PHY105M-Lab2

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105M

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Dec 6, 2023

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Part 1: Method: For this iteration of the pendulum experiment, we decided to try and find what amplitudes the model, T = 2 𝜋√ 𝑙 𝑔 , accurately predicts, and which ones it doesn’t. Thinking back to the last lab, we predicted that more extreme amplitudes, such as those over 30 degrees, would start to show some deviation from the predicted value, that could be due to a higher air resistance or some kind of friction because of the mass’s higher speed. To test this hypothesis, we once again set up the pendulum with a 50g mass and used the PASCO Photogate to measure the period of the pendulum 10 times. This time, however, the range of amplitudes we tested were: 5°, 10°, 25° and 50°. With these values, we can test at which range of amplitudes the mathematical model start to fail in accurately predicting period. We hypothesized that the model will fail to accurately predict periods for large amplitutdes which fall outside of the small angle approximation range. Data Mass: 50g Length: 32.5 cm Predicted periods: 1.14 s Pendulum Trials by Amplitude Angle = 5° Angle = 10° Angle = 25° Angle = 50° Trial 1 Period (s) 1.12 +/- .01s 1.13 +/- .01s 1.14 +/- .01s 1.19 +/- .01s Trial 2 Period (s) 1.12 +/- .01s 1.13 +/- .01s 1.14 +/- .01s 1.19 +/- .01s Trial 3 Period (s) 1.12 +/- .01s 1.13 +/- .01s 1.14 +/- .01s 1.19 +/- .01s Trial 4 Period (s) 1.12 +/- .01s 1.13 +/- .01s 1.14 +/- .01s 1.19 +/- .01s Trial 5 Period (s) 1.12 +/- .01s 1.13 +/- .01s 1.14 +/- .01s 1.19 +/- .01s Trial 6 Period (s) 1.12 +/- .01s 1.13 +/- .01s 1.14 +/- .01s 1.18 +/- .01s Trial 7 Period (s) 1.12 +/- .01s 1.13 +/- .01s 1.14 +/- .01s 1.18 +/- .01s Trial 8 Period (s) 1.12 +/- .01s 1.13 +/- .01s 1.14 +/- .01s 1.18 +/- .01s Trial 9 Period (s) 1.12 +/- .01s 1.13 +/- .01s 1.14 +/- .01s 1.18 +/- .01s Trial 10 Period (s) 1.12 +/- .01s 1.13 +/- .01s 1.14 +/- .01s 1.18 +/- .01s

Average 1.12s 1.13s 1.14s 1.185s Uncertainty (Random) 0 0 0 0.005270462767 Uncertainty (Systematic) 0.01 0.01 0.01 0.01 Compared T-values Angle = 5° Angle = 10° Angle = 25° Angle = 50° Predicted 1.414 0.707 0 3.182 Angle = 5° 0.707 1.414 4.596 Angle = 10° 0.707 3.889 Angle = 25° 3.182 Conclusion: Based on the data and the t-values, we can conclude that our hypothesis is mostly supported. Looking at the t-value comparison of the experimental periods for between the different amplitudes and their corresponding predicted values, the amplitudes of 10° and 25° had experimental periods which were indistinguishable from the model predictions (t < 1). The experimental period and predicted period for the 50° amplitude was distinguishable (t > 3). Surprisingly though, we saw the t-value for our 5° amplitude to be inconclusive (1 < t < 3), but this could likely be a result of human error when we were setting up the pendulum. Nonetheless, the overall results would seem to support our initial theory. If we were to do this experiment again, we would like to measure even more extreme angles, such as 70 degrees. In addition, another group next to us chose to do even more trials than us in order to get a more accurate average value and less uncertainty which is something we could also try and do to further strengthen our conclusion. Part 2: Methods

