The Period of a Pendulum (Part 1)

Welcome to experimental physics! Our goal with this lab sequence is for you to gain practice doing experimental science. This is separate from the lecture portion of the course, where the goal is to understand the inter-related and systematic nature of physics, and how to use physical models to mathematically predict the behavior of systems. As such there will frequently be times when the content in lab is noticeably different from the content in lectures; the difference in content does not mean that you've missed something in class.

In the following project, you will perform some simple experiments – dropping slips of paper from fixed heights and building pendulums – but we will use this data to develop ideas about experiment design, repeatability, systematic biases, statistical distributions, and model-testing.

Introduction

Experimental physics and model testing

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 already know.

Many factors go into designing, building and performing an experiment. Questions arise on which the experimenter must make decisions regarding how to proceed… questions such as the following:

How precisely must a measurement be made? (I.e. When is an experiment finished?)
How many data points should I take? (The answer is usually not “10” nor is it “ask the instructor”.)
Are my results good enough? (Think about what the goal was when you started.)

Figuring out on your own how to answer the above questions in the course of doing an experiment and the subsequent interpretation of the final result of the experiment are what we will refer to as the process of “Thinking through an experiment”. Learning how to think through an experiment is one of the main goals of the lab component of this course.

One way in which experimental physics can be done is through model testing:

A person makes observations and collects some data.
That person (or a different person) builds a mathematical model based on some physics which attempts to explain what's going on.
The model is used to make predictions about a different scenario.
The predictions of the model are tested in the new scenario to see if the predictions are correct.
- If the data are consistent with the predictions, we say the model is supported*. New predictions are made and more testing can be done.
- If the data are inconsistent with the predictions, the model is discarded or revised (or the experiment is checked for biases or errors).
- If the data is ambiguous – neither convincingly in agreement or convincingly in disagreement – then either more data is needed or a different experiment should be done.

*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.

We will get practice in model testing today by looking at the period of a simple pendulum.

Pendulum

The model we will test today is 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 $T = 2\pi\sqrt{\frac{L}{g}}$. (Do not worry if you have not yet encountered pendulums in lecture; you do not need to know any theory in order to test the model.)

Our model predicts that the period should not depend on the mass of the pendulum $m$ nor the angle $\theta$ from which it is released. We will specifically test whether the period depends on the release angle.

Part 1: Understanding statistics and building a pendulum

Lab notebook

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.

Lab Notebook Template

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.)

We will try to call out places where you need to write in your lab notebook by using the NOTEBOOK tag, like in the following:

NOTEBOOK: Fill out the top portion of the first page.

You should write down additional things in your notebook whenever you think it is useful – whether or not we specifically ask for it. These can include notes to yourself (e.g. to record/remember what you've done) or to your reader (e.g. to communicate an outcome or discuss a result). As the course goes on, you will begin to develop a feeling for what needs to go in your notebook and we'll use fewer reminders.

REMEMBER: You are going to be graded on the completeness of your lab notebook, not how pretty or perfect or correct it is. It should be neat (if the TA can't follow your work, they can't tell if it's complete), but you don't need to labor over it to make it perfect. It is a working document that will help you as you go along and as you write your conclusions for the post-lab assignment.

Dropping paper: a quick statistics lesson

You may be familiar with estimating uncertainties on measured quantities based on the resolution of your measurement device. (For example, using a ruler with tick marks at each millimeter, and then deciding that your uncertainty in measurement is plus or minus 1 mm or 0.5 mm.) This is a reasonable technique when you able to measure something only once, or when you do not expect fluctuations in a measurement. For this experiment, though, 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 (more exciting) pendulums.

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.

Determining uncertainty

Consider when you make a length measurement with a ruler. It is unlikely that the value you find is exactly on a tick mark of your ruler. And even it looks to your eye like it's on that tick mark, it may not be exactly that length. (If you zoomed in with a microscope would it still be perfectly on the line? Is the edge of the card perfectly flat? Is your ruler so perfect that the tick marks are exactly in the right place? The answer to all these questions is “no”.) Therefore, it's not sufficient to simply report a value for a length, you have to also provide your estimate of the uncertainty in that measurement.

