Table of Contents
Student Code Of Conduct
There are certain expectations which the department of physics holds regarding your conduct in the instructional labs. These expectations relate to Safety, Etiquette, Professionalism and Ethics. Your TA will review these expectations with you at the beginning of your lab session. Please note, that behavior which falls in the Unacceptable category can result in your removal from the lab. These types of behavior do happen in the lab every year, so please take a moment to reflect on them.
Now that that is out of the way, on to today's lab.
Precision Measurements and Model Testing (PHYS141) - 2025
Introduction
In physics, theory and experiment are fundamentally intertwined: you cannot have one without the other. Experimental discoveries open new avenues for theoretical development, while at the same time new theoretical predictions motivate experimental work. Experiments are performed for various purposes. Often experiments are designed to investigate phenomena for which there exists no theoretical understanding, while other experiments are designed to test theoretical predictions. Unexpected experimental results can lead to the discovery of “new physics”. Experiments are NOT done to confirm what we know from theory. It is Theory which has to be experimentally verified and tested.
In science, we never prove a theory. We can disprove a theory (by providing contradictory data which does not agree with the predictions of the model) or find data in support of a theory (by providing data which is consistent with the predictions), but we can never say that a theory or model is completely true. We might one day be able to use that model to make a new prediction that shows a flaw in the model, or repeated measurements with increasing precision may eventually lead to disagreement with a model that was previously in agreement with experimental results.
In the introductory lab, we made our own measurement device and used it to estimate the length dimensions of a card. When making these measurements, we had to estimate uncertainties on each value. Most students determined those uncertainties based on the limited resolution of their ruler. This week, we want to explore a completely different type of measurement – one where the uncertainty is better estimated through repeated measurements and statistics.
Lab notebook template
One member of the group should click on the link below to start your group lab notebook. (You may be asked to log into your UChicago Google account if you are not already logged in.) Make sure to share the document with everyone in the group (click the “Share” button in the top right corner of the screen) so each member has access to the notebook after you leave lab.
All members of the group are expected to contribute to all aspects of the experiment, including making notes in the lab notebook. If you brought a laptop or tablet to lab, you may want to open multiple copies of the notebook so that different group members can contribute simultaneously. If you did not, then it's OK to have just one group member typing in the document at a time (though you should regularly rotate record-keeping duties… both within the lab period and from lab-to-lab.)
The point of the lab notebook is to serve as a reference for when you later write up your individual summary of the work you did in lab. This gives you an opportunity to work together in lab to understand your data, figure out how to do the calculations and interpret the results. All of the work which you do as a group in the lab is legitimate content for your lab notebook. When it comes time to write up your own analysis and conclusions you will have the group notebook as a resource to draw on. You should be able to find all of your raw data in the notebook, as well as comments and reminders as to what you did in lab and anything else at the time which you thought would be useful to remember later for your conclusion.
Part 1: Comparing Measured Values
Something which is done all the time in physics is comparing experimental results with theoretical predictions and the results of other experiments. The seemingly simple act of comparing two numbers to see if they are in agreement is something you have likely taken for granted. You compare your measured value of the acceleration due to gravity $g$ to the “known” value of $9.8 m/s^2$ and if they are close you say they match and call it a day.
But what does it actually mean to say that two numbers are close enough to be considered in agreement with each other? Just saying they are close, or within 1% of each other is meaningless in science. How close is close enough depends on many factors. One of the more important considerations is how well do you know your measured values? Just because your calculator spits out a number to 14 significant figures, does not mean your measurement is accurate to that level of precision. This is where uncertainties in measured quantities becomes important. This lab will focus on how physicists approach the problem of determining how close is close enough. Specifically you will:
- Learn one convention* for quantitatively comparing two measured values with uncertainties for the purpose of assessing the degree of agreement between them.
- Learn how to quantitatively assess the accuracy and precision of your experimental technique.
Convention?
*A common misconception among students at this level is that there is some rigorous way to determine if your measured value is right. You will be learning specific mathematical techniques for calculating and propagating uncertainties, with well defined results indicating agreement or disagreement. However do not be fooled into thinking that these techniques are anything but conventions. Generally agreed upon standards which scientists use to try and communicate how well they know what they have measured.
