Name:
Lab Partners:
Measuring the Circumference of the Earth
Introduction
Eratosthenes of Alexandria (276 - 196 BCE) successfully measured the diameter of the earth using
knowledge of the sun’s position in the sky at two points in Egypt. On a certain day, he knew that
the sun was directly overhead in Syene, where the sun at noon shone directly into the bottom of
wells. On the same day at noon, the sun in Alexandria was not directly overhead, but cast shadows
which made angles from the vertical which were 1/50 of a circle. He then had the distance from
Alexandria to Syene measured by men pacing off the distance. The distance was measured to be
5000 stadia (800 km). He could then reason that if the sun was far enough away to make the rays
coming to the earth parallel, then 5000 stadia is 1/50 of the earth’s circumference. This result came
out within 1% of the accepted modern answer.
Figure 1: How Eratosthenes obtained the circumference of the Earth: The relationship of shadow
length to the arc and angle between two points on Earth.
As discussed in class, our procedure will mirror that of Eratosthenes. In addition, you will come up
with a method to measure the height of one of the two towers in the school or a light standard on the
football field. It is important that you tell the reader how you intend to make the latter measurement,
including all data and calculations. It is also important that you make all measurements to the
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proper number of significant figures!. Failure to do so will affect both the quality of your results
and ultimately your grade. Make sure you ask yourself how good your measurement can be before
you record a number or numbers.
Procedure
1. On a sunny day, take a plumb line and hold it with the weight just above the ground. Stand a
meter stick on the ground parallel to the plumb line. Measure its shadow length as accurately
as you can. In addition, record the exact time of the measurement.
Length of Shadow:
Time of Day:
2. Now decide what you want to measure, (circle one):
Bell Tower
Theater Tower
Football Light Stand
Flag Pole
In the space below write out your procedure for measuring this object. Before you go outside,
make a data table for each of the measurements you intend to make. Anyone reading your
procedure should be able to replicate your experiment. A diagram might also be useful in
illustrating your method.
Procedure for Measuring Height:
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Diagram Illustrating Method of Height Measurement
Data for Measuring Height:
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Analysis
Here, we want to calculate two things:
• The circumference of the Earth from our data and its error
• The height of the selected object you measured
Let’s first work on calculating the circumference of the Earth. In each step that follows, show a
sample calculation. You may do this in the margin of the lab, as most of these calculations are very
straightforward. However, it is critical that you report each of your answers to the proper number
of significant figures.
1. Calculate the angle the shadow of the sun makes with the vertical meter stick. This can be
done by first dividing the length of the shadow by 1.000 m to get the tangent of the angle.
Then find the inverse tangent of the angle. Note, I have left two blanks for you: One to report
the results of the calculation, and the other to report the number to the correct number of
sig. figs..
=
Angle of Shadow:
(SF)
2. Find out where the Sun was directly overhead at the time you made your measurements. To
do this go to the links page of the website (http://faculty.trinityvalleyschool.
org/pricep/phys/physlink.html) There you will find the link to the Sun-Earth
applet. As illustrated in class, use the applet to determine the Latitude and Longitude of the
of the position on Earth where the Sun is overhead.
Longitude of Sun:
Latitude of Sun:
3. We need to convert these degree measurements to the standard nautical measurements of
degrees, minutes, and seconds. Following the discussion in class, convert the latitude and
longitude to these units:
Latitude of Sun:
Longitude of Sun:
degrees
degrees
minutes
minutes
seconds
seconds
4. Record the Latitude and Longitude of TVS in degrees, minutes, and seconds:
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Latitude of TVS:
degrees
Longitude of TVS:
degrees
minutes
seconds
minutes
seconds
5. Now using the second applet on the links page, input the latitude and longitude of the Sun
and TVS to obtain the distance on the Earth between the two points. Record this distance to
the nearest kilometer.
Distance between the Sun and TVS:
6. As we discussed, the angle the Sun’s shadow made with the vertical of the meter stick is the
same angle as there is between the Sun’s zenith position and TVS. Hence:
Distance between Sun Zenith and Ft. Worth
Circumference of Earth
=
Angle of Shadow
360◦
From this relationship, calculate the circumference, and the radius of the Earth (assuming
the Earth is a sphere. Formula of the volume of a sphere: 4/3πr3 :
Circumference of the Earth:
=
(SF)
=
Radius of Earth:
(SF)
7. Finally, we need to address the accuracy of your results. In physics and chemistry this is
done by calculating the percent error, which is defined by:
Percent Error =
|Difference between your answer and the correct answer|
× 100
correct answer
What is the percent error in both the circumference and the radius of the Earth?
Percent Error in circumference:
=
Percent Error in radius:
=
(SF)
(SF)
Now, on the next page, please clearly outline and show the calculations, to the proper number
of sig figs, for the height of the object you selected. Also include a brief discussion of
how accurate your results are. In other words, are there assumptions you made in your
measurement that are not specifically true?
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Calculations to Determine Height of Object and Discussion of Accuracy
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Discussion
The use of angles to measure the position of objects may seem a little unfamiliar to you. What is
even more interesting is that angles can also be used to obtain distances, particularly in astronomy.
Let’s consider some quick calculations involving degrees, minutes, and seconds.
1. The Sun travels across the entire Earth in one day as the planet rotates. Based on this information, what distance across the Earth does the Sun traverse in one minute? Express your
answer in miles, meters, and degrees. Show your sample calculations below:
=
Distance Sun travels in miles =
meters =
degrees
2. Convert the degrees answer above into minutes and seconds.
Distance Sun travels in one minute =
minutes
seconds
3. Finally, how precise could we actually expect our answer for the circumference of the Earth
to be? Based on the meter stick we used, we argued that at best that we could estimate the
length of the shadow to 0.1 mm. Based on this, what is the minimum percent error in the
length of the shadow?
Minimum percent uncertainty in the shadow:
We should expect that the error in the final answer can be no better than this, and it is likely
that it is significantly higher. Based on this fact, list at least three sources of error that result
in the larger percent difference.
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