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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
Using the triply-reduced ruler will result in the more accurate measuring but the small size means that you can get very accurate measurements before multiplying by its scale. On a normal sized ruler, the lines would be much bigger, therefore leading to less accurate readings and more guessing of the length of the pencil rather than précised measurements.
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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
Size is the biggest factor in this measurement. Using something very small in order to get down to the finest and smallest measurement is the best option. That method yields the most accurate and precise measurements. Considering that, the triply-reduced ruler is by far the better option due to its smaller size. Although, the distortion that comes with the multiplying scale does cause some issues with the final measurement if using the triply 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
With no distortion and accurate measurements up to 4 decimal places, the triply ruler is the best option. Its small lines can lead to very accurate markings and measurements. With no distortion, when multiplying by its scale, the end result is extremely accurate. The regular ruler is not affected by these details so its measuring accuracy stays the same. In this case the triply reduced ruler is the better option under these conditions.
• 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 must understand that the triply reduced ruler is not going to be as accurate as it will be unless it is under these specific conditions. In reality there is some uncertainty in its multiplying scale and therefore the calculations may be slightly off. Again the regular ruler is left alone throughout the measurements so it has the same scale every time. You must also consider the possibility of human error on seeing the correct measurements and recording them.
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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
Using the singly reduced ruler is the better option because it is not as reduced and zoomed in as the triply reduced ruler. The further in you zoom to measure and then multiply, the better the chances get of a distortion error or measurement error with the data. Using the singly provides some zoom to get better readings and still doesn’t rely on the scale as much to get the final measurement. The singly ruler would give you accurate results and provide reliable measurements and precision.
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• Explain what factors you considered and how they influence your final answer.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
You must consider the potential for inaccurate measurements due to the multiplying scale in the rulers. A smaller ruler provides more accurate down to the point measurements, but they also have a larger multiplying scale, which could lead to issues if the initial measurement isn’t exact. A mistake using a triply reduced ruler is worse than a mistake using a singly reduced ruler. A small error multiplied multiple times can turn into a big error in the end.
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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
Using the triply reduced ruler and with a variation of 5 cm to 2 cm, there is a lot of room for uncertainty and error. As seen before, using a small measuring device and then multiplying by its scale can lead to big mistakes and miscalculations. With a variation of that size between intervals of water levels lowering, calculations done to determine its velocity and acceleration can be very misleading. Using the difference quotient method to obtain the velocity and acceleration involves the numbers from the original data. If these numbers are used multiple times and are incorrect, then the answer you receive will be far away from the actual answer. The error, even a small one, would be multiplied into a large separation from the correct calculation.
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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 first graph would be affected by a minor misaligning of data points. The graph would still be recognizable and you would still be able to determine the direction and slope of the line of best fit. However, once you calculate the second time and plot the points on a graph, it becomes less readable. The direction and position are harder to pick out and the data points seem to be scattered throughout. This a perfect example of a small mistakes growing after it is used twice in calculations.
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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 believe that even with some small uncertainties the first difference graph still shows reliable data. The graph would still be easy to read and you could use the information from the graph to gather accurate data on the slope and direction of motion. Based off of the calculations necessary for the first difference, the answers are off just a few percents if that much at all.
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• Answer the same for the second-difference graph.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
The second difference graph tells a different story though. Now, after using the uncertainty twice in back to calculations, because you use the first difference to calculate the second difference, the small error has grown into a larger one. After experimenting with this kind of data, we know that the second graph becomes much harder to distinguish and is less obvious about its features. It is difficult to add a best fit line in to average the points. The second difference shows more of the uncertainties than the first difference does.
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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 about the velocity of the system, and its slope. This comes from the calculations necessary for the difference which is position / time. This gives you the velocity. The second difference gives you the acceleration of the data. This goes hand in hand with physics and calculus definitions because the derivative of position gives you velocity(first difference) and the derivative of velocity gives you acceleration(second difference).
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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 behavior is linear, then I believe that you could estimate about 10% error unless it was stated that there were no uncertainties. If not, having uncertainties at the second difference would cause the graph and the data points to be off from the actual value by a good amount. Therefore if there is a linear form to the data points with uncertainty, then the slope calculated must be off from the actual slope.
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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
After reviewing through all the scenarios, and having previous background information on the topics at hand, I was able to give my opinion on the questions and provide what I believe to be accurate answers. Estimating error in problems can be challenging without actual numbers to look at and analyze, therefore knowledge about patterns of how things turn out and a sense of what is going to happen through experience with similar issues is the key to helping you answer these types of questions. After being exposed to more of these questions, I feel better prepared and more confident to answer future problems and to work out issues dealing with error and uncertainties.
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You clearly have a good understanding of this and some background with questions of this type (which is often but not always the case for students coming into this course).
Excellent insights and well-presented as well.
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