Sketch, plot and interpret load-extension graphs for an elastic solid and describe the associated experimental procedures
1.5.1 Effects of Forces – Load‑Extension Graphs for an Elastic Solids
Learning objectives
- Sketch, plot and interpret a load‑extension (force‑extension) graph for an elastic solid.
- Describe, step‑by‑step, the experimental method used to obtain the graph.
- Identify the features the Cambridge IGCSE Physics (0625) examiner expects.
- Know which points are core requirements and which belong to the supplementary (optional) material.
What the examiner expects (Core – 1.5.1)
- State a concise definition: A load‑extension graph plots the applied force (N) on the vertical axis against the resulting extension (m) on the horizontal axis.
- Identify and label on the graph:
- Origin (0, 0)
- Linear (Hooke’s‑law) region
- Proportional (elastic) limit – the last point that still lies on the straight‑line region
- Yield point (where the curve first deviates from the straight line)
- Plastic (permanent‑deformation) region
- Fracture point (if the specimen breaks)
- Explain that the gradient of the straight‑line region is the stiffness (spring) constant k (k = ΔF/Δx).
- State that Young’s modulus is supplementary – only mention it if the question is marked “supplementary”.
- Outline the experimental method, including measurement of the original length L₀ and cross‑sectional area A, and note any safety considerations.
Exam‑style sketch checklist
When drawing the graph in the exam, make sure you:
- Draw clear, labelled axes (F (N) vertical, x (m) horizontal).
- Mark the origin.
- Draw a straight line from the origin up to the proportional limit.
- Label the proportional limit, yield point, plastic region and fracture point.
- Write the gradient as k = ΔF/Δx beside the straight‑line portion.
- Keep the sketch neat – no extra curves or arrows unless required.
Theory (Core)
- Elastic solid: Returns to its original shape when the load is removed, provided the load does not exceed the elastic limit.
- Load (force) F: External force applied, measured in newtons (N).
- Extension x: Increase in length of the specimen, measured in metres (m).
- Hooke’s Law: F = k x where k is the stiffness (spring) constant (N m⁻¹). Valid only in the linear region.
Supplementary (optional) – Stress, strain, Young’s modulus and yield stress
These concepts are not required for the core IGCSE exam but may be asked in “supplementary” questions.
- Stress σ = F/A (N m⁻²)
- Strain ε = x/L₀ (unit‑less)
- Young’s modulus Y = σ/ε = (F/A)/(x/L₀) = k L₀/A (N m⁻²)
- Yield stress σyield = Fyield/A (used if a question asks for the material’s yield strength).
Typical load‑extension graph (Core)
Worked example
Test a copper wire of original length L₀ = 0.500 m and cross‑sectional area A = 1.0 × 10⁻⁶ m².
| Mass added (kg) | Load F = mg (N) | Extension x (mm) |
|---|---|---|
| 0.10 | 0.98 | 0.20 |
| 0.20 | 1.96 | 0.40 |
| 0.30 | 2.94 | 0.60 |
Steps to plot
- Draw the axes: F (N) vertical, x (mm) horizontal.
- Plot the three points (0.20 mm, 0.98 N), (0.40 mm, 1.96 N) and (0.60 mm, 2.94 N).
- Join them with a straight line – this is the linear (Hooke’s‑law) region.
Gradient (stiffness constant k)
\[
k = \frac{\Delta F}{\Delta x}
= \frac{1.96-0.98\ \text{N}}{0.40-0.20\ \text{mm}}
= \frac{0.98\ \text{N}}{0.20\ \text{mm}}
= 4.9\ \text{N mm}^{-1}
= 4.9\times10^{3}\ \text{N m}^{-1}
\]
Young’s modulus (supplementary)
\[
Y = \frac{kL_{0}}{A}
= \frac{4.9\times10^{3}\times0.500}{1.0\times10^{-6}}
= 2.45\times10^{9}\ \text{N m}^{-2}
\]
Interpretation of the graph (Core)
- Linear (Hooke’s‑law) region: Straight line through the origin – material behaves elastically.
- Proportional (elastic) limit: Highest point still on the straight line; up to this load the specimen returns to its original length.
- Yield point: First deviation from the straight line; a small increase in load produces a relatively large increase in extension.
- Plastic region: Beyond the yield point; deformation is permanent.
- Fracture point: Load at which the specimen breaks; the graph terminates.
Experimental procedure (Core)
| Step | Action | Purpose / Observation |
|---|---|---|
| 1 | Measure the original length L₀ and cross‑sectional area A of the specimen. | Required for stress, strain and Young’s‑modulus calculations. |
| 2 | Secure one end of the specimen to a rigid support; attach the other end to a hook that can hold masses. | Provides a vertical arrangement for applying a known load. |
| 3 | Place a vernier calliper or micrometer alongside the specimen to read the extension x. | Ensures accurate measurement of small changes in length. |
| 4 | Add a known mass (e.g., 0.10 kg) to the hook, wait until the system is at rest, then record the extension. | Creates one data point (F = mg, x). |
| 5 | Increase the mass in equal increments, recording the extension each time, until the elastic limit is reached. | Builds the linear portion of the graph. |
| 6 | Continue adding masses beyond the elastic limit, noting the larger extensions. | Shows the yield point and plastic region. |
| 7 | After the final load, carefully remove all masses and record the final length of the specimen. | Determines whether permanent deformation has occurred. |
| 8 | Plot the recorded loads (F) against extensions (x) on graph paper or using software. | Visualises the relationship and allows determination of k, elastic limit, etc. |
Safety note (AO3)
- Secure the specimen and support firmly before adding any mass.
- Never drop masses – add them gently to avoid dynamic overshoot.
- Wear safety glasses in case the specimen fractures.
Common pitfalls (AO3)
- Adding masses too quickly – the system may not be at rest before the reading is taken.
- Reading the extension from an angle (parallax error).
- Using a damaged or irregular‑shaped specimen, which gives an inaccurate cross‑sectional area.
- Neglecting to record the final length after removing the loads (you may miss permanent deformation).
Using the graph to find material properties (Core)
- Stiffness constant k: Gradient of the straight‑line region, k = ΔF/Δx.
- Elastic (proportional) limit: Load at the last point that still lies on the straight line.
- Yield strength (if asked): σyield = Fyield/A.
- Young’s modulus (supplementary): Y = k L₀/A.
AO3 – Experimental skills checklist
- Plan the experiment: choose a suitable specimen, decide on mass increments, and ensure measuring instruments are calibrated.
- Observe safety procedures (secure set‑up, wear goggles, add masses gently).
- Record data in a tidy table, including units.
- Plot the graph accurately and label all required points.
- Evaluate: identify sources of error, suggest improvements, and comment on the reliability of the results.
Common sources of error (Core)
- Parallax when reading the extension scale.
- Inaccurate measurement of the cross‑sectional area, especially for non‑cylindrical specimens.
- Friction in the support or hook adding extra resistance.
- Temperature changes causing expansion or contraction of the material.
- Sudden loading (dropping masses) leading to dynamic overshoot.
Summary
A load‑extension graph provides a concise visual representation of how an elastic solid responds to applied forces. The initial straight‑line region confirms Hooke’s law and yields the stiffness constant k. From k (and, if required, the supplementary material) Young’s modulus can be calculated. The point where the graph departs from the straight line marks the proportional (elastic) limit; beyond this lies the yield point, plastic deformation, and finally fracture. Accurate experimental technique—careful measurement of length, area, and extension; gradual loading; and attention to safety—ensures reliable data that meet the specific expectations of the Cambridge IGCSE Physics examiner.