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course Phy 232
6/4/12 submitted around 1:40 AM
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:
The measurement of the non-reduced ruler, since anytime you multiply by a scalar you are going to have rounding issues thus a greater percent error.
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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
Again you are going to have to worry about rounding off with your scalar value because, when you take your measurement you are probably going to have a decimal estimate then you are multiplying that by a scalar. So depending on how you round it off could change your final answer dramatically.
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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:
Your non-scaled ruler is still going to be more accurate because the scale factor is still going to change your answer slightly, all though not by much.
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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:
I thought that it would be more accurate since you have an accurate scale factor but then I thought that since you still are multiplying by an approximated decimal your answer is not going to be 100% correct. All you have done is take your inaccurate eyeballed measurement on the scaled lines of the scaled ruler out of the equation.
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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:
The reduced ruler will grant you a more accurate number since you can better fit it in the small space that you are measuring because it would be extremely hard to eyeball a distance of 1cm of a conventional ruler. The measurement you take with your normal ruler would be too small to approximate so you are left with no other option but to use the reduced ruler although I would use something more precise if available.
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• Explain what factors you considered and how they influence your final answer.
Your answer:
Again the reduced ruler will grant you a more accurate number since you can better fit it in the small space that you are measuring because it would be extremely hard to eyeball a distance of 1cm of a conventional ruler. You just have to consider what is going to give you the best chance at readability.
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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:
I would estimate that the uncertainty would be less than 50% quite a bit less in this case probably around 5-20%. I guess this because the water lever changed at 3/18 cm/sec and since we know that gravity is constant factors on working on areas of pressures like this (high volume to low volume) we know it has to be constant so it has to be pretty accurate. Depth vs. clock is easily measurable and converted into something we can use since it would be cm per second but doing it in reverse would change the uncertainty and make it higher since I think it would involve some kind of conversions
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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:
The graph of distance vs. time would be liner since it would have a lower uncertainty principle. But if you did clock time vs. midpoint difference you would still expect a liner line but it could be slightly more askew.
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• How reliably do you think the first-difference graph would predict the actual behavior of the first difference?
Your answer:
Since the uncertainty principle is low you would expect that the actual behavior would not be far off from what it would actually be.
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• Answer the same for the second-difference graph.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
Since the uncertainty principle would be higher this graph, it would reflect less accurately than the true graph, but if it still wasn’t too high (less than 45 %) then it will be quasi-accurate.
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• What do you think the first difference tells you about the system? What about the second difference?
Your answer:
The difference in the first one would tell you the distance the fluid travels over a specific time. The second would tell you the rate of what is acting on the system.
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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:
It would probably be in the ballpark of 30% slope since from your graph you could take two points that are on the most liner part of your graph and calculate its slope and then calculate the difference in slope.
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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:
Given a 30% ish uncertainty you would expect a 30% ish deviation from the actual slope.
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&#Very good responses. Let me know if you have questions. &#