SPC Ramuel Mendoza Raagas for Teaching Fellow Tri A. Nguyen

Physics E-1a
Classical Mechanics
Experiment #1:
Measuring from a Distance
Fall Term 2003
Submitted October 28, 2003
  1. Hypothesis:

    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 with 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 French Systeme Internationale. I will use both Imperial and SI units in this experiment. Imperial's great for carpentry and construction. SI is convenient to use when we deal with standard textbook values for planetary bodies, which are often posted in meters.

    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 89.44444║. 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 round-up-wise erratic but conceivable 89.5║. 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 89.44444║ is slurred into 89.4444║?

    The tangent of the angle value visibly drops from 103.1283 to 103.121.

    What happens when 89.4444║ is slurred into 89.444║?

    The tangent of the angle value visibly drops from 103.121 to 103.05.

    What happens when 89.444║ is slurred into 89.44║?

    The tangent of the angle value visibly drops from 103.05 to 102.31.

    What happens when 89.44║ is slurred into 89.4║, 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 89.4║ 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)!

    My Room's Seven-Foot Main Entry Door

    Conventional tape measurement readily ascertains that my door's carpenter did quite a pretty good job in trying to cut out a seven-foot height. I've had bad doors in the past--- years ago in the Philippines. I had an unpainted door which expanded by too much of a coefficient from the cold and uneven cement floor. Coming home timed from work, it was a royal pain to get to push that front door even enough to get myself in.

    how do i set about measuring the height of my door?

    Trust in the carpentry of my Lesley University campus boarding room allows for a quick review of my current such room's front door height by running a ruler tumbling flip-flop seven times. A person who would measure my room's main door of entry equppied with only a rules and no tape measure could run such an easy ruler-filipping cumulative measurement and(confidently disregarding the displacement fuzzed towards the ends of the kinematically jumping ruler--- ends with plastic" fat" ccarrying no useful length of gradations, not numbers---being the wear-away allowance of the measuring stick.

    1. 5.14% was my percentage error in my calculation of my door's height. I calculated m door's height to be 6.64 feet from my sighting distance of 7.88 feet. My door's actually a carpenter-neat seven-feet tall.

    2. This error sure was consistent with my plus or minus half degree indoor angle margin, considering my lucid explanation above.

    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.

    We must measure the height of the Science Center. Public Harvard internet information tells us flat-out that the Science Center is one hundred and sixty feet high. (

    ). 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 steps--- that is six strides or more--- an error of one foot may be incurred (like a fourth of a stride, or half a step). A stride, however, is quite a big thing. A stride is not a step! For us biped humans a step on one foot and then the other (a second step) comprises a stride. An equine stride comprises not only of four steps, but even includes whatever forward lift the horse kicks in by galloping forth.

    A human step may be thirty inches. A human stride may be sixty inches. At home I measured my step to be twenty three and a fifth inches. As such

    It is not enough however that we know the actual height of the Science Center. We are not in a TV game show where we get cash for buzzing the right answer. Awareness of a fact is nowhere as grand as the trial and process of knowledge. Galileo did know through Copernicus that the Earth actually moved around the Sun. What is of value to us is how that Italian expounded on the fact stated already by the Danesman--- that, and how he had to humiliatingly retract it centuries before his Heaven-resting soul was apologized to by the instituted Church.

    Disregarding the internet-posted fact about 160 ft., however, we may get at the approximate height of the Science Center via trigonometry, such as the secondary trigonometric function called a cotangent.

    4. How can we find L from our graphing?
    Could we start out not using graphing paper, but instead any lined writing paper such as yellow pad paper or notebook paper? This might show how arbitrary our x-axis is. I can't seem to get my range finder angle to steadily go up as I step back. I've got plus or minus two full degrees as error.

    There are lots of catches to graphing as directed, although we won't be doing the impossible. It's just that we've got to make decisions of what x and y intervals to take. Including the origin of cotangent equals zero (mathematically not an existing thing, but the limiting case for a very conspicuous asymptomatic value trend) and the hypothetical displacement away from dome tip plumb point seem attractive outsets, and I will seize on this approach, but the trade-off is including (0,0) makes the plot points of our actual data points push up into the upper-right corner, quadrant even on the graphing paper sheet, even landscape orientation of graph content may even be called for as when cotangent y-axis approaches almost 2, and x-axis may entertain up to only 82 strides. I want not to cut away the vast readingless pure-interpolative space standing for the Science Center interior, so that I could see if any geometric structure such as triangle may turn up with an area or other aspect key to deriving a guess value for L.
    By including the origin, however, I forfeit my right, however to plot even the point where I got sixty-six degrees on my range finder, when I had my back against the center gate (then locked) one sunny noon leading to the most popular diagonal walk-path from the Science Center to the MBTA bus stops and train tunnel. Empirically the data point at such gate is of more value than the zero point, or even the first dozen ghost strides where no Range Finder reading to be taken.

