Spring into Action

Spring into Action

Springs are everywhere: in pens, mattresses, car suspensions and bathroom scales. Push one too far, though, and it never recovers. Whenever a force changes an object's shape, the deformation is one of exactly two kinds:

Elastic or inelastic?

You bend a plastic ruler gently and it springs straight again. What kind of deformation is this?

• Elastic: it returned to its original shape
• Inelastic: it stayed bent
• It is neither: rulers cannot deform
• Both elastic and inelastic at once

Hooke's law

For a spring, the extension is proportional to the force. That is Hooke's law: F = k e F is the force in newtons (N), k is the spring constant in newtons per metre (N/m), and e is the extension in metres (m). A stiffer spring has a bigger k. This is a recall equation: it is not on the equation sheet, so you have to know it. You also have to be able to rearrange it yourself, because the exam is as likely to give you the force and the extension and ask for k as it is to ask the question the easy way round.

Find the force

A spring has a spring constant of 20 N/m and is extended by 0.5 m. What force is stretching it, in newtons (N)?

Match each quantity to its unit

Find the spring constant

A force of 12 N extends a spring by 0.3 m. What is the spring constant of that spring, in N/m?

Required Practical 6

Put the steps of the force-and-extension practical into the order you would carry them out.

• Clamp the spring at the top and measure its unstretched length
• Hang a known mass on the spring and measure the new length
• Subtract the unstretched length to get the extension
• Repeat for a range of masses, then plot force against extension

The limit of proportionality

Plot the results of RP6 and the graph tells its own story. At first you get a straight line through the origin: every extra newton adds the same extra extension, so the extension is proportional to the force and F = k e holds. Then, at the limit of proportionality, the line begins to curve. Past that point the spring stretches further for each newton you add, so F = k e no longer applies - and neither does the energy equation that depends on it. The gradient of the straight section is useful in itself: a steeper line means a stiffer spring, because it takes more force for the same extension.

Plot the spring

This is a force-extension graph for a spring obeying Hooke's law: a straight line through the origin at 2 N for every 1 cm of extension. Plot the point for an extension of 3 cm.

About the limit

Select ALL THREE statements that are TRUE about a spring's force-extension graph.

• Below the limit of proportionality the graph is a straight line
• Below the limit, F = k e applies
• Beyond the limit the line curves and F = k e no longer holds
• The limit of proportionality is where the spring first starts to stretch
• Beyond the limit the graph stays perfectly straight
• F = k e works for any extension, however large

Energy in a spring

Stretching a spring stores energy in it: elastic potential energy. Provided you stay below the limit of proportionality, it is given by: Ee = ½ k e² with Ee in joules (J), k in N/m and e in metres (m). Unlike Hooke's law, this one is on the equation sheet, so you are given it rather than having to recall it. Note the extension is squared, so doubling the stretch stores four times the energy - which is also why the units have to be metres before you start squaring anything.

Find the stored energy

A spring with a spring constant of 200 N/m is extended by 0.1 m, still below its limit of proportionality. Using Ee = ½ k e², how much elastic potential energy is stored, in joules (J)?

Spring summary

An object that returns to its original shape when the force is removed is _____ deformed. For a spring, force = spring constant × _____, written F = k e, but only below the limit of _____. The energy stored in a stretched spring is its _____ potential energy, Ee = ½ k e².