In part two we will manipulate the mass and length of the pendulum for a set of two angles (10° and 50°). We predict that the model T = 2 𝜋√ 𝑙 𝑔 should be able to predict periods accurately for small angles (θ ≤ 15°) regardless of the mass and length of the pendulum, however, the model will fail to accurately predict the period of pendulums with larger angles ( θ > 15°). Just as in Part 1, we will use a protractor to measure the θ of the pendulums and drop the pendulums manually. However, the periods of the pendulum will be recorded digitally with a PASCO photogate unit and corresponding softwares. Data Masses used: 20g, 200g Length of Pendulum: 0.315 m (for 20g mass), 0.330 m (for 200g mass) Model: T = 2 𝜋√ 𝑙 𝑔 Predicted periods: 1.12s (for 20g mass), 1.15s (for 200g mass) Amplitudes: 10°, 50° Angle = 10°, Mass = 20g Angle = 10°, Mass = 200g Angle = 50°, Mass = 20g Angle = 50°, Mass = 200g Trial 1 Period (s) 1.11 +/- .01s 1.15 +/- .01s 1.17 +/- .01s 1.21 +/- .01s Trial 2 Period (s) 1.11 +/- .01s 1.15 +/- .01s 1.17 +/- .01s 1.21 +/- .01s Trial 3 Period (s) 1.11 +/- .01s 1.15 +/- .01s 1.17 +/- .01s 1.21 +/- .01s Trial 4 Period (s) 1.11 +/- .01s 1.15 +/- .01s 1.17 +/- .01s 1.21 +/- .01s Trial 5 Period (s) 1.11 +/- .01s 1.15 +/- .01s 1.17 +/- .01s 1.21 +/- .01s Trial 6 Period (s) 1.11 +/- .01s 1.15 +/- .01s 1.17 +/- .01s 1.21 +/- .01s Trial 7 Period (s) 1.11 +/- .01s 1.15 +/- .01s 1.17 +/- .01s 1.21 +/- .01s Trial 8 Period (s) 1.11 +/- .01s 1.15 +/- .01s 1.16 +/- .01s 1.20 +/- .01s Trial 9 Period (s) 1.11 +/- .01s 1.15 +/- .01s 1.16 +/- .01s 1.20 +/- .01s Trial 10 Period (s) 1.11 +/- .01s 1.15 +/- .01s 1.16 +/- .01s 1.20 +/- .01s Average 1.11 1.15 1.167 1.207 Uncertainty (Random) 0 0 0.00483 0.004830458915

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Uncertainty (Systematic) 0.01 0.01 0.01 0.01 Compared T-values (mass = 20g, length = 0.315m) Predicted 1 (1.12s) Angle = 10° Angle = 50° Predicted 1 (1.12s) 0.707 3.324 Angle = 10° 4.031 Angle = 50° Compared T-values (mass = 200g, length = 0.330m) Predicted 1 (1.15s) Angle = 10° Angle = 50° Predicted 1 (1.15s) 0 4.031 Angle = 10° 4.031 Angle = 50° Conclusions In the 20g model, a comparison of the experimental period and predicted period for the 10° amplitude yielded a t-value of 0.707, indicating that they were statistically indistinguishable (t < 1). In contrast, the corresponding elements for the 50° amplitude exhibited a distinguishable difference, with a comparative t-value of 3.324 between them (t > 3). Moreover, when comparing the experimental periods of the two amplitudes, they were also found to be distinguishable from each other (t = 4.031). In the 200g model, when we compared the experimental period with the predicted period for the 10° amplitude, we obtained a t-value of 0, signifying that they were statistically indistinguishable (t < 1). This result also indicates that the experimental amplitude precisely matched the predicted value. Conversely, when we examined the corresponding elements for the 50° amplitude, a distinguishable difference was again evident as the comparative t-value equaled 4.031(t > 3). Again similarly as before, when we compared the experimental periods of the two amplitudes, they were distinguishable from each other with a t-value of 4.031. These findings supports our hypothesis which predicted that the model T = 2 𝜋√ 𝑙 𝑔 would be able to predict periods accurately for small angles (θ ≤ 15°) regardless of the mass and length of the pendulum, however, the model failed to accurately predict the period of pendulums with larger angles (θ > 15°).

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