    L can't be all that small. L may even need be necessarily slightly larger than D.

    Perhaps we may draw a line segment perpendicular to the slope emanating from the y-intercapt and get the area covered in the triangle under that cornered into the loew-left corner of our graph? Would that work?

    Why so? Since D of the dome is cerainly nothing of John Hancock or Prudential Tower magntitude, nor even of Five-Star Hotel magnitude, and considering the unique Lego-pyramidal/rice terrace shell structure of the Science Center, we cannot be right be beside the Science Center and go sight a PhiRangeFinder of 89.5 degrees aiming for the apex.

    I guess that L would be like two hundred feet at the least? But the lab handout does not prompt from us a ballpark figure for L in the way that one for D is asked for.

    So how do we get a good idea of L from the graph. First we set about with doing the graph, irregardless of yet-to-be-resolved abscissa issues. At least, we can ascertain a range for the cotangent function. Doing Yard-Gate to front side bushes distances, 54 to 66 degrees are the values I got. 54 degrees would then be for L. 66 degrees would be for L + xmax. What are the cotangent values for 54 and 66 degrees----- .726543 and 2.246, respectively. Kirkland street measurements yield much higher cotangent values--- up to 11.43. Beginning in Kirkland Street, however, our LKirkland proves to be larger, as 61 degrees is the least range finder obtained coming out of the Loker Commons pit stop into the elbow corner of Oxford and Kirkland streets.
    For a noticeable portion of our Dx interval, with Dq one or so degrees, Dcotangentf amounts to somewhat .03

    How bad was my error in ascertaining the angle on the Range Finder? It was bad enough with the door. What about outdoors doing the Science Center. Having like two to four tries measuring each stand-point, I get variations as extreme as give or take two full degrees. The filar reader of the Range Finder is quite vulnerable to the fall wind blowing.

    With such an extent of error, the range finder angle does not invariably increase from 54 degrees towards seventy-plus degrees. That's why a couple or even four measurements from the same standpoint prove useful. Nonetheless, I never got a range angle finder lower than 54 degrees. But moving backwards, after obtaining a measurement of 56.5 degrees, I might find an angle reverting to a lower value such as fifty-five. I guess, my error of non-steadiness figured into my vertical hold position of the piece of equipment (something already a matter to contend with doing the preliminary door measurements---- aggravated by the fact, that ---- no longer in my room --- I could not tape-mark an ideal device level (We don't have a tripod mount for a range finder, as may come handy even for young geodetic engineering students), although I tried to hold the range finder around shoulder level in a consistent manner.

    When I found varying angle readings out of the same stand-post, instead of averaging what I got, I just ditched out altogether the reading which seemed odd to the progression I was making.

    Should the steps adjust to the angle, as in the way our lab handout urged us to try all angles (as in all pertinent whole number angles towards ninety degrees, besides their half-degree increments)? Some angle within my 54 to 86 degree empirical range just didn't seem to register, continuous though be the function of relation between angle and stand-point displacement.

    After measuring first 54 degrees, I stepped back two full strides and from there got a reading of 55 degrees.

    The general picture is that for every stride back the cotangent rises by around 0.03 for an argument angle decreasing in like half-degree steps. The minutae of my data, however, shows error of plus or minus two degrees.

    The general slope, however, thus demonstrates a .06 slope per degree

    My data shows that the science center would be around 130 feet. I thus have an error of 18.75 percent.
    tangent f = D

    L + x
    cotangent f = L + x


    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 90║. Taking another, route, going down Kirkland Street, I got as high a value as 87.5║, after which Kirkland Street's own trees obstructed my view.

    When I go back five steps, I do 9.62 feet. I measured an angle of FRangeFinder=57 degrees. I took five more steps back (another two-and-a-half strides), and measured 58 degrees, still another five steps back I measured fifty-nine (59║) degrees.

    Each of my steps is worth is around 1.93 feet. I measured my step at home on my tiled floor. My tiles are squares with nine inch edges. My step took up more than two tiles. I measured the excess with a ruler. A such, it was a matter of eighteen inches across discretely counted tiles plus 4.25 inch on the ruler.

    I guess my L must have been 246 feet. A five-step displacement dx amounts to about 9.67 feet.

