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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 believe the original ruler reading will be more accurate, because when you end up multiplying the reading on the triply reduced ruler, you are magnifying any error in your reading.
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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
Factors I considered were the human error in measuring and the fact that you must multiply the triply reduced ruler reading by the necessary amount.
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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 believe the triply reduced ruler will give a better reading.
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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
Once again I considered the human error in measuring and the fact that you must multiply the triply reduced ruler reading by the necessary amount, but now that I am aware of the specific number of significant digits and accuracy, I think the ultimate reading can be better trusted.
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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 should ultimately give you the more accurate difference in the lengths.
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Explain what factors you considered and how they influence your final answer.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
This time, the only difference between rulers was the number of reducity, so therefore we should assume that the triply reduced will give us a better length.
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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
We should always keep in mind that uncertainty is always present in data collection and can therefore base our estimation from here. With depth changes varying over a 3cm difference over 6 three second intervals, we can assume there are many places to lose accuracy. With any data range, we need to determine if these number should have been the same or are increasing or decreasing in some time of pattern. Also, 3 second intervals involve correct timing and must be taken as such. Finally, with six data points, we should determine if these are trials or separate data collections. First difference calculations hold the least amount of error, as each time we differentiate the data we are magnifying any error and uncertainty present.
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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
The uncertainties would become much more apparent each time you differentiated the data so the first difference vs. midpoint clock time will be much more accurate and easy to read than will the second difference vs. midpoint clock time.
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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 the first difference graph would be reasonably accurate to determine the actual behavior, as long as the original data wasnt entirely inconsistent.
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Answer the same for the second-difference graph.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
The second difference graph would definitely be much harder to determine an actual behavior from and should therefore definitely not be taken into a certain aspect of the data. Any uncertainties present in the initial data have now been magnified twice so the graph is likely to look very off from the actual behavior of the system.
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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 tells you the rate at which the dependent factor is changing with the independent factor. The second difference is the rate at which the first difference is progressing. For example, if the initial data is distance vs. time, the first difference will be velocity and the second difference will be 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: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
I think you could probably rely on this slope to be fairly accurate is the behavior is definitely linear. If a graph is clearly a constant slope, we can assume there are no extremely significant outliers in the data and that nothing has been too magnified to gather appropriate 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
By analyzing a graph of second difference, it is fairly easy to determine whether the data has remained fairly accurate through differentiation or that it will not help determine the actual nature of the subject. If the data is clearly in a straight line after the uncertainties have been magnified, I think its reasonable to claim that the slope is a fairly good representation.
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&#This looks very good. Let me know if you have any questions. &#
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