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Comparison of Data to Models (PHYS141) - Fire Alarm Makeup Assignment
In Part 1, you built a pendulum and worked on refining the precision of your method for timing its period. You used a metric of being able to distinguish between two models where the difference between them is small as a way to define how precise your technique needed to be.
There is more to understanding the results of an experiment than simply concluding with a statement of agreement or disagreement. For example the two models we are using as standards in this lab become increasingly difficult to distinguish as you go to smaller angle. But there are many other possibilities. For example no measurement is perfect to infinite precision, you expect to see some deviations between your data and any model. However 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 this exercise you will:
- Know how to gain insight into your data by plotting their residuals.
- 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 used 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 cannot be done in closed form. It is however possible to calculate the periods to whatever degree of precision you need. The following link will open a Google Colab notebook which you can use to calculate a more precise value for the period of a simple pendulum. The notebook uses the Python programming language. This is not a programming course, but we do want you to see how simple programs can be used when doing experimental work. The notebook provided is a tool. You do not need to know how to program in Python to use it.
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.
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 containing some sample data and the model predictions and which generates a plot to help visualize the relationship between the data and the model:
Post-lab assignment
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.