laboratory -...

PHYSICS 51

LABORATORY REFERENCE

MANUAL J

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Thomas A. Moore

Alma C. Zook With contributions from

Torrin Hultgren ’00 and Alexandra Hui ‘01 Department of Physics and Astronomy

Pomona College Copyright © 2002

TABLE 5.1 TABLE OF STUDENT t-VALUES

N t-value N t-value 2 12.7 10 2.26 3 4.3 12 2.2 4 3.2 15 2.15 5 2.8 20 2.09 6 2.6 30 2.05 7 2.5 50 2.01 8 2.4 100 1.98 9 2.3 ∞ 1.97

Table of Contents

Section I: General Laboratory Recording and Reporting ............................................................... 1 Chapter 1: HOW TO KEEP A LAB NOTEBOOK.................................................................... 3 Chapter 2: PRESENTING DATA GRAPHICALLY............................................................... 13 Chapter 3: HOW TO WRITE A LAB REPORT...................................................................... 21

Section II: Dealing with Experimental Uncertainty...................................................................... 49 Chapter 4: THE STANDARD DEVIATION ........................................................................... 51 Chapter 5: EXPERIMENTAL UNCERTAINTY..................................................................... 57 Chapter 6: THE UNCERTAINTY OF THE MEAN................................................................ 65 Chapter 7: PROPAGATION OF UNCERTAINTY................................................................. 71

Section III: Fitting Functions to Data ........................................................................................... 83 Chapter 8: LINEAR REGRESSION ........................................................................................ 85 Chapter 9: LINEARIZING A NON-LINEAR RELATIONSHIP............................................ 95 Chapter 10: POWER-LAW FITTING AND LOG-LOG GRAPHS......................................... 99 Chapter 11: EXPONENTIAL CURVE FITTING.................................................................. 109

Section IV: Ray Optics ............................................................................................................... 117 Chapter 12: SINGLE THIN-LENS OPTICS.......................................................................... 119 Chapter 13: LENS SYSTEMS................................................................................................ 127

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Section I: General Laboratory Recording and Reporting

1. How to Keep a Lab Notebook 3

Chapter 1: HOW TO KEEP A LAB NOTEBOOK

“’But perhaps I keep no journal.’ ‘Perhaps you are not sitting in this room, and I am not sitting by you. These are

points in which a doubt is equally possible. Not keep a journal!’” -- Northanger Abbey

1.1 INTRODUCTION Keeping a lab notebook seems like a simple and obvious task, but it requires more care and thought than most people think. It is an important skill to learn since virtually every practicing scientist keeps a lab notebook of some type. Your lab notebook is your written record of everything you did in the lab. Hence it includes not only your tables of data, but notes on your procedure and at least some of your data analysis as well. You want all this information in one place for two main reasons, which continue to apply even after you leave the introductory physics laboratory. First, your lab notebook contains the information you will need to write a convincing report on your work, whether that report is for a grade in a course or a journal article. Second, your notebook is the source to which you turn in case someone questions the validity of your results. (You may be aware of the famous David Baltimore case of alleged scientific fraud, in which the lab notebook of one of Baltimore's collaborators was the subject of careful scrutiny. While the scientist in question was eventually exonerated, she spent an uncomfortable couple of years having others second-guess her lab records.) Your notebook therefore serves somewhat contradictory purposes. On the one hand, you need a complete and accurate record of your work in the lab, so you should write things down as they occur and before you have a chance to forget them. On the other hand, your notebook needs to be reasonably neat and well-organized, partly so you can find things later and partly so that if anyone questions your results, not only will they be able to find things, but the clarity of your notes will suggest that you investigated the problem carefully and systematically. However, it is hard to be completely organized when you are writing things down as you go. The purpose of this chapter is to suggest ways to balance these needs.

1.2 USE EVERY OTHER PAGE

As mentioned, a basic problem with taking notes on the fly is that it is hard to keep them organized. As the experiment progresses, you will be reminded of things you should have described in an earlier section, find that you should have taken a few more pieces of data, or discover mistakes in material you have already written. If you were simply to write your thoughts sequentially, you would never be able to find important bits of information later; no one’s “stream of consciousness” is sufficiently organized. Therefore, the cardinal rule of keeping a lab notebook is: give yourself plenty of space. Doing so makes extending tables or descriptions of procedure easy, and typically also makes your notebook easier to read. The best trick we have found for observing this rule is to use only one of the two facing notebook pages (probably the right page if you're right-handed, the left page if you're left-handed). If you have to go back later and add explanations, corrections,

1. How to Keep a Lab Notebook 4 calculations, or even new tables of data, you have a whole blank page available to record these things right next to the material you are supplementing or correcting. You can also use this blank page to record calculations as you reduce data (which allows you to look at the raw and reduced data at the same time), or to graph the data on the facing page. Give yourself plenty of space on the page, too; leave some blank space between explanations and around lists of equipment, sketches, and tables of data. This not only makes your notebook easier to read, it leaves you space to add short comments and/or corrections. The point is that initially using only every other page and leaving yourself plenty of space on the page provides you with the flexibility to go back and supplement or correct earlier material, without disturbing the organization and flow of your thinking in lab. Even if your “stream of consciousness” may not be organized, your lab notes can be if you give yourself this flexibility.

1.3 DIVIDE YOUR NOTES INTO LABELED SECTIONS

Anyone reading your lab notebook (including you, later on) will find following it much easier if you organize your presentation into clearly labeled sections. For most labs, the sequence of headings listed below works well. Each of these sections will be discussed in more detail later on.

(Title) Purpose Equipment Procedure Data Analysis

Any section heading listed above may be omitted or renamed if appropriate, and in some cases you may want to add other sections; you should consider this a “default” list that you start with, but modify if you have good reasons to do so. (If you compare this list to the outline for a lab report in Chapter 3, you will notice some remarkable similarities. This is not an accident. A well-organized lab notebook can really speed up writing a lab report later on.)

1.3.1 (Title)

This item should appear at the beginning of every lab description. You should begin each new experiment on a fresh page in your notebook, and start with a brief title for the experiment -- just enough to remind you what that section of your notebook is about. You should also record the name of your partner and the date you did the experiment. In fact, it's good practice to record the date at the top of each page and follow it with a page number for that experiment.

1.3.2 Purpose

This section can often be implied or even omitted, but you should always take a minute at the beginning of each lab to think about exactly what you are trying to measure in this lab and

1. How to Keep a Lab Notebook 5 why. If this takes more than a little thought, write something down here. This helps you to work through your thoughts clearly and completely, and if later in the lab you need to remind yourself of what you set out to do, you can refer back to this section.

1.3.3 Equipment

In this section, you should list and perhaps describe the equipment you use in the lab. This is particularly important if your apparatus differs from that described in the lab manual or from that used by other students. Any differences should be described carefully, in case this turns out to be relevant when comparing your results with those of other groups. In your equipment list, identify large pieces of equipment with manufacturer's name, the model, and serial number. (The definition of a “large” piece of equipment is “a piece of equipment large enough to have a serial number.”) Usually the serial number, if present, is marked on a small metal plate along with the model number on the front or the back of the unit. Smaller pieces of equipment and equipment made in-house will generally have a station number (1 through 12). With this information, you can repeat the experiment with the identical equipment if for some reason you are interrupted and have to return to the equipment much later. Furthermore, scientific equipment gets out of calibration. For example, a stopwatch could, in principle, run too fast or too slow; if you suspect that your time measurements are in error, you can check your stopwatch against another clock if you know which stopwatch you used! Identifying any pieces of equipment whose failure or misadjustment might influence your results is essential if you are to be able to track down possible sources of error. Also make a quick sketch of the setup, and/or a schematic diagram if appropriate. It is often easier to write notes about your procedure if you can refer to a sketch. Such a sketch is also an essential part of a formal written lab report, so a good sketch in your notebook can make writing the formal report easier. A sketch can also be useful for tracking down problems with your procedure or analysis. For example, suppose that you are doing an experiment with a simple pendulum (a ball swinging from the end of a string), and you draw a sketch of the pendulum that shows how you defined its length. If you find strange results when you analyze your data, you could refer to the sketch to determine if your original definition of the pendulum's length was the correct one.

1.3.4 Procedure

This section is very important, because most experimental errors can be traced to incorrect or inadequate experimental procedures. This section is especially important in labs in which you develop much of your procedure yourself. Start by writing a short paragraph or two outlining how you expect to carry out your measurements. This should not be too detailed, since you will probably modify your procedure as you go along, but this opening paragraph will help you settle in your own mind what you do to get started. Be sure to leave yourself plenty of space, because you will find yourself modifying your initial procedure, discovering additional variables that should be recorded, and revising your approach. Describe your procedure in complete sentences and complete paragraphs. Single words or phrases become mysterious very quickly as time passes. Start with a sentence or two about

1. How to Keep a Lab Notebook 6 what you're measuring, such as the period of the pendulum as a function of length. Give more details where necessary, if (for example) the lab manual does not give a more detailed procedure or if you find you need to depart from the procedure in the manual. We expect that people will get results consistent with physics as it is currently understood, since we will be dealing with well-explored physical principles. Poor results usually imply poor technique; your lab notebook is the place to start in establishing that your technique is good.

1.3.5 Data

In this section, record quantities that you measure directly, in an orderly table. Give each data table a title, unless its nature is clear from anything written immediately before it, and clearly label the rows and columns. It's also good to have an extra column, usually at the right-hand edge of the page, labeled “Remarks.” That way, if you make a measurement and decide that you didn't quite carry out your procedure correctly or you observe something unexpected, you can make a note to that effect in the “Remarks” column. For example, suppose that you realize in measuring the period of a pendulum that one of your measurements must have timed only nine swings instead of ten. If you indicate that with, say, “9 swings?” you could justify to a suspicious reader your decision to omit that point from your analysis. If you have room left on the page, you may also want to leave some room on one side or the other for data reduction later on. You will often be performing experiments in which you have two independent variables. Usually in such experiments you fix the value of one independent variable and make a series of measurements working through several values of the other variable. Then you change the value of the first variable and run through the measurements with the other variable again; then you change the first independent variable again, make another set of measurements, and so on. It's usually easier to set up this sort of sequence in your notebook as a series of two-column tables (or three columns, if you add “Remarks”) rather than a big rectangular grid. Label each table with the value of the independent variable that you're holding fixed, and keep the format of all the tables the same.

1.3.6 Analysis

This section should consist of notes regarding the processing of your data, “low-level” graphs (see Chapter 2) of your processed data, and conclusions you draw from your data and its analysis. In very simple labs, this section might be combined with the Data section, but in more complicated labs it's often better to keep them distinct.

1.4 WORK IN PENCIL (if you like) BUT DON’T ERASE DATA

We have heard of science labs in which students are required to record everything in ink, to make changing data impossible. We think this restriction is needlessly coercive. In this lab, then, use pencil if you prefer, and feel free to erase any short statements or blank tables and rewrite them to make things neater. (Entire paragraphs should simply be X’d out, though.) Use a pencil with a dark lead, though, so that your notebook is easy to read.

1. How to Keep a Lab Notebook 7 It is unwise, though, to erase any raw measurement data, since a measured value that seems to be in error at first may turn out to be correct when the analysis is completed. If something comes out screwy in your analysis, it is helpful to have a complete and absolutely honest record of all your raw data, to simplify tracking down the problem. Therefore, if you believe that a data point is in error, don't erase it. Instead, cross it out with a single line (so that it's still readable) and write the replacement above it or next to it.

1.5 A SAMPLE LAB NOTEBOOK

As an example, we give you some pages photocopied from a notebook Dr. Zook keeps for trying out possible experiments for the introductory lab. (Dr. Zook is the main author of this section, which is the reason she's allowing her lab notebook to be held up to public scrutiny for possible humiliation.) This experiment was suggested by a standard gee-whiz demonstration, in which you put a small ball like a superball on top of a more massive ball (a basketball is good) and drop them simultaneously, so the small ball winds up colliding with the larger ball immediately after the large ball has collided with the earth. If you drop the two balls right, the small ball bounces surprisingly high. You can see that, although her penmanship leaves something to be desired (a typed transcription follows) and she keeps her lab book in ink, she does follow the guidelines we gave earlier: there’s a short title, a list of apparatus and a couple of sketches. (The sketches are especially important because the apparatus is a home-built unit rather than a commercial one.) Then there’s some quantitative information about the two “superballs” used in the experiment and a brief summary of the planned procedure. On the second page of data collection (p. 31, really the third page on this experiment) there are some reasonably neat tables of data; in the lower left corner of those tables, there’s a comment about the large range in the bounce height and a suggested modification of the setup and procedure to reduce this range. Then there’s enough room left on the page to take more data with the revised setup (illegibly labeled “2nd try”), and to record the mean and standard deviation (see Chapter 5 on experimental uncertainty) of each set of five measurements. On the facing page (p. 30), which was originally left blank, there’s a graph of the results, which she drew after taking all the data and calculating the averages. It was originally drawn in pencil, although she went back over it in ink so it would show up on the photocopy. If you look at Chapter 2 (graphing), you can see that the graph satisfies the guidelines for the “low-level” graphs you would keep in your notebook: both axes are labeled, including units, and the graph has a brief title. In this experiment, Dr. Zook expected a linear relation between the initial height and the bounce height, and she has drawn in her eyeballed “best” line and calculated its slope from the two points marked with x’s. She initially left out the error bars, because this is only a low-level graph, but later put them in. If she were repeating this experiment now, she might have plotted the graph on a computer, printed it and the result of running LinReg (see Chapter 8 on linear regression) and stapled or taped the printout to p. 30 of her notebook.

1. How to Keep a Lab Notebook 8 [p. 29] 8/9/91 Colliding falling balls

Apparatus: Two superballs, one ~3/4” in diameter and the other ~2” in diameter. Each ball has a 1/4” hole through its center in which a teflon sleeve has been inserted, as shown below.

2 meters of nylon monofilament twine suspended from upper Unistrut in lab.

Wood box to which nylon cord is attached, w/ two 2-kg weights holding it down.

(sketch)

(another sketch)

The two balls have been threaded on the nylon cord. The nylon cord is marked every 10 cm w/ a magic marker.

Procedure: Raise both balls some distance, holding the lower and larger ball. Release from rest. Measure the maximum height of the upper ball as a result of the collision between the two balls after the larger ball collides w/ the floor. Repeat several times for a given height and also for several different heights. (Small data table) N.B. I didn’t unstring the balls for either of these measurements. [p. 31] 8/9/91(2) (Data tables) I can’t tell if the large range in heights is the result of random error or the result of my not starting the balls at the same mark. So I’ve added a (tall) ring stand w/ two rods to mark the initial location and approx. maximum height. The distance recorded under “initial position” is the location of the bottom of the larger ball. The center of the larger ball [must really refer to the smaller ball] is higher by the sum of the diameter of the large ball and the radius of the small ball, or 4.65 cm + 1.26 cm = 5.91 cm ≈ 6 cm

1. How to Keep a Lab Notebook 12 EXERCISES

Exercise 1.1

What are the two main reasons for only writing on one side of a set of facing pages in a lab notebook?

Exercise 1.2

Dr. Zook followed the suggested format reasonably closely but not perfectly. What are some differences between her notebook and the suggested format? In particular, what two sections are missing?

2. Presenting Data Graphically 13

Chapter 2: PRESENTING DATA GRAPHICALLY “ ‘A crowd in a little room -- Miss Woodhouse, you have the art of giving

pictures in a few words.’” -- Emma

2.1 INTRODUCTION

“Draw a picture!” is an important general principle in explaining things. Frank Churchill’s remark to Emma Woodhouse notwithstanding, “the art of giving pictures in a few words” is not nearly as useful as a good diagram or graph, because most people process visual information much more quickly than information in other forms. Graphing your data shows relationships much more clearly and quickly, both to you and your reader, than presenting the same information in a table. Typically you use two levels of graphing in the lab. A graph that appears in your final report is a higher-level graph. Such a graph should be done very neatly, following all the presentation guidelines listed at the end of this chapter. It's made primarily for the benefit of the person reading your report. A lower-level graph is a rough graph you make for your own benefit; they're the ones the lab assistants will hound you to construct. These lower-level graphs tell you when you need to take more data or check a data point, since any strange measurements really stand out in a graph. They're most useful when you make them in time to act in response to what you see. This means that you should graph your data roughly before you leave the lab so you still have the chance to make more measurements. (That's one reason we recommend that you leave every other sheet in your lab notebook free, so you can use that blank sheet to graph your data.) In graphing your data in the lab, you don't need to be too fussy about taking up the whole page or making the divisions nice, but you should label the axes and title the graph to remind yourself later what it shows. Graphing your data right after you have completed a set of measurements also flags regions in your data range where you should take more data. Typically people take approximately evenly spaced data points over the entire range of the controllable variable (the “independent” variable), which is certainly a good way to start. A graph of that “survey” data will tell you if there are regions where you should look more closely: regions where your graph is changing rapidly, going through a minimum or maximum, or changing curvature, for example. The graph helps you identify interesting sections where you should get more data, and saves you from taking lots of data in regions where little is happening. Graphing each point as you take it, though, is not a good idea. Doing so is inefficient and, worse, can prejudice you about the value of the next data point. So take five or six data points and then graph them all.

2. Presenting Data Graphically 14 For example, Figure 2.1 below shows the original data taken on a phenomenon called mechanical resonance. All you need to know about resonance for our purposes right now is that the “amplitude” (a measure of the response of an oscillating system) depends on the frequency at which that system is perturbed (or “driven”) by an external oscillating force. Notice that the experimenter initially chose driving frequencies in the first run that were approximately evenly spaced across the range shown in the graph. For this particular apparatus, the highest and lowest frequencies attainable with the equipment are easy to find, and the experimenter chose to space the frequencies evenly to get roughly 10 different frequencies over the range in frequencies. You can see from the graph of the original data that the response doesn't change very much at either very high or very low frequencies, but near some intermediate frequency, between 4 and 6 cycles/s, something strange and interesting happens.

Mechanical resonance

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Figure 2.1: First set of data (filled circles) for amplitude response versus driving frequency in an

oscillating system. Notice that the data points are evenly spaced. The experimenter noticed this, too, and went back to take more data in the interesting range of frequencies. The frequency spacing used in the second round is smaller than used in the first set by about a factor of ten, yielding 15 more measurements in the critical region. The result of adding the second set of measurements is shown in Figure 2.2. As you can see, the shape of the graph is now much better defined. Furthermore, the new data show that the anomalously high amplitude at 4.5 cycles/s is not a mistake (as one might think considering the other values). The experimenter could, of course, have taken data with the closer spacing over the entire frequency range, but that would waste time on measurements at both low and high frequencies where nothing much is happening. The strategy of taking coarsely spaced data and then backing up to take more data in interesting regions is a good compromise between completeness and efficiency. But remember that you usually can’t identify the “interesting” regions if you don't graph your data to begin with!


Mechanical resonance, more data

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2.2 ANALYZING YOUR GRAPH

Graphical data analysis is typically used as a euphemism for “find the slope and intercept of a line.” You will find this semester that you spend a lot of effort manipulating your data so that the resulting graph is a straight line. As you will find throughout the semester, the slope and/or intercept of such a line often gives useful information about the physical system under investigation. Determining the slope and intercept of a linear graph is such a common and important task that we have developed a computer program (described in Chapter 8 in this manual) to help you do it accurately. Many scientific calculators can also do this, although they almost never give the uncertainty in the slope and intercept. It is good to be able to estimate roughly the slope and intercept of a lower-level graph by hand, so that you can see if your measurements are at least roughly correct before you enter them all into the computer. Manual estimates of the slope and intercept also give you a check on the computer's results, allowing you to catch simple errors like entering the data in the wrong columns, for example. This process is so important that, although we have this fond hope that you learned how to do it in high school, we're going to review it anyway. Imagine that you have constructed a lower-level graph of your data by hand, and it looks pretty linear. Start by drawing in by eye the line that you think best matches your data. The analytic procedure called linear regression (described in Chapter 8) gives the optimum result, but in fact an eyeballed “best fit” line will generally be quite close to the line found by linear regression. Your job now is to find the slope and the intercept of that line you've drawn. The slope of a line is defined as the “rise over run,” the change in the vertical coordinate value divided by the change in the horizontal coordinate value. To determine the slope, you must

2. Presenting Data Graphically 16 first choose two points on your line. They need not be actual data points but they must lie exactly on your line. They should be about as far apart on the graph as possible to minimize the effects of the inevitable experimental uncertainty in their position. Mark each of those points with a medium-large dot or × and/or draw a circle around it. Read the coordinates ( ), yx 11 and ( ), yx 22 of each point off the graph. (As is the convention, the symbol x here represents the independent variable, plotted along the horizontal axis, and y is the dependent variable, plotted along the vertical axis.) With these two coordinate pairs, you can calculate the slope m using the equation

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y-intercept = (2.2) 1mx− Again, you can call either point ( as long as they both lie on the line. Since the y-intercept is defined as the value of y where a line intersects the y-axis (defined to be the x = 0 line), you can also read the intercept directly off the graph as long as the graph shows the x = 0 line. The graph in the sample lab notebook in section 1.5 illustrates the analysis of a lower-level graph. Note the use of x’s to mark the points used to compute the slope.

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2. Presenting Data Graphically 17 2.3 UNCERTAINTY BARS

Individual data points plotted on any graph should include uncertainty bars, sometimes called error bars, showing the uncertainty range associated with each data point. You should show both vertical and horizontal uncertainty bars, if the uncertainties are large enough to be visible on the graph. If they aren't large enough, you should mention this in your report so we don't think you've forgotten them. You draw uncertainty bars by indicating the “best guess” value (typically either a single measured value or mean of a set of measurements) with a dot, and then drawing an “I-bar” through the dot, whose length spans the 95% confidence range of that value. An example of such an uncertainty bar is shown in Figure 2.3 above. The single data point plotted corresponds to a measured pendulum period T of 1.93 s ± 0.03 s for an initial release angle θ of 20° + 2°. (The horizontal and vertical lines pointing to the error bars are not part of the graph, but are included to show you how the point and the uncertainty bars are related to the axes. Notice also that the T-axis does not begin at T = 0.)

2.4 PRESENTATION GUIDELINES for "higher level" graphs

You will create "higher-level" graphs for any written work that you submit for a full lab report. These graphs should be more carefully and formally drawn and labeled than the lower-level graphs that appear in your lab notebook. Here are some guidelines for constructing these graphs:

1. Draw your graphs in pencil; mistakes are easy to make. If you wish, go back later and touch them up in ink. High-quality graph paper may be purchased from Connie Wilson, the physics department secretary for 10¢ a sheet. Computer-drawn graphs are fine as long as they comply with the remaining guidelines. The program described in Chapter 8 of this manual makes it very easy to produce graphs that automatically comply with all these guidelines, but the graphs produced by other programs (such as Excel or Cricket Graph) may require extensive modification to fit the remaining guidelines.

2. Scale your axes to create as large a graph as possible consistent with the constraint that the divisions on the axes correspond to some nice interval like 1, 2, or 5 (times some power of 10). If you must make the graph smaller than full size to get nice intervals, OK, but check that you've picked the interval that gives you the largest possible graph (which will display your data in as much detail as possible). When using log-log or semi-log paper, choose paper with the number of cycles that gives the largest possible graph (see Chapters 10 and 11).

3. The lower left-hand corner need not be the point (0,0). Choose the range of values for each axis to be just wide enough to display all the data. If (0,0) does not appear on a hand-drawn graph, it is customary to mark the break in the axis or axes with two wavy lines (≈).

4. Mark the scale of each axis along each axis for the entire length of the axis. 5. Label both axes, identifying the quantity being plotted on each axis and the units being

used. 6. Give each graph a title that summarizes the information contained in the axes and provides

any additional information needed to distinguish this graph from other graphs in the report.


7. Give each graph a number (e.g., "Figure 2"), which you use in the body of the report or summary to refer quickly to the graph. (You can write such a number on a computer graph.)

8. Draw points and uncertainty bars as discussed in section 2.3. 9. If you calculate the slope and intercept of the graph from two points (rather than using the

method of linear regression described in chapter 10), indicate the two points you used on the graph. Draw the line through the two points, label it "Best-fit line" (or something similar), and give its slope and intercept on the graph in some large clear space.

2.5 CHECKLIST FOR EACH GRAPH IN A WRITTEN REPORT

Use this checklist to make sure that any higher-level graph that you include in a submitted written report (as opposed to your lab notebook) has the correct features and format.

The axes are scaled correctly with divisions equal to "nice" intervals. The graph is drawn as large as possible so that it fills the page. The scales on the axes have tick marks that run for their entire length. The axes have labels describing the variables they represent (including units). The measured data points are clearly plotted, including uncertainty bars. The graph has an appropriate title and figure number. The points used to calculate the slope and intercept are clearly marked (if that method is used).

EXERCISES

Exercise 2.1 Using the blank graph paper on the next page, create a higher-level graph of the data provided in Table 2.1. Use the checklist in section 2.5 to make sure that you have included everything. Distance fallen Kinetic energy 0.200 ± 0.003 2.00 ± 0.10 0.400 ± 0.003 3.78 ± 0.15 0.600 ± 0.003 5.65 ± 0.20 0.800 ± 0.003 8.01 ± 0.25 1.000 ± 0.003 9.82 ± 0.30 Table 2.1: Kinetic energy per unit mass of a falling object as a function of distance fallen. Exercise 2.2 Draw what you think is the best possible line through the data points in the graph you just created in the last problem, and find the slope and intercept of this line.

