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Fluids & Elasticity

Pressure, buoyancy, flow, viscosity, surface tension and elastic solids. 51 topics, from beginner to advanced. Open the interactive version with labs and quizzes

Fluids (liquids and gases) and elastic solids respond to forces in their own way. This area defines density and pressure, explains how pressure varies with depth and how Pascal's principle is used in hydraulics, and introduces buoyancy, viscosity, surface tension and flow. It also covers stress, strain and the elastic moduli that describe how solids deform.

How the topics connect

Fluids and elasticity describe how liquids, gases and solids respond to pressure and forces. Pressure and buoyancy explain floating and diving, flow and Bernoulli's principle explain lift and pipes, and surface tension and elastic moduli describe the behaviour of materials under load.

Real-world uses

Hydraulic machines

Explain how a car jack or excavator multiplies force using pressure transmitted through a fluid.

Ships and submarines

Calculate buoyancy and the depth at which a hull or dam must withstand pressure.

Medicine and biology

Understand blood pressure, the flow of blood through narrowed vessels, and the surface tension that lets insects walk on water.

Civil and materials engineering

Use Young's modulus and stress-strain data to select materials and check that beams and cables stay within safe limits.

Ships and aircraft

Naval architects check buoyancy and stability, while aircraft designers use Bernoulli and lift to shape wings.

Materials and structures

Engineers choose steel, concrete or rubber using Young's modulus and stress-strain data for each load.

Beginner

States of matter

Solid, liquid, gas and plasma differ in order and freedom.

Solids keep shape and volume, liquids keep volume but flow, gases fill containers, plasmas are ionised gases. The state depends on how strongly particles attract compared with their thermal energy.

Worked example

Water changes among ice, liquid and steam with temperature.

Common mistake

Glass and some plastics are amorphous solids, not true liquids.

Beginner

Density

Mass per unit volume.

Key relationρ = m/V

ρ = m/V, in kg/m³. It tells how tightly matter is packed. Water is 1000 kg/m³, air about 1.2, iron 7870.

Worked example

A 2 kg block of 0.001 m³ has ρ = 2000 kg/m³.

Common mistake

Check units: g/cm³ × 1000 = kg/m³.

Beginner

Relative density

Density compared with water.

Relative density (specific gravity) is the ratio ρ/ρ_water; it has no units. Values below 1 float on water.

Worked example

Ice 0.92, mercury 13.6.

Common mistake

A number, not a density in kg/m³.

Beginner

Pressure

Force per unit area.

Key relationP = F/A

P = F/A: force per unit area acting perpendicular to a surface. The same force on a smaller area gives a larger pressure; that's why knives cut.

Worked example

A 60 kg person on 0.02 m² of shoes: 29 kPa; on snow shoes of 0.4 m²: 1.5 kPa.

Common mistake

Pressure is a scalar, force is a vector.

Beginner

Units of pressure

Pascal, atmosphere, bar and mmHg.

Key relation1 atm = 1.013×10⁵ Pa

The pascal (N/m²), atmosphere (101,325 Pa), bar (10⁵ Pa), mmHg or torr (133.3 Pa) and psi (6895 Pa).

Worked example

Tyre at 32 psi ≈ 220 kPa.

Common mistake

Gauge vs absolute pressure differ by one atmosphere.

Beginner

Pressure in a fluid

Pressure increases with depth.

Key relationP = P₀ + ρgh

Pressure increases linearly with depth: P = P₀ + ρgh. It acts in all directions and depends only on depth, not on container shape.

Worked example

At 10 m under water: 2 atm total.

Common mistake

Pressure doesn't depend on the container's width.

Beginner

Pascal's law

Pressure change transmits equally through a confined fluid.

A pressure change applied to an enclosed fluid is transmitted undiminished to every part of it and to the walls. It underlies hydraulic systems.

Worked example

Pressing a brake pedal pushes fluid to all four brakes.

Common mistake

Only for incompressible fluids at rest.

Beginner

Hydraulic press

Small force on a small piston gives large force on a big one.

