Measuring PI PDF

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Measuring Pi...

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Measuring π: Introduction:

Many aspects of measurement and measurement uncertainty are unfamiliar to beginning students. Simple measurements made of circular objects will allow you to determine en experimental value for the number pi (π). This measurement will provide a setting in which to examine issues involving uncertainty and measurement.

Learning Objectives:

Conceptual: Distinctions between mean and true value, between error, uncertainty, and standard deviation and between accuracy and precision. Experimental: Measurement techniques. The use of vernier calipers. Identifying and distinguishing sources of uncertainty. Analysis and Reasoning: Reasoning from graphs. Proper reporting of significant digits. Original Data issues. Report Writing: Data Summaries. Proper reporting of significant digits.

Before Lab:

Read sections A -E in Measurement and Significant Figures of the Lab Supplements packet for this lab manual. You can also find this material on the web: http://www.scidiv.bcc.ctc.edu/Physics/Measure&sigfigs/Measure&sigfigsintro.html

Experimental Overview: Our goal is to reproduce the well known direct proportionality of circumference C to diameter D. This means the corresponding graph of the function C = f(D) should be a straight line with its slope being equal to π. We are going to apply graphical analysis to the experimental points on the circumference-versus-diameter coordinate plane. First we need to collect circumference data. Several aspects of measurement technique will be raised as you do this.

Equipment Setup:

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Gather the following items from the lab cart or the Community Center. • ruler and vernier caliper • carbon paper and string • typescript correction fluid (“white-out”) • cylinders, disks, and balls of different diameters

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Activity One: Determine the Circumference of a Penny 1. Lay down a blank sheet of paper with the carbon paper face down on top. Roll a penny across the carbon paper to make a trace of its edge for several complete revolutions. Tap the top of the penny at the ends to make a clear mark. Using a metric scale, measure the length of the trace. Label this trace. Record your data in the table Penny Data at the end of the report. 2. To determine the circumference, divide the length of the trace by the number of turns. To assign an uncertainty to the length measurement (column 4 of the table) you must take into account at least the following factors. • Precision of the device • Sighting uncertainty • Accuracy of your method for determining a complete turn. See Questions 1 - 3 below to help you with this. • Possible slippage of the coin. You will also want to determine the uncertainty of the circumference (column 6). This is just the uncertainty in column 4 divided by the number of turns. 3. Repeat this measurement four more times with different numbers of turns. Work in pairs with two pennies to speed data collection. Record all your data in the table. 4. Now find: • the mean circumference • the deviation from the average circumference for each measurement (column 7) • the standard deviation (see the appendix). In all your records retain the proper number of significant digits. You may want to take out the appropriate sections from the appendix for quick reference.

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Questions:

(Q1) How do you decide that you have made a complete turn?

(Q2) How accurately are you able to do this? (Q3) Estimate the uncertainty your method contributes to the length of your trace. (Q4) Which of your measurements has the smallest uncertainty for the circumference (column 6)? (Q5) What factors contribute to this uncertainty being small?

(Q6) Which has the largest uncertainty? (Q7) What factors contribute to this uncertainty being large?

(Q8) Given your method for finding the circumference, what one thing could you do to make a more accurate determina-tion?

(Q9) Which measurement is closest to the average value? This is the most reliable single datum but you could not know this without making all the measurements. (Q10) Compare your smallest uncertainty (the one you identified in (Q4) above) to the standard deviation. Which is larger?

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(Q11) Based on your work, will you get a more reliable result by rolling the penny through one turn ten separate times, or by rolling it through ten turns once?

Activity One Continued: Circumferences of Other Objects 5.

6. 7.

Repeat steps 1 & 2 for a nickel and a quarter (one trial each, not five). Use your most reliable method and enter the values in the table Circumference Data. Add your best value for the penny’s circumference to the first row of that table. Repeat steps 1 & 2 for a large ball bearing. Wrap a string several times around the large ball bearing and around cylinder(s) provided; determine their circumferences.

(Q12) In what ways, not already accounted for, does the string affect the uncertainty of the measurement?

8.

Assign uncertainty estimates for the string measurements with these factors in mind.

Activity Two: Measuring Diameter When it is necessary to make measurements which are more precise, you must have a better instrument. First let’s see why we want a better instrument. 1.

Measure the diameter of the penny using your metric scale and assign an appropriate uncertainty. Then find the percent uncertainty.

