Lab Report 1

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Report for Experiment #13 Simple Harmonic Motion Wyatt Poole Lab Partner: Owen Liang TA: Ahmed Fadul 9/19/23 Abstract This experiment investigated Simple Harmonic Motion (SHM) and Damped Harmonic Motion (DHM) by studying the oscillatory behavior of a glider on an air track. The objectives included determining key parameters such as amplitude, period, phase, and spring constant (k) for SHM, and measuring damping constants (α) for DHM under varying magnet configurations. Data collection involved recording position vs. time data, followed by extensive analysis. The results revealed the influence of damping forces on oscillations, with damping constants found to increase with the number of magnets. Comparisons with theoretical expectations showcased the interplay between theory and experiment in understanding the dynamics of oscillatory systems.

Introduction The purpose of this experiment was to explore the principles of Simple Harmonic Motion (SHM) and Damped Harmonic Motion (DHM) using a glider on an air track. The fundamental concepts under examination encompassed oscillatory motion, damping effects, and the interaction between magnets and conductive materials. This study aimed to provide insights into the behavior of mechanical systems undergoing oscillations, with a particular focus on the influence of damping forces introduced by magnets. In the context of SHM, the primary objectives were to determine key parameters, such as amplitude, period, phase, and spring constant (k). These parameters help characterize the motion of oscillating systems and are essential for understanding the underlying physics. For the investigation of DHM, the experiment sought to analyze the impact of damping forces on the oscillatory behavior of the glider. Specifically, the experiment aimed to measure damping constants (α) under different conditions, varying the number of magnets attached to the glider. By comparing the obtained α values with theoretical predictions, we could assess the extent to which damping affects oscillation. This study provides a comprehensive examination of oscillatory behavior in mechanical systems, shedding light on both ideal SHM and real-world DHM scenarios. The experiments involved the collection of position vs. time data, subsequent data analysis, and comparisons with theoretical expectations, ultimately facilitating a deeper understanding of the physical phenomena associated with oscillations and damping forces. Investigation 1: Simple Harmonic Motion Setup The experimental setup consisted of a glider placed on an air track, with two springs attached to either end. A PASPort motion sensor was used to measure the glider's position as it oscillated. The glider was initially displaced from equilibrium, and its motion was recorded and analyzed. Procedure The following steps were taken in the experiment: 1. The mass of the glider was measured and recorded using a scale. 2. The air track was set up, ensuring that the glider moved freely with minimal air pressure. 3. The air track was leveled and adjusted for stability. 4. The PASPort motion sensor was connected to the computer.

5. PASCO Capstone was opened, and data recording parameters were set up. 6. The motion sensor's position was adjusted to align with the glider's reflector. 7. It was confirmed that the motion sensor recorded smooth position changes as the glider was moved. 8. Data collection began for the equilibrium position. 9. Data was recorded for approximately 30 seconds to determine the equilibrium position. 10. The glider was slid towards the motion sensor, and oscillation data was recorded. 11. The data was inspected for smooth oscillation, and the setup was adjusted if needed. 12. Oscillation data was recorded for analysis. 13. The air track was turned off after data collection. Data (Raw and Derived Quantities) This section presents the raw and derived data obtained during Investigation 1, which focused on simple harmonic motion (SHM) of a glider on an air track. The primary objective was to determine key parameters such as amplitude, period, phase, and the spring constant of the system. Raw Data: The following raw data was collected during the experiment: 1. Mass of the Glider: The mass of the glider used in the experiment was measured and recorded. 2. Position vs. Time Data: Position measurements of the glider were recorded over time when it was at its equilibrium position with springs attached. These measurements were gathered using a motion sensor. (Raw data is available in the original data records.) 3. Position vs. Time Data - Oscillation: Position measurements of the glider were recorded as a function of time during oscillation when connected to two springs. The oscillatory behavior was observed for different initial positions. (Raw data is available in the original data records.) Derived Quantities: Several derived quantities were calculated to characterize the SHM of the glider: 1. Equilibrium Position x 0 : The equilibrium position x 0 was determined by averaging the position data recorded when the glider was at rest at its equilibrium position. (Refer to Computation of Derived Quantities section for details.) 2. Amplitude A : The amplitude A of the glider's oscillation was calculated by measuring the distance from the equilibrium position to the first positive peak in the position vs. time data. (Refer to Computation of Derived Quantities section for details.)

