Simple Harmonic Motion - Simple Pendulum

Experiment: Simple Harmonic Motion - Simple Pendulum Purpose The purpose of this experiment was to prove that the period of a simple pendulum is independent of both the mass of the hanging object and the angle of displacement of the pendulum. We also calculated an experimental value for acceleration due to gravity using the period of the pendulum. Theory A simple pendulum apparatus consists of a massed object connected to a massless string of a certain length. Simple pendulums are tools that demonstrate simple harmonic motion (if the angle of displacement is less than 30 degrees). When the pendulum is at rest, the displacement angle is equal to zero degrees. As the mass is released from some angle of displacement, the string swings back and forth, with gravity working as the restoring force. The period is the time it takes to complete one full cycle. Therefore, the period of a simpe pendulum is the time it takes to swing from one side to the other and back again. The period has units in seconds and is inversely proprortional to the frequency. The following equation can be used to find the measured period: Another useful calculation is the calculated period. The calculated period can be used as a GAV to determine how accurate the measured period is and can be found using the following equation: Another value that can be considered is angular frequency. Angular frequency is related to the physical properties of a system, but it is not dependent on all those properties. The physical properties present within the simple pendulum device include: length of the string, mass of the object, and the angle of displacement. We can use this information in order to find angular frequency with the following equation: In addition to finding results about the period, this experiment is also able to prove the acceleration due to gravity (due to gravity being the restoring force). By creating a graph of length vs. period^2 of the first three trials with the simple pendulum, a slope can be determined through excel. Once the slope is found, it will be possible to solve for acceleration due to gravity with the following equation: Procedure We first had to establish the pendulum apparatus by measuring the length of the string hanging from the pole with a meterstick to match the length indicated in the lab manual (1.2 m). I was responsible for this part. Then we attached the 50 g mass to the bottom of the string with a hook. At the top of the string where it was attached to the metal pole, we held up a protractor and measured the angle of displacement for the string from the vertical to be 10 degrees. I held up the protractor for most of the trials. A lab partner pulled the string out to match the 10 degree mark and let go as another partner started a stopwatch on her phone. Another lab partner counted 20 oscillations of the pendulum. At the exact moment the 20th oscillation was met, the timer was stopped. This process was repeated three times for each trial. The next trials included changing the string length to 1 meter and 0.75 meters at 10 degrees displacement. We also did trials with 0.75 meters of string and 20 degrees displacement, and then 0.75 meters, 20 degrees displacement, and a 100 g mass. Each of these situations was repeated a total of three times each, giving us three times (in seconds) for each. We then found the average time for each trial, experimental period (in seconds), GAV period, and percent error. Experimental period for the first three trials was squared and plotted against length of the string in Excel. The slope from the best fit line was used to solve for the experimental value of g. Here is a pendulum apparatus similar to the setup we used in lab to obtain our data. Data (attached) Analysis (attached) Conclusion The results of this experiment were very accurate. For trials 1-3 in which we only changed the length of the string and kept the mass and angle of displacement constant, the measured periods became smaller as we decreased the length of the string. For example, at a length of 1.2 m, our measured period was 2.251 s (with average time of 45.02 seconds for 20 oscillations). At a length of 1 m, the measured period was 2.0595 s (average time of 41.19 s for 20 oscillations). At 0.75 m, the measured period was 1.772 s (average time of 35.44 s for 20 oscillations). When comapared to the calculated (GAV) periods for trials 1-3, we obtained percent errors of 2.41%, 2.66%, and 2.01% for trials 1, 2, and 3, respectively. The fact that the period changed with the length of the string suggests that the period is dependent on length. For trial 4, the mass of the object was kept at 50 g, and the length of the string was kept at 0.75 m, but the angle of displacement was increased to 20 degrees. For this trial, our measured period was 1.769 s (average time of 35.38 s for 20 oscillations). Our calculated period was 1.737 s, giving us a percent error of 1.84%. The periods for trials 3 and 4 were very similar, which suggests that period is independent of the angle of displacement. For trial 5, the length of the string was kept at 0.75 m and the angle of the displacement was kept at 20 degrees, but the mass of the object was changed to 100 g. For this trial, our measured period was 1.7555 s (average time of 35.11 s for 20 oscillations). The calculated period was 1.737 s, giving us a percent error of 1.07%. The periods for trials 4 and 5 were very similar, which suggests that the period is independent of the mass of the object. From this experiment and the results, we learned that the period does not depend on the initial angle of displacement or the mass of the object hanging. The period in fact does depend on the length of the string. A graph was generated of length versus period squared for the first three trials. The slope of this graph from the best-fit line equation was used to find the acceleration due to gravity. With a slope of 0.233 m/s^2, we calculated the g to be 9.198 m/s^2. This gave us a percent error of 6.24% when compared to the GAV of 9.81 m/s^2, which is rather accurate but not as accurate as our results for period earlier. Sources of error include incorrect measuring of displacement angles and string lengths, delayed reaction times when started and stopping the stopwatch timer, and the string not being pulled tightly (with enough tension) when being released. This final source of error could have affected the path of motion of the mass, making it drop when being released instead of fall (simple harmonic motion).