3. How to Write a Lab Report 21

Chapter 3: HOW TO WRITE A LAB REPORT “… it was in plain, unaffected English, such as Mr. Knightly used even to the

woman he was in love with…” --- Emma

3.1 INTRODUCTION

Science is fundamentally a communal process, in which individual scientists develop ideas and then seek through the medium of scientific journal articles to convince the scientific community of their validity. Learning how to communicate your ideas effectively is therefore a crucial skill for a working scientist (and is useful in many other callings as well). Consequently, you need know how to describe the science that you do in a way that convinces the reader that your work is interesting and should be taken seriously. You may feel that comparing your lab work and the resulting report to “real” science that appears in journals is a bit pretentious, since we’re probably not going to have you do much cutting-edge physics in an introductory laboratory. The purpose of the lab reports, though, is not so much to see if you did bold, original work as it is to give you practice in writing scientific reports, so that you'll be able to do it well when you do do bold original work. Most articles in scientific journals (physics journals, at least) follow at least approximately a standard format, which looks something like this:

ABSTRACT I. INTRODUCTION A. Motivation B. Summary of the experiment II. THEORETICAL BACKGROUND III. EXPERIMENTAL DESIGN AND PROCEDURE A. Description of the apparatus B. Description of the experimental procedure IV. ANALYSIS A. Method of analysis B. Presentation of results C. Discussion of results D. (Optional) Suggestions for future improvements V. CONCLUSIONS A. Summary of the results B. Pertinence of the results to the questions raised in the introduction

This format has evolved to answer the general questions a potential reader will ask: What did you do? (Procedure) Why did you do it? (Introduction, Theoretical background) How did you do it? (Procedure, Analysis) What happened? (Analysis, Conclusions)


The format also provides some shortcuts for busy (or lazy) people. Most scientific prose tends to be fairly dense, and readers like to find out in a hurry if a paper is actually of interest or importance to them. The abstract section provides a concise summary of the article and its most important results, so the reader only has to read a few sentences to determine if the entire article is relevant. The introduction and conclusions contain a little more information; usually the reader goes to the introduction for more information about the motivation and the method of the experiment, and the conclusion for more detail on the results summarized in the abstract. Each of these report sections is discussed in a separate section of this chapter. You will probably find it helpful to read over the entire chapter the first time you are asked to write a lab-report section (to get some sense of how the pieces of a lab report fit together). At the end of the semester, when you will write a full report, you should go back and read the entire chapter again.

3.2 THE SHORT SECTIONS: The Abstract, Introduction, and Conclusions

Most published scientific papers are not read in their entirety by everyone who looks at them. It's not that they are poorly written (although some certainly are), and it’s not that scientists don't care; there are just so many hours in a day. The short sections of a technical paper -- the abstract, introduction, and conclusions sections -- identify the important results of your work, and persuade a reader that really reading the paper is worth the time. Typically a reader will look first at the abstract, to find out what the paper is about. If the abstract looks promising, the reader will look at the conclusions. If they look interesting (and especially if they're unexpected) the reader will then check the introduction to see if the experimental method looks good. If the introduction suggests that you knew what you were doing, then the reader will read the rest of the article for the details. 3.2.1 The Abstract An abstract is an extremely terse summary of the entire paper, about three to six sentences long, which in a journal appears in small print just below the article’s title and list of authors. (The abstract is also often published separately and distributed more widely than the article itself.) The purpose of an abstract is to provide readers with a brief glimpse into the subject of the article, to help them decide whether to read the whole thing. One of the first things that one does when beginning a research project is to search recent publications for articles that might be helpful: good abstracts make it possible to determine quickly which articles are relevant. The structure of the abstract is essentially a miniature version of the structure of the article, except that each of the five major sections (introduction, theory, experimental design, analysis, and conclusions) might be represented in the abstract by only a sentence or even a phrase. Often the theory section is omitted completely from the abstract unless the paper is theoretical (which will not be the case for your lab reports!). Even so, the outline for the whole article is a pretty good starting point for the outline of the abstract as well. The abstract should always summarize the introduction and conclusion sections; this means that it will always include a short summary of what question you were seeking to answer, what your results were and what they imply. Although the abstract is the first section of a lab report, you may want to write it last because it is a summary.


In particular, a physics abstract should include a summary of any quantitative results you report in your conclusions. Apparently including quantitative results in the abstract is not standard in chemistry and biology articles (or so some chem and bio majors say when we criticize them for omitting this information), but it is standard in physics. Remember, the abstract is the "hook" you use to get people to read the rest of the paper, and you can best capture their attention with a nice juicy quantitative result with a promisingly small experimental uncertainty. 3.2.2 The Introduction The introduction section is meant to provide the reader with the answers to two very important questions: What is the question that your experiment is supposed to answer, and why is answering this question interesting (and/or important)? In a published journal article, this section often begins with a brief summary of previous related research, a statement of a problem that this research has raised, and a brief description of the experiment in question and how it addresses the problem. Detailed descriptions are not appropriate in this section; the point is to provide a concise picture of your purposes and a broad survey of your approach. This section should capture the interest of your readers, provide them with some general orientation, and convince them that what you are doing is interesting and worth reading about. After you motivate the experiment, you should give a brief summary of the experimental method you will use. This need not be extensive; the detailed description goes in the procedure section, which is separated from the introduction only by the theory section. You need to give enough information so that a reader who is interested primarily in your method, perhaps to duplicate your experiment or apply it to a related problem, can see if that method is appropriate. 3.2.3 The Conclusions A conclusions section should, in one or two paragraphs, review the purpose of the lab and summarize the implications of your experimental results. That is, you should remind the reader of the basic question that the experiment was to address (as presented in the introduction), and then briefly explain how your results bear on that question or problem. This section should be a summary of information presented elsewhere rather than a place to present new information: the purpose of this section is to close the report with a review that highlights the most important results. As with the abstract, you should report quantitative results and their experimental uncertainties. Students often ask, “What's the difference between the conclusions and the abstract?” The answer is, “Not much.” Both are summaries of the rest of the report, and both contain quantitative results. The main differences have to do with location: the abstract is the “hook” at the beginning, and should contain hints of the wonders to come. It also summarizes the entire report. The conclusion comes at the end, and should give some sense of finality or closure. It will emphasize your deductions from your data analysis, describing them in more detail than is given in the abstract. Both the abstract and the conclusions should report comparisons between predictions, presumably made in your theory section, and your measurements or their consequences.


3.2.4 Appropriate Detail in the Short Sections By referring to the abstract, introduction, and conclusions sections as being the “short” sections, we imply that these three sections shouldn't be long enough to contain much detail. That's right. The abstract and conclusions sections in particular should be the least detailed, giving the broadest look at the purpose of the experiment and the implications of the results. The introduction should be a bit more detailed, but not much: its focus should be on a general statement of the problem to be considered and the experimental method used to study that problem. Too much detail in any of these sections will obscure the reader's view of the main issues in the report. 3.2.5 A Checklist for the Short Sections (All checklists in this chapter are summarized on the inside back cover of this reference manual.) In your short sections, you should Summarize the entire paper in the abstract Discuss quantitative results in both the abstract and conclusions State the problem or question under investigation in the introduction Summarize the experimental procedure in the introduction

3.3 THE THEORY SECTION

The theory section is meant to provide the reader with enough mathematical or theoretical background to understand how the experiment works, what assumptions have been made, and how the experiment is related to the physics being studied. This section may be very short (or even non-existent) if the theory is well-understood and the connections between the theory and the measurements are straightforward and obvious. It can be quite extensive, however, if the experiment is complex or the actual measurements being made are related in a complicated way to the results being compared to the theory. If, for example, you were measuring the average velocity over some interval for your experiment, your theory section would be very short: you measure a distance and a time, divide the first by the second, and there’s your average velocity. Suppose, on the other hand, that your experiment was the determination of an acceleration in a situation where you couldn’t be sure the object was starting from rest. It is still possible to find the acceleration, but you have to measure two time intervals over two distances, and the connection between those measurements and the final result involves a fair amount of algebra. In that case, you would be expected to derive the connection for your theory section, which you could expect to be one or two pages long. You don't need to show each algebraic step, but you should show some intermediate results, especially if they involve complicated algebra, a substitution, or some trick of manipulation. The amount of theoretical background that you provide also depends on the expertise of your intended audience. For the purposes of this course, you should imagine your typical reader to be a classmate (not a professor or a lab assistant), who for some reason has not done the lab in question and knows nothing about it. This situation is analogous to that of a researcher whose audience has quite a bit of general knowledge about physical principles and experimental techniques, but no experience with the specific experiment.


For this semester, at least, you should start your theory section with first principles, or at least the equation (such as the law of conservation of momentum or energy) that defines the phenomenon you'll be studying. In a journal article you wouldn't go this far back, because starting from first principles to get to the result would take up too much space. Doing so in the introductory lab is a good idea, though, because you're probably just learning how to write a theory section. And since you're doing experiments that are usually close to the basic principles, starting with those principles helps you to examine your assumptions carefully. 3.3.1 Writing down equations Theory sections tend to involve equations. There are three general rules about equations in text. Rule 1: Don't write equations in the body of the text. Give each equation a line of its own. (Set aside three or four lines in your printout if you write in equations by hand.) You may break this rule for very simple equations you will not need later. For example, “When L = 1.0 m, the period of a pendulum is about 2 s.” Setting equations apart from the text makes the text read more smoothly, and also signals to the reader that it's time to go into Math Mode. You also get more room for writing the equation. Rule 2: Give every equation a number (except the simple ones mentioned in Rule 1). This way you and the reader can find them easily later on. Rule 3: Don’t try to typeset equations without an equation editor. If your word processing software doesn’t have a integrated equation editor that will let you typeset equations in standard form, don't try to type in an equation or parts of it; instead write the entire equation in by hand, instead. Faked Greek letters are almost never recognizable, and the time required to get fractions to print out correctly isn't worth it. Most readers don't instantly recognize “**” or “^” as meaning exponentiation, either, and they look terrible. You can get "+" by underlining the "+" sign; don't use "+/-," because it looks terrible, too. Rule 4: An equation is a sentence and should be treated as such. Specifically, if you want to define the variables in an equation, use one of the two constructions below. The comments in brackets are, well, comments on the examples and would not appear in your report. (a) F = ma, [Notice the comma!] where F = net force, m = mass of the accelerated object, and a = acceleration. Or (b) F = ma. [Notice the period this time!] Here F = net force, m = mass of the accelerated object, and a = acceleration. If you do write in equations by hand, don't forget to enter them after you print out your report! Missing equations are a sure tip-off that you forgot to proofread your report. People seem especially prone to forgetting the Greek letters and special symbols in partially typeset equations, whereas they usually notice those big blank spaces set aside for entire equations. (Note: Recent versions of Microsoft Word, WordPerfect, and many other word-processing programs for both the Mac and Windows operating systems have integrated solid equation editors, and one can buy good stand-alone equation editors relatively cheaply. Dr.


Moore likes MathType, which is easy to use and can be used with any word processor: see www.mathtype.com. There is therefore no excuse any more for attempting to typeset equations without an equation editor. Writing equations in by hand, however, is perfectly acceptable and will not lower your report grade.) Look at the class text for examples of good style regarding equations: the book was typeset according to McGraw-Hill’s professional standards for science texts (as described in a long document that Dr. Moore has). Note in particular that variables should always be set in italics: this helps set them apart from the text and identifies them as variables as opposed to just letters. 3.3.2 Checklist for a Theory Section Your theory section should: Start with the basic defining equations Show all non-obvious intermediate algebraic steps Clearly describe any assumptions and/or approximations involved in the model Display each equation on its own line Give each equation an equation number

3.4 THE PROCEDURE SECTION

Your job in the procedure section is to convince your reader that you carried out an experiment carefully and knowledgeably enough that the reader should take your experimental results seriously. In describing your experimental procedure, you should think of the reader as someone who is unfamiliar with the particular experiment you are doing but who is familiar with the pitfalls of working with the equipment you will be using. Furthermore, to keep you on your toes, you should think of this reader as being someone who is inclined to be skeptical about your results and hence will be picky about your procedure. (This doesn't sound very friendly, but professional scientists act just this way reading other authors’ papers, especially about experiments they wish they'd thought of doing, or about experiments they were about to do themselves.) Consider, for example, an experiment you will do later, measuring the period of a pendulum as a function of several variables. Simply saying, “We measured the pendulum period as a function of mass hanging from the end” doesn't do justice to what's really a rather elaborate procedure. Making this measurement carefully requires multiple measurements, timing several periods for each measurement, and choosing a particular starting and stopping point in the swing, all to reduce the uncertainty in your results, and you should say so. You should also explain why you went to all that trouble; doing so enhances your credibility with the reader, providing evidence that you thought carefully about the experiment. (It also justifies going to all that trouble.) Most procedure sections have a fairly standard format, which (as usual) you should feel free to modify. A typical description of experimental procedure starts with a list and description of the equipment. The equipment description should state the precision to which measuring devices read. Anything that isn't a standard device should be described somewhat quantitatively.


(For example, in the pendulum experiment you would give the approximate length of the string, and say something that would tell the reader whether to look in the stockroom for lightweight fishing line or big, hefty twine for wrapping packages.) Identify your lab station by its number if it has one. Large pieces of equipment should be identified by manufacturer's name, model, and serial number, which you should have written down in your lab notebook. Giving this information in your report tells the reader what performance is possible from the equipment you used. It is very important that you also give the reader a sketch of the apparatus. A good and complete sketch may replace a text list of equipment, and if so, it should be used instead. Sometimes this sketch will be schematic in nature, like a block diagram or a circuit diagram; in that case, a computer-drawn sketch is fine. In cases where you need to show fine detail, or where it's important to show the geometry accurately, a carefully hand-drawn sketch is usually better (and takes much less time to do well). Unless you are very skilled or have very good drawing software, computer drawings don't normally look enough like the objects they represent to be useful. (The diagrams in the lab manual of the setup for the Speed of Light lab shows a funky but tolerable computer drawing.) The list and/or sketch of the apparatus tell the reader what equipment was available to you, and to some extent whether you set it up in an appropriate fashion. Next, you tell what you did with the equipment. You should do this in a logical order, but not be too "step-by-step" about it. Specifically, avoid a numbered list of steps, which are difficult to read and hence inappropriate except for the rare reader who intends to repeat your experiment exactly. At the other extreme, you should avoid narratives like this: "First we did (whatever), but that didn't work, so then we tried (something else) to fix the problem with the first measurements." Refine your procedure to remove these false steps, and present it in enough detail so that the reader can clearly understand what you did without being overwhelmed by irrelevant tiny details. If you've made some revision in some seemingly obvious procedure that significantly improves the accuracy of your results, though, make sure you take credit for it. For example: “At the longest pendulum lengths (L > 1 m), the pendulum frequently hit the wall before completing ten swings. For those lengths we only timed five swings. This gave satisfactorily consistent results." You can also refer to the lab manual if its description of the procedure is sufficiently detailed (many articles in professional journals refer to other papers for details regarding equipment or procedure), but be especially sure to include a complete description of any procedural details that do not appear in the lab manual! In referring to a lengthy source like the lab manual, state the author, title, year of publication, and page number. (For example, a reference to the lab manual should look like this: Moore and Zook, Laboratory Manual for Physics 51a, 2002, p. 16.) A reference to a journal article would state the author, journal name (but not the article title), volume number, number of the first page, and year of publication. Instructions from the lab instructor or lab assistant can be cited as A. C. Zook, 2002, private communication. (This format is used in journal articles to refer to a conversation, unpublished letter, or e-mail message from the person cited.)


You might also consider the following questions as you write this section: 1) How did you determine the experimental uncertainties that you chose? 2) What (if anything) did you do to reduce them? 3) Did you experience any difficulties with the apparatus? If so, how did you resolve them? 4) Did you encounter any problems or difficulties in following the lab manual's procedure?

If so, how did you resolve them? 5) Did you modify that procedure in any way, and, if so, how and why?

Standard techniques, such as the correct use of a stopwatch or a vernier caliper, need not be described in your procedure section. Unless you've given us some reason to be wary of your ability to use a device that you’ve presumably either used before (for example, the stopwatch) or received some instruction about (for example, a caliper) we'll assume that you used it correctly. One detail you should definitely include, at this stage in your career, is the number of times you repeated any given measurement. Every year, we’re surprised at the number of students who don't seem to remember the importance of repeated measurements. Remember that repeating repeatable measurements is the only way to determine the uncertainty of the measurement! Although you will formally calculate the experimental uncertainty in the analysis section, it's good to mention the uncertainty ranges of your basic, unprocessed measurements in the procedure section, or at least state whether a given measurement was repeatable or not. Finding the appropriate level of detail is difficult. You don’t need to tell the reader everything, but you do have to say enough. The ideal procedure section is one that provides just enough so that the reader to go into the lab stockroom, pick out the right equipment, repeat the experiment, and get results consistent with yours based only on the information in your report and the lab manual. Providing just the right amount of detail requires practice, and probably the most aggravating comments you'll get on your lab reports will be in this section. 3.4.1 A Few Comments on Style... Procedure sections are right up there with theory sections for putting the reader to sleep. In procedure sections, the culprit is usually excessive use of the passive voice. (“The ball was hit by the batter” rather than “The batter hit the ball.”) In the natural sciences, we have this fond hope that the identity of the experimenter should not affect the result of the experiment, except insofar as one person may be more skilled with equipment than another. Writing in the passive voice became standard in the scientific community partly to emphasize the universality of science by de-emphasizing the role of the individual experimenter. Unfortunately, the passive voice is really boring to read, partly because it is wordier and partly because it dilutes the sense of action. The place where the historical convention really requires the passive voice is the procedure section. In other parts of the report, the spring exerts a force, or some results suggest an inverse-square law; you're out of the picture, and you can avoid the passive voice without embarassment. But the procedure section is the place where you describe what you did, except that your identity isn’t supposed to important.


However, the times they are a-changin’! We here at Pomona are not the only people who have trouble staying awake reading technical literature, and we’ve noticed that authors of scientific papers are more often saying things like “We observed NGC 253 on seven consecutive nights looking for supernovae,” and even (gasp) “I measured the activity of the radioactive sample at 15-minute intervals.” So go ahead, be on the cutting edge -- every so often, admit that human beings with names and faces make those measurements. If you're describing a division of labor, the standard phrase seems to be “One of us (TAM) calibrated the Heisenberg compensators while the other (ACZ) carried out the tachyon-beam efficiency measurements.” 3.4.2 Checklist for a Procedure Section Your procedure section should: Provide a sketch or schematic diagram of experimental setup Provide a textual list and/or descriptions of equipment (when needed for clarity) Describe all measurements, in roughly the order in which they were made Describe any departures from procedure described in the lab manual, if any Describe any steps taken to reduce experimental uncertainty The last two descriptions should follow the description of the measurement in question.

3.5 ANALYZING YOUR DATA AND WRITING AN ANALYSIS SECTION

3.5.1 Data Reduction The general task you have to accomplish in an analysis section is this: You start with a bunch of numbers (your measurements). You want to wind up with a few numbers (maybe only one) that characterize those measurements. Those few numbers in turn presumably tell you something about a theoretical prediction you or someone else has made; typically you have to make a decision about the validity of a theory based on your results. You get to the few numbers from the many numbers through your data analysis. In your analysis section, you show the reader how you got from the many numbers to the few, in enough detail that the reader can decide if you used the appropriate methods and carried them out correctly. Then you present your case for the implications of your numerical results. For example, in the first lab you measured the time required for a mass on a spring to complete a fixed number of oscillations. Then you made another set of measurements for an identical gravitational mass made of a different material, calculated some means and standard deviations, and looked at the ratio of the two sets of measurements. Presumably you made several measurements of the time, and you must have made at least one measurement of each gravitational mass. Human nature being what it is, you probably compared your result to the accepted result. 3.5.2 Graphing Your analysis section could more accurately be called your “data presentation and analysis” section, because the first thing you must do in an analysis section is display the data you are analyzing. You should not, however, display your original or “raw” data (the numbers you wrote down in your lab notebook) in tables in your report, because it's very difficult to pick out data trends from a large table. Instead, you should present your data graphically, plotted on


Cartesian paper. Even this graph (or set of graphs) will probably not simply be a graph of your unprocessed data: you will more likely plot averages (or means) of sets of data with appropriate uncertainty bars. (See Chapter 2, Presenting Data Graphically, for details about setting up graphs.) Just drawing the graph isn't enough, though. You must tell the reader that it exists, what it’s about, and where it is. A typical first sentence in an analysis section reads something like this: “The dependence of falling time on distance from the initial position is given in Figure 1.” (Obviously you should give the dependent and independent variables for the experiment you're actually describing!) Notice that you have identified the graph both by the data being displayed and by stating a figure number. Identifying the graph by the data tells the reader why this graph is part of your logical argument about the meaning of your data and results. Identifying the graph by a number makes it easy to find, especially if you put all your graphs at the end of your report. If your word processor lets you display a graph on the same (or at worst the next) page as the text discussing the graph, then do that; the next best thing is to put all your graphs at the end. Either way, the reader knows exactly where to look for them, which is better than having a figure located at the nearest convenient empty space several pages away. (A word of caution about positioning graphs: you can use up an enormous amount of time trying to put a graphic in just the right location while keeping section and page breaks where you want them. If you find your word processor driving you mad while positioning figures, a particular problem with Word for Windows, put all your figures at the end. It is OK to do this, really!) You must, of course, show error bars on your graphs, unless they’re too small to be visible. If this is the case, say so explicitly so that your reader does not assume that you have simply forgotten about them (which could have deleterious effects on your grade). If your error bars are large enough to be visible, you should also state explicitly whether they represent one standard deviation, the 95% confidence interval, or some other range. (The 95% confidence range is standard.) The details of your analysis from here depend on exactly what question you are trying to answer with your data. Often in your theory section you have worked out an expected relationship between the variables that you are measuring. If the expected relationship is linear, you can check that the data you have graphed are consistent with that prediction. If the expected relationship is not linear, you will generally have to draw another graph of your data using one of the linearization techniques described elsewhere (Chapters 9, 10 and 11) to make the expected relation linear. If this second graph is necessary, refer to it by title and number in your report. It's usually a good idea to put the linearized plot right after the Cartesian plot, and comment briefly on the relation between the Cartesian and non-Cartesian plots in the report. For example, in a write-up of a pendulum lab, you might say something like this: “The curve in Figure 1 and the predicted L1/2 dependence suggest a power-law relation between pendulum length and period. Figure 2 shows a log-log graph of the data of Figure 1. The data in Figure 2 lie on a straight line, indicating that period and pendulum length are in fact related by a power law.” The result you are after in an experiment is often related to the slope and/or intercept of this final straight-line graph. Early in the semester you may find the slope and intercept by eyeballing the best-looking straight line. (You may also use this method later when you want a


quick estimate of the slope.) If you do this, indicate on the graph the two points you used for the slope and intercept calculations, and give the numerical results in your Analysis section. Later on, after you become familiar with a technique known as linear regression (see Chapter 8), you will use that method, usually with the program called LinReg. If you did some calculations to extract the value you want from the slope or the intercept of your final graph, please go through these calculations in enough detail that the reader can duplicate your work if necessary. If you have to do a series of very similar calculations (and they're more complicated than dividing by 2 or π), show one such calculation in some detail as an example and then state that the other calculations are similar. 3.5.3 Experimental Uncertainty An essential part of any analysis is a discussion of experimental uncertainty. Careful treatment of uncertainty is essential if you are to draw meaningful conclusions from your data. If you have to estimate the uncertainty of any measured quantities, describe how you did your estimate, unless you already did this in your procedure section. If you computed the uncertainty of a value, describe how you did that calculation and show an example calculation. Also make sure that you specify explicitly (where relevant) whether the uncertainty you are quoting is the uncertainty of a single observation or the uncertainty of the mean. Report uncertainties with units and in the same form and to the same precision as your results: for example, 3.98 ± 0.07 N, not 3.98 ± 6.8 × 10-2 N. If you are reporting a result (with uncertainty) whose magnitude requires the use of scientific notation, report both numbers written with the same exponent: (1.10 ± 0.06) × 10-6 meters, not 1.10 × 10-6 ± 6.2 × 10-8 meters. Comparing the precision of your uncertainty to your result is much simpler with the preferred format. This might be a good place to point out that “uncertainty analysis” or “error analysis” does not mean, “Explain what went wrong and how you'd do it differently next time.” Certainly, if in analyzing your data you realize that you carried out some part of the procedure in a way that gave poorer results than you had expected, and you don’t have the time do redo that part of the experiment, you should say so: thinking carefully about your procedure after you've done the experiment is an important part of improving your experimental technique, and can be critical for eliminating systematic errors from your results. The term “uncertainty analysis,” however, refers to the quantitative estimation of the experimental uncertainty in your numerical results. 3.5.4 Results Earlier we said that you should not give tables of your raw data in your analysis section (or anywhere else). There are occasions, however, when reporting processed results in tabular form is appropriate, when a graph is difficult or meaningless. Suppose, for example, that you repeated the Inertial Mass experiment with several other masses made of other materials. In this case, your independent variable is the nature of the material, not a numerical value. Consequently, a table, rather than a graph, giving the materials and the ratio of the periods would be a reasonable and clear way to present your results.


