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Unlock This CoursePull on a hair bobble and it gets longer. Let go and it snaps back. Pull the spring out of a ballpoint pen too far and it stays long and useless. This lesson explains both results.
When a force stretches an object, we do not measure its new length alone. We measure how much longer it has become. This is called the extension.
extension = stretched length − original length
Key terms:
A spring is 5.0 cm long with nothing hanging from it. With a 3 N weight on the end it is 8.5 cm long. Calculate the extension.
extension = stretched length − original length
extension = 8.5 − 5.0
extension = 3.5 cm
The extension is not 8.5 cm. That is the length of the whole spring.
Hang weights on a spring one at a time and measure the extension each time. A clear pattern appears.
| Force (N) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| Extension (cm) | 0 | 2.0 | 4.0 | 6.0 | 8.0 | 10.0 | 13.0 | 18.0 |
Up to 5 N, every extra newton adds the same 2.0 cm. Double the force from 2 N to 4 N and the extension doubles from 4.0 cm to 8.0 cm. Remember, this pattern is called directly proportional. The spring is obeying Hooke's law.
After 5 N the pattern breaks. The sixth newton adds 3.0 cm and the seventh adds 5.0 cm. The spring has passed its limit of proportionality and no longer obeys Hooke's law.
Key terms:
A spring obeys Hooke's law. A force of 2 N gives an extension of 3.0 cm. What is the extension with a force of 6 N?
The force is 3 times bigger (6 ÷ 2 = 3).
Extension is directly proportional to force, so the extension is 3 times bigger too.
extension = 3 × 3.0 = 9.0 cm
This only works if 6 N is below the limit of proportionality.
There is no drawn graph here, so follow these steps and sketch it yourself.
The fact to learn is this: the initial linear region of a force-extension graph is associated with Hooke's law. Where the graph is a straight line through the origin, Hooke's law is obeyed. Where it curves, it is not.
Some questions swap the axes and put extension on the vertical axis. The straight part still shows Hooke's law. The only change is that the curve then bends upwards.
Different materials give different graphs. Each one is described here with force on the vertical axis.
A long straight line through the origin, so it obeys Hooke's law over a wide range of forces. Past the limit of proportionality the line curves and gets less steep.
A very steep straight line at first. It obeys Hooke's law, but large forces give only tiny extensions. Past the limit the line curves over and becomes almost flat. The wire now stretches a lot for very little extra force, and then it snaps.
A curve almost all the way. There is no clear linear region, so a rubber band does not obey Hooke's law. The line starts at the origin and its steepness keeps changing. The band stretches easily in the middle, so the line is less steep there. Near its full stretch the band becomes stiff, so the line gets much steeper.
What happens when you take the force away? That depends on how far the material was stretched.
The material recovers its original shape after the forces causing the deformation have been removed. A trampoline spring stretches when you land and goes back when you bounce off. A squash ball flattens against the wall and becomes round again.
The material does not recover its original shape. It stays stretched, bent or squashed after the forces are removed. A paper clip bent straight stays straight. A crushed drinks can stays crushed.
Key terms:
A spring behaves elastically as long as the force is not too large. Stretch it too far and it is permanently deformed. Take the weights off and it is longer than when you started. The same is true of a metal wire.
A rubber band makes an important point. It does not obey Hooke's law, but it still shows elastic behaviour, because it goes back to its original length when you let go. So elastic and obeys Hooke's law do not mean the same thing.
A student hangs weights from a spring. The spring is 4.0 cm long with no weight on it. The student draws a force-extension graph. The graph is a straight line through the origin up to a force of 6 N. Above 6 N the graph curves.
(a) With a weight of 3 N the spring is 5.5 cm long. Calculate the extension of the spring. [2 marks]
(b) Explain how the graph shows that the spring obeys Hooke's law for forces up to 6 N. [2 marks]
(c) State what is meant by elastic behaviour. [2 marks]
(d) The student loads the spring to 9 N and then removes all the weights. The spring is now 4.8 cm long. Explain what this shows about the spring. [2 marks]
(a) extension = stretched length − original length = 5.5 − 4.0 (1)
extension = 1.5 cm (1)
(b) The graph is a straight line through the origin up to 6 N (1). This shows that extension is directly proportional to force (1).
(c) The ability of a material to recover its original shape (1) after the forces causing the deformation have been removed (1).
(d) The spring has not returned to its original length of 4.0 cm (1), so it has been permanently deformed and did not behave elastically at this force (1).
In part (b), 'a straight line' is not enough. Say that the line is straight and goes through the origin, then link it to extension being directly proportional to force. A straight line that misses the origin does not show direct proportion.