Linear and non-linear relationships

$\text{extension}=\text{stretched length}-\text{original length}$

A force–extension graph shows how these two variables are related. In the graph below, extension is on the horizontal axis and force is on the vertical axis.

Data for Linear and non-linear force–extension relationships

A linear force–extension relationship

A relationship is linear when its graph is a straight line. Equal increases in extension correspond to equal increases in force, so the graph has a constant gradient.

For an ideal spring in its directly proportional region, doubling the extension doubles the required force. Its graph is a straight line through the origin because zero applied force produces zero extension. In this region, the ratio of force to extension is constant:

$\frac{F}{x}=\text{constant}$

This is the region in which Hooke's law, $F=kx$, applies. Calculating $k$ is considered separately; here, the important feature is that the straight graph represents a constant relationship between force and extension.

A non-linear force–extension relationship

A relationship is non-linear when its graph is curved. Equal increases in extension then require unequal increases in force, and the gradient changes along the curve. Equivalently, equal additions of force do not produce equal additions of extension.

A spring may behave linearly at first but cease to do so beyond its limit of proportionality. Other materials, including elastic bands, can show a curved relationship over much of their range. A curve is still a meaningful scientific relationship: it should not be replaced by a straight line merely because it generally rises.

The gradient description depends on which variable is on each axis. With force on the vertical axis and extension on the horizontal axis, a changing gradient means the force required per metre of extension is changing.

Proportionality and elasticity are different ideas

The limit of proportionality is identified from the shape of the graph: beyond it, the graph stops being straight. The elastic limit concerns what happens after the force is removed. Below this limit, the object returns to its original length; beyond it, some deformation remains.

These limits are not definitions of the same event. A non-linear relationship tells you that force and extension are not proportional, but the loading graph alone does not prove that the object has been permanently deformed. To test elasticity, the load must also be removed and the object's length checked.

Investigating the relationship and the elastic limit

Clamp a spring securely beside a ruler and record its unloaded length. Add known masses one at a time, allowing the spring to become still before reading its new length. The applied force is the weight of the masses, measured in newtons. For each load, calculate the extension and record force and extension in a table.

Increase the load in regular steps and take several readings across a suitable range. Remove the load after successive stages and check whether the spring returns to its original length. A remaining extension indicates that the elastic limit has been exceeded. Keep the stand stable and avoid standing directly beneath the loaded spring.

Plot force against extension. A straight section shows a linear relationship; a curved section shows a non-linear relationship. Experimental points may scatter slightly, so use the overall pattern rather than deciding from a single anomalous reading.