The Young modulus of a wire

What you are trying to do

Determine the Young modulus of the material of a wire by loading it and measuring the extension produced.

Apparatus

• A long thin wire (2 m or more) of the test material, clamped at one end
• Pulley at the bench edge, with a hanger and slotted masses
• Micrometer screw gauge, for the wire diameter
• Metre rule (or travelling microscope) and a marker or sticky-tape flag on the wire
• Second (comparison) wire alongside, if using a Searle's-type arrangement
• Safety spectacles, and a padded floor or box beneath the masses

Variables

• Independent: the load (weight) hung on the wire
• Dependent: the extension of the wire
• Control: the wire itself — same material, original length and diameter throughout, and constant temperature

Method

1. Measure the diameter of the wire with a micrometer at several places along it and in two perpendicular directions at each place, then take a mean. The wire is rarely perfectly round or uniform, and the diameter is squared in the area, so this is where most of the uncertainty lives.
2. Measure the original length from the clamp to the marker with a metre rule.
3. Add the smallest load and record the new position of the marker; the extension is the change in that reading.
4. Repeat in equal steps, recording extension against load over as wide a range as the wire will take elastically.
5. Unload in steps and check the extensions retrace the loading values. If they do, the wire stayed elastic; if the wire does not return to its original length it has passed its elastic limit and those points must be discarded.

Analysis

1. Plot a graph of force (y) against extension (x). Take the gradient of the straight portion through the origin.
2. Compute the cross-sectional area from the mean diameter, A = πd² / 4.
3. The Young modulus follows from the gradient and the wire's dimensions: E = (F/A)/(ΔL/L) = gradient ×L / A.
4. Plotting stress against strain instead gives the Young modulus directly as the gradient, and lets you read off the limit of proportionality.

A worked set of readings

Typical readings for a 2.00 m steel wire of diameter 0.40 mm:

Plotting F against e gives a straight line through the origin with gradient 12566 N m⁻¹. The area is A = πd²/4 = π × (0.40 × 10⁻³)² / 4 = 1.257 × 10⁻⁷ m². Then E = gradient × L / A = 12566 × 2.000 ÷ 1.257 × 10⁻⁷ = 2.00 × 10¹¹ Pa, about 200 GPa — the right order for steel.

Where the uncertainty comes from

• Wire diameter: Much the largest contributor, because the area depends on d squared, so the percentage uncertainty in the diameter is doubled in the area. Repeat measurements along the wire and average.
• Extension: Small compared with the original length, so measure it as precisely as the apparatus allows; a travelling microscope or a vernier scale beats a metre rule.
• Original length: Comparatively easy to measure well; use a long wire so the percentage uncertainty is small and the extension is big enough to read.
• Temperature and sag: Keep the temperature steady, and use a comparison wire alongside so that any sagging of the support affects both equally and cancels.

What earns the marks

Safety