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Phy 231
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** Question Form_labelMessages **
Optical Distortion Percent Q
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For each level of reduction, give the difference between the two sides as a percent of the length of the hypotenuse. Give your results in the first line as a series of four numbers separated by commas, in order with the result with for the full-sized ruler first, the result for the triply-reduced ruler last. Use the appropriate number of significant figures in your results. Starting in the second line, give your explanation of how you got your results.
11.2%, 9.82%, 11.07%, 10.80%
I took the difference between the two sides, divided that result by the hypotenuse, and multiplied by 100%. ((hyp - side) / hype)*100%
I did this for each of the different sized measurements.
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According to your results, what would be the length of an object that measures exactly 1 cm on the full-sized copy, if measured using the singly, the doubly, and the triply-reduced copy? Give your answer in the usual comma-delimited format in the the first line, then starting in a new line explain how you got your results.
1.00 cm, 1.112 cm_s, 2.3332 cm_d, 4.92468 cm_t
I took the original at 100% and added the next percentage up to that result.
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cm_s = 0 + 1.00cm * 1.112 = 1.112cm_s
cm_d = 1.112 cms + 1.112*1.0982 = 2.3332 cm_d
cm_t = 2.3332 cm_d + 2.3332*1.1107 = 4.92468 cm_t
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I am having trouble using my percentage and using it to calculate the estimated size on the different sized rulers.
I know that each percentage is in reference to the measurement that preceded it, and that it should be added to that preceding measurement.
It is just that my numbers are not adding up correctly and I feel i made an error in my calculations.
I have been stuck for a few hours trying to figure this out, and think maybe there is an instruction that i am over looking. I will include a copy of my procedure in the next box, and If you could highlight a seen mistake in my calculation i will address it and take it from there.
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The beginning of the experiment actually occurs below this. I'll have to see your original data, and will keep the results you've reported here and your question in mind as I go through the document.
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The goals of this experiment are as stated above:
Understand how the different rulers have different degrees of precision and accuracy for different measurements.
Determine as accurately as possible any optical distortions in the copies. Related questions you should keep in mind and answer:
What is the margin of error in your placement of the markings?
Within what limits of accuracy can you place and measure the distance between two markings at each level of reduction?
Can optical distortion be detected within this margin of error?
If you have a ruler whose smallest division is a millimeter, then the position of a point on the ruler would be measured accurate to a millimeter, and you would also make your best estimate of where that point lies between the marking (e.g., a point between the 3.8 and 3.9 cm markings might lie halfway between those markings, in which case you would estimate the position as 3.85 cm; or it might lie closer to one marking than the other, so you might have an estimate of 3.82 cm or 3.86 cm; you should try to estimate the position between the smallest mark to the nearest tenth of that distance).
Using a piece of typing paper (actually any paper will do as long as its corners are not rounded), cut out a right triangle by trimming one of its corners in the manner indicated by the figure below (cut along the red line, remove the triangle, which will look like the triangle shown in the lower-right-hand corner of the figure). Make the triangle fairly short. The longest side should be between 1 and 2 inches long.
Measure the hypotenuse of your triangle, using each level of reduction. For each level of reduction you have a 'block' consisting of several rulers; for each level, measure in about the center of the middle ruler. Estimate each measurement to the nearest 1/10 of a division (you won't be accurate at 1/10 division and on the smallest reductions it will be difficult to estimate, but that's no excuse for not doing your best).
We'll make the following conventions for our units of measurement:
Let 'cm' stand for centimeters as measured with the full-sized ruler.
Let 'cm_s' stand for centimeters as measured with the singly-reduced ruler.
Let 'cm_d' stand for centimeters as measured with the doubly-reduced ruler.
Let 'cm_t' stand for centimeters as measured with the triply-reduced ruler.
Give your results for the hypotenuse below, separated by commas. A sample format, which gives a brief but complete (though not very accurate) answer, might be '3.14 cm, 5.37 cm_s, 9.48 cm_d, 13.25 cm_t'. Your numbers of course will differ from those given here.
5.35 cm, 8.15 cm_s, 12.65 cm_d, 19.45 cm_t
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Describe in words what you did to make your measurements as accurate as possible:
I measured the side with my regular ruler and recorded it as my first measurement. I then used this measurement on my ruler and placed it onto the other rulers and attempted to line up the lines as close as possible. This was done on a flat binder and was done for each different side measurement. This was the closest I could get to actual measured values, the ruler helped to flatten out the surface.
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Which of your measurements do you think would be the most accurate, in the sense of having the least uncertainty?
The measurement that has the least uncertainty would be the last measurement cm_t. Its 'mm divisions are the smallest and the distance between them are the smallest intervals.
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In the same way measure the shorter of the two legs of the triangle and give your results below:
2.45 cm, 3.85 cm_s, 6.15 cm_d, 9.35 cm_t
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Repeat for the longer of the two legs of the triangle and give your results below:
4.75 cm, 7.35 cm_s, 11.25 cm_d, 17.35 cm_t
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Consider the two sides whose lengths are closest. This might be the hypotenuse and the longer leg, or it might be the longer leg and the shorter leg, depending on how you cut your triangle.
