IMAT Physics · Fluids

Upthrust, Floating and Sinking

A fluid pushes up on anything in it with a force equal to the weight of fluid pushed aside; comparing densities tells you whether it floats.

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Why this matters

Four of the eight past questions in this chapter are about upthrust: its size on a submerged object, an object resting on the bottom, and a sphere that rises when the liquid around it cools.

Concept 1 of 2: Archimedes' principle: the size of the upthrust

Before the object went in, the water in its place was held up perfectly by the water around it. Put the object there and the surrounding water pushes on it in exactly the same way. So the upward push equals the weight of the water the object has replaced, and it depends on the liquid's density, not the object's.

Definition

Archimedes' principle: a body wholly or partly in a fluid feels an upward force, the upthrust (buoyant force), equal to the weight of the fluid it displaces.

Use the density of the fluid and the volume below the surface, never the object's density.
Once a body is fully submerged in a liquid, going deeper does not change the upthrust.
The apparent weight in the fluid is the true weight minus the upthrust.

Upthrust

F_B = \rho_{\text{fluid}}\, V_{\text{sub}}\, g

$\rho_{\text{fluid}}$ density of the fluid, in kg/m³
$V_{\text{sub}}$ volume of the body below the surface, in m³
$g$ gravitational field strength, in N/kg

Worked example

A metal block of volume 300 cm³ and mass 2.0 kg hangs from a spring balance, fully under brine of density 1.10 g/cm³. Taking

g = 10\ \text{N/kg}

, find the upthrust and the balance reading.

Practice this conceptself-check · 3 quick reps

The upthrust uses the liquid's density

The upthrust on a fully submerged body is

\rho_{\text{fluid}} V g

. Writing the object's own density there gives its weight, not the upthrust. An option of the form

(\rho_{\text{fluid}} - \rho_{\text{object}}) V g

is the net force, not the upthrust.

Concept 2 of 2: Floating and sinking by comparing densities

Push a fully submerged object down and let go. If it is less dense than the liquid, the upthrust beats its weight and it rises until just enough is under the surface for the two to balance. If it is denser, it sinks, and the bottom of the container takes the rest of its weight. Since a liquid's density changes with temperature, the same object can float in cold liquid and sink in warm.

Definition

Comparing the average density of an object with the density of the liquid:

$\rho_{\text{object}} < \rho_{\text{liquid}}$ : it floats, partly submerged, with upthrust = weight.
$\rho_{\text{object}} = \rho_{\text{liquid}}$ : it stays wherever it is put inside the liquid (it just floats).
$\rho_{\text{object}} > \rho_{\text{liquid}}$ : it sinks. Resting on the bottom, it still feels the upthrust, and the bottom pushes up with $W - F_B$ .
When floating, the fraction submerged equals $\rho_{\text{object}} / \rho_{\text{liquid}}$ .
Most liquids get denser as they cool, so a sphere that sank in the warm liquid can rise when it cools (the Galileo thermometer).

Fraction submerged when floating

\frac{V_{\text{sub}}}{V} = \frac{\rho_{\text{object}}}{\rho_{\text{liquid}}}

$V_{\text{sub}}$ volume below the surface
$V$ total volume of the object

Worked example

Ice has a density of 920 kg/m³ and sea water 1025 kg/m³. What fraction of a floating iceberg is below the surface? What fraction would be below the surface in fresh water of density 1000 kg/m³?

Practice this conceptself-check · 3 quick reps

A sunken object still feels upthrust

An object lying on the bottom is still surrounded by liquid, so the upthrust still acts on it. The bottom pushes up only with the weight minus the upthrust. Answering with the full weight leaves the upthrust out.

For a floating object, the upthrust equals the weight in every liquid

Move a floating block from water to oil and the upthrust does not change; it still equals the weight. What changes is how much of the block is under the surface. Options saying the upthrust is smaller in the less dense liquid are wrong while the block still floats.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Archimedes' principle: the size of the upthrust
Upthrust
$F_B = \rho_{\text{fluid}}\, V_{\text{sub}}\, g$
Floating and sinking by comparing densities
Fraction submerged when floating
$\frac{V_{\text{sub}}}{V} = \frac{\rho_{\text{object}}}{\rho_{\text{liquid}}}$

Watch out for (3)

The upthrust uses the liquid's density
→ Archimedes' principle: the size of the upthrust
A sunken object still feels upthrust
→ Floating and sinking by comparing densities
For a floating object, the upthrust equals the weight in every liquid
→ Floating and sinking by comparing densities

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