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3. Period of Oscillation T : The period T of the glider's oscillation was determined by analyzing the time coordinates of consecutive positive peaks and fitting them to a linear relationship with peak number. (Refer to Computation of Derived Quantities section for details.) 4. Phase : ϕ The phase was calculated using the time of the first positive peak and the angular ϕ frequency. (Refer to Computation of Derived Quantities section for details.) 5. Theoretical Spring Constant k’ : The theoretical spring constant k’ of the two springs acting on the glider was computed based on the known spring constant of a single spring. (Refer to Computation of Derived Quantities section for details.) 6. Experimental Spring Constant k : The experimental spring constant k was calculated using the measured quantities and Eq. (13.7) in the lab manual. (Refer to Computation of Derived Quantities section for details.) Table 1: Equilibrium Position Data The mass of the glider ( m g ): 0.1414 kg Table 2: Oscillation Data Computation of Derived Quantities

 The equilibrium position ( x 0 ) was found to be 0.618 m by averaging the equilibrium position data.  Centered positions ( x - x 0 ) for the oscillating glider were calculated.  A centered position vs. time plot for oscillation data was created (below).  The amplitude ( A ) of the first positive peak was estimated to be 0.2 m from the graph.  The times of the first 6 positive peaks were measured with uncertainty being half the spacing between adjacent time points. These times were plotted against their respective peak numbers (below).  The period ( T ) was calculated as the slope of the linear fit line. This is because each peak corresponds to one complete cycle of oscillation, and the time it takes to complete one cycle is precisely the period T . The change in time divided by the number of the peaks would therefore give us T , and the slope is y/x .  δ T was determined by using the IPL straight-line fit calculator.  Frequency ( f ) and angular frequency (ω) were calculated using T . f = 1 T ω = 2 π f  Both f and ω had their uncertainties calculated. δf = f δT T δω = ω δf f  The phase ( ) was determined using the time of the first positive peak. ϕ ϕ = ω∙t 1  The experimental spring constant ( k ) was calculated using the oscillation data and Eq. (13.7).

ω = √ k m This gave us a value of 2.135 N/m.  Uncertainty in this value was calculated to be 0.016 N/m δk = 2 ( δω ω ) ∙k The experimental and theoretical values of k proved to be a bit further apart than their expected error, which can most likely be put to an error in the calibration of the glider and sensor. Results  Amplitude ( A ) 0.233 ± 0.025 meters  Period ( T ): 1.617 ± 0.011 seconds  Phase ( ): -3.885 ± 0.014 radians ϕ  Spring constant ( k ): 2.135 ± 0.016 N/m Explanations for how these values were derived are addressed in the previous section (above). Comparison to Expected Value: In the context of simple harmonic motion (SHM), the theoretical relationships among the parameters were considered. While the theoretical values were not explicitly provided in this investigation, the relationships between these parameters were applied to assess the internal consistency of the measured values. The relationship between amplitude ( A ), period ( T ), and angular frequency (ω) in SHM is given by: A = 1 ω Therefore, the experimental value of amplitude ( A ) can be compared to the expected value derived from the period ( T ) as calculated above. Any significant discrepancies between these values may suggest systematic errors or limitations in the experimental setup. Plugging in our experimental value of ω gave an expected value of 0.257 m, which is within the expected error of the experimental A = 0.233 ± 0.025 m. The theoretical spring constant ( k’ ) of the two springs was calculated by multiplying the given spring constant for one spring by a factor of two, giving us 2.200 N/m.

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The experimental (2.135 ± 0.016 N/m) and theoretical (2.200 N/m) values of k proved to be a bit further apart than their expected error. Analysis of Unaccounted-for Errors: In this section, potential sources of error that may have affected the precision and accuracy of the measurements are discussed. While efforts were made to minimize known sources of error, there may still be unaccounted-for factors influencing the results. One possible source of error is air resistance, which can exert a damping force on the glider's motion. While damping effects were addressed in Investigation 2, minor damping in Investigation 1 may have influenced the amplitude and period measurements. Additionally, imperfections in the air track or slight misalignment of the motion sensor could introduce random variations in the data. These unaccounted-for errors may have contributed to the uncertainties associated with the measured parameters. Further examination of these potential sources of error and their quantification is required to refine the experimental procedure and enhance the accuracy of future measurements.

Investigation 1 Graphs Object 21 Object 23 Investigation 2 Experimental Setup: In Investigation 2, the phenomenon of damped harmonic motion was explored by introducing magnets to the glider in our experimental setup. The glider, situated on an air track, was equipped with two ring magnets attached symmetrically to its legs. The magnets interacted with the aluminum air track, inducing eddy currents and resulting in electromagnetic damping. The key elements of this setup included the glider, the ring magnets, and the air track.

