Determination of Young's Modulus

Two steel wires ( $1.5$ m each): One for testing and one as a reference to control for effects such as sagging.
Main scale and vernier scale: To measure the extension of the test wire accurately.
$1$ kg masses and holders: Used to apply force incrementally to the test wire.
Micrometer: To measure the diameter of the test wire for calculating the cross-sectional area.
Metre ruler: For measuring the initial length of the test wire.

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Arrange the apparatus as shown in the diagram, with both the test wire and the reference wire attached to the beam.
Measure the initial length $l$ of the test wire with the metre ruler.

Attach a 1 kg mass to both wires to make them taut, and record the initial reading on the scale for the test wire.

Add an additional 1 kg mass to the test wire only. Record the new scale reading and calculate the extension $e$ by subtracting the initial reading.
Repeat this process, adding 1 kg at a time up to around 8 kg, and measure the extension for each mass.

Repeat the experiment twice more for each mass to find the mean extension $e$ for each applied load $m$ .

Measure the diameter $d$ of the test wire at several points along its length using the micrometer. Take the mean value to calculate the cross-sectional area $A$ .

Using the average diameter $d$ , calculate the cross-sectional area $A$ of the wire:

A = \frac{\pi d^2}{4}

For each mass $m$ , calculate the force $F$ applied on the test wire by using $F = mg$ , where g ≈ 9.81 m/s².

Plot a graph of force $F$ on the y-axis against extension $e$ on the x-axis.
Draw a line of best fit. The gradient $G$ of this graph represents $\frac{F}{e}$ .

E = \frac{\text{stress}}{\text{strain}} = \frac{F/A}{e/l} = \frac{l \times G}{A}

Multiply the gradient $G$ by the initial length $l$ and divide by the cross-sectional area $A$ to determine $E$ .

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Wire Tension: The wire will be stretched tightly and could snap, which could injure eyes or other parts of the body. Wear safety goggles during the experiment.
Falling Weights: If the wire breaks, the weights may fall. Place a sand tray beneath the masses to cushion any potential impact.

Comparison Wire: The reference wire compensates for any sagging of the beam or thermal expansion. This ensures that only the test wire's extension is measured.
Length Accuracy: Use a long test wire (1.5 m or more) to minimise errors in measuring small extensions and to reduce percentage uncertainty in length measurements.

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Young Modulus $E$ : A measure of stiffness, calculated as the ratio of stress (force per unit area) to strain (extension per unit length).
Stress and Strain Relationship: Under small deformations, Hooke's Law applies, meaning stress is proportional to strain, and the constant of proportionality is $E$ .
Graphical Analysis: The gradient of the force vs. extension graph provides a proportional relationship, allowing us to calculate $E$ for the material.

Determination of Young's Modulus Simplified Revision Notes for A-Level AQA Physics