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 1
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: 2
Diagram Illustrating Method of Height Measurement Data for Measuring Height: 3
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. Latitude of Sun: Longitude 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: degrees minutes seconds Longitude of Sun: degrees minutes seconds 4. Record the Latitude and Longitude of TVS in degrees, minutes, and seconds: 4
Latitude of TVS: degrees minutes seconds Longitude of TVS: degrees 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 Angle of Shadow = Circumference of Earth 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: 4/3πr 3 : 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 correct answer What is the percent error in both the circumference and the radius of the Earth? 100 Percent Error in circumference: = (SF) Percent Error in radius: = (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? 5
Calculations to Determine Height of Object and Discussion of Accuracy 6
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. 7