#$&*
course phy 242
6/7/13 around 4:30 pm
Question: Suppose you measure the length of a pencil. You use both a triply-reduced ruler and the original ruler itself, and you make your measurements accurate to the smallest mark on each. You then multiply the reading on the triply-reduced ruler by the appropriate scale factor.
• Which result is likely to be closer to the actual length of the pencil?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
The one gained from using original ruler.
#$&*
• 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
The construction of the ruler, the reading of the ruler and consistence in keeping the measurement to further measure the rest of the pencil. Based on these factors, the original ruler is more appropriate to measure the pencil.
#$&*
*********************************************
Question: Answer the same questions as before, except assume that the triply-reduced ruler has no optical distortion, and that you also know the scale factor accurate to 4 significant figures.
• Which result is likely to be closer to the actual length of the pencil?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
The one gained from the triply-reduced ruler.
#$&*
• 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
The factors include: no optical distortion, scale factor accurate to 4 significant figures. These factors allow the pencil to have a more accurate reading than that of the original ruler.
#$&*
*********************************************
Question: Suppose you are to measure the length of a rubber band whose original length is around 10 cm, measuring once while the rubber band supports the weight of a small apple and again when it supports the weight of two small apples. You are asked to report as accurately as possible the difference in the two lengths, which is somewhere between 1 cm and 2 cm. You have available the singly-reduced copy and the triply-reduced copy, and your data from the optical distortion experiment.
• Which ruler will be likely to give you the more accurate difference in the lengths?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
The triply-reduced ruler will give a more accurate difference knowing that the difference is 1cm to 2cm.
#$&*
• Explain what factors you considered and how they influence your final answer.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
The factors that I considered include: accuracy of the triply-reduced ruler to within millimeter, the observer and the comparison among the other rulers.
#$&*
*********************************************
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
Without extensive mathematical analysis, there would be an uncertainty of 0.75 unit observed in depth. The reason for the estimate is that there are four time intervals in which the depth varies between 5 cm and 2 cm so therefore the uncertainty would be determined through dividing the difference in the depth values by the number of time intervals. In the first-difference calculation there would be a 0.01 uncertainty and in the second-difference calculation there would be an increase in uncertainty to about 0.1.
#$&*
• 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 have an obvious trend with minor data points 0.01 above or below the line but in the second , the uncertainty would be much more apparent with a less visible trend and data points having 0.1 uncertainty from the trend line.
#$&*
• How reliably do you think the first-difference graph would predict the actual behavior of the first difference?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
The first graph would predict the actual behavior of the difference quit well with its trend and line of fit.
#$&*
• Answer the same for the second-difference graph.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
The second difference graph will not be as reliable as the first due to not having clear trend as the first one.
#$&*
• What do you think the first difference tells you about the system? What about the second difference?
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
The first difference tells us about the slope of the original data . since the data was water depth vs clock time, the slope of that will be the velocity of the system. The second graph of the system will be the acceleration of the original graph.
#$&*
*********************************************
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
We could estimate the slope of the second difference graph to within 30% if the in fact the behavior of the graph is linear.
#$&*
Again no extensive analysis is expected, but give a brief synopsis of how you considered various effects in arriving at your estimate.
your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
I arrived at the approximation of 10 % from the following considerations: if the data gained from the beginning has about 10% uncertainty then the value gained for the first difference will be doubled that amount and so on. Therefore, when reaching to the value for the slope of the second difference graph the precision decreases and uncertainty increases.
#$&*
"
&#Your work looks good. Let me know if you have any questions. &#