course Phy 232
6/15 9pm
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?
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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?
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How accurate the triply reduced ruler is. If you cannot make an accurate reading on the reduced ruler, when you multiply by the scale factor, mistakes will be amplified.
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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?
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The triply-reduced ruler.
• 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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Since you know that the reduced ruler will be accurate when reduced, you can make a more accurate reading.
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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?
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Singly-reduced ruler
• Explain what factors you considered and how they influence your final answer.
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Since the length being measure is small, it would be easier to read the singly reduced ruler since the lines would not be as close together.
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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.
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The uncertainty for the observed depths would be about +-1.5 cm, and the uncertainty of the first-difference calculation would increase, with the second-difference calculations increasing even more since it is using values that are not completely accurate to calculate values, and then using these to calculate other values.
• 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?
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The first difference graph would show some discrepancy, but it would still be able to find a best fit line. The second difference graph may have enough discrepancy to make a best fit line almost impossible to find.
• How reliably do you think the first-difference graph would predict the actual behavior of the first difference?
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The first difference graph would not be completely reliable since there was some uncertainty, but it would more than likely be fine to use to find some more general observations.
• Answer the same for the second-difference graph.
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The second difference graph may be obscured enough that it is difficult to find anything useful from the graph. A line that may actually be horizontal may show on the graph as having a significant slope.
• What do you think the first difference tells you about the system? What about the second difference?
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The first difference tells you the velocity at which water is escaping (cm/s), and the second difference tells you the acceleration (cm/s^2).
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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)?
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No slope would be apparent.
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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Judging from previous assignments with actual data, the slope for the second difference graph using the data varied significantly from the actual graph. The data points were scattered to the point that a best fit line was not apparent.
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