Computer Applications Assignment 1: Hooke's Law experiment - an investigation of the behaviour(deformation) of three materials.

Name of Author: Gabriel Wong Yi Chong

Student ID: 31542042

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Hooke's Law experiment - an investigation of the behaviour (deformation) of three materials.

Table of Contents:
1. Objectives
2. Outline

3. Background Information of Robert Hooke
4. Biography on Robert Hooke
5. Introduction
6. Results
7. Conclusion
8. Reference list

Objectives:

1. To investigate the behaviour of 3 materials, of which the results were given.
2. To present and analyze the said results in a blog post.

Outline:

An experiment involving the deformation of three materials when force was applied unto them was carried.  The results obtained were analyzed with Microsoft Excel. Then, the experiment with its analysis were documented on a blog post.

In this blog post, we will discuss the behaviour of a material in detail.

The results of y₁ and y₂ are used to analyse 2 materials with elastic properties, which are still in their linear regions.

The results of z will be used to describe the behaviour of a material that has exceeded its elastic limit-  the plastic region.

Background Information of Robert Hooke - The person behind Hooke's Law:

Robert Hooke was an English physicist who made discoveries in various fields. These include and are not limited to elasticity, light, planets, and diffraction (Britannica, n.d.). Evidently, to call this polymath as merely a genius would be a huge understatement (TecQuipment, 2018). As a child, Hooke’s was plagued with poor health (TecQuipment, 2018). As such, he was unable to attend school (TecQuipment, 2018). Despite this, he managed to write some of the most significant scientific works of all time, Micrographia (Britannica, n.d.). He was also the pioneer who stated that all matter increase in size when heated (TecQuipment, 2018). Today, we look into one of his most reputable works, Hooke’s Law. 

A brief biography on the brilliant Robert Hooke.

Video 1: Simon Whistler narrates the biography of Robert Hooke. (Biographics, 2019).

Introduction:

Hooke's Law states that the force required to compress or extend (deform) an object to a distance is directly proportional to the said distance (Augustyn, n.d.).

Hooke's Law states that within the limit of proportionality, the extension of a material is proportional to the applied force (Bird and Ross, 2015, p.52).

The Law is demonstrated with the formula:

F = kx,

where F is the force applied, k is the constant of proportionality and x is the length of compression or extension of the material (BYJU'S, n.d.).

Figure 1.

As shown in Figure 1, when the magnitude of force applied on the material was increased by two, so will the extension produced by the material (BYJU'S, n.d.).

Figure 2.

The applied force is directly proportional to the straight line, as shown in the linear graph in Figure 2, as long as the limit of proportionality is not reached (Bird and Ross, 2015, p.52).

Figure 3

Figure 3 illustrates a force against deformation graph (Lumen, n.d.).

In Figure 3, it is shown that Hooke's Law is no longer obeyed when the limit of proportionality is exceeded; when the force no longer has a linear relationship with the extension and compression of the spring, Hooke's Law no longer holds (Bird and Ross, 2015, p.52).

Here are two videos that perfectly and concisely illustrate Hooke’s law:

1.

Video 2: Springs and Hooke’s Law are discussed in detail in this video. (Science Shorts, 2017).

2.     

Video 3: Ryan Barouki explains Hooke’s Law with diagrams. (DoodleScience, 2013).

Results:

The values obtained in relation to Hooke's Law:

Table 1.

The results obtained from the experiment were shown in Table 1.  The results were colour-coded to suit the corresponding graphs. The x values denotes the force applied (in Newtons) and y₁, y₂ and z are the deformation (in mm).

The results for y₁ and y₂ and z are examined to determine if they obey Hooke's Law.

Graph 1.

The results of deformation of 2 different elastic materials, which are  y₁ and y₂ are plotted on Graph 1. 

It is shown that there was 1 anomalous result on the graph of y₁ against x. This could be attributed to the random errors encountered when obtaining the results of the experiment.

If the trendline of y₁ and y₂ is extrapolated to the x = 0, it is shown that there are y- intercepts for both results. This could be attributed to either systemic errors when taking the results for y₁ and y₂ or the elastic material deformed under its own weight before any external force was applied (x). 

The graphs are linear and the gradient of the graph of y₁ against x and the graph of  y₂ against x are constant, which proves that the two elastic materials tested were still in their linear regions (obey Hooke's Law).

