## Introduction

Hooke’s Law is a theory of classical mechanical physics by Robert Hooke. It states that the force (F) needed to extend or compress a spring by a distance (X) is proportional to that distance. Hence, the equation: F=kX is deduced, where k is a constant dependent on the spring used, known as the spring constant. Simply, Hooke’s Law states that force is proportional to distance [of extension or compression]. However, after bypassing a spring’s elastic limit, Hooke’s Law no longer applies – force will no longer be proportional to distance.



Hence, the purpose of the experiment carried out was to analyse the property of three materials.

## Experimental Data

The following table presents the experimental data of the three materials tested. Using the data, three graphs were produced which are presented in the following section.

## Analysis

#### Graph #1 – y1 against x The graph of y1 against x is a linear curve, which shows that the first material obeys Hooke’s Law. However, based on the trend line, two points do not fall on it, that is at x = 7.00N and x = 9.00 N. This is likely to be attributed to errors made during measurement.

Generally, because the force applied, x, remained proportional to the deformation, y, throughout the whole curve, it is observed that force applied on the material did not exceed the elastic limit. Hence, after extension/deformation, the material returned to its original dimensions.

The equation of the graph is given to be y1 = ax + b. Via the graph, it is found that the equation is y1 = 1.56x + 1.38. Hence, the values of a and b are 1.56 and 1.38 respectively.

#### Graph #2 – y2 against x The graph of y2 is a linear curve, which shows that the material obeys Hooke’s Law.

Because force applied, x, is proportional to deformation, y, throughout the whole curve, it is observed that the force applied on the material did not exceed the elastic limit. Hence, after deformation, the material returned to its original dimensions.

The equation of the graph is given to be y2 = (a+0.5)x + b. Via the graph of y1 against x, the values of a and b were used to plot the graph of y2 against x.

#### Graph #3 – z against x The graph of z against x is a nonlinear polynomial curve of degree 3. Hooke’s Law states that force applied is proportional to extension/deformation, which is clearly not the case with this material.

The force applied on the material has passed the elastic limit, which means that the material underwent permanent deformation, losing its elasticity.

## Discussion & Comparison

1. Comparison of graphs:

Comparing the graphs of y1 and y2, several statements can be made. • The graph of y1 against x and y2 against x are similar, they are both linear curves. This denotes that both materials y1 and y2 obey Hooke’s Law, and that after extension, the materials return to their original dimensions.
• The physical characteristics of materials y1 and y2 are largely similar.
• The gradient of material y2 is greater than that of graph of material y1, which shows that the spring constant of material y2 is greater than that of material y1.
• When the force applied, x = 2.36 N, the extension of both materials y1 and y2 were the same, at y = 5.06 mm. This value was found via calculation, using matrices in Excel.
• The equations of y1 and y2 were rearranged in the form :
• Then, using matrices, the following workings were made:

Comparing the graphs of y against x and z against x, the following statements can be made.

• The graphs of y against x are completely different from the graph of z against x, as the latter is a nonlinear curve, compared to the former’s linear curve, showing that material z no longer conforms to Hooke’s Law, due to the force applied being greater than the elastic limit.
• Material z’s characteristics are greatly different from that of material y1 and y2, which could be due to a large variety of factors such as density, material composition, temperature, etc.

## Conclusion

The purpose of the experiment has been fulfilled, that is to analyse the properties of the three materials, y1, y2 and z. It has been shown that the first two materials, y1 and y2 obey Hooke’s Law, which states that force applied is proportional to extension. Furthermore, material z has been shown to be a material that has been stretched past the elastic region.

The Excel file can be obtained via the following link:

Assignment 1 – Hooke’s Law by Dennis Tan

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