    Going West down Kirkland Street

    I measured a FRangeFinder = 64 degrees. I went back ten steps (five strides) and got a 65 degree reading. Ten steps (5 strides) more back down I got a sixty six degree reading. Ten steps further back I got 66 degrees. Yet ten steps (5 strides) further back I got 67 degrees. Five strides more and 68 degrees registered on my Range Finder.

    two strides back59degrees║
    two strides back59degrees║
    two strides back59degrees║
    two strides back59degrees║
    two strides back57degrees║
    two strides back57degrees║
    two strides back57degrees║
    some feet behind54 degrees║
    some feet behind54 degrees║
    some feet behind54 degrees║
    some feet behind54 degrees║
    tangent f = D

    L + x
    cotangent f = L + x


    two strides back59.5║
    two strides back59degrees║
    two strides back59degrees║
    two strides back59degrees║
    two strides back59degrees║
    two strides back57degrees║
    two strides back57degrees║
    two strides back57degrees║
    some feet behind54 degrees║
    some feet behind54 degrees║
    some feet behind54 degrees║
    some feet behind54 degrees║
    tangent f = D

    L + x
    cotangent f = L + x

    4. What can L be? L can't be greater than five hundred feet, because the Harvard map page tells us that four hundred feet is the maximum floor length we get out of the Science Center. Behind the bushes and lawn-margins, I don't think I was that far away from the walled-in floor area of the Science Center.

    How may we find L from the graph? L may be considered an x0, past which Dx values emanate.

    5. One source of uncertainty is the measurement device itself. Another source of error is wind-blowing. Another source of error is the stride. I measured my own step at home as being twenty three point two inches--- hence, my stride 46.4 inches.
    6. I determine the height of the Science Center to be 128.9 feet, according to my data.
    7. I have an error of 18.75 percent, around as bad as it went with my door height measurement. All in all, I could even say I have an error nineteen percent, going by two significant digits.
    8. The size of the earth perhaps snugly fits right under Lovell's thumb. 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.

    9. The moon is
    L = 3.84 x 108 meters = three hundred eighty-four million meters

    Just this Saturday, October 11, elsewhere in America a Florida vs. Chicago baseball game telecast featured an orange moon in between playtime. I did not catch the moon orange over Cambridge this week, because I have not been in windowview or outdoors when sunset turns into early evening (considering the onset hours of my weekday Extension School classes)

    Although using old, crumpled, springless, caseless, paper tape measures were OK for doing my pre-Range Finder mesasurements of my seven-foot door at home, since like Galileo and his piano inclinato, there were ideal values to lapse to within arms' reach (Galileo in publishing his law of odd numbers as to the steadily-accelerated descent of a rolling ball, cared not so much for intricately recording any significant digits of experimental error. Tenth of a human pulse was all he noted down.)--- in measuring my arm's distension towards the moon way beyond my reach, I had deemed it necessary to utilize a brand-new, five-dollar button-spring cased tapes measure from a pretty stationery store.

    56.75 centimeters (.5675 meter) was the reading I got for the full-enough looking moon quarter past seven in the evening on October 9, 2003. This gave me a small angle of .00881 radians, which gave me a (full?) moon diameter of 3.38 x 106 meters, although going by the single significant digit solely warranted by my 5 millimeter paper quadrette edge measurement.

    I have never been good at pinning down myself just when the moon really is full. Was October 11 the night? That may seem plausible as I checked the http://weather.boston.com/ website tonight on October 11. Unfortunately, I could not archive my way a few days back to check whether the ninth or the tenth of October had a fuller moon. On the average, I have some idea of when the moon is full, give or take a day or two.

    We definitely do not want to have the strip of paper's measuring edge too wide. If the strip of paper is so much as a full centimeter wide, then you better have the reach of the boxing champion Evander Holyfield!

    55.2 centimeres gives a good value. 10.


    I missed the horizon thing. My Biology 100 class starts at 5:30 p.m. My Chemistry and Physics start at 6 p.m., so I haven't made it to the prime time of a full moon rising.