According to each ruler, what is the difference between these two sides? Give you answer in a format similar to that of the first question, as four quantities separated by commas.
0.600 cm, 0.800 cm, 1.400 cm, 2.100 cm
The above are the differences between the hypotenuse and the long side of the triangle for each measurement.
difference = Hypotenuse - long side
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For each level of reduction, give the difference between the two sides as a percent of the length of the hypotenuse. Give your results in the first line as a series of four numbers separated by commas, in order with the result with for the full-sized ruler first, the result for the triply-reduced ruler last. Use the appropriate number of significant figures in your results. Starting in the second line, give your explanation of how you got your results.
11.2%, 9.82%, 11.07%, 10.80%
I took the difference between the two sides, divided that result by the hypotenuse, and multiplied by 100%. ((hyp - side) / hype)*100%
I did this for each of the different sized measurements. dumm1arss3
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According to your results, what would be the length of an object that measures exactly 1 cm on the full-sized copy, if measured using the singly, the doubly, and the triply-reduced copy? Give your answer in the usual comma-delimited format in the the first line, then starting in a new line explain how you got your results.
1.00 cm, 1.112 cm_s, 2.3332 cm_d, 4.92468 cm_t
I took the original at 100% and added the next percentage up to that result.
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It isn't clear what you mean by 'the next percentage up', but there is nothing additive about this situation.
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If a 1 cm length on the full-sized ruler measured 1.112 cm_s, then every cm of your 5.35 cm hypotenuse measurement would measure 1.112 cm_s, and the entire 5.35 cm hypotenuse would measure 1.112 * 5.35 cm_s, somewhat short of 6 cm_s. However, it measured 8.15 cm_s. Similar inconsistencies would occur for the measurements of the sides.
The conclusion must be that either there is a flaw in your measurements of the three sides, or that 1.112 cm_s is not the length of a 1 cm object.
It is clear from your actual measurements, which are very reasonably consistent among themselves, that 1 cm measures somewhat more than 1.5 cm_s.
If you apply the same reasoning to the cm_t measurements, you will find that 1 cm measures somewhat less than 4 cm_t.
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cm_s = 0 + 1.00cm * 1.112 = 1.112cm_s
cm_d = 1.112 cms + 1.112*1.0982 = 2.3332 cm_d
cm_t = 2.3332 cm_d + 2.3332*1.1107 = 4.92468 cm_t
&&&& This calculation seems wrong, I know the original is 1 cm long, and the next ruler is 11.2% larger. So I multiply the 1 cm by 1.112 to get a measurement 11.2% larger.
It is after this lose focus, the next percentage is 9.82% larger, so does this mean I take my last measurement and add another 9.82 % to it like so.
1.112 cm + (1.112 * 1.0982) = 2.332 cm ?
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What would be the lengths, in units of cm of the full-sized ruler, of three objects, whose respective lengths measure 1 cm_t, 1 cm_d and a cm_s? Give the three lengths separated by commas in one line, then starting in a new line explain how you got your results.
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According to your present results what would be the length, on each of your rulers, of an object whose length on a the doubly-reduced ruler was determined to be 8.34 cm_d?
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You made your measurements in the middle of each 'block' of rulers. We might expect that, due to optical distortions in the copying process, there might be some difference in measurements made at different places on each ruler 'block'. Investigate this question.
Are there places on the triply-reduced copies where an object measured at one location gives a different result, due to distortions of the copy, than the same object measured at another location? If so, at what positions and at what level of reduction do you observe the most distortion?
Give your results and explain how you investigated this question.
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If you believe you did detect distortion, how much of the observed difference in measurements do you think you can attribute to actual distortion, and how much to limits on your accuracy and the precision of the markings?
If you did not detect them, this doesn't mean that there aren't distortions. There almost certainly are, but they might be too slight for you to measure. In this case, how small would they have to be before you would be unable to detect them? How big is the largest discrepancy you would be unable to discern? Give your results and explain your thinking.
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Don't actually do this, but if you were to write a 100-word paragraph with a #2 pencil, measuring the pencil before and after, which level of reduction do you think would allow you to determine most accurately the difference in the length of the pencil from eraser to point?
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Your instructor is trying to gauge the typical time spent by students on these experiments. Please answer the following question as accurately as you can, understanding that your answer will be used only for the stated purpose and has no bearing on your grades:
Approximately how long did it take you to complete this experiment?
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You have good, consistent measurements. However you appear to have the wrong idea about how the different measurements compare.
See if my note helps.
&#Please see my notes and submit a copy of this document with revisions, comments and/or questions, and mark your insertions with &&&& (please mark each insertion at the beginning and at the end).
Be sure to include the entire document, including my notes. &#
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