SPC Ramuel Mendoza Raagas | for Teaching Fellow Tri Nguyen |
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Physics E-1a Classical Mechanics | Experiment #1: Measuring from a Distance | Fall Term 2003 |
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People are capable of securing credible measurements of objects, even should these lie beyond the ready touch of our fingertips. Moreover the tools we use in measuring linear length can reach for orders of magnitude not apparent in the actual dimensions of the tool itself. With a metal ruler with gradations of only 1/64th of an inch apart, for example, we may in fact obtain reasonable values of microlength going as low as 10^{-7} meter (when we shoot a laser at an angle to such ruler). Similarly, with not even a full inch of unparticular paper, we may fetch for ourselves a distance tantamounting to hundreds of thousands of kilometers.
In science, we measure. When we don't write out our numbers and calculations involving them, what we end up with is NOVA. NOVA shows don't help kids with their MCATs. MCATs don't help test takers do Feynman integrals or reach the deep end of calculus.
The purpose of this experiment is to measure length along with any errors we must consider in our derivation of it.
This experiment involving measurement banks on the omnipresent underlying principle of proportion. Thanks to ratios, we can engage into conversion factors within and between different measurement systems such as the Ango-Imperial system and the Fench Systeme Internationale.
All our measurements for this experiment will be of either dimension type L or q. Length, either linear or angular, will be all we'll deal with here.
Error can get quite volatile in the door height calculational measurement. The range finder is an error-prone instrument. It is not an industrial-grade piece of equipment. A protractor is a reliable article of school supply, although the thickness of wood or plastic separates its gradation of etchings via elevation from its intersection with two ray-segment representations the user plants it on.
How error plays in here indeed proves quite interesting. Our Lab Manual tells us of adding percentage of errors for two data which figure into either the multiplication or division. But what of errors in data fed into trigonometric functions? Error and significant figures can get way more out of end, owing to the non-linear nature of trigonometry. A case in point is the tangent of 89.44444º. In this Experiment #1, we don't have to worry about such precise measurements of angle. Our Range Finder can only have us read the accurate 89 degrees, or else, the erratic but conceivable 89.5º. But, our rudimentary Range Finder aside, would not seven significant figures prove useful in astronomical measurements? I mean we're doing the moon now, but Hubble telescope sure wouldn't go by a mere four significant digits for its mirrors. What happens when 89.44444º is slurred into 89.4444º?
The tangent of the angle value visibly drops from 103.1283 to 103.121. What happens when 89.4444º is slurred into 89.444º? The tangent of the angle value visibly drops from 103.121 to 103.05. What happens when 89.444º is slurred into 89.44º? The tangent of the angle value visibly drops from 103.05 to 102.31. What happens when 89.44º is slurred into 89.4º, to make it a typical value for an Urone or Giancoli textbook value? The tangent of the angle value visibly whoppingly drops from 102.3 to 95.5! What happens when 89.4º is slurred into 89, as must be the case when we use our Range Finder? The tangent of the angle value visibly drops from 103.121 to fifty-seven (57)!The maximum convenient distance of the Range Finder r away from the door to be measured is 16.854î + 12.4583ĵ, where the k-component is conveniently taken as zero, being half-way up the door (Either moiety of the door's height-axis may be construed as being positive---- the other, negative---- along the z-coordinate.
3. Before I looked up the actual, official value in the Internet, I had estimated the height of the Science Center to be around one-hundred and forty feet.
The night I first got the lab handoutr I reckoned that the Science Center had perhaps one hundred forty feet more of it above from where I was, which was at the rim of our Lecture Hall B. With athlete's intuition, I asured myself that our Science Center must surely carry a height greater than a hundred feet.
We must measure the height of the Science Center. Although Harvard web, will tell us flat out that it's one hundred and sixty feet, we must go through the process of securing our own measurement, much like audience members of Greek Theater simply must go through the spectacle of viewing the play, although they well know the argument of the plot beforehand.
For every dozen or more strides, an error of one stride may be incurred. A stride may be thirty inches. At home I measured my stride to be twenty three and a fifth inches, which amounts to 1.933333 feet = 1.93 ft.
There's uncertainty in use of the RangeFinder device, even in the construction of it. I availed of the option to stick in a straw for sighting. I cut the straw a bit so that when it's unexposed end met up with the origin of my Range Finder, the whole assembly would fit into my zip-up organizer.
Public Harvard internet information tells us flat-out that the Science Center is one hundred and sixty feet high. http://map.harvard.edu/level4/LawSchool/sc.shtml Disregarding the internet-post fact, however, we may get at the approximate height of the Science Center via trigonometry, such as the secondary trigonometric function called a cotangent.tangent f = | D L + x | cotangent f = | L + x D | ||||||||
From what vantage point may the height of the Science Center be measured? Along Kirkland and Oxford Streets, trees get in the way--- but not always. The Science Center Dome is no major protrusion the likes of the Prudential Tower by which I had navigated my street-entangled friend out of Dorchester, where we had done some food-sorting as community service volunteers some years back. So, although the Science Center needs be no less than a hundred feet, its height doesn't add up as being hundreds of feet.
Beyond the corner of Kirkland Road anf its namesake street, the dome is not much worth sighting, but folding forward nt the Science Center from such boundary, F readings run fgfrom 87.5±5˚ to 79.5 =±5˚.F | measured on Septmeber 25 |
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readings | off the corner instersection of Kirkland Street and... |
87.5˚ | Kirkland Road |
84˚ | Kirkland Road (forward more) |
82.5˚ | Sumner Road |
81.5˚ | Sumner Road (forward more) |
80˚ | Kirkland Place |
79.5˚ | Kirkland Place (forward more) |
September 29 before noon | from bushy outskirt of Science Center just past which tip of Science Center dome may be sighted | |
some feet behind | 54 degreesº | |
many strides down back | the dome is hidden behind trees inside Yard leading to gate past Stoughton Hall, but can be peeped through air spaces, if equiradial movement shift of observer is applied | 71º |
The path striding back, plying along Stoughton Hall, may not be continued. Plying further back along the Yard-fronting panel of Stoughton Hall causes the dome to vanish behind the other building diagonally across Stoughton Hall.
If the path going backwards were unobstructed by buildings, then we could proceed and get angles more approaching the limit 90º. Taking another, route, going down Kirkland Street, I got as high a value as 87.5º, after which Kirkland Street's own trees obstructed my view.
When I go back five strides, I do 9.62 feet. I measured from there an angle of F = 57 degrees. I took five strides back, and measured 58 degrees, still another five strides back I measured fifty-nine degrees.
Each of my strides is worth is around 1.93 feet. I measured my stride at home on my tiled floor. My tiles are squares with nine-inch edges. My stride took up more than two tiles. I measured the excess with a ruler. A such, it was a matter of eighteen inches across two discretely counted tiles plus 4.25 inch on the ruler.
I guess my L must have been 246 feet. A five-stride dx is about 9.67 feet.
I measured a F_{RangeFinder}= 64 degrees. I went back ten strides and got a 65 degree reading. Ten strides more back down I got a sixty six degree reading. Ten steps further back I got 66 degrees. Yet ten strides further back I got 67 degrees. Ten strides more and 68 degrees registered on my Range Finder.
59.5º | |||||||||||
two strides back | 59degreesº | ||||||||||
two strides back | 59degreesº | ||||||||||
two strides back | 59degreesº | ||||||||||
two strides back | 59degreesº | ||||||||||
two strides back | 57degreesº | ||||||||||
two strides back | 57degreesº | ||||||||||
two strides back | 57degreesº | ||||||||||
some feet behind | 54 degreesº | ||||||||||
some feet behind | 54 degreesº | ||||||||||
some feet behind | 54 degreesº | ||||||||||
some feet behind | 54 degreesº | ||||||||||
tangent f = | D L + x | cotangent f = | L + x D | ||||||||
two strides back | 59.5º | ||||||||||
two strides back | 59degreesº | ||||||||||
two strides back | 57degreesº | ||||||||||
two strides back | 57degreesº | ||||||||||
some feet behind | 54 degreesº | ||||||||||
some feet behind | 54 degreesº | ||||||||||
tangent f = | D L + x | cotangent f = | L + x D | ||||||||
I've seen only snatches of the movie Apollo 13 when it was featured on Direct TV this summer--- I didn't get to the thumb sequences.
Serway tells us that the mean radius for [the oblate spheroid that is our planet] Earth measures to like 6.37 million meters. That would make for a 12.7 thousand kilometer diameter. The moon has a diameter 3.66 times smaller--- almost a percent, however, of its distance from us. The moon is not that far, considering its size. Hence, we know how it works on the ebb and flow of ocean tides.
