Hi do it again this time with this new information: Lab Report: Investigating the Period of a Pendulum
Authors: Christopher Lopez, Matteo Puccini, John Liu
Date: September 17, 2024
Course: PHY105M

Contents
Introduction	2
Part 1: Period vs. Amplitude	3
1.1 Materials and Methods	3
1.2 Results	4
1.3 Discussion	5
1.4 Conclusion	5
Part 2: Period vs. Amplitude Redux	5
2.1 Materials and Methods	5
2.2 Results	6
2.3 Discussion	7
2.4 Conclusion	7
Final Conclusion	7


Introduction
In this experiment, we aim towards understanding the relationship between the period of a pendulum and the amplitude of the pendulum. We want to know whether or not they are independent to one another by using the model introduced in the Prelab Activity:

(Where L is the length of the pendulum and g is the acceleration due to gravity)
We hypothesize that the period remains constant and is independent of the different amplitudes since the amplitude does not appear in this equation. However, we will conduct a 2-part experiment to test this. We justify the necessity of the experiment since it is foundational information on simple harmonic motion and split it into two for a more well-rounded approach.
Statistical Equations
Mean:

Sample Standard Deviation: Used to measure the precision of the period per angle category.


s = sample standard deviation
N = the number of observations (9)
Xi = the period
x̄ = the mean value of the period of certain degree
One-Sample T test: Used to compare the means against the population mean.

x̄ = sample mean 
μ0 = population mean
s = sample standard deviation
n = sample size
Two-sample t-test: Used to compare Part 1 and Part 2 to one another: 

x̄1 = Population mean of part 1
x̄2 = Population mean of part 2
Sx1 = standard uncertainties of part 1
Sx1 = standard uncertainties of part 2

Part 1: Period vs. Amplitude
Materials and Methods
Equipment:
PASCO Pendulum Setup
Hanging Mass: 50 grams
String: 51 cm length to hanging mass
Stopwatch: Manual
Protractor: Measuring angles
Tape Measure and Scissors: To create the string
Variables
Independent Variable: amplitude of displacement values are at 10°, 30°, 60°, and 80°.
Dependent Variable: period(s) 
Controlled Variables: length of the pendulum (51 cm), mass of the hanging object (50 grams)
Experiment Method
To set up the pendulum we attached the 50-gram mass to one end of the 51 cm string. We then conducted a quick check by moving the pendulum to various angles (10°, 30°, 60°, and 80°) to ensure there are no obstructions. For each amplitude, we released the pendulum and used the stopwatch to measure the period and recorded the time for nine trials per angle. To get the period we ensured that the pendulum went forward and back to its original location, then we logged each trial's period with uncertainties (0.01). 
After getting the data we calculated the mean, standard deviation, and T-scores for each amplitude. Planned to use the T-score criteria to compare each amplitude's results pairwise.
Results

Angle (θ) Degrees
Trial 1 (s)
Trial 2 (s)
Trial 3 (s)
Trial 4 (s)
Trial 5 (s)
Trial 6 (s)
Trial 7 (s)
Trial 8 (s)
Trial 9 (s)
80
1.29 ± 0.01
1.73 ± 0.01
1.54 ± 0.01
1.61 ± 0.01
1.64 ± 0.01
1.51 ± 0.01
1.48 ± 0.01
1.72 ± 0.01
1.54 ± 0.01
60
1.47 ± 0.01
1.51 ± 0.01
1.46 ± 0.01
1.61 ± 0.01
1.53 ± 0.01
1.60 ± 0.01
1.51 ± 0.01
1.44 ± 0.01
1.51 ± 0.01
30
1.49 ± 0.01
1.51 ± 0.01
1.44 ± 0.01
1.51 ± 0.01
1.46 ± 0.01
1.46 ± 0.01
1.46 ± 0.01
1.37 ± 0.01
1.41 ± 0.01
10
1.38 ± 0.01
1.28 ± 0.01
1.45 ± 0.01
1.44 ± 0.01
1.45 ± 0.01
1.42 ± 0.01
1.41 ± 0.01
1.37 ± 0.01
1.50 ± 0.01


Calculations (Formulas from 1.1):


Angle (θ) Degrees
T Scores
Sample Standard Deviation (s)
Sample Mean (s)
Population Mean (s)
80
1.684
0.135
1.562
1.486
60
1.500
0.058
1.516
1.486
30
-1.946
0.046
1.457
1.486
10
-3.588
0.063
1.411
1.486


