# 6.1.2 Hooke's Law

### Hooke's Law

• A material obeys Hooke’s Law if its extension is directly proportional to the applied force (load)
• The Force v Extension graph is a straight line through the origin (see “Extension and Compression”)
• This linear relationship is represented by the Hooke’s law equation

Hooke’s Law

• The constant of proportionality is known as the spring constant k

#### Exam Tip

Double check the axes before finding the spring constant as the gradient of a force-extension graph. Exam questions often swap the load onto the x-axis and length on the y-axis. In this case, the gradient is not the spring constant but 1 ÷ gradient is.

### The Spring Constant

• k is the spring constant of the spring and is a measure of the stiffness of a spring
• A stiffer spring will have a larger value of k
• It is defined as the force per unit extension up to the limit of proportionality (after which the material will not obey Hooke’s law)
• The SI unit for the spring constant is N m-1
• Rearranging the Hooke’s law equation shows the equation for the spring constant is

Spring constant equation

• The spring constant is the force per unit extension up to the limit of proportionality (after which the material will not obey Hooke’s law)
• Therefore, the spring constant k is the gradient of the linear part of a Force v Extension graph

Spring constant is the gradient of a force v extension graph

#### Combination of springs

• Springs can be combined in different ways
• In series (end-to-end)
• In parallel (side-by-side)

Spring constants for springs combined in series and parallel

• This is assuming k1 and k2 are different spring constants
• The equivalent spring constant for combined springs are summed up in different ways depending on whether they’re connected in parallel or series

#### Exam Tip

The equivalent (or effective) spring constant equations for combined springs work for any number of springs e.g. if there are 3 springs in parallel k1 , k2 and k3 , the equivalent spring constant would be keq = k1 + k2 + k3 .

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