The Young Modulus (CIE A Level Physics)

Stress

• Tensile stress is the applied force per unit cross sectional area of a material

$\sigma = \frac{F}{A}$

where σ is tensile stress in Pa, F is applied force in N and A is the cross-sectional area of the object in m²

• The ultimate tensile stress is the maximum force per original cross-sectional area a wire is able to support until it breaks

Strain

• Strain is the extension per unit length
• This is a deformation of a solid due to stress in the form of elongation or contraction
• Note that strain, ε , is a dimensionless unit because it’s the ratio of lengths

$\epsilon = \frac{x}{L}$

where x is extension in metres and L is the original length of the object, also in metres

Young’s Modulus

• The Young modulus, E , is the measure of the ability of a material to withstand changes in length with an added load ie. how stiff a material is
• This gives information about the elasticity of a material
• The Young Modulus is defined as the ratio of stress and strain

$E = \frac{\sigma }{\epsilon } = \frac{FL}{Ax}$

• Its unit is the same as stress: Pa (since strain is unitless)
• Just like the Force-Extension graph, stress and strain are directly proportional to one another for a material exhibiting elastic behaviour

Stress-Strain Graph

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

• The gradient of a stress-strain graph, when it is linear, is the Young Modulus

• To measure Young’s modulus of a metal in the form of a wire requires a clamped horizontal wire over a pulley (or vertical wire attached to the ceiling with a mass attached) as shown in the diagram below

 Experimental Equipment to Determine Young's Modulus

As the wire stretches, the tape marker moves along the ruler, showing the extension of the wire. This is based on the assumption that the wire stretches the same amount at all points along its length.

• A reference marker is needed on the wire. This is used to accurately measure the extension with the applied load
• The independent variable is the load
• The dependent variable is the extension

Method

1. Measure the original length of the wire using a metre ruler and mark this reference point with tape
2. Measure the diameter of the wire with micrometer screw gauge or digital calipers
3. Measure or record the mass or weight used for the extension e.g. 300 g
4. Record initial reading on the ruler where the reference point is
5. Add mass and record the new scale reading from the metre ruler
6. Record final reading from the new position of the reference point on the ruler
7. Add another mass and repeat method

Improving experiment and reducing uncertainties:

• Reduce uncertainty of the cross-sectional area by measuring the diameter d in several places along the wire and calculating an average
• Remove the load and check wire returns to original limit after each reading
• Take several readings with different loads and find average
• Use a Vernier scale to measure the extension of the wire

Measurements to determine Young’s modulus

1. Determine extension from final and initial readings

Example Table of Results

Table with additional data

2. Plot a graph of force against extension and draw line of best fit

3. Determine gradient of the force v extension graph

Force-Extension Graph

This gradient is neither stress nor strain, but can be used to calculate Young's modulus

4. Calculate cross-sectional area A from:

where d is the average diameter of the wire

5. Calculate the Young’s modulus from:

using the gradient of the force-extension graph, the initial length of the wire and it's average cross-sectional area