Fluid Mechanics and Properties of Matter: Pressure, Buoyancy, Flow, Elasticity
This domain mixes hydrostatics, fluid flow and the mechanical properties of solids. The exam typically sets a pressure or buoyancy calculation, a continuity or Bernoulli relation, and a stress–strain calculation on a wire. Marks are lost mostly through unit slips and through confusing gauge with absolute pressure.
Pressure and hydrostatics
Pressure is p = F/A, measured in pascals. In a static fluid the pressure at depth h is p = p₀ + ρgh, so pressure depends only on depth, not on the shape of the container. Gauge pressure is the excess over atmospheric; absolute pressure includes it. Pascal's principle underlies the hydraulic press: a small force on a small piston produces a large force on a large piston, with F₁/A₁ = F₂/A₂.
Buoyancy
Archimedes' principle: the upthrust equals the weight of fluid displaced, F = ρ_fluid V_displaced g. A floating body displaces its own weight, so the submerged fraction equals ρ_body/ρ_fluid. In a fully submerged object held by a string, the tension is weight minus upthrust; if the object is less dense than the fluid, the tension acts downward on the object.
Flow and viscosity
For an incompressible fluid, continuity gives A₁v₁ = A₂v₂, so a narrower pipe means faster flow. Bernoulli's equation, p + ½ρv² + ρgh = constant, holds for steady, non-viscous, incompressible flow: where the fluid speeds up, the pressure falls. Viscosity resists flow; Stokes' law F = 6πηrv leads to a terminal velocity where viscous drag plus upthrust balances weight.
Elasticity
Stress = F/A, strain = ΔL/L₀, and Young's modulus E = stress/strain. Hooke's law applies up to the limit of proportionality; beyond the elastic limit the material deforms permanently. The energy stored in a stretched wire is ½FΔx.
Worked example
A block of volume 1.0 × 10⁻³ m³ and density 600 kg/m³ is fully submerged in water (1000 kg/m³) and held by a string. Find the tension. Upthrust = ρ_water V g = 1000 × 1.0 × 10⁻³ × 9.81 = 9.81 N. Weight = 600 × 1.0 × 10⁻³ × 9.81 = 5.89 N. The block tends to rise, so the string pulls down: T = 9.81 − 5.89 = 3.9 N.
Traps that cost marks
- Using the object's volume where the displaced volume is different (a floating body is only partly submerged).
- Quoting gauge pressure when the question asks for absolute pressure, or vice versa.
- Applying Bernoulli to viscous or turbulent flow, or forgetting the ρgh term between two heights.
- Substituting the diameter for the radius when computing a cross-sectional area.
- Giving strain a unit; it is dimensionless, so Young's modulus carries the unit of stress.