# 3.7.6 Young's Modulus

### Young's Modulus

• The Young modulus is defined as

The measure of the ability of a material to withstand changes in length with an added load

• This gives information about the elasticity of a material ie. how stiff a material is
• The Young Modulus, E, can be calculated from the ratio of stress and strain

• 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

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

• The gradient of a stress-stress graph when it is linear is equal to the Young Modulus

#### 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’ ie. Stress ÷ strain.

### Determining the Young Modulus

#### Aims of the Experiment

• The aim of the experiment is to measure the Young Modulus of a metal wire
• This requires a clamped horizontal wire over a pulley
• This experiment can also be done with a vertical wire attached to the ceiling with a mass attached

Variables

• Independent variable = Force (or load) (N)
• Dependent variable = Extension (m)
• Control variables:
• The original length of wire
• The thickness of the wire
• The metal used for the wire

#### Equipment List

• Resolution of measuring equipment:
• Metre ruler = 1 mm

#### Method

This method is an example of the procedure for varying load and measuring the extension of a copper wire. This is just one way of measuring the relationship between them

1. Measure the diameter of the wire with a micrometre screw gauge or digital callipers. Take at least 3 readings and find an average
2. Set up the apparatus so the wire is taut. No masses should be on the mass hanger just yet
3. Measure the original length of the wire using a metre ruler and mark a reference point with tape preferably near the beginning of the scale eg. at 1 cm
4. Record initial reading on the ruler of the reference point
5. Add a 100 g mass onto the mass hanger
6. Read and record the new reading of the tape marker from the meter ruler
7. Repeat this method by adding a 100 g mass (at least 5 – 10 times) and record the new scale reading from the metre ruler
• An example of a table with some possible loads and extensions might look like:

#### Analysis of Results

1. Determine extension x 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

4. Calculate cross-sectional area from:

5. Calculate the Young’s modulus from:

#### Evaluating the Experiment

Systematic Errors:
• Use a vernier scale for more precise readings
• This is more likely to produce an accurate value for the extension
• If the wire is extended past its elastic limit, it will be permanently deformed
• To reduce the risk of this, remove the load and check the wire returns to its original length before taking any new readings
Random Errors:
• Parallax error from reading the marker on the ruler
• Random errors are reduced by repeating the experiment for all the loads and finding an average extension
• Reduce the uncertainty on the cross-sectional area by measuring the diameter in several places and calculating an average

#### Safety Considerations

• Wear safety goggles at all times in case the wire snaps
• Make sure a cushion or soft surface is kept directly below the mass hanger, in case it falls off

#### Exam Tip

Although every care should be taken to make the experiment as reliable as possible, you will be expected to suggest improvements in producing more accurate and reliable results (e.g. repeat readings and use a longer length of wire)

### Author: Katie

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.
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