Hooke's Law Experiment

Dominic Maddison 25542664

"I am aware of the requirements of good academic practice and the potential penalties for any breaches"

Introduction

Image 1: Robert Hooke [1]

For those who have never heard of Robert Hooke and/or his law of elasticity I will give a brief summary of who he was and what his law states.

             Robert Hooke was a scientist (predominately physicist) born on the Isle of Wight in 1635 and died at the age of 67 in London year 1703. He is best known for his law of elasticity, also known as Hooke’s law which is explained by Encyclopedia Britannica as; “for relatively small deformations of an object, the displacement or size of the deformation is directly proportional to the deforming force or load. Under these conditions the object returns to its original shape and size upon removal of the load” [2]

This law can also be expressed as the equation; 

                                                                       F=kx   [3]

             F=force (newton’s) 

k=constant

                       x=extension (meters)       
             

Figure:1 A diagrammatic representation of Hooke's law

If anyone is still having trouble with the concept of Hooke's law hopefully Figure 1 will help clarify what it states.

            
We carried out an experiment in the University of Southampton physics labs which investigated the behavior of three different elastic materials. y1,y2 and z. The aim of the experiment was to determine the relationship between load (force in newton’s) and the deformation of the material in millimeters.

Table 1: Results collected from first experiment.

Table 1 shows the data we collected in the first experiment. The data is show to 2dp. The x columns represents the load aplied in newtons and the y1 and y2 columns show the respective deformation in millimeters.

Figure 2 : Graph to show the linear relationship between force and deformation

I plotted the graph above from the data we collected from the experiment (Table 1). The graph not only allows us to see that there is a clear linear relationship between the force applied and the deformation of the materials but it also allows us to make a calculation for the k values of the two materials. Using excel the gradient of the two lines can be obtained in the form y=mx+c. The k values for the two materials are simply 1/m as the c values can be ignored because the graphs should in fact go through zero as when zero force is applied there should be zero deformation.

This is also the reason why the two 

graphs cross because if both the graphs started at zero deformation then they would not cross but simply go up at different gradients. This may be partly due to the force of gravity already acting on the materials which would deform them slightly but could also be due to the materials being damaged by exceeding there elastic limit in a previous experiment and so know longer return to the original length. As the force (N) was applied y2 deforms more than y1 and so quickly crosses it. The graph can be used to make a rough estimation of where the two lines cross which gave me a value of roughly (2.4, 5.0) (hard to be accurate with the scale as it is). A more Accurate answer can be calculated using the two equations on the graph and doing a simultaneous equation:

Y1=1.5583x + 1.375

Y2=2.0583x + 0.2

1.5583x + 1.375 = 2.0583x + 0.2

0.5x = 1.175

X = 1.175/0.5

X=2.35.

Then sub this value for x into either of the first equations to get a value for y which is- Y=5.03

So the actual co-ordinate that the two lines cross is (2.35, 5.03) 


The second experiment we conducted aimed to show what happens to a material when it passes its elastic limit. Would the deformation still be proportional to the applied load or would this no longer be the case.?

Table 2:Second experiment material z

Figure 3: Shows deformation of material Z when excessive force is applied

This second graph is again showing the relationship between applied force (N) and the deformation (mm). However in this graph we can clearly see that there is not a linear relationship. This is because this material has passed its “elastic limit” and so Force and deformation will no longer be directly proportional to one and other or show the linear relationship that the previous graph shows. Once a material passes this limit it behaves unpredictably and there is no real way to assume what it will do. Some materials will stop stretching and then snap whilst others will deform massively. 

Possible points of error


Most experiments that are carried out seem to develop possible errors through a variety of different reasons. The aim is to spot these errors when looking through the data and try and work out why they occurred. In this way we can help reduce errors in future experiments and so improve our results and findings. This experiment was no exception and some errors may have occurred. In figure 2 the line y1 helps identify a potential error. All the other points fall nicely on the line (or close too) where as this point falls some way of the trend-line. This could have been due to a number or reasons including; human error when taking the readings of deformation or the weights used (force in newtons) not being properly calibrated and so the 7N force may have actually been more like 7.5N of force applied.


In conclusion
The first experiment allowed a clear graphical representation of Hooke's law showing the linear relationship between force applied in newtons and deformation in millimeters. This therefore backed up Hooke's law which states they a proportional to one and other.
The second experiment allowed us to see that Hooke's law has its limits and it does not extend pass the elastic limit of a material. We can see this in the definition of Hooke's law where is states "for relatively small deformations".


[1] http://www.history-of-the-microscope.org/robert-hooke-microscope-history-micrographia.php
[2] http://www.britannica.com/EBchecked/topic/271336/Hookes-law
[3]http://www.physics247.com/physics-tutorial/hookes-law.shtml
[4] http://www.buzzle.com/articles/how-does-a-spring-scale-work.html

[5] http://www.britannica.com/EBchecked/topic/271280/RoberHooke