Buoyancy & Archimedes' Principle

A boat in a tank, and a block of cargo that is either aboard or over the side. The page works out how deep the boat sits, where the water level stands, and how far that level moves when the cargo leaves the boat.

Worked example — Buoyancy & Archimedes' Principle

Mass the boat is floating	160.0 kg
Liquid the boat pushes aside	160.29 L
Upthrust on the hull	1,569.6 N
How deep the hull sits	100.2 mm
Liquid the cargo pushes aside, through the hull	40.07 L
Height of the water surface above the tank floor	1.0267 m
How far the boat rises when it does	25.0 mm
Least water that would float the boat	440.00 kg

These are the values the calculator shows when the page opens. Change any input above to recompute them.

Loading interactive simulation...

Lesson

The theory — Buoyancy & Archimedes' Principle

Archimedes' principle: a body wholly or partly immersed in a fluid feels an upward force, the upthrust or buoyant force, equal to the weight of the fluid it pushes aside, F_b = ρgV. The force comes from pressure: the liquid presses harder on the bottom of the body than on its top, because the bottom is deeper. A body floats when the upthrust on its submerged part equals its weight, so a floating body displaces exactly its own mass of liquid. A body that sinks cannot displace more than its own volume, and whatever of its weight the upthrust does not carry rests on the floor. Every figure on this page is one of those two sentences.

What each symbol means

ρ: density of the liquid, in kg/m³: 998.2 for fresh water at 20 °C and 1,025 for seawater
V: volume of liquid pushed aside, in m³. A thousand litres is one cubic metre
F_b: the upthrust, ρgV, in newtons. For a floating body it equals the body's weight
d: draft, how deep the hull sits: its mass over ρ times its footprint
A_t: floor area of the tank. A volume added below the surface raises the level by that volume over A_t
A_b: footprint of the hull, the area of its flat bottom

Assumes: A box-shaped hull with a flat bottom and straight sides, and a tank with straight sides, so a volume divided by an area is a height. The liquid is still and incompressible and has one density throughout. A sunk block is a cube lying on the floor clear of the hull and fully under water; the verdict says so when the water is too shallow to cover it. Meltwater has the density of fresh water at 20 °C and keeps its volume when it mixes with the sea. Noerdlinger and Brower's figure is about 2.6% against this page's 2.68%; the densities and the mixing are where the two can differ, and this page treats both simply. Pine's density is a representative 500 kg/m³; any wood that floats gives the same zero.
Breaks when: Once the hull is not a box. A real boat's waterline area changes with depth, so its draft is found by integrating the hull shape, and the rise of the boat will not be a simple ratio; the level change of the pond is untouched, because it depends only on volumes. Surface tension breaks the snug-tank claim at the smallest scales, when the gap round the hull is a millimetre or two. Compressibility breaks nothing at these depths: a few metres of water squeeze it by far less than any figure the page prints. And a boat that takes water over its side stops being a boat: past the hull depth this page reports the boat as swamped rather than computing a sinking.

The anchor goes over the side and the pond goes down 🖖

Press Anchor in the boat, then Anchor on the bottom. In the boat, the anchor is floating along with the hull, and anything floating pushes aside its own mass of water: 40 kg of anchor, 40.07 litres. On the bottom it pushes aside only its own volume, which for iron at 7,874 kg/m³ is 5.08 litres. The missing 35 litres, spread over the 6 m² pond, lower the surface by 5.83 mm. At the same moment the boat rises 25.0 mm, and a rising boat is what the eye follows. The table under the results runs the same 40 kg in six materials: pine and ice change nothing, and gold, the densest, takes the level down by 6.33 mm, close to the 6.68 mm that would be all 40.07 litres gone.

Melting ice leaves a glass of water level and raises the sea 🖖

Ice floating in fresh water displaces its own mass of water, and when it melts it becomes that mass of water, so the level does not move: the tool prints 0.000 mm for any block of any size. In seawater the two stop matching. Ice in seawater floats 200 kg of ice by pushing aside 195.12 litres of sea, and melts into 200 kg of fresh water, which is less dense than the sea and takes up 2.68% more room. The level rises 0.873 mm. The density of the ice appears nowhere in that sum; only the two waters do. Noerdlinger and Brower worked this out for the oceans in 2007. They put the excess at about 2.6%, and reckoned that if all the sea ice and floating ice shelves melted, sea level would rise by about 4 cm.

