TAKE-HOME EXP. # 2 A Calculation of the Circumference and Radius of the Earth

Transcription

1 TAKE-HOME EXP. # 2 A Calculation of te Circumference and Radius of te Eart On two dates during te year, te geometric relationsip of Eart to te Sun produces "equinox", a word literally meaning, "equal nigt" or equal durations for nigt and day. Te Fall equinox occurs approximately Sept. 21, and te Spring equinox, approximately Marc 21. On tese two days, every year: Marc 21 and September 21 a) Te time between sunrise and sunset is approximately 12 ours everywere on Eart. b) Te Sun is directly overead at noon at te equator. As a consequence, te sadowterminator line is parallel to a line of longitude, and, terefore, te sadow of Eart bisects eac polar region. Tis experiment will use te second fact to directly measure your latitude angle. And, as a consequence, you can directly produce te actual circumference and radius of te Eart! Te circumference and radius of Eart elps to locate uman beings witin te context of a universe of galaxies. A. A Procedure To Find Te Circumference Of Te Eart Tis procedure was first used by Eratostenes, wo eaded te great Museum at Alexandria (or Cairo) in Egypt, around 200 BC Most of te ancient observers were completely convinced of te roundness of Eart. Te evidence was clear. Lunar eclipses were correctly interpreted as sowing te circular sadow of te Eart. Also, if one traveled nort, te nortern constellations got iger in te sky, and te soutern ones dropped toward te orizon. A similar effect occurred for soutward travel, wit te sout constellations getting iger in te sky, and even some new constellations, not seen from te nortern emispere would be visible.

2 TAKE-HOME EXPERIMENT #2 T.H.E. #2-2 Wit tat background, Eratostenes also ad some additional local information. On te summer solstice date, Eratostenes knew tat te Sun illuminated te bottom of a very deep well at Syene, about 500 miles directly sout of te Museum at Alexandria. In order to reac te bottom of suc a narrow, deep saft, te Sun at Aswan must ave been nearly directly overead at noon. If a stick ad been oriented vertically at Aswan on tat date, it would ave cast no sadow at noon on tat day. Note on te map tat Syene (Aswan) is just a sort distance nort of te 23.5 N latitude line, te Tropic of Cancer, wic is te fartest nort latitude at wic te Sun can be directly overead. Tat occurs tere about June 21, te solstice date. Formerly, Alexandria Formerly, Syene Nile Tropic of Cancer 23.5º N latitude Te Use of a Vertical Stick Called a Gnomon. Te use of a vertical stick perpendicular to a flat surface can tell one a great deal about te time of day and te time of year. Te ancient Greek word for tis device was "gnomon", and it as been directly taken over into Englis. It sould be found in any "collegiate" dictionary. a) At noon, te sadow of te stick (gnomon) is at its sortest and points directly nort in te nortern emispere. Watcing te sadow cange size and direction over te course of a day or two provides an accurate compass. b) If one watces te stick for an entire year, one can note oter interesting features of te rytm of te Sun's appearance. Imagine making a mark representing te sadow lengt on te plane at noon, say, once a week for a year. If one starts in te Spring, one observes te sadows at noon decreasing in lengt, getting sorter week by week, until late June. From tat time on te noon sadows get longer and longer... and longer until late December. Ten te sadow again reverses course. One sees two special dates were te sadow reverses course te Sun seems to "stand still", "solstice" in Latin. On te solstice date, wen Eratostenes knew te Sun was directly overead at Syene (Aswan), e did someting remarkably simple at Alexandria. He measured te lengt of a monument s sadow and te vertical eigt of te monument. Ten, wit is knowledge of te geometry of a circle and a measure of te distance between Alexandria and Syene, Eratostenes was able to calculate te circumference and radius of te Eart. SIDE VIEW L = lengt of sadow Sun's rays = eigt of vertical object

