Hooke’s Law

Plan:

            Tension and compression are forces which increase or decrease the dimensions of an object or structure. An object under tension has been extended. Its original length has been increased. Under compression, the original length is decreased.

            In our daily lives, we can find many situations in which there are materials which are under tension or compression. For instance, the suspensions of a car. The suspensions are really just a pair of springs which extend and compress under the tension and compression, so that the car bottom doesn’t scrap the floor. When a car hits a bump, at first the springs are compressed since the wheels are pushed upwards due to the rise in the ground. Once over the bump, the weight of the wheels extends the suspensions. If there weren’t any suspensions, car and bike rides would be very bumpy and wouldn’t be as pleasant.

            Another such example are the springs we find in our beds. Most beds contains springs in order to make the mattress as comfortable as possible. When we lie down on the bed, our weight is absorbed by the springs which are put under compression. This is why, with some beds, when we lay on them we ‘sink’ into it. The ‘sinking’ effect is produced by the compressed springs under your body and the other ones which are still extended normally, since no force is exerted on them. Naturally, with stiffer springs, the ‘sinking’ effect will be reduced since it will be harder to compress them.

            A last example of tensions and compression is one the latest sports: bungee jumping. People throw themselves off of buildings or bridges, naturally attached to an elastic rope, in order to have thrill of the experience of free-fall. When falling, the elastic cord begins to stretch until it reaches a point where the force acting on the rope, i.e. the weight of the person, cannot extend it further. As a result, due to all the elastic potential energy accumulated, the cord begins to return to its normal extension and therefore begins to pull the person back up again.

            Naturally, however, the ropes must undergo careful tests in order to assure their structural integrity. Additionally, the bungee jumpers also need to know its maximum weight load, since if any extra weight is put on it, then the rope will surpass its elastic limit, in which case it has been deformed and consequently will not return to its original extension. This is a very serious problem since if the cord does not stop stretching, the jumper may hit the ground or the water, and possibly die.

            This actually leads to the investigation which I will be carrying out. I will be analysing the behaviour of springs under loads, recording their extension and possibly trying to find out their elastic limit. In short, the experiment consists in proving, or disproving, as the case may be, Hooke’s Law. But first, we must understand what it actually is about.

Hooke’s Law:

              The simplest behavior under the action of a force is probably that of a steel spring. Robert Hooke, a British physicist, carried out a similar investigation and he saw that as the weight increased, so did the extension of the spring. However, what was exceptional was that, the extension of the spring augmented by the same amount as the force. This meant that the two factors were related: the extension of the spring is directly proportional to the weight of the load. This means that if the weight of the load is doubled, the extension of the spring will also double; if the weight is quadrupled, so is the extension. This could be written in other words as:

F α x or F = -kx

Where F is the weight of the load and x is the extension in centimeters of the spring. The k is the constant and the negative sign shows that the force exerted by the spring is in the opposite direction to the extension. This is because while the extension of the spring is downwards, due to the weight of the load, its force is upwards since it is trying to pull the load up in order to return to its normal extension. This is known as Hooke’s Law.

However, some of his further investigations have shown that this is true only up to a certain point. Beyond this limit, the extension increases more rapidly than expected, and the spring doesn’t return to its original extension: it has been permanently deformed. The limit at which this occurs is know as the proportionality limit. If the spring is stretched further, it reaches the elastic limit. Up until the elastic limit, the spring returns to its normal shape, however, beyond it, it becomes deformed and remains extended.

           Many materials obey Hooke’s Law to some extent. Two extremes of behavior are shown by copper wire and a rubber band. Copper suddenly gets easier to stretch and begins to flow until it breaks. Rubber, on the other hand, gets harder and harder to stretch and finally snaps when it breaks.

Now in this experiment, not only will I verify Hooke’s Law, but I will also investigate what happens when two or more elastic bands are put in parallel or in series. In addition to this I will also see how the thickness of the rubber band effects the extension. investigate the elasticity of materials other than the elastic bands, in order to see whether Hooke’s law is always followed.

            In order to make the experiment as fair as possible, I will use the same band while changing the weight loads. Then I will change the elastic material and use the same weight loads. Furthermore, I will wait for at least 10 seconds before taking the readings so that they can be as accurate as possible. If I recorded the result straight away, the band could still be stretching, or better, ‘bouncing up and down’, which would make recording difficult. Actually, to minimize problems with recording readings, I will attach a pointer to the band so that I can easily see how much it has stretched.

             Personally, I believe that the results of the experiment will support Hooke’s law. That is, I will see that the extension of the elastic band should be proportional to the weight load put on the elastic material. As a result I should get a graph with a straight line going through the origin, similar to the one shown below:

However, naturally beyond the elastic limit, we will see an increase in the extension, probably without any relationship with the increase of force.

            It will be interesting, though, to see what kind of a pattern we get when we use more than one elastic. Now, according to scientific knowledge, a large area of cross-section gives a small extension for a particular force. Consequently, this means that, if we put springs in parallel, then the extension should be smaller since the cross-sectional area should be larger. Nevertheless, if we put the bands in series, then we should get a bigger extension since the cross-section is smaller. Therefore in my opinion, the graphs should look something like this:

Results:

 Using one elastic band

Changing the thickness of the elastic band

Placing two elastic bands in parallel

Placing two elastic band in series

Changing the thickness of the elastic band

Placing three elastic bands in parallel

The elastic band of a pair of boxers

Analyzing Evidence:

             In this experiment, I investigated the properties of elastic bands. I tested most possible factors, such as thickness, length, elastic material, in order to see whether Robert Hooke was correct in his theories and I must say that he was.

