course Phy 232
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 think that the triply-reduced ruler will give a measurement that is closer to the actual length of the pencil. I think that
it will be three times more accurate that the regular ruler.
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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
In order to answer this question I had to consider whether or not the reduced ruler had the same number of tick marks on it,
but was the same size, which is what I ultimately assumed. Also, you have to consider that a ruler that is three times
smaller, but has the same number of tick marks, will actually have tick marks that measure distances less than a normal
ruler, thus being more precise.
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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 think that the reduced ruler is going to 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
You have to consider that since the reduced ruler will not have any distortion, it will be able to make measurements on a
finer scale.
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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 reduced ruler will give a more accurate difference in the lengths, because it measures on a finer scale.
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• Explain what factors you considered and how they influence your final answer.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
The factors that influenced my decision were the distance between the tick marks on each ruler, and my ability to read the
measurements on each ruler. Since the reduced ruler has tick marks that are closer together, and I can read the ruler, it is
more accurate to use.
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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 think that the overall trend of the data points would be clearly visible, but it would be relatively jagged, with certain
points not fitting in with the rest due to errors in reading the rulers.
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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
I think that the first difference vs. midpoint clock time graph would be a little bit more jagged, with the points
relatively scattered, but in a way that you can discern a general trend, and easily draw a line of best fit. I think that
that second difference vs. midpoint clock time graph would be even more scattered and very difficult to discern that trend
from.
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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 predict the behavior of the first difference relatively well, but not
necessarily at each exact point.
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• Answer the same for the second-difference graph.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
The second-difference graph will not be as accurate at predicting the 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
I think that the first difference will tell you about the velocity of the system, the direction that it is going. The second
difference will tell you whether or not the velocity of the system is increasing or decreasing.
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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 think that you could maybe estimate the slope of the graph within 30%, but it would depend greatly on the amount of
uncertainty in the observation of the data.
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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 think that the data points would be scattered about the axis, but in the right general area. I think that this would be a
result of the magnification of uncertainty as you move from the first difference to the second.
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&#Your work looks very good. Let me know if you have any questions. &#
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