WashingtonUniversityinSt.Louis* * MeasurementLab … · 2018. 8. 20. · WashingtonUniversityinSt.Louis* * MeasurementLab IntroductoryPhysicsLab* * Fall2015!! 3! Part)I:)Measurements)Have)Uncertainty)

Washington University in St. Louis Measurement Lab Introductory Physics Lab Fall 2015

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Measurement Pre-‐Lab: Introduction to Uncertainty

A Bit of History

You might never have thought about it, but the ability to measure accurately has had profound effects on the way humans (and not just scientists) see the universe. For example, the introduction of the first accurate clocks based on the motion of a pendulum radically changed the way humans thought about time. Even the meaning of the word hour was transformed by the introduction of accurate timekeeping.1 Before the pendulum clock, an hour was simply one-‐twelfth the time between sunrise and sunset, which means that the length of an hour varied throughout the year.

Our place in space has changed perhaps even more than our place in time. For starters, one of the great (and reasonable) objections to Copernicus’ heliocentric theory of the solar system was that the stars did not appear to move relative to each other throughout the year. Since the stars weren’t moving, it was absolutely logical to assume that they were all located on a sphere with the earth at the center. It was only with the introduction of better measuring devices that the stars finally moved and the earth was allowed to orbit the sun.2

As a third example, the problem of making precise measurements on very small scales (which comes up when studying quantum mechanics) makes one wonder what is predictable and what is completely up to chance, certainly one of the great overlaps between science and philosophy. Even the brilliant scientists that developed quantum mechanics in the first half of the twentieth century had trouble wrapping their minds around its implications. The list could go on. For the sake of brevity, we’ll stop there.

Reading from John R. Taylor’s Introduction to Error Analysis

Over the course of this term (and perhaps the next) you will be making lots and lots of measurements in lab. Even a measurement seemingly as simple as using a ruler can pose subtle problems. In this Pre-‐Lab, you will read an introduction to measurement from John R. Taylor’s An Introduction to Error Analysis after which you will complete a multiple choice quiz on Blackboard.

Do This: The reading can be accessed through the Olin library’s Ares service. You can find a link to Ares using the Pre-‐Lab Links page on the Measurements page of the lab website. Read the excerpt (about 18 pages) and answer the following questions. (You do not have to read the last half of the last page. Section 2.5 will not be addressed.) Note that the term “human error” is never used. We will never use the term “human error” because it doesn’t mean anything.


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PL1. What is the meaning of the term error in scientific settings?

PL2. Which of the following physical quantities can be measured with complete certainty?

PL3. When George measures the density of the crown (Section 1.3), what conclusion can he draw?

PL4. When Martha measures the density of the crown (Section 1.3), what conclusion can she draw?

PL5. What is the discrepancy between George’s and Martha’s values for the density of the crown (Section 1.3)? (The term discrepancy is defined on page 17 of Taylor’s book.)

PL6. According to the reading (Section 1.4), error analysis is important to which of the following people?

PL7. What is the term for the process of estimating positions between scale markings on an instrument such as a ruler or a voltmeter?

PL8. When using a digital stopwatch to perform an experiment, which of the following sources of uncertainty is greater?

PL9. If you were to perform an experiment using a stopwatch that consistently runs 5% too fast, then all of your measurements will be 5% too long. What is the term for an error of this sort?

PL10. Consider using a ruler with millimeter markings to measure the length of a pencil. Which of the following would be a reasonable estimate of the uncertainty in the measurement?

PL11. In most situations, the uncertainty that you assign to a measurement should be rounded to one significant figure. According to the reading (Section 2.2), what is the one significant exception to this rule?

PL12. According to the reading (Section 2.4), an interesting conclusion must…

PL13. According to the reading (Section 2.4), if there is an unexpectedly significant discrepancy between an experimental and theoretical value, there is reason to think that something must have gone wrong. Which of the following are possible explanations?

PL14. Which of the following terms should you never use due to its lack of meaning?

End of Pre-‐Lab


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Part I: Measurements Have Uncertainty

Getting Ready to Write a Report -‐ Extremely Important Information

This lab is designed to practice the terms and skills that you read about during the Pre-‐Lab. Another equally important goal of this experiment is to introduce you to the format that we would like to see in your reports. The In-‐Lab Links page on the Measurement page of the lab website gives example responses for questions that are very similar to the Synthesis Questions that you will be tackling today.

Do This: Go to the In-‐Lab Links page of the Measurement page of the lab website and find the example lab reports. Take a good look so that you don’t make the same mistakes that can be found in the poor lab report. There is also a file that describes in some detail how lab reports will be graded.

Read This: To respond well to the Synthesis Questions, you will have to become familiar with the tablet and the drawing software (SketchBook and/or Paintbrush, found on the dock), as well as how to incorporate the images you produce into Word. Instructions for all of the software can be found on the Reference page of the lab website.

Do This: There is a link to the Reference page on the red navigation bar on the homepage of the lab website. Find and read the instructions for Word, SketchBook, and Paintbrush. Your TA will be sad (L) if you ask a question that is answered in these short documents.

