# 4.7.2 Hooke's Law

### Hooke's Law

• When a force F is added to the bottom of a vertical metal wire of length L, the wire stretches
• A material obeys Hooke’s Law if:

The extension of the material is directly proportional to the applied force (load) up to the limit of proportionality

• This linear relationship is represented by the Hooke’s law equation:

#### F = kΔL

• Where:
• F = force (N)
• k = spring constant (N m–1)
• ΔL = extension (m)

• The spring constant is a property of the material being stretched and measures the stiffness of a material
• The larger the spring constant, the stiffer the material
• Hooke’s Law applies to both extensions and compressions:
• The extension of an object is determined by how much it has increased in length
• The compression of an object is determined by how much it has decreased in length

#### Force–Extension Graphs

• The way a material responds to a given force can be shown on a force-extension graph
• Every material will have a unique force-extension graph depending on how brittle or ductile it is
• A material may obey Hooke’s Law up to a point
• This is shown on its force-extension graph by a straight line through the origin
• As more force is added, the graph may start to curve slightly The Hooke’s Law region of a force-extension graph is a straight line. The spring constant is the gradient of that region

• The key features of the graph are:
• The limit of proportionality: The point beyond which Hooke’s law is no longer true when stretching a material i.e. the extension is no longer proportional to the applied force
• The point is identified on the graph where the line starts to curve (flattens out)
• Elastic limit: The maximum amount a material can be stretched and still return to its original length (above which the material will no longer be elastic). This point is always after the limit of proportionality
• The gradient of this graph is equal to the spring constant k

#### Worked Example

A spring was stretched with increasing load.
The graph of the results is shown below. What is the spring constant?   #### Exam Tip

Always double check the axes before finding the spring constant as the gradient of a force-extension graph.

Exam questions often swap the force (or load) onto the x-axis and extension (or length) on the y-axis. In this case, the gradient is not the spring constant, it is 1 ÷ gradient instead. ### 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.
Close Close