course Phy 122 The last two questions gave me a little trouble. I guess this was because the error analysis labs slightly confused me. What exactly do the first and second differences tell us? vx~֓e}bQassignment #001
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11:20:46 Most queries in this course will ask you questions about class notes, readings, text problems and experiments. Since the first two assignments have been experiments, the first two queries are related to the experiments. While the remaining queries in this course are in question-answer format, the first two will be in the form of open-ended questions. Interpret these questions and answer them as best you can.
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RESPONSE --> Ok.
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11:29:26 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? What factors do you have to consider in order to answer this question and how do they weigh into your final answer?
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RESPONSE --> Since you are using the appropriate scale factor for the triply-reduced ruler, this ruler should give you the most precise measurements. Precise and accurate are different, though. I believe a standard ruler would give a more accurate reading simply because there is no optical distortion. It should be taken into consideration that the errors on the singly and doubly reduced rulers were increasingly compounded on the triply reduced ruler. In order to insure I had the correct measurements I would measure with a standard ruler and the triply-reduced ruler to see if they agreed. Factors that may cause error include your ability to read the small marks, optical distortion of the paper ruler, ability to make a precise mark with what may be a dull pencil, etc. These factors will affect your ability to make an accurate reading and, therefore, I would measure with the standard ruler for accuracy and the triply-reduced ruler for precision. confidence assessment: 2
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11:33:37 Answer the same question as before, except assume that the triply-reduced ruler has no optical distortion and you know the scale factor accurate to 4 significant figures.
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RESPONSE --> If the triply-reduced ruler had no optical distortion and the scale factor was known to 4 significant figures, it would be more accurate and precise than a standard ruler. In this case, the same factors apply to both rulers and could cause error in either. A standard ruler can only measure accurately down to a millimeter. The triply-reduced ruler would, therefore, provide a better measurement because it can be estimated with the scale factor down to 4 sigfigs. confidence assessment: 3
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11:42:46 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? Explain what factors you considered and how they influence your final answer.
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RESPONSE --> Technically, the two rulers should give the same measurements as long as the scale factors are accounted for in the calculations. Again, since the triply-reduced ruler is a copy of a copy of the singly-reduced ruler, it will have compounded the error of the single by 3 times. Therefore, the singly-reduced copy may be more accurate. The triply-reduced copy, however, having smaller increments, will most likely be more precise. If you can accurately read and mark the lines and apply the scale factor, the triply-reduced ruler would be better to use for a measurement as small as 1 to 2 cm. The triply-reduced ruler is hard for me to see and the singly-reduced ruler is very easy to read. Therefore, my own error in reading would make the singly-reduced copy more accurate. It is also difficult to make an accurate mark with a pencil when using the triply-reduced copy. Again, I always double check my measurements with a different ruler. confidence assessment: 2
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12:01:38 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 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 in Assignments 0 and 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. {}{}How would these uncertainties affect a graph of first difference vs. midpoint clock time, and on a graph of second difference vs. midpoint clock time? {}How reliably do you think the first-difference graph would predict the actual behavior of the first difference? {}Answer the same for the second-difference graph. {}{}What do you think the first difference tells you about the system? What about the second difference?
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RESPONSE --> Uncertainties would increase as time passed due to the decreasing differences in depth. As change in depth decreases, it makes it more difficult for the reader to measure on the already distorted ruler in such quick increments. After referring back to my error analysis labs, it seems to me that the first differences were less precise than the second difference. Therefore, I believe the first difference calculations would be more spread out and have greater uncertainty. A graph of first difference vs midpoint clock time would be decreasing at a decreasing rate. The graph would first have a steep decrease and gradually level out. A graph of second difference vs midpoint time would also be decreasing at a decreasing rate, however, the slope would be nearly linear. This is because the differences do not vary as much as the first differences. I believe that the second-difference graph would be more reliable simply because there is not as much variation or uncertainty. It seems from my error analysis labs that the first difference is simply the differences between the running sums of depth changes and the second difference is the same as the standard deviation. When I calculated my standard deviations in error analysis 1 and then the second differences in error analysis 2, I realized that they were the same numbers. confidence assessment: 1
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12:06:59 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)? Again no extensive analysis is expected, but give a brief synopsis of how you considered various effects in arriving at your estimate.
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RESPONSE --> The slope of a linear graph is easy to calculate as long as you have 2 sets of points. Therefore, I would use the initial depth and time, 0 seconds, and my other point would be the final depth and time. Using the slope of a line equation, I could calculate the slope to within 1% of the correct slope. Honestly, I wasn't sure how to answer this question. I didn't know if you wanted an actual calculated percentage or simply an educated guess. confidence assessment: 1
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