Physics Lab 2

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Feb 20, 2024

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Lab 2 Part 1: Explaining and Investigating Method: For this lab experiment we tried to investigate potential reasons behind the observed inconsistencies between the period and amplitude of the pendulum. Building upon the previous conclusion drawn in Lab 1, which suggested that the period relied on the amplitude, our goal was to examine whether the mass played a role in causing one of the many discrepancies. By maintaining a consistent amplitude and length, our hypothesis predicted that possibly reducing or increasing the mass of the pendulum would lead to a decrease in the discrepancy and yield a lower t-score compared to a higher mass. For the purpose of keeping the controls of this experiment consistent, we decided to apply a lighter (less weighted) mass. To begin with, we consistently set the amplitude at 15 degrees as the release angle for the weight. Additionally, we measured the length of the string from which the mass was suspended to be 0.340 meters (34cm), and we maintained a gravitational constant of 9.8 m/s². With these initial conditions in place, we utilized the initial period formula to calculate the model period, which amounted to 1.17 seconds. To carry out the experiment, we tested 4 distinct weights as the independent variable: 10g, 20g, 50g, and 100g. For each weight, we used a PASCO pendulum given to us in the room, to precisely measure the period when released from the predetermined amplitude. To ensure accuracy, we set the PASCO device to record five time marks for each weight, and subsequently determined the average of the recorded times, which was considered as Trial 1 for that specific weight. All of the above steps were repeated five times for each (four) weights, and completed 5 trials per each weight. Data: 𝑀?𝑎?????????: ?𝑖?? (???????) + / − 0. 01 ??????? ● ( l ) - length of the string → 34.0 cm = 0.340 meters ● (g) - gravity → 9.8 m/s 2 ● Initial Period (T) = T = = = 1.17s 2π ? 𝑔 2π 0.340 9.8 If T>1 = data is distinguishable Mass (g) Trial 1 (sec) Trial 2 (sec) Trial 3 (sec) Trial 4 (sec) Trial 5 (sec) Average (μ) 10 1.31 ± 0.01 1.31 ± 0.01 1.31 ± 0.01 1.31 ± 0.01 1.31 ± 0.01 1.31 ± 0.01 20 1.32 ± 0.01 1.32 ± 0.01 1.32 ± 0.01 1.32 ± 0.01 1.32 ± 0.01 1.32 ± 0.01 50 1.33 ± 0.01 1.33 ± 0.01 1.33 ± 0.01 1.33 ± 0.01 1.33 ± 0.01 1.33 ± 0.01

Lab 2 100 1.34 ± 0.01 1.34 ± 0.01 1.34 ± 0.01 1.34 ± 0.01 1.34 ± 0.01 1.34 ± 0.01 10 g: σ = 0 | S = / 0/ = ±0 𝑥 5 20 g: σ =0 | S = / 0/ = ±0 𝑥 5 50 g: σ = 0 | S = / 0/ = ±0 𝑥 5 100 g: σ = 0 | S = / 0/ = ±0 𝑥 5 10 g: t = 2 → Inconclusive t = 𝑇1−𝑇0 | | δ𝑇1+δ𝑇2 t = = 2 | 1.31 − 1.17 | (0.01) 2 + (0) 2 20 g: t = 1→ Inconclusive t = = 1 | 1.32 − 1.17 | (0.01) 2 + (0) 2 50 g: t = 0 → Indistinguishable t = = 0 | 1.33 − 1.17| (0.01) 2 + (0) 2 100 g: t = 1 → Inconclusive t = = 1 | 1.34 − 1.17 | (0.01) 2 + (0) 2 Conclusion: Using the gathered data, we computed the average period, standard deviation, and uncertainty for each weight. As the systematic uncertainty surpassed the random uncertainty determined in each trial, we employed a systematic uncertainty of ± 0.01 for subsequent calculations. Utilizing these computations, we then determined the t-score for each weight and compared them to the model's t-score. By analyzing the t-values, we were able to ascertain whether a reduced weight contributed to diminishing the model's discrepancy. Based on our findings, drawing a satisfactory conclusion proves challenging due to the presence of three inconclusive t-scores and one indistinguishable t-score. These results indicate a lack of clear patterns that would either decrease or increase the discrepancy. Upon evaluating our random uncertainty, we determined that the systematic uncertainty surpassed the random uncertainty in each trial. As a result, we proceeded to employ a systematic uncertainty of ± 0.01 for the t-score calculations. After computing the t-scores for each weight and comparing them to the model's t-score, we discovered the following: the 10g weight had a t-score of 2 (inconclusive), the 20g and 100g weights had a t-score of 1 (inconclusive), and the 50g weight had a t-score of 0 (indistinguishable). Upon further examination, we can conclude that the ideal model accurately predicted the period for the 50g mass, as the t-score between the model and this mass was 0 and indistinguishable. However, we cannot draw the same conclusion for the 10g, 20g, and 100g masses, as the 10g weight increased the discrepancy while the 20g and 100g weights decreased

