Lab 1 - Measurement - Volume of the Library

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Apr 3, 2024

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Measurement - Volume of the Library Overview The basic activity of any experiment is making measurements. Scientists make measurements to make new discoveries. Engineers make measurements to build things. Physicians make measurements to diagnose and treat people. Understanding what a measurement is and how to make a measurement is central to the work many people do. There are some crucial characteristics of measurements to keep in mind: ● There is no such thing as an exact measurement. Every measurement has an error. ● Error does NOT mean mistake. An error is the limitation of the measurement. A range of values that the measurement is still valid. ● Error IS a part of the measurement. You can MEASURE error! Objectives The objectives of this experiment are: ● Practice how to design an experiment. ● Make measurements. ● Determine the error in the measurements. ● Learn techniques of propagating error of the results. Part 1 - Experimental Design What is the volume of the W.E.B. Du Bois Library? First, you will be designing a method to measure the volume of the library here on campus. When designing an experiment there are some basic questions you need to answer: ● What is the model of the system (library) you are measuring? Meter stick ● What physical quantities does a model have in common with the system? Meter. ● How will you measure these quantities? By rotating the meter stick to measure. ● What limitation (error) do you estimate in each of the measured quantities? It might not be accurate and while I am rotating the meter stick, it might not rotate properly, causing the result to me longer or shorter. University of Massachusetts - Amherst Spring 2023 Physics Department 1/8

By Ktr101 - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php? curid=17831764 Now with your lab partner, design a method to measure the volume of the library. You may only use a meter stick to measure the volume of the library. NOTE: You are not being graded on your accuracy. The variations from student to student is a crucial part of this lab! 1. Describe your methods of measuring the volume of the library. Include a picture or diagram of your method. I used the meter stick, and measured how many meter sticks there were on the length and width through rotating the meter stick. I also counted the bricks of one floor, and measured the length of one brick. Part 2 - Individual Measurement Now you and your lab partner will go to the W.E.B. Du Bois Library to measure its volume. The only thing you will have to measure with is a meter stick. EACH STUDENT MUST MAKE THEIR OWN INDIVIDUAL MEASUREMENTS! DO NOT SHARE YOUR MEASUREMENTS WITH ANYONE YET! University of Massachusetts - Amherst Spring 2023 Physics Department 2/8

WARNING: DO NOT ENTER THE FENCED IN SPACE AROUND THE LIBRARY! THIS IS RESTRICTED SPACE! 2. Is your method of measuring the library successful? Explain why or why not? If not, explain what new method to measure the volume of the library you used. Yes. I was successful at finding the overall length and width. However, for the height, I used the added height of each floor excluding the concrete floors’ height. 3. Record your INDIVIDUAL measurements of the dimensions of the library. Include your estimate of the error (+/-) of each dimension. Length (m) Width (m) Height (m) 33+/-0.2 33+/-0.2 63.7+/-10 4. Calculate the volume of the library using the dimensions you measured. What do you estimate is the error (+/-) in your value of the volume? The error of an individual measurement is an estimate. It is your best judgment of the limit you are able to make a measurement. Volume V (m) 73866.87+/-10000 Part 3 - Statistical Analysis - Lab Section Data You and your sectionmates have just measured the dimensions of the library. You all measure the same library, but do you all have the same measurements? The answer is, of course, no. Even if everyone used the same method to measure the dimensions of the library, and even if they used the same meter stick, there still are differences in the values of the measurements. Those differences are the errors in the measurements. There are good measurements, bad measurements, great measurements, and measurements that need improvement. But there is no such thing as a right or wrong measurement. But we make measurements assuming there is a true value to the measurement. We cannot measure the true value, but we make a measurement of the best value. The best value is the value we believe is closest to the true value. How do we determine the best value? If there is a true value to a measurement, some of the measurements you and your sectionmates made may be above the true value, while other measurements are below. The best value of a measurement, the value closest to the true value, is the average of all the measurements made. University of Massachusetts - Amherst Spring 2023 Physics Department 3/8

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How about the error? Again, some measurements are higher than the average, while others are lower. The range of measurements is the error. That range of measurements is the standard deviation of the measurements. ★ The average value of a set of repeated measurements is the best value of the measurements. ★ The standard deviation of a set of repeated measurements is the estimate of the error in the measurements. Let’s now compare all of the measurements made. From Moodle access the Google Sheet “Volume of the Library - Section Data.” You have to log onto Google Workspace (formally Apps) at UMass using your UMass NetID and password; NOT your personal email account. If you are already logged onto your personal Google account through your browser, you may have to log off. ● Open the spreadsheet Volume of the Library - Section Data. ● Click the tab at the bottom of the spreadsheet that corresponds to your lab section. Look for the two letter code that matches your Physics 151 lab section schedule in SPIRE. If you do not know what the two letter code is, ask your lab TA. ● Select a row in the spreadsheet and enter your UMass NetID ( NOT your 8 digit SPIRE number) and your measurements of the length , width , and height of the library. 5. Calculate the volume of the library using your values of , , and in your row of the spreadsheet. Remember spreadsheets are calculators, so DO the calculation in the spreadsheet. For example, if your measurements are in row 5, then in the column for volume (Column E) enter the equation =B5*C5*D5. ● When everyone in your lab section has entered their values of , , , and into their row of the spreadsheet, your lab TA will calculate the average =AVERAGE(), standard deviation =STDEV(), and percent error of the values starting in row 30. The percent error is the ratio of the error to the average value. For example, if the average value of the height of the library is and the error in the height is , then the percent error is: , NOT 6. Record the average +/- standard deviation of length , width , and height and volume of your lab sections measurements in the table below: University of Massachusetts - Amherst Spring 2023 Physics Department 4/8

