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 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?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv Well when working with the triply-reduced ruler you have to take into consideration the fact that the tick marks are so tiny your error in determining which one is actually the right measurement would be huge. You also have to consider the fact that you have to do a calculation with the triply reduced rule, which could produce an error.
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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 still believe that the original ruler would give the most accurate length.
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 Well with the new factors involved (no optical distortion and scale factor to 4 sig figs) it makes it hard to determine which measurement would be more accurate. However, you still have the small chance of mathematical error when dealing with the triply-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 Well, seeing no obvious advantage to using the triply reduced ruler I would have to say the single reduced.
Explain what factors you considered and how they influence your final answer.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv Well by looking at the triply reduced ruler it is obvious that the measure of possible error is huge and probable considering how difficult it is to actually measure something with this ruler.
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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 With this experiment a large measure of uncertainty would be expected. An uncertainty of at least half a cm would be likely. It would be very difficult to keep up with the timer and have good enough eyes to check the ruler with great accuracy. The first difference calculation would show great uncertainty and the second-difference calculation would multiple the uncertainty of the first difference calculation.
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 first difference vs. midpoint graph would have data points in a slightly random orientation. With a slightly jagged line connecting the points. A best fit line would be difficult to locate but possible. The second difference vs. midpoint graph would have data points in a completely random orientation. With a jagged line containing huge ups and downs between data points. The best fit line would be nearly impossible to locate.
How reliably do you think the first-difference graph would predict the actual behavior of the first difference?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv Even though the data would show slight variations, the first difference graph would give one a slight prediction (general positive or negative direction, slope, etc.) of the actual behavior of the first difference.
Answer the same for the second-difference graph.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv this graph however, would give one no direct indication of the behavior of the first difference.
What do you think the first difference tells you about the system? What about the second difference?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv The first difference would be able to possibly give you the slope of the graph as well as if it was increasing or decreasing and its rate or increase or decrease. The second difference might be able to show if it was 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 Well, if the depth vs. clock time graph is in fact linear (this would be highly unlikely), then I assume the slope could possibly be predicted to be with 30 % of the actual slope.
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 believe that with the given factors the amount of uncertainty in the second difference would be largely significant. The factors that are to be considered are as follows
Quality of copy provided
Distance between marks
Judgment error
Mathematical error
Timing error
Plus many more possible factors that are not listed.
However, if it did turn out to be linear, I am predicting that you could possibly come up with a slope within 30% of the actual slope.
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