Force and extension relationship

For example, if a spring is originally 12.0 cm long and becomes 15.5 cm long, its extension is 3.5 cm. It is not 15.5 cm, and it is not merely the increase since the previous load was added. Every extension is measured from the same original length.

The force–extension relationship

For many springs under moderate loads, doubling the applied force doubles the extension, while tripling the force triples it. Force and extension are then directly proportional. This behaviour is described by Hooke's law:

Here $F$ is the force in newtons, $x$ is the extension in metres and $k$ is the spring constant in newtons per metre. The spring constant describes stiffness: a spring with a larger $k$ needs a greater force to produce the same extension.

This proportional relationship applies only over the range in which the spring behaves in this way. If the load becomes too great, the relationship can stop being proportional. The detailed comparison of linear and non-linear behaviour belongs to the next lesson, but on any graph you should use only the straight proportional region when describing Hooke's-law behaviour.

Data for Force against extension in the proportional region

Reading a force–extension graph

A common convention is to put extension on the horizontal axis and force on the vertical axis. In the proportional region, the points follow a straight line through the origin. The origin represents zero applied force and zero extension.

With these axes, the gradient is

A steep force–extension line therefore represents a stiff spring: a relatively large force produces a small extension. A shallower line represents a less stiff spring. Always inspect the axis labels, however. If extension were plotted vertically against force horizontally, the gradient would instead be $1/k$, so the interpretation would reverse.

A graph can also be used to predict one quantity from the other. To find the force associated with a chosen extension, move from the extension axis to the line of best fit and then across to the force axis. The reverse process finds extension from force. Read from the line of best fit rather than joining individual points dot to dot.

Investigating a spring

A standard investigation uses a spring, clamp stand, boss and clamp, metre ruler, mass hanger and slotted masses. A pointer fixed to the lower end of the spring can make its position easier to read. The stand must be stable because falling masses or a toppling stand could cause injury.

1. Secure the clamp stand and suspend the spring beside a vertical ruler.
2. Measure the spring's original length using a clearly defined pair of reference points.
3. Add a known load gently. Allow the spring to stop moving, then record its new length at eye level.
4. Calculate the total extension from the original length.
5. Increase the load in equal steps, recording the force and extension each time. If masses are labelled in kilograms, convert their weight to force using $W=mg$.
6. Repeat readings where possible and calculate means, keeping the same reference points and ruler position throughout.
7. Plot force on the y-axis against extension on the x-axis, draw an appropriate line of best fit and judge the range over which the data form a straight line through the origin.

The independent variable is the applied force and the dependent variable is the extension. The same spring, ruler arrangement and reference points should be used throughout. Measurements should be taken only when the spring is stationary. Do not pull the masses down by hand or add loads beyond the safe range chosen for the spring.

Other simple spring systems

The same ideas apply to simple combinations of springs. Identical springs joined end to end in series extend more than one spring under the same force, so the combination is less stiff. Identical springs arranged side by side in parallel share the load and extend less, so the combination is stiffer. A force–extension graph lets these systems be compared: with force on the vertical axis, the stiffer system has the steeper proportional line.