Discussion
The results half support the prediction that the period of a pendulum is independent of amplitude. This is because the lower amplitudes (10 degrees) had a t-score of significant difference from the population mean, meaning greater deviations from the model. While higher amplitudes like 30,60,80 were closer to the population mean.
Uncertainty and Improvements
Uncertainties were accounted for in each measurement using 0.01s and we found that the. Potential sources of error include both random and systematic errors. For random, the human reaction time when using the stopwatch could introduce variability, and inconsistent release affecting each trial. For systematic, slight variations in the pendulum's length due to string elasticity and air resistance play some role. In addition, calibration inaccuracies of the stopwatch. To improve, using the photogate would be much better for calculating the times since we don’t have to rely on human reaction speeds. Increasing the number of trials would also help reduce the standard deviation of our results. Having a controlled automated releaser would be better to avoid the inconsistency of our releases which created some sway in the motion of the weight.
Comparison
When discussing with another group, we found that we both used the stopwatch for timing, showing how we have the same procedure for part 1. However, the angles measured differed between groups, with the other group perhaps exploring a wider or narrower range of amplitudes, and our results were a bit different in range and variability. 
Conclusion


Part 2: Period vs. Amplitude Redux
2.1 Materials and Methods
So, basing off the analysis of Part 1, we believe that random errors (due to human reaction time with the stopwatch) were the source of uncertainty, as evidenced by the relatively high standard deviations in most measurements.
To address the significant random errors in Part 1, we chose to use the PASCO Photogate since it offers precision, which we mostly needed.
Equipment 
PASCO Photogate – this replaces the stop watch
PASCO Software: needed to analyze the data
Same pendulum setup as in Part 1
Statistical:
Calculated the mean, standard deviation, and T-scores for each amplitude. Using theory found in part 1. Used the T-score criteria to compare each amplitude's results pairwise, similar to Part 1.
New Experiment Method
We first aligned the PASCO Photogate with the pendulum and made sure the data of the software is accurate or consistent. We used the same four amplitudes (10°, 30°, 60°, and 80°), and the same trials per angle (9 trials). We used uncertainties of 0.01s due to it being the last recorded decimal.
2.2 Results

Angle (θ) Degrees
Trial 1 (±0.01s)
Trial 2 (±0.01s)
Trial 3 (±0.01s)
Trial 4 (±0.01s)
Trial 5 (±0.01s)
Trial 6 (±0.01s)
Trial 7 (±0.01s)
Trial 8 (±0.01s)
Trial 9 (±0.01s)
10
1.44s
1.42s
1.43s
1.44s
1.43s
1.43s
1.44s
1.43s
1.43s
30
1.46s
1.46s
1.46s
1.46s
1.46s
1.46s
1.46s
1.46s
1.46s
60
1.54s
1.54s
1.53s
1.53s
1.53s
1.52s
1.52s
1.52s
1.51s
80
1.64s
1.63s
1.62s
1.61s
1.60s
1.59s
1.58s
1.58s
1.57s



Angle (θ) Degrees
Sample Mean (s)
Sample Standard Deviation (s)
T-Score
Population Mean (s)
10
1.432
0.007
−24.20
1.486
30
1.460
0.000
Undefined
1.486
60
1.527
0.010
12.080
1.486
80
1.602
0.024
14.250
1.486


2.3 Discussion
With the use of the PASCO Photogate, we greatly reduced the random errors associated with manual timing. The sample standard deviations decreased much more compared to Part 1. An example of this is the 10° amplitude had a standard deviation of 0.007 in part 2, while in Part 1 it was 0.063 s at the same amplitude. The great issue was that the T-scores were significantly higher than Part 1 because each amplitude was. However, we also saw that the 10° and 30° amplitudes showed greater standard deviations from the population mean, similar to Part 1, suggesting that the lower amplitude is more bound to systemic errors. 
2.4 Conclusion

Final Conclusion
Two Sample T-Test Results: (Part 1 compared to Part 2)



Angle (θ) Degrees
T-Score (2 Sample) (s)
10
0.99
30
0.20
60
0.56
80
0.88


Do the incomplete areas and add on to 2.3 

Also make sure all of this is done in part 2: Designing Your Experiment: Design an experiment like in Part 1 but using your new method. Record
your plan.
Check in with Another Group: Briefly discuss your plans with another group. They may provide you
with additional ideas and feedback. Note one similarity and one difference between your approaches.
Conducting Your Experiment: Conduct the experiment as planned. (Note: it might turn out that your
new method is no better than the old one, that’s perfectly fine, just try to be as precise as you can).
Check in with Another Group: Check back in with the same group you discussed things with before.
How do your results compare? Note two similarities and/or differences between what you found, and use
this to inform your proposals for future iterations.