Force–Extension Graphs

• Linear & Non-Linear Extension:

  • Hooke’s law is the linear relationship between force and extension
  • This is represented by a straight line on a force-extension graph
• • Materials that do not obey Hooke's law, i.e they do not return to their original shape once the force has been removed, have a non-linear relationship between force and extension
  • This is represented by a curve on a force-extension graph
• • Any material beyond its limit of proportionality will have a non-linear relationship between force and extension
• Linear and non-linear regions of a force-extension graph:

  
• Calculating 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: