PHYS 101 Lab Report 7

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Apr 3, 2024

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PHYS 101 Lab 7 Activity Noah Pogonitz Daniel Afshari James Biernacki 10/24/2023 1. Task 1: Develop a model Measure the equivalent spring constant of each of the three configurations shown above. Compare the parallel and series equivalent spring constants to that of the single spring and use this comparison to propose a physics model that will tell you the equivalent spring constant k for any number of springs N arranged in series or parallel. Your model should consist of two parts: one equation for the equivalent spring constant in terms of the number of springs combined in series, and another equation for the equivalent spring constant in terms of the number of springs combined in parallel 2. Task 2: Test your model Design and conduct experiments to test either your series or your parallel models using three or more springs. Test the model as fully as you can in the time you have remaining. Compare the equivalent spring constants predicted by your model to your measurements and decide whether your model is rejected or supported by your experiments. 3. Introduction: Introduce your experiment by briefly describing the question your group is answering or the phenomenon that you are exploring. If relevant, provide any physics background information needed to understand the experiment. In this experiment, we will be measuring and calculating the spring constant, k of three different configurations: a single spring, two springs in parallel, and two springs in a series. We will then compare the parallel and series configurations to the single spring configuration. Using the information obtained from these configurations to come up with a model that will ultimately allow us to calculate the spring constant, k for any number of springs, N arranged in either the parallel or series configurations. After we have developed our model, we will test this model and will be comparing the equivalent spring constants that are predicted by our model to the actual results we obtain during the experiment. The models for the spring constant equivalents for the parallel (Keff = NK) and series (1/Keff = N/K) along with their calculations are shown in the images below. To find the Keff, the equations of Hooke’s Law (F = -kx) and Newton’s 2nd Law (Fnet = ma) are required along with the equation T = 2π sqrt(m/k).

4. Methods: Describe the procedure of your experiment. It may be helpful to include diagrams or photographs. We first started by taking our IOLab and attaching a single spring to the end of the IOLab. We took the IOLab with the spring attached and hung it off the edge of a table. While the IOLab was hanging over the table, we pulled it, released it, and allowed it to oscillate. We used the accelerometer to find the period of the IOLab while it oscillated. To find the period, use the area of the graph between the area when the accelerometer is 0. We used the period to solve for the spring constant by using the equation k = m(2pi/T)^2. We then took 2 springs and placed them in series (can be seen in the image above) to each other and attached them to the IOLab’s force probe. We repeated the steps above to find the period which then can be used to find the spring constant. Lastly, we attached two springs parallel to each other (which can be seen in image above) to use the same steps as above to find spring constant.

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5. Results: Record the data from your experiment. This may include IOLab graphs or tables with numerical measurements. Make sure to include units on your measurements, as appropriate. You should present your data clearly before trying to describe or interpret it. Trial 1 Single: T = time/ number of oscillations k = m(2 π/T)^2 = 0.2(2π/ (4.60938/6)^2 = 13.38 N*m Trial 2 Single:

k = m(2 π/T)^2 = 0.2(2π/ (7.686/10)^2 = 13.37 N*m Trial 3 Single: k = m(2 π/T) = 0.2(2π/ (6.88216/9)^2 = 13.50 N*m Average k of single spring = 13.42 N*m Trial 1 (of 1) Parallel:

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k = m(2 π/T) =0.2(2π/(2.04416/4))^2 = 30.23 N*m Expected k for parallel according to model: 26.84 N*m Trial 1 (of 1) Series: k = m(2 π/T) =0.2(2π/(7.51278/7))^2 = 6.85 N*m Expected k for series according to model: 6.71 N*m 6. Analysis: Use words to describe the results of your measurement. What observations do you make about them? If relevant, do you see any patterns? Do not try to explain your data yet. For all of the trials, to calculate for the period we used the time highlighted in the graph divided by the amount of peaks in the highlighted sections. To find the spring constant for each trial, we

used the equation k = m(2 π/T)^2 where m is the mass of the IOLab and T is the period. Starting with the single spring, we found three spring constants to be 13.38 N*m, 13.37 N*m and 13.50 N*m which comes out to an average of 13.42 N*m. Moving onto the two parallel springs, we did one trial and found the spring constant to be 30.23 N*m. Last, we had the two springs in series and did one trial and found an spring constant of 6.85 N*m. 7. Discussion: Try to explain your result. Is it interesting or surprising? Why? Does it suggest any trends or physical properties? The results were not surprising as they followed our hypothesis. They confirm the assumptions we made that the forces of the springs are equal in series and that the change in distance is equal for parallel strings. Due to the assumptions made being confirmed, our hypothesis is supported by the experiment. Many of the values obtained were extremely close to the predicted values that we had, and the differences between the expected and actual values were 2.04% for series and 11.21% for parallel, which are extremely close considering that there was most likely error due to the spring not moving along the axis of the accelerometer at all times and the fact that we only performed 1 trial. This experiment suggests the trends that we listed in the introduction were in fact true. Conclusions: Briefly summarize your experiment and your findings. If you think there is more to explore about your measurement in further experiments, talk about it here. In our experiment, we found that our hypotheses that springs in a series have an equation of 1/Keff = N/K and that springs in parallel have an equation of Keff = NK are true. In the future, we can explore more by finding the relationship between springs with different K values and identical masses, since then the force or acceleration may not necessarily be the same. Roles: Daniel: Discussion and Conclusions Noah: Introduction and Results James: Methods and Analysis

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