Key relationF₂/F₁ = A₂/A₁

A small force on a small piston creates equal pressure that gives a large force on a big piston: F₂ = F₁(A₂/A₁). The small piston moves farther, so work is conserved.

Worked example

A 100 N push on 10 cm² lifts 1000 N on 100 cm².

Common mistake

You gain force but lose distance.

Beginner

Atmospheric pressure

Pressure from the weight of the air above.

The weight of the air column above gives 101.3 kPa at sea level, falling with altitude (about half at 5.5 km).

Worked example

A 1 m² table top bears a 10 tonne load of air, balanced by pressure beneath.

Common mistake

It decreases exponentially, not linearly.

Beginner

Barometer

Measures atmospheric pressure with a mercury column.

Key relationP = ρgh

A mercury barometer balances atmospheric pressure with a column of height h = P/ρg ≈ 760 mm. Changes in height indicate weather changes.

Worked example

With water the column would be 10.3 m.

Common mistake

The tube cross-section doesn't matter.

Intermediate

Manometer

Measures gauge pressure with a U-tube.

A U-tube partially filled with liquid measures pressure differences: ΔP = ρgh. One side connects to the gas, the other to the atmosphere or another gas.

Worked example

A 20 cm water difference means 1.96 kPa gauge pressure.

Common mistake

Gauge pressure relative to the atmosphere.

Intermediate

Gauge and absolute pressure

Gauge pressure is relative to the atmosphere.

Absolute pressure = gauge pressure + atmospheric pressure. Tyre gauges read the gauge value.

Worked example

A tyre at 220 kPa gauge has 321 kPa absolute.

Common mistake

Gas laws need absolute pressure.

Beginner

Buoyancy

Upward force on an immersed body.

A fluid pushes up on immersed bodies because the pressure at the bottom is greater than at the top. The net upward force is the buoyant force.

Worked example

You feel lighter in water.

Common mistake

Buoyancy acts on the part immersed only.

Beginner

Archimedes' principle

Buoyant force equals the weight of fluid displaced.

Key relationF_b = ρVg

The buoyant force equals the weight of the displaced fluid: F_b = ρ_fluid V_displaced g.

Worked example

A 0.002 m³ ball fully immersed in water: 19.6 N.

Common mistake

The fluid's density counts, not the object's.

Beginner

Floating and sinking

An object floats if its average density is below the fluid's.

An object floats if its average density is less than that of the fluid; a floating body displaces its own weight of fluid. Steel ships float as the hollow hull lowers the average density.

Worked example

Ice floats with 92% submerged.

Common mistake

Weight vs buoyancy, not mass vs volume alone.

Intermediate

Apparent weight in fluid

True weight minus buoyancy.

Apparent weight = true weight − buoyancy. It's how Archimedes tested the king's crown.

Worked example

A 5 N object showing 3 N in water has F_b = 2 N, V = 2.04×10⁻⁴ m³.

Common mistake

Reading changes with the fluid.

Advanced

Stability of floating bodies

Metacentre height determines stability of ships.

A ship is stable if the metacentre (where buoyancy's line of action crosses the centreline when tilted) is above the centre of gravity, giving a restoring torque.

Worked example

Heavy cargo on deck makes a ship top-heavy.

Common mistake

Wide and low-slung hulls are stabler.

Intermediate

Hydrometer

Measures liquid density using flotation depth.

A weighted glass float whose sinking depth indicates liquid density. It sinks deeper in less dense liquids.

Worked example

Checking battery acid or alcohol content.

Common mistake

Calibrated at a particular temperature.

Intermediate

Fluid dynamics

Study of fluids in motion.

Study of fluids in motion, with variables velocity, pressure, density and viscosity. Ideal fluids are incompressible, non-viscous and in steady flow.

Worked example

Water in a pipe or air around a wing.

Common mistake

Real fluids have viscosity.

Intermediate

Streamline and turbulent flow

Smooth layers versus chaotic eddies.

