Materials and Properties of Matter
Density, pressure and Hooke’s law
Density ρ = m/V. Pressure p = F/A. For a wire or spring, Hooke’s law states that extension is proportional to applied force up to the limit of proportionality: F = kx. The spring constant k is the gradient of a force–extension graph.
Stress, strain and the Young modulus
Tensile stress σ = F/A, with A the cross-sectional area. Tensile strain ε = extension ÷ original length. The Young modulus E = σ/ε and is valid only in the linear elastic region. A stress–strain graph shows elastic behaviour, yield, plastic deformation and fracture. Stiffness is resistance to extension; strength is resistance to breaking. Elastic strain energy is the area under the force–extension graph.
Worked example
A 2.0 m wire of diameter 0.50 mm carries a load of 20 N and extends by 1.2 mm. Radius = 0.25 × 10⁻³ 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. Keeping units in metres prevents a factor-of-1000 error.
Traps
Using diameter as radius; failing to convert mm to m; dividing extension by the stretched length instead of the original length; treating stress as force; applying the Young modulus beyond the limit of proportionality; confusing the elastic limit with the limit of proportionality.