Dropping paper: a quick statistics lesson
In the at-home experiment, you estimated uncertainties on measured quantities based on the resolution of your ruler. For this experiment, we're going to introduce a different way of estimating uncertainty based on making repeated measurements.
To get some practice – and to learn some definitions – we're going to first play with paper before moving on to the pendulum.
GOALS
At the end of Part 1 of this lab you will:
- Know how to calculate and use the $t^\prime$ criteria to quantitatively assess the degree of agreement between two measured values.
- Have performed a simple experiment and used the $t^\prime$ criteria to assess the degree of agreement between your measured value and the values obtained by the other groups in your lab.
Taking data
Your task is as follows: drop small paper squares from a height of 1 meter and determine how long they take to hit the floor.
- How will you measure the height?
- How will you measure the time?
Repeat the measurement several times (you will only have a few minutes) and record your data in your group notebook.
Once you have your data, discuss the question “How long did it take for the paper to hit the floor?” within your group. After a short time, your TA will lead a class discussion; be prepared to share your thoughts.
Partner up with another group in the class and compare your values. You both were trying to answer the question “How long does it take for a piece of paper to fall 1 m?” Do your two groups agree on the answer? What does agreement mean here? What criteria do you use to determine this?
Class discussion
Your TA will prompt a class discussion. Don't expand the following drop-down sections until instructed to do so.
Gaussian (normal) distributions
When we make observations of something which varies with random fluctuations, that data typically obeys a Gaussian (also called a Normal) distribution. Such a distribution can be characterized by parameters like the average, $\mu$, (or mean) and standard deviation, $\sigma$. The standard deviation tells us something about the typical “spread” in values. It tells us how wide the distribution is.
Suppose we a student who can collect data for a long time. The figures below show an example of how this student's Gaussian distribution evolves as they add more and more data.
 10 drops |  50 drops |
 100 drops |  500 drops |
As the number of drops increases, the distribution looks more and more like a smooth Gaussian and we get better estimates for both the average and the standard deviation. But notice that the standard deviation (the width of the distribution) doesn't change much; taking more data doesn't make the distribution more narrow.
For that reason, we need to define a different parameter… one that will scale as the number of data points increases to reflect the fact that our estimate of the uncertainty also improves with the number of counts. This parameter is called the standard deviation of the mean, $\sigma_{\mu}$, (or, sometimes called the standard error):
| $\sigma_{\mu} = \sigma/\sqrt{N}$. |
If we look now at those data distributions again, we can see that the standard deviation of the mean does decrease as we add more data. Hence, we can now more carefully state our average and its uncertainty as $\mu \pm d\sigma_{\mu}$.
 10 drops |  50 drops |
 100 drops |  500 drops |
Using the plot with 500 drops, we can now say that the average fall time is $T = 0.997 \pm 0.006$ seconds.
Revisiting your data
Now, return to your drop data and the data from the other group you spoke with. Compute the standard deviation of the mean and use the $t^{\prime}$ criteria you learned last week to look for agreement between your average value and that of the others. Are your results in agreement with the others? In disagreement? Inconclusive?
If you are in disagreement, what could be some reasons?
To help you with these calculations, we provide an online program here. This is a Google Colaboratory notebook running Python code. You do not need to know how to code to use the notebook… it is just a tool!
Criteria for establishing agreement – $t'$
Now suppose we have two measurements which we want to compare: $A \pm \Delta A$ and $B \pm \Delta B$. In order to determine whether they are in agreement, we will measure a quantity called $t'$: \begin{equation*} t' = \frac{A - B}{\sqrt{(\Delta A)^2 + (\Delta B)^2}}. \end{equation*}
In the case where only one value has an uncertainty (for example, when you want to compare a measured value $A \pm \Delta A$ to a predicted or literature value $B$), this simplifies to \begin{equation*} t' = \frac{A - B}{\Delta A}. \end{equation*}
If the values are within one uncertainty of each other, it is possible that the difference is due only random chance. We will consider this to constitute agreement: $\lvert t' \rvert \leq 1$. (Note that agreement might turn into disagreement if more data is taken and the size of the uncertainties shrink. Remember that we can never prove a model correct… we can only say that current data supports its.)
If the values are more than three uncertainties away from each other, it is statistically unlikely that the difference is due only random chance. We will consider this to constitute disagreement: $\lvert t' \rvert \geq 3$.
