Fluid Mechanics and Properties of Matter
Scope of the domain
This domain covers density and pressure in fluids, upthrust and floating, steady flow, and the elastic properties of solids. The exam asks you to calculate pressure at depth, apply Archimedes' principle, use the continuity equation, and determine stress, strain and the Young modulus.
Core rules
- Density ρ = m/V; pressure p = F/A, measured in pascals (N m⁻²).
- In a fluid at rest, the gauge pressure at depth h is p = hρg; absolute pressure adds atmospheric pressure.
- Pascal's principle: pressure applied to an enclosed fluid is transmitted equally throughout.
- Archimedes: upthrust equals the weight of fluid displaced, U = ρ_fluid V_displaced g. A floating body displaces its own weight of fluid.
- Continuity for an incompressible fluid: A₁v₁ = A₂v₂.
- Bernoulli's equation, p + ½ρv² + ρgh = constant, applies to ideal steady flow; faster flow is associated with lower pressure.
- Viscosity produces drag; terminal velocity occurs when drag plus upthrust equals weight.
- Hooke's law F = kx holds up to the limit of proportionality. Stress = F/A, strain = ΔL/L, and Young modulus E = stress/strain. Elastic deformation is reversible; plastic deformation is not.
Worked example
A wire of length 2.0 m and diameter 0.50 mm extends by 1.2 mm under a load of 20 N. Radius = 0.25 mm = 2.5 × 10⁻⁴ m, so A = πr² = 1.96 × 10⁻⁷ m². Stress = 20 ÷ 1.96 × 10⁻⁷ = 1.02 × 10⁸ Pa. Strain = 1.2 × 10⁻³ ÷ 2.0 = 6.0 × 10⁻⁴. E = 1.02 × 10⁸ ÷ 6.0 × 10⁻⁴ = 1.7 × 10¹¹ Pa.
Traps that cost marks
- Using the diameter as the radius, which makes the area four times too large.
- Failing to convert mm to m before calculating stress or strain.
- Using the object's density rather than the displaced fluid's density for upthrust.
- Thinking pressure depends on the shape or total volume of the container; it depends on depth and density.
- Confusing the limit of proportionality with the elastic limit, which is slightly beyond it.
- Quoting Young modulus in N rather than Pa.