A tonne of boat floats on 50 kg of water 🖖

The upthrust on a hull is the pressure under it times its area, and the pressure under it is ρgh: it depends on how deep the hull sits, not on how much water is about. So the boat needs only enough water to fill the gap round it up to its draft. A tonne in a snug tank puts a 1,000 kg boat with a 2 m² hull into a tank of 2.1 m², and the least water that floats it is 50.00 kg. Divide that by the boat's mass and you get 5.0%, which is the gap area over the hull area, 0.1 ÷ 2, with the density and the weight both cancelled out. Pressure at Depth shows the same thing from underneath: the force on a base is set by the depth over it, and a thin tube of water presses on its base with nearly ten times its own weight.

Problem solved in full

Lower the anchor on a rope. When does the pond go down? 5 steps
The rowing boat from the first preset, 120 kg with a 40 kg iron anchor, on a 6 m² pond. Instead of throwing the anchor, you lower it on a rope until it hangs under water, clear of the bottom, and then you let it settle. Does the level change while it hangs, and by how much does it change when it touches the bottom? Decide before reading on.

Open this problem in the tool
1. Start with everything aboard. The boat floats 160 kg, so it displaces 160 kg of water:
  
  $V_{1} = \frac{120 + 40}{998 . 2} = 0.16029 m^{3} = 160.29 L$
2. The anchor's own volume, and the upthrust on it once it is under water, the weight of 5.07 kg of water:
  
  $V_{a} = \frac{40}{7874} = 0.00508 m^{3} ρ V_{a} = 998.2 \times 0.00508 = 5.07 kg$
3. Hanging from the rope, the anchor is held up partly by the water and partly by the rope. The rope carries the rest:
  
  $T = (40 - 5.07) g = 34.93 kg \times 9.81 = 342.65 N$
4. That tension pulls down on the boat, so the boat is now floating 154.93 kg. Add what the hull displaces to what the anchor displaces beside it:
  
  $V_{2} = \frac{120 + 34 . 93}{998 . 2} + 0.00508 = 0.15521 + 0.00508 = 0.16029 m^{3} = V_{1}$
5. The total is the aboard figure exactly. Now let the anchor settle. The floor takes the 34.93 kg the rope was holding, the boat floats only its own 120 kg, and the water loses that share of its load:
  
  $V_{3} = \frac{120}{998 . 2} + 0.00508 = 0.12530 m^{3} Δ h = \frac{V _{3} - V _{1}}{A _{t}} = \frac{0 . 12530 - 0 . 16029}{6} = - 5.83 mm$
Answer

Nothing moves while the anchor hangs: the water is carrying all 160 kg, part through the hull and part through the anchor, just as it did when the anchor was aboard. The level falls by 5.83 mm at the moment the anchor touches the bottom, because from then on the floor holds up 34.93 kg that the water was holding. What decides the level is who carries the weight, and going over the side is not the event that changes it.

$V_{2} = V_{1} \Rightarrow Δ h = 0 V_{3} < V_{1} \Rightarrow Δ h = - 5.83 mm$

References (3)

The principle in Archimedes' own two halves: Proposition 5, a floating solid sinks until it displaces its own weight of fluid, and Proposition 7, a solid heavier than the fluid goes to the bottom and weighs less there by the weight of the fluid it displaces: T. L. Heath (ed.), The Works of Archimedes. Cambridge University Press, 1897. On Floating Bodies, Book I, Propositions 5-7, pp. 257-258.
Why melting floating ice raises the sea, and by about how much: meltwater takes up about 2.6% more room than the seawater the ice displaced, and melting all sea ice and floating ice shelves would raise sea level about 4 cm: P. D. Noerdlinger and K. R. Brower, "The melting of floating ice raises the ocean level." Geophysical Journal International 170(1), 2007, pp. 145-150. DOI 10.1111/j.1365-246X.2007.03472.x
The density of ice, 916.72 kg/m³ at its normal melting point, from the international equation of state for ordinary ice: International Association for the Properties of Water and Steam, Revised Release on the Equation of State 2006 for H2O Ice Ih, IAPWS R10-06(2009). Doorwerth, The Netherlands, September 2009. Table 6.

By the 42Math team · Reviewed against published sources. Corrections welcome.