On Saturday October 4, I could not do a measurement of the moon, as it was drizzling and the sky had an overcast of clouds.
What error do we consider? The Earth itself is a margin of error. It's radius, in fact.
The apparence of the moon itself lends to some error. I admit that even during the months I had savoured an orange summer moon in the Philippines, I could not ascertain which night the moon was indeed full---- I'd give or take a night or two.
There are some principles of Earth-moon behavior.
http://www.infoplease.com/ce6/sci/A0859764.html
Ignoring such and so several factors, we get s = rq = 3.84 x 10^{8} meters *
The bigger the portion of the moon is illuminated, the nearer so may the half-centimeter wide paper rectangle (or cigarette, or thumb, or snub antenna of low-end Nokia cellphone) be drawn into the observer's eye so as to eclipse the moon to a reasonable fit, rather than to eclipse in excess. Whereas, on the contrary, one would be harder put to eclipse any one of the stars with the such a foreground (of thumb or cigarette) except with such excess of foreground at hand.
Whereas for the door measurement and the dome measurement I had preferred to use the Anglo-imperial measurement system, I go metric for the moon measurement, because the meter indeed was conceived as a planet thing, just as Imperial length started out as a king thing.
Doctor Sarah (in the website http://mathforum.org/library/drmath/view/57566.html ) is one of many who recounts that our Earth's girth was orginally construed as being neatly forty million meters. Each of Earth's four corners perhaps poetically conceived as being ten million meters.
The earth is around 40070 kilometers in circumference. That's 40,070,00 meters.
http://math.rice.edu/~ddonovan/Lessons/eratos.html Could not the diameter of the moon be derived even from a crescent. Obviously, a New moon can't tell us anything, and the stars of the transition from September to October didn't point to any considerable portion of a moon, but I don't know why the first time I first saw the moon after getting the lab hand=-out , the moon was already a half-moon? How did the quarter moon pass me by?Must have been some inclement waether. How do we get data? Lifting it off the Internet is one way, but beyond merely retrieving the most accurately established values, we get reasonable values which themselves are of greater worth, considering the roads of reasoning rather than stationary set digits of their values. The earth's diameter is 12742 kilometers.On the surface, Experiment number one does not seem altogether to ask that much from us. Well, preparing for it in any case needs be at least thrice a burden as would be doing a Chem 5/7/E1 lab, although there would be ways to make it as arduous as doing a term paper.
If we take up the moral calling, however, not to slack off, and realizing our present-day material advantages, Experiment #1 can be quite a burden. Well all Physics E-1 labs don't have the fill in the blanks/boxes of Bio and Chem E-1 labs. Physics E-1 labs end up calling for more thinking, the discursive type, particularly.
Although pedagogical lab work for introductory science courses don't expect the discovering of anything new, the Physics E-1 lab would like to instill in us the glee of discovering something new for the individual neophyte student. Such newness no longer bears the grandeur of the new for mankind the likes of Newton's apple, Foucault's pendulum or Fermat's Last Theorem.
Thinking for ourselves what others have already thought out systematically (and tucked into Instructor's Solutions Manuals Whereas Eratosthenes had his wells of water, we have the well of the Internet. Although the Internet may be scoffed by some sector of old-school teachers as being Error is everywhere, but what is less pleasant is blunders. Blunders are like if you mistake the Earth's radius for its diameter. Blunders are what MCAT Physics try to trip the student on. Blunders such as mistaking a relation of proportion as being simple and direct whereas it is actually the square of one variable that is in ratio with another variable.0.54 meters +- .005 meters is the distance by which I put the little quadrette of paper away from my eye to fit reasonably snug such quadrette of paper's foreground over the moon. .005 meter (the quadrette's consequential width) divided by my .54 distension gived me a b = .00925925925... radians = .00926 radians. Multiplying such b value by L (the distance to the moon) so as to get a sector the arc length of which we'd approximate-interchange with a cord representing the moon's diameter. I get 3.5555... x 10^{6} meters = 3.5555... x 10^{6} meters = 4 x 10^{6} meters, which is what the moon any way is if we were to slur its diameter into a single significant figure. I can't have three significant figures to the fashion of 3.48 x 10^{6} meters, the official table value of the moon's diameter. I can't report neither 3.55 x 10^{6} meters nor 3.56 x 10^{6} meters, because my quadrette's width was measured using only a singe significant figure, being 5 millimeters, that is.