3 TAKE-HOME EXPERIMENT #2 T.H.E. #2-3 It almost seems like magic tat a simple lengt measurement locally could turn into a planetary measurement. B. Here s Te Reasoning Wen sunligt arrives in te vicinity of Eart, all te sunligt as te same angle of arrival. Te diagram to te rigt sows sunligt arriving directly overead at Syene at te summer solstice. At tat same moment, as we look to te nort and sout of Syene, all te arriving ligt is traveling parallel to te ligt rays at Syene. Nort pole London sunligt at Alexandria sunligt at Syene Because te Eart presents a curved surface to te arriving ligt, te incoming ligt makes different angles wit vertical objects at different locations along te surface. A vertical object at te Eart s surface can be defined as one tat is parallel to a radius of Eart at tat location. Te curvature of te surface causes vertical objects, like standing people, to ave different lengt sadows at different places at precisely te same moment. Eratostenes measured and L in Alexandria at te moment te bottom of te well was illuminated in Syene, likely to be noon on tat date. He also knew te distance d between te two cities of Alexandria and Syene. at te Nort pole radius, R, of Eart Sout pole at Alexandria L sunligt Vertical objects are located at various latitudes. Te sadow cast by eac can be obtained by using a line parallel to te incoming rays of te sun. Draw te line from te top of te object toward te orizon plane of te object. at Syene

4 TAKE-HOME EXPERIMENT #2 T.H.E. #2-4 Your Analogous Experiment. You will use an equinox date to accomplis te same ting tat Eratostenes did at a solstice date. Te "magic" in te equinox date is tat we know exactly were te sun is, and exactly wen: directly overead at te equator at noon. Te oter value tat is required will be your straigt-line distance, labeled d in te diagrams, from te equator to your location. An accurate statement about te equinox date is tat sunligt arrives directly perpendicular to te Eart's surface at te equator at noon (sun-time, not dayligt savings time). A vertical object of eigt,, placed perpendicular to a flat surface can be imagined to be a sligt extension of te radius of te Eart. Te diagrams to te rigt examines te geometry created by te placement of te vertical object and its sadow. Te sun's rays are everywere parallel at noon. W Tis view is a cross-section troug te Eart, sliced troug a line of longitude, so it splits te Eart in two. N equator d E Sunligt at noon sunligt is directly overead at equator at spring and fall equinoxes S Tere are two parallel lines, te equator and te incoming sunligt. Te radius of te Eart tat passes troug your location cuts across tese two parallel lines. Terefore, te two saded angles, labeled "A", are precisely equal, as in all suc cases of two parallel lines cut by a tird straigt line. equator A A E Notice tat te angle "A" formed by two Eart radii one from te center of te Eart to te equator, te oter from te center to your location fulfills te definition of latitude angle. Your location is "A" degrees nort of te equator. Te sun's rays are everywere parallel at noon. Tus, you ave two parallel lines, te equator and te incoming sunligt. Te radius of te Eart tat passes troug your location cuts across tese two parallel lines. Terefore, te two saded angles, labeled "A", are precisely equal, as in all suc cases of two parallel lines cut by a tird straigt line.

5 TAKE-HOME EXPERIMENT #2 T.H.E. #2-5 C. Here s Some Necessary Information About Circles a) As for any circle, once around te full circle of te cross-section of te Eart covers an angular distance of 360. b) Te circumference of any circle can be computed from te radius : circumference = 2 p r, were p = 3.14, and r = radius of Eart. Te circumference is te lengt a piece of string would ave by laying it carefully along te outline of a circle until you ave covered te outline exactly and completely. Te circumference of a circle is simply a lengt in meters. r radius, r Imagine a clock wit only one and pointing to te numeral "3" position. If you move tat and once around, returning it to te "3" position, you will ave moved te and troug an angle of 360º. D. Measuring Your Latitude Angle Te major requirement is to construct a vertical object of known eigt,, perpendicular to a flat, orizontal surface onto wic te sadow of te object is cast. A sloping reference level will not provide a sadow of "true " lengt. Sun Angle A = eigt of vertical object L = lengt of sadow A meter stick or a 12-inc ruler will work well, but any straigt object tat is vertical will work. (I've used fence posts, books, and even a pen, as my vertical object. Let's label watever is used as a "stick." ) Tis measurement needs to be done on te equinox date at noon, sun time, not dayligt savings time. If dayligt savings time is in effect, do te measurement at 1 PM. A time range of "noon ± 10 minutes" will make little difference in te outcome.