A measurement uncertainty is your best estimate as to how close your “measured” value is to the “true” value. It isn't a random guess or an arbitrary number, but it is judgement you have to make about how much you trust your value. If we say our best estimate for the measured value is $x$ and our estimate for the uncertainty is $\delta x$, then we'd report our final value as $x \pm \delta x$. This does not mean that the true value is absolutely within the range from $x - \delta x$ to $x + \delta x$, only that we think it is “likely” in that range. (We can get more specific on what we mean by “likely” and define, for example, the probability of being in a certain range… but let's save that for a later experiment.)

There are typically two ways you might estimate a measurement uncertainty – from looking at the “spread” of values after making repeated measurements, or from estimating the limit of your precision due to the resolution of your measurement device. Let's look at each in turn.

Repeating measurements: Suppose that the quantity you are measuring is difficult to pin down. Maybe it fluctuates with time (e.g. a length that is always varying), you have only one chance to measure it (e.g. the time it takes for a ball to drop), or it is inherently fuzzy (e.g. the diameter of a puffy cotton ball). In this case, if you make multiple measurements, each measurement you make might be slightly different than the last. The best estimate for the uncertainty in a case like this is therefore gotten by looking at the “spread” of values you obtain from repeated measurements.

If you are able only to make a couple measurements, a simple rule is to estimate the uncertainty as (“maximum value” - “minimum value”)/2. Put into words, you look at the range created by the most extreme values, and chose your uncertainty as half of that.
If you are able to dozens or hundreds of measurements, or if you know something about the statistical distribution of these measurements, then we can use more rigorous methods (like the standard deviation and standard error defined above.)

Resolution: The resolution of a measurement device is usually the smallest unit that the device can measure. On a ruler or analog dial, it is the distance between the two smallest tick marks. On a digital device, it is the smallest displayed digit. If you keep finding the same reading on the device every time you make the measurement (assuming you make the measurement the same way), then the “repeating measurement” technique above doesn't help you much. In this case, you can estimate your uncertainty based on how well you can read your scale on a single measurement. There is no fixed rule here; sometimes the uncertainty is equal to the smallest tick mark, whereas sometimes you can say it is closer to one line than the other, so the uncertainty is half a tick mark (or whatever your case may be.)

In addition to measurement uncertainties described above, we also have to be on the lookout for what are called systematic uncertainties (or sometimes just systematic biases). A systematic uncertainty is an uncertainty in your measurement that is due to some unaccounted for bias in your measurement or your assumptions. It could be that the measurement device you are using is in error (e.g. a thermometer that isn't calibrated correctly or a stopwatch that runs too fast or too slow), that your measurement technique is flawed (e.g. a start or stop signal to a stopwatch always arrives late), or that there is an incorrect assumption about what you are measuring (e.g. measuring the speed of an object that you believe to be at constant velocity, but which is actually accelerating).

Importantly, systematic uncertainties are different from “mistakes” (e.g. writing down a 4 when you meant to write down a 3). Systematic uncertainties are almost always present, but they may be difficult to identify or eliminate.

QUESTION: Think about your measurements today. What (if anything) could be biasing your values and how might you come up with a way to test for that? (You do not need to actually perform any additional tests. We're just looking for ideas.)

One final comment. We never determine an uncertainty by comparing our value to a “known” value. For example, if you measure the acceleration due to gravity to be $g_{\textrm{exp}} = 9.7~\textrm{m/s}^2$, the uncertainty on that value is NOT $\delta g = g-g_{\textrm{exp}} = (9.8 - 9.7)~\textrm{m/s}^2 = 0.1~\textrm{m/s}^2$. Some students may have learned this sort of calculation in high school as the “error” in a measurement, but that is the wrong way to think about uncertainties, and we will develop better ways of comparing two values below.

Significant figures

Once you have a value and its uncertainty, how do you report this final result?

This is where the question of significant figures appears. If you do some arithmetic and your calculator spits out an answer of 3.45679213, do all those digits matter? Probably not. In order to know how many digits do matter, you should look at the size of your uncertainty.

The rules for determining the number of digits to report can be summed up as follows:

Compute your uncertainty. Keep only one digit in the uncertainty, unless the leading digit is a 1 or a 2.
- Example: If your uncertainty is 0.543 units, then report the uncertainty as 0.5 units.
- Example: If your uncertainty is 0.0237 units, then report the uncertainty as 0.024 units.
Look at your value, and truncate your value to the same digit place as the final digit in your uncertainty.
- Example: If your value is 123.72 units and your uncertainty is 0.5 units, then you should truncate your value to 123.7 units.
- Example: If your value is 0.53325 units and your uncertainty is 0.024 units, then you should truncate your value to 0.533 units.
Put your value and uncertainty together
- Example: 123.7 ± 0.5 units
- Example: 0.533 ± 0.024 units

Uncertainties in calculated quantities (“propagation of uncertainties”)

Sometimes you directly measure a quantity and directly estimate the uncertainty. But what happens when you compute a new quantity using these measured values? That new quantity must have some uncertainty, and it must depend on the uncertainty of the individual measured components. This process of determining the uncertainty in a calculated quantity is called propagation of uncertainties.

