Refer to the pencil in Problem 1.31. Using the equivalent valu...

Question

Refer to the pencil in Problem 1.31. Using the equivalent values in Table 1.8 as conversion factors, convert the length measured in inches to centimeters. Compare the calculated length in centimeters to the length in centimeters measured using the metric ruler. How do the two values compare? Explain any differences.
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Accepted Answer

Welcome back. Everyone shown below is a strip of paper measured by a ruler, convert the measurement in centimeters to inches and compare the calculated value to the measurement in inches from the ruler. Explain the discrepancies between the values we have choices A through C which state that the discrepancies will arise from significant figure changes during calculations or inaccuracies in ruler measurements or inconsistencies in conversion factors. So we're going to begin by focusing on our measurement in centimeters where centimeters on the ruler are located in the superior portion of the ruler and given by the centimeters on the ruler, we have three sets of lines. The first type of line would be the larger divisions starting from zero centimeters to one centimeter where we would count a total of 10 lines from 0 to 1. So because we have 10 lines between then therefore one line we would say is equal to 1/10 which is equal to 0.1 centimeters. Now focusing on a quick glance of the length of our paper given in centimeters on the ruler, we would observe that the length of the paper is approximately five centimeters and extends a bit beyond past five centimeters at the fifth division or fifth line. And this fifth division is shorter than the larger divisions on our ruler for centimeters. And since it falls exactly on this line, the end of our paper falls exactly on this, we'll say a medium division representing half of one full line. We would say that this is representative of 10.5 centimeters which is half of 0.1 centimeters for one line. So to our five centimeters, we would additionally note down that the paper is at the fifth line which correlates to 50. centimeters. And we need to include one estimated digit of uncertainty, sorry of certainty, which we would observe since we agreed that the edge of our paper aligns just at 0.5 centimeters. Then our estimated digit of uncertainty is going to be 0.0 centimeters. And this would correlate to the measurement as 5. centimeters for the length of the paper. Now, we need to convert this to inches. So we will multiply by our conversion factor from our textbooks where we go from centimeters in the denominator two inches in the numerator and recall that for one inch, there is an equivalent of 2.54 centimeters. So canceling out our units of centimeters, we are left with inches and we will find that the length of the paper is equal to 2.17 inches when we round the value in our calculators to the hundreds place. So we would round it and get 2.17 inches. Now let's move forward and get the length of the paper using the ruler in inches. And we would find inches in the inferior portion or bottom portion of our ruler where from zero inches up to just before one inch, we can count a total marking that we can count a total of 16 lines from zero between one inch. And so then we can therefore say that one line is equal to 1/16 of an inch and observing our paper in reference to inches on our ruler. At a quick glance, we would say that it is approximately, it is approximately two inches but extends to the third smaller. Let's make it even more precise by drawing a line. But it looks like it almost extends to about the third smaller division of the ruler. But we can observe a bit of a gap and actually redrawing that line, the gap is a bit smaller, but it's still nonetheless a gap. So the edge of the paper is between the second smaller division and the third smaller division after two inches. So noting that down, we can say between the 2nd and 5th smaller lines, which we would. But since there is that very small gap from where the third smaller division is to the end of our paper, we would go with the second smaller line or second smaller division and say that the end of our paper is at the second line, which would equal approximately 2/16 of a full inch and 2/16 as a decimal recall is equal to approximately 0.1 inches. Now, at this point, you may be divided because you might want to say that the end of the paper is closer to the third line and that would be accurate too. But that would just change the our fraction to 3/16 which is closer to 0.2 inches. But for the direction I went in, we're going to go with 0.1 inches and to account for the difference between the second smaller division or second smaller line. And the end of the paper, we would say that the 0.1 inches plus one estimated digit of certainty would be based on about 0.8 inches. And I think that that's fair to say since half of the 2nd and 3rd division would be 0.5. So just beyond that, where the edge of our paper lies would be approximately 0.8. And so in full, our measurement for the paper and inches we can say is approximately 2. inches. And that is our full measurement using the ruler. So we do have different measurements between our calculated inches measurement to our actual interest measurement using the ruler and looking at our answer choices. It states that this discrepancy is explained from the changes in significant figures during calculations. And we would agree with that as a true statement. Since when we were converting from centimeters to inches, our value in our calculators be rounded to the 100th place to 2. inches, which is only three significant figures versus the value in our calculators, which was beyond three significant figures. So choice A looks like a good answer. Choice. But now let's consider choice B which states that the discrepancy between the values is from the inaccuracy of the ruler. And that would be a false statement because our ruler is a tool specifically used for accuracy of measurements. So choice B, we would rule out for that reason and then we have choice C which states that the discrepancy between the values is from the conversion factor. And because 2.54 centimeters equivalent to one inch is a standard conversion used in the metric system. We would rule out choice C as also a false statement. So that leaves us with choice. A as our final answer to explain why there is a discrepancy discrepancy between our measured and calculated value of inches for the length of the paper. I hope that this made sense. And let us know if you have any questions.

Key Concepts

Unit Conversion

Measurement Accuracy

Ruler Calibration

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