Young's modulus

For stretching or squashing along a length, the modulus is Young's modulus:

• Y = (F/A)/(ΔL/L) = FL/(AΔL), measured in N m⁻² (Pa), since strain has no unit.
• Turned round, the extension is ΔL = FL/(AY).

So a wire stretches more if it is longer, thinner (smaller A), more heavily loaded, or made of a material with a smaller Y. Metals have large Young's moduli (steel about 2 × 10¹¹ Pa), which is why a steel wire hardly stretches at all.

Example: a 2.5 m wire of density 8 × 10³ kg m⁻³ has mass 50 g, so its cross-section is A = m/(ρL) = 0.05/(8000 × 2.5) = 2.5 × 10⁻⁶ m². A load of 98 N (10 kg) stretches it by 2 mm. Then Y = 98 × 2.5/(2.5 × 10⁻⁶ × 0.002) = 4.9 × 10¹⁰ Pa.

Two common set-ups:

• Wires joined end to end carry the same tension; their extensions add up.
• Wires side by side, holding a rigid bar level, must have the same extension; they share the load.

A wire or rope that hangs carries the weight of everything below each point, so the stress is largest at the top. A hanging rope stretched only by its own weight extends as if the whole weight acted at its middle: ΔL = ρgL²/2Y. A 10 m rubber cord (ρ = 1.5 × 10³ kg m⁻³, Y = 5 × 10⁶ Pa) stretches by 1500 × 10 × 100/(2 × 5 × 10⁶) = 0.15 m.

Bones work mostly under compression, and they are very strong that way. A thighbone carrying a person's weight shortens by only a few hundredths of a millimetre.