Please copy and paste this embed script to where you want to embed

Sample Lab Report TITLE: The relationship between the circumference of a circle and its diameter.

circumference of a circle related to the diameter? diameter? The THEORY: How is the circumference relationship is accepted to be C = πD, where C is the circumference of the circle, D is the diameter of the circle, circle, and π is the constant constant pi = 3.141593. Thus, the circumference and the diameter diameter should be directly proportional. proportional. A plot of C vs. D should yield a line of slope π. The ratio Pi = C/D can can be calculated and compared compared to the accepted accepted value of π = 3.141593. The measured and accepted accepted values can be compared compared by calculating the percent error using . The values C and D can can also be compared by plotting plotting C vs. D (C on on the vertical and D on the horizontal). If the data follows follows the equation C = πD, then then the plot will be a line of slope π and y-intercept 0. PURPOSE: The purpose of this experiment was to gain laboratory experience by measuring the ratio of circumference to diameter two ways that agree with theory (using mean and standard deviation as well as graphical analysis), and then comparing the results. DATA TABLE: Object

Penny Battery Iron Cylinder Rubber tube Can

Diameter (cm) 1.90 3.30 4.23 6.04 6.6

Circumference (cm) 5.93 10.45 13.30 18.45 21.2

σD (cm)

σC (cm)

0.01 0.02 0.02 0.02 0.1

0.03 0.05 0.03 0.05 0.1

CALCULATIONS:

Pi = C/D,

Penny – Pi = C/D: Pi = 5.93/1.90 = 3.1211, 0.0228

=

Battery – Pi = 10.45/3.30 = 3.1667,

=

0.0245 Iron Cylinder – Pi = 13.30/4.23 = 3.1442,

= 0.0165

Rubber Tube – Pi = 18.45/6.04 = 3.0546,

=

0.0131

Can – Pi = 21.2/6.6 = 3.2121,

=

0.0510

,

= 3.14

= 0.0561

= 0.05073

Slope = π theo =

= 3.25

= 3.451% CALCULATIONS TABLE:

Object Penny Battery Iron Cylinder Rubber tube Can

= ___3.14_____

πtheo = ___3.25_____

Pi = C/D (no units) 3.12 3.17 3.14 3.06 3.21

= ___0.06_____

σPi (no units) 0.02 0.02 0.02 0.01 0.05

%Error Pi = ___0.051%_____

= ___3.5%_____

Note: The graph should fill an entire page in your notebook. This graph fills the entire page, but was shortened and rotated after scanning to be placed in this lab report.

CONCLUSION: The relationship between circumference and diameter of a circle was investigated in two ways using the same set of data. By theory, it is expected that the circumference is directly proportional to the diameter, and the proportionality constant is π, or C = πD for all circles. To investigate this, we measured the circumference and diameter of five different objects in a small range of circles. Each quantity was measured once for each object, implying the uncertainty of each measurement was directly related to the precision of the measuring tool and the techniques and abilities of the measurer. To get better precision, we should have measured each quantity a number of times and performed statistical analyses. The first part of the experiment analyzed the data using numerical methods. Over the five objects, we found an average ratio of Pi = C/D = 3.14 ± 0.06. This is well within the expected value of π = 3.141593, yielding a very small percent error of 0.051%. The second part of the experiment analyzed the data graphically by plotting the values of circumference versus diameter. The values were visually linear suggesting a constant slope. A best fit line was drawn without constraining a point through the origin, though it was expected that the line would pass through the origin. As seen by the graph, there is enough “wiggle room” in the line of best fit, that a good fit line could have been drawn through the origin, further supporting the claim that C = πD. The slope of the line of best fit was found to be πtheo = 3.25, yielding a percent error of 3.5%, which is noticeably larger than the numerical analysis, though still very small. A further analysis of uncertainty of the slope would give a better idea of the precision of the experiment, but as noted before, there is enough “wiggle room” in the line of best fit to reduce the slope to the accepted value, implying that our experiment is close to the accepted value of π = 3.141593. The uncertainties in the numerical analysis may have been recorded too low to effectively present the data gathering methods. The method of wrapping the paper around the object to get a value of circumference was likely less precise than initially noted. In conclusion, our measured values seem to be in agreement with the theory that the ratio of circumference to diameter is a constant value of π = 3.141593.

