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course Phy 122
6/9 11:11pm
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
I would think that using the reduced ruler to measure the pencil and then adjusting the result to real size would give a more accurate result because it narrows down the uncertainty by measuring on a smaller scale.
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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
You would have to know how much the reduced ruler was reduced by and how to convert it back to real size. You might want to consider that there is distortion in the reduced ruler due to photocopying for example. You would also have to consider how accurate the life size ruler was compared to the reduced ruler.
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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
I would think that using the reduced ruler to measure the pencil and then adjusting the result to real size would give a more accurate result because it narrows down the uncertainty by measuring on a smaller scale.
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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
You would have to make sure that the life size ruler was accurate in its own measurement compared to the reduced ruler.
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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 copy is more likely to give you the more accurate reading because it is measuring on a smaller scale which will be more precise.
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Explain what factors you considered and how they influence your final answer.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
If the triply reduced copy has already been calibrated so that you know the correct conversion factor to get it real-life size then I would trust that measurement more because it is initially a smaller distance between point so that you get a more precise measurement of your object.
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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
I would estimate that uncertainty in the depths of liquids using my photocopied ruler sets would average to about + or - 0.05cm. This is because I am using a ruler on the outside of a bottle that may not be of uniform size throughout and could therefore distort the actual volume of water possible at any given section in the bottle. There may be significant uncertainty in the brief-flow lab because I used the real-size photocopied ruler set without knowing the optical distortions of the rulers and because the bottle I used had ridges in it which made it a little difficult to see when the water drained to the exact mark I made on the bottle showing the interval.
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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
These uncertainties would not be as noticeable in the first difference vs midpoint clock time graph. However they would become more evident with the second difference vs midpoint clock time graph.
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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
I think that the first difference graph would do a decent job of predicting the overall trend or best fit line for the data.
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Answer the same for the second-difference graph.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
The second difference graph would not do as well of a job predicting the overall trend or best fit line for the data due to the magnification of uncertainty.
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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 will tell you that the velocity of the system decreases as the level of the water decreases and the second difference will tell you that the acceleration of the system will decrease as the water level decreases.
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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 think I could estimate the slope of the graph to within 10% of the correct slope only because the behavior is in fact 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 considered the fact that if I could not be as precise as I would like due to the variability in the shape of the bottle, etc, that I could at least be consistent with my measurements in that I could use the same section of the ruler for all measurements, I would measure as accurately as possible against the marks I made on my bottle and I would be as accurate as possible in hitting the timer button to record my measurements.
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Excellent insights. Very well done.
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