Young's Modulus

Young's modulus $Y$ measures how stiff a solid is under longitudinal stress. It tells us how much stress is required to produce a given strain along the axis of a wire or rod.

Concept

Definition. Young's modulus is the ratio of longitudinal stress to longitudinal strain in the elastic region:

$Y = \frac{\text{Longitudinal stress}}{\text{Longitudinal strain}} = \frac{F/A}{\Delta L/L} = \frac{F L}{A\,\Delta L}$

Units. $Y$ has the units of stress, i.e. pascals (Pa). Common materials have $Y$ in the range $10^9$ to $10^{12}\,\text{Pa}$, often expressed in GPa.

Typical values (approximate, at room temperature):

Uses. Selecting structural materials: skyscrapers, bridges, suspension cables, machine parts. High $Y$ is needed where rigidity is essential.

Derivation

Consider a wire of natural length $L$, cross-sectional area $A$, made of a uniform isotropic material. A tensile force $F$ at each end produces extension $\Delta L$.

Step 1 — Definitions:

$\sigma = \frac{F}{A}, \quad \varepsilon = \frac{\Delta L}{L}$

Step 2 — Hooke's law in the elastic region:

$\sigma = Y\,\varepsilon$

Step 3 — Substitute:

$\frac{F}{A} = Y\,\frac{\Delta L}{L}$

Step 4 — Solve for $\Delta L$:

$\boxed{\Delta L = \frac{F\,L}{A\,Y}}$

This is the basic working formula: extension scales linearly with force and length, inversely with area and modulus.

Step 5 — Effective spring constant of the wire:

$k = \frac{F}{\Delta L} = \frac{A Y}{L}$

Step 6 — Two wires of the same material in series behave like springs in series; in parallel they share the load.

Worked Example

A copper wire of length $3.0\,\text{m}$, diameter $1.0\,\text{mm}$, supports a $10\,\text{kg}$ load. Take $Y = 1.2\times 10^{11}\,\text{Pa}$, $g = 9.8\,\text{m/s}^2$.

Cross-sectional area: $A = \pi (d/2)^2 = \pi (5\times 10^{-4})^2 \approx 7.85\times 10^{-7}\,\text{m}^2$.

Force: $F = mg = 10 \times 9.8 = 98\,\text{N}$.

Extension: $\Delta L = FL/(AY) = 98 \times 3.0 / (7.85\times 10^{-7} \times 1.2\times 10^{11})$.

$\Delta L = 294 / (9.42\times 10^4) \approx 3.12\times 10^{-3}\,\text{m} = 3.1\,\text{mm}$.

Common Confusions

• Y depends only on the material, not on the size or shape of the sample.
• High $Y$ does not mean high strength. Diamond has very high $Y$ but breaks easily.
• $\Delta L$ depends on geometry; two wires of the same material but different lengths or thicknesses extend by different amounts under the same force.
• Young's modulus is not the same as the spring constant; their units differ.

Key Takeaways

• $Y = \text{stress}/\text{strain}$ in the elastic region; units Pa.
• Working formula: $\Delta L = FL/(AY)$.
• $Y$ is an intrinsic material property; geometry-independent.
• Typical $Y$: rubber $\sim 10^7$ Pa, steel $\sim 2\times 10^{11}$ Pa.