View more...
circumference of a circle related to the diameter? diameter? The THEORY: How is the circumference relationship is accepted to be C = πD, where C is the circumference of the circle, D is the diameter of the circle, circle, and π is the constant constant pi = 3.141593. Thus, the circumference and the diameter diameter should be directly proportional. proportional. A plot of C vs. D should yield a line of slope π. The ratio Pi = C/D can can be calculated and compared compared to the accepted accepted value of π = 3.141593. The measured and accepted accepted values can be compared compared by calculating the percent error using . The values C and D can can also be compared by plotting plotting C vs. D (C on on the vertical and D on the horizontal). If the data follows follows the equation C = πD, then then the plot will be a line of slope π and y-intercept 0. PURPOSE: The purpose of this experiment was to gain laboratory experience by measuring the ratio of circumference to diameter two ways that agree with theory (using mean and standard deviation as well as graphical analysis), and then comparing the results. DATA TABLE: Object

Penny Battery Iron Cylinder Rubber tube Can

Diameter (cm) 1.90 3.30 4.23 6.04 6.6

Circumference (cm) 5.93 10.45 13.30 18.45 21.2

σD (cm)

σC (cm)

0.01 0.02 0.02 0.02 0.1

0.03 0.05 0.03 0.05 0.1

CALCULATIONS:

Pi = C/D,

Penny – Pi = C/D: Pi = 5.93/1.90 = 3.1211, 0.0228

=

Battery – Pi = 10.45/3.30 = 3.1667,

=

0.0245 Iron Cylinder – Pi = 13.30/4.23 = 3.1442,

= 0.0165

Rubber Tube – Pi = 18.45/6.04 = 3.0546,

=

0.0131

Can – Pi = 21.2/6.6 = 3.2121,

=

0.0510

,

= 3.14

= 0.0561

= 0.05073

Slope = π theo =

= 3.25

= 3.451% CALCULATIONS TABLE:

Object Penny Battery Iron Cylinder Rubber tube Can

= ___3.14_____

πtheo = ___3.25_____

Pi = C/D (no units) 3.12 3.17 3.14 3.06 3.21

= ___0.06_____

σPi (no units) 0.02 0.02 0.02 0.01 0.05

%Error Pi = ___0.051%_____

= ___3.5%_____

Note: The graph should fill an entire page in your notebook. This graph fills the entire page, but was shortened and rotated after scanning to be placed in this lab report.

CONCLUSION: The relationship between circumference and diameter of a circle was investigated in two ways using the same set of data. By theory, it is expected that the circumference is directly proportional to the diameter, and the proportionality constant is π, or C = πD for all circles. To investigate this, we measured the circumference and diameter of five different objects in a small range of circles. Each quantity was measured once for each object, implying the uncertainty of each measurement was directly related to the precision of the measuring tool and the techniques and abilities of the measurer. To get better precision, we should have measured each quantity a number of times and performed statistical analyses. The first part of the experiment analyzed the data using numerical methods. Over the five objects, we found an average ratio of Pi = C/D = 3.14 ± 0.06. This is well within the expected value of π = 3.141593, yielding a very small percent error of 0.051%. The second part of the experiment analyzed the data graphically by plotting the values of circumference versus diameter. The values were visually linear suggesting a constant slope. A best fit line was drawn without constraining a point through the origin, though it was expected that the line would pass through the origin. As seen by the graph, there is enough “wiggle room” in the line of best fit, that a good fit line could have been drawn through the origin, further supporting the claim that C = πD. The slope of the line of best fit was found to be πtheo = 3.25, yielding a percent error of 3.5%, which is noticeably larger than the numerical analysis, though still very small. A further analysis of uncertainty of the slope would give a better idea of the precision of the experiment, but as noted before, there is enough “wiggle room” in the line of best fit to reduce the slope to the accepted value, implying that our experiment is close to the accepted value of π = 3.141593. The uncertainties in the numerical analysis may have been recorded too low to effectively present the data gathering methods. The method of wrapping the paper around the object to get a value of circumference was likely less precise than initially noted. In conclusion, our measured values seem to be in agreement with the theory that the ratio of circumference to diameter is a constant value of π = 3.141593.

Thank you for interesting in our services. We are a non-profit group that run this website to share documents. We need your help to maintenance this website.

To keep our site running, we need your help to cover our server cost (about $400/m), a small donation will help us a lot.