# 4.8.1 The Young Modulus

### The Young Modulus

• The Young modulus is the measure of the ability of a material to withstand changes in length with an added load
• This gives information about the stiffness of a material
• This is useful for engineers to make sure the materials they are using can withstand sufficient forces
• The Young Modulus is defined as the ratio of tensile stress and tensile strain

• Where:
• F = force (N)
• L = original length (m)
• A = cross-sectional area (m2)
• ΔL = extension (m)

• Since strain is dimensionless, the units of the Young Modulus is pascals (Pa)
• The Young Modulus of a material is typically a very large number, in the order of GPa

Table of the Young’s Modulus for Materials

#### Worked Example

A metal wire that is supported vertically from a fixed point has a load of 92 N applied to the lower end.
The wire has a cross-sectional area of 0.04 mm2 and obeys Hooke’s law.
The length of the wire increases by 0.50%.

What is the Young modulus of the metal wire?

A.    4.6 × 107Pa              B.    4.6 × 1012 Pa              C.    4.6 × 109 Pa               D.    4.6 × 1011 Pa

#### Exam Tip

To remember whether stress or strain comes first in the Young modulus equation, try thinking of the phrase ‘When you’re stressed, you show the strain’ i.e. Stress ÷ Strain.

### The Young Modulus from Stress-Strain Graphs

• The Young Modulus is equal to the gradient of a stress-strain graph when it is linear (a straight line)
• This is the region in which Hooke’s Law is obeyed
• The area under the graph in this region is equal to the energy stored per unit volume of the material

A stress-strain graph is a straight line with its gradient equal to the Young modulus

#### Worked Example

The graph below shows the stress-strain graph for a copper wire.

Use the graph to calculate the Young Modulus of copper.

Step 1: Determine the stress and strain where the linear region ends

The Young Modulus is the gradient of the linear region of a stress-strain graph

Step 2: Calculate the gradient of the graph in this region

### Author: Ashika

Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.
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