At some point you will draw some conclusions about whether the data you have obtained are consistent with the expected relationship between your variables. If you predicted a straight line in your theory section and your experimental results support your prediction, you should say so. You should, however, avoid comments like, “Our results prove that the theory is correct.” You can never prove a theory; to do so, you would have to perform all possible experimental tests of that theory, and you don’t have time for that in a three-hour lab period. On the other hand, it is possible to disprove a theory with a single contradictory measurement (provided that the experiment has been done correctly, which may be a matter of debate!). The accepted phrase in both cases is less rashly assertive: “Our results are consistent” (or inconsistent) with the theory. Often your discussion of the implications of your results will be straightforward; if you're working with a well-known physical system and you follow the treatment in a textbook to develop a theory, your results will be probably consistent with the theory. We have tried to slip in a few curve balls just to keep the lab from being “verify what's in the book,” though. Your discussion of the implications of unexpected results will show your strength as a physicist most clearly. You should be creative, but also very careful. Don't allow yourself to indulge in empty speculation about an unexpected result; test your speculations. If you come up with an explanation, try to show that it could indeed have caused an effect of the same magnitude and in the same direction as the effect you observed. That is, if your explanation predicts a greater-than-expected measurement, you'd better observe a greater-than-expected measurement if your explanation is to be valid. 3.5.5 Checklist for an Analysis Section Your analysis section should

Briefly describe the data Include a Cartesian (unlinearized) graph of data Include linearized graphs of data, if appropriate Discuss consistency or lack thereof with any theoretical predictions Discuss how you calculated the slope and intercept of any linear graphs Show the calculation of any derived quantities from slope or intercept Completely discuss all uncertainties involved, showing sample calculations if needed Discuss the results and their implications

3.6 PUTTING IT ALL TOGETHER

3.6.1 Proofreading In principle, if you write the various sections of your report using the guidelines above, you should be done. Before you turn in that masterpiece of scientific prose, though, you need to make sure that it all hangs together. That is, do the links between sections that you imply in one section actually appear in another? For example, did you test in your analysis section the equation that you derived in your theory section? If you made assumptions in your theory section, did you include tests of those assumptions in your procedure section? Did the measurements you describe in the procedure appear as graphs in analysis? Do your quantitative


results support your discussion and your conclusions? Is it clear that your theory and your procedure are about the same experiment? You should really read your report over twice. The first time through is for proofreading, a step we find people often omit. That word-processed output from the laser printer may look wonderful at first glance, but it has to stand up to a careful reading. Remember, the computer may not going to catch your mistakes in punctuation, and the spelling checker will probably not distinguish between “there” and “their,” or “it’s” and “its.” (Now is a good time for you to make sure that you know the difference between “it’s” and “its.”) It also won’t notice that you’ve left out the equations. (Indeed, using a spelling checker with technical writing can be pretty annoying, as it chokes on every technical word, symbol, and equation number.) Our experience with grammar checkers suggests that they are not up to college-level English, so don't slavishly follow every instruction your grammar checker makes, either. We're not suggesting that you turn your backs on some benefits of modern computer technology and not use your spelling and grammar checkers at all, but you should recognize that they have their limitations. The second reading is for sense and continuity. Do the steps of your procedure follow each other logically? Is the same true for your analysis? Do the sections of your report relate to each other as described above? If you can stand it, and if you can get yourself to write your report well ahead of time (a good intention with which the road to hell is no doubt liberally strewn), get someone else (preferably not your lab partner) to read your report. The lab assistants will be prepared to read over your reports for just such considerations as we've described above. 3.6.2 More on Good Writing Style The mechanics of your presentation are arguably its least important aspect. Nevertheless, a sloppy presentation can add to your reader’s difficulty in getting through your report, and hence lower your credibility. (If you didn’t care enough about your report to run it through your spelling checker, how much effort could you have gone to on the parts that needed some real work?) You are presumably already familiar with the need for correct spelling and punctuation; here are some mechanics of presentation that may be less familiar.

• Set apart the different sections of your report (abstract, introduction, etc.) with blank lines. • Avoid breaking a section between the heading and the first paragraph; that is, don’t leave a

section heading dangling at the bottom of a page with the text of the section beginning at the top of the next page.

You will be expected to write good, clear, English in your lab reports, using correct grammar in complete sentences. The days when someone in a science course could wail, “But this is a physics course, not an English course!” are, thanks to the concept of writing across the curriculum, long gone (if in fact they ever existed at classy liberal-arts colleges like Pomona). Remember that the point of any report is communicating with someone else. If you keep distracting your readers with grammatical mistakes or unclear prose, you will make it difficult for them to concentrate on the meaning. You will be graded partially on the quality and clarity of your writing. As a general guide to a good prose style, we recommend Strunk and White's The Elements of Style. It is a small paperback, usually available at the bookstore. We think it will be a useful investment for several of your classes. Also keep handy your copy of Hacker, A Writer’s


Reference from your ID 1 course; the lab staffers are likely to refer to it when pointing out grammatical mistakes. In spite of what Strunk and White say, however, you should use “inclusive” pronouns rather than the generic “he.” That is, you should use constructions like, “When physicists make measurements, they ...” rather than, “When a physicist makes a measurement, he ...” Strunk and White wrote their book before inclusive language became standard. It's the 21st century now, usage changes, and it's time to get with it. (You might also count the members of the lab teaching staff who are left out by the generic “he” and think of inclusive language as simple self-defense.) You should also avoid certain words and expressions. “Readings” (as in, “We took five readings for each distance”) belongs on Star Trek, where it's used to avoid using the technical terminology that a 23rd-century scientist would use, since the screenwriters don't have any idea what that terminology might be. You’re using 20th-century equipment (sorry, we didn’t replace everything at the turn of the millennium) and a 21st-century vocabulary, and you can describe exactly what you’re measuring: “For each distance between the source and the timer, we measured the time interval for the sound wave to travel that distance five times.” Other words and phrases that people often use incorrectly are:

• “Defined as,” in the sense of “found to be” or “may be described empirically by.” You can define the length of a pendulum as “the distance from the pivot to the center of mass of the bob,” if that is the correct definition, but you find or measure it to be 1 meter long.

• “Calculated value,” in the sense of “number we calculated from our measurements.” Usually the calculated value (or the theoretical value) is one you derive from some theoretical calculation, and the measured value (or the experimentally-determined value) is the one you calculate from your measurements.

• “Approximate” for “estimate” (as a verb). Estimates (as nouns) usually are approximations, in the sense that you typically know them to one significant figure. But you estimate a number (that is the process), and end up with an approximation (or better, an estimate [noun]) of its value.

• “Correlation” for “simple relation.” Saying that two quantities are “correlated” only means that they seem to be related in some way, so that if one changes, the other one changes as well. The relationship between variables in many disciplines of natural and social science can be extremely complicated, and although we often assume that some underlying cause is responsible for the relationship, this is often not the case: correlation does not imply causation. In physics, however, the variables that we generally will look at will be clearly related by some simple relation. Saying that two quantities are “correlated” in physics is usually too weak a statement: describe the relationship.

• “Calibration.” People really like this term, because it sounds so technical. It refers specifically to the comparison of one measuring instrument either against another or some reference standard, to make sure the instrument is working correctly. If this is not what you're doing (and you rarely will do this in this lab program), you are not “calibrating.”

• “Prove,” meaning “support.” We talked about this already, but it's worth repeating. You can't prove a theory with one experiment, although you can disprove a theory with one. Results can only support or be consistent with a theory.


• “Correct value” in the sense of “a value published in a book.” In some experiments, you might be measuring a value (like the speed of sound) whose value we can look up in a reference, and you may be tempted to call the value in the book the “correct” value. It is not the correct value: it is the (currently) accepted value. The values of physical constants published in books are summaries of experimental results, and new experiments can (and often do) lead to modifications in these accepted values.

An episode from the history of optics illustrates the last point. Albert Michelson (1852-1931) was the first American to win the Nobel Prize in physics, for his precision measurements in the field of optics. He invented the Michelson interferometer, used in the famous Michelson-Morley experiment to demonstrate (unexpectedly!) that the speed of light is the same in all inertial reference frames. He also made several measurements of the speed of light using a method very similar to the one you will use later on this semester, although with considerably longer baselines. (One of his measurements was made between Mt. Wilson and Mount Baldy [no lie!], and Baseline Road in northern Claremont was surveyed accurately as part of this measurement.) His last measurement, made in an evacuated tunnel about a mile long (on what was then the Irvine Ranch) was accepted as the standard for decades, and probably most physicists thought of his result as the “correct” one. A 1941 review of fundamental physical constants (R.T. Birge, “The General Physical Constants,” in Reports in Modern Physics, 8, 90, 1941) weights this result the most heavily in coming up with a weighted average of several contemporary measurements of the speed of light. You can guess what's coming. Later measurements, mostly made in the 1950s, consistently got results that disagreed with Michelson's. The disagreement wasn't very large, about 17 km/s (out of 300,000 km/s). Their result and Michelson's differed by more than the sum of the experimental uncertainties, though. Eventually a partial explanation for the discrepancy surfaced. Michelson died shortly before the experiment was actually performed, although he did see the apparatus installed. His collaborators made the measurements (almost 3,000 altogether) at night, to reduce temperature variations and human activity in the area as sources of experimental uncertainty. The baseline distance was measured during the day, though, and only two or three times. (It’s difficult to survey distances of more than a few tens of meters at night.) Apparently the thermal expansion and contraction of the ground itself with temperature was large enough to have a systematic effect on the speed of light they deduced from their measurements. Lest Michelson’s collaborators seem inept, we should mention that they were quite alert to some even more obscure possible sources of systematic error. In reporting their results, they mentioned an apparent weak dependence of the measured speed of light on the tides, but since they couldn’t identify the cause of this dependence, they couldn’t figure out how to correct for it, or even whether they should! The cause of this systematic effect is unclear even now. Michelson’s collaborators and the authors of the review article from which most of this historical summary is taken, mindful that “correlation doesn’t imply causation,” all hesitated to claim that the tides were directly responsible for the apparent variation in the measured speed of light. (For more details, see E.R. Cohen and J.W.M. DuMond, “Fundamental Constants in 1965,” Reviews of Modern Physics, 37, 537, 1965, and the references therein.)


The moral of this tale, of course, is that there are no “correct” results in science, only accepted ones. Even prominent scientists forget sources of systematic error, or run into systematic error where no one would have expected it, or someone comes along with better equipment. It is true that you're not likely to hit the frontiers of physics in an introductory laboratory, but you should get into the habit now of regarding every scientific result as only one carefully designed experiment away from revision. 3.6.3 How Long Should a Lab Report Be? (and Stuff to Leave Out) A typical scientific journal article might be about ten pages long. Your full lab reports will probably be shorter; try to limit yourself to the equivalent of four or five single-spaced typewritten pages of text, not counting graphs or diagrams. This means that few of the five major sections (the ones with Roman numerals on the outline) will exceed a page in length, and some may be shorter. There are also some items you should leave out of a lab report. Please don't complain about the equipment; we already know that if we had an infinite budget, we could buy really frictionless gliders and opto-electronic timers good to a microsecond. You won't have an infinite budget in real life, either. Even if your equipment budget is large, you will always be making measurements that require care and ingenuity to make; sometimes the equipment you would like doesn't even exist! Experimental physics isn't about making really precise measurements so much as it is about making the best measurements you can with the equipment you have. By practicing with the admittedly limited equipment available now, you prepare yourself for those later measurements when you can't improve the data simply by spending more money. Don't editorialize about an experiment being a “success” or “failure” in the context of agreement with accepted results or theories. It’s true that we have some expectation that your results will be in agreement with established laws of physics, because normally you won’t be dealing with particularly exotic (that is, poorly understood) physics in an introductory course. We also expect that in the full report, in which you do write a draft for which you have presumably analyzed your data, that if your results are in gross disagreement with established laws of physics, that you will make some attempt to figure out the cause of that disagreement and fix it. You will, after all, have part of an additional lab period to collect more data if that should seem appropriate, and that’s exactly why we arranged the lab schedule the way we did. In evaluating your work, though, we look primarily for evidence that you understood how the equipment worked, how the measurements you made were related to the theory discussed, and generally that you were thinking about what you were doing. Some real physical effect could be present that the designers of the lab overlooked, or have left in to keep you on your toes. (This happens more often than you might think.) If you have been careful about your work, be confident in presenting what you have observed. (The confidence should follow from being careful, though, and if the lab staff identify some systematic effect in data collection that you overlooked, go take more data!) 3.6.4 Example Lab Reports Two sample lab reports are provided as appendices to this chapter. Each is mostly well-written, but has problems with specific sections, as discussed in the exercises below. Except for these problem sections, though, you can use these reports as examples of good report style.


Note again that a summary of all checklists appears on the inside front cover of this manual.

EXERCISES

Exercise 3.1 Read the lab report entitled “The Speed of Sound.” This report is mostly well-written except for the abstract and procedure sections. See what you think is lacking in these sections (according to the checklists and other information in this chapter) and then compare with the comments on the last page of this chapter. (There is no penalty for not spotting everything: just do the best that you can.) Write your comments on the report itself. Exercise 3.2 Read the lab report entitled “Gravitational Potential Energy”. This report is mostly well-written except for except for the analysis section (where little superscripted numbers indicate problem areas). See if you can figure out what these numbered problems are, and then check the answers provided on the last page of the chapter. (There is no penalty for not getting everything just right: just do the best that you can.) Write your guesses in the margin of the report itself.


APPENDIX 3.1: FIRST SAMPLE LAB REPORT Torrin Hultgren Partner: Alix Hui 9/10/98 The Speed of Sound Abstract: In this lab we determined the speed of sound by timing the interval that it took for a loud bang to echo off a surface a known distance away. Our average time interval was 1.28 s, and the distance was 440 m, so our calculated value for the speed of sound was 343.8 m/s. This is consistent within our experimental uncertainty with the accepted value at 30°C, which is 349.7 m/s. Introduction: The speed of sound has many practical applications, such as determining the distance from lightning, knowing when jets will break the sound barrier, designing acoustical facilities like concert halls and auditoriums, and literally thousands of others. The phenomenon of an echo is familiar to most people, and it is a relatively easy way to measure the speed of sound. We used two blocks of wood to create a loud and sharp bang. We determined the distance using a counting wheel whose circumference we measured and we used hand stopwatches to time the echo. We repeated the time measurement 20 times to reduce experimental uncertainty. We calculated the speed of sound by dividing the distance measurement by the time measurement. In addition, because the speed of sound varies with the temperature of the air through which it propagates, we measured the temperature with a mercury thermometer in order to calculate the accepted value for the speed of sound. Procedure: We used the following pieces of equipment to do the lab. • Two small blocks of wood • 2 stopwatches • 1 measuring wheel • 1 meter stick • Thermometer • A small piece of masking tape We set up on the concrete bench closest to the grass on Marston Quad. We chose this spot because it lined up with the small wall at the end of Stover Walk (which we could see through the trees) which gave us an easy reference point for beginning our distance measurement. One of us held the stopwatch and the other hit the blocks together. Because we could see the blocks coming together we could anticipate when they would hit. Then we stopped the stopwatch when we heard the echo, without anticipating it. This gave us a slight delay in


timing the echo because of our reaction time, but we were able to correct for this as described below. Both of us made 10 time measurements and hit the blocks together 10 times. To account for the reaction time delay we devised this procedure. I started both stopwatches at the same time. I then handed one stopwatch to Alix and kept the other. Behind my back she simultaneously stopped her stopwatch and hit one of the blocks against the concrete bench. When I heard the sound I stopped my stopwatch. The difference between the two times on the stopwatches was my reaction time. We repeated this measurement for each of us five times. To calibrate the measuring wheel we put a small piece of tape at the edge of the wheel. We put the meter stick on the ground and lined this piece of tape up with one of the ends of the meter stick. We then rolled the measuring wheel along the ground next to the meter stick until the piece of tape had traveled one full revolution. The point that it lined up with was our value for the circumference of the wheel. For the distance measurement we began at the wall at the beginning of Stover Walk that lined up with the place where we had taken our time measurements. We walked the measuring wheel down the middle of Stover Walk, using the sidewalk lines to make sure we were traveling in a straight line and not zigzagging excessively. We continued across the street, and then used the sidewalk lines to line up perpendicularly so we could move over and roll the measuring wheel across the wood chips and right up to the face of Carnegie that we believed the sound was echoing off of. We then doubled this measurement to arrive at the total distance the sound had traveled. Analysis: The average of the measurements I took was 1.55 s, with a standard deviation of s = 0.05 s. The uncertainty of this measurement, using the Student t-value, is st = 0.05 s × 2.09 = 0.10 s (1) This measurement therefore had a fractional uncertainty of

%4.6064.0s 1.55s 10.0

== (2)

The similar values for Alix’s measurements, which were different because she had a different reaction time, were 1.43 s ± 0.13 s for a fractional uncertainty of 9.1%. Both of these fractional uncertainties seem reasonable for the type of measurements we were doing. My average reaction time was 0.27 s ± 0.02 s, and her average reaction time was 0.20 s ± 0.03 s. Our actual calculated times of flight were therefore 1.28 s ± 0.082 s and 1.23 s ± 0.11 s. Our measurement for the circumference of the wheel was 0.587 m. Our measurement for the number of rotations of the wheel was 374.3. The distance from us to Carnegie was therefore


m 220turn

0.587 turns m3. =×374 (3)

Doubling this we arrived at a total distance of flight measurement of 440 m. We generously estimated our uncertainty to be ± 1.0m. This gives us a fractional uncertainty for the distance measurement of 0.2%. Compared to the uncertainty of the time measurement, this is tiny. My calculated value for the speed of sound was

m/s 344s1.28

m 440= (4)

Propagating uncertainty using the weakest-link rule, my calculated uncertainty was ± 22 m/s. Alix's value was 355 m/s ± 32 m/s. The formula for the speed of sound as it varies with temperature is

Tvs ⎟⎠⎞

⎜⎝⎛

°⋅+=

Csm6.0

sm3.331 (5)

where T is measured in Celsius degrees. Our measured value for the temperature was 30°C. Plugging this into the above formula gives us an accepted value for the speed of sound of 349.3 m/s. This value lies well within both of our experimental uncertainties. Conclusion: We measured the time it took for an echo to travel a measurable distance. Using our separate time and mutual distance measurements we calculated two values for the speed of sound: my result was 344 m/s ± 22 m/s and Alix’s was 355 m/s ± 32 m/s. These values for the uncertainty are a reasonable fractional amount. Our calculated accepted value for the speed of sound based on the observed temperature was 349.3 m/s. This value lies well within the experimental uncertainty of both our measurements. COMMENTS ON THIS REPORT: Short sections: These are fairly good, except that the abstract should include an estimate of the uncertainty in their measurement of the speed of sound, not just their measured value. The introduction should provide a clearer statement of the particular experimental question to be resolved here (that is, that the goal of the experiment is to measure the speed of sound by measuring the round-trip time of an echo from a distant object and compare the result with an accepted formula for the speed of sound). Theory:


The Theory section is missing! This is obviously a simple experiment based on very simple theory, but at the very least the author should state explicitly that he is assuming that the speed of sound is constant, and give the appropriate equation for finding the speed from distance and time measurements. Procedure: The equipment list does not include their stopwatch number or the number of the measuring wheel. Consequently, if they needed to check their calibration of the wheel (or the accuracy of the stopwatch, which is less likely), they would have no way of identifying it. The procedure section does provide an equipment list but not a sketch or diagram. However, this lab is a case where an equipment list is probably more useful than a sketch for helping the reader understand how the lab works. Even though the guidelines strongly suggest that one should include a diagram, the guidelines should not be followed slavishly if a diagram does not really add much to the reader’s understanding. Do whatever makes things clearest to the reader! It might have been nice to briefly discuss that the author is assuming that the “actual” flight time of the echo that he will use to calculate the speed of sound is his measured flight time of the sound minus his reaction time. This is implicit but should be stated more explicitly. The calibration of the measuring wheel needs more discussion. For example, the piece of tape mentioned presumably has a finite width, probably about 1 cm. If they weren’t careful to identify a particular reference point on the tape (such as a pen mark on the tape, or one of the two edges), this would introduce a systematic error into their calibration, which would carry over into a systematic error in their value for the distance. The author also doesn’t state the precision of their measurement of the circumference of the measuring wheel. Without this, the reader has no way of knowing if the later estimate in the uncertainty of the distance is reasonable. It is also unclear if they repeated the circumference measurement or the distance measurement. Analysis: The main problem with this section is the uncertainty analysis. To begin with, the author mentions combining the uncertainties of their average time measurements for the echo time and the reaction time, but does not identify the method used to combine the uncertainties. Next, no uncertainty estimates are given for either the measurement of the wheel’s circumference or the number of revolutions of the wheel. Finally, the author invokes the weakest-link rule in finding the uncertainty in the final value for the speed of sound, but does not justify the use of the weakest-link rule by explicitly locating the weakest link in the calculation and then showing a sample calculation using that weakest link.


APPENDIX 3.2: SECOND SAMPLE LAB REPORT

GRAVITATIONAL POTENTIAL ENERGY Maria Goeppert-Meyer (Lab partner: Irene Curie) Sept. 26, 1995 ABSTRACT

mghVfi =−

fi V−

i f

mghV fi

In this experiment, we determined the change in the gravitational potential energy V of the system consisting of the earth and a dropped plastic slab as a function of the distance h through which the slab falls. We found this change in potential energy to be consistent with the expression V , where m is the mass of the object and g is the gravitational field strength. We found the value of g to be 9.81 ± 0.02 m/s2, consistent with results obtained in other laboratories. INTRODUCTION Consider the change V of the gravitational potential energy of a system consisting of the earth and a falling object, where V is the system’s initial potential energy, V is its final potential energy after the object has fallen a certain distance h. In section C7.4, the text claims that this change in potential energy is given by V =− , where m is the object’s mass and g is the gravitational field strength near the earth, a constant that is purportedly equal to 9.8 m/s2. This result, which is stated without justification in the text, is a basic and important result that subsequently used many times in the text. It would be valuable, therefore, to supply the empirical foundation for this assertion. Our goals in this experiment were to demonstrate for a specific object interacting with the earth that (1) for a given value of h, the value of V fi V− does appear to be proportional to m, (2) for a given value of m, the value V fi V− increases linearly with h, and (3) the value of g is what it is purported to be. In this particular experiment we dropped a plastic slab (released from rest at a known initial height) past a photodetector connected to a computer. A series of equally-spaced opaque bands painted on the slab interrupted the light falling on the photodetector, and the computer measured the time that it took each band to pass the photodetector. From this information, we could determine slab’s speed as each band passed the photodetector, and thus determine its kinetic energy after it had fallen whatever distance h was required to bring that particular band past the photodetector. Given the object’s kinetic energy as a function of h, we could find

as a function of h. By attaching various weights to the bottom of the slab, we could vary the mass of the falling object and thus check how V

fi VV −

fi V− depends on mass.


THEORY As the plastic slab drops under the influence of the gravitational interaction between it and the earth, the total energy of the earth-slab system must be conserved:

ffEfiiEi VKKVKK ++=++ ,,

(1) where Ki and Kf are the initial and final kinetic energies of the slab, KE,i and KE,f are the initial and final kinetic energies of the earth, and Vi and Vf are the initial and final gravitational potential energies of the system. According to the argument presented in section C7.3 of the text, we can consider the earth to be essentially at rest throughout the experiment (since it is so much more massive than the slab) and thus KE,i and KE,f are negligible. If we drop the slab from rest, then Ki = 0 also, and equation (1) becomes simply 2

21

fffi mvKVV ==−

fi V

(2) So, to measure the system’s potential energy change V − after the slab has fallen a distance h, all that we have to is measure the slab’s mass m and its final speed vf. We can easily measure its mass using a balance. We can measure its final speed as follows. Imagine that we paint an opaque band across the width of the slab perpendicular to the direction that the slab falls. As the slab falls, imagine that this band interrupts a horizontal beam of light between a light source and a detector. We can use a computer to register the time Δt that the beam is interrupted. If the height of the band is Δd, then the speed of the slab as the band crosses the beam is approximately given by: (3) tdv ΔΔ≈ /

mghV fi

This most closely approximates the slab’s speed halfway through the time interval and thus roughly as the center of the band passes the light beam. This speed, therefore, can be used to determine the slab’s kinetic energy after it has fallen a distance h equal to the change in the slab’s position from its release point to the position where the band is centered on the photocell beam. Finally, note that the claim is that V =− , where m is the slab’s mass and g is the constant gravitational field strength. If this is true, then plugging this into equation (2) yields 2

212

21

ff vghmvmg =⇒=

fi V−

(4) Therefore, if V is proportional to m as claimed, the slab’s final speed after falling through a given distance h should be completely independent of its mass, which should be easy to check. Also, if this is true, the slab’s mass is not really relevant and we do not need to measure it. PROCEDURE In this experiment, our falling object was a clear plastic slab about 1.1 m tall and 8 cm wide, with five opaque bands 5.0 cm tall and vertically separated (center to center) by 20 cm. We could vary the mass of the slab by attaching one to four weights to the bottom of the slab. We


dropped this slab past a photogate consisting of a paired infrared light source and a photodetector mounted on a lab table so that the line connecting the source and detector was horizontal (perpendicular to the motion of the slab). The output of the photodetector was connected to a small box which in turn was connected to a Universal Lab Interface (ULI) circuit board (sold by Vernier Software, Inc.), which processed the signal for the photogate before passing it on to a Macintosh Centris 610 (serial number 3255967). A program called ULI Timer (also from Vernier Software) monitored the output from the ULI and displayed time intervals on the computer screen (see Figure 1 for a sketch of our experimental setup.) The program was configured to display the length of time that each of the five dark bands on the slab blocked the photogate beam as the slab fell past it.

floor

receptacle forcatching the slab

slab

photogate (attached tothe desk with a clamp)

table

connector box

ULI Macintosh

FRONT VIEW

20 cm

20 cm

20 cm

20 cm

20 cm

5 cm

mark

holes for attaching weights

CLOSE-UP and SIDE VIEW of the slab:

Figure 1: Sketch of the apparatus After our lab instructor gave a brief demonstration of the equipment, each of the seven lab teams in our particular afternoon session did a run. When our turn came, one of us (M-G.M.) held the center of the upper end of the slab between his thumb and forefinger and adjusted its vertical position until a certain mark inscribed on the slab edge was aligned precisely in the middle of the photogate as reported by IC. We waited until the slab had stopped swinging back and forth and was completely at rest. I.C. then triggered the ULI Timer program to start taking data and M-G.M. dropped the slab. The computer then automatically recorded and displayed the time _t that it took each of the five opaque bands to pass the photogate. We wrote these five numbers on the blackboard, filling in a table already started by other teams. Once all the data was taken, each pair of lab partners calculated the means and uncertainties of the mean (using the techniques in chapters 3 and 5 of the lab reference manual)


of the results for Δt for each of the five bands. We discussed the results as a class and decided that these results appeared to be uncertain by very roughly ± 0.002 s. While we were calculating the means and uncertainties, our pair of partners took turns doing a total of seven more runs, three runs with two weights attached to the slab and four runs with four weights attached to the slab. These runs were also recorded on the blackboard. Finally, each pair worked individually to analyze the data. As we did this, we passed the slab from pair to pair so that each could check that the opaque bands were 5.0 cm tall and separated from center to center by 20 cm. We did this using an ordinary meter stick turned on its edge so that the scale was right next to and perpendicular to the bands. We estimated that the height of the bands was equal to 5.0 cm to within ± 0.05 cm and that the distances between the centers of the band was 20.0 cm to within ± 0.1 cm (we actually measured these from bottom edge to bottom edge). ANALYSIS A table of the mean values of the time intervals appears below:1

Band Number Δt

(no added weight)2Δt

(one added weight)2Δt

(four added weights)2

1 0.0252 0.0250 0.0256 2 0.0179 0.0181 0.0179 3 0.0146 0.0149 0.0144 4 0.0126 0.0124 0.0126 5 0.0112 0.0113 0.0110

It is clear from these results that the speed of the slab is independent of its mass3, so (as we argued in the theory section) V must befi V− 4 directly proportional to the slab’s mass m. From the values of Δt for the slab with no added weight, we calculated 2

21

fv

mghV fi

for each of the heights5. Figure 2 shows a graph of these results. According to LinReg,6 the slope of the line is 9.81023 and the intercept is 0.012865.7 This proves8 that V =− (though our value of g is a bit high due to experimental error).9 CONCLUSION In this experiment, we showed that the final kinetic energy per unit mass 2

21

fv

mghV fi =−

of a plastic slab dropped from rest through a distance h is independent of the mass of the slab and seems to be proportional to h (within experimental uncertainty), with the constant of proportionality being equal to 9.81 ± 0.03 m/s2. These results are completely consistent with the assertion made in the text that V , where g = 9.8 m/s2.


