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Investigating the Relationship Between Circumference and Diameter Purpose: to determine the relationship between the circumference and the diameter of cylindrical objects; to determine the accuracy of measurement; to compare accuracy and precision Procedure: Materials:five cylindrical objects, ie. metal tubing, salt containers, etc string (non-elastic) metric ruler graph paper or graphing program 1. 2. 3. 4. Obtain at least five cylindrical objects for measurement. Construct a data table that lists the name of the object, the circumference (cm) and the diameter (cm). C and d MUST be measured…not calculated. Make measurements using an estimated digit with the school metric ruler (Each measurement should have two decimal places in cm.) Carefully measure the diameter and the circumference of each object and record. Questions/Analysis: 1. 2. 3. For each object, divide the circumference by the diameter. Show your work and your answers. Do you notice any pattern or similarities between the resulting numbers? What unit does the resulting number have? Construct a graph by hand or by using a graphing program such as Graphical Analysis. (Excel can also be used, but gives no statistics on the resulting line formed.) To construct by hand, place the circumference on the y-axis and the diameter on the x-axis. To determine the scale for the axis, follow this procedure: For the y-axis, first count the number of boxes on the y-axis; then divide the highest circumference value in the data table by the number of boxes. Round UP to the nearest workable value. For example, if when divided the answer is 0.033…round to 0.05. If the value is 1.7…round to 4. 5. 6. 7. 8. 9. 10. 2. Usually multiples of 1, 2 and 5 work best for counting. This is the number that EACH BOX will represent. But, label the lines, not the box. DO NOT label every line since it will look crowded; instead label ~every 5th line. To construct the x-axis (diameter), count the number of boxes on the x-axis and then divide the highest diameter by the number of boxes. Finish as in Step 3. Using the method above will allow use of nearly all of the axis. Graphs do not look good if they are scrunched in one corner or if the data points go off the page. Plot the five points from the data table on the graph paper. Make a best-fit line by taking a ruler and drawing the BEST AVERAGE line through the points. You DO NOT have to start at zero, although for this graph the line should end up going through (0,0). End the line near the last data point. THE LINE DOES NOT HAVE TO TOUCH ANY POINT YOU HAVE COLLECTED! This is called a REGRESSION line, which is an average of all data points. Some points should lie above and some below the line. Give the graph a title; usually the format Y vs. X, in this case Circumference vs. Diameter. Line graphs in science are generally titled this way. Make sure that each axis is properly labeled and contains units: Circumference (cm) and Diameter (cm) Make an assessment of your graph using the following rubric: Criteria Variables 0 points The independent and dependent variables are not shown. Label The axes are not labeled. 1 point The independent and dependent variables are present, but they are improperly placed on the graph. The variable name is shown, but the unit is missing. 2 points The independent and dependent variables are placed on the correct axes. The correct variable name and unit are shown on each axis. Title The graph does not have a title. The graph has a title, but the title is not correctly written. Scale No scale is shown. Plots Data points do not accurately present the information. Graph Line The graph line is not shown on the graph. The scale is shown, but it is inappropriate for the data. Data points are shown, but one or more points is/are improperly placed. The graph line is shown, but the position of the line does not accurately represent the data. 11. 12. The title of the graph follows the generic form of "y versus x". The scale for each axis is appropriate for the data. All data points are clearly and accurately plotted. The graph line correctly represents the data. Find the slope of the line on the graph by using the formula y/x or Rise/Run. What are the units for this slope? Calculate a percentage error using the following formula: |EXP – THEO| THEO x 100 In this case, the EXPerimental value is the slope you found. The THEOretical value is 3.14. Absolute value is used so that all percentage errors are positive. 13. 14. Why is the theoretical value for this lab 3.14? What error is associated with measuring diameter? DO NOT INCLUDE HUMAN ERROR LIKE MISREADING THE RULER….WE ALWAYS ASSUME SOME HUMAN ERROR! 15. 16. 17. 18. What error is associated with measuring circumference? Only include error associated with instruments/tools used. The percentage error you found is called the ACCURACY…how close the experimental value is to the standard or theoretical value. PRECISION is the reproducibility of the measurements. When you divided C by d in question 1, were the values close to each other? If they were close, precision was good; if they were random, precision was poor. Describe your precision. Using a dartboard as an example, describe how a dart thrower, using three throws, could have the following: (A) good accuracy and good precision (B) poor accuracy and good precision (C) poor precision and poor accuracy In this lab, your accuracy could be good with the precision being poor. Explain using your graph to support your argument. Conclusion: Write a paragraph that summarizes the results and addresses the purpose of the lab. There are three parts of the purpose to address in this conclusion. DO NOT RESTATE THE PURPOSE AND DO NOT LIST ALL THE DATA!