Basic Thermodynamics ~ J. Pohl © | www.THERMOspokenhere.com (3-A111) |
Eratosthenes' Experiment
This physical experiment was
conducted around 200 BC.
Eratosthenes (~200 BC) worked as a librarian in the great library of Alexandria, Egypt. Those were times of greatly increasing travel from the East to West and back.
Eratosthanes sought to know the circumference of Earth. Knowing geometry and having observed eclipses of the moon, he believed Earth was a sphere of matter. To measure its circumference, he devised a method that used the rays of sunlight that hit Earth which he assumed arrive parallel.
Use Eratosthenes' data (shown in the figure) to calculate the circumference of Earth.
♦ Wells were common in Egypt. To dig straight down, a worker would place a plank across the well opening and at at its center he would lower a mass attached to a string. By Earth's gravity, the mass hung straight down; the line of the string established a "vertical" at that geographic location.
The mass hung in one well at Syrene showed something special. At precisely noon on the solstices the shadow of that masses fell directly to the well bottom, in near-perfect alignment with the rays of sunlight.
Elsewhere at noon (on days near the summer solstice) were one to look into the well one would see the full blinding reflection of rays of sunlight. Eratosthenes reasoned that this occurred because the rays of the sun struck Earth perpendicularly. Alexandria is approximately 500 miles due north of the well at Syrene. To measure the angle of incident sunlight there Eratosthenes lowered a plumb bob suspended by a string deep into a well. The sketch shows the angle of the rays at Aswan (zero degrees at noon on the say of summer solstice). Also depicted is the "1/50 th of a circle" angle measured at Alexandria at noon on the same day.
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(1)
The length of a radial arc equals its radius times the angle subtended (in radians). There are 2π radians in 360 degrees. |
We seek the radius (Earth's radius) associated with a circular arc length of 500 miles with the angle of arc subtended being 1/50 th of a full circle. Placing these numbers in the equation yields:
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(2)
Note: It would be a waste of time and to promote a bad habit to "do algebra" to get rEarth left of the equality. Leave equations as they are. Insert relevant numbers until there is only one unknown entity (all other terms are numbers with units ~ as in Eqn. 2). The idea is "always do algebra" as the last step, by putting the symbol of the unknown "left-of-equality" and the number (with units) right-of-equality. Finally, use the calculator. |
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(3) |
But the circumference of a sphere equals π times its diameter:
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(4) |
Today the measured radius of Earth is taken to be 3963 miles. Thus, supposing 3963 miles is the actual radius, the percent error of Eratosthenes' measurement was:
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(5) |
Thus his measurement of the Earth circumference (some 2000 years ago) was in error: Less than actual by only one-tenth of a percent. Very close, indeed!
In closing, notice that Eratosthenes "made some measurements" then entered those measurements into an equation to obtain a second, grander measurement. This is a common technique of engineering and science.
Eratosthenes(~200 BC) sought to know the circumference of Earth. Knowing geometry and having observed eclipses of the moon, he believed Earth was a spherical mass. To measure its circumference, he devised a method that used the rays of sunlight that hit Earth which he assumed arrive parallel. Use his data (shown in the figure) to calculate the circumference of Earth.