Hooke's Law (AQA A Level Physics): Revision Note

Hooke's law

• When a force is applied to each end of a spring or wire, it stretches

  • This phenomenon occurs for any material with elasticity, such as a wire or a bungee rope

• 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:

$F=k∆L$

• Where:

  • $F$ = force (N)

  • $k$ = spring constant (N m^–1)

  • $∆L$ = 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

• The extension, $∆L$, is the difference between the unstretched length and the stretched length

$\mathrm{extension}=\mathrm{stretched}\mathrm{length}-\mathrm{unstretched}\mathrm{length}$

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 starts to curve slightly as Hooke's law no longer applies

• The key features of the graph are:

  • The limit of proportionality:

    • Beyond the limit of proportionality, Hooke's law is no longer true 

    • The extension is then no longer proportional to the applied force 

  • Elastic limit:

    • Beyond the elastic limit, the material will no longer return to its original length

  • The gradient:

    • This only applies to the linear portion of that graph where Hooke's law is obeyed 

    • The gradient of a force-extension graph is equal to the spring constant

    • The gradient of an extension-force graph is $\frac{1}{k}$