0.20 0.40 0.60 0.80 1.000

0

2.0

4.0

6.0

8.0

10.0

Figure 2.10

COMMENTS ON THE ANALYSIS SECTION In general, the problem with this section is that it is far too short and thus does not provide us with some information that we need to understand the results and how the authors analyzed them. There are also several statements made that are not or cannot be supported by the data. Here are specific comments about the places in the analysis section where specific errors were flagged with numerical superscripts. (The simulated errors in this report reflect the most common types of errors that people make when writing analysis sections.)


(1) This table is nicely laid out, but a table of processed data like this should state the units of the quantities and state the uncertainties of the means as well as the means themselves. The writers should have also included a description of how many measurements went into calculating the mean and how the uncertainties were calculated. Also, if the uncertainties are really on the order of ±0.002 s (as stated in the procedure section), then the last digit in the tabulated data is totally meaningless. Are the uncertainties really more like ±0.0002 s? This would be consistent with the variation appearing in the table data. (2) One could include the units of the data in the column heading like this: “Δt in seconds”.

fi V−

(3) Is this really clear? Without knowing the uncertainties, the small variations in the values are impossible to interpret. (4) We really can’t say that V must be independent of m, only that our data are consistent with this interpretation. (5) This needs to be explained in much more depth. How did the writers calculate 2

21

fv from the data? What are the uncertainties of these speeds, and how were they calculated? One has to use something like the weakest link rule. How were the heights determined and their uncertainties estimated? It would have also helped greatly if the writers had listed the calculated values and uncertainties for 2

21

fv and h for each row of the table (or better yet, on a separate table). (6) A brief description of LinReg and what it does would be appropriate here. (7) The quantities quoted here have units and uncertainties: what are they? Also what is the significance or meaning of the slope and the intercept here? (8) An experiment can never prove that any theoretical assertion is true. The best that we can say here is that our results are consistent (or inconsistent) with this assertion. See the conclusion for better language. (9) How is g is related to something we have calculated in this lab? Also, the value is a bit high compared to what? Saying that the difference is due to “experimental error” says nothing. What kind of experimental error? Is the result within our uncertainties or not? If so, what does it mean to say that this is “a bit high”? (10) What are the experimental uncertainties of the data points? Are they not shown because they are too small to appear on the graph or did the writers simply forget about them them? What is being plotted against what here? (The axes should be labeled.) What is this graph about? (It should have a title!) What are the units of quantities displayed? What does the line mean? This graph is missing many of the features that a higher-level graph should have.

49

Section II: Dealing with Experimental Uncertainty

4. The Standard Deviation

51

Chapter 4: THE STANDARD DEVIATION “I may venture to say that his observations have stretched much farther…” -- Sense and Sensibility

4.1 INTRODUCTION

It is sometimes said that “physics is an exact science.” But physics is based on experiment, and experiment on measurement, and measurement is inexact by its very nature. Imperfect (or simply just finite) instruments, necessary approximations, and inevitable background noise make all measured values impossible to pin down exactly. Even so, clever experiments that coax even a very uncertain but crucial measurement from a sea of noise can lead to great advances in physics. A recent article in a science magazine trumpeted the fact that, with the help of data from the Hubble Space Telescope, the various scientific groups measuring the Hubble Constant (a quantity whose value determines the age and fate of the universe) were now getting results that agreed to within roughly 10% of each other. This was considered great news, partly because it was a dramatic improvement over the situation just a few years ago when measurements disagreed by more than a factor of two, but partly because the age of the universe computed from this quantity was finally settling down toward values that were consistent with other physical estimates of the age of the universe, soothing physicists’ worries that whole areas of physics might be poorly understood. The point is that, while the greatest precision is always desirable, even inexact measurements can lead to important scientific progress. But a crucial part of making good scientific use of inexact results is being able to quantify how inexact (the technical word is uncertain) a measurement is. We can draw scientific conclusions from an inexact value only if we know something about how sharp or fuzzy the value is. One of the major goals of this laboratory program is to teach you how to extract the maximum possible scientific meaning from uncertain measurement results. The next chapter of this reference manual will begin discussing the meaning of the concept of uncertainty in the context of measured quantities. The purpose of this chapter is to lay some mathematical foundations for that discussion by exploring ways that we might quantify the spread in a set of values that represent imperfect measurements of the same quantity.

4.2 POSSIBLE MEASURES OF SPREAD IN A DATA SET

Imagine that N teams of scientists measure a physical quantity, and each group ends up with a value that is somewhat different from the values obtained by other groups. This process gives us a data set consisting of N values, each of which represents an imperfect estimate of the “true value” of that quantity. As we will see in the next chapter, it is useful when discussing the physical meaning of these results to be able to quantify the “spread” in these measurement values. How might we quantitatively express the “spread” in a data set like this in a meaningful and useful way?


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The simplest way to express the spread in a data set is to use its range R, which in this context we define to be the difference between the highest and lowest values in the set. Clearly, the more “spread out” a set of data values is, the larger its range is likely to be. A potential problem with the range as a measure of the spread in a data set is that its value is determined entirely by the two most extreme values in the set, and all the other values are ignored! The extreme value may be the result of improbable flukes in the measurement process (or even outright errors), and are not representative of the whole by their very nature. One might imagine one data set where the values are mostly very closely clustered around some central value with only a handful of wildly different values at the extremes, and another data set having the same range whose values are fairly evenly spread out between the extremes. Would we really want to quantify the “spread” in these two data sets by the same number? A more meaningful way to quantify the spread is to compute the average deviation of the data points from the data set’s mean (= average) value. If we use the symbols to represent the first, the second, the third, etc. measurement values in our set of N values, and the symbol

Nxxxx K321 ,

x to stand for the mean of those values, then the average deviation is defined to be:

average deviation = ( )xxxxxxxxN N −++−+−+− K3211 (4.1)

You can see that this definition gives you exactly what the name implies, the average (summed over the entire data set) of the deviations of the measurements (which for a given measurement is the absolute value of how far that measurement is from the mean). Note that the absolute value is essential in this expression: since generally a given measurement is as likely to be above the mean as below it, if we did not take the absolute value of each difference, they would sum to zero. (In fact, the definition of the mean implies that if we remove the absolute value symbols in equation 4.1, the sum would be exactly zero, as you can check.) The average deviation, in contrast to the range, nicely takes each measurement value equally into account in its value, and thus is probably a better representation of the spread in a data set. It also has an easily understood meaning. However, it turns out that we often want to do some calculus with the quantity we use to characterize the spread in data. The absolute value presents a minor complication in doing calculus with the average deviation, because the derivative of |x| is undefined when x = 0. The standard deviation does not suffer from this problem. If we use the same symbols that we used in equation 4.1, we can write the standard deviation as follows:

standard deviation = ( ) ( ) ( ) ( )

1

223

22

21

−−++−+−+−

Nxxxxxxxx NK

(4.2)

If we ignore for a moment the fact that we are dividing by N–1 instead of N, the standard deviation is thus the square root of the average squared deviation. Dividing by N–1 instead of N is important for deep mathematical reasons beyond our level here, but note that for a data set consisting of a single measurement, the average deviation states that the “spread” in this data set is zero (suggesting that we know the measurement value perfectly), whereas the standard


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deviation states that the spread is undefined (0 ÷ 0), which more meaningfully suggests that we know nothing about the spread in a data set when that set only has one value. Equation 4.2, though it superficially looks more complicated than that for the average deviation, is actually simpler in a number of respects. Doing calculus with this expression is easier because squares and square roots are easier to deal with mathematically than the absolute value. Since many calculators are specially set up to calculate the standard deviation with a few button presses, it is generally much easier to calculate than the average deviation, and is really no more difficult to calculate even if one is reduced to using equations 4.1 and 4.2 directly. However, the main reason that the scientific community typically uses the standard deviation (rather than the range or the average deviation) to characterize the spread in the values of a data set is that it has an important property that the range and average deviation do not have. Imagine that we model each of the N measurements in our data set as being what happens when we add tiny random errors from fairly large number of sources to the measured quantity’s “true value” (whatever that might be). This “random error” measurement model is only a model of the measurement process (and a pretty simplified one at that), but it does seem to be useful and reasonably accurate in many cases. When this model does adequately describe a measurement process, then we find that if we plot the probability of getting a certain measurement value versus that value, we get the famous “bell-shaped curve” (which indicates that measurement values close to the quantity’s “true value” are common and extreme values are rare). A wide variety of measurement processes tend to yield data sets that are distributed this way. So here is the payoff. If the “random error” measurement model is a good model for the measurement process giving rise to the given data set, it can be shown mathematically that the value one gets by computing the standard deviation for this data set is fairly independent of N. This means that the standard deviation reflects the intrinsic scatter of the measurement process itself rather than the number of measurements. In contrast, the range of a data set typically increases as N increases (because the more measurements we take, the more likely we are, by chance, to encounter some really extreme measurement results), while the average deviation tends (more subtly and for more subtle reasons) to decrease with increasing N. It is this special relationship between the standard deviation and the most common and useful mathematical model for the measurement process that makes the standard deviation the most widely accepted measure of the spread of the values in a data set, and that’s why calculators are set up to find the standard deviation rather than the range or the average deviation.

4.3 CALCULATING THE STANDARD DEVIATION

One can fairly easily calculate the standard deviation of a list of measurement values directly from equation 4.2 by going through the following steps:

1. Write the values in a vertical column. 2. Sum the values and divide by the number of measurements N to find the mean. 3. Subtract this mean from each value and square the result. Write each “squared deviation” to the right of the corresponding measurement value. 4. Sum the squared deviations in this new column and divide by N–1. 5. Take the square root of the result to get the standard deviation.


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Almost any scientific calculator (which we recommend you have for this course) will provide an easier way for you calculate the standard deviation. In general, the process looks something like this:

1. Initialize the calculation somehow (varies from calculator to calculator). 2. Type in the first number. 3. Press a key that is usually marked Σ+ or M+. 4. Repeat steps 2 and 3 for each of the measurement values (after you press Σ+ each time, the

calculator typically displays the number of values that you have entered so far). 5. When all values have been entered, calculate the standard deviation by pressing one of the

following: s, sx, σ, 1−nσ (if the choice is between that and nσ ), xσ or 1, −nxσ (instead of

yσ or 1, −nyσ ). The choice, unhappily, varies from calculator to calculator. For example, on an HP-32S calculator, one initializes the calculation using the CLEAR button and selecting “clear all” from the choices presented. To get the final result, push the STAT button and select “s” and then “sx” from the choices presented. On a TI-36X calculator, initialize the calculation by selecting the STAT 1 function (to indicate we are going to do statistics of one variable), and get the final result by pressing the 1, −nxσ button. On a TI-82 or TI-83 calculator – apparently increasingly the calculator of choice if the selection of calculators turned into Connie’s Lost and Found is any indication – select the STAT key, use the arrow keys to get to EDIT if you aren’t there already and enter your data in one of the data lists (L1, L2, L3 …). Then press the STAT key again, use the right-pointing arrow key to choose CALC, the down-arrow key (if necessary) to get to 1-Var Stats, and then press ENTER to get the mean and standard deviation, as well as some sums used to calculate their values. You should look in the instruction booklet that came with your calculator to determine how to calculate the standard deviation on your calculator. (If you don’t have your booklet, do some experiments, ask a friend, or check your calculator manufacturer’s web site. If you come a bit early to lab, your lab instructor or a lab assistant may be able to help you.) To test that you are actually calculating the standard deviation as defined by equation 4.2, note that the standard deviation of the data set 2, 4, 6 should be 2.

EXERCISES

Exercise 4.1 Compute the standard deviation of the following data set using the direct method (the first method outlined in section 4.3). Show your work in the space below. (Be sure to keep track of units!) 0.56 s 0.52 s 0.59 s 0.48 s 0.51 s


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Exercise 4.2 Repeat Exercise 4.1 using whatever short-cut approach works on your calculator. This will serve as a check on both your work in Exercise 4.1 and your ability to use your calculator to find standard deviations.


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5. Experimental Uncertainty 57

Chapter 5: EXPERIMENTAL UNCERTAINTY “ ‘I am no matchmaker, as you well know,’ said Lady Russell, ‘being much too

aware of the uncertainty of all human events and calculations.’” --- Persuasion

5.1 UNCERTAINTY AS A “95% CONFIDENCE RANGE”

We generally assume in physics that any quantity we measure has a “true” value, which is the result that we would get if we had a perfect measuring apparatus. Fifteen minutes in any laboratory, regardless of the sophistication of the equipment, will rapidly disabuse you of the notion that any measurement apparatus is perfect. Real measurement devices suffer from a variety of imperfections that limit our knowledge of the “true” value of any measurement. Devices may be poorly made, out of adjustment, subject to noise or other random effects, or hard to read accurately, and all devices read to only a finite number of digits. These problems mean that the exact value of any measured quantity will always be uncertain. Uncertainty is therefore an unavoidable part of the measurement process. We will (of course) always seek to reduce measurement uncertainty whenever possible, but ultimately, there will remain some basic uncertainty that cannot be removed. At this point, our task is to estimate thoughtfully the size of the uncertainty and clearly communicate the result. How can one quantify uncertainty? In this course, we will define a value’s uncertainty in terms of the range within which we are 95% confident that the “true value” would be found if we could measure it perfectly. This means that we expect that there is only one chance in 20 that the true value does not lie within the specified range. This range is called the 95% confidence interval. The conventional way of specifying this range is to state the measurement value plus or minus a certain number. For example, we might say that the length of an object is 25.2 cm ± 0.2 cm: the measured value in this case is 25.2 cm, and the uncertainty U in this value is given as to be ±0.2 cm. The uncertainty thus has a magnitude equal to the difference between the measured value and either extreme edge of the uncertainty range. This statement means that we are 95% confident that the measurement’s true value lies within the range 25.0 cm to 25.4 cm. It may be helpful to point out that there is nothing magic about our choice of the 95% confidence interval for stating the uncertainty. We could just as properly decided to define the uncertainty as the 50% confidence interval (half the measurements lie within the confidence interval) or the 0.1% confidence interval (only one measurement in 1000 lies outside the confidence interval). If you think about the proposed uncertainties just given, though, we think you will agree that the 50% confidence interval excludes too many reasonably likely measurements, while you would need to make 1000 measurements to find the 0.1% confidence interval in the first place, at least using the method described in Section 5.5. We will also see in Section 5.6 that the 95% confidence interval is particularly easy to estimate from the standard deviation, another reason for choosing it for reporting the uncertainty of a measurement.


Now (as you may have already noticed), this definition of uncertainty is rather fuzzy: one person may be more confident about a value’s precision than the next. In fact, the definition seems almost devoid of objective meaning except as a description of the experimenter’s frame of mind. We will see that the definition is not as subjective as it seems, particularly in certain cases to be discussed shortly, where it is possible to make an educated and generally agreed-upon estimate of the uncertainty. But the fact remains that “uncertainty” is itself an uncertain concept, and uncertainties should only be taken to be rough estimates, good to one or at most two significant digits. In spite of this problem, knowing the uncertainty of a measured value is essential if one is to correctly interpret the meaning of a measured value. For example, imagine that you measure the period of a (very long) pendulum to be 12.3 sec. Imagine that someone’s theory predicts that the period should be 11.89275 sec. Is your result consistent with that theory or not? The answer to this question depends entirely on the uncertainty of your result. If your result has an uncertainty of ±0.5 s, then the true value of your measured duration could quite easily be the same as the theoretical value. On the other hand, if the uncertainty in your result is ±0.1 s, then it is not very likely (less than one chance in 20) that the true value behind your measurement is the same as the predicted value, meaning that the theory is probably wrong. What a measurement means, therefore, can depend crucially on its uncertainty!

5.2 SYSTEMATIC ERRORS

Why aren’t measurements perfect? The causes of measurement errors can be divided into three broad classes: systematic problems, limited precision, and random effects. The focus of this chapter will be on the last of these, but the first two causes need to be discussed briefly. Systematic errors occur when a piece of equipment is improperly constructed, calibrated, or used, or when some physical process is going on in the experiment that you haven’t thought of in your experimental design. As a somewhat contrived example of problematic equipment, suppose that you measured lengths with a meter stick that you failed to notice had been cut off at the 5 cm mark. This would mean that all of your measured values would be 5 cm too long. Systematic errors resulting from equipment problems are relatively easy to identify once you have some reason to suspect they exist; you compare your equipment to two other, similar, pieces of equipment and see if they all give the same result for the measurement. If your device doesn’t agree with the other two, probably your device has a problem. As an (also somewhat contrived) example of a systematic error due to unexpected physics, suppose you were trying to measure the acceleration of gravity by timing the motion of a falling object. But – and this is the contrived part, at least after the invention of Newtonian physics – suppose you didn’t know about air resistance, so you used for your object a wadded-up ball of paper. You would find a much smaller value for g than the accepted value! You don’t normally include systematic errors in the uncertainty of a measurement; if you know that a systematic problem exists, you should fix the problem. In the meter-stick example above, you would use a complete meter stick, or add 5 cm to all your measurements. Systematic errors arising from unanticipated physics are harder to find, although they’re the source of many Nobel Prizes in physics. Unfortunately, no well-defined procedures exist for finding systematic


errors: the best that you can do is to be clever in anticipating problems and alert to trends in your data that suggest their presence. You might come to suspect the short meter stick, for example, if you noticed that your data would agree with theoretical predictions if all of your length measurements were about 5 cm shorter. In the second example, you would come to suspect the presence of air resistance if you let the ball fall through several different heights and observed that your calculated acceleration got systematically smaller with increasing height. To notice this, though, you do have to think of carrying out those measurements at different heights. In some cases, it is appropriate to estimate the magnitude of possible systematic errors and include them in the uncertainty of a measurement result. For example, it is possible to read most automobile speedometers to a precision of about 1 mph. But it is well known (to some people, at least) that variations in the manufacture and calibration of speedometers mean that the reading of a typical speedometer may be off by as much as 5%. Therefore, the uncertainty of a speedometer reading of about 60 mph would have to be taken to be ± 5% of 60 mph, or about ± 3 mph. The uncertainty in this case is called a calibration uncertainty. We will deal with calibration uncertainties only rarely in this course.

5.3 LIMITED PRECISION

No measurement device can read a value to infinite precision. Dials and linear scales, such as meter sticks, thermometers, gauges, speedometers, and the like, can at best be read to one tenth of the smallest division on the scale. For example, the smallest divisions on a typical metric ruler are 1 mm apart: the minimum uncertainty for any measurement made with such a ruler is therefore about ± 0.1 mm. This statement is not an arbitrary definition or convention: rather, it is a rule based on experience. If you try using a ruler to make as precise a measurement as you can, you should be able to see that it is really quite difficult to do better than the stated limit. For measuring devices having a digital readout, the minimum uncertainty is ± 1 in the last digit. For example, imagine that you measure a time interval with a stopwatch, and find the result to be 2.02 s. The measurement’s “true value” in this case could be anywhere from 2.010...01 s to 2.0299...99 s. We cannot narrow this range without knowing details about the design of the stopwatch: does it round up to 2.02 s just after the true elapsed time exceeds 2.01 s, or does it round to the nearest hundredth of a second, or does it not register 2.02 s until at least 2.02 s have passed, or what? So in this case, we must take the uncertainty to be at least ± 0.01 s. In both of the cases described above, these rules are meant represent the minimum possible uncertainties for a measured value. Other effects might conspire to make measurements more uncertain than the limits given (as we shall see), but there is nothing that one can do to make the uncertainties smaller, short of buying a new device with a finer scale or more digits.

5.4 RANDOM EFFECTS

This chapter is mainly focused on the analysis of random effects. It is commonly the case that repeated measurements of the same quantity do not yield the same values, but rather a spread of values. For example, you might determine the speed of sound by standing at a fairly large distance (like the width of Marston Quad) away from an object that simultaneously emits a


flash of light and a loud sound when someone flips a switch. After measuring the distance to the source, you would measure the time between seeing the flash and hearing the sound. If you measure this interval five times, you are almost certain to get five different results (for example, 0.54 s, 0.52 s, 0.55 s, 0.49 s, and 0.53 s). Why are these results different? In this case, the problem is that it is difficult for you to start and stop the stopwatch at exactly the right instant: no matter how hard you try to be exact, sometimes you will press the stopwatch button a bit too early and sometimes a bit too late. These unavoidable and essentially random measurement errors cause the results of successive measurements of the same quantity to vary. Random perturbing effects, which are sometimes human and sometimes physical in origin, are a feature of almost all measurement processes. Sometimes a measuring device is too crude to register such effects: for example, a stopwatch accurate to only one decimal place might read 0.6 s for each of the measurements in the case described above. But laboratory instruments are often chosen to be just sufficiently sensitive to register random effects. Granted, you’d like as precise an instrument as possible, but there is no point in buying an instrument much more precise than the limit imposed by unavoidable random effects. For example, a hand-held timer like a stopwatch that reads to a hundredth of a second is better than one that registers to only a tenth, because (as we’ll see in a couple of weeks) it’s possible to pound down your random experimental uncertainty down to a few hundredths of a second by making enough measurements. But there would be no scientific point, although there might be a marketing advantage, in making a stopwatch that reads to a thousandth of a second, because the added precision of the watch would be swamped by the scatter in the measurements resulting from its operation by a human being. The point is that random effects will be an important factor of many of the measurements that you will encounter in any scientific experiment. Now, it should be clear that such effects increase the uncertainty in a measurement. In the stopwatch case, for example, that different trials lead to results differing by several hundredths of a second implies that the uncertainty in any given measurement value is larger than the basic ± 0.01 s uncertainty imposed by the digital readout.

5.5 THE DISTRIBUTION OF VALUES DUE TO RANDOM EFFECTS

The first step towards describing the magnitude of the uncertainty due to random effects is to understand more precisely what these effects do to a set of measurement values. Consider, for example, a simple experiment where 25 different people measure the length of a soft-drink can with a ruler. Assume that the “true” length of the can is 8.51293... cm. None of the 25 people will measure the object to have exactly this length, of course, because people will view the ruler and can from slightly different angles, make different judgments about the exact reading, and so on.


Nevertheless, we would expect the measurements to cluster around the value 8.51 cm, with most of them agreeing with that result within a few hundredths of a centimeter or so (that is, tenths of a millimeter). Results different from the true value by roughly 0.1 cm will be less common, but not really rare. Results that differ much more dramatically from the true value will be rarer, and the more that a measurement value differs from the true value, the less likely it is to occur. If one were to plot the number of measurements obtained versus the measurement value, we might obtain a graph looking something like the graph shown in Figure 5.1. Note that the range of measurement values has been divided into “bins”, each 0.02 cm wide, so, for example, measurement values of 8.50 cm and 8.51 cm would both be counted as being in the central bin. (The purpose of grouping values into bins like this is to show more clearly the characteristic shape of the distribution: a graph where each bin was only 0.01 cm wide would be flatter and less revealing.) A graph of this type is called a histogram.

F req

uenc

y of

occ

urre

n ce

Histogram of length measurements, N = 25

1

2

3

4

5

6

7

8.42

8.4 4

8 .46

8.48

8.5 0

8.52

8.54

8.56

8.5 8

8.60

Length of can (cm)Fig. 5.1: Distribution of 25 measurements of the length of a can

This graph roughly sketches what is often called a “bell-shaped curve.” If we were to plot 100 or 1000 measurements on this graph instead of just 25, the curve would be more smooth, symmetrical, and bell-like. Measurement values subject to random effects are almost always distributed in such a pattern. In fact, it is possible to show that a bell-shaped distribution of values having specific and well-defined characteristics is the mathematical consequence of perturbing effects that are truly random in nature and continuously variable in size. We call the specific bell-shaped distribution of values caused by such random influences a normal or gaussian distribution. Simply by looking at this graph, we can make a rough estimate of the uncertainty of any individual measurement value. The definition of “uncertainty” that we have adopted implies that the uncertainty range should enclose the true value (in this case 8.5129... cm) about 19 out of 20 times. In the case shown above, a range of ±0.06 cm attached to any of the measurements would include the true value, except for the one case in the rightmost bin. One out of 25 is roughly equal to one out of 20, so ±0.06 cm would a reasonably good estimate of the uncertainty of a given measurement in this (hypothetical) case. As mentioned in Section 5.1, we could have chosen to define experimental uncertainty with a 50% confidence interval or a 0.1% confidence interval. It’s pretty clear that we don’t have enough measurements to find a 0.1% confidence interval! Finding the 50% confidence interval for the data above should be easy: we just have to throw out the 12 or so measurements that lie farthest from the mean. Doing this for the given set of measurements does raise a


somewhat troubling point, though. The 12 measurements to eliminate that seem most obvious are probably the eight measurements in the bins labeled 8.44, 8.46, and 8.48, and the four measurements in the bins labeled 8.54, 8.56, and 8.58. But this tosses out measurements rather unsymmetrically, which doesn’t seem like a good idea. This problem could be caused by our choice of bin size, but smaller bins would spread out the histogram and make the distribution less clear. So the 95% confidence interval seems to be a good compromise between keeping enough data points for a symmetric distribution of data, and having to make huge numbers of measurements.

