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. If no force is applied, 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.

The elastic limit is the maximum point to which an elastic material can stretch and still 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:

The equation: Force equals Spring constant multiplied by Extension

Equation: F equals k times e

  • 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.

This relationship can be represented by a force-extension graph, where force is plotted 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.

Graph depicting the relationship between Force (N) on the vertical axis and Extension (m) on the horizontal axis. The curve starts linearly and then begins to curve upwards. A point on the curve is labelled 'Limit of proportionality' with a blue box.

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, beyond the limit of proportionality, the relationship is no longer linear. Beyond 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.

You’ve used 10 of your 10 free revision notes for the month

Sign up to get unlimited access to revision notes, quizzes, audio lessons and more

Sign up