Force–Extension Graphs (AQA GCSE Physics)

Revision Note

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Linear & Non-Linear Extension

  • Hooke’s law is the linear relationship between force and extension
    • This is represented by a straight line on a force-extension graph

  • Materials that do not obey Hooke's law, i.e they do not return to their original shape once the force has been removed, have a non-linear relationship between force and extension
    • This is represented by a curve on a force-extension graph

  • Any material beyond its limit of proportionality will have a non-linear relationship between force and extension

Linear and Non-linear Relationship, downloadable IGCSE & GCSE Physics revision notes

Linear and non-linear regions of a force-extension graph

Ashika, Physics Project Lead

Teacher tip

Ashika

Physics Project Lead

In my experience, students can find the Hooke's law graph a bit tricky, because it changes shape and it has specific points that you need to be able to name and explain. I like to break it down for my students and tell the story of the graph.

At the origin (0, 0) the spring is unstretched. As the straight line slopes upwards on the graph, the spring is stretching under the influence of two opposing forces. If you remove the forces from the spring at any point on that straight line part of the graph, the spring will go back to its original length. At this part of the graph, force and extension are directly proportional. If you double one, the other also doubles. However, once you stretch the spring so far that its extension falls in the curved region of the graph, it will no longer go back to its original length. The point at which this happens is the limit of proportionality, that's the point where the graph starts to curve. Beyond this point, the relationship is no longer directly proportional.

You do need to understand this graph, because Hooke's law questions are very common in exams. 

Calculating Spring Constant

  • The spring constant can be calculated by rearranging the Hooke's law equation for k:

  • Where:
    • k = spring constant in newtons per metres (N/m)
    • F = force in newtons (N)
    • e = extension in metres (m)

  • This equation shows that the spring constant is equal to the force per unit extension needed to extend the spring, assuming that its limit of proportionality is not reached
  • The stiffer the spring, the greater the spring constant and vice versa
    • This means that more force is required per metre of extension compared to a less stiff spring

Large and Small Spring Constant, downloadable IGCSE & GCSE Physics revision notes

A spring with a larger spring constant needs more force per unit extension (it is stiffer)

  • The spring constant is also used in the equation for elastic potential energy

Worked example

A mass of 0.6 kg is suspended from a spring, where it extends by 2 cm. Calculate the spring constant of the spring.

Step 1: List the known quantities

    • Mass, m = 0.6 kg
    • Extension, e = 2 cm

Step 2: Write down the relevant equation

Step 3: Calculate the force

    • The force on the spring is the weight of the mass
    • is Earth's gravitational field strength (9.8 N/kg)

W = mg = 0.6 × 9.8 = 5.88 N

Step 4: Convert any units

    • The extension must be in metres

2 cm = 0.02 m

Step 5: Substitute values into the equation

Examiner Tip

Remember the unit for the spring constant is Newtons per metres (N/m). This is commonly forgotten in exam questions

Interpreting Graphs of Force v Extension

  • The relationship between force and extension is shown on a force-extension graph
  • If the force-extension graph is a straight line, then the material obeys Hooke's law
    • Sometimes, this may only be a small region of the graph, up to the material's limit of proportionality

Force Extension Graph, downloadable IGCSE & GCSE Physics revision notes

The Hooke's law region on a force-extension graph is where the graph is a straight line

  • The symbol Δ means the 'change in' a variables
    • For example, ΔF and Δe are the 'change in' force and extension respectively
    • This is the same as rise ÷ run for calculating the gradient

  • The '∝' symbol means 'proportional to'
    • i.e. F e means the 'the force is proportional to the extension'

Spring Constant Gradient, downloadable IGCSE & GCSE Physics revision notes

The spring constant is the gradient, or 1 ÷ gradient of a force-extension graph depending on which variable is on which axis 

  • If the force is on the y axis and the extension on the x axis, the spring constant is the gradient of the straight line (Hooke's law) region of the graph
    • If the graph has a steep straight line, this means the material has a large spring constant
    • If the graph has a shallow straight line, this means the material has a small spring constant

  • If the force is on the x axis and the extension on the y axis, the spring constant is 1 ÷ gradient of the straight line (Hooke's law) region of the graph
    • If the graph has a steep straight line, this means the material has a small spring constant
    • If the graph has a shallow straight line, this means the material has a large spring constant

Worked example

A student investigates the relationship between the force applied and extension for three springs K, L and M. The results are shown on the graph below:Force Extension Worked Example, downloadable IGCSE & GCSE Physics revision notesWhich of the statements is correct?

A      K has a higher spring constant than the other two springs

B      M has the same spring constant as K

C      L has a higher spring constant than M

D      K has a lower spring constant than the other two springs

ANSWER:   D

    • The graph has the extension on the y axis and the weight (force) on the x-axis
      • This means that the spring constant is 1 ÷ gradient

    • Therefore the steeper the straight line, the lower the spring constant
    • K has the steepest gradient and therefore has a lower spring constant than L and M

Examiner Tip

Make sure to always check which variables are on which axes to determine which line has a larger or smaller spring constant, as well as the units for calculations

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Ashika

Author: Ashika

Expertise: Physics Project Lead

Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.