What is the relationship between radius and circumference?
Purpose: To learn about graphing and linear relationships using circles.
Materials:
 Cylinders of varying sizes
 Meter stick or ruler (you MUST use metric quantities)
 String
 Computer with Logger Pro (later in the lab)
Procedure:
1) Go to lab and select one cylinder to measure.
2) As a group, decide on a process to accurately measure the radius and circumference of your cylinder.
3) Make your data table in the space below, be sure to label each column. If you decided to make
multiple measurements of each cylinder, make sure that your data table has space for repeating
measurements.
4) Take measurements of your cylinder.
5) No two measurements are exactly the same. This is the uncertainty in the experiment. If you make a
single measurement, you can estimate the uncertainty based on the tool you used (e.g. the ruler). If
you make multiple measurements, a good estimate of the uncertainty is to take the range of values you
measured (biggest minus smallest) and divide by the number of measurements taken. Record an
uncertainty with your measurements.
6) If you made multiple measurements, calculate the average radius and circumference for the cylinder.
7) Get a new cylinder. Repeat steps 3 to 6 to find the new cylinder’s radius and circumference.
8) Repeat the process for at least 4 cylinders.
a) On your completed average data table with all 4 cylinder information circle the largest
experimental uncertainty value. You are only as good as your worst measurement so the circled
value will be the uncertainty value you use on the computer.
9) Log on to the network. Open Logger Pro.
a) Double click column x and name it: Radius. Short name: R Units: cm. Press done.
b) Double click column y and name it: Circumference. Short name: C Units: cm. Press done.
c) Now, fill in your data in the approriate field. The graph should be made as you enter data.
Save this document as “circumference graph”
Describe the general shape of the graph. Is it shaped like a bowl, line, curve, nothing?
d) Add the uncertainty bars by double clicking on any data point. The dialog box that opens has an
options tab. In options, check the box by “error bar calculations”. Select “fixed value” and “error
constant +/-“ and enter the uncertainty value you found in 8a.
e) Save the graph as “circumference graph”
f) You should notice the graph is linear. Logger pro will draw the linear best fit. On a linear graph
all of the points of data should be exactly on one line!
Physics
Circumference Lab (white) [Type text]
g) Click on the linear fit (r =) icon
line appears on the graph.
. You now have a line on our graph. The equation for the
h) To view the graph better, select the auto scale icon
.
i) The equation for a linear graph is y = mx + b Define each of the following variables:
(a) The y is for the y axis, what is the label on the y axis of your graph?
(b) The x is for the x axis, what is the label on the x axis of your graph?
(c) The b is for the y intercept also known as the starting point on the y axis. At what y
value does the line you drew cross the y axis?
(d) The m is for the slope of the line of data. The slope of a line can be calculated from
rise divided by run.
(e) Write a y=mx +b equation where you replace the m with your calculated slope
value and the b with your estimated y intercept value.
Show your equation to the instructor at this time and have it signed.______________________ if
this is not signed you will lose credit.
Data Analysis:
10) Copy and paste the graph you made into a Word document.
11) Here we will see what the data actually means. It was useful to learn how to make a trendline, but
now we want to know more! What is the equation for the circumference of a circle?
a) Compare the circumference equation to our equation for a line. Write the equation for the
circumference of a circle right above the y = mx + b below.
b) Notice that the C and y are alone on the left side of the equation.
Circle the C and y together in one circle.
y =mx+b
Notice that the r is right above the x. Circle the r and x together in one circle.
c) Notice that 2π is right above the m. Put a box around the 2π and m. This must mean that the slope
of our line on the graph is equal to 2π. Is it? Of course 2π is equal to 6.28. Was your slope 6.28?
d) What about b? There is no value above the b in the circumference equation. Answer in your
notebook what was the b value from the graph we printed? What is the value of b in the
circumference equation?
e) Your linear fit line gives you an uncertainty for your m value (you may need to right click on the
fit box, then turn on uncertainty in the options menu). If you add and subtract that value from the
slope value given on Logger Pro, does it overlap with 6.28?
Conclusion:
Write a brief (3-5 sentence) conclusion in the same Word document as your graph. Your conclusion
should address whether your results agreed or disagreed with the expected result, sources of uncertainty in
your lab, and changes that you would make if repeating this experiment in the future. Submit the
conclusion to the instructor using Google Drive or by email.
Physics
Circumference Lab (white) [Type text]