Table of Contents
Comparison of Data to Models (2024)
In Part 1, you built a pendulum and worked on refining the precision of your method for timing its period with the goal of being able to distinguish between two models where the difference between them becomes smaller with decreasing angle. You investigated two angles and used the $t'$-test to evaluate the degree of agreement or disagreement between your measured values and the two models.
There is more to understanding the results of an experiment than simply concluding with a statment of agreement or disagreement. For example the two models we are “testing” in this lab become increasingly difficult to distinguish as you go to smaller angle, so you expect to see increasing deviation between your data and the small angle approximation. But there are many other possibilities. For example no measurement is perfect to infinite precision, you certainly expect to see some deviations between your data and a correct model. But what if all of your data have t-prime values that indicate agreement, but every single data point is slightly larger than predicted? What if you get good agreement over some range of your data, but see increasing disagreement outside of that range?
Understanding what your data are telling you is crucial to the process of scientific investigation. This is the primary focus of today's lab.
GOALS
At the end of todays lab you will:
- Know how to interpret your data over a range of t-prime values by plotting their residuals.
- Understand how plots of residuals can be used to identify potential systematic effects in your experiment.
- Understand how to draw appropriate conclusions when comparing measurement to a model over some range of an independent variable.
Pendulum Period Models
As discussed in the previous lab, we will use the small angle approximation and the results of a numerical calculator to illustrate the process of evaluating how well your data distinguish between two models.
Small Angle Approximation
One model is the small angle approximation which states that the period $T$ of a pendulum should depend only on the length $L$ of the pendulum and the acceleration due to gravity $g$ according to the formula $T = 2\pi\sqrt{\frac{L}{g}}$.
A More Complete Model
Although the simple pendulum is a very straight forward system, solving the equations describing its motion is mathematically complicated and in fact cannot be done in closed form. It is however possible to use computational methods to calculate the periods to whatever degree of precision you need. Programs like Mathematica are helpful for this, but to save time you can use either the Colab notebook or the online calculator provided below.
Collect a new set of data
You will use the period measurement technique which you developed in the previous lab. Open up your lab notebook from last week. You can continue writing in this document as this is an extension of that experiment. No need to start a new Google Doc (unless you are part of a new group)! Just put in a new section heading. Note that you should NEVER alter the notes from previous work in a lab notebook. If you discover that previous work is incorrect you leave the old work alone. Your lab notebook is a record of what you did in the lab including the good, the bad and the ugly.
Rebuild your pendulum from the previous lab.
Your goal is perform precise measurements of the period of your pendulum at the angles; 5°, 10°, 15°, 20°, 25° and 30°, and use this data to more fully evaluate your experimental results.
For your individual assignment you will be expected to discuss the following.
Discuss the degree of precision you were able to obtain using your technique for measuring the period of a pendulum.
As part of your discussion make sure that you:
- Include a statement about how consistent your measurements were with data taken the previous week. Include data values with uncertainties to backup your statement. [EP]
- Don't forget to include or comment on uncertainties in measured values. For example when a bunch of measurements all have the same uncertainty you can simply state something like “All time measurements had an uncertainty of +/-0.3 seconds” if that was indeed the uncertainty you decided upon in the lab. [EP]
- Include a plot of the data and the predictions of both models.[DA]
- Include an xy plot of the residuals for your data vs both the small angle approximation and the results of the numerical computation for $T$. [DA]
- State what the plots of the residuals tell you about your data for both models. [SC] [DC]
- Make a plausible statement of what may have contributed to any systematic trends that show up in your residual plots. [SC] [DC]
- Make a statement about the ability of your measurement technique to distinguish between the small angle approximation and the numerical calculations. [SC] [DC]
How much data do you need?
Two questions which come up a lot in physics labs:
- How much data should I take?
- Is my data good enough?
