Simple Harmonic Motion EXP13

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Charles Williamson Experiment 13: Simple Harmonic Motion Abstract This experiment seeks to measure simple harmonic motion by calculating the amplitude, period, phase and frequency of an object under oscillations.The effects of added damping forces are also investigated. The experiment uses a frictionless air track with springs and added weights to measure its position over time. Objectives 1. Study oscillations of a glider held by a spring 2. Understand concepts of oscillating motion: amplitude, frequency, and phase 3. Explore harmonic motion that includes damping forces Introduction Newton’s 2 nd Law: acceleration only occurs when there is a non-zero net force. If an object in equilibrium is at rest initially, it remains at rest indefinitely. In this experiment, the glider is in equilibrium by the forces of two springs pulling on it with equal forces. These forces cancel each other. When the glider is moved from the equilibrium position and under the influence of spring forces, the glider will seek to return to its equilibrium position by a total force known as the restoring force. However, the glider will have velocity across the equilibrium position causing it oscillate around this position. This oscillation around the equilibrium position is called simple harmonic motion. This investigation also explores the frictional forces acting on the glider. Through the equation below, the damping constant (b) acts upon the amplitude and frequency of the motion. Investigation 1 The basic apparatus is a glider resting on an air track attached to springs. When the glider is moved from its equilibrium position, its position as a function of time is measured by a PASPort motion sensor that relays data to the computer. This investigation’s goal is to determine the initial amplitude of motion A, the period T, the phase ꝟ, and the spring constant k. The first step is to determine the equilibrium

position connecting the 7.5cm springs to the glider and the track and record the data as it rests. The next set of data was the motion of the glider after it was moved to the 40cm mark on the track. To begin, the glider position at rest was recorded to establish the equilibrium position. The graph below displays this data. 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Position (m) Run #1 Time (sec) Glider Position (m) Graph 1: Glider position at rest Next, the glider was moved to the 40cm mark on the track and released. The graph below displays the glider position data that was record. 0 100 200 300 400 500 600 700 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Position (m) Run #2 Time (sec) Glider Position (m) Graph 2: Glider position after released from 40cm on the track

Using the glider’s position data from run 1 and 2, the centered position of the glider was calculated using the equation: x – x 0 . This new data was plotted in graph 3. 0 5 10 15 20 25 30 35 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Centered position vs. time Investigation 1 Time (sec) Glider Center Position (cm) Graph 3: Glider Centered Position as a function of time The amplitude of the first peak is found to be 0.565 cm. The estimated amplitude from graph 2 was found to be 0.57cm. Next, the times at which the first six peaks appeared in graph 2 were recorded and used to generate graph 4. The slope of the line is the period T and found to be 2.59. This conclusion can be made because the period T is a function of time and peak formation. Next a straight fit line calculator found the ꝟT to be 0.00597. 0 1 2 3 4 5 6 7 0 2 4 6 8 10 12 14 16 f(x) = 2.6 x − 1.62 Period T of Run #2 Peak Number Time (sec) Graph 4: The first six peaks and their corresponding time to generate period T Using the experimentally derived period T, the frequency f and angular frequency ⴍ , and their respective errors using the equation below. The frequency was 0.386 with an error of 0.000672. The angular frequency was 2.42 with an error of 0.00521.

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Next, the value of the phase ꝟ was found to be -0.386 using the equation below. Lastly, the spring constant was determined both theoretically and experimentally in order to be compared. The experimental value of k was found to be 2.199 N/m using the equation below. Because the nominal spring constant of one spring is 1.1 N/m, the theoretical spring constant of both springs would be 2.2 N/m. Investigation 2 Investigation 2 seeks to understand the role of friction within the data. To begin, a magnet is placed on either side of the glider. The glider and air track are both made of aluminum and thus not attracted to the magnet. The magnets serve to induce currents or an electromagnetic braking force that is proportional to the velocity of the glider. The magnets were added as pairs to the glider up to a total of six magnets. The amplitude, angular frequency, and spring constant were experimentally determined to understand the damping forces within the experiment. Similarly, to Investigation 1, the center position of the data was determined by subtracting the equilibrium position from the position data of the glider in motion. This generated a similar centered position graph that featured the three glider plus (2,4,6) magnets position data sets and the data from investigation 1 (graph 3). The glider position of the first six peaks was found from graph 5 and can be seen in Table 1.

