Hooke’s Law Analysis

Hooke’s law is found under a part of physics known as elasticity. Elasticity is the ability of a material, usually a spring, to stretch to a certain length and then return to its original length when the force that caused the deformation itself is removed[1]. Hooke’s law was discovered by Robert Hooke in 1660, and the law itself states that for small deformations, the displacement of the deformation is directly proportional to the load/force causing the deformity[2]. This discovery by Hooke laid the foundation for the understanding of different elastic materials and the foundation for stress and strain, 2 other very important concepts in engineering mechanics[3].

The equation for Hooke’s law is the following:

f = kx

Where f is the force

Where k is the spring constant

Where x is the displacement of the string from the original position

Figure 1: Hooke’s Law Demonstration [2]

Analysis of Graph #1: Y=ax+b

Using the following data points, a graph can be generated that should show a linear correlation. This is due to the direct proportionality between the displacement and the force.

X here is the force applied in Newtons, while y is the deformation of the spring in millimeters. The linear correlation can be seen through each additional Newton added. The string stretches vertically by 1.5mm. This is in the data point of x=1 to x=6. However, in data points x=6 and x=7 there is a clear difference as the data point itself is no longer within the boundaries of the line of best fit. Here, the additional Newton between 6N and 7N caused the spring to stretch between 10.50mm to 13.00mm. This could be construed as an error made when the data was collected, most likely more weight was added than realized. This can also be shown through the data points after where each additional newton only stretched the string by 1mm. This is a systematic error that caused the graph to be inaccurate. Although, this can be construed to be a change in pattern. It is possible for the string to begin to stretch at a different rate through each additional Newton. It is possible that the string itself could have reached its elastic limit, meaning that it can no longer be set back to its original length and shape due to the induced stress. Furthermore, Hooke’s law is only an approximation, thus when measured the string may display a different length than what was originally calculated.

The y intercept calculated was b = 1.375. In Hooke’s Law, the y intercept represents the original length of the string prior to the displacement. Here there is 0 force applied to the string, according to the graph and data, thus this is the value of the string prior to stretching. Using the numbers provided and some calculations, Y(9) = 1.5583(9) + 1.375 = 15.3997 = 15.4 ,the string has stretched a total of 14.03mm.

Analysis of Graph #2: z=x3+b

By cubing x, an exponential or logarithmic curve is produced. Here this represents a rate of increase, as more force is added onto the spring, the longer it gets. This continues until the spring has reached its elastic limit. At this point the curve itself has reached its vertical asymptote. Here, y will continue to increase without the need of additional forces, until the spring itself reaches its material limit and tears or seizes. This follow Hooke’s law and the law of elasticity quite well. The material will continue to increase vertically without the need of excessive force until the point where the spring snaps. This breakage occurs when the gradient of is equal to 0.

Analysis of Graph 3: Y1 vs Y2

The graph above shows the two Y’s, Y1 and Y2, side by side. It shows the relationship between force and extension. The graph of Y2 (orange line) appears to have a higher spring constant compared to Y1 (Blue line). This can be seen when the same amount of force is applied, at 4N, the extension is greater by over 1mm compared to Y1. The graph also shows that the two springs meet at a certain point, meaning that at one point both the extensions are equal. The intercept of both x and y were calculated using the following formulae: [4]

The x intercept and y intercept were calculated via simultaneous equations, the following is the process of calculation:

By simplifying and solving for x

Then by substituting back into the first equation

The same value is received when x is substituted back into the second equation

The table above displays the x and y intercepts calculated from both the simultaneous equations and through excel. The data calculated from both methods are very close to each other. As can be seen, the excel formulae produced much more accurate results, usually to the hundredth decimal place. This is because excel does not round the numbers as the calculations are made, while when doing them by hand rounding occurs due to the functions of the calculator. Thus, the data provided by excel is much more accurate than the one by hand.

Conclusion:

Hooke’s law suggests that the strain applied to a spring is proportional to the stress that has been applied. This law is shown to be true within the 3 graphs above. All the 3 graphs displayed above are very closely related. They each demonstrate the displacement due to the force at different points in time. The first graph shows the displacement increase over a period of increased force and graph 2 shows the limit the spring itself can take before reaching it elastic limit. The final graph compares 2 different springs with different spring constants and how that affects the displacement with variable force. As shown in the graph, the spring with the higher spring constant has greater displacement per force than the one with the smaller spring constant due to the direct proportionality of Hooke’s Law.

Hooke’s law applies greatly to real life. A demonstration of this is on a trampoline. As an individual jumps on a trampoline the springs holding the elastic mat stretch and then revert back to their original shape[5]. This follows Hooke’s law and the elastic limit of those springs can be seen when the trampoline itself breaks as the springs snap.

Another example of Hooke’s law is within mechanical engineering and the crumple zone of vehicles. The crumple zone essentially analyses the maximum amount of stress a vehicle can take within a certain location before being deformed permanently. The exact applications are followed within Hooke’s Law.

Works Cited:

1) Encyclopedia Britannica. (2016). Elasticity. [online] Available at: https://www.britannica.com/science/elasticity-physics [Accessed 5 Nov. 2018].

2) Encyclopedia Britannica. (2017). Hooke’s law. [online] Available at: https://www.britannica.com/science/Hookes-law [Accessed 4 Nov. 2018].

3) Encyclopedia Britannica. (2018). Robert Hooke. [online] Available at: https://www.britannica.com/biography/Robert-Hooke [Accessed 4 Nov. 2018].

4) Bhalla, D. (2015). Excel : Intersection of two linear straight lines. [online] ListenData. Available at: https://www.listendata.com/2012/10/excel-intersection-of-two-linear.html [Accessed 7 Nov. 2018].

5) S, M. (n.d.). Hooke’s Law Research – Melody’s AS Physics Portfolio. [online] Sites.google.com. Available at: https://sites.google.com/a/seq.org/melodys-as-physics-portfolio/home/hooke-s-law-research [Accessed 9 Nov. 2018].