Procedure: 1. Initially, the mass and color of two identical ring magnets were measured and recorded, ensuring that they were of the same color to maintain consistency in their properties. 2. An experimental apparatus was set up, including the glider on the air track, PASCO Capstone software, and motion sensor, as outlined in the provided instructions. 3. The first set of position-versus-time data was collected with the two ring magnets attached to the glider, with the glider being released at the 40 cm mark. The apparatus was adjusted if necessary to ensure that the resulting graph displayed smooth oscillations. 4. Subsequently, the mass of two additional magnets was measured, and they were added symmetrically to the glider's legs. The air flow was adjusted as needed to maintain an adequate air cushion between the glider and the track. A second set of position-versus-time data was recorded. 5. Finally, the process was repeated, with two more magnets added to the glider, ensuring symmetry, and capturing a third set of position-versus-time data. The data collection was organized to investigate the effect of the number of magnets on the glider's oscillation, with increasing mass and electromagnetic damping. The damping force was expected to increase with the number of magnets, leading to a change in oscillation frequency. Data (Raw and Derived Quantities) This section presents the raw and derived data obtained during Investigation 2, which focused on damped harmonic motion (DHM) of a glider with added magnets. The data was collected to investigate how the presence of magnets affected the glider's oscillatory behavior. Raw Data: The following raw data was collected during the experiment: 1. Mass of Magnets: The masses of individual ring magnets used in the experiment were measured and recorded. (Refer to Table 3 for details) 2. Position vs. Time Data: Position measurements of the glider were recorded as a function of time for different configurations with varying numbers of magnets attached. These measurements were obtained using the motion sensor. (Raw data is available in the original data records.) Derived Quantities: Several derived quantities were computed to analyze the impact of magnets on the DHM of the glider: 1. Period with Added Mass T : The period T of the glider's oscillation was determined by analyzing the time coordinates of consecutive positive peaks and fitting them to a linear relationship with peak number. (Refer to Computation of Derived Quantities section for details.)

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2. Constant α: The constant α was determined for each set of data by fitting an exponential trendline to the amplitude vs. time data and extracting the decay constant from the equation. (Refer to Computation of Derived Quantities section for details.) 3. Constant b : The constant b was calculated to quantify the strength of electromagnetic damping acting on the glider-magnet system. It was determined through an equation relating it to α. (Refer to Computation of Derived Quantities section for details.) 4. Angular Frequency ω: The angular frequency ω was calculated using the formula ω = 2π / T, where T is the period. (Refer to Computation of Derived Quantities section for details.) 5. Theoretical Angular Frequency ω' : The theoretical angular frequency ω' was calculated using the Equation 13.15, where 'k' is the spring constant and 'm' is the total mass of the glider system. (Refer to Computation of Derived Quantities section for details.) 6. Theoretical Period T': The theoretical period T' was determined using the formula T = 2π / ω’, where ω' is the theoretical angular frequency. (Refer to Computation of Derived Quantities section for details.) Table 3 - Mass of Ring Magnets: The above table provides the recorded masses of individual ring magnets used in the experiment. The values of mass (in kilograms) were obtained using a digital scale. The subsequent sections will delve into the calculations and analysis of the derived quantities, shedding light on the influence of added mass and electromagnetic damping on the glider's DHM behavior. Computation of Derived Quantities In Investigation 2, various derived quantities were computed to gain a deeper understanding of the glider's behavior in damped harmonic motion (DHM). These derived quantities helped assess the impact of added mass and electromagnetic damping on the system's oscillations.

 Period of Oscillation with Added Mass: To investigate the effect of added mass (from magnets) on the period of oscillation, the period was measured for different numbers of magnets. The period T was calculated using the slope of the linear fit obtained when plotting the time coordinates of peak positions against their respective peak numbers (below). The derived periods allowed us to analyze how the added mass influenced the oscillatory behavior.  Constant α: The constant α was determined using α from the exponential decay formula associated with DHM. This decay formula was the exponential trendline (given by Excel) of the graph of amplitudes of peaks vs the time of those peaks (below). The formula was: A = A 0 e − αt Where: A 0 represents the y-intercept of the graph, A represents the y-values on the graph and t is time in seconds.  Constant b : Isolating and solving for α allowed the use of Equation 13.14 to solve for b : α = b 2 m The computed value of b provided insight into the strength of electromagnetic damping experienced by the glider-magnet system.  Angular Frequency ω: The angular frequency ω was calculated using the equation: ω = 2 π T  Theoretical Angular Frequency ω' : The theoretical angular frequency ω’ was calculated using Equation 13. 7: ω’ = √ k m − b 2 4 m 2  Theoretical Period T’ : The theoretical period T’ of the damped oscillation was calculated using Equation 13.9: T ' = 2 π ω' The computed values of b and T for various numbers of magnets provided critical insights into the behavior of the glider-magnet system under the influence of electromagnetic damping and added mass. These derived quantities were essential for assessing the experiment's goals and comparing the results to theoretical expectations.