You have most likely asked your lab instructor these questions in previous lab courses. However professional researchers cannot simply ask these questions of someone else. They have to determine for themselves when they have enough measurements of sufficient quality. A goal of this lab is to illustrate how to go about this.
The answer to these questions is based on what you are trying to accomplish with your data. Many experiments are done to attempt to distinguish between two models of a phenomena, often to a very high degree of precision. For the purposes of illustrating the process, assume that your experiment requires you to be able to distinguish between the two methods given above for calculating the period of a pendulum at angles of 5°, 10°, 15°, 20°, 25° and 30°. This is the metric you shall use to determine if your data are “good enough”. The $t^[\prime]$ test can be used to tell when your measurement for a given angle is sufficiently precise to distinguish between the two.
- Begin by measuring the period for the same angles as you used last week. In principle you should obtain the same results. If you do not obtain the same result you should come up with a plausible explanation for why they are in disagreement. Do Not Spend Time trying to get your results from today to match those from last week. For todays lab you need to obtain data over a range of angles.
- Deciding how long to collect data at any given angle is an exercise in evaluating tradeoffs. You have a finite amount of time to collect data and a specified range of angles to cover. You may not be able to get the precision you desire for each angle. This is ok, real experiments have similar constraints and learning how to make good choices about how to spend your time is part of learning how to do science.
Use the first 2 hours of the lab to work on collecting the best possible data you can for all 6 angles. Do not get stuck on one angle. If you have spent 15 minutes collecting data at 30° and still do not have agreement, go ahead and move on to the next smallest angle. Once you have collected data at all 6 angles you can go back and continue trying to reduce the size of your statistical uncertainty for earlier measurements until time for this part of the lab runs out.
Since this lab is not an experiment to test the pendulum models you are not being graded on getting some correct final result. We are using the pendulum models as a way of illustrating the teaching points for this lab.
If your final data set is not in agreement with either model, or even in disagreement with both models, this is OK! So long as you have made a good faith effort in lab to collect the best data possible you will not be marked off for this. Doing an experiment to test these models to a high degree of precision would actually require many days of work in the lab. The measurement is not as simple as you might expect. Factors such as energy lost to air resistance, friction in the pivot point, and any flexing of the support structure would all have to be investigated and accounted for. The effect of the mass of the string would have to be taken into account. Even something as seemingly simple as a “simple” pendulum quickly becomes a complex problem the moment you start working with it in the lab. This is a very important point. Understanding how nature really behaves is complicated, and there is no such thing as a perfect experiment or even the “right” answer. In lecture you learn physics theory and how to apply it in a very controlled environment where a lot of simplifying assumptions are made such as; massless strings, frictionless surfaces, spherical cows, etc. But doing science is about pushing our understanding of how nature behaves, so you have to learn how to identify and account for the real impact of all of these simplifying assumptions. This is what we mean when we say the lab component of the course is about How We Know What We Know!
Residuals
After about an hour and a half of data collection, your TA will lead a class exercise in a discussion of Residuals and how to use them to gain insight into the question of how well do your data agree with a model. More specifically you will learn:
- What a residual is.
- How to interpret a plot of your residuals.
Misinterpreting what your data are telling you about your experiment one of the most common mistakes which students make when analyzing their experimental results. It is important that you understand both what conclusions you can and cannot draw from your data in comparison to a model.
Just to make things simpler, here's a link to a colab notebook where you can enter your data and the model predictions and generate a plot to help visualize the data:
Don't forget to log out of both Google and Canvas, and to close all browser windows before leaving!
Post-lab assignment
Expectations for your individual report were provided earlier in the section titled Collect a new set of data.
REMINDER: Your individual assignment is due 48 hours after lab. Submit a single PDF on Canvas.
Don't forget to include the Attestation
Copy the following text and paste it in at the end of your individual report just above your name. This will serve as your signature for the report.
By submitting this lab report I attest that all of the work contained is my own, or is properly attributed to another source. This includes all text, calculations and plots.