Forces · Investigation
Springs and Hooke's law
A spring is the only instrument in the lab that turns a force into a length you can read with a ruler. It works beautifully — right up until it does not.
Start here
1 N stretched it 20 mm. 2 N stretched it 40 mm.
Two readings, one spring, a ruler clamped alongside it. The pattern looks obvious, so make it do some work: predict the extension when you hang 10 N on the same spring.
What will the extension be at 10 N?
For the first few newtons the pattern holds exactly: every newton adds the same 20 mm, so the extension is proportional to the load. But every spring has a load beyond which that stops being true, and after that each newton adds more than the last. Two readings can tell you the rule. They cannot tell you how far the rule goes — only more readings can do that, which is what the bench below is for.
Extension is how much longer the spring has become, not its total length: extension = stretched length − natural length. While the extension is proportional to the load, the spring is obeying Hooke's law, and a graph of extension against load is a straight line through the origin. The load where that stops is the limit of proportionality.
At the bench · loading a spring
Take the readings. Plot them. Find where it bends.
No readings plotted yet
One spring, a clamped ruler, and loads from 0 to 10 N. Set a load, record the reading, and the point goes on the graph. The dashed line is the straight line the first readings make.
Commit first. Which two readings would you take to test whether the extension is proportional to the load?
2 N
Load
—
Extension
—
Extension per newton
—
The spring
—
The relationship · a beam and a graph, not a triangle
Each newton adds the same amount, until it does not.
This is a beam and a graph rather than a triangle, because what is being taught is a proportionality read off a straight line: two quantities that keep the same ratio. The bars show it as a part–whole — three newtons of extension is three equal helpings of one newton’s worth — and the graph shows the same fact as a straight line through the origin, with the bend where the law gives out.
While a spring obeys Hooke's law, extension is proportional to load.
extension in mm = stretched length in mm − natural length in mm
extension ÷ load stays the same for every reading on the straight line
past the limit of proportionality, that ratio stops being constant
Worked example · one step at a time
A spring extends 40 mm under a 2 N load. Staying on the straight line, what is the extension under 5 N?
Step 0 of 5
Convert
40 mm stays 40 mm · 2 N stays 2 N
Both extensions are in millimetres and both loads in newtons, so there is nothing to convert.
Formula
extension ÷ load is the same for every reading
That is what proportional means, and it is why the graph is a straight line.
Insert
extension ÷ load = 40 mm ÷ 2 N
Using the one reading given, and extension means the increase in length.
Fine-tune
40 ÷ 2 = 20 mm for each newton
Millimetres divided by newtons leaves millimetres per newton.
Answer
5 N × 20 mm/N = 100 mm
A hundred millimetres, as long as 5 N is still on the straight line.
Worked example · one step at a time
A spring extends 6.0 cm under 3 N. Staying on the straight line, what is the extension under 8 N, in millimetres?
Step 0 of 5
Convert
6.0 cm × 10 = 60 mm
The answer is wanted in millimetres, and a centimetre is ten of them, so multiply by 10.
Formula
extension ÷ load is the same for every reading
Proportional, so one ratio describes every point on the line.
Insert
extension ÷ load = 60 mm ÷ 3 N
The converted extension goes in. The 6.0 never does.
Fine-tune
60 ÷ 3 = 20 mm for each newton
Millimetres divided by newtons leaves millimetres per newton.
Answer
8 N × 20 mm/N = 160 mm
Leave the 6.0 in centimetres and the answer reads 16 — ten times too small for the unit asked for.
Your turn · the same five steps
Your reading: 40 mm under 2 N. Predict the extension at 4 N.
Write all five lines before you check, then set the bench to 4 N and see whether the prediction held.
Put a load on the spring first — a zero reading has nothing to scale up.
The five lines, marked
Convert
40 mm stays 40 mm · 2 N stays 2 N
The extension is already in millimetres and the load already in newtons, so there is nothing to convert.
Formula
extension ÷ load is the same for every reading
True while the spring is on the straight line.
Insert
extension ÷ load = 40 mm ÷ 2 N
Both figures come from the reading on your own bench.
Fine-tune
40 ÷ 2 = 20 mm for each newton
Millimetres divided by newtons leaves millimetres per newton.
Answer
4 N × 20 mm/N = 80 mm
Double the load, so the prediction is double the extension.
The five lines predict 80 mm, and setting the bench to 4 N gives exactly 80 mm. The proportionality held, because both loads are on the straight line.
A spring stretches 2.4 cm under a 4 N load. Staying on the straight line, what is the extension under 10 N, in millimetres?
This one needs the Convert line to do some work.
The five lines, marked
Convert
2.4 cm × 10 = 24 mm
The answer is wanted in millimetres, so multiply the centimetres by 10.
Formula
extension ÷ load is the same for every reading
Proportional, so one ratio describes the whole line.