Diameter

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, Percent uncertainty

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Questions:

(Q13) How does this compare to the percent uncertainty for your circumference measurements for the penny.? (Q14) How accurately must you measure diameter for both measurements to have the same percent uncertainty? One instrument that can measure more accurately is the vernier caliper. It is a slide type instrument used to take inside, outside, and depth measurements. The vernier caliper has two metric scales and two English scales. We will consider the two metric scales. One is fixed. It is located on the lower part of the beam. It is divided into centimeters and subdivided into millimeters, like on a usual plastic scale. The other one is called the vernier metric scale and is the lower scale on the slide. This scale is nine millimeters long and is subdivided into ten equal parts, so that each division on the vernier scale equals 0.9 mm. The part of the reading from the vernier scale is in tenths of a millimeter, which means that the precision of this instrument is 0.1 mm or 0.01 cm. How to read a vernier caliper in metric units: When measuring with an ordinary scale, the object itself indicates the endpoints of the measurement. When using a vernier scale, the first mark of the sliding scale indicates the endpoint. This is shown by the small arrow in the figure. Interpreting this mark the reading is D = 1.1 cm (to two digits). Now the indicator is past the 0.1 mm mark. But by how much? It seems further than half way, but it is not up to the next mark. 1 Scale 2 0 Fixed Here is where the vernier scale comes in to play. Sliding Scale Find the graduation on the sliding part of the beam which is most nearly in line with a graduation on the fixed scale. Inspection shows that the seventh mark seems to line up best with a line on the fixed beam and none of the other graduations of the sliding scale line up as well as the seventh. This makes 7 the next significant digit of our measurement. This represents 7/10ths of a millimeter, so the measurement should be D = 1.17 cm.

If the first graduation on the lower part of the beam is directly in line with a graduation on the fixed scale, then there are no tenths of millimeters to add to the initial two digits. Had that been the case D = 1.10 cm is the measurement.

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2. Using a vernier caliper, find diameter of a penny in centimeters. Record your measurements (in centi-meters) in the Diameter Data table including the uncertainty. Retain the proper number of significant figures. 3. Repeat step 2 for a nickel, a quarter, the ball, and, the cylinder, completing the table. Switch people for each object. If it is convenient for you during graphing, copy the circumference data to the table in the spaces provided. 4. Plot the results of your measurements as points on a coordinate plane of circumference versus diameter. A blank graph has been provided. Include error bars for each point. 5. Determine, by careful visual judgment, the straight line that fits the data best. The smaller the measurement uncertainty for a datum (hence the smaller its “error bar”), the more “power” it should have in determining the placement of the best-fit line. Because the linear regression programs built into most graphing calculators and spreadsheet programs do NOT make this distinction between high precision and low precision data, they are not suitable for our work. Data Sheets:

All of the paper you used today for tracing the circumference constitutes original data. You must never throw this away. Go over each page and label each measurement if you have not done so. Are there any numbers written anywhere on these pages? Circle any values recorded on these pages and write a brief caption. The caption should include the name of the quantity and the activity during which it was recorded. Do not erase any values during the course of the lab. The standard for all sciences is that you strike through values you do not need and write a correction nearby.

Cleanup:

Some of the items you used today came from the lab cart. Some came from storage places around the room or from the Community Center. You are responsible for returning items back to their proper place. This means you must remember where it was taken from at the beginning of lab.

Conclusion:

Compare the obtained experimental value of π with its true value 3.14159... . Quote the corresponding numerical values as part of your response.

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Data Tables. Answer the following questions below your graph: (Q15) The slope of the best-fit straight line has a meaning that can be related to measured values of π. Give a clear statement of that meaning. (Do you know how to translate a ratio as an English sentence? If you do not, please ask for advice.) (Q16) Estimate the accuracy of your measured value of π. Give the absolute error and the corresponding percent error for it. (Q17) Estimate the precision of your measured value of π. Give the uncertainty (obtained from the graph) and the corresponding percent uncertainty for it. (Q18) Is your measurement of π more accurate than precise or more precise than accurate? Choose the phrase (See Q18 below the graph) that best describes your result. Penny Data. 1

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Trial

Number of turns

Length of trace (cm)

Uncertainty (cm)

Circumference C = L/N (in cm)

Uncertainty (cm)

Deviation C-C (cm)

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Mean C --

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Circumference Data. Obje ct

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Number of turns

Length of trace or string (cm)

Uncertainty (cm)

Circumferenc e C = L/N (cm)

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Q

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BB(6 )

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C1

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Diameter Data. Obje ct

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Diameter

Uncertainty (cm)

Circumference C = L/N (cm)

Uncertainty (cm)

P

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N

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Q

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Circumference vs. Diameter for Several Objects. 30.0 28.0 26.0 24.0 22.0 20.0 18.0 16.0 14.0 12.0 10.0 8.00 6.00 4.00 2.00 0.00 0.00

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5.00 Diameter (cm)

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(Q15) Meaning of the slope (Q16) Accuracy:

Absolute error Percent error (Q17) Precision: Uncertainty Percent uncertainty (Q18) "Our measurements are of 1. high precision and high accuracy" 2. high precision and low accuracy" 3. low precision and high accuracy" 4. low precision and low accuracy"

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