Otis Ozone
George Washington HS
Submitted April 18, 1984
MEASURING THE VALUE OF PI
Goal
Experimentally determine the value of pi by finding the ratio between the circumference and the diameter of several cylindrical solids.
Theory
The ratio of the circumference of a circle to its diameter is a number, named pi. The value of pi can be calculated using a variety of purely mathematical techniques, yielding a series of digits that begin with 3.14159 and continue in no apparent pattern.
The value of pi can be verified experimentally to 2 or 3 digits by measuring the circumference and diameters of cylindrical objects, since the cross section of a cylinder is a circle. To obtain a truly experimental result for pi, a measurement of circumference, independent of diameter, must be obtained. One method, for example, would be to wrap a string around the cylinder and then measure the string’s length.
List of Apparatus
30-cm long ruler, ruled in millimeters
Soft-lead pencil
Paper
5 cylindrical objects (identified in data table)
Method
The circumference and diameter of five cylinders were measured. To find the circumference of each cylinder, a pencil mark was placed on one end of the cylinder. Then the cylinder was rolled without slipping across a piece of paper until 2 marks appeared on the paper. The distance between the marks was measured with the ruler.
To find the diameter of each cylinder, the ruler was placed on a table. Then the cylinder was placed on the ruler in such a position as to give the greatest distance from one side of the circular cross section to the other.
To take accurate readings from the ruler, the ends of the ruler were never used as endpoints for the distances measured. Also, lines of sight to the ruler scale were as nearly perpendicular to the scale as possible.
Data
|
Cylinder ID |
Circumference |
Diameter |
Circumference ÷ Diameter |
| Skinny S2 | 0.0398 | 0.0120 | 3.32 |
| Slim B3 | 0.0505 | 0.0163 | 3.10 |
| Just Right | 0.0612 | 0.0195 | 3.14 |
| Plump B6 | 0.0797 | 0.0253 | 3.15 |
| Pipe Coupling | 0.1250 | 0.0390 | 3.21 |
Legend of Symbols
C is the circumference of the cylinder.
D is the diameter of the cylinder.
Sample Calculations
For cylinder Just Right, the ratio of circumference to diameter is:
C/D = 0.0612 m/0.0195 m = 3.14.
Graphical Analysis
A graph of circumference versus diameter for the five cylinders is shown below. A linear least-squares fit was applied.
Matching (or Correspondence) Table
|
Math |
maps to |
Physics |
Value |
Units |
|
y |
--> |
C |
variable |
m |
|
m |
--> |
pi |
3.20 |
none |
|
x |
--> |
D |
variable |
m |
|
b |
--> |
0 |
-0.0004 |
m |
Equation of Fit: C = 3.20D – 0.0004 m.
The absolute uncertainty in the measurements of both circumference and diameter is estimated to be half a millimeter or 0.0005 m. Using the measurements for Just Right, the percentage uncertainties in circumference and diameter measurements are:
% uncertainty in C = 100(0.0005 m/0.0612 m) = 0.8 %
% uncertainty in D = 100(0.0005 m/0.0195 m) = 3 %
Total % uncertainty in C/D = 0.8 % + 3 % = 4 %
The uncertainties for Just Right are expected to be representative of the set of cylinders, since that cylinder is intermediate in size. Smaller cylinders will have larger uncertainties and larger cylinders will have smaller uncertainties.
The percentage error between the value of the slope of the fit and the accepted value of pi is:
Error Analysis (qualitative)
The measurement of diameter may be in error because the method of determining which points of the cross section to measure between was strictly by sight. It was necessary to alternate sighting down one side of the cylinder and then down the other side in order to approach the actual diameter. The scale was partially covered during this process.
The measurement of circumference may be in error because the cylinder may have slipped while being rolled across the paper. The width of the pencil marks is large enough to cause uncertainty in reading the end points on the ruler scale.
A source of error that could affect the measurement of both circumference and diameter is the extent to which the cross section deviates from a perfect circle.
An improvement in the method of measuring diameter would be to place the cylinder in the jaws of a pair of vernier calipers so that the diameter could be determined without sighting. Measurements of both circumference and diameter could be improved by taking several measurements of each for every cylinder.
Conclusion
The diameter and circumference of five cylinders was measured. Circumference was measured by wrapping a string around each cylinder, while diameter was measured by placing the end of each cylinder on a centimeter scale. A linear fit to circumference versus diameter gave a value for the ratio of circumference to diameter of 3.20. The intercept for the fit was -0.0004 m, which is close to the estimated uncertainty in the measurement of distance.
The percentage difference between the slope of the fit and the accepted value of pi is 2%. This is less than the total measurement uncertainty estimated for an intermediate cylinder. This indicates that measurement uncertainty is the primary source of error.