Measurement & Estimation 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. 1
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 2
Part I: Measurements Have Uncertainty Getting Ready to Write a Report The first part of this lab is designed to practice the terms and skills that you read about during the Pre- Lab. Another 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 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), as well as how to incorporate the images you produce into Word. Instructions for all of the software can be found on the In- Lab Reference page. Do This: There is a link to the In- Lab Reference page on the red navigation bar on the homepage of the lab website. Find these files so that you can refer to them as you create your report today. 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! 3
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 a digital caliper with a readability of 0.01 mm. Equipment Gray Case containing o Machine Key o Centimeter ruler o Millimeter ruler o Calipers STOP 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 the 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 either 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. 4
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. S1 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. STOP 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. 5
Part II: The Intro Physics State Fair The Story The 2015 Missouri State Fair will be held in Sedalia from August 13 to August 23. In addition to deep fried candy bars, guessing games are a staple of state fairs. If you don t want to go home empty- handed, you need to be an esteemed estimator. Part II is your chance to improve your estimation skills! Equipment Mark Twain s The Adventures of Tom Sawyer Jar of jellybeans (one per classroom) 2. Estimation, Not Guessing As you may know by now, you will have to make reasonable estimates time and again throughout this semester. The key word here, which is unfortunately often overlooked, is reasonable. What that means is that you should be able to demonstrate a clear line of reasoning that led you to make your estimate. In that sense, a reasonable estimate is very different from a guess. Checkpoint 2.1: Estimate how many words there are in The Adventures of Tom Sawyer. Show your work and your reasoning. Do This: An example of an excellent response to Checkpoint 2.1 can be found on the In- Lab Links page of the lab website. Check it out! The actual number of words is quoted at the end of the example response. S2 Synthesis Question 2 (40 Points): Estimate the number of jellybeans in the big jar. You are NOT allowed to use any standard measuring devices (like a ruler) in this estimation. Nor are you allowed to move the jar or remove the lid. Be sure to show your work and your line of reasoning. Further, please include a colorful drawing of the jellybean jar as part of your response. Throughout the semester you will be asked to draw many diagrams. Using color well can make these diagrams more successful. 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. 6