# Measuring the Earth Using a GPS M. Scott Wilkerson & M. Beth Wilkerson, DePauw University, May 10, 2007

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2 Measuring the Earth s Polar Circumference 1. Students should organize into teams of two sharing one GPS unit. 2. In each team, you will mark waypoints using the UTM (Universal Transverse Mercator), coordinate system, where the units are in meters. The numbers for UTM represent eastings (relative to the Central Meridian in the UTM zone, and northings (relative to the Equator). Important: Set each unit to the datum WGS 84 so that all students are working with the same datum (model of the Earth s shape). To obtain decimal degree values (latitude-longitude coordinates) for each point, you can set the units to decimal degrees once you are back in the lab and access the waypoint information page for that waypoint to obtain the coordinates in decimal degrees. Alternatively, you could simply change the GPS units in the field and gather the waypoints in decimal degrees (or teams could use two GPS s, one in each coordinate system). 3. Locate a safe place away from buildings, cliffs, trees, power lines, and other obstacles that allows you to walk in a north/south direction for at least 200 meters - the longer the better. You can use the GPS coordinates to determine a true north-south track. 4. Take a GPS reading at the north end of your chosen track (Note: If you can average the points and then mark the waypoint, please do so (and make sure that WAAS correction is enabled). Name the waypoint a meaningful name so that you can identify it later. Record the UTM coordinates in Table 1. You can record the latitude and longitude from this stored waypoint later in the lab or you can switch units in the field and gather a new waypoint in decimal degrees (see step 2 above). Enter the value for decimal degrees in Table Repeat for the south end of your chosen track. NOTE: Do not round off! Please retain a minimum of 5 places to the right of the decimal point. Table 1. Measuring the Earth s Polar Circumference North Point (Latitude; decimal degrees) a. North Point (Longitude; decimal degrees) b. North Point (UTM northing; meters) c. North Point (UTM easting; meters) d. South Point (Latitude; decimal degrees) e. South Point (Longitude; decimal degrees) f. South Point (UTM northing; meters) g. South Point (UTM easting; meters) h. Distance in decimal degrees [a-e ] i. Distance in meters [c-g meters] j. Knowing the ratio: k. Distance in decimal degrees [i] 360 = Distance in meters [j], [PC] meters we can solve to find the Earth s polar circumference: ([j] meters * 360 ) 1 [PC] km = * [i] 1000 To calculate % error (assuming a true polar circumference of 40,008 km): [PC] % error = [k] km - 40,008 km 40,008 km l. Measuring the Earth Using a GPS Page 2

3 Measuring the Earth s Equatorial Circumference 1. Follow steps 1 & 2 for Measuring the Earth s Polar Circumference above. 2. Locate a safe place away from buildings, cliffs, trees, power lines, and other obstacles that allows you to walk in an east/west direction for at least 200 meters - the longer the better (Note: you may want to use one of the points used for the polar circumference calculation). You can use the GPS coordinates to determine a true east-west track. 3. Take a GPS reading at the east end of your chosen track (Note: If you can average the points and then mark the waypoint, please do so (and make sure that WAAS correction is enabled)). Name the waypoint a meaningful name so that you can identify it later. Record the UTM coordinates in Table 2. You can record the latitude and longitude from this stored waypoint later in the lab or you can switch units in the field and gather a new waypoint in decimal degrees (see step 2 above). Enter the value for decimal degrees in Table Repeat for the west end of your chosen track. NOTE: Do not round off! Please retain a minimum of 5 places to the right of the decimal point. Table 2. Measuring the Earth s Equatorial Circumference West Point (Latitude; decimal degrees) a. West Point (Longitude; decimal degrees) b. West Point (UTM northing; meters) c. West Point (UTM easting; meters) d. East Point (Latitude; decimal degrees) e. East Point (Longitude; decimal degrees) f. East Point (UTM northing; meters) g. East Point (UTM easting; meters) h. Distance in decimal degrees [b-f ] i. Distance in meters [h-d meters] j. Knowing the ratio: k. Distance in decimal degrees [i] 360 = Distance in meters [j], [LC] meters we can solve for the circumference of the Earth along this line of latitude [LC] in kilometers: [LC] km = ([j] meters * 360 ) 1 * [i] 1000 and to determine the equatorial circumference [EC] in kilometers (use degrees not radians): [EC] km = [LC] cos (latitude [a]) To calculate % error (assuming a true equatorial circumference of 40,075 km): l. [EC] % error = [k] km - 40,075 km 40,075 km Measuring the Earth Using a GPS Page 3

4 Polar Circumference & Equatorial Circumference 1. Discuss how your calculated polar circumference compares with both the accepted polar circumference and with Eratosthenes measured circumference. Give multiple reasons for why your answer is not exactly the same as the accepted circumference of the Earth. 2. What do your polar and equatorial circumference calculations tell you about the shape of Earth? 3. Using your calculated values, how long would it take to walk around the Earth along the polar circumference (in days)? Along the equatorial circumference (in days)? Along the circumference of the line of latitude in Table 2 (in days)? To determine this, set the receiver to the page where you can determine how fast you are moving (km/hr). Show your work. Polar circumference days Equatorial circumference days Latitudinal ( ) circumference days (use the latitude from Table 2a) Measuring the Earth Using a GPS Page 4

5 Determining the Mass and Volume of the Earth Determine the length of the polar circumference (Table 1k). Knowing that the circumference of a circle is: a. b. 2 *! * R where R=radius of the Earth. [R km ] km = [a] km (2 *!) [R m ] meters = [R km ] 0 Mass Knowing that mass is: c. Mass = (acceleration * R 2 ) G where acceleration due to gravity = 9.8 meters/s 2, and G, the constant of proportionality, is 6.67 x10-11 m 3 / kg second 2. You can determine the mass (kg) of the earth using the Earth s radius [R m ] from [b]. To calculate the mass % error (assuming a true mass of 5.98x10 24 kg): d. % error = [c] kg x10 24 kg 5.98x10 24 kg Volume Knowing that volume is: e. Volume = 4/3 *! * R 3 You can determine the volume (km 3 ) of the earth using the Earth s radius [R km ] from [b]. To calculate the volume % error (assuming a true volume of x10 12 km 3 ): f. % error = [e] km x10 12 km x10 12 km 3 Measuring the Earth Using a GPS Page 5

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USING VECTORS TO MEASURE ANGLES BETWEEN LINES IN SPACE Consider a straight line in Cartesian D space [x,y,z]. Let two points on the line e [x,y,z ] and [x,y,z ]. The slopes of this line are constants and

### Figure 1: A backpackers solar panel and rechargeable batteries.

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