Formula of Young's Modulus: Y = FL/(AΔL), Unit, Examples

The formula of Young's modulus is $Y = \frac{\text{tensile stress}}{\text{tensile strain}} = \frac{F/A}{\Delta L/L} = \frac{F L}{A\,\Delta L}$ . It tells you how stiff a material is when you pull or push along its length. Steel, with Y close to 2.0 × 10¹¹ Pa, barely stretches; rubber, with a value thousands of times smaller, stretches easily.

Young's modulus is a property of the material, not of the particular wire. A thick steel cable and a thin steel wire have the same Y, although the cable stretches less under the same load because its area is larger. It is the most used of the three elastic moduli collected on the fluids and elasticity hub.

Quick facts

Formula
$Y = \frac{F L}{A\,\Delta L}$
As a ratio
$Y = \frac{\sigma}{\varepsilon}$
SI unit
pascal (Pa); tables often use GPa
Dimensional formula
[M¹ L⁻¹ T⁻²]
Steel
about 2.0 × 10¹¹ Pa
Graph
slope of the straight part of the stress strain graph

Formula of Young's modulus: symbols and meaning

Quantities in Y = FL / (A ΔL)
Symbol	Meaning	SI unit
$Y$	Young's modulus of the material	Pa
$F$	Load pulling the wire (often mg)	N
$L$	Length before loading	m
$A$	Cross section, $\pi r^2$ for a round wire	m²
$\Delta L$	Increase in length	m

The formula is simply stress (F/A) divided by strain (ΔL/L), with the fractions tidied up. Because strain has no unit, Y is measured in pascals, just like stress. It only applies inside the elastic limit, where the wire returns to its original length when the load is removed.

Worked examples

Example 1: finding Y

A wire 3.0 m long with cross section 1.0 × 10⁻⁶ m² stretches by 3.0 mm under a load of 200 N. Find Young's modulus.

$Y = \frac{F L}{A\,\Delta L} = \frac{200 \times 3.0}{1.0\times10^{-6} \times 3.0\times10^{-3}} = \frac{600}{3.0\times10^{-9}} = 2.0 \times 10^{11}\ \text{Pa}$

This value is typical of steel.

Example 2: extension of a steel wire, with diameter and mass

A steel wire (Y = 2.0 × 10¹¹ Pa) is 2.0 m long and 0.80 mm in diameter. A 10 kg mass hangs from it. How much does it stretch? Use g = 9.8 m/s².

Weight: $F = 10 \times 9.8 = 98\ \text{N}$ . Radius 0.40 mm, so $A = \pi (4.0\times10^{-4})^2 = 5.03\times10^{-7}\ \text{m}^2$ .

$\Delta L = \frac{F L}{A Y} = \frac{98 \times 2.0}{5.03\times10^{-7} \times 2.0\times10^{11}} = \frac{196}{1.005\times10^{5}} = 1.95\times10^{-3}\ \text{m}$

The wire stretches by about 1.95 mm.

Halve the diameter before squaring. Using 0.80 mm as the radius gives an area four times too large and an extension four times too small.

Typical values of Young's modulus

Approximate Young's modulus of common materials
Material	Y in Pa	Y in GPa
Steel	2.0 × 10¹¹	200
Copper	1.2 × 10¹¹	120
Aluminium	7.0 × 10¹⁰	70
Glass	6.5 × 10¹⁰	65
Wood (along the grain)	about 1 × 10¹⁰	about 10
Rubber	about 10⁶ to 10⁸	0.001 to 0.1

Exact figures vary with the alloy and the test, so use the value your question gives. A higher Y means a stiffer material, not necessarily a stronger one: glass is stiff but breaks easily.

Measuring Y and reading the graph

In the school lab, Searle's apparatus hangs two identical wires side by side from the same support. One carries a fixed weight as a reference, the other carries increasing loads, and a spirit level with a micrometer screw reads the tiny extension. Using a reference wire cancels any sag of the support and expansion from temperature. Plot load against extension: the straight line slope is F/ΔL, so Y = slope × L / A.

On a stress strain graph the slope of the linear part is Y directly. The same slope idea covers the other moduli on the modulus of elasticity formula page.

Exam slips with Young's modulus

Forgetting to convert mm to m in both the diameter and the extension.
Using mass for F. Multiply by g first.
Thinking Y depends on length or thickness. Those change the extension, not Y.
Applying the formula past the elastic limit. Hooke's law no longer holds there.
Quoting Y for a liquid. Fluids have only a bulk modulus.

Frequently asked questions

What is the formula of Young's modulus?

Y = (F/A) / (ΔL/L) = FL / (AΔL): tensile stress divided by tensile strain.

Which SI unit is used for Young's modulus?

The pascal (N/m²). Because values are large, tables use gigapascals, 1 GPa = 10⁹ Pa.

Does Young's modulus change with the length of a wire?

No. A longer wire of the same material stretches more under the same load, but the ratio of stress to strain, and so Y, stays the same.

Why is steel more elastic than rubber?

In physics, more elastic means it takes more stress to make the same strain. Steel's Young's modulus is far larger than rubber's, so it is more elastic in this sense.