Wednesday, 19 November 2014

Hooke's Law Experiment

Hooke's Law Experiment

Konstantinos Fragkiskos kf4g14
"I am aware of the requirements of good academic practice and the potential penalties for any breaches"

1. Hooke's Law


According to Robert Hooke, an English scientist of the 17th century, for a range of values, the magnitude of an object's deformation is directly proportional to the force applied to it.

F = k Δx 

where F is the tensile or compressive force applied to the object, k is a constant that differs for every material and indicates how stiff it is and how easy or hard it is to be deformed, and Δx is the deformation. The greater the elasticity constant, the less the deformation for the same value of the force applied (in other words, the greater the constant the stiffer the material).

Under these conditions, if the object is released it returns to its original shape (it is within the elastic region).

The maximum value for which Hooke's Law is obeyed and the applied force (F) is directly proportional to the deformation (Δx or ΔL) is called limit of proportionality.
The maximum value for which the object if released returns to its original shape is called elasticity limit and beyond that the object enters the plastic region and it starts becoming permanently deformed and after a point it can even break.

It must be mentioned that when the elastic limit is exceeded, the applied force causes a bigger deformation than it would cause within the elastic region, and of course they are no longer directly proportional to each other.


The following graph can help us comprehend the correlation between the applied force (F) and the deformation (ΔL):



(Figure 1)



2. Experiment 


After carrying out a Hooke's Law experiment that investigates the behaviour of three materials we were given some results and were asked to interpret and analyze them.
The results are tabulated in the following table:

(Table 1)


The two first materials y1 and y2 are two different elastic materials that are still within the limit of proportionality, whilst the third material z has gone beyond the elastic region and has entered the plastic region which means it has started deforming permanently.



Materials y1 and y2


Plotting the deformation y against the applied force x for the first two materials we get the following graph:

(Graph 1)

  • The graph confirms that these two materials obey Hooke's Law and that the deformation is proportional to the force applied. This means that the materials are still within the elastic region and if they are released they will return to their initial form.


As expected we have linear trendlines for both materials, as the deformation is proportional to the applied force for both of them. 



  • We can notice that the slope of each trendline is equal to the constant k of the material raised to the power -1 because:
          k = F / Δx
           but  slope = Δx / F,
          thus slope = 1 / k.


  • By looking at the graph we can make an estimate of the value of x, for which the two lines meet: 2.3.
          Using the equations given by the graph and solving them simultaneously for the interception               point we have:

          y1 = 1.5583 x + 1.375
          y2 = 2.058 x + 0.2

          1.5583 x + 1.375 = 2.058 x + 0.2

          0.4997x = 1.175

          x = 2.3514 , which is very close to our estimate, thus the graph is pretty accurate.


  • Looking at the graph 1 we can also understand that the material y1 is stiffer than the material y2, as in average y1 deforms less than y2 when we apply the same force.


Possible Errors for materials y1 and y2:

From Graph 1 it is obvious that not all points of the material y1 are on the trendline. This is probably due to some random errors in making measurements during the experiment.


Another thing that indicates a random error in measurements is the fact that, if we try to calculate the constant k for each point of the graph seperately (either for material y1 or for material y2) , the constant will not always be the exact same number, although it should be.


In addition, we notice that when the force applied is equal to zero, the trendlines of both materials show us that there is still a deformation (initial deformation). The initial deformation of y2 is smaller than the initial deformation of y1, which is also the reason the lines intersect at the point x = 2.35.
It would be expected that when no force is applied, no deformation is caused because:
 Δx = F / k 
  and when F = 0,
 Δx = 0 / k = 0 .

There are some possible explanations for this fact:
-Maybe the random errors in the measurements affected the slopes of the trendlines such that they start from an initial value of deformation instead of zero.
-Another possible explanation is this: the initial length of the springs used to conduct the experiment was measured before the springs were hung. Therefore, when the springs were hung, even before any force was purposely applied on them, another force was applied to them that probably wasn't taken into account: their weight. The weight caused the springs an initial deformation which was not included to the initial length (the fact that the stiffer material has a greater initial deformation than the less stiff material, probably means that it is also heavier). Thus, this means that when the force applied by the conductors of the experiment is 0, there is still a value for the deformation which is the deformation caused by the weight of the springs.




Material z


This is the table including only the values of the force applied x and the corresponding values of deformation z for the third material:

(Table 2)


Plotting the deformation z against the applied force x for the third material we get the following graph:

(Graph 2)



From the graph it is obvious that the deformation of the material z is not directly proportional to the applied force x. As the force increases the deformation increases dramatically. For little force applied we have a much bigger deformation. 
This means that the material z has gone past the proportional limit, under which Hooke's Law is obeyed. It has also exceeded the elastic limit and is now in the plastic region, which means that the deformation it undergoes is permanent and once it is released it will not return to its initial shape.

All these confirm the given fact that the third material has gone past its elastic region and as expected the trend is no longer a straight line but a curve.




Possible Errors for material z:

Like in Graph 1, in Graph 2 not all points of the material z are on the trendline. This is also probably due to some random errors that occured in making measurements during the experiment and/or because of some systematic errors that are inherent to the apparatus.




3. Conclusion


In general, the results of the experiment and the graphs presenting them confirm Hooke's Law and the fact that when the deformation of a material exceeds the elastic limit then the deformation is much greater than the applied force.

Graph 1 regarding the behaviour of materials y1 and y2 shows that they obey Hooke's Law, as the two similar to each other trendlines of both materials are straight and the deformation is directly proportional to the force. Additionally, when the force will no longer be applied on the materials, they will return to their initial shape because they did not exceed the elastic limit.

Graph 2 concerned with the behaviour of material z confirms that it has gone past its elastic region into the plastic region, given that its trendline is a curve of a sharp increase meaning that the deformation equals the force raised to the power 3 (i.e. the deformation is much greater than the force applied). Thus, the deformation, which the material z is subject to, is permanent and it will not return to its initial shape if it is released. 

Therefore, on the whole the experiment was successful and supports the theory despite the experimental errors (inaccurate measurements due to human error or non-precise calibration of the apparatus).







4. References:


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