Read these instructions: Lab 2
Period of a Pendulum Continued
Physical Models: Anharmonic Oscillator, Physical Pendulum
Analysis Tools: Mean, Standard Deviation, Standard Deviation of the Mean, t-score, Regime of Validity,
Scatter plots with uncertainty bars
Experimental Systems: PASCO Pendulum Setup, Hanging Clips, Weights
Equipment: Stopwatch, Photogate, Cell Phone Holder, Tape Measure, Protractor, Scissors, Precision Scale
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
Last week we investigated the following model for the period of a pendulum::
T = 2π
√
l
g (2.1)
Where T is the period, l is the length (from the pivot to the 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. But (if you measured
precisely enough) you found that in at least some cases the model was falsified! Now what?
Your TA will begin lab by leading a discussion of possible explanations for the discrepancy. After contributing
your own point of view and hearing others, you’ll need to decide which you think is mostly likely and and
determine a way to investigate it.
1
Part 1. Explaining and Investigating
Like last time, you’ll need to set up a pendulum to experiment with. Configure it as closely as possible to
the way you had it last week. Record which mass you chose and the length l of the pendulum for later
reference.
Deciding on a New Hypothesis: To further help you formulate a new prediction, here are some examples
of new hypotheses students came up with last semester and how they investigated them.
• Some students thought that the model is accurate only for some ranges of data and not others. Based
off their data from last week, they made a precise statement which could be tested by e.g. collecting
new data both inside and outside of the range to see if the pattern continued.
• Some students thought the model is simply incorrect, and proposed a different model, for example
T = 2π√l/gθ2 or T = 2π√l/g(1 + θ). Students used e.g. excel to find a best fit function to provide
a new prediction, then collected additional data and checked if the new model predicted the results
accurately.
• Some students thought that the issue was an extra dissipative force (like friction or air resistance) that’s
unaccounted for in the model. They performed new trials that would be expected to increase/decrease
these effects and checked if it increased/decreased the descrepancy.
• Some students thought the mass of the pendulum, or shape, or material, played a role, or that the
length mattered. They changed these variables around and tested if this increased/decreased the
discrepancy.
• There were many other, completely different and good, ideas. These are just examples!
Last week we compared data for different amplitudes pairwise. This week, depending on how you approach
things, you may need to compare data to a model directly (like T = 2π√l/g). But how do you determine
the uncertainty in the model? We’ll learn more about this next week, for this week you can treat δTmodel = 0.
Check in with your TA: BEFORE PROCEEDING, check in with your TA to discuss your proposal. They
may offer you some additional ideas or feedback.
Designing Your Experiment: Now, write out a plan for a high-precision test of your hypothesis. You
should plan on collecting at least as much data as last week, all entirely new (so different amplitudes or for
different pendulum configurations etc. as appropriate).
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 or hypotheses.
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. Just make sure to record any
changes. Make sure all your data is labeled clearly with units, and that you keep track of uncertainty and
try to minimize it. Keep working until you’re confident in your conclusion.
Forming a Conclusion: What did you find? Was your prediction for the cause of discrepancy borne
out? List a few improvements you could make, or alternative explanation you could investigate, for a next
iteration.
2
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. Were either of your hypotheses able to account for the
discrepancy?
Part 2. Exploring Alternatives, New Systems or Other Variables
Any investigation we complete, whether we falsify a given model or not, can be continued simply by com-
paring the model to new systems or by exploring other aspects of the model.
Your approach to Part 2 will depend on what you found in Part 1. If in Part 1 you identified the cause of
the discrepancy with the model to your satisfaction, proceed with the instructions below. If not, follow the
instructions for Part 1 a second time for a new possible explanation.
Either option is worth full points as long as its logical given your results and conclusions thus far. Think of
the process as following the flowchart provided in the module.
Deciding on a New Hypothesis: If you now have a reasonable account of the pendulum for at least some
range of data, consider the following options for a next iteration of your investigation:
• Perhaps the model works better for some values of l than others?
• Perhaps the period actually depends on the mass m of the pendulum, even though the model says it
doesn’t?
• Perhaps the discrepancy in the model came from the type of pendulum we used, and would be more
accurate for a different one? Configure a different pendulum type and compare with the original model
T = 2π√l/g. Make sure you have a well thought out reason for trying this though.
• Something else you’re interested in? Discuss with your TA.
Designing Your Experiment: Design an experiment to test your new hypothesis.
Conducting Your Experiment: Conduct the experiment as planned.
Forming a Conclusion: What did you find? How could you improve your experiment?
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.

