Exploring Hooke's Law: Experimental Validation and Analysis of Spring-Mass Oscillations

Categories: Physics

An oscillating body, such as a spring-mass system, exhibits repetitive motion back and forth or up and down. The time required to complete one cycle of this motion is known as the period. The force responsible for this oscillatory behavior is called the restoring force. In the context of a spring with an attached mass, the restoring force is directly proportional to the displacement from the equilibrium position.

Theoretical Background:

The potential energy function for a spring-mass system near its equilibrium position is characterized by a quadratic term:
U(x)=21​kx2(1.1)(1.1)

Where is the potential energy, is the spring constant, and is the displacement from the equilibrium position? The potential energy equation is derived by substituting the energy expression into the general form [1.2]:

(1.2)

The differentiation of the potential energy function with respect to displacement yields the force acting on the system, known as Hooke's Law [1.3]:

(1.3)

Hooke's Law is valid for small displacements where the restoring force is linear [3].

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When considering a vertically hanging spring supporting a mass against gravity, the gravitational potential energy must also be accounted for [1.4]:

(1.4)

Differentiating this expression gives the gravitational force acting on the system [1.5]:

(1.5)

Combining Hooke's Law and the gravitational force, the spring constant () can be calculated in equilibrium [1.6]:

(1.6)

The experimental setup involved using a ruler, loads of varying masses, and a Hooke's Law apparatus consisting of an iron stand and a metallic spring. The spring was hung vertically, and masses were incrementally added to test Hooke's Law.

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The elongation of the spring was measured using a laser pointer and a camera [see Figure 1.1].

Figure 1.1 shows the initial set-up of the experiment with a load holder attached to a spring in front of the ruler. As loads are added, the spring elongates (Figure 1.B), and the increase in length is measured carefully.

Results and Discussion:

Multiple trials were conducted to ensure precision and validity of results. Potential sources of error, such as measurement judgment and rust on the spring, were considered and minimized through various techniques, including the use of a laser pointer, a camera, and rust removal.

The measurements were recorded for different loads, and the elongation of the spring was calculated. The calculated spring constant () using Equation (1.6) was compared with the calibrated values for accuracy.

Calculations and Formulas:

The key formula used in the experiment is Hooke's Law (), where is the force exerted by the spring, is the spring constant, and is the displacement from the equilibrium position.

Additionally, the formula for the spring constant () in the equilibrium state was utilized:

(1.6)

Tables:

The collected data for loads, corresponding displacements, and calculated spring constants were organized into tables. The tables provided a clear representation of the relationship between the attached load and the increase in length of the metallic spring within the elastic regime.

Pearson Product Moment Correlation:

To test the relationship between the attached load and the elongation of the spring, a Pearson Product Moment Correlation was performed. This statistical analysis provided insights into the strength and direction of the linear relationship between the variables.

The experiment successfully investigated Hooke's Law by determining the maximum load at which the law is obeyed. The mathematical relationship between the attached load and the increase in length of the metallic spring within the elastic regime was demonstrated. The calculated spring constants were compared with calibrated values for accuracy.

Despite potential sources of error, the use of precise measurement techniques and data analysis methods contributed to the reliability of the results. The laboratory experiment effectively combined theoretical principles with practical applications, reinforcing the understanding of oscillatory motion and Hooke's Law.

Displays the computed values for the given exercise, revealing that a similar trend was observed compared to the experiment. In the case of a 3 kg load, the spring constant "k" became unnecessary to compute, as its mass exceeded the maximum load capacity of the metallic spring for Hooke's law.

The Pearson Product Moment Correlation Coefficient analysis indicates a strong positive correlation between the mass of the loads and the increase in the spring's length. This suggests that as the mass increases, the length of the spring also increases, considering all gathered measurements. Furthermore, when examining each regime separately, the correlation remains strongly positive in the elastic regime, demonstrating a linear proportional relationship between the mass and length variables.

In conclusion, the experiment establishes that the load mass is linearly proportional to the increase in the spring's length. Hooke's law holds true within the elastic regime for loads with the same spring constant "k" measurements. However, beyond the elastic limit, Hooke's law is no longer applicable, and alternative principles are required for measuring the constant "k."

It is highly recommended to use high-quality metallic springs for accurate results. Additionally, conducting the experiment with multiple trials and smaller increments in load mass is advised to gain a deeper understanding of the relationship between load and spring length.

Acknowledgments: We express our gratitude to those who contributed to the success of the experiments. Special thanks to Mr. Rogelio Dizon, our Advance Laboratory I professor, for his time, knowledge, and guidance. Our appreciation goes to our parents, colleagues, and classmates for their valuable support, advice, and resources. Lastly, we acknowledge God for providing us with the strength, wisdom, and grace to complete both the experiment and this laboratory report.

Updated: Feb 26, 2024