# 5.3.3 Force–Extension Graphs

### 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 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

#### 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 #### Exam 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 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 steep 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: Which 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

• 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

#### Exam 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 ### Author: Ashika

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.
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