# Because the slope is, a slope of 5 would mean that for every 1cm increase in diameter, the circumference would increase by 5cm.

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1 Measurement Lab You will be graphing circumference (cm) vs. diameter (cm) for several different circular objects, and finding the slope of the line of best fit using the CapStone program. Write out or paraphrase the questions below before answering them. Pre-lab: 1. Given that we will be plotting circumference (cm) vs. diameter (cm), a. What variables will be on each axis? EYA Circumference is on the y and diameter is on the x, because the name always is in the order of y vs. x b. What units should the slope and y-intercept of the line have? EYA The slope will be in units of cm/cm (unitless because the cm cancel), because the slope is rise/run, and both the rise and run are in cm. The y-intercept is in cm because the y-int. is the y-value when the x-value is 0, hence it has to have the same units as the y-axis. c. What is the meaning of the y-intercept in terms of this graph? EYA The y-int. is the y-value when the x-value is 0, so in this case, it is the circumference of a circle when the diameter is 0. d. If a group found the slope of this graph to be 5 (note, that this might not be the actual relationship you d expect), what would that mean? EYA Because the slope is, a slope of 5 would mean that for every 1cm increase in diameter, the circumference would increase by 5cm.

2 2. What do you already know about the relationship between the circumference of a circle and that circle s diameter? Be as specific as possible in your answer, if you don t know what the relationship should be, make sure you state that. Any specific relationship you know about the relationship between circumference and diameter or stating you don t know. 3. Write out a hypothesis (remember a hypothesis is an educated guess, not just any guess, so you must say why you think that it doesn t have to be correct, it just needs to be rationally explained why you think that). a. What do you expect to find as a slope of the line of best fit for the circumference vs. diameter graph? Any hypothesis for slope with numbers and reason b. What do you expect to find as a y-intercept for the line? Any hypothesis for y-intercept with numbers and reason

3 Experimentation: 1. Write out a data table in the form as shown below (you and your lab partner will measure each item twice to make sure of accuracy, if both measurements are close to each within a couple mm then take the average, if not measure a third time and take the average of the two closest numbers). Diameter (cm) Circumference (cm) Object description Meas 1 Meas 2 Avg Meas 1 Meas 2 Avg a. Your table will have 21 rows for data entry (23 total with headings as shown) 2. You will be measuring the circumference and diameter of 7 objects, and trading data with two other groups to get the 21 total, measured, data points. a. When measuring, some things to keep in mind. i. CAREFULLY collect data, to at least the mm precision. ii. Be smarter than the paper, if your measurements and your partner s measurements are more than a few mm off from each other, someone likely measured incorrectly, measure a third time to see which one is inaccurate. b. The 21, measured objects will consist of: i. 7 objects you measure, try to get objects that span a fairly large diameter range 1. 6 different objects with different diameters from the provided objects.. 2. One of your own circular objects, again with a different diameter (coin, pen, pencil, water bottle, etc.) ii. 14 other data points are from 2 other groups 1. Try to pick groups that do not have many of the same objects that you or the other group has a couple duplicates are fine per group, but the overall fewer duplicates, the better out data set will be. 2. Record the averages from the 2 other groups. Make sure you record them into the correct columns (sometimes groups accidentally flip the columns when collecting data), and enter all 21 points in your data table. Completed data table with the 21 data points

4 Analysis of Data: 1. Using CapStone, enter data of Diameter and Circumference Averages. You DO NOT have to put what the object was on the graph. 2. Determine if (0,0) would be an unmeasured, but valid, data point or not, if (0,0) is a valid data point, add it to the data in CapStone so it will be included in calculations and show up on your graph. 3. Add a line of best fit (linear fit) then print out two copies, one for each person. 4. Make sure the graph looks presentable (no connected lines, labeled axes, title, has an annotation with your names and class period, no legend, no uncertainties shown in the fit data etc.) 5. On the print of the graph, write out the equation for the line of best fit in terms of the circumference, diameter, slope and y-intercept with units. Graph with: No connected lines y-axis label of circumference (cm) x-axis label of diameter (cm) line of best fit your names on the graph point (0,0) on graph Example equation with m = 3.09, b = 0.15 ( ) ( ) ( ) ( )

5 Conclusions: (Again paraphrase the question) 1. Consider point (0,0)? a. What would that point actually mean? It means if the diameter of a circle is 0cm then its circumference would be 0cm. b. Did you plot (0,0) on the graph or not? (0,0) is plotted because it is a valid data point. c. Why is it or is it not a valid data point? (0,0) is valid because if a circle has no diameter (a single point), it has no distance around the dot either. 2. Interpret the equation of the line of best fit using the correct variables, slopes, and y-intercept, explaining the significance Don t just rewrite the equation in different variables and units, you already did that on the graph itself. Explain what the equation and relationship really means. Example: m = 3.09, b = 0.15 When the diameter is 0cm then the circumference would start at 0.15cm. Then for every increase of 1 cm in diameter, the circumference increases by Compare the results to your hypothesis, how close were the results to your hypothesis values? Account for discrepancies (did you not know, did you make a mistake thinking about the equation, was data collected incorrectly, etc) and similarities (why do you think it s similar). Compare your hyp. to your data. As long as you put how you knew or what you didn t consider, then it s correct.

6 4. Look up the mathematical relationship between the circumference and the diameter, if you didn t know it before. So, if data was measured perfectly, the circumference vs diameter graph would have a relationship of C = m D + b a. What are the theoretical values for the slope and y-intercept? What source did you use to find them (or did you already know)? Theoretical: m = π (~3.14); b = 0cm, state where you got numbers (or if you knew them) b. Find the magnitude of the percent difference between the experimental and theoretical slope values comparing the slope you found to the theoretical value for the slope (the closer your percent difference is to 0%, the closer your values are to the theoretical values) (show your work). The magnitude percent difference equation is below (notice the absolute value): ( ) ( ) ( ) Our measurements: m = 3.09; b=0.15 c. Comparing your experimental data to the theoretical values, what does the equation you found experimentally for the line of best fit mean in about the accuracy of your data measurements? Consider both the slope percent difference and the relative y-intercept values. Why does it tell you that? Example: 3.09 is only 1.59% off of 3.14, and 0.15 is close to 0, so our measurements were fairly accurate, not perfect, but close. 5. If you were to perform the same experiment but plot a diameter vs. circumference graph, how would your values and units change in terms of y, x, m, and b variables? EYA Because it s y vs. x in the name, the y variable would be diameter and the x variable would be circumference. The y-intercept wouldn t change, because the line passes through (0,0), so x and y intercepts are the same point. The slope would now be the ratio of diameter to circumference, so instead of 3.14, it d be 1/3.14 or

7 6. Explain some sources of error that could have affected this experiment. At least 2 sources of error. Example: hard to measure diameter precisely (might only have a cord, not the diameter) String hard to put around the circular object accurately Most objects weren t perfect cylinders, so diameter and circumference may have been measured at different spots.

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