We'll introduce the general formula for uncertainties in a minute, but first let's look at the most common cases.

Products and quotients

If our quantity is computed by multiplying or dividing variables, then the final uncertainty is given by the sum of the fractional uncertainties (i.e. the uncertainty divided by the value, $\delta A/A$) of each variable added up in quadrature (which is just a fancy way of saying square each term, add them, and take the square root).

So, if we have a quantity $f(x,y,z) = \dfrac{xy}{z}$, then the uncertainty in $f$ is $\delta f$, with $\delta f$ defined by,

$\dfrac{\delta f}{f} =\sqrt{\left( \dfrac{\delta x}{x}\right)^2 + \left( \dfrac{\delta y}{y}\right)^2 + \left( \dfrac{\delta z}{z}\right)^2}$.

Notice that this equation is the same whether a given variable is multiplied or divided.

Sums and differences

On the other hand, if our quantity is computed by adding or subtracting variables, then the uncertainty is given by the sum of absolute uncertainties, again added in quadrature.

So, if $f(x,y,z) = x - y + \dots +z$, then the uncertainty in $f$ is

$\delta f = \sqrt{\vphantom{|}(\delta x)^2 + (\delta y)^2 + \dots + (\delta z)^2}$.

Notice that the same formula applies whether you add or subtract; the uncertainty in the final sum or difference is always bigger than the uncertainty in any individual term.

Exponents

When our calculated quantity includes an exponent (power), $f(x) = x^n$, then the uncertainty in $f$ is

$\dfrac{\delta f}{f} = n \dfrac{\delta x}{x}$.

General formula

Those formulas above handle many (or maybe even most) situations, but if you are calculating a more complicated equation, we must turn to the general formula for calculating uncertainties.

For a generic function $f(x_1,x_2,...x_n)$ with measured values and uncertainties $x_1 \pm \delta x_1$, $x_2 \pm \delta x_2$ ,… $x_n \pm \delta x_n$, the final uncertainty is

$\delta f = \sqrt{\left(\dfrac{\partial f}{\partial x_1}\delta x_1 \right)^2 + \left(\dfrac{\partial f}{\partial x_2}\delta x_2 \right)^2 + \dots + \left(\dfrac{\partial f}{\partial x_n}\delta x_n \right)^2 }$.

We see that each term is the slope of the function with respect to a given variable multiplied by the uncertainty in that variable, and that these terms are squared, summed, and square-rooted (i.e., added in quadrature). You can check for yourself that this formula reproduces the formulas above when we have products/quotients or sums/differences.

We will return to the propagation of uncertainties again in later labs and go into more depth on what these equations mean and where they come from. For now, though, we can just use these equations as tools. If you are interested in where this formula comes from, you can read some more details here.

Statistics definitions

Let's define some useful statistical measures. For the following, we will assume that you measured a quantity $N$ times and got a list of values $x=[x_1, x_2, \ldots, x_N]$.

The mean (or average) of a list of numbers is given by $\mu = x_{\textrm{avg}} = \dfrac{1}{N}\sum_i^N x_i$.
The standard deviation of a list of number is given by $\sigma = \sqrt{\dfrac{\sum_i^N (x_i-\mu)^2}{N-1}}$.
The standard deviation of the mean (or standard error) is given by $\mu_{\sigma}=\sigma/\sqrt{N}$.

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.

Are your results in agreement with the others? In disagreement? Inconclusive? How can we tell?

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!

We will use this same program again later in the project to help us plot and visualize data.

Google Colaboratory Notebook

Criteria for establishing agreement – $t'$

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 values and your conclusions in the group notebook! Communicate what you think the take-away messages are – both for your future self and for any other scientist who might read your work.

The period of a pendulum

You have been provided a variety of rods, clamps and other apparatus with which you can make a pendulum. Your task is to test whether or not the period of a pendulum is independent of the initial angle as predicted by the model: $T = 2\pi\sqrt{\frac{L}{g}}$. Using the apparatus provided it is possible to make very precise measurements if your experimental technique is good. Since we want you to learn how to think your way through these experiments, we are not going to specify how to best use the apparatus at your disposal. Instead it is up to you to figure out how to get the most out of the equipment in the time available to you.

