4.2.2 Young modulus

infoNote

The Young modulus (E) quantifies the stiffness of a material. It is a measure of a material's resistance to deformation under stress.
For materials that obey Hooke's law (up to the limit of proportionality), stress is directly proportional to strain. This constant ratio of stress to strain is what defines the Young modulus.

E = \frac{\text{Tensile Stress}}{\text{Tensile Strain}}

Breaking down this relationship:
- Tensile Stress: The force per unit cross-sectional area applied to stretch or compress the material.
- Tensile Strain: The ratio of the extension of the material to its original length.
Using formulas from stress and strain:

E = \frac{F \cdot L}{A \cdot \Delta L}

Where:

On a stress-strain graph, the gradient of the straight-line portion (where Hooke's law is obeyed) represents the Young modulus.
- Gradient = Young modulus
- This part of the graph typically shows a linear relationship between stress and strain, indicating the material's proportionality limit.

infoNote

Suppose we have a material with the following data:

Force applied ( $F$ ) = $200$ N,

Original length ( $L$ ) = $2$ m,

Cross-sectional area ( $A$ ) = $0.01\ m²$ ,

Extension ( $\Delta L$ ) = $0.005\ m$ .

To find the Young modulus ( $E$ ):

E = \frac{F \cdot L}{A \cdot \Delta L} = \frac{200 \cdot 2}{0.01 \cdot 0.005}

E = \frac{400}{0.00005} = 8 \times 10^6 \, \text{Pa}

So, the Young modulus of the material is $8$ MPa (megapascals).

Young modulus helps in assessing whether a material is suitable for specific structural applications where resistance to deformation is crucial.
A high Young modulus indicates a stiff material that does not easily deform, like steel.
A low Young modulus suggests a flexible material, such as rubber.

Young modulus Simplified Revision Notes for A-Level AQA Physics