   

Figure 4

In the cells above (Figure 4), it is shown that the equations of y₁ and y₂ are:

1. y₁ = 1.5583x + 1.375
2. y₂ = 2.0583x + 0.2

The equations are compared with the formula F = kx. 

The values of k corresponds to the inverse of the gradient of the graph of y₁ against x and inverse of the gradient of the graph of  y₂ against x.

The gradient of the graph of y₁ against x is a = 1.5583, 

while the gradient of the graph of y₂ against x is a +0.5 = 2.0583.

The value of k for graph of y₁ against x is:

The value of k for graph of y₁ against x is:

It is found that the k, the constant of proportionality of  y₁ is greater than that of y₂ . This proves that the elastic material that provided results (y₁) is  stiffer and more force is required to deform the said material as compared to the material that provided results (y₂).

The estimate of the value of x where the two lines of y₁ and y₂ intersect is shown in the cell is shown on the graph. This estimate is the value of x( force applied) where both elastic materials have the same deformation.

The estimate: 2.3N

The true value can be calculated with the command bar function in Microsoft Excel (as shown in Figure 5): 

=MMULT(MINVERSE(B47:C48),D47:D48)

Figure 5

Alternatively, the true value where the two lines meet could be solved via substitution:

Equation 1:

y₁ = 1.5583x + 1.375

Equation 2:
y₂ = 2.0583x + 0.2

Equation 1 is substituted into Equation 2:

y₁ = y₂ 

The value of x of the coordinate where the two lines meet is substituted into equation 1:

 From Equation 1:

The true value where the two lines meet is (2.35 N, 5.037 mm).

Graph 2.

The results of deformation of 2 different elastic materials, which are  y₁ and y₂ are plotted on Graph 1. 

The stiffness of the material appears to increase as the force applied increases.

The equation of the graph of deformation, z against force applied, x is a third power polynomial.

As the object is in the plastic region, it no longer returns to its original length when the force is removed (Bird and Ross, 2015, p.52).

Conclusion:

The experiment has successfully shown the linear relationship between deformation and applied force, as long as the object has not exceeded the limit of proportionality. The graphs of y₁ and y₂ against the applied force, x both have linear trendlines. 

Hooke's Law is shown to hold true when an object is in its elastic region.

However, the graph of z to the applied force, x has a dissimilar pattern as compared to both of the graphs mentioned. 

The equation of the graph of z to the applied force, x  is a cubic function. 

As the results (z) does not have a linear relationship with the force applied, x, Hooke's Law is not obeyed.

Hooke's Law does not hold true when an object is in its plastic region.

Reference List:

1. Augustyn, A. (n.d.). Hooke's law. [online] Britannica. Available at: https://www.britannica.com/science/Hookes-law [Accessed 31^st October 2020].
2. Bird, J. and Carl, R. Mechanical Engineering Principles 3^rd edition (Routledge Taylor and Francis Group, 2015), p. 52
3. BYJU's The Learning App, (n.d.). Hooke's Law -Stress And Strain. [online] Available at: https://byjus.com/physics/hookes-law-equation-experiment/ [Accessed 31^st October 2020].
4. Lumen, (n.d.). Hooke’s Law. [online] Available at: https://courses.lumenlearning.com/boundless-physics/chapter/hookes-law/ [Accessed 31^st October 2020].
5.  TecQuipment, (2018). Robert Hooke: Hooke’s Law. [online] Available at: https://www.tecquipment.com/knowledge/2018/robert-hooke-hookes-law [Accessed 16 November 2020].
6. Britannica, (n.d.). Robert Hooke. [online] Available at: https://www.britannica.com/biography/Robert-Hooke [Accessed 16 November 2020].
7. Biographics, “Robert Hooke: The Leonardo of England” YouTube, published November, 2019, [https://www.youtube.com/watch?v=chtxPnS1_GQ], accessed November 2020.
8. Science Shorts, “Springs & Hooke's Law - GCSE & A-level Physics” YouTube, published March 5, 2017, [https://www.youtube.com/watch?v=S3qjFfXXUkI], accessed November 2020.
9.  DoodleScience, “Hooke's Law | GCSE Physics | Doodle Science” YouTube, published December 29, 2013, [https://www.youtube.com/watch?v=dnebaW-a338], accessed November 2020.