3.2MechanicsFoundation

Density and Pressure

A steel ship floats, while a single steel nail sinks — both made of the exact same material. The explanation has nothing to do with the metal itself and everything to do with two related ideas: how tightly packed that metal’s mass is, and how a fluid pushes back against anything submerged in it.

35 min Video by Chris Gozzard (That Physics Guy) Module 3.2 Forces in action

Pressure and Density — A level PhysicsWatch the full walkthrough before the notes below.

Open on YouTube

What you'll be able to do

Define density and use $ρ = \frac{m}{V}$
Define pressure and use $p = \frac{F}{A}$
Derive and use the pressure–depth relationship in a fluid, $p = h ρ g$
State Archimedes’ principle and calculate the upthrust on a submerged object

Density

$Density$ , $ρ$ , is the mass per unit volume of a substance, measured in kg m⁻³. It is an intrinsic property of the material itself — a large block and a small block of the same pure substance have the same density, even though their masses and volumes differ enormously.

ρ = \frac{m}{V}

Density = mass ÷ volume.

1Use

ρ = \frac{m}{V}

ρ = \frac{2.7}{1.0 \times 1 0 ^{- 3}}

Answer

ρ = 2700

kg m⁻³ (this is aluminium).

Pressure

$Pressure$ is the force acting perpendicular to a surface, per unit area, measured in pascals (Pa), where $1$ Pa $= 1$ N m⁻². For a given force, spreading it over a larger area reduces the pressure — exactly why snowshoes stop you sinking into snow, and why a sharp knife (small contact area) cuts far more easily than a blunt one with the same applied force.

p = \frac{F}{A}

Pressure = force ÷ area.

1Use

p = \frac{F}{A}

p = \frac{120}{0.40}

Answer

p = 300

Pressure in a fluid

The pressure at a depth $h$ below the surface of a fluid arises from the weight of fluid above that point pressing down. Considering a column of fluid of cross-sectional area $A$ and height $h$ : its weight is $ρ \times A h \times g$ , and dividing by the area $A$ (since pressure = force/area) leaves the area cancelling out entirely.

p = h ρ g

Pressure due to depth h in a fluid of density ρ.

1Use

p = h ρ g

p = 8.0 \times 1000 \times 9.81

Answer

p = 7.85 \times 1 0^{4}

Pa (about 0.77 atmospheres of additional pressure)

Tip — Pressure due to depth depends only on the depth and the fluid’s density — never on the shape or width of the container, which is why a narrow tube and a wide swimming pool have exactly the same pressure at the same depth.

Upthrust and Archimedes’ principle

Because pressure increases with depth, the fluid pressure on the bottom of a submerged object is always greater than the pressure on its top — producing a net upward force called $upthrust$ . $Archimedes’ principle$ states that the upthrust on a submerged (or floating) object exactly equals the weight of fluid it displaces.

F_{upthrust} = ρ_{fluid} V_{displaced} g

Upthrust = weight of fluid displaced.

1Use

F_{upthrust} = ρ V g

F_{upthrust} = 1000 \times 2.0 \times 1 0^{- 4} \times 9.81

Answer

F_{upthrust} \approx 1.96

Tip — An object floats when upthrust can equal its full weight without needing to be fully submerged — this is exactly why a hollow steel ship floats (displacing a large volume of water) while a solid steel nail (much smaller volume, same density) sinks.

Equation recap

ρ = \frac{m}{V}

Density.

p = \frac{F}{A}

Pressure.

p = h ρ g

Pressure due to depth in a fluid.

F_{upthrust} = ρ_{fluid} V_{displaced} g

Archimedes’ principle.

Common mistakes to avoid

Believing pressure at a given depth depends on the width or shape of the container.

p = hρg depends only on depth and fluid density — the container’s shape or width plays no part.

Thinking a denser object can never float.

What matters is whether the object can displace enough fluid for the upthrust to equal its weight — a hollow, less densely-packed shape (like a ship’s hull) can float even when made of a material denser than water.

Calculating upthrust using the object’s own density instead of the fluid’s.

Upthrust depends on the density of the DISPLACED FLUID, not the density of the submerged object itself.

Key takeaways

Density ρ = m/V; pressure p = F/A.
Pressure due to depth in a fluid: p = hρg, independent of the container’s shape.
Archimedes’ principle: upthrust on a submerged/floating object = weight of fluid it displaces.
An object floats if it can displace enough fluid that upthrust equals its own weight.

Test yourself

Ready to lock in Density and Pressure? Pick a mode and earn XP & Dobloons.

Density and Pressure

35 min Video by Chris Gozzard (That Physics Guy) Module 3.2 Forces in action