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course Phy 232
6/18 14
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 result likely to be closer to the actual length of the pencil is the result calculated using the original ruler itself. However, both should be very close.
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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 factors you must consider are how accurately you take the measurement and how accurate the scale factor is. They weigh into the answer, because using the original measurement is only affected by the accuracy of the measuring and not the scale factor as well.
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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 this is the case, then the results should be exactly the same to 4 significant figures. However, the result using the original ruler should still be more accurate.
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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 factors you must consider are the same as before. Even with 4 significant figures, the original ruler will still be more accurate because, theoretically, and infinite number of significant figures can be calculated.
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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 data from the singly-reduced copy should be the most accurate, followed by the triply-reduced copy and then the optical distortion experiment.
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• Explain what factors you considered and how they influence your final answer.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv I considered the human error by the eye, the accuracy of the scale factor and the accuracy of the weight of the apples.
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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 There would be some uncertainty in the observed depths. When calculating the first-differences calculations, there would be more uncertainty and in the second-difference there would be a great deal of uncertainty. When starting out with a small amount of uncertainty, in every experiment that amount grows exponentially with each calculation.
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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 first difference vs. midpoint clock would be somewhat jagged but should still be able to find the line of best-fit. In the graph of the second difference vs. midpoint clock time there should be a great deal of jaggedness and there should be practically no way to see the line of best-fit without a computer.
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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 The first-difference graph would be somewhat accurate in predicting the actual behavior of the first difference. There would be some uncertainty causing for the connecting lines slopes to be a little off.
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• Answer the same for the second-difference graph.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv The second-difference graph would be much worse and would almost not be able to predict the actual behavior of the second difference.
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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 us the velocity of the water flowing out. The second difference tells us the acceleration of the water flowing out.
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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 would say within 30% of the actual slope, if the graph of the second-difference behavior was linear.
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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 would consider the amount of uncertainty at the beginning of the experiment and given the fact that it is the second-difference. If it were the first-difference I may have said 10% simply because there is less uncertainty to be considered.
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&#This looks very good. Let me know if you have any questions. &#