In streamline (laminar) flow, layers slide smoothly along fixed paths; in turbulent flow velocity fluctuates chaotically and mixes the fluid. High speed or large size leads to turbulence.

Worked example

Smoke from a candle goes up smoothly then breaks into eddies.

Common mistake

Turbulence increases drag.

Intermediate

Equation of continuity

Mass is conserved, so speed increases in narrower pipes.

Key relationA₁v₁ = A₂v₂

For incompressible steady flow A₁v₁ = A₂v₂ (volume flow rate is constant). The fluid speeds up in narrow sections.

Worked example

A pipe narrowing from 10 cm² to 2 cm²: speed ×5.

Common mistake

Blocking a hose end with a thumb speeds up the jet.

Intermediate

Bernoulli's principle

Faster flow means lower pressure along a streamline.

Key relationP + ½ρv² + ρgh = const

Along a streamline P + ½ρv² + ρgh = constant (energy conservation per unit volume). Where speed is high, pressure is low.

Worked example

Water speeding up through a constriction has lower pressure.

Common mistake

Valid for steady, incompressible, non-viscous flow.

Intermediate

Applications of Bernoulli

Aerofoil lift, atomisers and Venturi meters.

Key relationbernoulli

Aerofoil lift, atomisers (perfume sprayers), Venturi meters, carburettors, curving balls, and chimney draught.

Worked example

A fast wind above a roof lowers pressure and may lift it.

Common mistake

Don't over-simplify lift to equal-transit-time.

Intermediate

Torricelli's theorem

Speed of efflux from a tank depends on depth.

Key relationv = √(2gh)

Speed of liquid leaving a hole at depth h below the surface is v = √(2gh), the same as free fall through h.

Worked example

Hole 1.25 m below the surface: 5 m/s.

Common mistake

Range of a jet is maximum at mid-depth.

Advanced

Venturi meter

Flow speed from pressure difference between wide and narrow sections.

A pipe with a narrow throat; the pressure difference between wide and narrow sections gives the flow speed: v₁ = A₂√(2ΔP/ρ(A₁² − A₂²)).

Worked example

Used for measuring flow in pipes and carburettors.

Common mistake

Needs calibration for real fluids.

Advanced

Pitot tube

Measures flow speed from stagnation pressure.

Measures flow speed from the difference between stagnation pressure (fluid stopped) and static pressure: v = √(2ΔP/ρ). Used on aeroplanes.

Worked example

ΔP = 1 kPa in air: v ≈ 41 m/s.

Common mistake

Icing can give wrong readings.

Advanced

Lift on a wing

Pressure difference and downwash give lift.

An aerofoil deflects air downward (Newton's third law) and creates lower pressure above than below (Bernoulli). Lift ≈ ½ρv²SC_L, increasing with angle of attack up to stall.

Worked example

A 747 requires about 250 km/h to take off.

Common mistake

Stall occurs when flow separates.

Advanced

Magnus effect

A spinning ball curves because of pressure differences.

A spinning ball in air drags the air along its surface, creating a pressure difference that curves its path. Used in football free kicks and topspin.

Worked example

Backspin on a golf ball adds lift.

Common mistake

The spin deflects air to one side, so the ball is pushed to the opposite side.

Intermediate

Viscosity

Internal friction in fluids.

Key relationF = ηA dv/dy

Viscosity η measures internal friction between fluid layers: F = ηA(dv/dy). Honey is more viscous than water; viscosity of liquids decreases as temperature rises, gases the opposite.

Worked example

Water 1 mPa·s, glycerol 1.4 Pa·s.

Common mistake

Not the same as density.

Advanced

Poiseuille's equation

Flow rate through a pipe depends on the fourth power of radius.

Key relationQ = πr⁴ΔP/8ηL

Volume flow rate through a pipe is Q = πr⁴ΔP/(8ηL): very sensitive to radius. Halving the radius reduces flow by 16.

Worked example

Narrowed arteries raise blood pressure significantly.

Common mistake

Valid for laminar flow only.