5.6 THE MEANING OF THE STANDARD DEVIATION

In chapter 4 of this manual, we defined the standard deviation s of a set of N measurements with mean Nxxxx K,,, 321 x to be given by the expression

1))( 2222

1 −++−−=

xxxxxs NK (()() 32

−+−+N

xxx (5.1)

Let the symbol xi stand for an arbitrary “ith” measurement in our set. In chapter 4, we said that the standard deviation was a measure of how much any one data point was likely to differ from the mean of the set of measurements, so it’s a good candidate for describing the experimental uncertainty of a set of measurements. In this chapter, though, we’re introducing a different definition, one that looks at all the measurements and characterizes the uncertainty by the range of all but possibly a few of the measurements. You probably won’t be surprised to find out that the two ways of estimating the uncertainty are related to each other. Recall that we have defined the uncertainty U of any measurement xi to be the value such that we are 95% confident that the “true value” of the measured quantity lies within the range xi ± U. If we have happened to take a large number of measurements of this quantity, our otherwise somewhat subjective “95% confidence” can be given a directly quantitative meaning: the measurement’s true value (which should correspond to the value at the peak of the bell curve) should lie within the range xi ± U for 95% of the measurements xi. Given a set of measurement values, then, we can use this criterion to determine the value of U. The only problem is that we need hundreds (if not thousands) of measurements to accurately estimate U this way; to accurately determine U, N must be large enough that the number of measurements in the 5% that exclude the true value is more than just a handful. Fortunately, mathematicians have shown that it is possible to accurately estimate the value of U that would have this property for a very large set of measurements from a much smaller set. The uncertainty U of any given single measurement can be estimated using the standard deviation of a small set of similar measurements as follows: U ≈ ts (uncertainty of a single measurement) (5.2) where t is the so-called Student t-factor, a number that depends somewhat on N, the number of measurements in the set used to calculate s. A table of t-values as a function of N is given below, and is also reproduced in the inside front cover of this manual.


TABLE OF STUDENT t-VALUES

N t-value N t-value 2 12.7 10 2.26 3 4.3 12 2.2 4 3.2 15 2.15 5 2.8 20 2.09 6 2.6 30 2.05 7 2.5 50 2.01 8 2.4 100 1.98 9 2.3 ∞ 1.97

And this is the other reason to define the experimental uncertainty as the 95% confidence interval, and not the 50% confidence interval or the 0.1% confidence interval. The t-value for a reasonably but not onerously large number of measurements (10 or more) is pretty close to 2. Two is easy to remember, and so is 95%. (Two is almost too easy to remember: see the next paragraph.) The appropriate factor for the 50% confidence interval would be about 0.675; that for the 0.1% confidence interval would be 3.29. So once again, the 95% confidence interval is chosen for utility, not because there’s anything magic about 95%. While uncertainties are generally accurate only to one significant digit, this table states values to two or three significant digits to show clearly the difference between adjacent values. Note that for N > 30, the t-value is within a few percent of being 2.0: for this reason, some books will tell you that the 95% confidence range for a given measurement xi is simply xi ± 2s. However, this is not a good estimate of that range for the small values of N that we will commonly encounter. In using the table, you should also keep in mind that it is really only valid for measurements that are randomly distributed. Specifically, if the uncertainty of your measurement is limited by the precision of the apparatus rather than random effects, you should not use equation 5.2: you should instead use one of the strategies outlined in section 5.3. Please note that equation 5.2 estimates the uncertainty of any given single measurement in the set. As we’ll see later, though, if we have already bothered to take a set of measurements required to determine s, we might as well compute the mean of the set, which is a better estimate of the measurement’s true value (that is, it has a smaller uncertainty) than any arbitrarily chosen single measurement is. Note in addition that the table is telling you indirectly something about the number of measurements you need to get a good estimate of the uncertainty. In particular, two measurements are not enough! Three measurements are a bare minimum, and five is a good compromise between getting a good estimate of the uncertainty and spending lots of time on a single measurement.


EXERCISES

Exercise 5.1 A standard household thermometer has one mark for every two °F. What is the minimum uncertainty that you should assign to the temperature that you read from such a thermometer? What do you think is the best uncertainty to assign to this reading, do you think larger than this? Explain your reasoning. (Be aware that there is no absolute right answer to this last question.)

Exercise 5.2 According to your digital bedside clock, it took you exactly 12 minutes to dress for class some morning. What uncertainty should you assign to this result? Explain your reasoning.

Exercise 5.3 Imagine that you are one of ten different people who measure the time of flight of a thrown baseball. Assume that these ten measured speeds are as listed below and to the left. On the grid below and to the right, plot a histogram of this data. (Choose a “bin” size that displays this data as a pseudo bell-curve rather than scattered data or only one or two columns.) 2.53 s 2.58 s 2.67 s 2.63 s 2.59 s 2.60 s 2.62 s 2.56 s 2.66 s 2.61 s

Exercise 5.4 Compute the standard deviation of the data from the previous problem, and estimate the uncertainty of your particular measurement.

6. Uncertainty of the Mean 65

Chapter 6: THE UNCERTAINTY OF THE MEAN “ ‘Now, for instance, it was reckoned a remarkable thing that at the last party in

my rooms, that upon an average we cleared about five pints a head.’” --- Northanger Abbey

6.1 USING A SET’S MEAN AS A “BEST GUESS”

Imagine that we are doing an experiment, and we want to make the best possible guess as to the true value of a quantity whose measured values are subject to some kind of random error, either due to human estimation error (as in the case of the stopwatch measurements in the speed of sound lab) or random perturbing physical effects (noise), or both. In order to get a sense of the uncertainty of the measurement, you have already taken N individual measurements of the quantity. If the effects that perturb the measured value away from its “true value” are truly random, then they are equally likely to cause any given measured value to be higher than the true value as they are to cause it to be lower than the true value. This means that if you take a set of N measurements of the same quantity and those measurements are subject to random effects, it is likely that the set will contain some values that are too high and some that are too low compared to the true value. If you think of each measurement value in the set as being the sum of the measurement’s true value and some random error, what we are saying is that the error of a given measurement is as likely to be positive as negative if it is truly random. One way to get a good estimate of the true value of a quantity is therefore to calculate the mean (average) of the measurement values in the set. When you calculate the mean of a set of measurements, you sum the measurements and then divide by N, the number of measurements in the set. If we imagine each measurement to be equal to the true value plus some random error, then the sum of N measurements will be equal to N times the true value plus the sum of all the random errors. Since the random errors are as likely to be negative as positive, they will tend to cancel each other out when summed, meaning that the sum of N measurements is likely to be very close to N times the true value. Dividing the result by N to get the mean thus yields a number that is likely to be close to the true value. The mean thus represents a better guess as to a quantity’s true value than any of the individual values in set of measurements. The assumption that the errors cancel each other out gets better and better as the number of measurements in the set increases. For example, if there are only three measurements in a set, there is one chance in eight that all three measurements are too high, implying that the mean of these measurements will not be a good estimate of the quantity’s true value. As the number of measurements in a set increases, though, the probability that there is an approximately even mix of measured values in the set that are too high and too low becomes larger and larger.


6.2 THE UNCERTAINTY OF THE MEAN

The mean of any finite set of measurements is not going to be exactly equal to the quantity’s true value: the random errors are not likely to perfectly cancel (especially if the number of measurements is relatively small. Saying that the mean is a “better guess” than any single given measurement is essentially saying that the uncertainty range of the mean is smaller than the uncertainty range of any given single measurement. But how much smaller is it? How can we quantify the uncertainty of the mean of the set of measurements? We define the uncertainty of the mean Um of a set of N measurements to be that value such that we are 95% confident that ± Um encloses the “true value” of the measurement (where x is the mean of the set). If were to take enough data so that we have many sets of N measurements, (as we did in the Inertial Mass lab), then this definition means operationally that if Um has the correct value, the range ± Um should enclose the true value for 95% of the sets’ means. According to the mathematicians, we can estimate the uncertainty Um of the mean of a normally distributed set of measurement using just one set of N measurements as follows:

N

tsm ≈U (6.1)

where s is the standard deviation of the set and t is the Student t-factor. In the Uncertainty of the Mean lab, you should have verified that this expression was at least approximately consistent with the definition of uncertainty just described. Note that the estimate of the uncertainty of the mean given by equation 6.1 has two properties that we know (or at least intuitively expect) to be true. First, it implies that the uncertainty of the mean is indeed smaller than the uncertainty of a single measurement (by the factor N/1 ), as we would expect from the argument given in the previous section. Second, the more measurements we take, the smaller Um becomes, implying that the mean becomes a better estimate of the measurement’s true value as the number of measurements it embraces increases (as we should expect).

6.3 ESTIMATING UNCERTAINTIES FROM THE RANGE

Calculating uncertainties of the mean through the standard deviation can quickly become very tedious, particularly when an experiment involves a number of sets of measurements. It would be nice to have a method that allows us to determine the uncertainty of the mean of a set of measurements at a glance. Here is such a method. The first step is to compute the range R of your set of data, which is defined to be the largest value minus the smallest value in your set. (You can often compute R to one or two significant digits in your head.) Then look in Table 6.1 for the number N of the measurements that you took. An estimate of the value of the uncertainty (expressed as a multiple of R) appears in the next column, and again, you can probably do the multiplication (to one or two significant digits) in your head. For example, if you have 10 measurements of a period of


time ranging from 2.05 s to 2.22 s, the range is 0.17 s and the uncertainty of the mean of these measurements is (according to the table) Um = 0.23R = 0.23(0.17 s) = 0.04 s. Easy! The downside is that the value one reads from this table is only an estimate of the value that we would get by calculating

Ntsm /≈U , and not a particularly great one for small N, because it turns out that when N is small, the range of a set of measurements fluctuates from one set to another even more than the calculated value of s does. Still, if we are after a quick and dirty estimate that is still good to about one significant digit, this method is hard to beat. We will call this method the range method of determining the uncertainty of the mean and the method of directly computing the uncertainty of the mean using Ntsm /≈

U m

TABLE 6.1 ESTIMATING FROM THE RANGE

N U m U m N 2 6.40R 10 0.23R 3 1.30R 15 0.15R 4 0.72R 20 0.12R 5 0.51R 25 0.09R 6 0.40R 30 0.07R 7 0.33R 40 0.05R 8 0.29R 50 0.04R 9 0.26R

U the formal method. You may freely use the range method for quick estimates and for checking formal calculations, but you should be sure to use the formal method when doing the final calculations that you will report in an analysis summary or full report.

6.4 REPEATABLE AND UNREPEATABLE MEASUREMENTS

The discussion in the previous sections suggests that if you want to determine a measured value as precisely as possible, it is better to take a set of measurements of that quantity and compute the mean than it is to take just one measurement. IMPORTANT: In this lab program, when a measurement is repeatable, you should always (automatically!) take a set of four to twenty measurements of the quantity in question, compute the mean of the set to get a best guess for the measurement value, and then compute the uncertainty of that mean. But when is a measurement “repeatable”? There are three different kinds of measurements that are not repeatable. The first are intrinsically unrepeatable measurements of one-time events. For example, imagine that you are timing the duration of a foot-race with a single stopwatch. When the first runner crosses the finish line, you stop the watch and look at the value registered. There is no way to repeat this measurement of this particular race duration: the value that you see on your stopwatch is the only measurement that you have and could ever have of this quantity. In the second category are subjectively influenced measurements where a subjective human judgment is required to make the measurement and such a judgment can be influenced by knowledge of past results or what the answer “should” be. In such cases, while we can technically repeat the measurement process, our knowledge of previous results, or our expectation of a certain outcome influences our judgment to such an extent that we cannot be sure that those measurements will remain randomly distributed. For example, imagine measuring


the size of an object using a ruler. The first measurement that you take of this quantity may be subject to random effects (for example, the way that you line up the ruler on the object, the orientation of your eye with respect to the ruler, and so on). The second and subsequent measurements that you take of this quantity will be strongly influenced by your knowledge of the first value that you obtained. It is very difficult to be completely unbiased about such measurements after the first, particularly when it comes to reading linear scales; after reading it once, it is hard to avoid looking at the scale the same way subsequently. This is not generally a problem with digital readouts (where you do not make a subjective judgment about the measurement) or when the random influences overwhelm the influences of subjectivity. For example, if you are using a gauge to read the pressure of steam in a boiler and clearly the reading is fluctuating from moment to moment by substantially more than one division on the scale, your memory of a previous measurement is not likely to have much influence on subsequent measurements. The final kind of unrepeatable measurement is an unvarying measurement whose value always comes out to be exactly the same if we do try to repeat it. This is a particular problem with digital readouts, and is an indication that any random errors that might influence the measurement are smaller than the precision of the instrument. In such a case, equation 6.1 cannot be used because it would clearly yield a zero uncertainty: instead, we have to estimate the uncertainty from our knowledge about the instrument’s precision. Here is a general rule for distinguishing a repeatable from unrepeatable measurements: A measurement is repeatable if (1) you are sure that you are really measuring the same quantity each time you repeat the measurement, (2) your memory of the earlier measurement will not affect your later measurement, and (3) your measurement device is sufficiently precise that you will see variations in its readout when you repeat a measurement. In general, if you are making a type of measurement that you have never made before, you should repeat the first measurement at least three times: once to estimate the measured value, a second time to see if the measurement varies, and a third time as a check if the first two came out the same. (If the first two didn’t come out the same, you still want the third measurement because it will improve your statistics.) As you build up your repertoire of measurements, you will learn what sorts of measurements tend to be repeatable and what how large typical uncertainties are for non-repeatable measurements. For example, you have presumably made enough measurements with millimeter-scale rulers to know that you probably will be biased by your first measurement, but that the uncertainty of such measurements is between 0.1 mm and 0.5 mm, depending on your skill at interpolating between the millimeter marks.

6.5 THE BOTTOM LINE

In this laboratory program, you should assign an uncertainty to every measured quantity as a matter of course. If the measurement of that quantity is repeatable, take a set of measurements (at least three, usually five) and compute the mean and the uncertainty of the mean. Use the mean as the measurement value in any subsequent calculations. If the measurement is not repeatable, estimate the uncertainty of your single measurement based either on previous experience, your common sense, or on these two principles: (1) the


uncertainty of a linear scale or dial is greater than ± one tenth of the smallest division and (2) the uncertainty of a digital readout is ± 1 in the final digit. For example, you may remember that the uncertainty of stopwatch measurements in the Inertial Mass experiment was a certain value: this would be a good estimate of the uncertainty of an unrepeatable stopwatch measurement. Remember that all uncertainty estimates (calculated or otherwise) are estimates good only to about one significant figure. You will probably typically report calculated uncertainties to two significant figures, to avoid roundoff error. Also check that the reported precision of a quantity’s uncertainty is consistent with the reported precision of the quantity itself.

EXERCISES

Exercise 6.1 Which of the following measurements are repeatable? Explain your response.

(a) Measuring the length of a sheet of paper with a ruler. (b) Measuring the width of a building with a meter stick. (c) Measuring the outside temperature with a thermometer. (d) Measuring how long it takes a rock to drop 5 ft using a watch with a sweep second hand. (e) Measuring the duration of a speech with a single watch.

Exercise 6.2 Consider the set of time measurements given in exercise 5.3 in chapter 5 of this lab reference guide. What is the mean of these measurements, and what is its uncertainty? Calculate the latter using both the formal method and the range method. Are the results about the same?

7. Propagation of Uncertainty 71

Chapter 7: PROPAGATION OF UNCERTAINTY “She drew up plans of economy, she made exact calculations…” --- Persuasion

7.1 INTRODUCTION In many kinds of physics experiments, one would like to know the uncertainty in a quantity (call it f) that is calculated from directly measured and uncertain quantities a, b, c,… ; that is, f is a function f(a,b,c,...) of the measured quantities a, b, c, ... . For example, this problem would arise in an experiment where we want to determine the uncertainty in an object’s speed if that speed is calculated from uncertain time and distance measurements. The general problem of determining a calculated quantity’s uncertainty is called the problem of propagation of uncertainties, expressing the idea that uncertainties in measured quantities beget uncertainties in quantities calculated from them. The goal of this chapter is to explore means for intelligently addressing this problem.

7.2 SOME NOTATION AND TERMINOLOGY

We will use the symbol U[f] to refer to the experimental uncertainty in any quantity f, whether that uncertainty has been directly measured or calculated from the uncertainties in other measured quantities. If the quantity f depends on measured quantities a, b, …, then its uncertainty U[f] should be related to the uncertainties in a, b,…, that is, on U[a], U[b], and so on, in some way that we should be able to calculate knowing how f depends on these variables. It turns out that the most useful quantity to know when dealing with the problem of propagation of uncertainty is a variable’s fractional uncertainty, which is defined to be ratio of the variable’s uncertainty to its measured or best guess value:

fractional uncertainty of f ffU ][[ =fQ ] (7.1)

(The symbol Q is meant to make you think “quotient.”). This quantity is very closely related to the concept of percent uncertainty: to get the percent uncertainty from the fractional uncertainty, simply multiply by 100. These ideas are so closely and simply related that we will often treat “fractional uncertainty” and “percent uncertainty” as if they were the same. As an example, say that the measured value of f is (5.96 ± 0.60) cm. The fractional uncertainty of f is then Q[f] = 0.6 cm/5.96 cm = 0.10, and its percent uncertainty is 10%; that is, its uncertainty is equal to 10% of its best-guess value. Note that whatever units a quantity f might have, U[f] has the same units, so the ratio Q[f] of these quantities (and thus the fractional or percent uncertainty) will always be a unitless number. This observation is also a reminder that a quantity’s uncertainty U[f] and its fractional uncertainty Q[f], while related, are not the same thing: if they were, they would have the same units.


7.3 A GENERAL APPROACH TO PROPAGATION OF UNCERTAINTIES

),,,( Kcbaf

a

Think of the function as a machine that has a handle (like a control stick) corresponding to each of its input variables a, b, c, …, and a big dial with a pointer that indicates the output value f. Each of the input variables affects the final value shown on the dial, so adjusting the positions of the handles individually or in combination will change the value shown on the dial. Now if the input value a has an uncertainty U[a], then we can wiggle the handle corresponding to the variable a back and forth from its most probable value a by a positive or negative amount δ in the range ][aUa ≤δ

af

and still be consistent with the experimental data. This wiggling will cause the value of f indicated by the dial to wiggle back and forth from its central value by a certain amount as well. Let us define δ to be the (presumably small) change in the value of f from its central value when a is moved from its central value by ][max aUa = +δ , corresponding to the upper extreme limit of a’s uncertainty range, while the other handles are held constant. Similarly, let bfδ be the change in f when b is moved from its central value by an amount ][max bUb +=δ while the other variables are held constant, and so on. Now, what is the uncertainty in f when all of its variables are free to wiggle around within their uncertainty ranges simultaneously? The maximum distance f could be from its central value is K+++ cba fff δδ=f δδ max

( ) ( ) ( )

if all of the input values happen to be simultaneously at whichever edge of their uncertainty range causes them to shift f in the same direction. But this is fairly unlikely, because there is only roughly a 5% chance that any single variable will be at or beyond either limit of its uncertainty range; the likelihood that all of the variables are simultaneously at or beyond their limits on the correct side to push the value of f in the same direction is quite small: (0.05)2 = 0.0025 if there are two independent variables, (0.05)3 = 0.000125 if there are three independent variables, and so on. Because of this, a statistically more accurate estimate of the uncertainty of f due to the uncertainties in all of its variables is

K+++≈ 222][ cba ffffU δδδ

K,, ba ff

(7.2) (The proof is beyond our scope here.) Quantities whose effects are “added” by squaring, adding, and then taking the square root like this are said to be added in quadrature. Calculating U[f] thus reduces to the problem of finding the changes δδ due to each variable separately. This can be done easily in simple cases. Consider the special case where f(a,b) = a - b. If we increase a to aa δ+ while keeping b fixed, then f changes to ][max aUafafbaaff a =⇒=⇒−+=+ = +δ δ δ δ δδ


][bUfbafter subtracting f = a - b from both sides. Similarly, you can easily see that = −δ (negative because when b goes up, f goes down). Therefore the total uncertainty in f in this case is

( ) ( )22 ][][][ bUaUfU += when f = a - b (7.3) Therefore, if a and b have the same uncertainty, then the best estimate of the uncertainty in f is not 2U[a] (as one might naively expect) but rather ][4.1][2 aUaU ≈ . On the other hand, if U[a] is more than about 3 times larger than U[b], then ( )2][aU

jnm cbkacbaf =),,,( K

11/ −== TDTD

jnm cbkacbaf =),,,( K

is more than 9 times larger than U[b], and thus will dominate the expression for U[f] in equation 7.2.

7.4 THE WEAKEST-LINK RULE

Most calculated quantities f that arise in physics experiments can be put in the form (7.4) where k is a constant and m, n, and j are exponents that may be positive or negative, and are usually integers or simple fractions. A dependence of this form on the variables a, b, c, …is called a power-law dependence. For example, an object’s calculated speed v depends on the distance D it had to travel and the time T that it took to travel that distance according to the power-law relation v . In this case, the constant k = 1. If equation 7.4 is true, then the weakest-link rule provides a fast and simple way of estimating the uncertainty U[f] in the calculated quantity f: The fractional uncertainty Q[f] of

is approximately equal to the largest of the values ],[aQm ],[bQn

],[cQj and so on. The fractional uncertainties Q[a], Q[b], Q[c], … of the variables are typically quite different in a real experiment, and so doing a few rough divisions in your head can quickly guide you to the variable whose fractional uncertainty is largest. We will look at why this rule is correct in a moment. First note that this rule says two interesting things. The first is that the “weakness” (that is, the fractional uncertainty) of a calculated quantity f is determined primarily by the “weakest” of the quantities on which it depends, the “weakest” here being the quantity whose fractional uncertainty times its exponent is largest. The rule’s name emphasizes this by bringing to mind the old saying “the strength of a chain is determined by its weakest link.” (Note that contrary to the currently popular game show, the “weakest link” is the one we keep in our calculation!) The second interesting thing that the rule says is that it is the fractional uncertainty in f that is related in a simple way to the fractional uncertainties of its variables. This was not the case when we are talking about a simple sum or difference of variables, but as we will shortly see, it is the most natural way to deal with power-law relations.


11 −= TDvavg

m/s8.14)s82.0/()m12.12( ==avg

Let’s see how this rule might work in a given situation. Imagine that we are computing the magnitude of an object’s average velocity , where D is the distance it travels during a time T. Say that we have measured D = (12.12 ± 0.02) m and T = (0.82 ± 0.05) s. The best guess value of v . The fractional uncertainties in D and T are:

0017.0m12.12m02.0][][ ===

DDUDQ , 061.0

s82.0s05.0][ ==TQ

The fractional uncertainty in T is more than 35 times larger than that for D so it dominates. According to the weakest link rule, the fractional uncertainty in is thus given by 11 −= TDvavg

m/s9.0)m/s8.14)(061.0(061.0][061.0][1[ ===⇒=−≈ avgavgavg vvUTQv

bkabaff 2),( ==b

]Q Note that the calculations here are quick and simple: that is the beauty of the weakest-link rule. The general “proof” of the weakest-link rule is somewhat beyond our mathematical means here, but let’s see how we might “prove” it in the simple case where . If b changes to b δ+

bkafbkafbkabkabbkaff bb δδδδδδ 22222 )( =⇒+=+=+=+

bkaf 2= ][bUb

while a remains the same, then f changes to If we now divide both sides of this by and set =δ , we find that

][][2

2

bQbbU

bb

bkabka

====δδ

ffbδ

aa

(7.5)

If we change a to δ+ ba 2)δ+

bkaf 2=

while b remains the same, then f changes to . Writing out the square and subtracting from both sides, we get

akff a (δ =+

( )

2

22

)(2

2)(2

akbakabf

kabfbaaaakf

a

a

δδδ

δδδδ

+=

+=++= 2)( akbaf δ

⇒

++

bkaf 2=

Dividing both sides of the result by yields:

22

2

2

)(22aa

aa

bkabkaakab

ff a δδδδ

+=+=2)( akb δ (7.6)


aNow, if we can assume that the variation δ due to a’s uncertainty is much smaller than the value of a itself, then , and we can ignore the second term in comparison to the first. Then, if we set

aaaa /)/( 2 δδ <<Ua = ][aδ , we find that

][2][2 aQaUafa =≈

aδ

(7.7)

If we now divide both sides of equation 7.2 by f and substitute in the results of equations 7.5 and 7.7, we find that

( ) ( )2222

][][21][][ bQaQff

ff

fffUfQ ba +=⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛==

δδ

If one of 2Q[a] or Q[b] is larger than the other by a factor of 3 or more, that term will dominate inside the square root and thus be essentially equal to Q[f]. Thus we have seen that the weakest link rule does indeed adequately summarize the more exact calculation in this case as long as (1) the fractional uncertainty in a is fairly small, so that we can ignore the complicating term in equation 7.5, and (2) one of 2Q[a] or Q[b] is larger than the other by a factor of 3 or more.

7.5 WHAT IF THE WEAKEST-LINK RULE DOESN’T APPLY?

The weakest-link rule does not apply to situations where f’s dependence on its variables is not a power-law relation (for example, the simple sum f(a,b) = a+b). The weakest-link rule is also not very accurate in situations where the fractional uncertainties in the variables are large fractions of 1, or when two fractional uncertainties are nearly the same. What do we do in such situations? The first level of approximation is to use the weakest link rule anyway, and simply recognize (and state in your lab notebook) that the estimate of the uncertainty might well be inaccurate. The weakest-link rule will almost always yield estimates good to within a factor of two or so unless your formula for f involves logarithms or exponentials. In situations where one is not interested in high degree of precision this may be acceptable, as long as you recognize situations when the rule might not be expected to give accurate results and factor that into your conclusions. A better way to determine the uncertainty of f would be to calculate many values of f using values of its variables a, b, c, … that are randomly chosen from the raw data for these variables. Then one can determine the uncertainty of f in the usual way by evaluating the standard deviation of the set of values for f and so on. This method almost always gives an excellent estimate of U[f] as long as the number of values of f that you generate is reasonably large (more than 20 at least!). However, because this method is so tedious, we cannot recommend it unless you have a computer program to do the work. An approach of last resort is to apply the general method outlined in section 7.2. Calculate (by hand) the variation in f when you vary each variable from its central position to the upper edge of its uncertainty range while leaving the other variables constant. Then use equation


7.2 to compute the total uncertainty in f from these individual variations. This will generally be pretty tedious compared to the weakest link method, but does yield reasonably accurate answers in all cases. This is the method that you must use if f involves a logarithm or exponential, unless you have a computer program that can do the calculation outlined in the previous paragraph. See section 7.6 below for a description of just such a computer program. Of course, if f involves the simple sum or difference of two variables, one can apply equation 7.3, which we derived especially for the simple difference case. (You should be able to convince yourself that equation 7.3 also applies to the case of a simple sum.)