            As a matter of fact, if we look at he first graph, in which I displayed the results for just one elastic band, we can easily see that there is a straight line. Hence this proves Hooke’s law, in which the British scientist stated that the extension of the elastic material is proportional to the weight load applied on it. This can be written with variables as:

F  α  x             where F = the force applied on the elastic material and

x = the extension of the material

To put it in simple words, this means that the two factors increase, or decrease, in the same ratio. So if I put twice as much weight on the elastic band, then the extension of the latter will be twice as much; if I treble the force, the extension increases just as much. As a result, I can say that my predictions were also correct.

            In addition to this, we can see that this pattern is seen in all of the graphs thus suggesting that despite the factors that I had changed, Hooke’s Law is still followed. Naturally, however, in a different way. In fact, we can see that if the thickness is increased, we could say, the ‘resistance’ or its tensile strength increases. That is, while for one elastic band a force of 7 N would break it, an elastic band with twice the thickness of the original one will not snap until a force twice as large is applied, i.e. 14 N. This consequently, suggests that there is also a relationship between the thickness of the elastic material and the tensile force. In fact, there seems to be a behavior similar to that between force and extension. So I believe that I would be correct in saying that the thickness of the elastic band is proportional to the tensile force applied. The tensile force per unit area is called tensile stress and is measured in N m^-². Consequently, if we divide the latter by the former we can get the tensile stress of the elastic material:

Tensile stress  =          tensile force    

                           Area of cross-section

Regarding the thickness of the bands, we can also say that if we use three elastic bands of one small thickness (i.e. 0.2 cm), the tensile strength, and hence the tensile stress, will be the same as that of one elastic band with a thickness three times that of the small one (i.e. 0.6 cm).

            On the other hand, when I tested elastic bands in series, I noticed a much different behavior. In fact, if we look at the results of the two elastic bands placed in series, we can straight away see that it extended twice as much as just one elastic band, for the same amount of force. This suggests, therefore, that the longer the elastic material the more it will stretch. Actually we can be more precise and say that if the elastic band is twice as long, it will stretch twice as much; if it is three times as long, it will stretch three times as much. The extension per unit length is called tensile strain, and can be found by dividing the extension by the original length:

Tensile strain  =          extension        

                               original length

Naturally, however, even though there is a great extension, the tensile strength is a lot smaller. In fact, an elastic band twice as long has the same tensile strength as that of just one band.

            Last, but not least, we can see that even different types of elastic material obey Hooke’s law, thus suggesting that most materials do follow it, maybe even for sufficiently small tensile strains. Anyhow, we can see that the material used is much more elastic since it stretches just as much as three elastic bands in parallel and has the same tensile strength, even though its thickness is half of the latter. Consequently, this tells us that even though most materials follow Hooke’s law, their tensile stress and tensile strain may vary, depending on the material used.

            On the whole, I would consider this experiment quite a success, since I managed to prove Hooke’s Law as well as my predictions, and even though I did encounter some difficulties, which I will mention in the following section, I did not have may accuracy problems since I did not find errors in my results.

Evaluating Evidence:

             Throughout this experiment I did not meet many problems with accuracy. However, I did have some slight difficulties in recording the results. In fact, despite the use of a pointer to facilitate taking down the reading, I still was not sure at times whether it was a millimeter more or less. As a matter of fact, this is shown by the error bars drawn on the graphs. Hence this resulted in a minor inaccuracy, however, it did not affect my results that much since they still followed Hooke’s Law.

            Anyhow, in my opinion there is only one way to solve this problem and that is to resort to electronic apparatuses. In fact, by using some sort of laser beam, the computer to which it would be attached to could easily calculate the extension, and do so probably more accurately than the human eye. In addition to this, the computer could also come up quickly with graphs and other results in order to minimize the scientist’s workload and maximize the efficiency and accuracy of the experiment.

            Some of the further information that could be given could be the work done on the elastic band. As a matter of fact, this can be easily found by calculating the area under the force-extension graph. It can also be calculated easily through mathematical means. As I mentioned in the plan, knowing that the force exerted by the spring for a given extension x, is –F, then the energy stored in the stretched spring is equal to the work done on it as it is stretched. This can be written as the following formula:

Work done on spring = ½kx²

With this formula, as well as the area under the graph we can easily calculate the work done by the elastic material until a certain extension.

From the table we can see that the most work was done by the last three elastic bands. This makes sense since they were the ones to stretch the most and to have the greatest tensile strength, thus allowing them to do greater quantities of work.

            Another such variable which could be found by the computer could be the Young’s modulus. Samples of all materials are found to obey Hooke’s lay for at least small tensile strains. Under these circumstances, the ratio of tensile stress over tensile strain is constant. This quantity is called the Young modulus E:

E  =      tensile stress     =  F/A

            Tensile strain         x/l

Measured in Pascal, the Young Modulus of a material is one piece of information used by engineers in selecting materials for particular uses.

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