Advanced

Stokes' law

Drag on a small sphere in viscous fluid.

Key relationF = 6πηrv

A sphere moving slowly in a viscous fluid feels F = 6πηrv. It lets you measure viscosity from a falling-ball experiment.

Worked example

Used by Millikan to find oil-drop radius.

Common mistake

Fails if flow becomes turbulent.

Advanced

Terminal velocity in fluids

Balances weight, buoyancy and viscous drag.

A falling object reaches steady speed when weight = buoyancy + drag; for a small sphere v_t = 2r²(ρ − σ)g/9η.

Worked example

Dust settling slowly in air.

Common mistake

Larger drops fall faster.

Advanced

Reynolds number

Predicts the onset of turbulence.

Key relationRe = ρvd/η

Re = ρvd/η compares inertial and viscous forces. Below ~2000 in a pipe the flow is laminar, above ~4000 turbulent.

Worked example

Water at 1 m/s in a 2 cm pipe: Re ≈ 20,000.

Common mistake

Depends on the geometry, so critical values vary.

Intermediate

Surface tension

Surface acts like a stretched membrane.

Key relationT = F/L

Molecules at a liquid surface are pulled inward, so the surface behaves like a stretched membrane with tension T = F/L. It allows insects to walk on water.

Worked example

Water: 0.072 N/m.

Common mistake

Soap lowers surface tension.

Intermediate

Surface energy

Work needed to enlarge a surface.

Key relationW = TΔA

Increasing the surface area costs energy W = TΔA. Liquids minimise area, which is why drops are spherical.

Worked example

Making a 2 cm² soap film with two surfaces needs 2TΔA.

Common mistake

Count two surfaces for films and bubbles.

Advanced

Excess pressure in drops and bubbles

Smaller bubbles have greater pressure.

Key relationΔP = 2T/r

A curved surface has excess pressure ΔP = 2T/r (drop), 4T/r (soap bubble with two surfaces). Smaller bubbles have higher pressure and the smaller one empties into a larger one.

Worked example

A 1 mm soap bubble: 4·0.025/0.001 = 100 Pa.

Common mistake

Number of surfaces matters.

Advanced

Angle of contact

Angle between liquid surface and solid wall.

The angle between the liquid surface and the solid at the line of contact. Water on glass is below 90° (wets), mercury is above 90° (doesn't wet).

Worked example

Water forms a meniscus curved up; mercury curved down.

Common mistake

It depends on the solid, liquid and gas.

Intermediate

Capillary rise

Liquid climbs in narrow tubes.

Key relationh = 2T cosθ/ρgr

In a narrow tube a liquid rises by h = 2T cosθ/(ρgr) if it wets the wall and falls if not. It helps water move up plants and in paper towels.

Worked example

Water in a 0.5 mm radius tube: about 3 cm.

Common mistake

Narrower tubes give larger rise.

Beginner

Cohesion and adhesion

Attraction between like and unlike molecules.

Cohesion is attraction between like molecules (surface tension), adhesion between unlike (wetting). Their balance determines the contact angle.

Worked example

Water sticks to glass; mercury sticks to itself.

Common mistake

Both are electromagnetic in origin.

Beginner

Elasticity

A body's ability to return to shape after force removal.

The ability of a solid to regain its shape after deforming forces are removed, within limits. It arises from interatomic forces acting like springs.

Worked example

A steel spring returns to its length.

Common mistake

Beyond the elastic limit deformation is permanent.

Beginner

Stress

Force per unit area inside a deformed body.

Key relationσ = F/A

Stress is the internal force per unit area, σ = F/A, in pascals. It may be tensile, compressive or shear.

Worked example

A 1000 N load on a 1 mm² wire: 1000 MPa.

Common mistake

Stress is not force.

Beginner

Strain

Fractional deformation.

Key relationε = ΔL/L

Strain is the relative deformation: tensile ε = ΔL/L, volume strain ΔV/V, shear angle. It's dimensionless.