7.6 THE PropUnc PROGRAM

bkaf 2=

PropUnc is a computer program that uses the “calculate many values” approach to generate an accurate value of uncertainty of f in all cases involving five or fewer variables. A screen shot of the program set up to calculate the uncertainty of the function is shown in Figure 7.1. All that you have to do to use the program is to type symbols, values, and uncertainties for your basic variables in the “variables” section and the symbolic expression for f in the “expression” section and punch the “Evaluate” button. The program then calculates a randomly-chosen value for each variable that lies within the uncertainty range you specified for that variable and calculates the value of f using randomly-perturbed variable values using the formula you supply. It is therefore much like LinReg, which generates 19 more data sets with measured values consistent with your measured values and their uncertainties. The computer repeats this process N times, where the default value is N = 100, but you may vary it if you wish. Finally, the computer calculates the standard deviation of the N values of f it has generated and the uncertainty in f from that. In other words, the computer simulates having N teams of experimenters like your team who have measured the same variables and have used them to calculate values of f. The uncertainty in the value of f is clearly related to the spread in the values obtained by the N fictitious teams. In the case shown in the figure, the fractional uncertainty in f is 10%. Note that the quantity a has by far the largest fractional uncertainty, 5% compared to 1% for the other variables, so the weakest link rule would say that Q[f] ≈2Q[a] = 2⋅5% = 10%. Thus the program agrees with the weakest-link rule in this case. If you press the “Evaluate” button again, however, you may get slightly different results, because of the random nature of the simulation. Choosing larger values of N will make the calculation more accurate, but could be slow on an old computer. We actually would rather you use the weakest-link rule whenever you can; you will not always have PropUnc handy in real life, so it is good to practice using the weakest-link rule, which is simple and usually gives good results. You may use PropUnc (1) to check a weakest-link calculation, or (2) whenever the weakest-link rule or equation 7.4 does not apply. PropUnc is installed on all the lab computers and also may be freely downloaded from the Physics 51 web site.


Figure 7.1. The PropUnc screen.

7.6 SOME EXAMPLES

Example 7.6.1: Suppose that you want to find the uncertainty in the volume of a cylinder when you have measured its diameter and height. The volume V of a cylinder in terms of its diameter d and height h is given by hd 2

41π=V . Here the volume has power-law dependences on the

variables d and h, so we should be able to apply the weakest-link rule. Suppose that our measurements are d = (0.200 ± 0.002) m and h = (0.600 ± 0.003) m. The fractional uncertainties in d and h are:

005.0m600.0m003.0][,01.0

m200.0m002.0][ ==== hQdQ

2d∝

(7.8a)

Note that even though the absolute uncertainty of d is smaller than that for h (0.002 m compared to 0.003 m), the fractional uncertainty of d is larger. Moreover, since V , the weakest link rule tells us that we should be comparing 2Q[d] to Q[h]: we see that in this case the first is four times larger than the second. Therefore, according to the weakest-link rule,


02.0][2}[ =dQV ≈Q (7.8b) The 1% uncertainty in d thus leads to a 2% uncertainty in V. Now that we have the fractional uncertainty in V we can find the absolute uncertainty pretty easily. The central volume value that we calculate from our best-guess estimates of d and h is

32241 m0188.0)m600.0()m200.0(

4===

ππ hdV

333 m0004.0m000377.0)02.0)(m0188.0(][][ ≈==⋅= VQVVU

Nf ln

(7.9)

This value is uncertain to 2%, so its absolute uncertainty must be where I have rounded the uncertainty to one significant digit. An uncertainty this size means that it is pointless to include more digits than we already have in equation 7.9. So a statement of this value and its uncertainty would be V = (0.0188 ± 0.0004) m3 or (1.88 ± 0.04) x 10-2 m3. Note that in both cases we have written the two values so that they are multiplied by the same power of 10. This makes the values much easier to compare. Example 7.6.2: Imagine that the number of bacteria in a certain colony at a certain time is N = 305,000 ± 15,000. What is the uncertainty in = ? (You might need to know the uncertainty of the logarithm if you want to draw an uncertainty bar for this data point on a log-log graph.) Since is not a power-law relation, we cannot use the weakest-link rule. If we can’t use PropUnc, we can fall back on the general method. In this case, if we change N from its central value of 305,000 to the upper limit of its uncertainty range which is 320,000, the value of ln N changes from ln(305,000) = 12.6281 to ln(320,000) = 12.6761, so the change in f due to this change is

Nf ln=

480.0+=Nfδ . Since f only depends on N in this case, equation 7.1 implies that

( ) 05.0048.02 ≈== NfU δ][ = Nff δ (7.9) where we again have rounded to one significant digit. The result from PropUnc is shown in Figure 7.2. Note that in “computerese” ln N becomes “log(n)”. We use a small n to distinguish it from the number N of trials the computer evaluates.


Figure 7.2: PropUnc’s check of equation 7.9.

If we were to naively apply the weakest-link rule anyway we would estimate that since the fractional uncertainty in N is 15,000 / 305,000 ≈ 0.05, the fractional uncertainty in Nf ln= would also be 5%. This would lead us to estimate the uncertainty of f to be 0.05(12.63) ≈ 0.63, which is more than 10 times larger than the more correct calculation given by equation 7.9. This illustrates our earlier statement that the weakest-link rule does poorly when f involves logarithms.

7.7 THE BOTTOM LINE

You will be expected to state uncertainties of all calculated quantities in this lab program. Use the weakest-link rule to estimate these uncertainties whenever that rule applies; when it doesn’t, either use one of the methods discussed in section 7.5 or use PropUnc to calculate the uncertainty.


EXERCISES

Exercise 7.1 A person is measured to run a distance of 100.00 m ± 0.05 m in a time of 11.52 s ± 0.08 s. What is the person’s speed and the uncertainty of this speed according to the weakest link rule? Exercise 7.2 A spherical balloon has a radius of 0.85 m ± 0.01 m. How many cubic meters of gas does it contain, and what is the uncertainty in your result?


Exercise 7.3 Imagine that you want to estimate the amount of gas burned by personal cars every year in the U.S. You estimate that there are an average of about 0.7 ± 0.4 cars per person in the U.S., that there are 275 million ± 30 million people in the U.S. currently, that a car is driven on the average about 15,000 mi ± 3,000 mi a year, and that the average number of miles per gallon that a car gets is about 23 mi/gal ± 5 mi/gal. What is the approximate amount of gas burned and what is the approximate uncertainty of this estimate? Exercise 7.4 Equation 7.10 suggests that rather than dropping the other uncertainties entirely (as the weakest link rule suggests) perhaps we would get a more accurate estimate of the fractional uncertainty in a power-law relation by multiplying the fractional uncertainty of each variable by its power, squaring the result, adding the squares and taking the square root of the sum. Do this for the case described in Exercise 7.3 above. Is the answer you get from doing this careful way much different from just using the weakest link result? Suppose that you do some research that enables you to reduce the fractional uncertainty in the all quantities but the worst one to 1%. Does reduce the uncertainty much? If you really want to improve the uncertainty, what would be the variable to focus on, and why?

83

Section III: Fitting Functions to Data

85

Chapter 8: LINEAR REGRESSION “ ‘It is a point of great delicacy, and you must assist us in our endeavours to

choose exactly the right line of conduct.’” --- Mansfield Park

8.1 INTRODUCTION Many physical systems have the happy property that the effect you’re investigating has one primary cause and a nice, simple dependence on that cause. The simplest interesting dependence is a linear one; that is, one in which the cause (described by x) and the effect (described by y) are related in this way: y = mx + b This is known as a linear relationship because (as we’re actually pretty sure you already know) if you were to graph y against x on a set of coordinate axes, the resulting graph would be a straight line. Imagine that you’re going on a longish road trip (by car), during which you expect to be doing a lot of highway driving at an essentially constant speed. In this case, if you were to measure the odometer reading as a function of the amount of time you’ve been on the road, you would find that a graph of your results was a straight line. (If the odometer reading weren’t a linear function of the time, you could quite rightly conclude that you hadn’t been driving at a constant speed.) In this example, we can rewrite that general linear equation to fit our specific physical situation and give physical interpretations of all the symbols. We might choose letters to remind us of exactly what each symbol represents and write our equation as 0dvtd += Here t refers to the time on the road (at constant speed), d refers to the distance traveled in that time, and v is the speed (the magnitude of the velocity) of the car. What about that d0 term? That’s the initial reading on the odometer. If you have a trip odometer on your car that you reset to zero at the beginning of a trip, d0 on the trip odometer would be zero. But unless you buy a new car every time you set out on a road trip, d0 will almost never be zero if you’re using your main odometer for this exercise.

It’s possible to work the speedometer/odometer problem in reverse. Suppose your speedometer isn’t working properly in that the number the needle is pointing to (OK, most of the physics faculty drive old clunkers with analog speedometers) is not really the speed of the car. It’s working well enough that if you keep the needle pinned at 60 miles/hour, your car is traveling at some constant speed; you just can’t be confident that the constant speed is in fact 60 mph. This could be pretty important if you’re driving through an area dense with highway patrolfolk with quotas of speeding tickets to fill. You could still determine your speed, and

8. Linear Regression

86

calibrate your speedometer at the same time, by holding your speed constant (as registered on your uncalibrated speedometer) and recording the value that the odometer registers at several different times. If both your odometer and your clock were ideal measuring devices, able to register displacement and time without experimental uncertainty, a graph of the odometer values versus time would lie along a perfectly straight line. The slope of this line would be the true speed corresponding to your chosen constant speedometer setting. In real life, of course, you don’t have perfect equipment, and thus your measurements will always include some experimental uncertainty. Therefore, though a graph of real measurement data (such as the data you might get with a real car odometer and clock and an unsteady foot on the gas pedal) might still give you a pretty straight line, but your data points would probably not all lie exactly on the line. In fact, you might well be able to draw more than one line that looks like a good match to your experimental results. The purpose of this chapter is to describe a procedure for finding the slope and intercept of the straight line that “best” represents your data in the presence of the inevitable experimental uncertainty of such measurements. Statisticians call the process of determining such a “best fit” line linear regression.

8.2 FINDING THE “BEST-FIT” LINE

Before we can determine the line that “best fits” a set of experimental data points, we must first agree on a quantitative criterion for deciding what a “best” fit would be. Several different kinds of criteria might be used, but the most common criterion uses the least squares best fit. This criterion says that the “best-fit” line will be the one that minimizes the sum of the squared vertical distance between each point and the line. This means that (in some sense at least) that the line is as close as possible to each data point. To be more specific, suppose that the “true” equation describing your data has the form y = mx + b (8.1) The least squares best fit line is the line with slope m and intercept b that minimizes the sum

( ) ( )∑∑==

−−=−≡N

iiobsi

N

icalciobsi bmxyyyS

1

2,

1

2,,

(8.2)

Here xi is the value of the independent variable for the ith data point, yi is the actually measured value of the dependent variable for the ith data point, N is the number of data points, and yi,calc is the value for y that you would calculate from xi using a given pair of values for m and b. If we were to simply pick some values of m and b out of a hat, then we could calculate yi,calc from these values for each value of xi, subtract it from the actually measured value yi, square that difference, and then sum over all the data points to find S. We could then pick another pair of values for m and b, compute S for that pair, and so on. After a while, we would


87

find that certain choices of m and b make S smaller, while others make S larger. The “best fit” values of m and b would be those values that make S the smallest. The trial-and-error method for determining m and b just described would be very tedious to carry out in practice. Happily, application of some multivariable calculus makes this all unnecessary. With some work, one can show that the values of m and b that yield the smallest possible values of S are:

( )( ) ( )( ) ( ) ( )( )

( ) ( ) ( ) ( )22

2

22,

∑∑∑ ∑ ∑∑

∑∑∑∑∑

−

−=

−

−= iiiiiiiii

xxN

yxxyxb

xxN

yxyxNm

iiii

(8.3)

where the sum should be performed from i = 1 to N (that is, over all data points). The derivation of these formulas is somewhat involved and employs calculus (at the level of Math 32) that is somewhat beyond the level of this class, so it is available in an optional section (8.8) that you can look at if you are interested. The point to understand here is that we can use calculus to find the values of m and b that minimize the value of S without doing a trial-and-error search.

8.3 THE UNCERTAINTY OF THE SLOPE AND INTERCEPT

If each measured value yi were exactly equal to its “true” value, then equation 8.3 would yield THE best fit line. But if the measured values yi (not to mention the measured values of xi) are uncertain, then it is not clear what yi ought to be for a given xi and therefore what the best-fit line really is: uncertainties in yi and xi therefore make m and b uncertain, too. Now in many cases, m and/or b are interesting physical quantities whose values we would like to determine in this experiment: if this is so, we would like to know the uncertainty of each of these quantities. How might we determine this uncertainty? One way to compute the uncertainty would be to repeat the whole experiment, say, 20 times (either sequentially or with 20 different lab teams). If we did this, each data set that we collect would be somewhat different (due to experimental uncertainties), and therefore the best-fit values for m and b would be somewhat different as well. We could then estimate the uncertainty of m in a very straightforward way by taking our 20 different values of m (one for each trial of the experiment), finding the standard deviation s of these values, and then computing the uncertainty by multiplying s by the Student t-factor for N = 20. We could then calculate the uncertainty of b in the same way. This method would yield honest experimental estimates of the uncertainties in m and b that are based directly on what the uncertainties of these quantities mean at the fundamental level. The problem is that actually doing this would be a very tedious and time-consuming way to answer the question. Fortunately, we have a shortcut. While we may not have time or person-power to actually do the experiment 20 times, a computer program can easily simulate the process of running the experiment 20 times (and thus compute the uncertainties in m and b as described above) using data from a single real experiment.


88

8.4 USING THE LINREG APPLICATION

LinReg is a computer program that does this simulation for you. It accepts your experimental data from one run of the experiment you performed, and calculates the best-fit slope m and intercept b for your actual data. But with the push of a button, you can make the program simulate the efforts of 19 other imaginary lab teams, who gather data with uncertainty ranges matching yours but with actual measured values varying randomly within the uncertainty range specified for each quantity. By computing m and b for each of the 20 slightly data sets (your actual data and the data from the 19 imaginary teams), the program can easily generate estimates of the uncertainties of m and b that follow from the uncertainties of your measurements. LinReg is available for both the Macintosh and Windows operating systems, so to use the program, select a lab computer with your favored operating system and launch the program in the way that is appropriate for that operating system. You can also download either flavor of program from the Physics 51 web site. When the program starts, it displays a window with cells on the left-hand side where you will enter information about your data set. Enter your names(s), the horizontal variable name and unit, and the vertical variable name and unit, and press the “Done” button. The program will not let you proceed until you entered something in each text box. (If one of your variables is unitless, type “none” in the units box.) After you have successfully done this, the program displays a table in the previously empty part of the window into which you can enter your data. When you type a number, it is entered into the active cell in the table (the white cell). You can change the active cell in the table by pressing the tab key (to scan forward across the table), shift-tab (to scan backwards), return (to move down one row), or the arrow keys. When you have entered all your data, you can display a graph of your data by pressing the “Graph” button in the “Display Modes” box. If you also press the “Do Linear Fit” button, the program computes and draws the best fit line to your data. The program also uses your table of data as the basis for creating 19 new tables of data that are the simulated experimental results of 19 lab teams like your own. In creating the invented data tables, LinReg simulates the measurement process by randomly choosing each measurement value from a bell-shaped distribution of values having the same mean and uncertainty as the corresponding measurement in your table. This creates sets of data that are essentially like yours except that each measurement value has been perturbed from your corresponding value by an amount consistent with your uncertainty, exactly as if a different lab team had made the measurement and come up with a bit different value. By pressing the up or down arrow buttons on the screen, you can scan through the data from the 19 other imaginary lab teams one team at a time. LinReg displays the data for the selected team either as a table (if you are in the table display mode) or a graph (if you are in the graph display mode), just as it would display your data. The number of the team appears at the top of the table or the top left of the graph. Pressing the “Yours” button takes you immediately back to your own (actual) data set.


89

Also, after you have pressed the “Do Linear Fit” button, LinReg will automatically display the values of the slope and intercept for the best-fit line to the displayed data as well as the uncertainties in these quantities. It computes the uncertainties of these quantities in the straightforward manner described in the previous section. To compute the uncertainty in the slope, for example, it simply computes the standard deviation of the 20 best-fit slopes generated from the 20 lab team data sets (your set and the 19 simulated sets) and multiplies by the Student-t factor for N = 20. (You can easily verify this by checking the calculation by hand if you like.) This is exactly the way that we would calculate the uncertainty of any given team’s slope if we had access to the data from 20 actual lab teams. LinReg computes the uncertainty in the intercept the same way. You cannot change any of the invented data sets, but you can change your own data at any time by displaying your own data using the table display mode, selecting the cell to be changed and retyping the data. If you do this, however, you will find that you have to press the “Do Linear Fit” button again to display the new best-fit line, because LinReg now has to recompute the best fit line to fit your modified data. LinReg also will recompute all of the invented data sets from the imaginary lab teams to be consistent with your new data. You can also clear all your data and start over by pressing the “Clear Data” button. You can also at any time change the variable labels and/or units and/or the names of your team members by pressing the “Edit Names and Units” button. You can also print any table or graph shown on the screen by selecting “Print…” from the File menu. LinReg will change the size of the graph to fill a full sheet of paper, but otherwise, what you see on the screen will appear on your printout. The lab computers will be set up to print automatically to the printer in the lab room. If you use LinReg on your personal computer, it should automatically print to whatever print destination you have set up for your computer. The current version of LinReg (as of this writing) cannot save your measurement data or read data from a disk file. This was a deliberate design decision (so that the lab computer’s hard disks did not get filled with people’s old LinReg data files), but it does mean that if you clear your data or quit LinReg by accident, you will have to enter all of your data again by hand. In addition to being found on the computers in the lab, LinReg can be downloaded from the Physics 51 web site (www.physics.pomona.edu/phys51.html). LinReg is freeware: you may make a copy for yourself and/or distribute it to your friends.

8.5 USING LINREG TO CREATE LOG-LOG PLOTS

If you check the “Show Ln” check box for either the horizontal or vertical variable, LinReg displays and/or plots the natural logarithm (and automatically computes and displays the uncertainty in that logarithm) of each measured value of that variable. This feature allows you to create log-log plots (see Chapter 10) or semilog plots (see Chapter 11) very easily: to create a log-log plot of your data, for example, simply check both boxes! You will find, however, that you cannot edit the value of the logarithm or its uncertainty. To make changes to your data, uncheck the “Show Ln” box and then modify the data that you


90

originally entered. This is because LinReg uses your originally-entered values as the master data set: the logarithms are treated as just a different way to display this master data.

nkxy =

xnky logloglog += xnky lnlnln

You will also see that LinReg computes natural logarithms instead of base-ten logarithms. We will use base-ten logs in Chapter 10 because it makes it a bit easier to explain and understand how the logarithms are related to the original values. However, if you think about it, the base one chooses to work in is irrelevant: given a power-law relationship , we have and also += (8.4) where log() is the base-ten logarithm and ln() is the natural logarithm. Therefore if we plot ln y versus ln x, we still get a straight line with slope n. The only thing that is different is that the intercept is now equal to ln k, not log k.

8.6 OTHER TOOLS FOR DOING LINEAR REGRESSION

Many scientific calculators and a number of commonly available spreadsheet or statistical analysis programs can also do linear regression (that is, compute a best-fit line to entered data). However, such programs rarely calculate the uncertainties in the slope and intercept of this line, and even when they do, they usually do it in a way that makes incorrect assumptions (such as assuming that the uncertainties in the horizontal variable are zero and/or the uncertainties in the vertical variable are all the same). We think that you will find that LinReg is easier to use than these alternative tools and also will give you better and more useful results in the context of this lab. We might mention in passing that many of these calculators and/or programs compute a quantity called “correlation coefficient” (usually given the symbol R or r2) for your data when they compute a best-fit line. R is a measure of closely your data fit a straight line, and is thus an (indirect) expression of the uncertainty we should have about how good the line is. (R = 1 indicates that your data fit a line perfectly; R = 0 indicates that there no correlation between your data and any straight line.) While the correlation coefficient R can be very useful in helping one discern whether there is a meaningful linear relationship buried in the noisy data sets common in the social sciences, the data in physics experiments usually fit a straight line so well that R is always essentially 1, making it pretty useless as a measure of the quality of the best-fit line. Therefore, if you do compute R for any data in this lab program, don’t brag about how close to 1 it is!

8.7 LIMITATIONS OF LINEAR REGRESSION

It is important to recognize that the technique of linear regression and the program LinReg have their limitations. In particular, using LinReg makes linear regression so easy that you are likely to forget an important point: the equations and the LinReg program will always try to find the “best” line that fits your data, even when your data do not resemble a line at all! There is no substitute for actually looking at the graphed data to check that it looks like a reasonably straight line. The pre-lab exercises at the end of this chapter illustrate this point.


91

8.8 (OPTIONAL) DERIVING EQUATION 8.3

You do not have to understand where equation 8.3 comes from, but its derivation is not all that complicated if you have had Math 32, and understanding the derivation can help you understand the least-squares method better. Let us rewrite the sum of the squared differences S (see equation 8.2) by working out the square in the sum explicitly, as follows:

( ) ( )∑∑==

++−+−=−−=N

iiiiiii

N

iii bmbxbyxmymxybmxyS

1

2222

1

2 222

mS ∂/bS

(8.5)

We can minimize S the usual way of setting its derivative to zero. Since S depends on two unknown quantities (m and b) here, we must calculate partial derivatives of S with respect to m and b, and set each derivative separately equal to zero. Remember that to calculate the partial derivative ∂ of S with respect to m, we calculate the ordinary derivative of S while treating b as a constant. Similarly, to calculate the partial derivative ∂ ∂/ of S with respect to b, we treat m as a constant. Therefore, we want:

0 21

= = −=

xi

N

2 22 2

11

+ + ⇒ + ===

∑ ∑∑∂∂

Sm

y mx bx bx mx x yi i i i i ii

N

i

N

( ) ( i i) (8.6a)

0 2 2 2= = − + + ⇒ + =∑ ∑∑∂∂Sb

y mx b b mx yi i

N

i i

NN

( ) ( )

i

N

b m x y Nb m x yi

N

ii

N

ii

N

ii

N

ii

N

11 1 1 1 1= = = = =∑ ∑ ∑ ∑ ∑+ = ⇒ + =

A xi≡

(8.6b) 1 11= ==i ii

Though they are unknown, we can factor m and b out of the summation (since their values are the same for all values of i). Equations 8.6a and 8.6b then become (respectively):

, (8.7a) b x m x x yi ii

N

i ii

N

= = =∑ ∑ ∑+ =

1

2

1 1

(8.7b)

∑ , since the sum of 1 from i = 1 to N is simply N. Now, let us define B yi≡ ∑ ,

, and D : these are known quantities since they can be computed from our experimental data. We can rewrite equations 8.7 in terms of these quantities as follows. C i≡ ∑ 2 x yi i≡ ∑x

Ab + Cm = D, Nb + Am = B (8.8) These represent two equations that we can solve the two unknowns m and b in the usual way. (For example, to compute m, multiply the top equation by N and the bottom by A, add the equations to eliminate b, and then solve for m.) The results are:


92

( )

mN x y x y

=− ∑∑∑ND AB

NC A N x x

i i i i

i i

−−

=− ∑∑

2 2 2 (8.9a)

( )

b CB AD x y x x yi i i i i=−

=−∑ ∑ ∑∑ 2

NC A N x xi i− − ∑∑

2 2 2 (8.9b)

which are equivalent to equations 8.3.

EXERCISES

In the following exercises, plot the data given in the box on the left in the graph area provided to the right. The slope and intercept for the “best” line fitting each data set (according to the least-squares method) appear below the box for the data. Sketch this “best-fit” line on the graph. Then comment: does this line really appear to fit the data? (You will see how important drawing a graph is for successfully interpreting how much real meaning these slope and intercept values have!) Exercise 8.1

x y 0.0

1.0

–0.10

0.31

2.0

3.0

4.0

5.0

1.05

1.55

1.80

2.41

x y 6.0

7.0

2.69

3.52

8.0

9.0

10.0

3.86

4.51

5.02

m = 0.51, b = –0.11 Exercise 8.2

x y 0.0

1.0

2.0

3.0

1.22

3.25

4.97

3.68

4.0

5.0

5.44

4.48

x y 6.0

7.0

8.0

9.0

6.38

6.13

7.28

6.93

m

10.0 5.30

= 0.44, b = –2.83


93

Exercise 8.3

x y –10

–8

–6

–4

–2

0

0

–3.60

4.80

–5.50

5.88

–6.00

x y 0

2

4

6

8

6.00

–5.88

5.50

–4.80

3.60

= 0.0, b = 0m

10 0


94

9. Linearizing a Non-Linear Relationship

95

Chapter 9: LINEARIZING A NON-LINEAR RELATIONSHIP

“ ‘It does not come to me in quite so direct a line as that; it takes a bend or two,

but nothing of consequence.’” --- Persuasion

9.1 INTRODUCTION

The purpose of many physics experiments is to explore how one measured quantity depends on another. For example, in this course, you will explore how the force exerted by the end of a spring depends on the length of the spring, how the period of a pendulum depends on its length, and so on. Research physicists at present explore things such as the temperature at which a material becomes superconducting depends on the strength of the magnetic field permeating the material, how the period of orbiting neutron stars depends on time, how the number of subatomic particles scattered from a certain kind of target depends on the angle of scattering, and so on. The most useful way to present the results from an experiment like this is to plot a graph showing how one variable depends on the other. In some cases, the relationship between the variables is linear (meaning that the graph will show a straight line), as we have seen in several experiments in this course. The method of linear regression discussed in Chapter 8 provides a powerful tool for extracting information about such linear relationships. But in many cases, the relationship between the experimental variables is not linear. As we will see in the next section, a straightforward graph of such a nonlinear relationship does not tell us very much, and we have no easy way to determine the “best fit” to a nonlinear graph the way that we do with a linear graph. The purpose of this chapter is to open your mind to possible tricks for creating a linear graph of quantities that we expect to be related nonlinearly. If we can artificially create a linear graph of a normally nonlinear relationship, we can use linear regression to extract information about the relationship that would otherwise be hard to obtain.

