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course PHY 202
2/7 about 1:40 pm
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 triple reduced should be more accurate because the marks are closer together and you can distinguish easier where the end of the pencil lies in between the two marks. This all depends on how clear the copy is and the accuracy of the scale factor used to multiply.
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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
I noted these in the previous answer. The distortion would influence this greater than the scale factor, as long as the scale factor is at least accurate to a hundredth.
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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
My answer is still the triple reduced 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
With the marks being closer together, it is typically easier to tell where the end of the pencil falls between the two marks. Once it is reduced too much though, it would actually be more beneficial to have spacing farther apart. Once they are too close together, you would not be able to tell the difference between, for example, .25 cm and .5 cm.
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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
Because the difference in length is so small, it might be easiest to use the single reduced. It would be hard to distinguish much difference in the triple reduced. I remember in the lab for PHY 201 for the rubber band calibration, it was very difficult to hold the ruler up and get an accurate measurement while the bag of dominoes was dangling off the edge of the table.
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• Explain what factors you considered and how they influence your final answer.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
Again, because of the small change in length, I believe, the single reduced ruler would help to be more accurate. When I hold the two rulers together, 1 cm on the single reduced ruler is about 3.2 to 3.4 cm on the triple reduced. After multiplying by the scale factor, with the variation in being able to detect the correct measurement by eye, and the optical distortion, the result could be far off from actual difference.
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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
If you use the triple reduced ruler, and do not get an accurate reading, the results will continue to get worse. First difference will be off, but it will continue to compound and get farther and farther from the actual result.
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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 data points in the first difference would be slight scattered. For the second difference graph, the variation would be much greater and you may not be able to as easily detect the slope of the curve.
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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
This graph would still be somewhat reliable. You would be able to fit a curve to the date and be able to distinguish if the curve is increasing at an increasing or decreasing rate.
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• Answer the same for the second-difference graph.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
This graph would likely be very difficult to fit a date curve to. It might be possible, but most likely the points will be scattered too much.
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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 give you the velocity and the second difference will give you 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: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
If the second difference is linear, then I would think you could estimate the correct slope within about 10%. If there are uncertainties though and some scattered points, then the actual slope will definitely not be the same as calculated from the date curve.
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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
From some of the labs in PHY 201, I was able to understand the patterns and problems pertaining to this exercise. Without actual numbers, it is difficult to determine with certainty.
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&#Good responses. Let me know if you have questions. &#