understand that the area under the force–extension graph represents the work done

In physics, materials can stretch or deform in two main ways: elastic and plastic. Think of a rubber band that snaps back to its original shape (elastic) versus a bent metal ruler that keeps its new shape (plastic).

📐 Elastic behaviour follows Hooke’s law: $F = kx$, where F is the force, k the spring constant, and x the extension.

🧪 Plastic behaviour occurs when the material is stretched beyond its yield point. After this point, it does not return to its original shape.

When a material is stretched within its elastic limit, the stress (force per unit area) and strain (relative deformation) are directly proportional.

📏 Hooke’s law can be written as a graph: $F$ vs. $x$ is a straight line with slope $k$.

🔢 Example: If a spring has a constant $k = 200\,\text{N/m}$ and is stretched by $0.05\,\text{m}$, the force is $F = 200 \times 0.05 = 10\,\text{N}$.

Once the yield point is exceeded, the stress–strain curve bends and the material deforms permanently.

🔧 The yield strength is the maximum stress the material can withstand without permanent deformation.

💡 Analogy: Imagine bending a paperclip. The first few bends are reversible (elastic), but after a few more, it stays bent (plastic).

The area under the $F$–$x$ curve represents the work done on the material:

$$W = \int_{0}^{x} F(x')\,dx'$$

For a linear elastic region, this area is a triangle:

$$W = \frac{1}{2}kx^2$$

📊 In the plastic region, the curve is no longer linear, and the area includes both elastic and plastic work.

1. Identify the elastic limit on the graph.
2. Calculate the area of the triangle (elastic part).
3. For the plastic part, approximate the area using trapezoids or integrate the actual curve.
4. Add both areas to get the total work.

📝 Tip: When given a graph, sketch the area manually if the exact curve is not provided.

• 🔍 Read the question carefully. Look for words like “elastic limit”, “yield point”, or “work done”.
• 📐 Use the correct units. Force in N, extension in m, work in J.
• 📏 Show all steps. Even if you know the final formula, writing the integration or area calculation helps you earn partial marks.
• 🧮 Check your answer. For elastic work, confirm that it matches the triangle area formula.
• 🎓 Practice sketching graphs. Being able to draw a force–extension curve from data is a common exam skill.