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

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
• 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.

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