Precision Measurements and Model Testing (Part 2)

In Part 1, you built a pendulum and tested it against a model which predicts $T = 2\pi \cdot \sqrt{\frac{l}{g}}$. In this second part, you will continue your investigation of the dependency of the period of a pendulum on the initial angle.

Although we did not point it out you may already know that the above model is often referred to as the small angle approximation. Thus it is expected to give reasonably good results for small angles, and become progressively less accurate for increasingly large angles. The reason why most introductory textbooks emphasize the small angle approximation is that a more complete treatment of the problem is mathematically complex and in fact cannot be solved in closed form. However the fact of the matter is that we have two models, one of which is claimed to be more accurate. Testing a claim like this is an important part of experimental science. Just because one model is more mathematically complete than another does not mean it is a more accurate description of the phenomena. A lot of time and effort is spent by physicists not just testing such models, but pushing the precision of their measurements to higher levels of precision, because if one can show experimentally that the data are in conflict with the predictions of a model, no matter how strongly that model is accepted as being correct, then you will have shown that there is something more to learn about the phenomena in question. This is how science progresses. But how do you know if the disagreement between your data and an accepted model is an indication that the model is not correct, as opposed to your data being inaccurate?

The goal for todays lab is to expand upon your work from the previous week and determine to what degree can you distinguish between these two models. Showing disagreement between the two models should be relatively easy. But for smaller angles you will need to push your technique and make sure that you understand sources of bias in your data. The challenge is to determine to the greatest degree of precision you can, using the apparatus at your disposal and within the time of the lab period, the accuracy of the two models over the widest range of angles possible.

The following two questions should guide your work today.

  • What is the smallest angle for which your measurements can discriminate a difference?
  • How small can you reduce your error bars (i.e. uncertainties) on your measured periods?

Pendulum Period Models

Small Angle Approximation

The model you were given last week was 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 the online calculator provided here.

Pendulum Period Calculator.

Continuing Your Experiment

This week you will continue working on your pendulum.

  • Open up your lab notebook from last week. You will continue writing in this document. Do not start a new Google Doc (unless you are part of a new group)! Just put in a new section heading to indicate to your TA where your second day notes begin. 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.
  • Select a new member of the group to be the record-keeper. Remember that everyone will rotate through this role throughout the quarter, and that everyone – including the record-keeper – should have a chance to get their hands on the equipment.
  • You will need to rebuild your pendulum… did you keep good notes from Part 1?

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:

Remember that the TA will come around to talk to each group and to check lab notebooks, so remember to record and think about your data as you go!

Submit your lab notebook

Make sure to submit your lab notebook by the end of the period. Download a copy of your notebook in PDF format and upload it to the appropriate spot on Canvas. Only one member of the group needs to submit to Canvas, but make sure everyone's name is on the document!

Don't forget to log out of both Google and Canvas, and to close all browser windows before leaving!

Post-lab assignment

After the lab is over, each student in your group will write up their own conclusions and submit them to their TA via Canvas. Your individual conclusions are due no later than 48 hours before the start of your next lab. Your conclusions will be graded for completeness and quality according to the rubric on the PHYS141 main wiki page.

Even though you worked as part of a group in the lab, and submitted one group notebook, your individual conclusions must be your own work.

Your conclusions should not require more than one or two pages of text, though the final document may be longer if you include plots of data. The focus of your writeup should be on the final conclusions which you are able to draw based on your work in the lab. Assume that the reader, i.e. your TA, knows what the lab is about and has access to your groups notebook. As such you do NOT need to write about the following:

  • What you did in the lab.
  • Background and motivation for the experiment.
  • Theory.
  • Details of apparatus used.

You are expected to clearly articulate your conclusions and discuss how your data support those conclusions.


Take a moment to think a bit about the learning objectives for this lab course. These were listed on the lab homepage, but as a reminder we provide them again here.

For the introductory physics laboratories here at the University of Chicago, we have adopted a set of learning objectives. By the end of this course, you should be able to do the following:

  • collect data and revise an experimental procedure iteratively and reflectively;
  • evaluate the process and outcomes of an experiment quantitatively and qualitatively;
  • extend the scope of an investigation whether or not results come out as expected;
  • communicate the process and outcomes of an experiment; and
  • conduct an experiment collaboratively and ethically.

Put succinctly, the goal is to understand how we know, not what we know.*

* These goals were first outlined by the Physics Education Research Lab at Cornell University for labs at all levels, but especially for introductory labs. You can read more about the philosophy behind these learning goals  here.)

Drawing scientifically appropriate and meaningful conclusions is not easy. It is a skill which is learned and which you develop over time. Here are some thoughts to guide you in drawing conclusions from your lab work.

The conclusion is your interpretation and discussion of your data.

  • What do your data tell you?
  • How do your data match the model (or models) you were comparing against, or to your expectations in general? (Sometimes this means using the $t^{\prime}$ test, but other times it means making qualitative comparisons.)
  • Were you able to estimate uncertainties well, or do you see room to make changes or improvements in the technique?
  • Do your results lead to new questions?
  • At the end of the lab period were there things you learned about what you did and how you did it that you could use to improve your experiment and obtain more precise results?

In about a few paragraphs, draw conclusions from the pendulum data you collected today. Address both the qualitative and quantitative aspects of the experiment and feel free to use plots, tables or anything else from your notebook to support your words. Don't include throw-away statements like “Looks good” or “Agrees pretty well”; instead, try to be precise.

Remember… your goal is not to discover some “correct” answer. In fact, approaching any experiment with that mind set is the wrong thing to do. You must always strive to reach conclusions which are supported by your data, regardless of what you think the “right” answer should be. Never should you state a conclusion which is contradicted by the data. Stating that the results of your experiment are inconclusive, or do not agree with theoretical predictions is completely acceptable if that is what your data indicate. Trying to shoehorn your data into agree with some preconceived expectation when you cannot support that claim is actually considered to be fraudulent, don't do that.

In the next lab, we will return to the pendulum to take more measurements with the goal of increasing the precision of your measurements based on what you learned in this lab period. This is a common theme in experimental physics; making measurements, analyzing your data, gaining experience and learning how to improve your experimental technique, going back into the lab to refine your experiment and improve your results… We assure you that no one plans and executes an experiment all in one go.

REMINDER: Your post-lab assignment is due 48 hours before your next meeting. Submit a single PDF on Canvas.