid — so trust this sheet for anything larger than a few millimetres and fully wetted.