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course Phy 242
Question: Suppose you measure the length of a pencil. You use both a triply-reduced ruler and the original ruler itself, and you make your measurements accurate to the smallest mark on each. You then multiply the reading on the triply-reduced ruler by the appropriate scale factor.
· Which result is likely to be closer to the actual length of the pencil?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv The original ruler itself is likely to measure the pencil closer to the actual length because the triply-reduced ruler could be distorted or the scaling factor could be rounded and these both would potentially skew the measurements significantly. Also, if it is distorted and you're multiplying by a scaling factor, you've increased the distortion by multiplying by the scaling factor.
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· What factors do you have to consider in order to answer this question and how do they weigh into your final answer?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv You have to consider whether or not the triply-reduced ruler has any kind of distortion and whether or not the scale factor is a rounded value that could skew the final measurement. This would weigh into whether or not the smaller scale of measurement would be more precise or if the effect of scaling would skew it to be inaccurate.
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Question: Answer the same questions as before, except assume that the triply-reduced ruler has no optical distortion, and that you also know the scale factor accurate to 4 significant figures.
· Which result is likely to be closer to the actual length of the pencil?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv Measurements using the triply-reduced ruler is likely to be closer to the actual length of the pencil because you are using a smaller scale of measurement and since there is no optical distortion and the scale factor is somewhat accurate your measurements will be more precise.
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· What factors do you have to consider in order to answer this question and how do they weigh into your final answer?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv You have to weigh in the fact that there is no optical distortion that could possibly skew your measurements. With this, and the fact that the scaling factor is accurate to four decimal places, the other variable to consider is the visual accuracy of your measurements which should be more accurate using the triply-reduced ruler because of the smaller scale.
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Question: Suppose you are to measure the length of a rubber band whose original length is around 10 cm, measuring once while the rubber band supports the weight of a small apple and again when it supports the weight of two small apples. You are asked to report as accurately as possible the difference in the two lengths, which is somewhere between 1 cm and 2 cm. You have available the singly-reduced copy and the triply-reduced copy, and your data from the optical distortion experiment.
· Which ruler will be likely to give you the more accurate difference in the lengths?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv I would think the singly-reduced copy would be best in case there is any distortion, because you are only introducing distortion one time in stead of three times (like the triply reduced ruler would). ??? I don't know what you're talking about when you say ""and your data from the optical distortion experiment"" because we haven't had an optical distortion experiment???
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You're right. I moved that experiment and neglected to modify this query.
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· Explain what factors you considered and how they influence your final answer.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv I considered the distortion factor and how that could skew my results. If you use the triply-reduced ruler and there is any distortion, this distortion has now been magnified 3 times, compounding the distortion. Where as the singly-reduced ruler would only be distorted once.
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Question: Later in the course you will observe how the depth of water in a uniform cylinder changes as a function of time, when water flows from a hole near the bottom of the cylinder. Suppose these measurements are made by taping a triply-reduced ruler to the side of a transparent cylinder, and observing the depth of the water at regular 3-second intervals.
The resulting data would consist of a table of water depth vs. clock times, with clock times 0, 3, 6, 9, 12, ... seconds. As depth decreases the water flows from the hole more and more slowly, so the depth changes less and less quickly with respect to clock time.
Experimental uncertainties would occur due to the optical distortion of the copied rulers, due to the spacing between marks on the rulers, due to limitations on your ability to read the ruler (your eyes are only so good), due to timing errors, and due to other possible factors.
Suppose that depth changes vary from 5 cm to 2 cm over the first six 3-second intervals.
Assume also that the timing was very precise, so that there were no significant uncertainties due to timing.
· Based on what you have learned in experiments done through Assignment 1, without doing extensive mathematical analysis, estimate how much uncertainty would be expected in the observed depths, and briefly explain the basis for your estimates. Speculate also on how much uncertainty would result in first-difference calculations done with the depth vs. clock time data, and how much in second-difference calculations.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv I think there will be some uncertainties with the observed depths because of distortion and limitations of my ability to read the ruler. Assuming the uncertainty caused by limitations in the ability to read the ruler is minimal, and that the distortion of the ruler is constant from mark to mark, I think the uncertainty will decrease after the first-difference and even more after the second-difference calculations.
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· How would these uncertainties affect a graph of first difference vs. midpoint clock time, and how would they affect a graph of second difference vs. midpoint clock time?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv These uncertainties are going to somewhat skewed in the first difference vs. midpoint clock time. But in the second difference vs. midpoint clock time they will become less skewed.
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· How reliably do you think the first-difference graph would predict the actual behavior of the first difference?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv I don't think it would predict the actual behavior very reliably. Since the graph will be skewed, the behavior will not be very well represented.
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· Answer the same for the second-difference graph.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv I think the second difference graph will be better and hopefully predicts the actual behavior of the second difference more acurately.
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· What do you think the first difference tells you about the system? What about the second difference?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv The first difference tells you the difference in depth between the six, three second intervals. The second difference tells you the difference between the first differences. In other words, it's the difference in the change of depth.
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Question: Suppose the actual second-difference behavior of the depth vs. clock time is in fact linear. How nearly do you think you could estimate the slope of that graph from data taken as indicated above (e.g., within 1% of the correct slope, within 10%, within 30%, or would no slope be apparent in the second-difference graph)?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv I think I could estimate the slope of the graph within 10%.
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Again no extensive analysis is expected, but give a brief synopsis of how you considered various effects in arriving at your estimate.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv I considered the fact that my original data was somewhat skewed. However, I also considered the fact that when I did the first-difference calculation, some of the error was cancelled out because it existed in each measurement. Then, even more of it was cancelled out from doing the second-difference calculation.
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&#This looks good. See my notes. Let me know if you have any questions. &#