Lab 1_ Springs and Elastic Potential Energy

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Apr 3, 2024

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Lab 1: Springs and Elastic Potential Energy Group #3 Ann McGinty Olivia Kawalec Lysa Korusenge Monday, January 22, 2024 iOLab Unit #4 Introduction: The purpose of this lab was to use Excel to calculate the kinetic and potential energies of a small spring, knowing the mass of the iOLab, to better understand the Law of Conservation of Energy and Hooke’s Law. This is done by experimentally measuring the mass of the iOLab in Exercise 1, graphing a parametric plot of the iOLab’s Force and Wheel position, velocity, and acceleration when a small spring is attached to it in Exercise 2, and using this data to determine how efficient the system is in Exercise 3. Exercise 1: Measure the Mass of the iOLab Remote: Data Collection In this exercise, we calibrated the iOLab, inserted the eye hook, and then selected the accelerometer and force sensors on the iOLab software. We began measuring the data, picked up the iOLab device by its toggle for a few seconds, then set it down and stopped the collection. Afterwards, we highlighted the section of the graph on the force sensor that indicated the iOLab was being held in the air. This represents the force of gravity on the iOLab. The experimental quantity for acceleration was also indicated in the accelerometer sensor’s graph. This allowed us to solve for the mass using the formula: 𝐹 𝑔 = 𝑚𝑔.

Figure 1.1 - Image of iOLab being held in the air to calculate the force due to gravity Exercise 1: Measure the Mass of the iOLab Remote: Data Analysis Figure 1.2 - Highlighted region is showing acceleration due to gravity while iOLab is stationary on table Based on the first graph, we found that Ay = -9.824 m/s^2 +/- 0.0360 m/s^2 This is our value of g (acceleration due to gravity).

Figure 1.3 - Highlighted region is showing the force of gravity when the iOLab is being held stationary in the air Based on the second graph, we found Fy = -1.992 N +/- 0.0310 N This was our value of the force due to the Earth’s gravitational pull. 𝑚 = 𝐹 𝑔?𝑎𝑣 /𝑔 𝑚 = − 1. 992 𝑁 / − 9. 824 𝑚/? 2 𝑚 = 0. 203 𝑘𝑔 𝑚 = 203 𝑔 σ 𝑚 = 𝑚 (σ 𝐹 𝑔?𝑎𝑣 /𝐹 𝑔?𝑎𝑣 ) 2 + (σ 𝑔 /𝑔) 2 σ 𝑚 = 0. 203 𝑘𝑔 (0. 0310 𝑁 / − 1. 992 𝑁) 2 + (0. 0360 𝑚/? 2 / − 9. 824 𝑚/? 2 ) 2 σ 𝑚 = 0. 00325 𝑘𝑔 Therefore, the mass we obtained is m = 0.203 kg +/- 0.00325 kg.

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Figure 1.4 - Image of iOLab on triple beam balance The mass value obtained from the triple beam balance is approximately 195 grams + 4.8 grams (199.8 grams, or 0.1998 kg). The uncertainty associated with this value is +/- 0.05 grams (half the smallest value that can be detected on the scale, which is 1 gram). The value calculated from the triple beam balance scale and our calculations are very similar, and also very close to the expected value of 200 grams. Exercise 1: Measure the Mass of the iOLab Remote: Conclusion The mass of the iOLab was 0.203 kg +/- 0.00325 kg, as calculated from the equation: Fg = mg. We took the force of gravity and the acceleration due to gravity from the force and the accelerometer graphs, respectively. This agrees with the expected value of 200 g for the iOLab. The mass we obtained from the triple beam balance was 199.8 g +/- 0.05 g. One potential

source of error includes holding the iOLab unsteadily in the air, causing the force of gravity measurements to be off. To minimize this error, we can keep our elbow on the table to assure that the iOLab is wobbling as little as possible. Exercise 2: Measure the Spring Constant, k, of the Short Spring: Data Collection In this exercise, we added the small spring onto the force sensor of the iOLab. The iOLab data was reset, and the force and wheel sensors (including position, velocity, and acceleration) were selected. The iOLab was pushed back and forth relative to a vertical barrier (the wall). Next, a parametric plot, plotting Force on the x-axis versus Wheel Position on the y-axis. Afterwards, two points were selected at opposite ends of the plot, and the slope of the line was obtained. This was done five times. Finally, the five values of the slope were averaged to determine the mean and corresponding standard deviation. Figure 2.1 - Image of spring being pushed against the vertical barrier (wall)

Exercise 2: Measure the Spring Constant, k, of the Short Spring: Data Analysis Figure 2.2 - Screenshot of our first trial with the cursor showing the Force and Wheel Position at the beginning. Figure 2.3 - Data table of raw data from the five trials we conducted x start (m) y start (N) x end (m) y end (N) slope = rise/run -0.0069 1.8148 0.001 0.5516 -159.90 -0.012 1.8344 -0.0011 0.2464 -145.67 -0.011 1.9454 -0.0001 0.1458 -165.10 -0.011 1.9427 -0.001 0.3054 -163.73 -0.007 1.7487 0.003 0.2867 -146.2 average = -156.12 standard deviation = +/- 9.49276

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Slope calculation for the first trial: Slope = rise / run Slope = (Y_end - Y_start) / (X_end - X_start) Slope = (0.5516 N - 1.8148 N) / (0.001 m - (-0.0069m)) Slope = -159.90 N/m We followed the same process of calculations for trials 2-5. After averaging the five values of our slope through Excel, our average value and its uncertainty was: ų = -156.12 N/m with an uncertainty of σ = +/- 9.49276 N/m Therefore, k short = 156.12 +/- 9.49276 N/m Exercise 2: Measure the Spring Constant, k, of the Short Spring: Conclusion The short spring’s constant was k short = 156.12 N/m +/- 9.49276 N/m, which was determined by averaging five trials values of slope values where we pushed the cart with the short spring in the force sensor into a vertical barrier and allowing it to compress fully. To measure the spring constant, we generated a parametric plot while using the cursor tool to determine two points (one at each end) to calculate the slope of the linear parametric plot. Finally, we averaged the five values of the slope which resulted in values of average (ų) and standard deviation (σ) for the spring constant, k.