Read This: Your report will be created in Microsoft Word. Every report will start with the same template so that it is easy to include all of the necessary information.

Do This: Go to the In-‐Lab Links page of the Measurements page of the lab website. Download and open the Lab Report Template. Then rename the file using the format:

Partner1_Partner2_LabTopic.docx

For example, if Dan Flanagan and Drew Osterhout were completing the Measurement lab, they would title the file

DanFlanagan_DrewOsterhout_Measurement.docx

Read This: You are expected to save and back-‐up your lab report often. These computers will not save files after you log off or shut down, whether it happens intentionally or not. Periodically saving your report on a thumb drive or using web-‐based storage is highly recommended.

Read This: Now you’re ready to go!


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1. Length (At Length)

In the gray plastic case, you have a machine key (that hunk of steel) and three instruments for measuring its length: a ruler with 1-‐cm divisions, a ruler with 1-‐mm divisions, and digital calipers with a readability of 0.01 mm.

Equipment

• Gray Case containing o Machine Key o Centimeter ruler o Millimeter ruler o Calipers

Equipment Note: Please see page 2 of the Cleanup! Slideshow before you put the rulers back in the case. The rulers can be damaged if you put them back improperly.

Read This: As you make these measurements, you have to keep in mind the properties of the measuring device as well as the properties of the machine key. With the two rulers, you will have to perform interpolation. That is, you’ll have to estimate where between the lines the end of the object lies. With the digital calipers you’ll have to be careful to make sure that they have been properly zeroed. An improperly zeroed instrument can introduce systematic error to an experiment.

Read This: How do the properties of the machine key come into this? Notice that the machine key has rounded edges. Maybe the last millimeter or so is rounded. The rounded edges won’t likely make it difficult to use the ruler with 1-‐cm markings. Nor will the rounded edges affect how you use the calipers because they can clamp onto the flat ends of the machine key. However, the rounded edges make it difficult to use the ruler with millimeter markings. This might mean that you estimate the uncertainty to be larger than the 0.5 mm that was used with the pencil in the Pre-‐Lab reading.

Checkpoint 1.1: Measure the length of the machine key using the ruler with 1-‐cm divisions. Further, estimate the uncertainty in your measurement. That is, how confident are you in your measurement? Record the length of the machine key using the form 𝑥!"#$ ± 𝛿𝑥.

Read This: Remember that your lab report will consist of responses to Synthesis Questions. Checkpoints should not be answered directly in your report. However, you should write responses to Checkpoints in a notebook and/or discuss the Checkpoint thoroughly. These notes and discussions will help you when you get to the Synthesis Questions.

Checkpoint 1.2: Measure the length of the machine key using the ruler with 1-‐mm divisions. Further, estimate the uncertainty in your measurement. Be honest with this estimate! The rounded edges make this measurement a little difficult. Record the length of the machine key using the form 𝑥!"#$ ± 𝛿𝑥.

STOP


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Read This: When working with digital instruments, the instruction manual should tell you what the uncertainty in the measurement is. The instruction manual for these digital calipers states that their uncertainty is 0.02 mm.

Checkpoint 1.3: Measure the length of the machine key using the digital calipers. Further, estimate the uncertainty in your measurement. Record the length of the machine key using the form 𝑥!"#$ ± 𝛿𝑥.

Checkpoint 1.4: Was your best guess value the same every time? Does this mean some of your measurements were wrong? Discuss.

Read This: Your machine key is supposed to be 3 inches long. However, as you know, making something exactly a given length is impossible. The company that sold us the machine keys knows this as well and reported a tolerance for that length. The company states that the machine key might be 0.01 inches longer or 0.03 inches shorter than the quoted length.

Checkpoint 1.5: Convert the quoted length of the machine key into metric units. In addition, convert the maximum and minimum lengths (as given by the tolerance) into metric units.

Read This: In Synthesis Question 1, you will be asked to make a diagram that displays several measurements graphically, a technique that was shown in the Pre-‐Lab reading. In order to make your task easier, you can download a premade scale from the In-‐Lab Links page. The scale can be pasted into either Paintbrush or SketchBook. Then you can draw over it. As you make your diagram, it’s possible that the uncertainty of certain values will be too large or too small to show up well on the scale. That’s okay! Do the best you can.

Read This: Before you type your response, be sure to check out the example reports on the In-‐Lab Links page.

Synthesis Question 1 (60 Points): Do any of the measurements that you made strongly suggest that the length of your machine key is within the tolerance quoted by the manufacturer? Respond by creating a graphical representation of your three measurements as well as the range of lengths allowed by the manufacturer. Then write a paragraph analyzing the diagram that you have created. (Note: See the In-‐Lab Links page to find a pre-‐made scale that you can use to help create your diagram.)

Do This: Save a copy of your report on a thumb drive or using some web-‐based storage. Remember that you are responsible for having a back-‐up copy of your report. These computers will not save files after you log off or shut down, whether it happens intentionally or not. Periodically saving your report on a thumb drive or using web-‐based storage is highly recommended.

Equipment Note: Please see page 2 of the Cleanup! Slideshow before you put the rulers back in the case. The rulers can be damaged if you put them back improperly.

