Springs and things
Springs, rubber bands and wires are all stretchable items: if you apply a force to them they get longer and, conversely, if you oblige them to take up a longer length, a tension will arise within them. The tension and the associated elongation are related, although not always in a simple way. It is best to take the view that the extension causes the tension force rather than the other way round. The relevant equation in simple Hookean cases is F = k x.
Here are two graphs.
The arrows indicate whether we are currently stretching the item or releasing it. Note that the horizontal axis is the extension - i.e. how much the specimen has stretched. If we were to put the overall length of the specimen on the horizontal axis, then the graphs would be shifted bodily to the right by an amount equal to the original length of the specimen.
Look at the right hand graph, for a rubber band.
There are no straight-line portions on this graph.
The extension for a 2 N force is twice as much if you are currently releasing it as when you are stretching it. So the history of your experiment matters. This is called hysterisis (but I don't think any pun is intended).
When you finally release the rubber band, it ends up at the same length as it started, in spite of the curious states it has been through. This is called elastic behaviour (and allows us legitimately to call the specimen an 'elastic band').
Look at the left hand graph. Springs and wires both exhibit this behaviour, although the extensions are very different: if graphs for both were plotted on the same axes, the wire graph would look almost vertical by comparison with the spring graph.
For low tensions the graph is straight. Tension is proportional to extension: the specimen is obeying Hooke's law (F = k x)
P is called the limit of proportionality. For lengths and tensions greater than the co-ordinates of P, Hooke's Law is no longer obeyed.
Y is called the yield point (it sometimes coincides with P). Before this point the material behaves elastically, and the graph simply tracks backwards along itself when you release the tension. After point Y, it takes a parallel path upon release, like the one shown, resulting in a permanently deformed specimen. Y is sometimes called the elastic limit. When the specimen has gone past the elastic limit it is said to be behaving plastically.
Oscillations
This discussion includes all the really important ideas of dynamics at IGCSE so, although it seems complicated, it may repay study. If you just look at one diagram at a time, it may not prove too indigestible.
In what follows, bear in mind that a = Fresultant / m. We will take downwards to be positive. g is both 9.8 N/kg and also 9.8 m/s2.
Diagram A
The spring is unstretched. The only force acting on the block is the weight W, which arises from the earth's gravitational pull on the mass m. This weight acts at the centre of gravity of the block.
The
resultant force on the block is W,
so the acceleration is a = W/m = mg
/m = g = 9.8 m/s2
downwards.
The block is not moving at present, but it is accelerating. Therefore it will set off downwards, picking up speed quite quickly.
Diagram B
The spring has a small extension x1. Therefore there is a small tension T1 (which arises from Hooke's Law). This force acts at the top of the block. It is represented by an upward arrow, shorter than the arrow representing the weight.
The resultant of W and T1, acting in opposite directions, is a small downwards force.
The acceleration is (W - T1)/m, which is still downwards, but is now less than 9.8 m/s2.
The block has been accelerating downwards ever since it set off and has by now picked up a moderate speed. It is still accelerating downwards.
Diagram C
Because the block has been moving downwards all this time, it has now reach such an extension x0 that the tension T0 is exactly equal to the weight.
The resultant force is now zero, and the acceleration is therefore zero too.
The block has by now picked up even more speed and is in fact moving at its maximum speed.
Observe that if we catch the block at this point, and carefully 'place' it in this position but in such a way that it is not moving, then it will stay where we put it. This is because the acceleration is zero (by the argument presented in the first two bullet points of this section), which means that the velocity does not change from the zero velocity with which we have placed it. But ignore this observation when considering diagram D
Diagram D
The block has continued moving downwards and has now produced such a large extension that the tension T2 is bigger than the weight W.
The resultant force is now upwards, so the acceleration is also upwards and equal to (T2 - W)/m.
An upwards acceleration counts as negative, and this has been in operation ever since the block passed through the position of diagram C. So by now the downwards speed has diminished, and the block is moving quite slowly.
What happens next
Shortly the speed of the block will reach zero.
By then the tension will be even bigger, and the resultant force will be an even bigger upward force.
The upward acceleration will be quite big, so the block will start to move upwards.
Etc, etc. . . .