course Phy 232
6/10 10:30 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: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv The original ruler will get the most accurate measurement.
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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: You have to consider which ruler is more accurate and you also have to consider the appropriate scale factor. The triply-reduced ruler could have distortion when you try to take the measurement. You have to also consider how close the scale factor is to being accurate.
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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 The triply reduced ruler will get 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: This time you know there is no optical distortion and you know the scale factor goes to 4 sig figs so it will now be more accurate.
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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 triply-reduced rubber band would be better because it is more reduced to read than the singly-reduced ruler and you can get a more precise number then you know the optical distortion for both of them.
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Explain what factors you considered and how they influence your final answer.
Your answer: You have to think about the measurement have more significant figures and then also being able to convert it back with a proper optical distortion value.
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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: When you draw the best fit line to the curve some points will lie above the line and some below. This can be the uncertainty in the measurements due to limitations in reading the ruler, optical distortion, and taking the right measurement at the right time. There is a smaller amount of error in the first difference calculations than the second and third. Each time you take a difference calculation there is more and more error.
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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 first difference vs midpoint would be lines connected between each two points that resemble a curve. The second difference will be somewhat jagged and the third is even more jagged.
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How reliably do you think the first-difference graph would predict the actual behavior of the first difference?
Your answer: The first graph will be pretty reliable because it doesnt have too much error.
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Answer the same for the second-difference graph.
Your answer: The second difference graph will be less reliable because it will have acquired more error from the next difference calculation.
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What do you think the first difference tells you about the system? What about the second difference?
Your answer: The first difference tells the velocity and the second the 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: I think that you could estimate the slope within 30% if graph was actually linear. The one that I looked before was so jagged that it was hard to even guess what the best fit line would look like.
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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: I considered the example we considered before this one where the second-difference acceleration graph had a best fit line that was still linear but did not resemble the true slope because there was so much error by the time you had calculated two differences.
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&#Good responses. Let me know if you have questions. &#
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