Hooke’s Law and Force-Extension Graphs

Hooke’s Law

Hooke’s law states that the extension of an elastic object is directly proportional to the force applied to it. This means that if the force applied to an object is doubled, then the extension of the object also doubles. This also means that if there is no force applied, then there is no extension. However, this only applies within the limit of proportionality of the elastic object, such as a spring.

The limit of proportionality is the point at which additional force will not be supported by the linear relationship of Hooke’s law. This linear relationship states that extension is directly proportional to the applied force.

The elastic limit is the furthest point an elastic material can stretch while still being able to return to its original shape. Once the elastic limit is exceeded, the elastic object will not return to its original shape. At this point, the object becomes permanently deformed and the object will no longer show the original elastic behaviour.

Hooke’s law can be expressed in the equation:

  • F = Force in newtons (N)
  • k = Spring constant in newtons per metre (N/m)
  • e = Extension of the elastic object in metres (m)

Spring Constant

The spring constant, denoted by the letter k, is how much the spring extends with a given force. This is a measure of how stiff a spring is up until its limit of proportionality or elastic limit. It tells us how many newtons it would take to stretch an elastic object, such as a spring, by one metre. This means, that the higher the spring constant is, the stiffer the elastic material is, as it requires a greater force to stretch it.

We can show this relationship with a force-extension graph, which plots force against extension. As the force increases, so does the extension. This is seen as a straight line that passes through the origin.

  • The extension is the increase in length of an object

Force-Extension Graphs

After this, you can plot the extensions on a force-extension graph, with force on the vertical (y) axis and extension on the horizontal (x) axis.

Below the limit of proportionality, force and extension are directly proportional and we call this relationship Hooke’s law. The deformation will be elastic, meaning that if the force is removed, the object will return to its original shape.

However, above the limit of proportionality, the relationship becomes limited. Above this point, deformation is inelastic. The gradient of the force-extension graph below the limit of proportionality is equal to the spring constant.

Interpreting a Force-Extension Graph

  • The steeper the force-extension line is, the stiffer the spring is.
  • The flatter the force-extension line is, the more flexible the spring is.

After you read through this, go back and look at elastic potential energy, as it should now be easier to understand.