This is the rubric:
Part 1: Method
• 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
Part 1: Data
• 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
Part 1: Conclusion
• 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
Part 2: Method
• 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
Part 2: Data
• 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
Part 2: Conclusion
• 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

Things to remember while writing your lab notes:

Mention the names of all group members in the beginning.
Divide your notes into the following sections:
Part 1
Method
Data
Conclusion
Part 2
Method
Data
Conclusion
Follow the rubric and answer questions asked in the manual.
Represent your data with tables and graphs. Provide a caption explaining the contents of each table, graph, figure, etc.
Report quantities with uncertainties and units. Remember to round off numbers to the systematic uncertainty of your measuring device.
Some factors that impact your grade:
Proper explanation of what you did, how you did it and why you did it.
Properly reporting quantities.
Showing your analysis. Mention math formulas, spreadsheet functions, etc.



For graphs just say *insert type of graph here* and explain what the graphs shows cause u cant do it since your a text model.

Now i want you to look at lab 1 which is previous to this lab to understand how to do lab 2 cause u need to use our conclusion in lab 1 to do lab 2:

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.............................................................................................................6
Part 2: Period vs. Amplitude Redux.................................................................................. 6
2.1 Materials and Methods.......................................................................................... 6
2.2 Results.................................................................................................................. 7
2.3 Discussion............................................................................................................. 8
2.4 Conclusion.............................................................................................................9
Final Conclusion............................................................................................................... 9
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: used to represent the average value and give us an idea of where the “center” of
the period dataset is.
Sample Standard Deviation: Used to measure the precision of the period per angle
category trials.
● s = sample standard deviation
● N = the number of observations (9)
● Xi = the period
● x̄ = the mean value of the period of certain degree
Two-sample t-test: Used to compare the periods 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
Reading T:Scores (this holds base to our conclusions):
Part 1: Period vs. Amplitude
1.1Materials 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 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 will
release the pendulum and use the stopwatch to measure the period and record 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
(0.1s).
After getting the data we will get the mean, standard deviation, and T-scores for
each amplitude. Each data gives us an understanding of the relationship between the
amplitude and period. Planned to use the T-score criteria to compare each amplitude's
results pairwise.
We hypothesize that this experiment due to the nature of a stopwatch will
conclude a high standard deviation and T-score, meaning that the inaccuracies could
challenge the relationship between the period of a pendulum and the amplitude of the
pendulum.
1.2Results
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 ±
0.1
1.73 ±
0.1
1.54 ±
0.1
1.61 ±
0.1
1.64 ±
0.1
1.51 ±
0.1
1.48 ±
0.1
1.72 ±
0.1
1.54 ±
0.1
60 1.47 ±
0.1
1.51 ±
0.1
1.46 ±
0.1
1.61 ±
0.1
1.53 ±
0.1
1.60 ±
0.1
1.51 ±
0.1
1.44 ±
0.1
1.51 ±
0.1
30 1.49 ±
0.1
1.51 ±
0.1
1.44 ±
0.1
1.51 ±
0.1
1.46 ±
0.1
1.46 ±
0.1
1.46 ±
0.1
1.37 ±
0.1
1.41 ±
0.1
10 1.38 ±
0.1
1.28 ±
0.1
1.45 ±
0.1
1.44 ±
0.1
1.45 ±
0.1
1.42 ±
0.1
1.41 ±
0.1
1.37 ±
0.1
1.50 ±
0.1
Calculations (Formulas from Introduction):
Angle (°) Mean (s) Sample Standard
Deviation (s)
Standard
Uncertainty (s)
80 1.562 0.135 0.045
60 1.516 0.058 0.019
30 1.457 0.046 0.015
10 1.411 0.063 0.021
T-test
Comparison
T-value
80 vs 60 0.952
60 vs 30 2.382
30 vs 10 1.755
10 vs 80 3.042
30 vs 80 2.22
10 vs 60 3.651
1.3Discussion
Uncertainty and Improvements
Uncertainties were accounted for in each measurement using 0.1s 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.
1.4Conclusion
In Part 1, we found some sort of relationship between the period of a pendulum
and its amplitude using manual timing. The t-scores revealed that some comparisons,
like 10° vs. 80° (t = 3.042) and 10° vs. 60° (t = 3.651), were distinguishable, indicating a
