Experiment 13 - Declan Rogers

Report for Experiment #13 Simple Harmonic Motion Declan Rogers Lab Partner: Aidan Kaneshiro TA: Zhuyao Wang 5-12-22 Abstract This experiment examined harmonic motion. The experiment was done with a glider attached to two springs on opposite ends while sitting on a track that produced an air buffer between it and the glider. The oscillation was examined with no motion to determine the equilibrium position and then it was observed with no added forces as it oscillated to determine a control motion for the glider. Then there were three trials with increasing weight added by two magnets to observe the effect that added weight had on the motion and in what manner it changed as the weight increased. Following all four trials it was determined that the period increased with the number of magnets attached. This result supports the theory that an increase in weight of an object in harmonic motion will directly decrease the period of motion of that onject.

Introduction This experiment analyzed simple harmonic motion in the real world. In most instances of motion in the real world, friction is also involved and can be quantified. However, this experiment attempted to demonstrate simple harmonic motion uninhibited by outside factors like friction. To this end, the glider, to which the springs of the experiment are attached, is set upon an air track to create a separation between the surfaces and ensuring that there is no dry friction present. This glider would oscillate along the track due to the two springs and data based on this oscillation was recorded. Then the weight of the glider was increased in increments of two magnets while repeating the oscillation and data collection. This was continued until trials had been completed for no magnets, two magnets, four magnets, and six magnets. Although the friction was limited, the data returned indicated that there was still a certain resistance to the motion of the glider, likely caused by the magnetic field of the magnets which create an electric field and attracts the track which is made of metal. The intention of the experiment was to quantify this resistance and correlate it to the magnetism of the magnets, ensuring that the simple harmonic motion, at its base, performed as theoretically expected. The data collected was used to create multiple graphs to observe the amplitude of oscillation and the compare it to the height of each peak recorded by the ultrasonic sensor. An exponential relationship between the ultrasound distance data and the amplitude of the oscillation. Investigation 1 The first investigation focuses on simple harmonic motion without external factors, setting the control for the overall experiment. Data is recorded while the glider is in stasis and when it is pulled to the 40 cm location along the track. To setup this investigation, first take the mass of the glider with the digital scale. Take down this value on a spreadsheet. Activate the air in the track so that the valve is three quarters open. Next ensure that the glider does not move along the track by balancing out the end with the screw lifting or lowering the track. Connect the PASPort USB to the computer and ensure that it is also connected to the ultrasonic motion sensor. Open the PASCO Capstone application on the computer and open the table and graph option. Create one column that records the position in meters and one column that records the time in seconds. Then, select the hardware setup and ensure that the sensor is set to Hz. Move the glider right in front of the sensor and turn the sensor until it is facing the glider perfectly by turning the knob on the side. Once the main setup has begun, attach the springs to the glider. One should be fastened to each side of the glider and then to their respective ends of the track. Ensure that all collected values are to the thousandths place. The first trial will document the state of equilibrium. Leave the glider as is and run the PASCO Capstone application for thirty seconds. Copy the table of

This average was utilized against the second trial points to graph the gliders position in relation to its equilibrium rather than in relation to the location of the sensor as the program collects. The graph below shows this relationship. Object 7 Fig. 1: Position vs Time of the glider in relation to its equilibrium position Through this graph the amplitude, phase angle, and period were determined and recorded in the table below: Table 2: Estimated Amplitude, Period, and Phase Angle A (m) 0.20 period (s) 2.60 phase ϕ -0.99

Both the amplitude and the period can be found by analyzing the first peak of the graph and then put into the equation below to determine the phase angle: Equation 3: x − x 0 = Acos ( ϕ ) Then, utilizing these estimated values, the frequency, angular frequency, spring constant, and actual phase angle can be determined. Table 3: Frequency calculated using estimated data (with respective errors) f (s) δf δf/f 0.3840 0.6232 1.6248 The frequency is found by calculating the reciprocal of the period: Equation 4: 1 T The propagated error can then be found using the absolute error in frequency of 0.6232 in the following equation: Equation 5: δf f Table 4: Angular Frequency calculated with error ω (s) δω δω/ω 2.4100 0.6232 0.2586

Table 6: Spring Constant calculated with error K (N/m) δk δk/k 2.2280 0.5761 0.2586 The spring constant is determined using the mass of the glider and the angular frequency in the manner shown below: Equation 11: ω 2 m glider Then the propagated error of the spring constant is then found with Equation 12: Equation 12: √ ( δω ω ) 2 + ( δ m glider m glider ) 2 Then the absolute error of the spring constant is found with the expression below: Equation 13: δk ∗ k Then the visualization of the linear decay in the motion as it approaches equilibrium is done via graphing. This included each peak plotted against the time at each peak which was recorded in Table 7.

Table 7: Each numbered peak and the time at that peak: Peak Time (s) δ period 1 1.00 0.0005 2 3.65 0.0005 3 6.25 0.0005 4 8.85 0.0005 5 11.45 0.0005 6 14.05 0.0005 0 1 2 3 4 5 6 7 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 f(x) = 2.61 x − 1.58 R² = 1 Time (s) vs. Peak Number Peak number Time (s) Fig. 2: Time (s) vs Peak Number Fig. 3: IPL Straight Line Fit Calculator

With the absolute error of the mass of 0.00005 kg, the propagated error is determined with the equation below: Equation 14: δ m magnet m magnet From the data of the program ran for each trial as well as the data gathered in the first investigation and the equilibrium point, the following graph was produced: Fig. 4: Position vs. Time graph of all three trials and control trial centered about equilibrium point

With this graph a table with each of the peaks and their times can be found alongside the error Table 9: Peaks and their times for two magnets with error Peak Time (s) δ period δT/T 1 1.10000 0.00050 0.00046 2 3.75000 0.00050 0.00013 3 6.40000 0.00050 0.00008 4 9.10000 0.00050 0.00005 5 11.8000 0.00050 0.00004 6 14.5000 0.00050 0.00003 Table 10: Peaks and their times for four magnets with error Peak Time (s) δ period δT/T 1 2.50000 0.00050 0.00020 2 5.20000 0.00050 0.00010 3 8.00000 0.00050 0.00006 4 10.7500 0.00050 0.00005 5 13.4500 0.00050 0.00004 6 16.2000 0.00050 0.00003 Table 11: Peaks and their times for six magnets with error Peak Time (s) δ period δT/T 1 2.60000 0.00050 0.00019 2 5.45000 0.00050 0.00009 3 8.25000 0.00050 0.00006 4 11.0500 0.00050 0.00005 5 13.9000 0.00050 0.00004 6 16.6000 0.00050 0.00003

0 1 2 3 4 5 6 7 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 f(x) = 2.74 x − 0.25 R² = 1 Time vs. Peak Number Peak Number Time (s) Fig. 7: Time vs. Peak Number graph for four magnets Fig. 8: IPL straight line fit calculator for four magnets 0 1 2 3 4 5 6 7 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 f(x) = 2.8 x − 0.17 R² = 1 Time vs. Peak Number Peak Number Time (s)

Fig. 9: Time vs. Peak Number graph for six magnets Fig. 10: IPL straight line fit calculator for six magnets With each of the graphs and the IPL straight line fit calculation that follows, it is shown that most of the data is without error. The first trial has a slope of 2.6814 s with a y-intercept of -1.6100 s, as compared to the IPL values of 2.6814 ± 0.6410s and -1.6100 ± 2.4962s. The second trial has a slope of 2.7429 s with a y-intercept of -0.2500 s, as compared to the IPL values of 2.7429 ± 0.6557s and -0.2500 ± 2.5534s. In the final trial the slope is 2.8043 s with a y-intercept of -0.1733 s, as compared to the IPL values of 2.8043 ± 0.6704s and -0.1733 ± 2.6107s. As is evident, each of the calculated values falls withing the range of each respective IPL error and is consistent with what is expected of the experiment. The data from the fourth figure was then used to produce a table with each peak and the amplitude at that point for each trial of investigation two. Table 12: Peaks and amplitudes for two magnets with error Peak A (m) δA δA/A 1 0.2060 0.0005 0.0024 2 0.1780 0.0005 0.0028 3 0.1480 0.0005 0.0034 4 0.1250 0.0005 0.0040 5 0.1060 0.0005 0.0047 6 0.0840 0.0005 0.0060 Table 13: Peaks and amplitudes for four magnets with error

