Hooke's law

Extension and compression

Extension happens when an object increases in length, and compression happens when it decreases in length. The extension of an elastic object, such as a spring, is described by Hooke's law:

force = spring constant × extension

F = k~e

This is when:

  • force (F) is measured in newtons (N)
  • spring constant (k) is measured in newtons per metre (N/m)
  • extension (e), or increase in length, is measured in metres (m)

Example

A force of 3 N is applied to a spring. The spring stretches reversibly by 0.15 m - the fact that the string stretches reversibly means that it will go back to its normal shape after the force has been removed. Calculate the spring constant.

First rearrange F = k \: e to find k:

K = \frac{F}{e}

Then calculate using the values in the question:

k = 3 \div 0.15

k = 20~N/m

Limit of proportionality

Spring constant is a measure of the stiffness of a spring up to its limit of proportionality or elastic limit. The limit of proportionality refers to the point beyond which Hooke's law is no longer true when stretching a material. The elastic limit of a material is the furthest point it can be stretched or deformed while being able to return to its previous shape. Once a material has gone past its elastic limit, its deformation is said to be inelastic.

The higher the spring constant, the stiffer the spring. The spring constant is different for different elastic objects. For a given spring and other elastic objects, the extension is directly proportional to the force applied. For example, if the force is doubled, the extension doubles. This works until the limit of proportionality is exceeded.

When an elastic object is stretched beyond its limit of proportionality, the object does not return to its original length when the force is removed. In this instance, the relationship between force and extension changes from being linear, or directly proportional, to being non-linear.

Non-linear extension occurs more in some materials than others. Materials like clay or putty usually show non-linear extension.

Force-extension graphs

Linear extension and elastic deformation can be seen below the limit of proportionality.

Non-linear extension and inelastic deformation can be seen above the limit of proportionality. The limit of proportionality is also described as the 'elastic limit'. The gradient of a force-extension graph before the limit of proportionality is equal to the spring constant.

A force extension graph. Linear section drawn from origin to occupy half of graph area. Non linear section has decreasing gradient. Change from linear to non-linear is marked and labelled.