physical science need it within 7 hours small lab work

Data is provided. Read and follow the directions carefully!

Introduction

An estimate of pi was made by the ancient Chinese and Egyptians who used a value of 3. But the Greeks were probably the first to make a more accurate determination. Archimedes found the value of

pi as a number between 31/ 7 and 310 / 71• A more accurate value was not possible until the decimal system was introduced in the 17th century. Today modern computers can determine pi with more than a mil­ lion digits after the decimal.

In today's experiment we shall determine the value of pi by examining the relationship between the radius of a circle and its circumference.

The direct proportionality between any two entities, y and x, is represented by the generic equation of a straight line:

y=mx+b

in which m and bare constants. The value of mis called the slope of the line. The value of bis called they-intercept because it becomes the value of y when x = 0.

If we assign the symbols c for circumference and d for diameter, the equation can be written as

c = md + b

In this equation the slope, m, of the line shows the direct relationship between circumference and diameter, which is the value of 7t.

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10 Experiments in Physical Science

Equipment

Several circular objects (Assortment of PVC pipes) Ruler, 30 cm

Procedure

I. Measurements. Obtain a variety of circular objects from the instructor's desk. (Return them to the desk when you have finished using them.) Measure the diameter, d, and the circumference, c, of six circular objects. Make sure that you are measuring the outer diameter. Include the largest and smallest object. You will find it easier to use the decimal scale. Your measurements should be pre­ cise to a tenth of a centimeter (or millimeter). Record your measurements in the Data Table on the lab report. Pay attention to significant figures. If your measurement happens to fall exactly at the centimeter level, then you need to show that there are no millimeters beyond that point and you place a decimal point followed by a zero. For example, if your measurement happens to be exactly 12 cm and nothing more (or less) than 12, you would report 12.0 cm.

Note: In a data table, if units are given at the head of a column, you should not repeat them after each number in the column.

2. Plotting. You will now determine the pattern that exists between diameter and circumference by graphing the data that you have obtained. Using the graph paper provided, the horizontal (x) axis will represent the diameter, d, and the vertical (y) axis, the circumference, c. For each measured circular object, plot the value corresponding to its diameter and circumference as a single point surrounded by a small circle on the graph (0). Use a ruler and a lead pencil to draw the best straight line through all the points. The "best straight line", also called the trendline, is a single line that passes through, or as close to, as many points as possible. This line represents an average trend for all the points, although it may not pass through any point.

3. Slope of the line. You will now determine the slope, m, of the line on your graph. Locate any two points on the line, fairly wide apart, and find their values, cl' d1 and cz, dz. The points should be taken from the line on your graph and not from any of the measurements. Mark each point with

an arrow, but do not draw a circle around the points because they are not actual measurements.

On your data sheet, calculate the ratio (cz - c1) I (dz - d1 ) . This is the slope of the line, which is your experimental value of 7t.

4. They-intercept. Use a ruler to extrapolate (extend) the line all the way to the vertical axis. The point where the line meets the vertical axis gives the value of the constant, b, called the y-intercept. The value of b obtained from your graph can be positive if the line meets they axis above zero, negative if it meets they axis below zero, or zero if it exactly meets the origin.

Now consider this: As the circumference of a circle gets smaller, what happens to the diameter? It gets smaller too, of course. This means that as c approaches zero, so does the value (md + b) and therefore the value of b will also be zero. Theoretically therefore, the equation of your line should be:

c= md+ 0 or c = md

Determining the Value of Pi 11

5. The percent error is a measure of how close your results are to an accepted value. It is determined as follows. Note that only the absolute value of a difference is used. Calculate your percent error for pi.

6. Note that the accepted value of pi = 3.14159

7. Calculate your percent error using the following equation.

Percent error = Accepted value – experimental value x 100

Accepted value

= ___________%

NAME PARTN ER. .SECTIO N _

Data and Results for Determining the Value of Pi

DATA TABLE

Description of Measured Object	Diameter (cm)	Circumference (cm)
Small pipe	3.7	13.0
Small to medium pipe	4.5	16.1
Medium pipe	5.3	18.0
Large Pipe	9.7	32.0
Extra Large pipe	11.1	34.9
2XL Pipe	14.2	40.0

1. Use a lead pencil to plot the data in the table on the graph paper provided, and use a ruler to draw a best-fit line through your data.

2. Locate two convenient points on your line, and determine the slope, m, of the line on your graph.

Show your work:

This is your experimental value of pi. Give its value to three significant figures

3. Give the value of they-intercept, b, obtained from your graph cm Theoretically, what should this value be? cm

Is your value reasonably close to what it should be theoretically? [Yes] [No] (Circle the correct answer.)

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14 Experiments in Physical Science

4. Now you can use the values that you obtained for your slope and y-intercept to complete the equation of your straight line:

c= d+ _

5. Calculate the percent error from the actual value of 7t. (See page 11 .) Show your work:

Sources of Error

1. While determining the circumference of a cylindrical object, suppose that you wrapped the measuring tape around it at a slight slant. Comment, using complete sentences, on how this would affect the following:

a. Would your circumference measurement be larger, smaller, or would it make any difference?

b. Would the value you obtain for pi be larger, smaller, or would it make any difference?

2. While measuring the diameter of a circular object, suppose that your ruler did not pass exactly through the center of the circle. Comment on how this would affect the following:

a. Would your diameter measurement be larger, smaller, or would it make any difference?

b. Would the value you obtain for pi be larger, smaller, or would it make any difference?

3. In determining the slope of your line, why is it more precise to select points on the line and not from any of the measurements?

1-5.0

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