Introduction: We estimate the difference in time between sunsets at different positions. In the first position, we set our clock to t=0 and lie down. Once we see the sunset, we immediately stand up and start the timer. Now, we are at an elevated position and can see the sun above the Earth's shadow. We wait for the Earth to rotate a tad more until we are again in Earth's shadow and the sun sets again.
Data and Calculations:
The time we calculated between both disappearances of the sun was t=9 seconds.
We can convert this time to the angle the earth rotates meanwhile.
t/(60*60*24)=[Theta]/360
where we compare the ratios of seconds in a day to the Earth's rotation in degrees [Theta] in a day, and calculate
[Theta]=.0375 degrees
We can use the trigonometric definition of cosine to calculate Earth's radius.
After 9 seconds, the Earth has rotated ~.0375 degrees. Assuming my height as 1.5 meters, the sun is out of my line of sight while standing at this time.
In the above picture a is the Earth's radius, and I am at point A standing tall and high with my head in the skies. Define [Theta] as Angle ABC, and AC is tangent to the sphere with angle [Gamma]=90 degrees.
We see that Cos[Theta]=a/c, where a is the Earth's radius and c is the radius a plus my height h.
so Cos[.0375]=a/(a+h).
(a+h)*Cos[.0375]=a, so a=.0015*Cos[.0375]/(1-Cos[.0375])=7003.32 kilometers.
According to Wikipedia, Earth's radius is 6371km so this is a surprisingly accurate estimate.
Note:
Assume our stopwatch had read 8 seconds. Then a=.0015/(1-Cos[1/30])=8863.58 kilometers
or 10 s: a=5672.99 kilometers
A small error in timing can indicate a huge error in calculated radius because a seemingly negligible change in time will be compounded by virtue of its smallness.
A possible source of error is that timer's judgment reaction time from when the person lying/standing sees the sunset. Also, the jumper's own judgment might not be consistent when lying and when sitting regarding when the sunsets. Atmospheric affects also tampered with our observation since the clouds made it difficult to directly observe the sun set.
To check for personal and random errors we performed the experiment in two different locations, at the sandy beach and atop the pier (the lifeguard didn't let us stand on his station for a third location). This method of multiple testing also put a limited check on some systematic errors (faulty timers, uneven ground) but not all (clouds remained).
Conclusion:In the 3rd century B.C., Eratosthenes had measured the radius of the Earth when the sun was directly overhead on the day of the summer solstice. One popular story goes about how he measured the angle of his shadow with respect to him (which would be 0 degrees with the sun overhead) in Syene and had a friend to the same simultaneously in Alexandria. Then, by using the ratio delta[Theta]/360=Distance between Alexandria and Syenee/(circumference of earth)
he extrapolated the radius of the earth.
This experiment had a sublime and humbling effect. From one small spot on Earth we were able to measure the size of our entire world.
Acknowledgements: Thanks to John for his hilarious dry puns which one-upped mine, to Mee for posing by a picture of my back, to Iryna for accepting her role as a reclining sand model, and to the sun for bothering to set.
"sublime and humbling effect" - I love that!
ReplyDeleteHow did people go and forget that the world was round after Eratosthanes? Or is all that stuff about "people thought the world was flat before Columbus and Magellan" not actually true?