# 1.27 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:

ΔF = kΔx

• Where:
• F = applied force (N)
• k = spring constant (N m–1)
• Δx = 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 Stretching a spring with a load produces a force that leads to an extension

#### Force–Extension Graphs

• The way a material responds to a given force can be shown on a force-extension graph
• 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?

Step 1: Rearrange Hooke's Law to make the spring constant the subject Step 2: Compare the gradient to the equation in Step 1

• This graph is length - extension, so the gradient gives: • Therefore k is the reciprocal of the gradient  Step 5: Calculate the spring constant by finding the reciprocal of the gradient Step 6: Write the answer, including units

• Spring constant, k = 8.0 N m−1

#### 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. ### Get unlimited access

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