Length (m) Width (m) Height (m) Volume (m 3 ) 32.6+/-2.25 32.59+/-2.24 93.19+/-16.64 98937.13+/-19350.63 7. Is your personal value of the volume higher or lower than the average ? Which of your measurements of , , and do you think contributed to your value of being higher or lower than the average value of ? Explain your answer. Mine is lower. My measurement of height contributed the most as it is much lower than the average height. This is because I measured the actual height within each floor instead of the total height of the library. Part 4 - Physical simulation of the Monte Carlo Method Why is there a range of values of a measurement? Again, even if everyone used the same method and the same meter stick to measure the dimensions, there still would be a range of measurements. Making a measurement ultimately means making a judgment. Where to place the meter stick? Which mark on the meter stick is closest to the length of the library? The answer to these questions are judgments you have to make. Inherent in making these judgments is randomness . Sometimes you judge too short or too long. This randomness in judgment of a measurement is the origin of the error in the measurement. Again, you did not make a mistake; the error is a part of the measurement. If randomness is inherent in making a measurement, then we want to use a method of determining the best measurement that has randomness built into it. This method is called the Monte Carlo method. The method gets its name from the city of Monte Carlo, which is famous for its casinos (think randomness of rolling dice or shuffling a deck of cards). The basics of the Monte Carlo method is to make a random selection of a set of values, then use that random selection to compute a quantity of interest; like the volume of the library. Now form teams of 4 students at each station in the lab. At each station you will have 3 cups: red, green, and blue. On small pieces of paper each teammate will write down their measurements of the dimensions of the library. Place the pieces of paper into the 3 cups; length in the red cup, width in the green cup, and height in the blue cup. Now take turns shaking each cup and randomly drawing a piece of paper with the values of the dimensions of the library. Record the random selected values in the Google Sheet “Volume of the Library - Team Data.” Each teammate must make a spreadsheet in order to edit the spreadsheet. click the menu File > Make a copy to make your own personal copy. You will submit this spreadsheet as part of the assignment for this experiment. University of Massachusetts - Amherst Spring 2023 Physics Department 5/8

Put the pieces of paper back into the cups and allow your teammate to shake the cups and draw 3 new values of the dimensions of the library. Yes, it is possible that pieces of paper may be drawn more than once. Do this a total of 24 times (6 times for each of the 4 teammates). 8. For each draw, calculate the volume from the randomly selected dimension of the library. 9. Calculate the average =AVERAGE() and standard deviation =STDEV() for each dimension of the library. 10. Calculate the percent error on each of the three dimensions. 11. Using the percent error, which of the three dimensions will have the largest contribution to the uncertainty of the volume? Justify your answer. The height. Because it has the highest percent error which means it will affect the volume more than the other factors. Part 5 - Propagation of Error By now you understand that any measurement you make has an error associated with it; the error is a part of the measure. You also understand how to determine error in a set of measurements using average and standard deviation. The question now is how do the errors in the measurements of the dimensions of the library connect to the error in the volume of the library. It is not too hard to imagine if the length, width, and height all have error, then the volume of the library, which depends on these dimensions, must have an error too. The method for determining the error in volume of the library based on the library’s dimensions is called propagation of error . When you calculate the actual volume the errors in the dimensions are propagated to the error of the volume. We will look at a few different techniques of error propagation. The first technique is algebraic. In the previous part you calculated the percent errors for each of the dimensions of the library. The percent error of the volume of the library is simply the sum of the percent errors of the dimensions: This technique is called provisional error propagation. A slightly more sophisticated technique adds the percent errors in quadrature . University of Massachusetts - Amherst Spring 2023 Physics Department 6/8

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12. Use the technique of adding percent errors in quadrature and calculate the percent error of the volume. Use the average and standard deviation of , , and from the spreadsheet “Volume of the Library - Section Data.” Another technique of error propagation uses the Monte Carlo method. In Part 4, you and your teammates randomly selected values of , , and of the library. Combining those values will give you a random value of the volume . 13. In the spreadsheet “Volume of the Library - Team Data” calculate a value of the volume for each of the 24 randomly selected values of , , and . 14. Calculate the average and standard deviation of the 24 values of . 15. Calculate the percent error of the volume using the average and standard deviation of . 16. Tabulate the dimensions , , , and for the three methods of determining the volume of the library: Individual measurement, Section data, Monte Carlo method. Method (m) (m) (m) (m 3 ) Individual 33 33 67.83 73866.87 Section Data 32.6 32.59 93.19 98937.13 Monte Carlo 33.3 33.4 89.6 99378.51566 17. Finally tabulate all of the percent errors for the three methods of determining the volume of the library: Individual measurements, section data, Monte Carlo method. Method Individual 0.03 0.03 0.24 0.27 Section Data 6.91 6.88 17.86 20.35 Monte Carlo 0.9 0.9 20.87 19.55 18. Ask your lab TA for the value of the volume of the library. Which of the three methods agree with the given value of ? Explain how you judge agreement. The volume my TA measured is 64288m^3. The closest measurement is my individual method. The reason might be that the individual method was taking the concrete and the solid space into consideration. University of Massachusetts - Amherst Spring 2023 Physics Department 7/8

19. How does your individual estimate of the error in the volume compared to the other methods (Section Data and Monte Carlo)? Was your estimate too high, too low, or consistent with the other methods? My individual estimate of the error was too low compared to other methods. University of Massachusetts - Amherst Spring 2023 Physics Department 8/8

Lab 1 - Measurement - Volume of the Library

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