Results: The measured values of the period (T) and damping constant (b) for each data set are outlined below, accounting for their respective uncertainties:  No Magnets:  Period ( T ): 1.617 ± 0.006 seconds  Damping Constant ( b ): 0.070 kg/s  Two Magnets:  Period ( T ): 1.735 ± 0.011 seconds  Damping Constant ( b ): 0.148 kg/s  Four Magnets:  Period ( T ): 1.601 ± 0.006 seconds  Damping Constant ( b ): 0.301 kg/s  Six Magnets:  Period ( T ): 1.860 ± 0.011 seconds  Damping Constant ( b ): 0.386 kg/s Comparison to Expected Value: The primary focus in Investigation 2 was to examine how the addition of magnets affected the period of oscillation and whether the damping constant increased proportionally with the number of magnets. The period for 2 magnets was within the expected range of uncertainty with the theoretical value, and the rest differed very slightly more (see tables in Appendix). The angular frequencies had no calculated error, so it is hard to say how accurate they were, but they were all very near to their expected values. Given the range of uncertainties, the consistency of the results with the expected outcomes based on the physical principles involved was assessed. Analysis of Unaccounted-for Errors: While efforts were made to minimize errors during data collection, several potential sources of unaccounted-for errors could have influenced the results. These included imperfections in the air track, variations in the magnetic properties of the ring magnets, and inaccuracies in positioning the magnets symmetrically.

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Furthermore, the effects of air resistance and friction may have played a role, even though damping was a controlled variable. Identifying and quantifying these potential sources of error would provide a more comprehensive understanding of the experimental limitations. The investigation into damped harmonic motion with magnets allowed for the exploration of the relationship between added mass and damping effects on oscillatory motion. By examining these factors, insights were gained into the interplay of mass, damping, and frequency in harmonic oscillations, contributing to the understanding of fundamental physical phenomena. Investigation 2 Graphs 0 2 4 6 8 10 12 14 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Centered Position vs. Time (0 magnets) Time (s) Centered Position (m)

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Centered Position vs Time (2 Magnets) Time (s) Positon (m)

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 -0.3000 -0.2500 -0.2000 -0.1500 -0.1000 -0.0500 0.0000 Centered Postion vs Time (4 Magnets) Time (s) Position (m) 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 -0.3000 -0.2500 -0.2000 -0.1500 -0.1000 -0.0500 0.0000 Center Position vs Time (6 Magnets) Time (s) Position (m)

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0.00 2.00 4.00 6.00 8.00 10.00 12.00 -0.3000 -0.2000 -0.1000 0.0000 0.1000 0.2000 0.3000 Overlay of all 4 Trials 0 Weights 2 Weights 4 Weights 6 Weights Time (s) Position (m)

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9 10 f(x) = 1.62 x − 0.61 Peak Times vs. Peak Number (0 magnets) Peak Number Peak Time (s)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 1 2 3 4 5 6 7 f(x) = 1.74 x − 1.18 Peak Times vs. Peak Number (2 magnets) Peak Number Peak Time (s)

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0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9 10 f(x) = 1.6 x − 0.28 Peak Times vs. Peak Number (4 magnets) Peak Number Peak Time (s)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 1 2 3 4 5 6 7 f(x) = 1.86 x − 0.95 Peak Times vs. Peak Number (6 magnets) Peak Number Peak Time (s)

0 1 2 3 4 5 6 7 8 9 10 0 0.05 0.1 0.15 0.2 0.25 f(x) = 0.34 exp( − 0.25 x ) Peak Amplitude vs Peak Time (0 magnets) Peak time (s) Peak Amplitude (m) 0 1 2 3 4 5 6 7 0 0.05 0.1 0.15 0.2 0.25 f(x) = 0.34 exp( − 0.45 x ) Peak Amplitude vs Peak Time (2 magnets) Peak time (s) Peak Amplitude (m)

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0 1 2 3 4 5 6 7 8 9 10 0 0.05 0.1 0.15 0.2 0.25 f(x) = 0.89 exp( − 0.81 x ) Peak Amplitude vs Peak Time (4 magnets) Peak time (s) Peak Amplitude (m)