If the values are between one and three uncertainties of each other, we cannot say with certainty if the difference is random chance or a real disagreement. We will consider this to be inconclusive: $1 < \lvert t' \rvert < 3$.
Remember to record your data, notes about how it was taken, plots of the data and the results of calculations performed while working in the lab. You will need this information when for your individual summary, so be thorough.
Individual Assignment - Part 1
In the lab you worked as part of a group. Your group may well have produced plots of data and performed calculations which will be included in your individual assignment.
You may not however, include work which was done by someone else as part of your individual assignment. You are expected to go through all of the calculations, and produce all of the plots that you include in the assignment, even though someone else in your group may have already done all of this in the lab.
Part of the reason for this is of course that doing the work is part of the learning process. So even if your partner ran the python code in lab because they already know python, it is still important that you work through the scripts and gain experience with them. You will not receive full credit for the report if you have not done the work yourself.
Additionally there is the matter of academic honesty. By handing in an assignment with your name on it you are making the statement that the work is your own. If for some reason you choose to include work which was done by someone else, you must give attribution to the person who did the work. Not doing so is considered plagiarism, which is a serious matter.
For your individual assignment you will be expected to discuss the following.
Assess the agreement between your measured value of the time it takes for the paper square to fall, and the values obtained by the other groups in your lab?
As part of your discussion make sure that you:
- State the degree of agreement or disagreement, as determined by $t^{\prime}$, for each group that you compared your results to. [DA] [SC]
- Show your calculation of $t^{\prime}$. Write out the full calculation, do not skip steps. If made comparisons with more than one group in class, it is only necessary to show this calculation once. [DA]
- For each group that you compared your results to, summarize your conclusion regarding whether or not you were in agreement as determined by your $t^{\prime}$-test. For results where you were in disagreement, was there any plausible reason for the disagreement? Plausible does not include “human error”, “If we had better equipment or more time”. This should require no more than about a paragraph for each comparison. [DC] [SC]
The description of how your individual assignments will be graded can be found here.
Part 2: Precision measurement - Developing A Measurement Technique
The focus of this part of the lab is to learn about what goes into making a precision measurement.
By the end of Part 2 of this lab you will:
- Develop a method for timing the period of a pendulum to a high degree of precision.
- Identify, investigate, and assess possible sources of systematic bias in your experimental technique.
- Quantitatively assess the ability of your timing technique to reach a specified degree of precision.
Precision measurements are common in physics research. One example can be found in high energy particle physics experiments at places like CERN, which continue measuring the mass of fundamental particles to higher and higher precision. Why do they do this? Because the Standard Model, which is the theory that describes fundamental particles like the Higgs boson, makes very precise predictions for their masses. If experimental results disagree with these predictions, even by the tiniest amount, it could be an indication of new physics. So no matter how precisely a particles mass has been measured, there is always the possibility that pushing the precision one more decimal place could lead to a break through.
The key to making precision measurements is not having the “best equipment”, whatever that means. The key is understanding what factors are limiting how well you know your measured quantities. This understanding comes from a process, a process which involves identifying and investigating possible sources of bias which are inherent in your experimental technique. In a very simplified form this process can be broken down in the following parts:
- Testing for self consistency. I.e. does your measurement technique consistently yield the same results.
- Identifying possible sources of bias.
- Conducting experiments to investigate and quantify possible biases.
- Determining whether or not something needs to be changed in your experimental technique.
These steps are repeated until you have reduced the bias to an acceptable level. It is this process which we are focusing on for this lab.
We will use the period of a simple pendulum as the phenomena we are investigating. Although the physics involved in the theoretical description of this problem is straight forward, the resulting equation for the period cannot be solved in closed form. You may have encountered this already in lecture, but having already seen it is not necessary for this lab. Typically an approximation is made, assuming that the starting angle of the pendulum is small, this simplifies the math so that it can be solved in closed form. The result is the Small Angle Approximation
$T = 2\pi\sqrt{\frac{L}{g}}$,
where $T$ is the period, $L$ is the length of the pendulum, and $g = 9.81m/s^2$ is the acceleration due to gravity.
The small angle approximation gives good results when the angle is small. But how small is small enough depends on how precisely you need to know the period. For larger angles you can resort to more complex numerical methods for determining $T$, but how small does the angle need for the difference to be noticable? That depends on how precisely you need to know the period.
