Force–Extension Graphs (AQA GCSE Physics)

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 force and extension are no longer directly proportional; they have a non-linear relationship

  • This is represented by a curve on a force-extension graph

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

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

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

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

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'

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