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0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0 2 4 6 8 10 12 f(x) = NaN exp( NaN x ) Peak Amplitude vs Peak Time (6 magnets) Peak time (s) Peak Amplitude (m) 0 1 2 3 4 5 6 7 0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 f(x) = 0.05 x + 0.06 b vs Number of Magnets number of magnets b

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Conclusion In this experiment, we aimed to investigate the behavior of a damped harmonic oscillator, specifically a glider subjected to damping forces induced by the presence of ring magnets. We followed a detailed procedure, which included the attachment of ring magnets to the glider and the collection of position vs. time data. The analysis involved determining the amplitude, period of oscillation, damping constant, and other derived quantities. Our main results indicate that as the number of ring magnets increased, the amplitude of oscillation decreased, and the period of oscillation also exhibited variations. The damping constant 'α' was determined, and we found that it increased with the number of magnets, indicative of a stronger damping effect. These outcomes align with the theoretical expectations outlined in the Introduction. The theoretical expectations were based on the principles of damped harmonic motion, where the amplitude decreases exponentially, and the frequency is affected by the added mass of the magnets. Our experimental data consistently supported these expectations, demonstrating the significant impact of damping forces on the glider's motion. However, in any experimental endeavor, there are potential sources of error. Small discrepancies between our results and theoretical predictions could be attributed to factors such as air resistance and uncertainties in measurements. Further investigation and error analysis would be required to quantify these potential sources of error accurately. To improve the procedure for future experiments, one could consider more precise measurements of the ring magnets' masses and their distribution on the glider. Additionally, controlling air resistance more rigorously and minimizing external disturbances could enhance the accuracy of the data collected. Overall, this experiment provided valuable insights into damped harmonic motion and the influence of damping forces on oscillatory behavior. Questions 1. What period of oscillation would you expect if the mass of the glider were doubled? Assume no damping. T = 2 π √ m k If m was doubled to 2 m , the new T would be equal to T √ 2 . With no damping our calculated T was 1.617 s, so double mass would be T = √ 2 ∙ 1.617 = 2.287 seconds

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2. How big is the frictional force (i.e., the damping force due to the magnets) at the moment when the glider reaches its first peak position? At what instant does the glider experience a maximum of the frictional force? Explain! The frictional force due to the magnets is at its maximum when the glider is at its maximum displacement (amplitude) from the equilibrium position and is momentarily at rest before changing direction. This occurs at the peak position of the glider's motion. At this point, the velocity of the glider is momentarily zero, and the damping force, which is proportional to velocity, is 0. ( F damping = - bv) . Since b is constant, damping force will be maximum when velocity is maximum, at the mean position of the glider. 3. What is the fraction of energy lost between peaks 1 and 2, i.e., during the first full oscillation of the glider which carries no magnets? If there are no magnets, thus no damping force, there is no energy lost since energy loss occurs due to damping. 4. Compare the spring term k/m to the frictional term α 2 in Eq. (13.15). Does this justify the statement that friction really doesn’t affect the frequency of oscillation in this experiment? In Eq. (13.15), the spring term k/m represents the ratio of the spring constant to the mass of the system, while the frictional term α 2 represents the square of the damping constant. When comparing these terms, we find that the spring term is much larger than the frictional term in most cases, particularly when the mass of the glider is significantly greater than the damping constant squared. This justifies the statement that friction does not significantly affect the frequency of oscillation in this experiment. The frequency of oscillation is primarily determined by the mass and the spring constant, while damping primarily affects the amplitude and rate of decay of the oscillations. 5. What would the physical picture of the process be if α were bigger than ω ? If the damping constant α were larger than the angular frequency ω , it would imply that the damping force is dominant and strong compared to the restoring force provided by the spring. In this scenario, the glider's motion would exhibit strong damping effects, causing the oscillations to quickly decay and come to a stop. Physically, the glider would experience a rapid decrease in amplitude, and the motion would approach a state of equilibrium without exhibiting significant oscillations. This situation represents overdamped motion, where the damping force dominates the behavior of the system, preventing it from exhibiting typical harmonic motion characteristics. Essentially, the glider would move towards its equilibrium position without overshooting it or undergoing oscillations. Honors Questions

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Appendix A Link to Excel Spreadsheet Acknowledgments I would like to thank my TA for providing equations during class time and helping set up the experiment, my lab partner for meeting with me outside of class time to review results, and my roommate Kyle who already did the report for helping me to understand some of the questions. References No outside sources used

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