dependence of period on amplitude. Other comparisons fell into the inconclusive range,
suggesting uncertainty about their independence. The standard deviations were
relatively high due to manual timing errors, which increased the variability in our
measurements. The standard uncertainties, calculated for each angle, reflected this
variability and highlighted the limitations of using a stopwatch for precise
measurements.
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. We now hypothesize that
this experiment will be with less standard deviation and be much closer to proving the
independence of the amplitude and period in simple harmonic motion.
Equipment
● PASCO Photogate – this replaces the stop watch
● PASCO Software: needed to analyze the data
● Same pendulum setup as in Part 1
Variables (Same as Part 1)
● 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)
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. With our data, we will then calculate the
mean, standard deviation, and T-scores for each amplitude. This will be done using
theory/formula found in the introduction.
2.2 Results
Angle
(θ)
Degree
s
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
Calculations (Formulas from Introduction):
Angle (°) Mean (s) Sample
Standard
Deviation (s)
Standard
Uncertainty (s)
80 1.432222222 0.006666667 0.002222222
60 1.46 0 0
30 1.526666667 0.01 0.003333333
10 1.602222222 0.024381231 0.008127077
T-test
Comparison
T-statistic
80 vs 60 12.5
60 vs 30 20
30 vs 10 8.601395
10 vs 80 20.17704
30 vs 80 23.57476
10 vs 60 17.4998
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.
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 as
part 1 but to a much lesser degree. For random, the inconsistent release of the weight
in the air each time was greatly affecting each trial. For systematic, air resistance plays
a role in which the period will never be consistent in different amplitudes.To improve
further our experiment, if we start combining photogate timing with some sort of
automated release mechanisms, we will reduce the natural “sway” that is created in the
pendulum's path after release. If we wanted to even go further, removing air resistance
would be the most ideal scenario, but may require extensive equipment. And the final,
natural improvement, is to increase the range of amplitudes tested to find any
discrepancies.
Comparison
When comparing our study with other groups, we found that both of our groups
with the photogate managed to decrease the variability of the data from part 1. There
was a kind of similar trend in which the higher the amplitude the higher, the slightly more
period. However, differently, some scenarios their periods were similar regardless of
amplitudes leading to questions about systemic errors in our conduction. Another
difference was that they used different amplitudes which made the data difficult to
compare accurately.
2.4 Conclusion
In Part 2, we used the PASCO Photogate to reduce random errors and improve
precision. The t-scores for all amplitude comparisons were above 3, confirming a clear
dependence of period on amplitude across all angles tested. This was consistent with
our findings from Part 1 but with greater accuracy due to reduced standard deviations.
The use of the photogate significantly lowered standard uncertainties, indicating more
reliable data collection and minimizing human error.
Final Conclusion
The experiments demonstrate that the period of a pendulum is mostly dependent on the
amplitude, especially at larger angles. Part 2's improved precision with the photogate
provided more reliable data, showing that all amplitude comparisons were
distinguishable with high t-scores. The T-Scores in part 2 were all distinguishable (Ex. T
score: 12.5, 20, 8.6), meaning that there is a very unlikely probability that it is
independent. This suggests that real-world factors like air resistance and deviations
from small-angle approximations significantly influence pendulum behavior. Future
experiments could explore these effects further by testing additional amplitudes and
minimizing external influences. Therefore, we could not support our hypothesis and
believe that our experiment has systematic errors. In part 1, the t-scores did include
some inclusive t-scores(Ex. T score = 1.75, 2.22) and one indistinguishable score (Ex. T
score = 0.952), which is different from part 2, and we believe that this stems from the
nature of human reaction time, which actually helped the model be inaccurate enough
to ignore air resistance, which is notable in part 2.
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
In addition, we can compare part 1 to part 2 and their population means and see that they are
indistinguishable, showing that even though they settle different results, they do not differ from
each other statistically, leading to similar understandings.

Now do Lab 2 fully and make up data that is reasonable along with the methods: