Sketch, plot and interpret load-extension graphs for an elastic solid and describe the associated experimental procedures

Published by Patrick Mutisya · 14 days ago

Cambridge IGCSE Physics 0625 – Effects of Forces

1.5.1 Effects of Forces – Load‑Extension Graphs for an Elastic Solid

Learning Objective

Students will be able to sketch, plot and interpret load‑extension (force‑extension) graphs for an elastic solid and describe the experimental procedures used to obtain such graphs.

Key Concepts

  • Elastic solid – a material that returns to its original shape when the applied load is removed, provided the load does not exceed its elastic limit.

  • Load (Force) \$F\$ – the external force applied to the solid, measured in newtons (N).

  • Extension \$x\$ – the change in length of the solid under load, measured in metres (m).

  • Hooke’s Law: \$F = kx\$ where \$k\$ is the spring (or stiffness) constant (N m\(^{-1}\)).

  • Stress \$\sigma = \dfrac{F}{A}\$ and strain \$\epsilon = \dfrac{x}{L0}\$, where \$A\$ is the cross‑sectional area and \$L0\$ the original length.

  • Young’s Modulus \$Y = \dfrac{\sigma}{\epsilon}\$ – a measure of the material’s stiffness.

Typical Load‑Extension Graph

The graph below shows the relationship between load \$F\$ (vertical axis) and extension \$x\$ (horizontal axis) for an elastic solid.

Suggested diagram: Sketch of a load‑extension graph showing a straight‑line region (Hooke’s law), a yield point, and a region of permanent deformation.

Interpretation of the Graph

  1. Linear region (Hooke’s law): The graph is a straight line through the origin. The gradient \$k\$ equals the stiffness constant. Within this region the material is elastic.

  2. Elastic limit (or proportional limit): The highest point on the straight‑line portion. Up to this load the material will return to its original length when the load is removed.

  3. Yield point: The load at which the graph deviates from linearity. Small increases in load produce larger extensions.

  4. Plastic region: Beyond the yield point the material undergoes permanent deformation. The graph may become curved and does not pass through the origin when the load is removed.

  5. Fracture point: The load at which the material breaks; the graph terminates.

Experimental Procedure

To obtain a load‑extension graph for an elastic solid (e.g., a metal wire), follow the steps below.

StepActionPurpose / Observation
1Measure the original length \$L_0\$ and cross‑sectional area \$A\$ of the specimen.These values are needed to calculate stress and strain later.
2Attach one end of the specimen to a fixed support and the other end to a hook that can hold masses.Provides a vertical arrangement for applying load.
3Place a vernier calliper or a micrometer ruler alongside the specimen to read the extension \$x\$.Allows accurate measurement of small changes in length.
4Add a known mass \$m\$ (e.g., 0.1 kg) to the hook and record the corresponding extension \$x\$.Load \$F = mg\$ is applied; record one data point.
5Repeat step 4, increasing the mass in equal increments (e.g., 0.1 kg each time) until the specimen reaches its elastic limit.Builds the linear portion of the graph.
6Continue adding masses beyond the elastic limit, noting the larger extensions.Shows the yield point and plastic region.
7After the final load, remove all masses and record the final length of the specimen.Determines whether permanent deformation has occurred.
8Plot the recorded loads \$F\$ (vertical axis) against extensions \$x\$ (horizontal axis) on graph paper or using software.Visualises the relationship and allows determination of \$k\$, elastic limit, etc.

Using the Graph to Find Material Properties

  • Stiffness constant \$k\$: Gradient of the straight‑line portion, \$k = \dfrac{\Delta F}{\Delta x}\$.

  • Young’s Modulus \$Y\$: From the linear region,

    \$Y = \frac{F/A}{x/L0} = \frac{k L0}{A}.\$

  • Elastic limit: Load at the last point that still lies on the straight line.

  • Yield strength: Stress corresponding to the yield point,

    \$\sigma{\text{yield}} = \frac{F{\text{yield}}}{A}.\$

Common Sources of Error

  • Parallax error when reading the extension.

  • Inaccurate measurement of the cross‑sectional area (especially for non‑cylindrical specimens).

  • Friction in the support or hook causing additional resistance.

  • Temperature changes affecting material dimensions.

Summary

A load‑extension graph provides a visual representation of how an elastic solid responds to applied forces. The initial straight‑line region confirms Hooke’s law and allows calculation of the stiffness constant and Young’s modulus. The point where the graph deviates marks the elastic limit, beyond which permanent deformation occurs. Careful experimental technique—accurate measurements, gradual loading, and proper recording—ensures reliable data for analysis.