Each preceding table and its data was then used to produce a graph with a visual on the relationship between amplitude and peak. 0 1 2 3 4 5 6 7 0.000 0.050 0.100 0.150 0.200 0.250 f(x) = 0.25 exp( − 0.18 x ) R² = 1 Amplitude (m) vs. Peak Number Peak Numbers Amplitude (m) Fig. 11: Amplitude vs. Peak Number graph for two magnets 0 1 2 3 4 5 6 7 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 0.160 0.180 0.200 f(x) = 0.23 exp( − 0.26 x ) R² = 1 Amplitude (m) vs. Peak Peak Number Amplitude (m) Fig. 12: Amplitude vs. Peak Number graph for four magnets

0 1 2 3 4 5 6 7 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 0.160 0.180 f(x) = 0.24 exp( − 0.37 x ) R² = 1 Amplitude (m) vs. Peak Peak Number Amplitude (m) Fig. 13: Amplitude vs. Peak Number graph for six magnets These graphs show that there is an exponential relationship between the amplitude and peak number in simple harmonic motion. With this data the b value can be determined by observing the decay constant of each graph and plugging it into Equation 17: Equation 17: α = b 2 m magnet With the b values of each trial and the magnet number of each trial, the following table was produced: Table 15: Number of magnets and corresponding b value number of magnets b 2 0.00392 9 4 0.02353 2 6 0.04941 7

The T’ values were found using the b value, the spring constant, and the total mass of each trial. All of these variables were compiled into Equation 18 to find the T’ values for every trial with its respective data. Equation 18: 2 π √ ( k m ) − ( b 2 4 m 2 ) The T’ values that were calculated can be compared to the T values directly from the data. The slope the graph with time versus the peak can be compared to the T’ in that trial as shown in Table 16. In the two-magnet trial the T was 2.6814 seconds and the T’ was 2.6985 seconds. In the four-magnet trial the T was 2.7429 seconds and the T’ was 2.7715 seconds. In the six-magnet trial the T was 2.8043 seconds and the T’ was 2.8432 seconds. Each measured period is incredibly close to the following calculated value, both sets of data showing that increases in the mass of the glider results in increases in the period. Conclusion This experiment included finding the equilibrium of the glider connected to two springs and then the behavior of its oscillation with no added weight, followed by adding increments of two magnets to the weight for three added trials. The overall purpose was to determine the effect that added weight on the glider had on its motion. The first trial consisted of finding the position of equilibrium as well as the control trial with zero added magnets. Then the second trial had the three trials including added weight, with two, four, and six magnets respectively. The distance was measured with an ultrasonic sensor and then this data was used to calculate the period of each trial and determine the impact that the added magnets had on the simple harmonic motion. The first investigation found the spring constant of 2.228 N/m which is incredibly close to the theoretical value of 2.200 N/m. This comparison was necessary to determine the basis of the following trials would be compared to a control that performed correctly. The second investigation found the different periods of the added weights through a time versus peaks graph. The experimental period was found and then compared to the period found through the slope. For the two magnets the T was 2.6814s and the T’ was 2.6985s, for the four magnets the T was 2.7429s and the T’ was 2.7715s, and for the six magnets the T was 2.8043s and the T’ was 2.8432s. Each of these pairs of periods were very close to one another, signifying a limited error

in the overall experiment. As the weight of the glider increased, the period increased proportionally, supporting the idea that the magnets had a dampening effect on the glider’s motion. Despite the extremely limited error, the experiment overall could be improved to close the small gaps. The major area where human error was allowed was during the release of the glider. The timing of the PASCO program was done by the human experimenter and allowed for some delay in the time between beginning the program and releasing the glider, adding a discrepancy to the data. With a synchronized release system the program could be activated the moment the glider was let go for near perfect data. Questions 1. The period would increase by the square root of two as it is inversely proportional to the frequency which is in turn inversely proportional to the square root of the mass. 2. The force is proportional to the velocity, therefore the point at which the kinetic energy is highest is where the greatest friction is located. The position with the highest kinetic energy is as the glider is speeding past the equilibrium point. 3. In the first complete oscillation, there should be no friction force on the glider with no magnets as it is floating on an air track. 4. The friction does not impact the magnitude of frequency as it is not part of the equations for k/m or the α 2 . 5. The glider would not move as the decay constant would be greater than the angular frequency if α were greater than ω.