Your task for this lab is to develop and refine an experimental technique for measuring $T$ precisely. The metric you will use for deciding when your precision is good enough is the ability to distinguish between the small angle approximation and a more complete calculation for an angle of 20°.
Assuming a value for the local acceleration due to gravity of $9.81m/s^{2}$, a pendulum length of $1m$, and a starting angle of $20$ degrees, a more complete calculation of the period of a simple pendulum predicts a period of
$T = 2.012s$.
Note that the metric we are using is not inherently interesting as an experiment. We are simply using it for the purpose of illustrating the process of making precision measurements. It is however analogous to the example of the mass of the Higgs boson.
Experimental Setup
Other labs in this sequence will focus on the process of designing your own experimental setup. For this lab we will specify a setup which we know is capable of making measurements of $T$ with sufficient precision. There are four measured quantities in this experiment, the angle $\theta$ of the pendulum, its length $L$, its mass $M$ and the time it takes to complete one period of oscillation $T$. $M$ and $L$ will remain constant throughout the experiment*.
Use the equipment provided in the lab to construct the setup shown in figure 1 below. You want to make sure that the assembly is rigid enough that it will not flex as the pendulum swings, as this would affect the period of your pendulum.
Figure 1
Make your pendulum 1m in length.
Measurements of the Mass (M) and length (L) of your pendulum should be recorded in your group lab notebook. Include units and an estimate of the uncertainty in your measurement.
For this lab we will not investigate the uncertainties in the measurement of the mass and length of the pendulum. It is however important to get in the habit of always including units and an uncertainty for any and all measured values. For this lab you can simply estimate how well you “think” you know the mass and the length. Your estimate for these quantities just needs to be plausible. For example if you use a digital scale to measure the mass, you cannot claim you know that value any better than +/- 0.5 in the last digit of the display. That is plausible. For the length of the pendulum you might decide that you only trust your reading of the meter stick by eye to within 0.5mm. This is also plausible.
Yes, these are judgement based estimates and not precise measurements. There are plenty of ways you could more quantitatively estimate the uncertainties in these measured quantities, you could test the accuracy of the scale using a calibrated mass standard for example. But for this lab we are focusing specifically on the time measurement $T$. Judgement based estimates of uncertainty are however a legitimate part of experimental science research. In fact judgement comes into play a lot in research, both theory and experiment. One of the overarching goals of the first year lab sequence is to give you practice exercising your judgement as a scientist.
Measuring the angle $\theta$
In order to make a meaningful and consistent study of the timing, you need to be able to set the initial angle both precisely and repeatably. Additionally you need to be able to release the mass from the starting angle consistently and without imparting any momentum. It is reasonable to start out with the idea of using a protractor to set this angle. However this turns out to be a challenging method to use. Since we are focusing on the measurement of $T$, we will specify the following procedure for setting the initial angle $\theta$. This procedure is easy to implement and can produce consistent results.
Instead of a protractor, use trigonometry and the measured length of the pendulum to calculate how far the mass should be from its resting position (in the horizontal plane) at the desired angle. As illustrated in figure X, a simple ring stand can then be used to mark this starting position. By lightly taping the ring stand to the floor you can fix it in place well enough to gently hold the mass against it and release by letting go. With a little bit of practice you should be able to consistently and accurately set and release the mass.
Developing Your Timing Method
Now it is time to work on developing a technique for timing the period of a pendulum to a specified degree of precision. There are a lot of ways you could measure the period of a simple pendulum. For the purpose of this exercise you will simply use a stopwatch to time the period of multiple swings of the pendulum, and then divide this total time by the number of swings. This technique has the advantage that although there may be some uncertainty associated with when the stopwatch operator starts and stops the timer, there will be no such uncertainty in the intervening periods thus reducing this effect.
For a stopwatch use either an app on your phone or find an online stopwatch.
Note that there are a lot of ways in which you could measure the period of a simple pendulum. You could use lasers and photodetectors, or record video and use particle tracking software. So why do we have you use a simple and less precise technique?
Because the purpose of this lab is NOT to measure the period of a pendulum to confirm theoretical predictions.
