Density, Pressure and Archimedes' Principle

Density and pressure are deceptively simple quantities with profound consequences. They explain why ships made of steel can float, why aeroplanes can fly, why your ears pop when you dive, and why a single drawing pin supports your full weight without breaking your skin. This lesson covers OCR Module 3.2.3 (Equilibrium) and includes fluid pressure and the magnificent insight of Archimedes, discovered — legend has it — in a Syracusan bath more than 2,200 years ago.

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Density

Density is the mass per unit volume of a material. It is a scalar, with SI unit kg m⁻³:

ρ = m / V

Material	Density (kg m⁻³)
Air (at sea level, 15 °C)	1.23
Cork	240
Ice	917
Water (pure, 4 °C)	1000
Seawater	1025
Aluminium	2700
Steel	7850
Copper	8960
Lead	11 340
Mercury	13 534
Gold	19 320
Platinum	21 450
Osmium (densest natural element)	22 590

Water's density of 1000 kg m⁻³ is a useful benchmark; anything denser sinks in water and anything less dense floats (before accounting for shape).

Worked Example — Mass of a Copper Block

A cuboid copper block measures 5.0 cm × 4.0 cm × 2.0 cm. Find its mass.

• Volume V = 0.050 × 0.040 × 0.020 = 4.0 × 10⁻⁵ m³
• Density ρ (copper) = 8960 kg m⁻³
• Mass m = ρ V = 8960 × 4.0 × 10⁻⁵ = 0.358 kg (358 g)

Common Exam Mistake: Using cm³ as volume but kg m⁻³ as density — mixed units. Always convert to SI before substituting.

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Pressure

Pressure is the force per unit area acting perpendicular to a surface. It is a scalar:

p = F / A

SI unit: pascal (Pa) = N m⁻². Other common units:

Unit	Value in Pa
pascal (Pa)	1
kilopascal (kPa)	10³
bar	10⁵
atmosphere (atm)	1.013 × 10⁵
millimetre of mercury (mmHg)	133.3
pound per square inch (psi)	6895

At sea level, atmospheric pressure ≈ 101 kPa or about 1 atm. This is the weight of a column of air, one square metre in cross-section, extending from ground level to the top of the atmosphere — roughly 10 tonnes per m². We do not feel it because it acts equally in every direction and is balanced by the pressure inside our bodies.

Why a Sharp Knife Cuts

A sharp knife has a very small contact area. The force you apply (a few newtons) is spread over perhaps 10⁻⁸ m², giving pressures of hundreds of MPa — more than enough to exceed the yield stress of soft materials like bread or flesh. A blunt knife spreads the force over a larger area, giving much lower pressure and hence no cutting.

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Pressure in a Fluid at Depth

Imagine a column of fluid of cross-sectional area A, depth h, and density ρ. Its volume is V = Ah and its mass m = ρAh. The weight of this column is mg = ρAhg. This weight presses down on the base of area A, exerting a pressure:

p = F / A = ρAhg / A = ρgh

So fluid pressure increases linearly with depth:

p = ρ g h

where p is the pressure due to the fluid alone (gauge pressure). The total absolute pressure at depth h is:

p_total = p_atm + ρ g h

Worked Example — Pressure at the Bottom of a Swimming Pool

A swimming pool is 2.5 m deep. Find (a) the gauge pressure at the bottom and (b) the absolute pressure.

(a) ρ_water × g × h = 1000 × 9.81 × 2.5 = 24 525 Pa ≈ 24.5 kPa

(b) Absolute = 101 325 + 24 525 = 125.9 kPa

So the pressure at the bottom of a pool is about 25% more than atmospheric. Deep-sea divers experience far more: at 100 m depth, gauge pressure is 981 kPa — about 10 atmospheres.

Worked Example — Hydraulic Press

A small piston of area 1.0 × 10⁻⁴ m² is pushed with force 50 N. The fluid is incompressible and connects to a larger piston of area 2.5 × 10⁻³ m². What force does the larger piston exert?

Pressure is the same throughout the fluid (Pascal's principle): p = 50 / 10⁻⁴ = 5.0 × 10⁵ Pa Force on large piston: F = pA = 5.0 × 10⁵ × 2.5 × 10⁻³ = 1250 N