Materials Tutorial 2 - Hookes Law

 

If we load a spring, we find that the extension (code e) or stretch is proportional to the force (code F).  If we double the force, we double the stretch.  This is called Hooke's Law.

   F e

F = ke

The constant of proportionality is called the spring constant (or force constant) and is measured in newtons per metre (N/m). 

 

Remember to convert the extension into metres and the load into Newtons before working out the spring constant.

 

In the syllabus, the extension is given the code Dl, i.e. the change in length.  In some text books, you may see it given as x.

 

Question 1

When a 500 g mass is placed on a spring, it stretches by 12 cm. 

 

What is its spring constant in N m-1?            

Answer

 

We can plot this as a graph:

We can see that the graph is a straight line and that the gradient gives us the spring constant.  That is why we have the extension on the horizontal axis.

The same is true if we apply a squashing force.

 

Springs in series and parallel

It may seem strange to bring in terms more familiar in electric circuits, but we can have springs arranged in series and parallel.  The picture below shows a single spring, then two springs in series.

 

If we load the first spring with a weight W, we see that it extends by e.  Now we attach a second identical spring in series, and put on the weight W.   The same force acts through each spring, so each spring stretches by e.  Therefore the total stretch is 2e.

 

Since k = F/e, we can easily see that the spring constant halves.

 

Question 2

When a 500 g mass is placed on a spring, it stretches by 12 cm.  A second identical spring is now placed below the first.

 

How far do the two springs in series stretch?

 

What is the new spring constant in N m-1?            

Answer

 

 

The picture below shows the same two springs in parallel:

The springs are identical to the ones before.  This time the weight W is shared out equally between the two.  Since each spring has 0.5 W acting on each one, the stretch is 0.5 e.  The spring constant of the parallel springs is therefore k = W 0.5e = 2k.

 

Question 3

When a 500 g mass is placed on a spring, it stretches by 12 cm.  A second identical spring is now placed alongside the first.

 

How far do the two springs in parallel stretch?

 

What is the new spring constant in N m-1

Answer

 

Cars can be modelled as a block of mass m, placed on 4 springs each of spring constant k.

 

Question 4

A car of mass 1600 kg is placed on four identical springs.  Each spring is seen to be compressed by a distance of 5 cm.  What is the spring constant of each spring?

Answer

 

 

Elastic Strain Energy

The energy is the area under the force-extension graph.  How do we achieve this result?

 

If we stretch the spring by a tiny amount dl, we do a tiny job of work:

 

 

This is shown by the little rectangle.

 

Do it again, we get another little rectangle:

Now fill in all the little rectangles:

All the little rectangles give area under the graph to give:

 

 

A neater graph is shown here:

 

So we can use this result to say:

This result can also be obtained using the  process of integration, which is part of the branch of mathematics called calculus.  You are not expected to know this for AS, although you will be if you study Physics at University.  All we need to say here is that the energy is the area of the triangle.

We can derive two further expressions from this result:

and

Question 5

Show that E = ke2.                                                                                   

Answer

Question 6

Use a similar method to show that E = 1/2 F2/k

Answer

Question 7

A car of mass 1600 kg is placed on four identical springs.  Each spring is seen to be compressed by a distance of 5 cm.  What is the energy in each spring?

Answer

 

When we stretch the spring, we have to do a job of work.   If we release the spring, we can recover that energy, which is called the elastic strain energy.  Ideally we recover all of it but in reality a certain amount is lost as heat.  This lost energy is called hysteresis.  This is shown on the graph in the yellow triangle.

Hysteresis, please, not hysteria.

 

 

Elastomers

A rubber band is an example of a material that does not obey Hooke's Law.  The force extension graph looks like this:

Rubber is an example of an elastomer.  It is made up of long polymer chains that are tangled up with cross-links (disulphide bridges) between the chains.  The chains with no load are rather like spaghetti.  When the load is small, a large extension is seen because the chains are simply untangling. 

Once the chains are untangled, then the bonds between atoms are stretched.  This requires a greater force with a small extension.

We can see from the graph that there is a large hysteresis.  This is because work has to be done to break cross-links as the elastomer is stretched.  When the elastomer is relaxed, the cross-links reform, and give out heat as they do so. 

Natural rubber is quite soft and flexible as there are relative few disulphide bridges.  Hard rubber is vulcanised, which means that more disulphide bridges are added.