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STOP


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Part II: A Clever Measurement

The Story

Mark Twain’s immortal character Tom Sawyer is known for being a clever kid. (Just check out Chapter 2 of The Adventures of Tom Sawyer, the famous whitewashing scene.) In the upcoming experiment, you’ll have to be equally clever in order to measure the thickness of a page of Tom’s text.

Equipment

• Mark Twain’s The Adventures of Tom Sawyer • Millimeter ruler • Digital calipers

2. The Thickness of a Sheet of Paper

Checkpoint 2.1: Try to directly measure the thickness of a single page of The Adventures of Tom Sawyer using your millimeter ruler. Discuss why this is impossible in terms of uncertainty.

Read This: While directly measuring the thickness of a single sheet of paper is impossible with the millimeter ruler, an indirect measurement can be easily accomplished. By measuring the thickness of many pages at once, you can reduce the uncertainty in your measurement so that the measurement becomes meaningful.

Checkpoint 2.2: Determine the thickness of a page in The Adventures of Tom Sawyer by measuring the thickness of many pages at once. (We will address the uncertainty in the next Checkpoint).

Read This: How can we assign an uncertainty to your response to Checkpoint 2.2? All you have to do is divide the uncertainty in the total thickness by the number of pages that you measured. For example, let’s say you measured the thickness of 100 pages to be 0.78 ± 0.05 𝑐𝑚. Then the thickness of a single page would be 0.078 ± 0.005 𝑐𝑚 or 78 ± 5 𝜇𝑚. See Appendix A if you are interested in the rigorous proof of this extremely useful trick.

Checkpoint 2.3: Write your response to Checkpoint 2.2 in the form 𝑥!"#$ ± 𝛿𝑥.

Checkpoint 2.4: Use the digital calipers to directly measure the thickness of a single page of The Adventures of Tom Sawyer. Write your response in the form in the form 𝑥!"#$ ± 𝛿𝑥.


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Synthesis Question 2 (40 Points): Evaluate the success of the clever measuring trick that you just learned and practiced. A successful evaluation will include

• the thickness of a page as determined using a millimeter ruler written as 𝑥!"#$ ± 𝛿𝑥 • a brief (two or three sentences) description of how that thickness was found • the thickness of a page measured directly using digital calipers written as 𝑥!"#$ ± 𝛿𝑥 • an analysis of the discrepancy between the two measurements (a figure is NOT

required, though you may certainly include one if you’d like) • conclusion about the success of the trick

Time to Clean Up!

Please clean up your station according to the Cleanup! Slideshow found on the lab website.

References

[1] Dohrn-‐van Rossum, Gerhard. (1996). History of the Hour. University of Chicago Press, Chicago, IL.

[2] Danielson, Dennis and Graney, Christopher M. (2014). The Case Against Copernicus. Scientific American, January 2014, Volume 310, no. 1, 72-‐77.

S2


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Appendix A: Reducing Uncertainty by Measuring Many Pages at Once

The Problem at Hand

Consider a measured thickness, 𝑥!"!#$, that represents the total thickness of some number of pages 𝑁. Mathematically, the thickness of a single page, 𝑥!"#$, can be given by

𝑥!"#$ =𝑥!"!#$𝑁

Here the thickness of a page is a function of only one uncertain measurement (𝑥!"!#$). Before we continue with this specific case, let us investigate the general form of this problem.

The General Case

Consider a quantity 𝑞 that is a function of a single measured value 𝑧. We could write this in a very general way as

𝑞 = 𝑓(𝑧)

When we have such a relationship (a function of a single variable), the uncertainty 𝛿𝑞 in the calculated value 𝑞 is given by

𝛿𝑞 =𝑑𝑓𝑑𝑧𝛿𝑧

where 𝛿𝑧 is the uncertainty in the measurement of 𝑧.

Back to the Problem at Hand

In our problem, 𝑥!"#$ is like 𝑞, 𝑥!"!#$ is like 𝑧, and !!"!#$!

is like 𝑓 𝑧 .

That means to find the uncertainty in 𝑥!"#$, we just take the derivative of !!"!#$!

with respect to 𝑥!"!#$

and then remember to multiply that by 𝛿x!"!#$. We find

𝛿𝑥!"#$ =1𝑁𝛿x!"!#$

Since 𝛿x!"!#$ does not change as x!"!#$ increases or decreases, this equation shows that increasing 𝑁 will reduce 𝛿𝑥!"#$.

Note that this calculation relies on the assumption that there is very little variation in thickness from page to page.

Documents

Washington*University*in*St.*Louis* * Measurement*Lab* … · 2018. 8. 20. · Washington*University*in*St.*Louis* * Measurement*Lab* Introductory*Physics*Lab* * Fall*2015*!! 3! Part)I:)Measurements)Have)Uncertainty)

WashingtonUniversityinSt.Louis* * MeasurementLab … · 2018. 8. 20. · WashingtonUniversityinSt.Louis* * MeasurementLab IntroductoryPhysicsLab* * Fall2015!! 3! Part)I:)Measurements)Have)Uncertainty)