At the end of the last project, you did some qualitative comparisons to decide whether your paper drop data better supported one model over another. Model building and model testing are important parts of experimental physics, and they are some of the chief ways that we test (and ultimately *improve*) theories.

In this project, we will push this idea further, investigating a model point-by-point in order to make *quantitative* comparisons and investigate how to fold in systematic uncertainties in addition to the statistical ones.

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 one) 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
- If the data are inconsistent the predictions, the model is discarded or revised.
- 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.

You had some exposure to model testing at the end of the paper drop experiment. We will get more practice in this project by looking at the period of a simple 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 where this model comes from in order to test it.)

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

Here is the lab notebook template for this project. Remember to fill this out **as you go along**. Do not wait until you have completed the experiment.

*This section is to be completed individually BEFORE your first meeting with the TA and your lab group.*

We intend for this section to just be a _short _(30 minutes or less) exploration of what it's like to physically build and play with a pendulum. You will not use this pendulum later in the lab, so it doesn't need to be perfect. Make sure you leave yourself plenty of time to use and take data with the simulation below, since that is going to form the bulk of what you will discuss at the group meeting.

Many, many objects in nature undergo periodic motion and you can make a simple pendulum at home from *nearly anything!*

Cut a length of string (or thread, yarn, a belt, sturdy tape, a neck tie, a USB cord, a plant vine,…) and attach to it some object to serve as your mass. Try to find something round (or at least symmetric) and not too big and heavy. (We don't want you to hurt yourself if it flies off!) Maybe try a small rock, a Cheerio, or a a paper clip.

NOTEBOOK: Take a picture of your pendulum and describe how it moves in your notebook. Some things to consider… Do you hold the top of the string in your hand or do you tie it to something to keep that point fixed? How do you attach the mass? Does it swing smoothly or is disjointed? Does it move in a flat plane or does it swing around in 3-dimensions after you release it?

NOTEBOOK:Pull the pendulum to moderate angle – say 10 or 20 degrees (it doesn't have to be exact) – and let it go. Once it starts swinging, does it swing forever, or does it eventually come to a stop? What are some physical reasons that it might slow down?(No right or wrong answers here… just brainstorming!)

NOTEBOOK: Use a stopwatch (either the one built into your phone or something on the internet) to time the period of the pendulums motion. Do you measure the time it takes to go from one extreme to the other and back, or the time it takes to go from the minimum through one cycle and back to the minimum? What can you do to estimate the uncertainty in this period?

NOTEBOOK:Is the period you measure “roughly” the same as the predicted period from the model introduced above, $T = 2\pi\sqrt{\frac{L}{g}}$? (Don't worry about uncertainties. Don't even worry about measuringLorTto great precision. We're looking more for a statement like “the model predicted about 0.45 seconds and I measured about 0.5 seconds.”)

As you build and play with your pendulum, you will probably find that it's actually a bit harder to make precise measurements with it than you expect. Real experiments are usually messier than theories!

We could keep making measurements with the pendulum you built, but it would be difficult to have everyone in the group construct *exactly* the same one so that you can compare results with your fellow students. Instead, we will now switch to using an interactive PhET simulation of a simple pendulum. (PhET originally stood forPhysics Education Technology, but the site has now expanded to include other science disciplines.) This simulation should run on most modern web browsers (including phones), and we also include downloadable backup version of the simulation that you can run offline if needed.

(If the above does not work, or if you need to download the simulation for later use offline, click here.)

When you open the link, you should see a selection screen. Choose **Intro**.

Play around with the options and get the pendulum oscillating.

*The following questions are just for your own understanding. They do not need to be recorded in your lab notebook.*

- How do you get the pendulum moving?
- Qualitatively (that is, without using numbers), what parameters (e.g. mass, length, angle, friction strength, acceleration due to gravity) seem to affect how fast the pendulum swings?
- How could you measure the time it takes for the pendulum to complete one period?
- How can you estimate the uncertainty of your measurement?

