Stretch a wire: ΔL = FL/AY.

A loaded wire (or two joined in series) stretches by ΔL = FL/(AY), shown on a magnified gauge; materials include steel, copper and rubber.

Choose the diameter of a 2 m steel wire so a 500 N load stretches it by no more than 1 mm; then compare rubber at the same stress.

Using diameter as radius; calling rubber more elastic than steel; tension 2F in a hanging wire.

• The ratio of stress to strain within the proportional region OA is a property of the material, called a modulus of elasticity.
• Young's modulus Y = tensile (or compressive) stress σ ÷ longitudinal strain ε = (F/A)/(ΔL/L) = FL/(AΔL). Experiment shows the same strain for equal tensile and compressive stress.
• Strain has no unit, so Y has the unit of stress, N m⁻² or Pa. Metals have large Y; wood, bone, concrete and glass have rather small Y.
• To stretch a wire of cross-section 0.1 cm² by 0.1% takes 2000 N for steel, and 690 N, 900 N and 1100 N for aluminium, brass and copper. Steel is therefore the most elastic of the four, which is why heavy machines and structures use it.
• Steel rod, radius 10 mm, length 1.0 m, pulled with 100 kN, Y = 2.0 × 10¹¹ N m⁻²: stress = 3.18 × 10⁸ N m⁻², elongation = 1.59 mm, strain = 1.59 × 10⁻³ ≈ 0.16%.
• Copper wire 2.2 m and steel wire 1.6 m, both 3.0 mm in diameter, joined end to end: same tension and area, so ΔLc/ΔLs = (Ys/Yc)(Lc/Ls) = 2.5 with Yc = 1.1 × 10¹¹ and Ys = 2.0 × 10¹¹ N m⁻². A total stretch of 0.70 mm splits as 0.50 mm + 0.20 mm, and the load is about 1.8 × 10² N.
• Human pyramid: 280 kg in all, the bottom performer 60 kg, so his legs carry 220 kg, or 2156 N, 1078 N per thighbone. With L = 0.5 m, radius 2.0 cm (A = 1.26 × 10⁻³ m²) and Y(bone) = 9.4 × 10⁹ N m⁻², each bone shortens by 4.55 × 10⁻⁵ m, a fractional change of 0.0091%.

Teaching this idea? Put the simulation on your own page or LMS with this code.