School kids measure distance to the Moon(technologyreview.com)
technologyreview.com
School kids measure distance to the Moon
http://www.technologyreview.com/blog/arxiv/23205/
7 コメント
Shure, I think I could work it out, but I would still like to see your process.
PS: A pendulum and a calendar and a stopwatch or clock?
PS: A pendulum and a calendar and a stopwatch or clock?
Suppose the moon is in a circular orbit with radius R and angular velocity w. Then its position looks like R exp(iwt), so its acceleration is the second derivative of this = -Rw^2 exp(iwt), whose magnitude is -Rw^2. (That's a standard formula but I always have to derive it from scratch.) So GM = R^3w^2.
Now, what do we know? If we have a calendar that marks full moons, we know how often the occur and therefore we know w. What about the stuff involving GM? Well, now construct a pendulum. Standard formulae again but let's do it from scratch again. Length L, angular displacement a: vertical displacement is negligible to first order, so for small displacements the restoring force just comes from the tension in the string. To first order again, that equals the gravitational force on the mass, so the horizontal component of the acceleration is GM/r^2.La where r is the radius of the earth. So, measure the period of the pendulum (and of course its length); that tells us a and therefore GM/r^2. So provided we know r we now have GM and therefore R, and we're done.
So, one remaining question for RiderOfGiraffes: are you happy to look up r in a reference book for this exercise, or do you have a classroom-friendly way of getting that by simple observation and calculation too?
Now, what do we know? If we have a calendar that marks full moons, we know how often the occur and therefore we know w. What about the stuff involving GM? Well, now construct a pendulum. Standard formulae again but let's do it from scratch again. Length L, angular displacement a: vertical displacement is negligible to first order, so for small displacements the restoring force just comes from the tension in the string. To first order again, that equals the gravitational force on the mass, so the horizontal component of the acceleration is GM/r^2.La where r is the radius of the earth. So, measure the period of the pendulum (and of course its length); that tells us a and therefore GM/r^2. So provided we know r we now have GM and therefore R, and we're done.
So, one remaining question for RiderOfGiraffes: are you happy to look up r in a reference book for this exercise, or do you have a classroom-friendly way of getting that by simple observation and calculation too?
You mean apart from the original definition of the meter as one ten millionth of the distance from the north pole to the equator passing through Paris? That means the circumference of the Earth is 40*10^6 meters, so the radis is that over 2pi.
I guess, then, in addition to the pendulum, stopwatch and calendar we also need a ruler. We can use these to measure the size of the Earth directly using either the method of Eratosthenes (who also created the sieve for finding primes, and invented the system of latitude and longitude) or using the method used by Abu al-Rayhan al-Biruni.
That last, by the way, is cool, and I intend to use it in my next maths masterclass field trip.
I guess, then, in addition to the pendulum, stopwatch and calendar we also need a ruler. We can use these to measure the size of the Earth directly using either the method of Eratosthenes (who also created the sieve for finding primes, and invented the system of latitude and longitude) or using the method used by Abu al-Rayhan al-Biruni.
That last, by the way, is cool, and I intend to use it in my next maths masterclass field trip.
Yes, I do mean other than apart from the original definition which, after all, is not the current definition. (You could say "but everyone knows ...", but then you could equally say that "everyone knows" g ~= 9.8m/s^2.) The Eratosthenes method is lovely but surely rather difficult in a classroom. (Am I missing a trick?) All I know about the al-Biruni method is a couple of sentences from Wikipedia, but it sounds as if it effectively needs some external (geographical) data that you can't measure in a classroom. Hmm.
OK, I've done some more homework ...
The pendulum method gives the equation
So R/D = (T/2)(2pi/L) where T is the transit time, and turns out from observations to be around 12500 seconds.
Combining these equations we get:
The pendulum method gives the equation
l L^2 R^2 = P^2 D^3
l = pendulum length (assume 1m)
P = pendulum period (roughly 2s)
L = Lunar orbital time (27.32 * 86400)
R = Radius of the Earth
D = Distance to the Moon
So R/D = (T/2)(2pi/L) where T is the transit time, and turns out from observations to be around 12500 seconds.
Combining these equations we get:
D = l*(pi*T/P)^2
R = l*pi^3*T^3/(L*P^2)
Bother - forgot that asterisks have a special meaning. Sorry about the formatting.
Yes, pendulum and stopwatch.
More later.
More later.
[deleted]
For really accurate measurements check out http://www.physics.ucsd.edu/~tmurphy/apollo/apollo.html
By bouncing a laser off several reflectors that were left on the moon, they are able to get accuracies of one millimeter.
I'm reminded of Clifford Stoll's TED talk where he dicusses (among other things) how he teaches high schoolers to calculate the speed of sound.
http://www.ted.com/index.php/talks/clifford_stoll_on_everyth...
http://www.ted.com/index.php/talks/clifford_stoll_on_everyth...
If you've got two OLPC XO laptops you can measure the distance between them using a similar techique:
http://wiki.laptop.org/go/Acoustic_Tape_Measure
http://wiki.laptop.org/go/Acoustic_Tape_Measure
Measuring the height of a building using a barometer:
http://www.snopes.com/college/exam/barometer.asp
http://www.snopes.com/college/exam/barometer.asp
They should try to measure it the way Ptolemy did! Or the distance to Sun...
REALLY smart school kids would go to wolfram alpha to get the distance to the Moon, and let the suckers analyze echo's from old NASA clips.
This deserves a separate discussion.
See http://news.ycombinator.com/item?id=802856
See http://news.ycombinator.com/item?id=802856
If NASA really did fake the Moon landings, they did a bloody good job!
If you want to know how to calculate the distance, let me know and I'll write something about it.