    Table[Cot[x*Pi/360]*160.000, {x, 7, 72}]
    In[2]:= Table[Cot[x*Pi/360]*160.000, {x, 7, 72}] Out[2]= {2615.98,2288.11,2032.99,1828.81,1661.66,1522.3,1404.3,1303.1,1215.32,1138.46,\ 1070.58,1010.2,956.122,907.405,863.283,823.129,786.425,752.741,721.713,693.\ 036,666.448,641.725,618.674,597.128,576.941,557.986,540.151,523.336,507.455,\ 492.429,478.19,464.674,451.826,439.596,427.939,416.814,406.184,396.014,386.\ 274,376.936,367.975,359.366,351.088,343.121,335.447,328.049,320.91,314.018,\ 307.357,300.916,294.683,288.648,282.799,277.128,271.626,266.285,261.096,256.\ 054,251.15,246.378,241.734,237.21,232.801,228.504,224.312,220.221} {2615.976876175948`,2288.1066010739087`,2032.992757787953`,1828.8083684418148`\ ,1661.6635328221055`,1522.2983126756137`,1404.3019770991932`,1303.\ 095428475935`,1215.3206580360243`,1138.4591555814734`,1070.5849981307856`,\ 1010.200242348007`,956.1222983092904`,907.4050911388335`,863.282747891062`,\ 823.1286425552497`,786.425124971393`,752.7408175165527`,721.7133605859292`,\ 693.036139885465`,666.4479632144668`,641.7249493657351`,618.6740951837982`,\ 597.1281292110203`,576.9413614017399`,557.9863110145454`,540.1509476145994`,\ 523.3364189574626`,507.455168378114`,492.4293659480404`,478.1895940388629`,\ 464.6737404281316`,451.82606169612814`,439.5963871127396`,427.93943902829176`,\ 416.8142503510083`,406.1836633062892`,396.0138965466074`,386.2741699796952`,\ 376.93637853180053`,367.97480755780106`,359.3658838246746`,351.08795698640614`\ ,343.12110728152936`,335.44697585410796`,328.04861465268743`,320.\ 91035332144315`,314.0176808808241`,307.3571403153866`,300.91623445541313`,294.\ 6833417653533`,288.6476408434278`,282.7990425988626`,277.12812921102034`,271.\ 6260990921743`,266.2847171760829`,261.09626994060636`,256.05352464656806`,251.\ 14969233879845`,246.37839421033334`,241.73363097838416`,237.2097549620385`,\ 232.80144458759113`,228.50368107873834`,224.31172711621377`,220.\ 22110727538774`} In[3]:= Table[Cot[x*Pi/360]*1.000, {x, 7, 72}] Out[3]= {16.3499,14.3007,12.7062,11.4301,10.3854,9.51436,8.77689,8.14435,7.59575,7.\ 11537,6.69116,6.31375,5.97576,5.67128,5.39552,5.14455,4.91516,4.70463,4.51071,\ 4.33148,4.1653,4.01078,3.86671,3.73205,3.60588,3.48741,3.37594,3.27085,3.\ 17159,3.07768,2.98868,2.90421,2.82391,2.74748,2.67462,2.60509,2.53865,2.47509,\ 2.41421,2.35585,2.29984,2.24604,2.1943,2.14451,2.09654,2.0503,2.00569,1.96261,\ 1.92098,1.88073,1.84177,1.80405,1.76749,1.73205,1.69766,1.66428,1.63185,1.\ 60033,1.56969,1.53986,1.51084,1.48256,1.45501,1.42815,1.40195,1.37638} In[4]:= Table[(Cot[x*Pi/360]*160.000)-220.221, {x, 7, 72}] Out[4]= {2395.76,2067.89,1812.77,1608.59,1441.44,1302.08,1184.08,1082.87,995.1,918.\ 238,850.364,789.979,735.901,687.184,643.062,602.908,566.204,532.52,501.492,\ 472.815,446.227,421.504,398.453,376.907,356.72,337.765,319.93,303.115,287.234,\ 272.208,257.969,244.453,231.605,219.375,207.718,196.593,185.963,175.793,166.\ 053,156.715,147.754,139.145,130.867,122.9,115.226,107.828,100.689,93.7967,87.\ 1361,80.6952,74.4623,68.4266,62.578,56.9071,51.4051,46.0637,40.8753,35.8325,\ 30.9287,26.1574,21.5126,16.9888,12.5804,8.28268,4.09073,0.000107275} In[6]:= Table[((Cot[x*Pi/360]*160.000)-220.221)/3.8666666, {x, 7, 72}] Out[6]= {619.592,534.798,468.82,416.014,372.787,336.744,306.228,280.054,257.353, 237. 475,219.922,204.305,190.319,177.72,166.309,155.924,146.432,137.721,129.696, 122.28,115.404,109.01,103.048,97.476,92.2553,87.3531,82.7405,78.3919,74.2847, 70.3987,66.716,63.2205,59.8979,56.735,53.7203,50.8431,48.0938,45.4637,42.9448,40.5298,38.2122,35.9857,33.8449,31.7845,29.7998,27.8865,26.0404,24.2578,225352,20.8695,19.2575,17.6965,16.184,14.7174,13.2944,11.913,10.5712,9.26703,7.9988,6.76484,5.56361,4.39364,3.25356,2.14207,1.05795,0.0000277436}
    • http://www.amazon.com/exec/obidos/tg/detail/-/0879759801/ref=lib_rd_ss_TT01/103-1069484-6090244?v=glance&s=books&vi=reader&img=4#reader-link
    • http://www.thelatinlibrary.com/newton.html
    • http://members.tripod.com/~gravitee/booki1.htm
    • vorlon.ces.cwru.edu/~ames/principia/principia.pdf
    • http://www.maths.tcd.ie/pub/HistMath/People/Newton/Principia/Bk1Sect1/PrL1S1.pdf