Insert
extension ÷ load = 24 mm ÷ 4 N
The converted extension goes in. The 2.4 never does.
Fine-tune
24 ÷ 4 = 6 mm for each newton
Millimetres divided by newtons leaves millimetres per newton.
Answer
10 N × 6 mm/N = 60 mm
Leave it in centimetres and the answer reads 6 — right number, wrong unit, no marks.
The five lines give 60 mm, provided 10 N is still on the straight part of the line.
Key fact
While a spring obeys Hooke's law, its extension is proportional to the load: a graph of extension against load is a straight line through the origin, and every newton adds the same extension. Past the limit of proportionality the line bends, and the spring may not return to its original length.
Think again
“Extension is how long the spring is.”
It is how much longer it has become. A spring with a natural length of 50 mm holding a load at 90 mm has an extension of 40 mm, not 90 mm, and a graph plotted with total length on the axis does not go through the origin — it starts at 50 mm with no load on it at all. That is why the first measurement in this investigation is taken before anything is hung on the spring, and why the sentence extension = stretched length − natural length is worth writing out every time until it is automatic. Nearly every wrong Hooke's law graph is this mistake.
“Past the limit, the spring snaps.”
The limit of proportionality is not a breaking point, and passing it is not dramatic. What stops is the neat arithmetic: each extra newton now adds more extension than the newton before it, so the graph curves away from the straight line, and the spring no longer springs all the way back. Take the load off and it is left permanently longer than it started, which is why an overstretched spring balance reads wrongly for ever afterwards even though it looks perfectly all right. Snapping happens much later, if at all.
Mastery ladder
Not started yet.
Rungs 3 and 4 you mark yourself.
Rung 1 · Calculate
A spring extends 30 mm under a 3 N load. Staying within the straight line, what is its extension under 7 N?
Rung 2 · The one that catches people
A spring is loaded past its limit of proportionality, then the load is taken off completely. What happens?
Rung 3 · Explain
A newton meter has a scale of equally spaced marks alongside its spring. Explain why the marks can be equally spaced, and why the instrument becomes useless if it is badly overloaded.
Rung 4 · Take it somewhere new
A set of kitchen scales weighs up to 5 kg and its dial is marked in equal steps of 100 g the whole way round. Someone stands on it with one foot. Explain what must be true of the spring inside for the dial to work, and what the owner should check afterwards.
Key note
Forces deform objects as well as move them: they stretch and squash. For a spring, extension — the increase in length, not the length — is proportional to the load while the spring obeys Hooke's law, so the graph is a straight line through the origin and equal loads add equal extensions. That is what makes a spring a usable measuring instrument. Beyond the limit of proportionality the line bends and the spring stops returning to its original length.
Going further
Robert Hooke published this relationship in 1678, and he published it first as a scrambled anagram — a way of claiming a discovery without giving it away while he checked it. Unscrambled it says, in Latin, as the extension, so the force: nine words, no equation, no graph, because neither had been invented as a way of presenting results. Reading it now, the striking thing is how narrow the claim is. Hooke did not say all materials do this; he said that within a certain range, springs and wires do. Every honest law in physics comes with a range attached, and the interesting science usually starts at the edge of it.
Stretching a spring also stores energy. That is why a wound clock runs, a mousetrap goes off, a bow fires an arrow and a trampoline throws you back up — the work you do pulling it out of shape comes back when it returns. At GCSE you will find that the energy stored is the area under that straight line, which is a neat piece of reasoning: the further you stretch it, the harder each extra millimetre becomes, so the energy grows faster than the extension does. Squashing works the same way, which is what the springs in a car's suspension, the foam in a running shoe and the crumple zone of a car all rely on — though a crumple zone is designed to deform permanently and never come back, absorbing the energy instead of returning it.
Before this lesson
Connects to
At GCSE this becomes
- Hooke's law with a spring constant, elastic and inelastic deformation, and the energy stored in a stretched spring as the area under a force–extension graph.
Where to next
Ask Mr Badmus AI
Got a set of readings that will not make a straight line?
Before this one is done for real: this is the one investigation in the unit that needs a risk assessment, because finding the limit of proportionality means loading a spring until it stops behaving. Eye protection for everyone at the bench — an overloaded spring can let go, and it leaves at speed. A tray of sand or a padded box directly under the load, so a falling mass lands on something soft. Nobody's hands, feet or knees under the hanger at any point. The stand clamped or its base weighted, so the whole set-up cannot topple towards anyone. Loading to destruction is a demonstration, done once, behind a safety screen — not a class activity.
The spring bench is a teaching model. This spring is given 20 mm of extension for each newton, a limit of proportionality at 6 N, and permanent deformation above 9 N; all three are chosen so that the bend in the graph is reachable, not measured from a real spring. Real readings scatter by a millimetre or two, and a real spring's limit is not a sharp line. Loads are treated as exact newtons.
Lesson content © MrBadmusAI.