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course Phy 202
2/2/2015 11:30pm
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
Assuming that the triply-reduced ruler is completely accurate, I would still think that the original ruler itself would be closer to the actual length of the pencil.
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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 The more calculations that are done to come up with an answer, the larger the potential for error. The triply-reduced ruler has been reduced from the original 3 times, and has to be multiplied 3 times to return to the same number as the original. By the time these two calculations are through, the second answer may be just the slightest bit different from the reading on the original ruler.
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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 If the ruler is actually accurate to 4 significant figures, then this time the triply reduced ruler will probably be more accurate. Especially if you are just using the actual reading and not multiplying to find the length in terms of the original ruler.
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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 the factor of optical distortion, which was negated in this instance. You also have to consider the number of calculations and potential for error, which was also negated by using the one reading. You also have to consider the given accuracy level, which in this case was high, at 4 figures.
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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 The triple reduced copy will give the most accurate length measurement.
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Explain what factors you considered and how they influence your final answer.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv We would have to consider optical distortion, but I am assuming that we are using the same ruler with no optical distortion. We also have to consider how minute the distance is that we are trying to measure. In this case it between 1-2 centimeters, which is very precise. The triply reduced version is more precise.
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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 Without doing an extensive mathematical analysis, it is still evident that the uncertainty of the first-difference calculations with respect to depth vs. clock time data would be less than that of the second-difference calculations for the same set.
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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 The graph of the first difference calculations would be more accurate than the second graph. The second graph would take the minor uncertainty errors and magnify them, as we saw in our previous graphing exercise with deterioration of difference quotients.
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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 dont think that the first difference graph would be 100 percent accurate, but that we could take the points and use the best fit line to make it as accurate as possible.
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Answer the same for the second-difference graph.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv We could use the same technique for the second graph, although it would not be as accurate because once again the errors would be magnified.
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What do you think the first difference tells you about the system? What about the second difference?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv Graph one is a calculation of velocity
Graph two is a calculation of acceleration
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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 Based on the estimates we did by ourselves on deterioration of difference graphs previously, I would say that in this case we would still be able to estimate the slope within 30%.
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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 mostly just thought about what we learned while looking at the deterioration of difference graphs. It is apparent that by the second-difference, the error would make the graph more scattered, but possibly still enough to estimate a best fit line and regain some accuracy.
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Self-critique (if necessary):
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Self-critique rating:
#$&*
course Phy 202
2/2/2015 11:30pm
Question: Suppose you measure the length of a pencil. You use both a triply-reduced ruler and the original ruler itself, and you make your measurements accurate to the smallest mark on each. You then multiply the reading on the triply-reduced ruler by the appropriate scale factor.
Which result is likely to be closer to the actual length of the pencil?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
Assuming that the triply-reduced ruler is completely accurate, I would still think that the original ruler itself would be closer to the actual length of the pencil.
#$&*
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 The more calculations that are done to come up with an answer, the larger the potential for error. The triply-reduced ruler has been reduced from the original 3 times, and has to be multiplied 3 times to return to the same number as the original. By the time these two calculations are through, the second answer may be just the slightest bit different from the reading on the original ruler.
#$&*
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Question: Answer the same questions as before, except assume that the triply-reduced ruler has no optical distortion, and that you also know the scale factor accurate to 4 significant figures.
Which result is likely to be closer to the actual length of the pencil?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv If the ruler is actually accurate to 4 significant figures, then this time the triply reduced ruler will probably be more accurate. Especially if you are just using the actual reading and not multiplying to find the length in terms of the original ruler.
#$&*
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 the factor of optical distortion, which was negated in this instance. You also have to consider the number of calculations and potential for error, which was also negated by using the one reading. You also have to consider the given accuracy level, which in this case was high, at 4 figures.
#$&*
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Question: Suppose you are to measure the length of a rubber band whose original length is around 10 cm, measuring once while the rubber band supports the weight of a small apple and again when it supports the weight of two small apples. You are asked to report as accurately as possible the difference in the two lengths, which is somewhere between 1 cm and 2 cm. You have available the singly-reduced copy and the triply-reduced copy, and your data from the optical distortion experiment.
Which ruler will be likely to give you the more accurate difference in the lengths?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv The triple reduced copy will give the most accurate length measurement.
#$&*
Explain what factors you considered and how they influence your final answer.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv We would have to consider optical distortion, but I am assuming that we are using the same ruler with no optical distortion. We also have to consider how minute the distance is that we are trying to measure. In this case it between 1-2 centimeters, which is very precise. The triply reduced version is more precise.
#$&*
*********************************************
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 Without doing an extensive mathematical analysis, it is still evident that the uncertainty of the first-difference calculations with respect to depth vs. clock time data would be less than that of the second-difference calculations for the same set.
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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 The graph of the first difference calculations would be more accurate than the second graph. The second graph would take the minor uncertainty errors and magnify them, as we saw in our previous graphing exercise with deterioration of difference quotients.
#$&*
How reliably do you think the first-difference graph would predict the actual behavior of the first difference?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv I dont think that the first difference graph would be 100 percent accurate, but that we could take the points and use the best fit line to make it as accurate as possible.
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Answer the same for the second-difference graph.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv We could use the same technique for the second graph, although it would not be as accurate because once again the errors would be magnified.
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What do you think the first difference tells you about the system? What about the second difference?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv Graph one is a calculation of velocity
Graph two is a calculation of acceleration
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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 Based on the estimates we did by ourselves on deterioration of difference graphs previously, I would say that in this case we would still be able to estimate the slope within 30%.
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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 mostly just thought about what we learned while looking at the deterioration of difference graphs. It is apparent that by the second-difference, the error would make the graph more scattered, but possibly still enough to estimate a best fit line and regain some accuracy.
&#This looks good. Let me know if you have any questions. &#