Graphical Analysis and Techniques ​
Procedure
The goal of this exercise is for you to determine therelationship and constant of proportionality between the radius andarea of a circle. You may already know what this relationship is,but here you will attempt to “prove†it to yourself. You'll beprovided the diameters of several circles, from which you can findthe respective radii. The areas of the circles will be found by anindependent method. If we then plot a graph of area vs radius forthese circles, hopefully the shape of the curve generated willsuggest what the relationship is a allow you to “zero in on itâ€just like in the above example.
We will be using some data collected from circles of varyingsize cut out from rigid sheets of paper. If we first determine thearea of the rectangular sheets of paper and measure their mass, wecan compute the density of the paper. Thus the area of the cut outcircles can be determined by measuring their mass and using thesame density value.
Let's define the two-dimensional (or surface) density as: D =m/A
where m is the the mass, and A is the area it covers. Since acut out of this same paper will have the same density of the entiresheet, we can solve for the area by using the same density andmeasured mass. Thus we have: A = m/D.
Below is a set of data collected for two sheets of paper used togenerate the circles we'll use. That is followed by the dimensionsof the cut out circles.
Table 1: Measurements of Paper Sheets
| Mass (g) | Length (cm) | Width (cm) | Area (cm2) | Density (g/cm2) |
Sheet 1 | 9.198 | 27.93 | 21.63 | | |
Sheet 2 | 9.104 | 28.01 | 21.62 | | |
Average =
Table 2: Measurements of Paper Circles
Diameters (cm) | Mass (g) | Area (cm2) | Radius (cm) | Radius2 (cm2) |
4.88 | 0.308 | | | |
6.19 | 0.481 | | | |
7.09 | 0.624 | | | |
7.89 | 0.768 | | | |
9.15 | 1.012 | | | |
10.35 | 1.271 | | | |
11.75 | 1.667 | | | |
15.63 | 2.889 | | | |
1. Complete the area and density values in Table 1. Be sure toprovide one sample calculation of each here and remember to limitthe digits appropriately. The area of a rectangle is length timeswidth. Also, fill in the average density at the bottom of thetable.
2. Using the average density found for Table 1, use the massesof the circles in Table 2 to determine their respective area.Please provide one sample calculation here. Also compute the radiivalues from the diameters in Table 2.
