Force–Extension Graphs (AQA GCSE Physics: Combined Science)

• 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 regions of a force-extension graph

• 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

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
  • g 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

• 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

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