Read this incomplete lab report: 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	2
1.1	Materials and Methods	2
1.2	Results	4
1.3	Discussion	4
1.4	Conclusion	5
Part 2: Period vs. Amplitude Redux	6
2.1 Materials and Methods	6
2.2 Results	6
2.3 Discussion	7
2.4 Conclusion	7
Overall Conclusion	7
References	7
Appendix	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 hypothesis 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.
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
Statistical 
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
T-Score: Used to compare the means against the population mean

x̄ = sample mean 
μ0 = population mean
s = sample standard deviation
n = sample size
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








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


2.4 Conclusion
Overall Conclusion
References
Appendix


I want you to complete the unfinished portions using this rubric: 
• Summarize analysis and draw appropriate conclusion
• Cite quantitative results in support of conclusions
• Properly qualify results in terms of uncertainties or the limits of the experiment
• At least one new actionable possibility for a next iteration of the experiment is proposed and motivated
• Summarize assessment of the model, compare with prediction and Part 1, answer other questions posed in the lab manual
• Data is presented in appropriate tables and/or graphs with units, labels, and titles
• Collected data is of sufficient quality and is in accordance with proposed method
• Uncertainty is evaluated/estimated appropriately
• Uncertainty is sufficiently minimized, through the number of trials or choice of instrument, as needed
• Analysis performed correctly and in accordance with method, sample calculations included as appropriate
• Goal explained and justified, includes prediction, referencing Part 1 as appropriate
• Choice of equipment and measuring devices is clear and justified
• Choice of statistical and analysis tools is clear and justified
• Choice of which variables to vary, measure, hold constant, and range of variable(s) explored is clear and reasonable
• Overall clarity of method
• Summarize analysis and draw appropriate conclusion
• Cite quantitative results in support of conclusions
• Properly qualify results in terms of uncertainties or the limits of the experiment
• Multiple actionable possibilities for a next iteration of the experiment are proposed and motivated
• Summarize assessment of the model, compare with prediction, answer other questions posed in the lab manual
• Data is presented in appropriate tables and/or graphs with units, labels, and titles
• Collected data is of sufficient quality and is in accordance with proposed method
• Uncertainty is evaluated/estimated appropriately
• Uncertainty is sufficiently minimized, through the number of trials or choice of instrument, as needed
• Analysis performed correctly and in accordance with method, sample calculations included as appropriate
• Goal explained and justified, includes prediction
• Choice of equipment and measuring devices is clear and justified
• Choice of statistical and analysis tools is clear and justified
• Choice of which variables to vary, measure, hold constant, and range of variable(s) explored is clear and reasonable
• Overall clarity of method

Here is the assignment:

Page
1
of 3
Lab 1
Period of a Pendulum
Physical Models: Simple Harmonic Oscillator
Analysis Tools: Mean, Standard Deviation, Standard Deviation of the Mean, t-score
Experimental Systems: PASCO Pendulum Setup, Hanging Clip
Equipment: Stopwatch, Photogate, Cell Phone Holder, Tape Measure, Protractor, Scissors
Safety Concerns: Careful not to allow your pendulum to swing and hit people and, correspondingly, watch
out for swinging pendulums while walking around the room!
Introduction
The goal of lab this week and next will revolve around investigating the period of a pendulum. In the Prelab
Activity for this week, you were introduced to a model that predicted that the period obeys the relation:
T = 2π
√
l
g (1.1)
Where T is the period, l is the length (from the pivot to the hanging object’s center of mass), g ≈ 9.81
N/kg is the strength of the Earth’s gravitation field. Most importantly the amplitude, θ, doesn’t show up
anywhere in the formula and therefore according to the model has no effect on the period.
This means that according to the model, the period of a given pendulum is the same regardless of the size
of a pendulum’s “swing”. It’s this aspect of the model we’ll seek to test. We will emphasize obtaining as
precise a test of this as possible.
1
Part 1. Period vs. Amplitude
First you’ll need to set up a pendulum to experiment with. You’re welcome to choose any length of string
and any of the four hanging masses. Just make sure to record which of the hanging masses you choose and
the length l of the pendulum for later reference.
Quick Check : Once your pendulum is set up, try displacing the pendulum a bit, letting it go, and watching
it oscillate a few times. Are you able to do this? Ask your TA if you’re having trouble with your setup. Try a
few different amplitudes (different sized displacements). Does it seem like there’s any difference in the period?
Designing Your Experiment: Now, write down a plan for a high-precision measurement of the period
of your pendulum for 4 different amplitudes of your choice. Make sure to include at least one amplitude
≤ 10◦ and at least one ≥ 30◦ Use last week’s lessons on measurement with the stopwatch and statistical
uncertainty to inform your decisions. Plan to use the t-score criteria described by your TA to compare
results from different amplitudes. Compare all 4 results pairwise (6 total comparisons).
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: Go ahead and give it a shot. You can change your method around while
you’re at it, you don’t need to treat your previous decisions as a contract. Make sure all your data is labeled
clearly with units, and that you keep track of uncertainty and try to minimize it.
Forming a Conclusion: Does your data support the model that the period is independent of amplitude?
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.
Part 2. Period vs. Amplitude Redux
Options for Improvement: Regardless of what you found in Part 1, your goal will be to improve your
confidence in your measurements so that you can correspondingly increase your confidence in your conclu-
sions.
How you improve the precision of your measurements will depend on whether the limiting factor in your
Part 1 was systematic error or random error. Make an assessment of which was larger. Then consider the
following. Pick the option which makes the most sense given your assessment:
• According to the manufacturer the stopwatch systematic uncertainty is actually much lower than the
digital display leads you to believe. Rather than .01 seconds, it’s actually accurate to within .01 %.
So whereas normally there would be no value in taking additional measurements once σ/√(N ) < .01
second, you actually can keep reducing σ/√(N ) all the way down to .0001 ×  ̄x. For example if your
mean value is 5 seconds you can keep taking additional measurements until the uncertainty is .0005
seconds. This is a good option to reduce systematic uncertainty if it was the limiting factor in Part 1
(if you got the standard deviation of the mean down to, or below, .01s in most of your measurements
in Part 1).
2
• On your table you have a PASCO photogate. You may use this to measure time instead of the
stopwatch. This is a good option if you’d like to reduce random uncertainty associated with e.g.
your reaction times or if your standard deviation of the mean was larger than .01s for nearly all
measurements in Part 1. To use the photogate, first make sure the PASCO interface is on. Then
go to “Desktop/PHY105M/” and move PhotogatePeriod.cap” to the desktop. Then open Capstone
from the bottom menu. After Capstone is open, select ”open experiment” from the menu and open
“Desktop/PHY105M/PhotogatePeriod.cap”. Then, all you need to to is hit “record” But ask your TA
if you need help. (If you don’t see the folder, log out then log in again and make sure to select ”Yes”
when prompted about connecting to the server).
• In the Prelab, you were introduced to the app Phyphox. This has a “Pendulum” setting (under
“Mechanics”) which allows you to use your phone as a pendulum. You may use this along with a
phone mount obtained from your TA. Just make sure to record any changes in l. This is also a good
way to reduce random uncertainty.
• You can try recording a video of the pendulum using your cell phone and determining the period that
way. Which kind of uncertainty could this eliminate? Make sure to explain.
Pick the measurement tool you think has the best chance of improving upon your results from Part 1.
Quick Check : Try it out and make sure your method works.
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.
Forming a Conclusion: What did your new method reveal about the dependence of a pendulum’s period
on the amplitude of the swing? Did it agree with what you found in Part 1? Which method, Part 1 or Part
2, was more precise? How confident are you in your conclusion?
Writing Up Your Lab Notes for Submission
Write up your notes for Part 1 and Part 2 in an organized way according to the rubric provided.
3


So do the quick checks, and all categories i havent done except the final conclusion