Determining Young modulus

Practical method

Aim

Determine the Young modulus of a metal wire by measuring extension for increasing loads.

Variables

Independent variable: load (tensile force)
Dependent variable: extension
Control variables: the same wire, original length and diameter; temperature; stay within the elastic region.

Apparatus

• Long thin copper or steel wire
• Clamp and pulley at the end of the bench
• Slotted masses
• Metre rule
• Micrometer
• Marker and vernier scale (or Searle's apparatus)
• Safety goggles

Method

1. Measure the original length L from the clamp to the marker (about 2.5 m) with a metre rule.
2. Measure the diameter with a micrometer at several points and in two perpendicular directions; take the mean and calculate A = \(\dfrac{\pi d^{2}}{4}\).
3. Add a small load to straighten the wire and note the marker position.
4. Add masses in steps, recording the marker position each time; remove them to check the wire returns to its original length.
5. Plot force against extension. Young modulus E = \(\dfrac{\text{gradient} \times L}{A}\).

Sample results

L = 2.50 m, d = 0.30 mm, so A = 7.07 × 10⁻⁸ m².

Analysis

Force against extension is a straight line through the origin with gradient \(\dfrac{40}{0.00707}\) = 5.66 × 10³ N m⁻¹. E = \(\dfrac{\text{gradient} \times L}{A}\) = \(\dfrac{5.66 \times 10^{3} \times 2.50}{7.07 \times 10^{-8}}\) = 2.0 × 10¹¹ Pa, typical of steel.

Uncertainty

The diameter (0.30 ± 0.01 mm) has a 3.3% uncertainty, which doubles to 6.7% in A — the largest contribution. Length (±1 mm in 2.50 m) and extension (±0.05 mm in 7 mm, 0.7%) contribute much less, so E = (2.0 ± 0.2) × 10¹¹ Pa.

Errors and improvements

Risk assessment

Common mistakes

• Using the diameter instead of the radius in A = πr².
• Leaving extension in millimetres.
• Using the total length after loading instead of the original length.

Worked analysis

At 40 N the extension is 7.07 mm. Stress = \(\dfrac{40}{7.07 \times 10^{-8}}\) = 5.66 × 10⁸ Pa; strain = \(\dfrac{0.00707}{2.50}\) = 2.83 × 10⁻³; E = \(\dfrac{5.66 \times 10^{8}}{2.83 \times 10^{-3}}\) = 2.0 × 10¹¹ Pa.

Common misconception

Young modulus is a property of the material, not the wire: a thicker wire of the same metal extends less but has the same E.