Hooke’s Law

I. Abstract

The experiment is purely on Hooke’s Law and the concept of elasticity. It is performed simply by putting weights on the mass hanger. Using Hooke’s Law, apparatus weights and 2 types of spring was computed for the spring constant. By performing more than three trials and increasing the weights 10g per trial, the behavior of the spring constant was observed with varying force and displacement. With the spring constant obtained, the work done on the spring can also be calculated.

II. Introduction

The experiment defines elasticity which is the ability of an object to return to its original length and shape after the deforming forces are removed. It also makes use of the Hooke’s Law to observe the elasticity of a spring. The experiment is to prove if the amount of deformation is directly proportional to the magnitude of the applied force. Understanding these concepts is essential for innovation since it has many applications in real life especially in engineering and other sciences.

III. Theory

An example of elasticity is a rubber band, as we stretch the rubber band, it changes its shape, removing the force applied to it, it will return to its original form.

Hooke’s Law shows the relationship between the forces applied to a spring and its elasticity. The equation: W = kx

Derivation:
W = ∫XoXf Fdx
W= FXf – FXo or W = ½ kx2
Where:
F = Force applied
x = displacement of the stretched object
k = Force constant (dynes/cm; N/m)

The Modulus of Elasticity is stated as the rate of change as a function of stress. For solid materials, Moduli of Elasticity in tension and compression are approximately equal and are known as Young’s modulus. It’s expressed by: Y = ?/ , S*(?/?), where ε =?/?0

Therefore:
Y = ??0/??

IV. Methodology
Part I: Setting up of the Equipment
Before doing the experiment, the equipment was set up by hanging the spring from the notch of the support arm. The stretch indicator was placed at the bottom of the spring and was aligned at exactly zero. Attached to the bottom of the stretch indicator was the mass hanger that will be used in placing different masses in the experiment. B. Determining the Force Constant of the Spring

A mass was placed on the mass hanger. The change in displacement of the spring and the weight of the hanging mass was recorded. Using the equation F=kx, the force constant of the spring (k) was computed, where F is the weight of the hanging mass and x is the change in displacement of the spring. The procedure was repeated for 3 more trials by adding 10g in each trial. Using the values determined in each trial, the average value o the force constant was computed. A graph was plotted using force vs. displacement and the slope of the line was determined. The percentage difference of the average value of the force constant and the slope was calculated. All procedures were repeated using the other spring. C. Determining the Work Done on the Spring

Using the gathered data, the total work done in stretching the spring was computed using the equation: ?=12(??2−?02) where xf is the displacement of the spring in the 4th trial and x0=0. The area under the graph was determined using the concepts in mathematics. The total work done and the area of the force vs. displacement graph was compared. V. Results and Discussion

Part I: Setting-up of the Equipment

On the first part, a spring was hung from the notch on the support arm, and a Stretch Indicator was connected to the bottom of the spring. A mass hanger was then connected to the bottom of the Stretch Indicator.

Part IIA: Determining the Force Constant of the Spring

In the experiment proper, certain mass was placed on the hanger. The spring then elongated with a certain displacement, and comes back to its normal length every time the mass was removed. This illustrated the elastic property of a spring to return to its original length once the deforming force was removed.

From the data gathered, the force constant of the spring was computed as well as the slope of the line and the percent difference.

The graph above depicts a direct proportionality of the force and displacement which obeys the principle of Hooke’s Law. It states that “Within the elastic limit of a body, the deforming force is directly proportional to the elongation of the body.”

Some errors were recorded on the first part of the experiment brought by: Human error
Stretch Indicator not levelled properly’

These errors can be avoided if the Stretch Indicator was levelled properly with the assistance of the members for the displacement to be recorded in accurate value.

Part IIB: Determining the Force Constant of the Spring

Using the same procedure, the force constant, average force constant, and slope of the line were computed and obtained only with a different spring.

Based on the gathered data, the values obtained for the force constant are similar for each trial, making the percent difference zero.

Just like the first graph, the relationship between the deforming force and the amount of spring stretches shows direct proportionality which satisfies the Hooke’s Law.

For this part of the experiment, the group acquired a zero percent difference, which means that no error was made during the experiment.

Part III: Determining the Work Done on the Spring

In the last part of the experiment, same data were used to compute for the work done in stretching the spring.

VI. Conclusion
It has been observed in the experiment that as force was applied to the string, its length increased. The stiffness of the spring is defined as k in the Hooke’s Law as the spring constant.[4] By stiffness, k represents the maximum amount of force taken by the spring to resist deformation per unit length. In order to determine the force constant, the group manipulated the formula of the Hooke’s law. Different trials were done in order to compute for the elongation (x). To solve k, recorded values were substituted to this formula k = mg/x. It is stated in the Hooke’s Law, that the deforming force is directly proportional to the elongation of the body. This was observed during the experiment. Also, the work done by the springs were determined by substituting the data gathered from the experiment to the formula W = ½ kx2.