Hooked on Hooke’s law? An investigation into Robert Hooke’s law of elasticity and plasticity of materials.

Hersh Khatri, Student ID : 27842908

Hooked on Hooke’s law? An investigation into Robert Hooke’s law of elasticity and plasticity of materials. 

Introduction: 

In this experiment, 3 different materials were used to measure the spring constants of the respective springs. According to Hooke’s law, the material constant is equal to the force exerted on the material divided by the change in length of the material or the extension of the material from the original position (k=  F/∆x , where k = to the material constant, F = to the force applied, and ∆x = to the change in length or extension) (Ross,2015).

Aim:

The aim of the experiment is to record the extensions of the 3 different materials and investigate the relationship between the force applied to the material to the extension of the material or change of length of the material.
For 2 materials, the materials would be extended within their elastic limits and the remaining material, the material would be extended into its plastic limit.

Abstract: 

During this experiment, 3 materials were put under a force that increased linearly by 1 N every test. The results showed that at 9 Newtons ( the maximum force applied in the experiment), Material Y1 had the least deformation at 15.00mm, Material Y2 had the second least deformation at 18.72mm and finally Material Z had the most deformation at 730.38 mm. It could be concluded that Materials Y1 and Y2 underwent linear elastic deformation which abides by Hooke's law (Ross,2015), whereas Material Z underwent plastic deformation.

Table of results:

This is the table of results that was collected during the experiment. The force exerted on the different materials was controlled, and was increased by 1 Newton every test on each material.

Table 1: Raw Data 

What do these results represent and what was observed?

From these results that have been gathered, we can see that Material Y1 has a lower rate of increase in comparison to Material Y2. This would indicate that Material Y1 requires more force in order for it to stretch in comparison to Material Y2. From this, we can say that the material is stiffer than Material Y2 as it requires more force to stretch and deform.
During the experiment, it was identified that the two strings that remained in their elastic limit returned back to their original state as to agree with Hooke’s law of the elastic and plastic limits (Ross, 2015)of materials. It was also observed that the third material surpassed its elastic limit and did not return back to its original state, hence staying in the plastic state.

Graphing the data:

Graph 1:

Graph 1 - Material Y1

Graph 2:

Material Y2 and Material Y1

Solving the simultaneous equation to find the intersection:

Graph 3:
 

Material Z

Analysis of the graphs:

Analysis of Graph 1:

Graph 1 is the representation of the data of the Force Applied, measured in Newtons, and the deformation of Material Y1. The graph indicates that there is a linear relationship between the force applied and the deformation of Material Y1. This fits into Hooke’s law and Young’s Modulus of Elasticity (Ross, 2015) as it indicates that there is a positive linear relationship between the Force Applied and the Deformation meaning that the material is still in the elastic phase.
It can also be noted that the trend line on the graph does not begin from the origin, which represents that there is a systematic error in the experiment. This could have been caused by a measurement error during the process of the experiment.

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Analysis of Graph 2:

Graph 2 is the representation of both data sets of the deformation of Material Y1 against Force Applied and the deformation of Material Y2 against Force Applied. The graph is a comparison between the two results which makes it easier for the reader to compare the two sets of data. The two graphs represent Hooke’s Law which is ‘that up to the elastic limit, the extension, x, of a spring is proportional to the tension force, F.’ (Kirk, 2007)
As you can see from the graph, the two lines represent the individual data sets and you can see where they intersect. This intersection point signifies the point at which the deformation of both materials are the same length at which the same amount of force applied to them. This length is at an extension of 5.04mm and with a force of 2.35N. The working out of this is below, which was solved using simultaneous equations.
The two materials display linear graphs which means that we can conclude that these two materials have not gone under any plastic deformation and only elastic deformation. It can also be noted that Material Y2 displays to deform more than Material Y1 as the force is increased. The difference between the two materials is always increasing as the line of Material Y2 has a steeper gradient.

Analysis Graph 3:

Out of the three graphs, graph 3 is the irregular graph. The two previous graphs displayed functions that were linear and that created a line of best fit of a straight line. However, graph 3 represents an exponential function as the values for the deformation increase exponentially as the force applied to

http://www.1800trampoline.com/ProductImages/stretchedspring.jpg

Material Z increases. As Hooke’s Law only applies to materials up to the elastic limit. This means that graph 3 does not coincide with Hooke’s law as it has broken through the elastic limit and now in the plastic limit.

Improvements to the experiment:

The experiment could be improved by having more results and repeats of each test. However, this could be a problem as for a different spring, the spring constant would be different, so therefore a repeat of the test could be difficult.
Another way that the experiment could be improved is by having more frequent forces being tested on the spring, for example, testing the spring with increases in the force by 0.5 N. This would allow the data to be more accurate as well as precise.

References

• Kirk, T., 2007. IB Study Guides Physics for the IB Diploma, Standard and Higher Level. 2nd ed. Oxford : Oxford University Press .
• Ross, J. B. &. C., 2015. Mechanical Engineering Principles. Third ed. Oxon: Routledge.