UChicago Instructional Physics Laboratories

User Tools

Site Tools

You are here: start » phylabs » lab_courses » phys-107-wiki-home » introduction-to-experimental-phys-107

Table of Contents

Introduction to Experimental Physics (PHYS 107)

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.

This first lab will be an introduction to experimental physics in general and to the specific things you will be expected to do during each lab period in this class.

The structure of the lab

One of the main purposes of the physics labs in this course is to learn how scientists study the world around them. With a focus on the scientific process of asking questions and doing an experiment, students can learn how to think like a scientist.

Each lab has a similar structure.

  • Students will work in groups of 3.
    • Designate one person to be the official record-keeper. (This role will change week-to-week.)
    • All students are expected to contribute ideas to the notebook, but the record-keeper will serve as something like the “group leader” (keeping records, making sure that the group has enough data, and asking questions to the group when things aren't clear).
    • All students are expected to work with the equipment and get experience handling apparatus and taking data.
  • Each group will keep an electronic lab notebook.
    • The group should work on the notebook throughout the period, and the TA may ask to look at as they walk around checking on groups.
    • It is not formal, but it should be neat and organized.
    • It is a record of work done, and it is a place to make sketches, do quick calculations, or to write thoughts or questions.
    • Do not delete or erase. Instead, carefully indicate sections that you want to change (e.g. by making them a different color and adding a note); the notebook is a record of everything, including mistakes.)
  • The group notebook is also the report for the group which will be graded. In addition to text, feel free to include photos or drawings in the report.
    • Did you build something? Make a sketch.
    • Did you observe something? Record what you see.
    • Did you ask a question? Write it down.
    • Did you collect data? Make a table.
    • Did you calculate quantities? Show your work.
    • Did you make a plot? Put a copy in the report.
  • Groups also have access to a whiteboard to share thoughts and ideas with each other and with other groups.

Lab notebook template

Click the link below to create a copy of the template you will use for this first lab. You will be prompted to log into your UChicago account (if you aren't logged in already) and it will ask you if you to create a copy of the template in your personal Google Drive.

Lab Notebook Template

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.

It is OK to make mistakes in the notebook! It isn't meant to be perfect.

Activity 0: Getting warmed up

Your TA will ask you a question. Use the whiteboard to work out ideas or to present your thoughts. After a few minutes, you will be asked to share your thoughts with the class.

NOTEBOOK: Take a photo of your whiteboard and include it in your notebook.

Activity 1: How long does it take for a piece of paper to hit the floor?

Taking data

For our first activity, we're going to measure how long it takes for a piece of paper to fall through a distance of 1 meter.

  • 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 personal notebook and in the group report.
NOTEBOOK: Record your data in your lab notebook as you take it!

Now that you've got many numbers, how will you answer the question, “How long did it take for the paper to hit the floor?” Discuss this within your group and be prepared to share you thoughts with neighboring groups.

NOTEBOOK: Record notes from your discussion in your group report, too!

Analyzing the data

In order to do data processing and analysis in this course, we will use Google Colaboratory “notebooks” (sometimes just shortened to Google Colab), which run on the Python programming language. These notebooks are collections of text (to provide you with instructions or information) interspersed with code (to do calculations or to make plots). You do not need to know how to program in Python to use these notebooks; instead, we have made these notebooks beginner friendly so that they only require you to enter your data and execute prewritten snippets of code. Consider these notebooks to be tools (like a fancy calculator).

Below is the notebook we will use for this project.

Google Colab link

Run Part 1 of the notebook using the default data already included there in order to get used to how it works. When you are comfortable, edit the first cell with your values of $t$ and complete the calculations for your data.

NOTEBOOK: Think about the values calculated and plotted in the Google Colab notebook. What values or plots are relevant to put in the group report?

Class discussion

Your TA will prompt a class discussion. Don't expand the following drop-down sections until instructed to do so.

NOTEBOOK: When the discussion is finished, your TA will give you a few minutes to write down your thoughts. Up until now, your notebook has been used for recording data and observations… now you want to try to draw “conclusions”. Communicate what you think the take-away messages are – both for your future self and for any other scientist who might read your work.
Determining uncertainty

Consider the length measurements you made in today's experiment. When you make such a measurement, 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, its 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.)

QUESTION: Now think about the measurements you made today. Which is the appropriate type of uncertainty for a single measurement of paper displacement? Which is the appropriate type of uncertainty for the average paper displacement?

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
Comparing quantities

Often in experimental physics, we want to compare our number to someone else's.

  • Maybe we have a model which makes a prediction for a value. How does our value compare to the prediction?
  • Maybe two groups are trying to measure the same thing. Do the two groups agree?
  • Maybe we actually expect a result to differ from another known result. How different is different enough? 