A word of warning about simulations

Simulations can play an important role in real physics research. Sometimes, they are used as a visualization or calculation aid. (For example, they can help us “see” something that is difficult to grasp or understand only through the math or can help us solve an equation that has no analytical closed form.) Other times, they can be used to generate new, *real* insights. (For example, by showing that an unexpected or complex phenomenon *emerges* due to a set of simple fundamental interactions. This method is especially common in astrophysics – where the simple law of gravity gives rise to complex galaxy and solar system motion – and in the study of liquids and solids – where the forces between individual particles lead to crystallization, electrical resistance, or other material properties.)

However, because a simulation is *programmed* by a human, we have to be careful and think about the assumptions that went into its creation. What are the underlying physics equations? How are these used to generate results? What is programmed into the simulation to be *exact*? What is instead *calculated* from other equations?

In the case of the pendulum simulation we use above, we don't have access to the underlying code, so we can't say for sure what's going on under the hood. Does this mean our above simulation is * invalid*? No… it just means we have to be careful what conclusions we draw from it.

Testing angles up to 10°

For this section and all future sections, adhere to the following settings and instructions. These settings have been chosen to mimic the typical pendulums that you would build if you were on campus for this lab.

- The pendulum mass should be set to 0.10 kg.
- The pendulum length should be set to 0.50 m.
- The friction should be set to the 20% mark (the second tick mark from the bottom of the scale).
**The stopwatch tool should be used for all timing measurements.**An important part of the experiment is allowing for natural variations in human reaction time and determining how to account for them.

You are welcome to adjust the speed while exploring how the simulationworks, but while making measurements, it should be run at **Normal** speed, not Slow; do not use the **Pause** or **Advance** buttons. We want this simulation to be realistic, and the challenges inherent in making a real-time measurement is part of that.

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 simulation, 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 simulation; instead it is up to you to figure out how to get the most out of the equipment in the time available to you.

**NOTEBOOK:**
<

- Begin by calculating the predicted value for the period given your pendulum length.
- Measure the period for $\theta = 5^{\circ}$ and for $\theta = 10^{\circ}$. How will you estimate the uncertainty in each period?
- Record your data, observations, and thoughts in your lab notebook. Include information about your procedure (including pictures).
- For each of your final period values, compare them to your predicted value. Do they agree, do they disagree, or is it inconclusive?

At the group meeting, you will show off your homemade pendulums and discuss the values you measured in the simulation. In particular, be prepared to talk about how you estimated the uncertainty in your periods and what sort of biases or problems that different procedures may introduce.

After comparing technique and values, your TA will help the group develop a plan for extending this experiment to larger angles.

NOTEBOOK: Take notes during your meeting.

*This section is to be completed AFTER your first meeting with the TA and your lab group.*

Based on the discussion with the your group, extend your experiment to angles greater than 10 degrees (and up to about 40 degrees).

NOTEBOOK: Describe you measurement method(s) and record your data and uncertainties.

Before the meeting, share your results with the rest of the group and (if everyone shares in time) try to plot the data yourself to see how it looks. At the meeting, the TA will collect your data, plot it, and lead a discussion on the results. Be prepared to discuss your methods and share data, and be ready to participate in a discussion about how your group's data compares to the simple pendulum model at each angle.

After your second meeting, you will again need to write up your summary and your conclusions. Include any data tables, plots, etc. from the experiment or discussions as necessary in order to show how your data support your conclusions.

This part doesn't need to be long; one or two pages should be sufficient. What is important, however, is that your writing should be complete and meaningful. Address both the qualitative and quantitative aspects of the experiment, and make sure you cover all the “take-away” topics in enough depth. 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 exact 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, under any circumstances 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.

**REMEMBER**: Since we are at the end of the quarter, your last report is due **only 24 hours** after your lab meeting so that the report doesn't conflict with your final exam.

