Lab 4 PHY105M (1)

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University of Texas *

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105M

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Chemistry

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Apr 29, 2024

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5

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Introduction: This week our objective was to examine the elastic properties of a chosen extensible material to determine whether Hooke’s law applied. For this experiment, we opted to investigate the behavior of a rubber band that was provided in the lab. To investigate the mechanical properties of a band under varying loads, we utilized a meter stick to measure the amount of stretch in meters. The experiment involved applying masses of weights 1 kg, 0.7 kg, 0.5 kg, 1.7 kg, and 1.2 kg to the band. We aimed to determine the spring constant, k, in the equation F=-kx, where F represents force in newtons and x denotes the displacement in meters. The uncertainty that was noted for force due to the multiplication of mass and gravity was +/- .0098N and the uncertainty for the displacement based on the use of the meter stick was +/- .0005m. To determine k, we first calculated the force exerted by each mass using the formula F=mg, where m is the mass and g is the acceleration due to gravity that is 9.8 m/s ^2. Then, we divided the force by the change in length of the band when it stretched, yielding the spring constant k in newtons/meter. We also calculated the uncertainty propagation as there are uncertainties in both force and displacement which contribute to the uncertainty of k. Subsequently, we plotted the data obtained from the experiment into two different graphs of k vs.x and f vs. x. The resultant graphs allowed us to observe the relationship between force and displacement and verify if it followed a linear trend as described by Hooke's Law. Despite the diverse range of masses utilized, our hypothesis was that the relationship between force and displacement would maintain a linear trend, consistent with Hooke's law. However, it's important to note that the uncertainty in our experiment primarily stems from measurement errors in determining both the masses and the corresponding displacements of the rubber band.. Conducting Your Experiment: ΔF (N) Δx (m) k 𝑖 K δ 𝑖 9.8 +/- 0.0098N 0.035 +/- 0.0005m 280 +/- 4.01N/m 4.01 6.86 +/- 0.0098N 0.19 +/- 0.0005m 361.05 +/- 9.52 N/m 9.52 4.9+/- 0.0098N 0.01+/- 0.0005m 490 +/- 24.52N/m 24.52 11.76+/- 0.0098N 0.048 +/- 0.0005m 245 +/- 2.56 N/m 2.56 16.66+/- 0.0098N 0.0705 +/- 0.0005m 236.3 +/- 1.68 N/m 1.68 Note: We used 5 different masses to determine the linearity of the rubber band with force in N, displacement in m, and the spring constant in N/m.
Conclusion: Upon analysis of the force versus displacement plot, it was evident that Hooke's law applied consistently across the entire range of forces tested. Additionally, the plot exhibited a linear relationship between force and displacement, further confirming the validity of Hooke's law within the tested regime. This outcome aligns with our hypothesis that the behavior of the rubber band would adhere to a Hookean regime, thereby failing to reject our initial prediction. However, the experiment could be enhanced in future iterations by implementing more precise measurements of force and displacement, as well as considering environmental factors that may influence the material's behavior. Additionally, upon examination of the k versus displacement graph, a downward slope was observed. This suggests that as the rubber band stretches further, it may be experiencing a decrease in stiffness or an increase in elasticity, possibly due to material fatigue. Though we viewed a linear regime as plotted in the scatterplot of f vs. x, the uncertainties in the spring constant ki varied, ranging from approximately +/-4.01N/m to +/-24.52 N/m. This variation of uncertainties for the spring constant can be attributed to the uncertainties of both force and displacement, which is why the uncertainty propagation formula was used as they affect the determination of k. Reducing these uncertainties through improved measurement techniques or instrumentation could lead to even
clearer validation of Hooke's law within the tested masses. Alternative ranges that could be explored could be higher ranges in the 2,000 kg range, and a possibility of improvement could be using a different measuring device instead of a meter stick as well as using a different material that could possibly stretch more as ours was very sturdy. Part 2: Introduction: In the initial phase of the experiment, we successfully validated Hooke’s law across the entire range of forces applied to the rubber band. Considering this, we sought to explore a different range of forces in search of potential non-linear behavior, indicating a deviation from Hooke's law. For this purpose, we utilized masses of 0.4 kg, 1.9 kg, 2.5 kg, 3.0 kg, and 3.4 kg, extending our investigation beyond the previous range. Following the established methodology from the first part, we employed a meter stick to measure the stretch in meters as we applied the varied masses to the rubber band. Our objective remained to determine the elastic properties of our material and we first had to find the spring constant, k, in the equation F=−kx, where F represents force in newtons and x signifies displacement in meters. The uncertainties for Force and displacement were +/-.0098N and +/- .0005m as we used the meterstick to determine the length the rubber band stretched. To derive k, we utilized the formula F=mg to calculate the force exerted by each mass, where m represents the mass in kg and g denotes the acceleration due to gravity which is approximately 9.8 m/s^2. We made use of the uncertainty propagation equation as there is an effect on the value of k based on the uncertainties of force and displacement. Subsequently, we determined k by dividing the force by the change in length of the rubber band during stretching, or displacement. Following the data collection, we plotted the force that we obtained from multiplying the mass and gravity, against the corresponding displacement of the rubber band. This plotting enabled us to observe the relationship between force and displacement, allowing for the identification of any potential non-linear trends in the elastic response of the rubber band. We anticipate that as the applied forces increase within this extended range, the relationship between force and displacement will maintain linearity, following the principles outlined by Hooke's law. It's essential to note that the primary source of uncertainty in our experiment stems from measurement errors in determining both the masses and the corresponding displacements of the rubber band. Conducting Your Experiment: ΔF (N) Δx (m) k 𝑖 k δ 𝑖 3.92 +/- 0.0098 N 0.08 +/- 0.0005m 490 +/-30.65 N/m 30.65 18.62 +/- 0.0098 N 0.079 +/- 0.0005m 235.7 +/-1.5 N/m 1.5 24.5 +/- 0.0098 N 0.087 +/- 0.0005m 281.01 +/- 1.62 N/m 1.62
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