In order to be quantitative about these sorts of questions, we need to establish some criteria. For this course, we will use a measure called $t^{\prime}$ . (This is related to – but not equal to – something called the Student's t-test. If that doesn't mean anything to you, don't worry about it.)

Suppose we have two quantities with uncertainties which we want to compare: $A \pm \delta A$  and $B \pm \delta B$ . I this case, $t'$  is defined as

$t' = \frac{A - B}{\sqrt{(\delta A)^2 + (\delta B)^2}}.$

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

$t' = \frac{A - B}{\delta A}.$

Agreement

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:  $ |t'| \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 something to be true… we can only say that the current data supports agreement.

Disagreement

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: $ |t'| \geq 3$.

Inconclusive

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:  $1 < |t'| < 3$.
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.

Activity 2: How quickly does water travel along paper?

In the first activity, we quantified how certain we were about the value of one measured quantity by repeating the measurement. But what do we do when the thing we are measuring isn't as clear cut? Or when repeating many times isn't an option?

Taking data

In this activity, we're going to measure how quickly a piece of paper absorbs water. This sort of information might be useful if you wanted to know how effective a paper towel is, how well a treatment protects books from water damage, or how well a new building material would perform in a flood. While there are many possible ways to do this experiment, for this activity part of the goal is for everyone to have the same sort of data to work with. To this end, the plan is to place one end of a paper towel under a damp sponge and time how long it takes the water to travel distances of 1, 2, 3, and 4 cm.

An example of water uptake in a paper towel.

Two students doing this experiment have come up with two different models for what is going on.

  • Alicia believes that this scenario is an example of flow: the sponge continually pushes new water into the paper, which pushes the old water further along. Since the flow rate is constant, Alicia predicts that the distance the water travels will increase linearly with time: $x = Rt$, where $R$ is the flow rate.
  • Riley believes that this scenario is an example of diffusion: the water molecules aren't being pushed, but instead move about randomly and bump into things as they flow outward. Since the water isn't traveling in a straight line, it will take longer to move the same distance, so Riley predicts that the distance the water travels will increase as the square-root of time: $x = D\sqrt{t}$, where $D$ is called the diffusion coefficient.

You will now set the experiment up and measure for yourself. From the figure above, you should notice that the edge of the water isn't uniform. Your group should plan ahead of time on how you'll measure progress.

  • How will you measure time?
  • What will be your criteria for saying the water has “reached” a line?
  • How will you estimate the uncertainty in your time?

When your TA gives you the go-ahead, place one end of your sponge over the edge of your paper and start timing. Record the times for each distance as you take them, and include uncertainties in the time to reach each mark.

Spend a moment talking within your group about what you saw. Some things to consider are as follows:

  • How precise were your time measurements?
  • Do you think you would get the same times if you were to repeat the experiment?
  • At first glance, does it seem like the water was absorbed at a constant rate (Alicia's model) or did the water slow down the further away from the sponge it got (Riley's model)?

Your TA will lead a short class discussion; be prepared to talk about what you saw in your data.

Analyzing the data

Return to the Google Colab notebook and move on to Part 2.

Plotting the data

In order to investigate whether the data better support Alicia's model or Riley's model, we can plot it.

Since you likely found a much greater uncertainty in time than in distance, we will invert the two models so that time is on the y-axis:

Alicia $x = Rt \Rightarrow t = (1/R)x$ or $ t = Ax$ (where $A = 1/R$).
Riley $x = D\sqrt{t} \Rightarrow t = (1/D^2)x^2$ or $t = Bx^2$ (where $B = 1/D^2$).

Input your data to create a plot and adjust the values of $A$ and $B$ to find the best “fit” for the data for each model.

NOTEBOOK: Which functional form fits the data better? Do you believe the data support Alicia's model or Riley's (or neither)? How could we make our argument more quantitative?

Repeating the experiment

If there is time, try repeating the experiment.

  • What would you change this time about how you make the measurements or set up the experiment?
  • Are your new data consistent with the previous data? (Why or why not?)
  • Do you come to the same conclusions?
NOTEBOOK: Remember to include conclusions for this activity. What take-away message do you want a reader to know about the water uptake experiment?

Conclusions

At the end of the lab, you will need to record your final conclusions (about 1 or 2 paragraphs) in your lab report summing up the important results and take-away points from your experiment. Remember that you should only draw conclusions which are supported by the data, so be ready to back up any statements you make!

When you're finished, save your file as a PDF and submit it to the appropriate Canvas assignment. (Only one student needs to submit the notebook, but make sure everyone's name is on it!) If you make a mistake, you can re-submit, but work done after the end of the lab period will not be accepted.

Remember to log out of all your accounts after you submit!