#$&*
course PHY 202
1/28/15 at 10:20PMRe-submission. Original submission was on 1/26/15 but re-submitting to be on the safe side.
Question: Suppose you measure the length of a pencil. You use both a triply-reduced ruler and the original ruler itself, and you make your measurements accurate to the smallest mark on each. You then multiply the reading on the triply-reduced ruler by the appropriate scale factor.
Which result is likely to be closer to the actual length of the pencil?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
The original ruler itself is likely to be closer to the actual length.
#$&*
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
You must consider that the triply-reduced ruler has smaller intervals in between measurement marks, so it is more difficult for the human eye to perceive the differences in between them. Then, multiplying by a scale factor only magnifies the error by that factor.
#$&*
*********************************************
Question: Answer the same questions as before, except assume that the triply-reduced ruler has no optical distortion, and that you also know the scale factor accurate to 4 significant figures.
Which result is likely to be closer to the actual length of the pencil?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
It certainly helps to remove a source of error by removing optical distortion, but if I am understanding correctly what a triply-reduced ruler is, it would still be more difficult to discern the differences in between measurements. In a normally-sized ruler, for example, if the length of the pencil falls in between two marks on the ruler, the student can approximate this measurement in between these hash marks. In a reduced ruler, it is more difficult to discern between minute spaces between hash marks and errors in approximation then become magnified when the measurement is multiplied to scale.
#$&*
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 do have to consider that the scale factor is accurate to 4 significant figures, which lends credibility to measuring with the triply-reduced ruler, but it seems to me that it would still lead to an unnecessary source of error.
#$&*
*********************************************
Question: Suppose you are to measure the length of a rubber band whose original length is around 10 cm, measuring once while the rubber band supports the weight of a small apple and again when it supports the weight of two small apples. You are asked to report as accurately as possible the difference in the two lengths, which is somewhere between 1 cm and 2 cm. You have available the singly-reduced copy and the triply-reduced copy, and your data from the optical distortion experiment.
Which ruler will be likely to give you the more accurate difference in the lengths?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
I feel as though the singly-reduced copy will still give the most accurate results, for the same reasoning as I explained above - that human error in perception between the hash marks will be more pronounced on a smaller scale.
??? At this point I have found the triply-reduced ruler in the initial lab kit you sent me, but I am not sure I understand the need for the differently scaled rulers. ????
#$&*
@&
Some are better for certain purposes than others, and which is better in a given circumstance depends on the user.
But mainly they are there to illustrate a number of points about measurement and precision.
*@
Explain what factors you considered and how they influence your final answer.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
Human perception for measurements falling in between hash marks will be more accurate on a larger-scaled ruler than a smaller-scale.
#$&*
*********************************************
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
Since the rulers measure to the 0.1cm, I think that human error in perception of measurements between the 0.1cm marks would lead to an uncertainty between 0.1 and 0.01cm. In an original copy ruler, the measurements would likely be accurate at least to the 0.1cm, since the hash marks for these points are clearly labeled; however, in a triply-reduced ruler, the necessity for speed reduces the accuracy possible in reading such small ruler marks. Therefore, the students readings may stray from the actual by 0.1cm or even more, but most likely no more than 0.3cm for any given reading.
#$&*
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 clock time would perhaps show some deviation, but the differences would not be so pronounced here. In the graph of the second difference vs. midpoint clock time, the uncertainties and deviations would be much more pronounced and show greater discrepancies from the actual/expected.
#$&*
How reliably do you think the first-difference graph would predict the actual behavior of the first difference?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
The first-difference graph would have a fairly reliable prediction of the actual behavior, because at this point, the uncertainties are not so magnified that they result in any major deviations; at worst, the general trends hold true even if the data points deviate individually.
#$&*
Answer the same for the second-difference graph.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
The second-difference graph would have a marked decrease in reliability as the uncertainties in the data would be significantly magnified. It may be difficult to see any clear trend in the data.
#$&*
What do you think the first difference tells you about the system? What about the second difference?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
The first difference shows the overall system (velocity), whereas the second difference shows a more zoomed in view of the system that is more sensitive to uncertainty (acceleration).
#$&*
*********************************************
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
Considering the magnitude of the uncertainties, as we saw in the exercise on difference quotients where the weakly-fitting line comes to mind, I think the slope would be correct within 10% at best.
#$&*
Again no extensive analysis is expected, but give a brief synopsis of how you considered various effects in arriving at your estimate.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
Even with discrepancies in the data, there should be some general trend shown in the system, but this becomes markedly difficult by the second difference quotient. Thinking back to the exercise on difference quotients and how difficult it was to determine the line of best fit for the second difference graph, it becomes difficult to expect much accuracy from such results. Therefore I think I am being somewhat optimistic in thinking the slope is accurate within 10% error.
#$&*
&#This looks good. Let me know if you have any questions. &#