Worked example

A 2 m wire stretching 1 mm: ε = 5×10⁻⁴.

Common mistake

Use the original length.

Beginner

Hooke's law for solids

Stress is proportional to strain in the elastic region.

For small deformations stress ∝ strain; the constant is the elastic modulus. It holds below the proportional limit.

Worked example

Rubber doesn't obey it well.

Common mistake

It's a limit, not a universal law.

Intermediate

Young's modulus

Stiffness for tension or compression.

Key relationY = σ/ε

Y = stress/strain for tension or compression. Steel 200 GPa, aluminium 70 GPa, rubber ~0.01 GPa. A stiff material has high Y.

Worked example

1 mm² steel wire (L = 2 m) under 100 N stretches 1 mm.

Common mistake

ΔL = FL/AY.

Intermediate

Bulk modulus

Resistance to uniform compression.

Key relationB = −VΔP/ΔV

B = −VΔP/ΔV measures resistance to uniform compression; its inverse is compressibility. Water 2.2 GPa; gases much lower.

Worked example

Pressure of 100 atm compresses water by 0.45%.

Common mistake

Related to speed of sound.

Intermediate

Shear modulus

Resistance to shearing.

Key relationG = shear stress/shear strain

G = shear stress/shear strain, resistance to sideways deformation. Liquids have zero shear modulus.

Worked example

Steel G ≈ 80 GPa.

Common mistake

Gases and liquids can't support static shear.

Intermediate

Stress–strain curve

Shows elastic limit, yield point and fracture.

The curve shows the proportional region, elastic limit, yield point, plastic region, ultimate tensile strength and fracture. Ductile metals show necking.

Worked example

Steel wire stretched until it breaks.

Common mistake

Area under it is energy per volume.

Advanced

Poisson's ratio

Lateral to longitudinal strain.

Key relationν = −ε_lat/ε_long

When stretched, a material thins sideways by ν times the axial strain. ν ≈ 0.3 for most metals, 0.5 for rubber (incompressible).

Worked example

A rod 1% longer becomes 0.3% thinner.

Common mistake

Cork has ν ≈ 0, so wine corks fit bottles.

Advanced

Elastic potential energy density

Energy stored per unit volume.

Key relationu = ½σε

The energy stored per unit volume is u = ½σε = ½Yε². Total energy is u × volume.

Worked example

A steel wire stretched to ε = 10⁻³: 100 kJ/m³.

Common mistake

Spring energy ½kx² follows.

Advanced

Bending of beams

Deflection depends on Young's modulus and the second moment of area.

A beam bends under load; the deflection depends on Young's modulus and the second moment of area I. Deep beams (I-beams) resist bending as they put material far from the neutral axis.

Worked example

A ruler is stiffer on edge than flat.

Common mistake

Deflection scales with L³.

Advanced

Torsion of a cylinder

Twisting a wire needs torque proportional to angle.

A twisted cylinder develops a restoring torque τ = (πGr⁴/2L)θ. Hollow shafts are almost as strong as solid ones.

Worked example

Drive shafts in cars are tubes.

Common mistake

Torque scales with r⁴.

Frequently asked questions

Why does pressure increase with depth in water?

Each layer of water above supports the weight of the layers above it. Pressure grows linearly with depth, at a rate of rho times g for a fluid of density rho.

What is the difference between stress and pressure?

Pressure is force per unit area acting equally in all directions in a fluid. Stress is force per unit area inside a solid and can act in different directions, so it is described more generally.

Why does a steel ship float when a steel nail sinks?

The ship's hull encloses a large volume of air, so its average density is less than water. Buoyancy depends on the weight of water displaced, not the weight of the steel alone.

Why do ships float if steel sinks?

A ship's average density, including the air inside its hull, is less than water. The buoyant force equals the weight of the water it displaces, which supports the ship.

Does a faster fluid always have lower pressure?

Bernoulli's principle says the pressure is lower where the speed is higher, along a streamline and in steady, non-viscous flow. Real flows add other effects, such as viscosity and turbulence.

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