Below are some resources for TAs to use. If you're a student… you can look, but you don't need to go through or understand anything here (unless your TA asks you to explore any of these things during your meeting.)

Period of a Pendulum -- Python script

While this Python notebook is intended for use by the TAs, students may also use it if they wish. Teaching Python is not a part of this course, though, so your TA may not be able to help you if you have trouble using or editing this script.

*See the Wikipedia page for more details on the derivation of these forms: https://en.wikipedia.org/wiki/Pendulum_(mathematics).*

Now we can use a more sophisticated model for the predicted period. This model *depends* on angle, and therefore needs to be calculated at each discrete angle in order to make comparisons.

We won't derive the formula here, but the full solution comes from solving a differential equation that involves all the forces acting on the pendulum mass. The solution (exactly) is

$T = 2\pi\sqrt{\frac{L}{g}} \bigg(\frac{K(\sin(\theta/2))}{\pi/2}\bigg),$ |

where

$K(x) = \int_0^{\pi/2} \frac{\textrm{d}u}{\sqrt{1-x^2\sin^2 u}}$ |

is a so-called *elliptical integral*. This integral has no closed-form solution, so the best we can do is to rewrite it in terms of Legendre Polynomials and perform a Taylor expansion around zero angle. This gives the following approximation:

$T = 2\pi\sqrt{\frac{L}{g}}\bigg[1 + \frac{1}{4}\sin^2(\theta/2) + \frac{9}{64}\sin^4(\theta/2)+\ldots\bigg].$ |

This approximation has an infinite number of terms. We can see whether adding the additional two terms shown here improves the agreement between the experimental data and the model.

**Significant figures**

If we are talking about exact numbers, then you know that 1.4 = 1.40 = 1.400 (and so on) for any number of zeros after the four. However, in an experimental physics setting, most numbers are measurements made about the physical world, and therefore the number of digits you present has a meaning; you write down *only* those digits that are significant. When the value is an experimental measurement, it must have an uncertainty associated with it, and then determining which digits are significant is an easy task: we should only present one or two digits in the uncertainty, and the value of the measurement is significant only up to that precision. A few examples are shown in Table 1.

$7.40 \pm 0.3$ | wrong |

$7.4 \pm 0.32$ | wrong |

$740.345 \pm 32.189$ | wrong |

$7.4 \pm 0.3$ | right |

$6.0 \pm 0.05$ | wrong |

$6.00 \pm 0.05$ | right |

Table 1: How many digits to keep after uncertainty starts |

Note that these rules should be obeyed *whenever* you present a value – be it in the text, in a table, or on an annotation on a plot. This doesn’t mean, however, that all other digits should be thrown away. For intermediate values entering calculations, we should always keep as many digits as possible, even if we don’t show them. If all your analysis is made in the same platform (for example, a python script) this should happen automatically.

**Rules for determining significant figures**

Two rules for using significant figures are given in the *An Introduction to Error Analysis* by John R. Taylor on page 15:

**Rule for Stating Uncertainties:**“Experimental uncertainties should almost always be rounded to one significant figure.”

*Exception*: If the leading digit in the uncertainty is a 1 or a 2, then it may be better to give two digits in the uncertainty, since, for example, rounding, from 1.4 to 1 would be a large proportional reduction of the uncertainty.

**Rule for Stating Answers:**“The last significant figure in any answer should usually be of the same order of magnitude (in the same decimal position) as the uncertainty.”

For example, the expression $1.234 \pm 0.06$ is WRONG! If the uncertainty is 0.06, then the 4 in the value is meaningless. Contrast this with the expressions $1.234 \pm 0.006$ and $1.23 \pm 0.06$ which both have the correct number of significant digits.
**Units**

Every measured quantity has a unit, and therefore whenever you present a value, you need to include the unit.

- Do not say a length is 0.2, but instead that it is 0.2 cm.
- Do not give a period as 1.25, but instead 1.25 seconds.