0 5 10 15 20 25 30 35 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Centered position vs. time Investigation 1 Investigation 2: 2 magnets Investigation 2: 4 Magnets Investigation 2: 6 Magnets Time (sec) Glider Centered Position (m) Graph 5: Centered Position of Run #2 of investigation 1 and Run #3, 4, 5 of investigation 2 Table 1: Glider Position at initial 6 peaks 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 f(x) = − 0.09 x + 0.68 f(x) = − 0.05 x + 0.36 f(x) = − 0.04 x + 0.37 f(x) = − 0.04 x + 0.39 Peaks vs. Peak number Investigation 2 2 Magnets Linear (2 Magnets) Linear (2 Magnets) 4 Magnets Linear (4 Magnets) 6 Magnets Linear (6 Magnets) No Magnets Linear (No Magnets) Linear (No Magnets) Peak Number Glider Position Graph 6: Glider Position at initial 6 peaks. Series 1= 2 magnets, Series 2= 4 magnets, Series 3= 6 magnets Peak # 2 Magnets 4 Magnets 6 Magnets 1 0.359198 0.344098 0.339298 2 0.309198 0.282698 0.262598 3 0.265398 0.236798 0.207098 4 0.231698 0.196498 0.165098 5 0.197498 0.163898 0.134498 6 0.170098 0.136398 0.107698

0 2 4 6 8 10 12 14 16 0 0.1 0.2 0.3 0.4 0.5 0.6 f(x) = 0.74 exp( − 0.11 x ) f(x) = 0.36 exp( − 0.08 x ) f(x) = 0.38 exp( − 0.07 x ) f(x) = 0.38 exp( − 0.06 x ) Peak amplitude vs. time 2 Magnets Exponential (2 Magnets) 4 Magnets Exponential (4 Magnets) 6 Magnets Exponential (6 Magnets) 0 Magnets Exponential (0 Magnets) Time (sec) Peak Amplitude (cm) Graph 7: Graph of Peak Amplitude vs time Decay constants were derived from the exponential trendline using the equation below (13.13). Using this value the damping constant was determined using the equation below (13.14). The b values for no magnets, 2 magnets, 4, magnets, and 6 magnets were 0.083, 0.0443, 0.0559, 0.07112 respectively. 0 1 2 3 4 5 6 7 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.07 0.06 0.04 0.08 b vs. number of magnets 0 Magnets 2 Magnets 4 Magnets 6 Magnets Number of Magnets b value Graph 8: b values and the number of magnets Graph 8 demonstrates the b values of the data and shows that the b value increases proportionally with the addition of magnets. # of Magnets Angular Frequency Frequency Period T (Theoretical) Period T (Experimental) Uncertainty T 0 Magnets 2.4227 0.3855 2.593457 2.5997 .0057 2 Magnets 2.35487 0.37479 2.66817 2.6814 .0057 4 Magnets 2.2392356 0.36484 2.74093 2.75 0.0058

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6 Magnets 2.234855 .355688 2.811451 2.8186 0.0057 Using the equations below to find the frequency and angular frequency, the theoretical period T was calculated. This value was compared to the experimental period T and was found to be within the uncertainty of the experimental period T. Lastly, the spring force constant was compared using the theoretical and experimental values. The theoretical value was again 2.2N/m as in Investigation 1. Using equation below, the experimental spring force constant was found for each glider mass (table 2). As seen below, the experimental spring force constant values are all well within a 10% confidence. # of Magnets Spring Force Constant (Theoretical) Spring Force Constant (Experimental) % Difference 0 2.2 2.199 0.0004% 2 2.2 2.196 0.15% 4 2.2 2.09 4.7% 6 2.2 2.195 0.22% Conclusion The experiment sought to study the oscillations of a glider and the effects of added mass. From the data, the experimental and theoretical spring force constants were compared and found to be similar. Investigation 2 also compared the theoretical and experimental period T values. The theoretical value was found to be within the experimentally determined period T uncertainty. There are two points of error to consider in the data. Systematic errors such as an incorrectly set up apparatus that skewed the glider position data. Next, random errors could be present in the data collection. These errors are inherent but can be reduced by having a more precise glider position tracking software that is able to produce a larger data set. Questions 1. The period T would increase by T*Sqrt(2) 2. The damping force is 0 at its first peak position. As the glider pass the equilibrium position the damping force is greatest because the glider has its greatest amount of kinetic energy. 3. Ideally there would be no energy lost since there is no damping force due to the magnets and the properties of an air track.

4. Since the frictional term a^2 represents the damping force in the equation, it does not show that friction affects the oscillations. 5. If the angular frequency increases this is because the damping constant has increased. If the damping constant becomes too big the frequency of oscillations will also decrease until eventually their will be no oscillations.