Lab 2 it. Consequently, there is no clear correlation between a smaller weight resulting in a smaller t-score. For future iterations, to address more discrepancies in our experiment, we can repeat the lab using smaller weight increments and a reduced amplitude value. By adjusting these two variables, we may be able to achieve a smaller discrepancy and consequently obtain lower t-values for each weight increment used.Some other alternative explanations we could investigate for the next iteration could be changing the mass of the weight by either making it heavier. Another possible explanation could be changing the length of the string by making it shorter or longer. These explanations could be the reason for the discrepancies in the model. We checked our results with the same group and both groups got distinguishable t-scores and decided that for the next iteration we could change the mass of the weight by either going lighter or heavier. Both of our hypotheses were able to account for the discrepancy. Part 2: Exploring Alternatives, New Systems or Other Variables Method: In the second part, we are examining whether using smaller and more consistent weight increments, as well with a reduced amplitude, will result in a decrease in discrepancy and t-score. Building upon the conclusions from the first portion of the lab, which consisted of 1 indistinguishable result and 3 inconclusive results, we decided on the same procedure. To begin with,the length of our string remained the same 0.340, but decided to apply a lower amplitude of 10° and smaller weight increments of 10g: 10g, 20g, 30g, and 40g. For each weight increment, we used a PASCO pendulum to precisely measure the period when released from the specified amplitude. We conducted five trials for each weight, allowing the PASCO device to record five time marks for each trial. We then calculated the average of the recorded times, which represented Trial 1 for that specific weight. This entire process was repeated a total of five times for each weight.We calculated the respective random uncertainties and t-scores after obtaining our average period values at each weight, and then compared them to the model period of 1.17 seconds. Data: 𝑀?𝑎?????????: ?𝑖?? (???????) + / − 0. 01 ??????? ● Amplitude: 10 ° ● ( l ) - length of the string → 34.0 cm = 0.340 meters ● (g) - gravity → 9.8 m/s 2 ● Initial Period (T) = T = = = 1.17 s 2π ? 𝑔 2π 0.447 9.8

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Lab 2 Mass (g) Trial 1 (sec) Trial 2 (sec) Trial 3 (sec) Trial 4 (sec) Trial 5 (sec) Average (μ) 10 1.33 ± 0.01 1.33 ± 0.01 1.33 ± 0.01 1.33± 0.01 1.33 ± 0.01 1.33 ± 0.01 20 1.33 ± 0.01 1.33 ± 0.01 1.33 ± 0.01 1.33 ± 0.01 1.33 ± 0.01 1.33 ± 0.01 30 1.35 ± 0.01 1.35 ± 0.01 1.35 ± 0.01 1.35 ± 0.01 1.35 ± 0.01 1.35 ± 0.01 40 1.36 ± 0.01 1.36 ± 0.01 1.36 ± 0.01 1.36 ± 0.01 1.36 ± 0.01 1.36 ± 0.01 10 g: σ = 0 | S = / 0/ = ±0 𝑥 5 20 g: σ =0 | S = / 0/ = ±0 𝑥 5 50 g: σ = 0 | S = / 0/ = ±0 𝑥 5 100 g: σ = 0 | S = / 0/ = ±0 𝑥 5 10 g: t = 2 → Inconclusive t = 𝑇1−𝑇0 | | δ𝑇1+δ𝑇2 t = = 2 | 1.35 − 1.17 | (0.01) 2 + (0) 2 20 g: t = 2 → Inconclusive t = = 2 | 1.33 − 1.17 | (0.01) 2 + (0) 2 30 g: t = 0 → Indistinguishable t = = 0 | 1.35 − 1.17 | (0.01) 2 + (0) 2 40 g: t = 1 → Inconclusive t = =1 | 1.36 − 1.17 | (0.01) 2 + (0) 2 Conclusion: We discovered that lower weight increments and a smaller amplitude value did not reduce the disparity for all weight increments and resulted in lower t-scores. Because we initially generated a random uncertainty of zero for all average period values, we had to use the systematic uncertainty of 0.01 to calculate our t-scores. Three inconclusive t-score results and one indistinguishable t-score value were found when compared to our theoretical period value of 1.34 seconds. The t-score obtained at 10g and 20g was 2 (inconclusive), 30g was 0 (indistinguishable), and 40g was 1 (inconclusive). We can see from our results that the ideal model successfully predicted the period for the 30g weight because the t-score between the model and this weight was 0 and indistinguishable. However, given our prior unsatisfactory

Lab 2 t-scores, we cannot conclude that smaller mass' and lower amplitude will result in less disagreement and a lower t-value. When compared to Part 1 of our lab, the same 10g weight resulted in equal t-score values, however the same 20g resulted in a huge variance.The 20g weight resulted in a lower t-score of 1, while the t-score grew to 2 in Part 2. It's difficult to pinpoint the causes of these variances, although it could be related to a smaller amplitude; nonetheless, more testing would be required to rule out the amplitude size as a factor. We proposed that we improve our lab by researching a wider range of weights, both little and large, because our results offered contradicting data that prevented us from identifying clear significant differences between mass and period. With that being said, we could possibly look further into these larger weight ranges at different string lengths to see when the discrepancy increases and/or decreases or instead of looking at the impacts of mass on period, we could perhaps look at the effects of amplitude on oscillation.