9.2 AN EXAMPLE

For example, imagine an experiment where we want to determine an object’s acceleration as it slides down a frictionless incline by measuring its displacement as a function of time. We will find in Unit N that such an object should experience a constant acceleration of magnitude

θsinga =aax

(where θ is the angle the incline makes with the horizontal) directed down the incline. Let’s define the x-axis to point down the incline, so that += . If the object is released with zero initial velocity, then theoretically the displacement d of the object should be related to the time t since it was released as follows: 2

212

21

0)( attaxtxd x ==−= (9.1)


96

Now, if one were to draw a graph of d versus t, one would expect a parabola (as shown in Figure 9.1) instead of a straight line. This is fine, except that (particularly with the uncertainties shown) it is difficult to distinguish a graph of a relationship from one showing or

or any one of many other relationships. Since there are a large number of relationships between d and t that could produce similar-looking curves, it is difficult to verify by just looking at the graph that our assumption that the object has constant acceleration is reasonable. Moreover, there is no simple way to compute the value of a from such a graph.

2td ∝ 3td ∝3/4td ∝

Figure 9.2.)

Figure 9.1: Graph of displacement vs. time for a hypothetical experiment involving an object sliding down a frictionless incline. Is the curve sketched by these data a parabola or not? What is the value of a consistent with these data?

Figure 9.2: the same now lie onintercept oalso that thbe calculat

rresponding t values.) arithm of both sides of equation 9.1, getting:

We can address both of these problems by plotting d as a function of t2, not t. (See Figure 9.2.). In this case, if the object’s acceleration really is constant, then the graph will be a straight line, and if the object’s acceleration is not constant, the graph will be curved. Therefore, a mere glance at the graph helps us check whether our basic assumption is correct. If the graph does turn out to be a straight line, then the slope of such a graph, whose value and uncertainty can be easily determined using LinReg, will be a/2, making it easy to determine the value and uncertainty of a. (Note that according to Chapter 7, a will have the same percent uncertainty as a/2. One also can use the weakest-link rule discussed in that chapter to find the uncertainty in t2 given the uncertainty in t. We need to know the uncertainty in t2 to draw the uncertainty bars in

0

0.4

0.8

1.2

1.6

0 1 2 3 4 5

t (seconds)

d (m

eter

s)

0

0.4

0.8

1.2

1.6

0 5 10 15 20

t^2 (seconds^2)

d (m

eter

s)

Graph of displacement vs. t2 for experimental data. Note that the data a straight line, the slope and f which can be calculated. (Note e uncertainties of the t2 values must ed from the uncertainties of the

co

Alternatively, we could take the base-ten log ( ) ( ) taatd log2logloglog 2

1221 +== (9.2)


97

If we were to plot log d versus log t, then we would get a straight line with slope = 2, which is the power to which t is raised in equation 1, and an intercept equal to ( )2/log a . This means that we can calculate the value and uncertainty of a from the value of the intercept and its as calculated by LinReg. This is the power-law fitting technique discussed in more detaChapter 10. The point here is that we have come up with two different methods of creating a linear graph of the inherently nonlinear relationship expressed by equation 9.1.This does not exhaust the possibilities by any means (we could plot

uncertainty il in

d vs. t or d2 versus t4 for example), though the two methods discussed are the most natural in this case. Having a linear graph makes it possible interpret and analyze the experimental data more easily.

The only real cost is some added work in computing the size of uncertainty bars. If we

e

RIZ

So a general procedure for creating linear ns might be umma

ction of one variable that when plo of that second variable) will yield a straight line if the hypothetical relationship is true.

d

/or

tions may have been in error.) 4. If

s,

were to plot d versus t2 (for example), we have touncertainty in t using one of the methods describproblem. In the case at hand, the weakest link rulsimply twice the fractional uncertainty of t.

9.3 A GENERAL APPROACH TO LINEA

compute the uncertainty of t2 from the d in Chapter 7. This is usually not much of a

e tells us that the fractional uncertainty of t2 is

ATION

graphs of nonlinear relatio s rized as follows. Assuming that you have a hypothesis about the nature of the relationship b n the experimental quantities being graphed, etwee

1. Find a fun tted against the other variable (or a function

2. Compute the uncertainties in the plotted quantities (if different from the measurequantities) using the appropriate method described in Chapter 7.

3. Draw a rough linearized graph of your experimental data to make sure that the plotted quantities really do seem to lie on a roughly straight line. (If not, try to determine whetheryour original hypothesis is correct or whether some of your measurements andcalcula

the rough graph does look pretty linear, then enter the plotted quantities (with their uncertainties) into LinReg, and have it compute the slope and intercept of the best-fit line.

5. If the value of the slope and/or intercept of the graph are linked to interesting quantitiecompute the value and uncertainty of any such quantities using the value and uncertainty ofthe slope or intercept computed by LinReg.


98

EXERCISES

d above. If we determined the acceleration for various angles θ, how might we display the relationship between these quantities in an

apg of t

EImagenerg their separation r (see chapter C7). Describe at least two ways that you could create a graph that would display the expected relation between these quantities as a straight line.

Exercise 9.1 Consider the hypothetical incline experiment discussea

propriately linearized graph? How would the slope of your graph be related to the magnitude he gravitational field vector?

xercise 9.2 ine you are doing an experiment to measure the magnitude of the electrostatic potential y V(r) between two charged objects as a function

10. Power-Law Fitting and Log-Log Graphs

99

Chapter 10: POWER-LAW FITTING AND LOG-LOG GRAPHS

“She had taken up the idea, she supposed, and made everything bend to it.” --- Emma

10. DEALING WITH POWER LAWS

Although many relationships in nature are linear, some of the most interesting elationships are not. Power-law dependences, of the form

nkxxy =)( (10.1)

y common. In many cases, we might suspect that two experimental variables are

we don’t know n, it would seem at first glance that the best we could do would be simply try different possible values of n to see what works. This could get old fast. We can, however, take advantage of the properties of logarithms to convert any relationship of the type given by equation 10.1 into a linear relationship, even if we do not know either k or n! The most basic property of logarithms (for any base, but let’s assume base-ten logs) is that baab loglog)log(

1

r are particularlrelated by a power-law relationship, but are unsure of what k or n are. For example, we mightsuspect that the period T of a simple harmonic oscillator might depend on the mass m of the oscillating object in some kind of power-law relationship, but we might be unsure of exactly what the values of either n or k. If we knew n, then we could plot y vs. xn to get a straight line; the slope of that line would then be k. But if

+= 10.2a) From this it follows that n times n times ( ) anaaaaaaan loglogloglog)log(log =+++=⋅⋅⋅= KK (10.2b) (This is actually true so, in the case of base-ten logarithms, this means that

even for non-integer n.) Al

( ) bbbb =⋅== 110log10log (10.2c) which shows that ra

ising 10 to a power is the inverse operation to taking the (base-ten) log.


100

With this in mind, let us take the (base-ten) logarithm of both sides of equation 1use the properties described by equation 10.2. If we do this we get

0.1 and

( ) ( ) xnkxkkxy nn logloglogloglog +=+== (10.3) Now d et

nuv += (10.4)

ow, t log y vs.

e slope of We can of n by c

log

efine ux ≡log and vy ≡log . Substituting these into 10.3 and rearranging, we g

klog N his is the equation of a straight line. This means that if we graph v vs. u (that is, log x), we should end up with a straight line, even if we do not know what n and k are. Furthermore,th this line is the unknown exponent n in equation 10.1.therefore find the value alculating the slope in the usual way. That is,

12

12

12

12

loglogloglog

xxyy

uuvv

uvn

−−

=−−

=ΔΔ

= (10.5)

The value of the intercept (which is the value of yv log= when 0log == xu ) is klog , so if wcan find the intercept and its uncertainty, we can find k and its uncertainty. In summary

e

the form given in equation 10.1, take the garithm of both sides, and convert it to a linear relationshi e and interlated ing

0.2 AN EXAMPLE OF A LOG-LOG GRAPH

As an example, consider a hypothetical experiment testing how the period of an object spring depends on the object’s mass. Table 10.1 gives a set of

this experiment, Figure 10.1 shows a graph of period vs. the mass. he me ata, while Figure 10.2 shows a graph of

Period (sec) log(Mass) log(Period)

, we can take any relationship oflo p whose slop cept are re to the unknown values of n and k. This is, therefore, a very powerful way of learnabout unknown power-law relationships (and displaying them). A graph that plots ylog versus

xlog in order to linearize a power-law relationship is called a log-log graph.

1

oscillating at the end of a

easurements taken during mTable 10.2 shows the base-10 logs of t asurement d

log T vs. log M. Table 10.1 Table 10.2 Mass (kg) 0.125 2.42 -0.903 0.384 0.325 3.76 -0.488 0.575 0.525 4.75 -0.280 0.677 0.725 5.52 -0.140 0.742 0.825 5.87 -0.084 0.769


101

00.10.20.30.4

log(

Pe

0.5riod) 0.6

0.70.8

re 10.1: Graph of the period of oscillation s a function of mass.

Figure 10 Graph log of the period as a function of the log of the mass.

8 0.9

6

2

Perio

d 4 (sec

)

00 0.2 0.4 0.6 0.8 1

Mass (kg)-1 -0.8 -0.6 -0.4 -0.2 0

log(Mass)

Figu .2: of the a We can see that the graph of the period vs. the mass does not yield a

ne; if the uncertainties are smaller than the dots representing the data points, w very good straight

e would have to ay tha

athe gr ares in Figure

lis t a straight line is inconsistent with the data. On the other hand the plot of log T vs. log Mis a very nice straight line, suggesting that the period T and the mass M have a power-law relationship. What are n and k according to this experiment? We can easily get a quick estim te from

aph. For the sake of round numbers, consider the points marked with squ10.2. The slope of this graph (rise over run) is thus

48.020.080.043.072.0

loglogloglog

12

12 =−−

=−−

=MMTTn (10.6)

The two points that you choose to compute the slope need not correspond to actual data points: simply choose convenient points on your drawn line near the ends of the line. I chose the points o that the denominator of the expression above would be simple. Since exponents in physical

lue of n is . (This turns out apter N12 in the class

t e intercept is t e where the line c s g M = 0 grid li ording to the g is is roughly wh g T = 0.82 (note th t t = 0 line is the ge of the g re, not the left!) e intercept is 0.82 l ich means that

k = 100.82 = 6.6 s/kg1/2.

ssituations are most often integers or simple fractions, we might guess that the actual va½ to be the theoretical value as well, as we will see chext.)

Th he plac ro ses the lo ne. Accraph, th ere lo a he log M right edraph he . So th = og k, wh


102

10.3 A COMMENT ABOUT LOGARITHMS AND UNITS

ng in and out of these calculations ra l rule about special functions: the log fun supposed to have a dimensionless ar s supposed to be a pure number, with no ed to it. What's going on? uct of the number part and the units part; th it of centimeters. When you take the log logarithm of this product, which beh sum of the logarithms of the two

is therefore log (10) + log (cm) = hich may seem rather disturbing.

se you typically find the logarithm of a

uantity with units only in the process of finding the difference between two logarithms of m

e attach to each logarithm of log s here s is the unit of seconds here), we see that two log s terms cancel out when we subtract,

g

knn log)kglog(loglog(sec)log

You may also have noticed that units seem to sprither haphazardly. This behavior conflicts with a genera

ction, like sine, cosine, and the exponential function, is gument. That is, the number you take the logarithm of i units or dimensions (like seconds or centimeters), attach

One can think of a number with units as the prodat is, 10 cm is 10 (a pure number) multiplied by the un

arithm of a number with units, then, you are taking the aves like the logarithm of all products: the result is the

numbers making up the product. The logarithm of 10 cm1 + log (cm). You can't give a numerical result for log cm, w

Fortunately, this all works out fine anyway, becauqqquantities with the same units. Consider, for exausing equation 10.6. If wquantities with the same units. Consider, for exausing equation 10.6. If w

ple, the situation where we compute the slope in the numerator the appropriate term ple, the situation where we compute the slope in the numerator the appropriate term

(w(wleavin a unitless number in the numerator. The same thing happens in the denominator. Therefore, the slope ends up being a unitless number, as it should be. To find the units of k, note that equation 10.4 in this situation really ought to read

leavin a unitless number in the numerator. The same thing happens in the denominator. Therefore, the slope ends up being a unitless number, as it should be. To find the units of k, note that equation 10.4 in this situation really ought to read +


102

10.3 A COMMENT ABOUT LOGARITHMS AND UNITS

You may also have noticed that units seem to spring in and out of these calculations rather haphazardly. This behavior conflicts with a general rule about special functions: the log function, like sine, cosine, and the exponential function, is supposed to have a dimensionless argument. That is, the number you take the logarithm of is supposed to be a pure number, with no units or dimensions (like seconds or centimeters), attached to it. What's going on? One can think of a number with units as the product of the number part and the units part; that is, 10 cm is 10 (a pure number) multiplied by the unit of centimeters. When you take the logarithm of a number with units, then, you are taking the logarithm of this product, which behaves like the logarithm of all products: the result is the sum of the logarithms of the two numbers making up the product. The logarithm of 10 cm is therefore log (10) + log (cm) = 1 + log (cm). You can't give a numerical result for log cm, which may seem rather disturbing. Fortunately, this all works out fine anyway, because you typically find the logarithm of a

uantity with units only in the process of finding the difference between two logarithms of m

e attach to each logarithm of log s here s is the unit of seconds here), we see that two log s terms cancel out when we subtract,

g

knn log)kglog(loglog(sec)logτ +=+ μ + , (10.7)

e crosses the vertical grid line corresponding log μ = 0: we see from the graph that logτ has the value log τ0 = 0.82 there. Therefore

plie

kg)log(log(sec)logloglog)kglog(0log(sec)log 00 nkknn

where in this expression we are considering τ and µ to be the unitless parts of T and M respectively. The intercept is the point where the lintoequation 10.7 im

s that

(10.8) τ τ⇒++⋅=+ = + − When we take the antilog of (that is, 10 to the power of) both sides of this, all the items in logs get multiplied together, so we get (assuming that n is really 1/2):

1/21/2

82.0log sec6.6sec10sec10 0

=⋅

=⋅

=τ

k (10.9) n kgkgkg

tities, and fill in the og (as we did with k in the last section) to make them

Keeping track of these unit terms when working with logarithms involves a lot of work, however, and less often pays off the way that keeping track of units in normal equations does.

herefore, people generally ignore the units associated with logarithmic quanTunits of quantities after taking the antil


103

consistent across the master equation 10.1. But if you ever get confused about units and want to make sure that things work out correctly, this is how to do it.

rediction, sometimes a previous low-level Cartesian plot. Figures 10.3(a) through 10.5(a) are pical

t

nce you have plotted the points, you should use a aw th ht line that you

ink best fits your data. You can then use this line to estimate the slope n using equation 10.6. straight

est fit d line where 0log =x . Note that this vertical line may not correspond

the left edge of your graph! In Figure 10.2, for example, it happens to be at the right edge of e graph, and on a general log-log graph, it could be almost anywhere. (If the line log

10.4 A PROCEDURE FOR EXPLORING POWER-LAW RELATIONSHIPS

Log-log graphs are most useful when you suspect your data has a power-law dependenceand you want to test your suspicion. Sometimes your suspicion is based on a theoreticalpty Cartesian graphs that could be power laws. Whatever the source of your suspicion, your next step is to plot the logarithms of your data as a low-level graph. If this graph looks like a pretty good straight line (within your experimental uncertainties) you can proceed to the nexsteps.

Fig. 10.3(a) Fig. 10.3(b) Fig. 10.3(c) O ruler to dr e straigthYou can also estimate the value of the constant k in equation 10.1 by extrapolating yourline back to the vertical grid line where the value of the independent variable (let’s call it x) is equal to 1 (and thus 0log =x ): the value of log k is the vertical scale reading where your bline crosses this vertical gritoth 0=x is off the edge of your graph, you can often bring it onto the graph by changing the units of x. For example, if x is a distance ranging from 20 cm to 200 cm, the place where 0log =x is when x =1 cm, which will probably be off the left side of your graph. But if

we change the units of x to

eters, then the place whe is where x is right in the middle of your ata.)

To estimate the uncertainties of these quantities, draw a new line with the largest slope at yo

of k.

m re 0log =x = 1 m, which d th u think might be consistent with your data, and another line with the smallest slope consistent with your data, and find the slope and intercept for each of these lines. The greatest and least slope will then bracket the uncertainty range of n and you can use the greatest and least values of klog determine the greatest and least values of k, which bracket the uncertainty range

y=x̂ a, a > 1

40

0

10

20

30

0 2 4 6 8 10

x

y=x̂ a, a > 1

0

10

20

30

0 2 4 6 8 10

x

40

y=x̂ a, 0 < a < 1

3

0

0.5

1

1.5

2

2.5

0 2 4 6 8

x

10


104

To go further than crude estimates, one needs the help of a computer. The program LinReg, which is discussed in Chapter 8 of this manual, makes it very easy to plot log-log graphs

ertainty).

radius). Plot a log-log graph of the versus the distance on the graph paper provided as Figure 10.6 on the next page. Table 10.3: Planetary Periods vs. Mean Orbital Distances Planet Distance (AU)

and find the best-fit slope (with its uncertainty) and the best-fit intercept (and its unc

EXERCISES

Exercise 10.1 The table below gives the orbital periods T (in years) of the planets known to Newton as a function of their mean distance R from the sun in AUs (where 1 AU = the earth’s mean orbital

period

Period (yr) log(Distance) log(Period)

Mercury 0.39 0.24

Venus 0.72 0.62

Earth 1.00 1.00

Mars 1.52 1.88

Jupiter 5.20 6 11.8

S 9.54 29.46 aturn

Exercise 10.2 Assuming that the pe d distance are related by a power-law of the form nkRT = , where n is an integer or simple fraction, what does your graph suggest is the likely value of n?

Exercise 10.3 Find the value of k (with appropriate units) for the data of Table 10.3 from the intercept of yourlog-log graph. Combine this w esult of exercise 10.1 to find the power-law equation (othe typ

rio and

ith the r f e given in equation 10.1) that seems to fit this data.


105

Figure 10.6


106

10. L) USING LOG-LOG PAPER

If you have more than five or ten data points, calculating the logarithms quickly gets tedious even with a calculator. To reduce this tedium (which would have been particularly gruesome in the era before computers and calculators), someone invented a special kind of graph paper called log-log paper. In effect, this kind of graph paper calculates the logarithms for you. Imagine that you have data for an independent variable x that ranges from, say, 0.01 m to about 10 m. The values of xlog would then range from about –2.0 to 1.0, and a useful horizontal scale might look something like this:

Now, imagine that we were to also draw a scale immediately above this scale that showed the corresponding values of x. The two scales together would look like this:

(Note how the pattern of the spacing between marks on the upper scale is identical for each power of 10.) Now, note that if we had graph paper that had its axes pre-labeled as shown in the upper scale, then we could locate points on the plot directly according to their value of x rather than having to compute the value of xlog for each data point. The way that the pattern repeats for each power of 10 makes it possible to create general and flexible graph paper with essentially pre-labeled scales. You can repeat the pattern as frequently as necessary in each coordinate to cover the range of your data. The graph paper for Exercise 10.4 is an example of log-log paper which has 3 cycles of the pattern horizontally and vertically, making it possible to display data po whose x and/or y values span up to three powers of ten or decades. If you compare this g aper to the double-axis shown above, you will see that only the equivalent of x scale is displayed on this paper: the xlog scale has been suppressed for the sake of clarity, but should be considered implicit. Also you will note that the publishers of the graph paper do not commit you to particular powers of 10: each decade is labeled as if it spans from 1 to 10. You can cross out the numbers shown to adapt the graph paper to the particular ranges of your data points. Figure 10.7 shows how you would do this for the harmonic oscillator data given in Table 10.1. The point is that with just a little relabeling you can use graph paper like this to conquickly a log-log graph without having to do any actual calculations of logarithms. This is gfor doing low-level, quickie graphs of a set of data that you think might reflect a power-law

5 (OPTIONA

-2.0 0.5-1.5 -1.0 -0.5 0.0 1xlog

0.01 0.02 0.1 0.2 0.5 1.0 2.0 5.0 10.00.05x

log x-2.0 0.5-1.5 -1.0 -0.5 0.0 1

ints raph p

struct reat


107

relation nkxy = . You can even read the value of k directly from the graph by finding the point here

hangeth

Using log-log paper is optional in this course; normally you will be able to use a computer program. But you can purchase sheets of log-log paper from Connie (the department secretary) for 10¢ per page if you would like to use it for quick low-level graphing.

igure : A log-log plot of the harmonic oscillator data in Table 10.1. Notice that the paper

w your best-fit line crosses the vertical line corresponding to x = 1 unit and reading the vertical coordinate of this point according to the vertical axis. But how can you compute the slope n of data drawn on such graph? Remember each axis

as an implicit linear scale that reflects the log of the value displayed. “Linear” means that the hc in the value of xlog or ylog is proportional to the physical distance on the sheet of paper. So to find the slope, all at you have to do is measure the rise of your line (in cm on the sheet of paper!) and divide by the run (in cm).

F 10.7has only one decade in each coordinate to match the range of the data.

Mass (kg)

1.0

10.0

0.10 1.00

T

ime(sec)


108

( ONAL) Exercise 10.4 Plot the planetary orbit data in Table 10.3 on the log-log paper on the next page, and use the methods described in the previous section to find k and n assuming that nkRT = . Check that these agree the values you found before.

OPTI

with

11. Exponential Curve Fitting 109

Chapter 11: EXPONENTIAL CURVE FITTING 11.1 INTRODUCTION Many processes in nature have exponential dependences. The decay with time of the amplitude of a pendulum swinging in air, the decrease in time of the temperature of an object that is initially warmer than its surroundings, and the growth in time of an initially small bacterial colony are all processes that are well-modeled by exponential relationships. To better consider the issues involved in dealing with such relationships, let’s consider a very specific case. The absorption of radiation by a given thickness of some material can be modeled by the following simple exponential relationship: xeRxR β

0)( = (11.1) Here R(x) is the count rate of radiation particles (typically measured as the number of clicks on a Geiger counter that take place in some fixed time such as one minute), R0 is the count rate with no shielding present, x is the thickness of the shielding material, and β is a negative constant that describes how rapidly the count rate decreases as the shielding thickness increases. Some measurements on the rate at which radiation particles emitted by 55Fe are detected when a Geiger counter is shielded one or more thin sheets of aluminum foil appear in Table 11.1 and Figure 11.1. Note that the count rate decreases as the thickness of the aluminum shielding increases. (Note also that the error bars in this case are just barely large enough to be visible.)

Table 11.1: Counts/min vs. thickness of Al shielding

Al thickness (cm)

Count rate (counts/min)

0.00162 1850 0.00324 1250 0.00486 800 0.00648 450

0

500

1000

1500

2000

Cou

nt R

ate

(1/m

in)

0 0.002 0.004 0.006 0.008 0.01

Thickness of Al (cm)0.0081 350

0.00972 165

Figure 11.1: Count rate of radiation particles vs. thickness of aluminum shielding


While the count rate is clearly not a linear function of the shielding thickness x, you could not ion

e diffic u an R(x) curve for each pair of values, and then

ee which pair best matches your experimental data, but this approach would clearly be very tedious. Exponential curve fitting, like power-law fitting, is a good example of a technique in which linearization would work if you already knew the exponent – but you don’t.

11.2 LINEARIZING EXPONENTIAL RELATIONSHIPS

Fortunately, a better strategy exists. If we take the (natural) logarithm of both sides of equation 11.1, we get 00 ln)(ln)(ln RxxyxRxR

(just by looking at the graph) tell the difference between the exponential dependence of equat11.1 and certain power laws. Finding the value of the constant β would b ult as well. Yocould try different values of β and R0, calculates

= + =⇒ +β β (11.2) if we define )(ln)( xRxy ≡ . This is the equation of a straight line. Therefore, if you graph

)(ln xR vs. x, you sh ows such a graph.

Figure 11.2: Plot of the natural log of the count rate vs. aluminum thickness

Furthermore, the slope of this line is β, the value of the constant in the original exponential equation. You find the value of β by calculating the slope in the usual way. That is,

ould wind up with a straight line. Figure 11.2 sh

5

6

7

0 0.002 0.004 0.006 0.008 0.01

Thickness of Al (cm)

ln(c

ount

rate 8

9

)

12

12

12

12 )(ln)(lnxx

xRxRxxyy

xy

−−

=−−

=ΔΔ

=β (11.3)

Now, the data graphed above don't lie perfectly on a straight line, which is the result of experimental uncertainty. The line, however, looks like a good approximation to the data. The best fit line, however, seems to pass through the points (0 cm, 8.0) and (0.01 cm, 5.1). [Remember that, strictly speaking, both the y values should have the unit terms of


ln(counts/minute) added, but those terms will cancel when you do the subtractithis line is

on.] The slope of

1-cm 2900.8−=

− (11.4)

f log

to eterm

n ab cm-1. If you ure x in mm ins

am will then alculate and plot the natural logarithms (and the uncertainties in those logarith of the easur

People have also invented special graph paper that can be used to create semi-log plots r or computer. This paper is called semi-log paper, and we have

g plot of the radiation shielding data of Table 11.1 in Figure 11.5, and o aper at the end of this chapter.