The purpose of this lab is to learn how to think about and investigate a measurement technique. More sophisticated measurement techniques still have potential biases and other factors which limit exactly how well they work. So the same questions about inherent biases need to be investigated. The method we are having you use allows you to practice the process in the context of a technique which is straight forward and and whose limitations are easier to grasp.
Your goal is to develop a timing technique which allows you to distinguish between the small angle approximation and the full calculation of the period of your pendulum for an angle of 20°. In order to accomplish this you will need to:
- Practice your technique until you are able to get consistent results.
- Determine how many swings of the pendulum you will time, how will you release the pendulum and how will you start and stop the timer.
- How do you define consistent? You must come up with a scientifically plausible criteria for establishing consistency.
- Your experimental results should not depend on who does the timing and who releases the pendulum. You must establish this.
- Document your procedure in your lab notebook, you will need to duplicate it in next weeks lab.
Once you have developed a technique which produces consistent results, you need to consider possible systematic biases inherent in your technique. Specifically we want you to investigate the possible effects of air resistance and friction at the pivot point on your technique. Spend a bit of time thinking about how these factors might bias your measured periods, and what you can do to investigate and quantify them. Then discuss your thoughts with your TA before proceeding to the next task.
- Develop a method for assessing and quantifying this potential systematic bias.
- You need to establish not just whether there is a bias in the measurement, you also need to quantify it in a scientifically plausible manner.
- Use this data to inform a decision on how you will minimize the effect of this bias in terms of modifying your technique.
Out Of Lab Assignment - Part 2
For your out of lab assignment you will be expected to discuss the following.
Timing Consistency.
- Describe the timing procedure you choose to use. [EP] [SC]
- Describe how you established the consistency of this procedure and quantify how consistent it was. [EP] [SC]
- Use your data to quantify the degree of consistency you established for this procedure. Show your calculations. [SC]
Measurement Bias.
- Describe the source of the bias you investigated. Why and how would you expect this effect to your measured periods. [SC]
- Describe how you investigated this bias. What procedure did you use to test for and quantify the effect. [EP] [SC]
- Quantify the impact of this bias on your period measurements. Include data and calculations as necessary.[DA]
- Describe how you decided to account for this bias in your procedure. [DC] [SC]
Finally…
- State (quantitatively) how well you were able to distinguish between the predictions of the small angle approximation and the result obtained from a more complete calculation. [DC] [SC]
- Show your data and calculations which support your statement. [DA]
Lab Report Attestation
Copy the following text and paste it in at the end of your individual assignment just above your name. This will serve as your signature for the report.
By submitting this assignment 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.
Important Note
Through out all of your physics lab courses you will repeatedly hear “NEVER evaluate the success of your experiment by comparing your results to known values or theory”! The point of experimental investigation is to test these things, and to discover new phenomena for which there is currently no explanation.
What you are being asked to do for this lab might seem like it violates this edict. It does not!
To be clear, this lab is NOT about doing an experiment to test either the small angle approximation nor the more complete calculation! This lab is specifically focusing on the process of establishing that your experimental technique can attain a required degree of precision. This distinction may be subtle, but it is important.
Why would someone do this? Imagine that your bigger goal is to perform a series of very precise measurements of the acceleration of gravity $g$ at different locations around the earth, and that you have chosen to use a pendulum for determining $g$ by measuring its period at these locations. You may have reason to expect the difference to be one part in 10,000. You would need to establish that you are able to attain this minimum degree of precision with your timing method. For THIS reason you might choose to use these two calculations of the period of the pendulum as a known standard. Similar to how you might test the accuracy of a scale using a known mass.
Note that you would also have to test your method at a location where you independently know the local value of $g$, otherwise you'd have a circular argument.
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!
Individual Assignments
After the lab is over, each student in your group will write up their own individual assignment and submit them to their TA via Canvas. Your individual assignment is due no later than 48 hours after your lab session.
Specifications for the individual assignment are specified in yellow boxes, like this one, where appropriate in the body of this wiki page. For your convenience the two assignments for this lab can be found at these links. Both assignments should be put into a single report which you submit to your TA via canvas.
Out of lab assignment for part 1.
Out of lab assignment for part 2.
Each assignment includes a list of specifications that indicate what to include to satisfy the assignment.
Do not forget to include the attestation statement at the end of your report which indicates that the work you are handing in is your own.
Grading
The description of how your assignments will be graded can be found here.