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 exercise will be an introduction to experimental physics in general and to the specific things you will be expected to do during each lab period this quarter. It is to be completed on your own by the end of Week 2 of the quarter. Submit your lab notebook (described below) to Canvas by Friday, October 11 at 5:30 pm CDT. You will not meet in-person for this lab; your first in-lab session will be during Week 3.

This on-your-own lab has the following parts:

• a project you can do at home that will introduce you to some ideas related to measurement and uncertainty; and
• an opportunity to compare your results to the results of some hypothetical classmates in order to draw conclusions.

Unlike future labs this quarter, – which will have both an in-lab notebook (due at the end of the period) and an out-of-lab report (due 24 hours after your next lab session) – for this lab you will only need to complete the lab notebook.

If you haven't done so yet, read over the lab course homepage (PHYS 121/131 here or PHYS 141 here) to make sure you understand how the labs run overall.

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.

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.

Much of experimental physics revolves around making measurements. In an in-person lab, we can ensure that everyone uses the same measurement tools, but when you are working on you own, that may not be the case. Think about what you have at your disposal in your house or dorm room. Maybe you have a ruler or tape measure (that measures in inches? centimeters? meters? yards?) and maybe you have a stopwatch app on your phone (with sensitivity to 1 second? 0.1 seconds? 0.001 seconds?) But do you have a scale for measuring mass? A hygrometer for measuring relative humidity? A spectrometer for measuring the wavelength of light? A particle detector for measuring the mass of the Higgs boson?

Sometimes the first step of any project is considering what tools you have available and then determining the best experiment you can perform with those tools.

For this exercise, we will create our own length measurement device (a “ruler”) and use it to measure the three dimensions of a common item: your University of Chicago ID card (or, if you don't have that, an equivalent plastic card like a phone card, membership card, credit card, or state/national ID card).

You will make your own ruler that measures in a made-up unit which we will call the charta, and which we will abbreviate with the letter $c$. (“Charta” is the Latin word for “paper”.) We will define one charta (1 $c$) as the length of the long side of a “standard” piece printer of paper. (“Standard” may mean something different depending on what country you are in or what you have access to. That's okay.)

The simplest ruler you could make is a single piece of paper marked “0” on one end, and “1” on the other. Think about the small card you are trying to measure the dimensions of, though… is a very coarse ruler like that sufficient for estimating it's dimensions? (Probably not.) A ruler that would be more useful would be one that has fractions of a charta marked on it. But how many fractions and which ones?

NOTEBOOK: Using a standard piece of paper, make a ruler with however many tick marks you think are appropriate. Include a picture of your ruler, and describe how you determined where to place the tick marks. Try to come up with a more rigorous method than “hmmm… that looks about right”.

Now take your card and measure all three length dimensions as best you can with your ruler. (Yes, we want you to measure the thickness, too!)

When you make a measurement, it is unlikely that the value you find is exactly on a tick mark. And even if 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 do dozens or hundreds of measurements, or if you know something about the statistical distribution of these measurements, then we can use more rigorous methods (and we'll teach you those in time.)

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.)

Now think about the measurements of the three dimensions of your card. Which of the above two methods is more appropriate here? Will your uncertainty on all three values be the same? (Probably not.) How will you estimate uncertainty on that smallest dimension, the thickness?

NOTEBOOK: Measure all three length dimensions of your card and estimate the measurement uncertainties on each. Describe how you determined your uncertainties. Is one of the two methods above more appropriate here?

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.

NOTEBOOK: Think about your three measurements. 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.

You now have three length dimensions for your card and three uncertainties on those lengths. How do we report those values?

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

NOTEBOOK: Look back at your dimensions and uncertainties… do you need to adjust how you presented your results? (Go ahead and edit your previous answers. This doesn't need to be a separate entry.)

Each of your lengths was directly measured and the uncertainties were directly estimated. 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 two 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,

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

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.

General formula

Those two 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

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.

Let's see how this works with your own measurements.

If we call the two longer sides of the card $x$ and $y$, then the area of the face of your card is the product, $A = xy$. From the formulas above, the uncertainty in the area should therefore be $\delta A = A \sqrt{(\delta x/x)^2 + (\delta y/y)^2}$.

NOTEBOOK: Compute the area of the face of your card, $A = xy$, and determine the propagated uncertainty, $\delta A$. Do you need to propagate the uncertainties in both $x$ and $y$, or can one be neglected?

Now look at the volume. If we multiply by the thickness $z$ to find the volume $V = xyz$, how do your compute the uncertainty in the quantity?

NOTEBOOK: Compute the volume of your card, $V = xyz$, (where $z$ is the thickness of your card). What is the uncertainty, $\delta V$? Do you need to propagate the uncertainties in $x$, $y$ and $z$, or can one or two uncertainties be neglected?

Now that you have finished your measurements and uncertainty calculations, it is time to draw some conclusions. This is the point in the scientific process where you present your final results, and discuss what they mean in the broader context. It can be a chance to claim agreement with prediction (if that's the case) or to discuss why there may be disagreement or why the results remain inconclusive.

Today's activity was a very simple measurement and not a full lab where you were testing a model or investigating a phenomenon. Therefore, there isn't really much to “conclude”. However, we can at least compare your measurement to those of some hypothetical classmates, and try to discuss why you agree or disagree with those students (if either of those are the case).

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

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

Now let's return to your measurements.

Suppose three of your classmates made the following estimates for the area and volume of their card:

NOTEBOOK:  Compare your area and volume measurement to the three classmates. For each, is your value in agreement, is your value in disagreement, or is it inconclusive?

You may find that you are only in agreement with one or two of your classmates (or maybe none, or maybe all three!) What does this mean? Are you correct and they are wrong? Are you wrong and they are correct? Could someone have a systematic bias that isn't corrected for?

It particular, you must always be on the lookout for your own expectation biases and your desires to get the “right” answer. If we said that Wynn used A4 paper to define her charta unit and Omar used US Letter-size paper, would that change your mind about whose data you agree with? (It should not.) Going further, it is unethical to change data, drop points, or inflate uncertainties in order to make your value “agree” or “disagree” with another number.  

Agreement or disagreement invites us instead to ask questions about why?

• If there is disagreement, what could be different about the two measurements?
• If there is agreement, does this agreement continue to hold as we improve measurement techniques and make the uncertainties smaller?
• Is disagreement due to differences in measurement technique or actual physical differences?

Most experiments do not have a clear ending. There are always more improvements that can be made to shrink uncertainties or test for systematic biases. In future projects, proposing improvements (and sometimes following through with them) will be an important part of the process.

NOTEBOOK:  What is one improvement you could make to your ruler or to the way in which you made measurements that you think would reduce the uncertainty in your final measurements?

NOTEBOOK:  You likely did not find agreement or disagreement with all three of your hypothetical classmates. If you had the opportunity to talk to them, what are one or two questions you would ask them? What would it be important for you to know (or to see) if you wanted to better understand why you were in agreement or disagreement with them?