Testing $\theta \leq 10^{\circ}$

Begin by constructing a pendulum and determine the predicted value for the period given your pendulum length. Measure the period for angles up to and including $\theta = 10^{\circ}$.

Use the calculation and plotting resources provided in the Google Colab notebook above to look at your data.

Remember to update your group notebook as you go, including information about (and pictures of) your pendulum and technique, preliminary data or plots, and your first thoughts.

Your TA will be walking around the room talking with groups. Be prepared to share your techniques and results when they visit, and be ready to show that all group members are participating.

Tips

Below are some tips to get you started:

You are going to be making multiple measurements of periods for different angles. How are you going to ensure that each trial for a given angle starts off the same? How will you ensure that you begin at the same angle each time? How will you make sure that the mass is released the same way each time? You may have to spend a little bit of time trying different techniques to determine an appropriate procedure. Don't forget to document your work in your notebook, even the attempts that you decide against using.
How many and at what angles will you need to take data? The only way to know the answer to this sort of question is to look at your data as you take it. Within your group, designate at least one person to record the data, compute averages, and compare with predictions from the model in real time. Your data will tell you what you need to be doing in order to accomplish the stated goal of the experiment. For example, you might try taking some quick data at a relatively small angle and at a relatively large angle and look at the results to gain insight into what range of angles are needed. Using your data like this to inform your experimental procedure is an acquired skill; no one starts off especially just good at it. We cannot tell you how to do it, the only way to develop the skill is to do it yourself.
For a given angle, how many periods do you need to measure and average? Again, you have to look at your data to determine how much is enough. Ultimately your goal is to make the most precise measurement possible in the time available to you.

Note that there is no “right” answer to questions like “how much data do I take?”. The answer is not necessarily “I will take 10 measurements at 10 angles.” Ten is a nice round number, but it's not special; you need to balance the time it takes to collect data with the the range of data you need to have by the end of the period. Even professional research scientists do not have infinite time and resources to conduct a “perfect” experiment – whatever “perfect” may mean.

You (as well as professional scientists) are limited by the equipment and resources available. You have a finite amount of time to produce a result. Experimental physics is an iterative process of taking some data, evaluating that data to see what needs to be improved, and making adjustments to your apparatus and technique based on these intermediate results. Keep in mind what it is that you are trying to accomplish at the end of the day (or in this case, the end of the lab period).

Ask your instructor questions and bounce ideas off of them or other students in the lab; everyone learns from interacting with one another. The instructors are there to help guide you through the experiment, but they will not tell you exactly what angles to measure or how many measurements to take. They will not tell you if your data are “good enough”. These are things you have to decide for yourself.

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

We will return to the pendulum you have built again in the next lab, but let's take a moment to reflect on what we've done so far.

Answer the questions/prompts below in a new document (not your lab notebook) and submit that as a PDF to the appropriate assignment on Canvas when you are done. You should write the answers to these questions by yourself (not in a group), though you are welcome to talk to your group mates to ask questions or to discuss.

These post-lab assignments are not meant to be long assignments. Typically, your responses will be one or two pages of text total, and we expect you to spend about 1 hour or so working on it. Ask your TA for clarification if you don't understand what a question is asking or what a prompt is suggesting.

Conclusions

Let's begin by getting some practice drawing conclusions.

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?
Can you think of other ways to extend or improve the experiment?

In about one or two 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 fraudulent.

Questions

Next, we want you 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.)

Consider the following questions:

The objectives of the lab course are intended to benefit students regardless of their future careers.
- How do these objectives differ from your expectations of a typical lab course?
- Choose at least two of the objectives from the five above. How do these objectives apply to your current career interests?
The overarching principle of these labs is to understand how we know, not what we know. Reflecting back on today's experiment, identify at least one thing that your group did that that helped you understand how instead of what.
In the next lab, we will return to the pendulum to take more measurements. What is one improvement you could make – to your apparatus, to your measurement technique, to your data collection strategy, etc. – that will help you reduce your statistical uncertainties? What is one improvement you could make to help you study (or eliminate) systematic uncertainties?

REMINDER: Your post-lab assignment is due 24 hours after your lab concludes. Submit a single PDF on Canvas.