6 TAKE-HOME EXPERIMENT #2 T.H.E. #2-6 Doing it at noon, sun time, a few days on eiter side of te equinox date will also make little difference. If te weater doesn't cooperate, do it as soon as it does. Te angle labeled "A" is "te angle of te incoming sunligt" measured from te vertical. sunligt THE REQUIREMENTS: Te eigt must be at 90 to te surface, and te surface on wic te sadow is cast must be orizontal. It cannot be sloped up or down at all. Te eigt and te lengt, L, of te sadow needs to be measured. Angle A is te same as latitude angle. L = lengt of A 90º, eigt of vertical object HELPFUL HINTS: You can measure te vertical alignment of be comparing it to a "plumb line", wic is simply a string tied to some weigt (your keys, for example) and te weigt is allowed to ang freely. Te string is ten vertical by definition. Also very important: Te sadow s lengt must be carefully considered. Note in te diagram to te rigt tat te sadow's lengt L is measured from te leading edge of te object, not from its center or back edge. Measure "L" from front edge of extended object. L

7 TAKE-HOME EXPERIMENT #2 T.H.E. #2-7 NAME (print) ID# PARTNERS (if any) DATA SHEET Measuring Your Latitude Angle Time and date of measurement: Measure te sadow lengt L and te vertical eigt, including an estimated uncertainty: = ± cm Sadow L = ± cm Divide L by. L = Ten compute angle A by doing te following using your calculator. A = angle = tan 1 ( L )= degrees. HERE S HOW: Wit te result of L in your calculator window, use te inverse tangent operation (tan 1 ) on your calculator. To do so usually requires itting te 2nd function button on te calculator to access tis function, wic is located above te tangent button. [Angle "A" sould be very close to te nortern latitude angle at wic you made te measurement. If it's not, eiter someting is wrong wit your measurement, or your calculator is putting out some oter angular measure, like radians or grads, bot of wic are directly proportional to degrees. Tere is a button somewere on your calculator wic will allow you to switc modes between degrees, radians, and grads. Switc to degrees and try again.] C. Estimating A Relatively Accurate Circumference And Radius Of Eart Please assume tat te distance d from te equator to your location somewere near Los Angeles is 2350 ± 50 miles. If you are more tan 100 miles nort or sout of Los Angeles, please adjust te distance value. Next, from te information you ave in tis experiment, please obtain te circumference and te radius of te Eart. Sow your reasoning and work. Explain ow you obtained te values. W N equator A S d E at noon Sunligt Circumference of te Eart = ± miles Radius of te Eart = ± miles % difference between measured value of te radius r and te accepted value (See Endnote 1 ) rexp-raccepted x 100% = % raccepted

8 TAKE-HOME EXPERIMENT #2 T.H.E. #2-8 Appendix 1. Te Reason Te Ligt Arrives In Parallel All Over Te Eart Some people tink of te Sun as occupying a relatively small portion of te sky, more akin to a point source of ligt tan to an extended "wall of ligt" covering te entire sky. Te apparent disk of te Sun is no bigger tan te apparent disk of te Moon, eac occupying about 0.5 in te sky. Tis perception bears some examination. Te geometric reality of sunligt is someting quite different. In geometric fact, te actual diameter of te Sun is about 100 times te diameter of te Eart. Tis 100-to-1 geometry governs te pysical pat of te ligt. Even toug te Eart is a distance of about 200 sun-diameters away from te Sun, te ligt from te Sun arriving at Eart is ligt from a very large extended source. Te Eart intercepts only a tiny portion of te ligt emitted. Below are two circles wit a ratio of diameters 100 to 1. Eart Sun 1 Te "accepted" average value for Eart's circumference is C = 24,906 miles, and for te radius r = 3964 miles. A careful measurement on te appropriate dates sould get one to witin about 5% of tese values.