Elastic Energy & Measuring Young's Modulus
Stretching a wire stores energy; a careful experiment measures how stiff it is.
Lesson at a glanceShow the key points
- Stretching a wire slowly by l needs work ∫(AY/L)x dx = AYl²/2L (14.11), stored as elastic potential energy.
- U = ½ (maximum force)(extension) = ½ × stress × strain × volume (14.12).
- Lab method: an experimental wire beside a reference wire (to cancel temperature and support sag), loads added in steps, extensions read with a vernier.
- Plot extension against load: a straight line through the origin; Y = gL/(πr² × slope).
- Use loads below half the breaking load; read on loading and unloading.
A drawn bowstring stores the energy that later launches the arrow. Any stretched wire does the same, and measuring how much it stretches is how we find Young’s modulus in the lab.
Energy stored in a stretched wire
In its natural shape a body’s molecular potential energy is at its minimum; call it zero. Deforming it means doing work against the internal forces, which raises its elastic potential energy.
Fix one end of a wire (natural length L, cross-section A) and stretch it slowly, so the applied force always equals the tension. When the extension is x, Hooke’s law gives , so . Stretching it from 0 to l takes work
A steel wire 1.5 m long with cross-section 2.0 mm² is stretched by 1.5 mm. How much elastic energy does it store? (Y = 2.0 × 10¹¹ N/m²)
Measuring Young’s modulus in the laboratory
A simple, accurate arrangement uses two wires hanging side by side from the same rigid support:
- A reference wire (2–3 m long) carries a main scale and a fixed heavy weight that keeps it straight. Any sagging of the support or expansion due to temperature affects both wires equally, so it cancels out.
- The experimental wire carries a vernier that slides against the main scale, and a hanger for slotted weights.
- Measure the wire’s radius at several places with a screw gauge; from the known breaking stress, find the breaking load and stay below half of it.
- Hang a small initial load (1–2 kg) to remove kinks; note the vernier reading.
- Add equal loads (0.5 or 1 kg) one at a time, waiting a minute each time, and note the reading. Then remove them in the same steps, reading again.
- Plot extension against load: a straight line through the origin.
If the slope of the line is (extension per unit mass),
In such an experiment with a 2.0 m wire of radius 0.25 mm, the extension–load graph has slope 0.50 mm per kg. Find Y.
- 1Elastic PE of a stretched wire: = ½ × stress × strain × volume.
- 2Lab method: experimental wire beside a reference wire, loads in steps, vernier readings on loading and unloading.
- 3, from the slope of extension against load.
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