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?
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Your Answer: ->->->->->->->->->->->->
The original ruler would be closer to the actual length of the ruler itself. The smaller ruler would be harder to read and this would cause more uncertainties.
• What factors do you have to consider in order to answer this question and how do they weigh into your final answer?
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Your Answer: ->->->->->->->->->->->->
You would have to take into account the fact that the rulers are different sizes. The size and accuracy of a measure device greatly affect its accuracy.
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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?
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Your Answer: ->->->->->->->->->->->->
The smaller ruler would give a closer measurement of the actual pencil, it would be easier to read and has a precision to 4 significant figures.
• What factors do you have to consider in order to answer this question and how do they weigh into your final answer?
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Your Answer: ->->->->->->->->->->->->
You have to take into account that at this point, the smaller ruler is more accurate and easier to read.
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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?
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Your Answer: ->->->->->->->->->->->->
If we were dealing with the triply-reduced ruler that is accurate to 4 significant figures and easier to read, I would say that the more accurate difference in lengths would be given by the triply-reduced ruler.
• Explain what factors you considered and how they influence your final answer.
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Your Answer: ->->->->->->->->->->->->
As was stated earlier, the triply-reduced would be easier to read and give a result accurate to more significant figures.
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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.
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Your Answer: ->->->->->->->->->->->->
First I would graph the data so that I could see if there were any noticeable uncertainties. Looking at graphical trends I would see if the rate was decreasing as predicted. By taking the derivative of the data, rise over run, we would also be able to see uncertainties in the data.
• 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?
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Your Answer: ->->->->->->->->->->->->
The uncertainties would make the graphs look jagged as we went to the first and second difference calculations.
• How reliably do you think the first-difference graph would predict the actual behavior of the first difference?
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I think that the first difference graph would not be smooth, it would be somewhat jagged. You might be able to see that actual trend of the first difference.
• Answer the same for the second-difference graph.
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Your Answer: ->->->->->->->->->->->->
The second difference graph would be very jagged and it would be very hard to see any trends.
• What do you think the first difference tells you about the system? What about the second difference?
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Your Answer: ->->->->->->->->->->->->
The first difference tells you the velocity of the system since the derivative of position is velocity. The second difference would tell you the acceleration since the derivative of velocity is 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)?
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Your Answer: ->->->->->->->->->->->->
If the second difference graph or the plotted second derivative showed linear behavior I think that the slope of that data could easily be estimated within 10% of its actual value.
Again no extensive analysis is expected, but give a brief synopsis of how you considered various effects in arriving at your estimate.
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If the data points given for the second derivative were nearly linear, as a best fit line was drawn through those points all the points would come very close to that line. This would indicate that it is a fairly accurate model.
&#This looks very good. Let me know if you have any questions. &#