CIRCUMFERENCE VS



CIRCUMFERENCE VS. DIAMETER

DOING AN EXPERIMENT

© 2005 Robert A. Wilson

For this experiment you will need twelve to fifteen objects, each of a different diameter. You can use a variety of household objects: cans, plates, etc.

Note: Even though you may know the “answer” that is the desired relationship, approach this as an experimental determination of the relationship between circumference and diameter. Do not calculate the circumference of the circle from your knowledge of its diameter.

1. THE EXPERIMENT

In beginning an experiment you must first determine your objectives - what you are trying to accomplish. When you have a clear picture of your objectives, write them down in a concise statement.

You are trying to find, experimentally, the relationship between the diameter and size of a set of circles. What is meant by "size"? Size could mean the circumference (the length of the perimeter) or the enclosed area of the circles. Suppose we finally come to an agreement that by "size" we mean the circumference of the circle. You might state your objective as follows:

In this experiment we will determine how the circumference of a circle depends on its diameter.

After determining your objectives, you must decide what measurements to make to reach those objectives.

In the present experiment it is quite clear what quantities should be measured. The objectives state that the relationship between circumference and diameter is to be

determined. This can be accomplished directly by a straightforward measurement of both quantities. You might write: “For each circle in the set, I will measure its diameter and circumference.”

2. PRELIMINARY OBSERVATIONS AND MEASUREMENTS

After deciding what to measure, you must decide on the appropriate measuring instruments. E.g., it would be inappropriate to measure the mass of a marble using a bathroom scale, or to measure its diameter using a yardstick.

You should make notes on the measuring instruments used. (In an academic or commercial laboratory, most instruments have an identifying name, model number, and serial number - these should all be recorded. What appears to be errors in your data might be traced to a faulty or incorrectly calibrated instrument.)

Determine the details of your measuring procedure. Many measurements can be made in several ways. Your procedure should be simple and accurate. Make trial measurements at this point - try different methods. You will find some methods to be difficult or inaccurate. Your trial measurements should extend over the full range of possibilities. Make notes on your trials.

In this experiment it would be appropriate to measure the diameter with a ruler. To find the diameter of a circle you want to measure it at its widest point. How can you find the "widest point"? Is the circle perfectly round? If it is not perfectly round, what would be the meaning of "the diameter of the circle"? Decide on the answers to these questions and describe (in writing) how you are going to determine the diameter of the circle.

How will you find the circumference of the object? The ruler cannot be wrapped around the circumference. Some possibilities: (a) Wrap a string around the circle and measure its length. Are you sure the string was not stretched? (b) Roll the circle on the ruler. Did it slip? Are you sure you rolled it exactly the circumference? (c) Use a measuring tape. Is the calibration of the tape the same as the ruler? (Wood and cloth expand and contract differently for changes in temperature and humidity.) Does the measuring tape stretch? If the tape must be laid over itself at the end of the measurement, does the increase in thickness of the tape itself introduce an appreciable error?

Measure the smallest and largest rings. Are your measuring instruments appropriate to the full range of measurements?

Try these and other methods. Record your observations and decide on the best method of measuring the circumference. (Do not calculate the circumference using a math formula, measure the circumference directly.)

Decide on the methods you are going to use and describe them in your notes.

Sketches are often valuable in describing the apparatus and the quantities being measured. Sketches should be neatly drawn (with the aid of a straightedge). Label important parts of the sketch.

The preliminary observations should not be viewed as a well defined - follow the steps - procedure. The order in which the preliminary observations are made is a mix of the above. You might make a sketch of the first rough measurements and jot down the instruments used. These measurements suggest a variation, which requires a new sketch and the description of an additional measuring instrument. Etc.

3. COLLECTING DATA

Accuracy is important in everything you measure. Choose the most accurate measuring techniques you can. Always read your instruments to the highest precision possible, usually by estimating to one-tent h the smallest scale division. Record the data with the maximum precision measured. You can always round off a value later, but you can not reconstruct the greater precision.

Your data taking should be organized. This is usually accomplished by setting up data tables in which to record your measurements as they are made. A well-organized data table can speed up the data collection process and contribute to the correct numbers being recorded.

|ring |trial |diameter |circumference |

|# | |(cm) |(cm) |

|1 |1 | | |

| |2 | | |

| |3 | | |

| |4 | | |

| |avg. | | |

|2 |1 | | |

| |2 | | |

| |3 | | |

| |4 | | |

| |avg. | | |

All data tables should, however, have a title and labels and units that describe the measured quantities. (You do not need to include the units after each entry, the unit in the heading column is applied to all the entries in the column.) If other numbers are to be calculated directly from the original data, room should be made for adjacent rows or columns to hold the calculated values. A possible data table arrangement is shown above.

Choose a format for your data table and lay it out in your notebook. Use a straight edge to make the table neat. Title the data table, and be sure you have included appropriate labels and units.

As you collect your data, make a rough graph of the average values. This graph may show you where additional measurements are needed. Any data point, which deviates from the pattern formed by the other points on the graph is a subject for closer investigation.

From your preliminary measurements you should know the smallest and largest values to be plotted. Set up the axes fo r the graph of circumference and diameter in your notebook. As you compute each pair of average values, plot them on the graph. (Note that for the preliminary graph, you n eed not calculate the average value. You can estimate the average value of the set of numbers. After collecting all of your data, you can then resort to a calculator or computer to calculate the average values.) If a point does not fit the pattern, check it out immediately.

4. GRAPH THE DATA

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In this laboratory you will learn:

a) how to take and record experimental data,

b) how to treat this data to find the average value of a set of measurements,

c) how to display the data graphically,

d) how to interpret the graph,

e) how to write a summary of the experiment. You will learn these techniques while experimentally investigating the relationship between the diameter and size of a set of rings.

This lab spells out more detail than those to come. Yo u may notice that it is very directed in what to write in your notebook. A large part of the lab is de signed to reinforce the ideas of the discussion in the Physics Laboratory Notebook. The idea is to write down what you do so that you can refer back to the book later and understand what you did. This is a key component of any laboratory science work.

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