Exercise 3: Calculate Kinetic Energy of the Remote and Elastic Potential Energy in the Spring: Data Collection In this exercise, we investigated the relationship between the kinetic energy and potential energy of the spring. The iOLab was reset, and the Force and Wheels sensors (including position, velocity, and acceleration) were selected. The iOLab, with the small spring attached to the force probe, was pushed into the vertical barrier (the wall) where it proceeded to rebound after it hit the surface. The hand pushing the iOLab was released after it pushed it into the vertical surface. The kinetic energy was calculated before and after it hit the wall using the velocity measurements and the mass from Exercise 1. The maximum spring elastic potential energy was also calculated. This was repeated for five collisions. Figure 3.1 - Image of spring and the vertical barrier (wall) before it is let go and pushed

Exercise 3: Calculate Kinetic Energy of the Remote and Elastic Potential Energy in the Spring: Data Analysis Figure 3.2 - Screenshots of our first trial with the cursor showing initial velocity (Vy), final velocity (Vy), and force at the peak.

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Figure 3.3 - Screenshot of data showing appropriate highlighted regions from our first trial

Table 3.3 - Data table showing the raw data from five trials Trial Number Initial Velocity (m/s) Final Velocity (m/s) F peak (N) x = F / k (m) Uspring ½kx^2 (J) KE initial ½mv^2 (J) KE final ½mv^2 (J) 1 -0.36 0.18 1.281 0.008205 0.005255 0.013154 4 0.01827 2 -0.303 0.2201 1.138 0.007289 0.004147 59 0.009318 6 0.022533 3 -0.281 0.201 0.973 0.006232 0.003032 0.008014 5 0.020401 5 4 -0.24 0.2 0.962 0.006162 0.002964 0.005846 4 0.0203 5 -0.2 0.18 0.775 0.004964 0.001924 0.00406 0.01827 Exercise 3: Calculate Kinetic Energy of the Remote and Elastic Potential Energy in the Spring: Conclusion In this exercise, we pushed the iOLab against the wall five times, collecting three pieces of information from each trial: initial velocity, final velocity, and force at the peak (maximum force). Afterwards, we used Excel to help us calculate the energy of the spring, and the initial and final energy. From looking at our five trials we noticed that our initial kinetic energy and final kinetic energy values were not the same. This suggests that energy was not conserved and was being lost. There are several things that can explain this loss of energy, such as friction, the compression (deformation) of the spring against the wall, or thermal energy from the sliding of the table. The spring energy was not very close to the initial kinetic energy and the final kinetic energy (i.e. from trial 1: kinetic energy initial + kinetic energy final = spring energy would be 0.0131544 J + 0.01827 J = 0.0314 J, not 0.005255 J, the spring’s energy). Therefore, this must be an inelastic collision, and energy was transferred to the factors previously stated, found in the environment. It is also important to note that our spring compressed and became slightly

twisted sideways when hitting against the wall, creating a potential source of error in our initial and final velocity values, and thus, our initial and final kinetic energy values. Overall Conclusion & Error Analysis In this lab, we investigated Hooke’s Law by placing a spring under pressure and measuring the force applied to the iOLab (spring) and the distance traveled by the iOLab. Using the plotted data, we were able to find the spring constant by finding the slope of the values (spring constant = force/displacement). Additionally, we investigated the conversion of kinetic energy to elastic potential energy stored in the spring. In exercise 1, the mass of the iOLab was 0.203 kg +/- 0.00325 kg. The mass of the iOLab was 0.203 kg +/- 0.00325 kg, as calculated from the equation: Fg = mg, taking the force of gravity and the acceleration due to gravity from the force and the accelerometer graphs, respectively. This agrees with the expected value of 200 g for the iOLab. In exercise 2, the short spring’s constant was k short = -156.12 +/- 9.49276 N/m. When stretching the short spring, it feels very stiff (difficult to stretch and/or compress). This relates to the high spring constant (156.12 N/m with an uncertainty of +/- 9.49276 N/m), which results from the greater force needed to stretch/compress a smaller distance. This proves that the spring constant is in agreement with the stiffness the spring feels when under pressure. In exercise 3, our calculated energy values for our trials were not similar. Therefore, we concluded that energy was being lost due to friction, thermal energy, or the compression of the spring. As a result, it can be labeled as an inelastic collision. There are various sources of error that can impact the accuracy of our measurements: (1) There is the possibility that friction between the spring and the surface fluctuated our calculated values, but it is not considered in our calculations because it is difficult to

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quantify. Additionally, the continuous friction between moving parts can generate heat, leading to a conversion of kinetic energy into thermal energy. This conversion represents a loss of energy from the kinetic energy of the system. In such cases, the total energy of the system is not conserved. (2) While stretching the iOLab, there is a possibility that the spring's motion is not perfectly linear, which is a necessity for Hooke’s Law calculations (assuming that x-value of distance is in one direction only). To minimize these sources of error, we could ensure we are properly and accurately calibrating the device and using controls such as the surface where iOLab is stretched. By the end of this lab, we gained a deeper understanding of Hooke’s Law, the effects of stretch and compression on springs, and energy conservation.