Fundam ntally, the vertical axis on a piece of semi-log graph paper shows the values of y cor g o an it linear scale for ln y. To illustrate this, we have drawn the implicit line the r ge of Figure 11.5. Thacc e va shown on the left hand scale, we are really implicitly plotting the val display the right hand scale. This makes it easy to plot measured values dir out having to compute the logarithms. er such logarithmic scales that the pattern of y-marks on the left-hand sca de to at exactly after y increases by a power of 10. The printers of semi-log

h 0, and make the paper more flexible by he ttom edge of the axis. (The numerical

bels also simply apt the paper to the range of your own ata by relabeling ’s on the axis by a sequence of increasing powers of 10 that span our particular data. This is illustrated in Figure 11.5.

cm 0-cm 01.0

The point (0 cm, 8.0) where the line intersects the y axis, is (of course) the y intercept othe line. So the value of y(0) is the constant ln R

1.5=β

0 in equation 11.4. Therefore, R0 is the antiexponential) of 8.0, or 2980 counts/minute. Therefore, we can use the ln R vs. x graph(

d ine both the constants in equation 11.2; that equation now reads:

-1cm 290 where,)counts/min 2980()( −== ββxexR (11.5)

In the expressio ust be measured in cm, since the value of β has units ofanted to meas tead, β would have to have the value –29 mm-1.

ove, x mw

11.3 CREATING SEMI-LOG GRAPHS USING LINREG

On can easily create a semi-log graph using the program LinReg: simply enter the basic data into the data table and then check the “Show Ln(Vertical Data)” box. The progrc ms) m ements displayed on your vertical axis. (See Chapter 8 for a more detailed description of LinReg.)

11.4 SEMI-LOG GRAPH PAPER

without the help of a calculatoincluded a sample semi-loa blank sheet f semi-log p e

trespondin implicar scale onording to th

ight edlue of y

erefore when we plot a point on the graph

ue of ln y ed onectly with

It is a prople can be ma

ty of repe

paper do repeanot specifying t

t t e pattern of marks for each power of 1 power of 10 that you will use at the borepeat for each power of 10.) You adthe 1’s and 10

lady


It really doesn’t matter what power of 10 you assign to the bottom edge of the axis. Changing the definition of the power of 10 at the bottom simply shifts the logarithmic scale up or down. Since when we compute the slope we are only really interested in the differences between logarithmic values, shifting the logarithmic scale up or down doesn’t affect anything.

11.5 A PROCEDURE FOR EXPLORING EXPONENTIAL RELATIONSHIPS

Semi-log graphs are most useful when you suspect (for one reason or another) your data our suspicion, do the following:

ing

plotting your results using Ca paper. If the resulting graph is e a pretty good straight line, this reinforces your suspicion that the data might

exponential relationship, and it is worth e n2. Find the constant multiplier k, by extrapolating your best fit line back to x = 0 and reading

e value ln k o lue of k itself (if you used semi-log graph paper).

3. Calculate the value of β from the slope xy

has an exponential dependence of the form xkey β= . To test y

1. Plot a low-level semi-log graph of your data either using semi-log paper or by calculatvalues of ln y and rtesian graph in either casreflect an continuing to th ext step.

either th ff the vertical axis (if you used Cartesian graph paper) or the va

Δ Δ/)(ln as discussed in section 11.2. (If you used semi-log graph paper, you will have to compute ln y by hand for the two points that you use to define the slope.)

4. Finally, refine these values for intercept and slope by entering the data into LinReg. Be sure to check your r

EXERCISES

Exercise 11.1 Verify that if β = 29

xercis

er than Figure 11.2, your new estimates of β and R0 will probably be better an the earlier estimates summarized in equati i u calculate ln R for

two points on the line to accurately determine the slope, but you can read R0 right from the iagram.]

esults against what you found using your low-level plot.

0 cm-1, then β also equals 29 mm-1, as claimed below equation 11.7.

E e 11.2 Figure 11.5 on the next page shows a semi-log graph of the radiation data, with error bars. Draw

u think best fits the data, and then find values of β and R0 from your line. Since the line that yoFigure 11.5 is largth on 11.7. [H nt: Yo will have to

d


Figure 11.5: Semi-log plot of data from Table 11.1.

0.002 0.004 0.006 0.008 0.010

Thickness of Al (cm)


E e 11.3 On the blank semi-log paper provided in Figure 11.6, plot the data given in the table to the right. Determine whether this data seems to reflect an exponential relationship

time t (min) Number of bacteriaN

xercis

tβ

o

ainty

10 149,000 ± 15,000 eNN 0= , and if so, find the values of β and N0 that best fit this data from both graphs. Also, plot in your lab 20 215,000 ± 20,000

30 335,000 ± 35,000 40 477,000 ± 45,000 50 769,000 ± 75,000

n k a graph of ln N versus t on ordinary graph paper and do the same analysis. (You can use equation 9.16 in Chapter 9 of this manual to compute the uncert in

teboo

ln N.)


Figure 11.6: Blank piece of 3-cycle semi-log paper

117

Section IV: Ray Optics

12. Single Thin-Lens Optics 119

Chapter 12: SINGLE THIN-LENS OPTICS “ ‘Will not your mind misgive you, when you find yourself in this gloomy

chamber, too lofty and extensive for you, with only the feeble rays of a single lamp to take in its size…?’”

--- Northanger Abbey

12.1 PRELIMINARY COMMENTS

The purpose of this chapter and the next is to provide some background information about the optics of thin lenses that you need for the optics labs and the Speed of Light lab. We are providing instruction in optics in this course in context of the laboratory because studies have suggested that this is a topic that students learn much better from laboratory experiences than from classroom instruction.

12.2 THE RAY MODEL OF LIGHT

The behavior of light in most optical systems whose components are larger than few a tenths of a millimeter can be most easily understood in terms of a particle model of light. (We will discuss various models of light and the reason for the particle model’s size limitation at the end of Unit E and the first few chapters of Unit Q.) The particle model of light imagines that light consists of infinitesimally tiny particles that we call photons, which travel in straight lines through a vacuum, (essentially) straight lines through air (or any other uniform medium), and either rebound from or are absorbed by solid opaque objects. We call a possible photon trajectory through an optical system a ray of light and can represent it on a diagram of the system by a thin line.

Photons (say from the sun) falling on any point on the surface of an opaque object are typically scattered in all possible directions. If the surface is particularly shiny or glossy, there may be more photons scattered in some directions than others, but usually there are at least some photons scattered in any given direction. Since light travels in straight lines in air, the rays representing the possible paths of photons scattered from such a point therefore look like straight lines pointing radially away from the point as shown in Figure 12.1. When we look at such an object, our eyes collect some of the scattered photons and our brains construct a mental image of the object based on the number, color, and trajectory of the photons collected.

rock

sunlight

point

eye

F i g u r e 6 . 1 : A n i l l u s t r a t i o n o f h o w w e c a n r e p r e s e n t p h o t o n p a t h s o n a d i a g r a m u s i n g r a y s . T h e s o l i d r a y s o n t h i s d i a g r a m r e p r e s e n t p a t h s o f p h o t o n s c o m i n g f r o m t h e s u n , w h i l e t h e d a s h e d l i n e s r e p r e -s e n t s o m e p o s s i b l e p a t h s f o r p h o t o n s s c a t t e r e d f r o m a s p e c i f i c p o i n t o n a r o c k . ( T h e r a y s i n d i c a t e o n l y t h e p o s s i b l e d i r e c t i o n s t h a t p h o t o n s c a n t r a v e l : t h el e n g t h o f a r a y o n a d i a g r a m , u n l i k e a v e c t o r , h a s n o particular physical meaning.)


Note that the light that scatters from a given point on an illuminated object radiates away from the point in much the same way that it would if we were to replace the scattering point with a point source of light. In ray diagrams, we will typically ignore the light illuminating an object and treat points on the object as if they emit light. Ray diagrams also typically illustrate the rays emanating from only one or two points on an object, and even then just a handful of possible photon trajectories radiating from those points. You should always keep in mind that every point on an opaque object radially scatters light in an infinite number of possible directions.

12.3 THE FOCAL POINT OF A CONVERGING (CONVEX) LENS

A lens is a transparent object that bends the trajectories of photons moving through it. (We will discuss why materials like glass bend light in the third chapter of Unit Q). A converging lens is constructed with just the right shape so that it bends parallel rays of light to a single point, which we call the focal point of the lens, as shown in Figure 12.2. We call the distance between the center of the lens and the focal point the lens’s focal length f. Note that such a lens actually has two focal points: parallel rays going through the lens from the left will converge at the focal point on the right, while parallel rays coming from the right would converge at the focal point on the left. (Such rays are not shown in Figure 12.2.) Both these focal points lie on a line that goes through the lens perpendicular to the plane of the lens: we call this line the optic axis of the lens.

Note that in Figure 12.2, we have drawn the rays as if the lens bends them suddenly at the exact center of the lens. Actually, each ray is bent a certain amount when it enters the lens and a

optic axisfocal point

Figure 12.2: The focal points and focal length of a converging lens

focal length f

converging lens

focal point


certain amount more when it leaves the lens. If the lens is very thin comp(or other important distances in an optical system), details like this can b

ared to its focal length e ignored and we can

use the imple model that each ray is bent at the center. If a lens satisfies this criterion in a given contex It e a converging lens in the sense described above if its two sur es of a spherical surface and the lens is wider at the center than at the edges. In the Lens Systems experiment later in the semester, you will encounter a lens

herical surfaces, but which is narrower at the center than at the edges; this lens is a diverging lens. Lenses formed from sections of spherical urface

nly bring to a focus those rays that happen to lie in the plane of the

ject

-

other points on the object, you would see those rays emanating from any given point on the object being focused at a corresponding point in the plane on the right. Notice that to an eye farther to the right of the plane, photons from the object that go from A and B through the lens look exactly as if they had been originally emitted by points A´ and B´ respectively. Therefore, instead of the object appearing a distance p behind the lens, it will look to the eye as if it is floating in space upside down and a distance q in front of the lens. YoYet regth

en

st, we call it a thin lens.

turns out that a thin lens will bfaces are both small patch

whose surfaces are also small sections of sp

s s, whether converging or diverging are called spherical lenses. In the initial demonstrations, you will also work with some cylindrical lenses, whose front and back surfaces are patches of a cylindrical surface instead of a spherical surface. While a cross sectional view of such a lens (drawn perpendicular to the front and back curved surfaces) looks just like Figure 12.2, a cylindrical lens will opaper.

12.4 FORMING AN IMAGE WITH A CONVERGING LENS

Imagine that we place an object near the axis of a thin converging lens at a position a distance p from the lens such that p > f. Some of the rays scattered by a given point on the obencounter the lens, and it turns out that these rays become focused at another point to the right ofthe lens’s focal point. Figure 12.3 below shows how rays from two points A and B on an arrowshaped object are collected and focused at two corresponding points A´ and B´ on a plane a distance q from the center of the lens. If we could show you the trajectories of rays emanating from

u might want to try this during the first optics exploration lab: it is actually a bit tricky to see. the object is not really at that location, as you can verify by passing your hand through the

ion where the object appears to be. What we have in front of the lens is instead an image of e object.

An especially vivid example of an image like this is a trick vase that you might have sethat uses curved mirrors to project an image of a coin so that it looks exactly as if the coin werefloating at the mouth of the vase. When you move your hand to pick up the coin, however, you find nothing there! In the case shown in Figure 12.3, if you move your head just a little bit so that your eye no longer intercepts photons going through the lens, the image disappears. Cues like this let you know that the image that seems to be there is not in fact real. The designers of floating-coin vase have cleverly masked or eliminated most of these kinds of visual cues and masked scattered light from other sources: as a result, the floating coin image looks unusually bright and realistic.


converging lens

Figure 12.3: Rays that emanate from a given point on the object a distance p > f from the lens are focused by the lens to a corresponding point lying on the plane perpendicular to the lens’s axis and a distance q > f from the lens. Dashed lines are rays from point A and solid lines are rays from point B. Note that not all rays coming from the object are captured and bent by the lens; most of these rays miss the lens. (The diagram shows only two such rays for each point.)

Dsimpliforientatirays emaradiatingthat the t

1. Thth

thi

arallel to the axis of the lens. According to the definition of the focal point of the le

12.5 PRINCIPAL RAYS

rawing a diagram like Figure 12.3 is difficult and time-consuming. We can use a ied version of Figure 12.3, however, to quantitatively determine the location and

on of the image formed by a converging lens. We do this by considering three special nating from a given point A on the object, rays that we can call the principal rays from that point. (If you ever take another optics course, though, you should be aware erm “principal rays” is used somewhat differently in advanced optics.)

e first principal ray is the ray that goes through the center of the lens. Near the center of e lens, the front and back surfaces of the lens are essentially parallel, like an ordinary flat

piece of glass. Just as photons going through a flat pane of glass are not significantly deflected, so photons going through the center of the lens are not significantly deflected:

s principal ray is therefore simply a straight line. 2. The second principal ray is the particular ray that moves away from the point in a direction

initially pns, that ray will be bent so that it goes through the focal point on the other side of the

lens.

optic axis

focal point

imagefocal point

B

A'

object

A

B'

object distance p image distance q

focal length f


3. The third principal ray is that particular ray that moves away from the point in a directioninitially toward the focal point on the near side of the lens: again according to the definitiof the focal point, the lens will bend this ray so that it becomes parallel to the axis of thelens.

on

e are n

of

ret. Note

the right triangle ΔA´B´O (since they have the same acute angle). This means that

W interested in these special rays because they are much easier to construct accurately thaany of the other rays emanating from the point. Figure 12.4 shows that constructing these three rays from any given point A on the objectuniquely locates the corresponding point A´ on the image. (In a pinch, we can use any twothese rays to locate A´.) Such a diagram is relatively simple to draw and easy to interpthat if we know the focal length f and object distance p, we can use a diagram like this to determine quantitatively the image distance q.

second principal ray

12.6 THE THIN-LENS EQUATION

We can actually use Figure 12.4 to determine the quantitative relationship between the object distance p, the image distance q, and the focal length f. Note that the right triangle ΔABO is similar tothe ratio of the lengths of these triangles’ short legs should be equal to the ratio of their long legs:

pq

ABBA=

'' (12.1)

The triangles ΔDOF and ΔA´B´F are also similar: this means that

third principal ray

focal point

focal point

object distance p

image

DA

OF

B

A'

B'C

C'first principal rayobject

optic axis

focal length

image distance q

Comment [ACZ1]:


1−=−

=fq

ffq ''

DOBA (12.2)

Since

AB = DO, we can set these equations equal, yielding

1−=fq

pq (12.3)

vide both sides of this equation by q and then add 1/q to both sides, we get the thin-lens , a very important and useful equation that links p, q, and f.

If we diequation

fqp111

=+ (12.4)

CISES

e 12.1 re 12.4 on the previous page, draw in the principal rays for the object point C. (The rays

converge on the image point C´.)

EXER

ExercisOn Figushould

e object’s image?

ExercImagthe riA

Exercise 12.2 Imagine that an object is 22 cm to the left of a converging lens whose focal length is 15 cm. Where would you look for thANSWER:

ise 12.3 ine that an object is 18 cm to the left of a converging lens, and you find its image 12 cm to ght of the lens. What is the focal length of the lens?

NSWER:


EConsray mpositi sistent with the thin-lens equation.

xercise 12.4 ider the drawing below. Assume that the focal length of the lens is 7 cm. Use the principal ethod to locate the position of the object’s image. Then verify that your constructed on is con

converging lens object

optical axis

0 10 cm 20 cm 30 cm 40 cm

13. Lens Systems 127

Chapter 13: LENS SYSTEMS 13.1 INTRODUCTION

This chapter presents a discussion of more complicated issues in ray optics that builds on and extends the ideas presented in the last chapter (which you must read first!)

13.2 VIRTUAL IMAGES

Consider a situation where we place the object closer to a converging lens than its focal length. Figure 13.1 shows a ray diagram for such a situation.

third principal ray

second principal ray

first principal ray

focal point

object distance p

image distance q

focallength

object

image

A

A'

focalpoint

Figure 13.1: Formation of a “virtual” image by placing the object within the focal point.

In this situation, the rays from point A on the object never actually converge anywhere to right of the lens. On the other hand, these rays will appear to an observer exactly as if the

were not there and the rays were diverging from the point A´. Therefore, the object will look e observer as if it were magnified in size and set farther back from the lens than it actually

thelens to this. (This is the principle behind the operation of a magnifying glass.) This is a qualitatively different kind of image than the images formed by the lens in Figures 12.3 and 12.4 in the last chapter. We call an image where the rays of light radiating from a point on the object actually converge to a physical point in space (as in Figures 12.3 and 12.4) a real image. When the rays from an object point never actually converge toward a different physical point in space but simply appear to the eye as if they were radiating from an image point, we call the image a virtual image.


Note that to the eye, a real image and a virtual image look the same: in each case, the photons radiated from a specific point on the object and refracted through the lens look as if they were radiated from some other point in space than they actually are. Even so, these kinds of images are physically different. In the lab, with a lens system in front of you, you can decide if an image is real or virtual with the aid of an opaque white card. If you can a location for the card that gives you an in-focus image of the object, that image is real. If you can’t find such an image no matter how close to or far from the object you put the card, and you don’t see anything better defined than a bright blotch, the image is probably virtual. (We say “probably” because you might simply be having trouble finding the image.) Deciding from a ray diagram if a given image is real or virtual is more difficult but it can be done, and here’s how. If two or more actual rays traveling away from the source, not rays extended backwards (approximately toward the source), intersect, the image will be real. If you can get two rays to intersect only by extending one or both of them back toward the source, the image will be virtual. Even though we did not derive the thin-lens equation with this situation in mind, we can still use it to determine the image distance of the virtual image. Now, however, since p < f, 1/p > 1/f, so 1/q in equation (12.4) must be negative. In this case, a negative “distance” q means that the image lies on the other side of the lens than it does in the normal situation shown in Figure 12.4.

13.3 A DIVERGING (CONCAVE) LENS

Unlike a converging (convex) lens, a diverging (concave) lens is thinner in the middle than it is at the edges. If its surfaces are small patches of a spherical surface, then a thin diverging lens has the property of diverging initially parallel rays so that they appear to the eye as if they were radiating from a point that we call the focal point of the diverging lens. (See Figure 13.2.)

focal pointfocal point

diverging lens

focal length f

optic axis

re 13.2: A diverging lens bends initially parallel rays so they appear to an observer to be away from a point that we can define to be the lens’s focal point.

Figuradiating


an construct a ray diagram to locate the image produced by a diverging lens, as shown in Figure 13.3. Note that a diverging lens bends

hich initially is moving parallel to the axis of the lens), so that it ears to

y thaual.

describes the behavior of diverging lenses as well as converging hoices of signs of p, q, and f). As an optional exercise (that would

ally t

Just as in the case of a converging lens, we c

the second principal ray (wapp be radiating from the focal point of the lens. The third principal ray in this case is a ra t originally is going toward the focal point on the far side of the lens: this ray is bent parallel to the axis. The image produced by a single diverging lens is always virt The thin-lens equationlenses (with an appropriate cre est and develop your understanding), you might try to prove this using an argument similar to the one we used to get equations 12.1 through 12.4 in the last chapter.

firstprincipal ray

third principal ray

secondprincipal ray

image distance q

focal length f

focal point focal point

optic axisBA'

B'

A

object distance p

Figure 13.3: Using a ray diagram to locate the virtual image formed by a diverging lens. The ted lines show the trajectories that the photons appear to follow.

dot

3.4 MULTI-LENS SYSTEMS

We can use both the ray diagram technique and the thin-lens equation to analyze optical ystem

), we can do the same

ind of analysis even if one or both images are virtual.

1

s s consisting of two or more lenses. The basic technique is to take the object and use the equation and/or diagram to predict the characteristics of the image produced by the first lens inthe sequence. We then use this image as the object for the second lens in the sequence. See Figure 13.4 for an example of a two-lens ray diagram. Though Figure 13.4 shows a sequence ofreal images (mostly because the diagram is cleaner and easier to followk


object distance p focal length f

image distance qobject distance P

focal length F

A

B

A'

B'

A"

B"

object

firstimage

secondimage

image distance Q

Figure 13.4. A ray diagram for a two-lens system.

ject must actually be closer to the lens than the lens’s focal point to be magnified in this

a distant object, we need a more complicated

is

e. The objective creates an inverted real image of the distant object exactly at its focal point. If we arrange things so that this real image is just inside the focal length of a second lens (the eyepiece), the second lens then creates an apparently larger inverted virtual image of the object for the viewer, as shown in Figure 13.5. The angular size θ of this image can be much larger than the angular size of the original object θ0. The angular size of the final image will appear to the eye to be larger by a factor of θ/θ0. If we define the distance between points A´ and B´ to be h´, and assuming that the angles θ and θ0 are very small so that tanθ ≈ θ and so on (this will be true for realistic situations), then

magnification factor m =

10.5 CONSTRUCTING A TELESCOPE (optional)

The simple magnifier shown in Figure 13.1 works well increasing the apparent angular size of nearby objects, but is not suitable for increasing the angular size of distant objects, sincethe obway. To increase the apparent angular size ofsystem. We can construct a suitable telescope with two lenses as shown below. Imagine that we have an object very far from the telescope. Rays coming from any specific point A on the object will thus be approximately parallel when they reach the first lens of the telescope (which we callthe objective of the telescope) and will make a common small angle θ0 with respect to the axof the telescop

dFdf

dfhdFh

−=

+−

≈)/(')/('

0θ+θ (13.1)

Since the real image is typically very close to the focal point of the eyepiece in real telescopes, d is very small compared to either f or F, and thus the magnification factor is usually m ≈ f/F.


parallelrays fromobject's

objective lensfocal length of objective f

θ 0

θpoint A

eyepiece

focal lengthof eyepiece F

second objectdistance P

d

A'focal pointof eyepiece

B'B"

A"

second image distance Q

Figure 13.5: A simple refracting telescope. The objective creates a real image of the distant object, and the eyepiece creates a virtual image with a larger angular size than this real image, as if it were an actual object placed at the same position. A two-lens microscope is similar in construction: the microscope objective creates a real image of the object, and the eyepiece creates an inverted virtual image of larger angular size from this real image. A two-lens microscope gives larger and higher-quality images than are possible with a single-lens magnifier.

Exercise 13.1 s with a focal point of 20 cm. While working with the optical

bench, you are moved to place the object only 10 cm from the lens, as shown below. ) Wh

EXERCISES

Imagine that you have a convex len

(a at does a ray diagram look like for this lens arrangement? 20-cm lens

object

focal point

focal point

0 10 cm 20 cm 30 cm 40 cm

(b) Where does the thin-lens equation predict the image will land?


(c) Explain how we can reconcile the predictions of the ray diagram and the thin-lens equation. How would the eye interpret the rays coming from the second lens as forming an image? (d) What kind of image is this? Can you project it on an opaque screen? Explain. Exercise 13.2. (A system with two lenses, but no curve balls.)

se focal length is 20 cm and one whose focal m and then place an object 50 cm from the

cm lens will form an image?

s is e?

Imagine that now you are using two lenses, one wholength is 10 cm. You separate the two lenses by 50 c20-cm lens (see the drawing below). (a) Where does the thin-lens equation predict that the 20- (b) If you treat this image now as the object of the second lens, how far from that second lenthe imag (c) Where will the image of this “object” form, according to the thin lens equation? (d) What does the ray diagram from this setup look like? (You will have to draw in the focal points of the two lenses. Note the scale! Remember that you can get the principal rays for the second object from the location of the image formed by the first lens.)

20-cm lens10-cm lens

0 50 cm 100 cm 150 cm

(e) Is this ray diagram consistent with your prediction using the thin-lens equation?


xercise 13.3. (A virtual object.) You are using your 20-cm lens and your 10-cm lens again. This time you separate the two lenses by 30 cm and you put the object 40 cm in front of the 20-cm lens. (The scale has changed again since the last drawing.) (a) Draw the ray diagram for this situation. (Go as far as you can.)

E

20-cm lens

0 50 cm 80 cm

10-cm lens

20 cm

) Where does the thin-lens equation predict the first image will form this time?

) How far will this image be from the second lens? ) Wh

(e) Your result for exercise 13.1 should suggest how the eye will interpret the diverging rays lens as forming an image. Describe how to locate this second image on your

diagram.

(f) Check that the location of the second image is consistent with the thin-lens equation.

(b (c(d at problem do you encounter trying to use this image as the object for the second lens?

from the second

Checklists for writing a lab report Your short sections should:

Summarize the entire paper in the abstract Discuss quantitative results in both the abstract and conclusions State the problem or question under investigation in the introduction Summarize the experimental procedure in the introduction

Your eory section should: Th Start with the basic defining equations Show all non-obvious intermediate algebraic steps Clearly describe any assumptions and/or approximations involved in the model Display each equation on its own line Give each equation an equation number

Your cedure section should: Pro Provide a sketch or schematic diagram of experimental setup Provide an equipment list with model numbers and/or brief descriptions (if needed

for clarity or completeness when the sketch is insufficient) Describe all measurements, in roughly the order in which they were made Describe any departures from or details expanding on the procedure described in the

lab manual, if any (should follow the description of the appropriate measurement) Describe all steps taken to reduce experimental uncertainty (should follow the

scription of the appropriate measurement) Your Analysis section should

de

Briefly describe the data Include a Cartesian (unlinearized) graph of data Include linearized graphs of data, if appropriate Discuss consistency or lack thereof with any theoretical predictions Discuss how you calculated the slope and intercept of any linear graphs Show the calculation of any derived quantities from slope or intercept Completely discuss all uncertainties involved, showing sample calculations if

eeded n Discuss the results and their implications

Your graphs should have Axes scaled correctly with divisions equal to “nice” intervals Axes chosen so that the displayed data fills up as much of the graph as possible Tick values on each axis for the entire length of the axis Descriptive labels for each axis, including units Data points clearly plotted with uncertainty bars An appropriate title and, if there is more than one graph, a figure number early marked points used to calculate slope and intercept, if appropriate

Your ting should: Cl

wri Use correct spelling, punctuation, and grammar Use complete sentences Be clear, vivid, and concise Be typeset so that the report is legible and easy to read

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