Optical Distortion of Paper Rulers

 

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Many students in the past found this experiment to be tedious and didn't see the point of the exercise.  Repetitive measurement can tedious, especially if one has not done a lot of measurement in previous mathematics and science courses and lacks experience with the process.  In the Spring 2010 revision we now measure the legs of a triangle rather than measuring one ruler against the others.  These measurements should be inherently quicker and should provide the necessary experience in a shorter time frame.

There are two main points to this lab:

The majority of students report spending between 1 and 2 hours on this experiment.  A few report times of less than 1 hour.  A few report times in excess of 4 hours.

In your lab kit you will find several sheets of plain white paper, which you should not discard.  In general, never discard anything in your kit.

You will also find copies of meter sticks. Some will be full-sized, some will be reduced.  There are three levels of reduction.

If you are working ahead and don't yet have your Standard Kit, if there has been a delay in the availability of Standard Kits, or if the paper copies have been left out of your kit, you can download and print these images.  The best quality image files are nearly 10 mB each at this point, so if you have a dial-up connection allow sufficient time to download them, and consider using the smaller images listed under Acceptable Resolution.  If your connection allows you to reliably download larger files, those under Best Resolution are a little sharper; this is especially so for the doubly-reduced document.

If you choose to print out the documents, you might need to click on 'Scale to Fit' in order to get the document at its intended size.  The details for versions of XP and Vista (current as of July 2008):  Right-click on the link, choose Print, click on Preferences, click on the Layout tab and check the Scale to Fit option.

Acceptable Resolution will be found in the following:

Best Resolution Files

 

The full-sized copy looks like this:

Note that the top copy is upside down and backwards.  Orient your copy the same way and in small letters near the top of the page, but positioned so as not to interfere with any of the marks on the ruler, write the word 'top'.

We will refer to the four levels of reduction sizes as 'full-sized;, 'singly-reduced', 'double reduced' and 'triply reduced'.  In addition to the 'full-sized' copy shown above, there is one sheet of 'singly-reduced' rulers, and another sheet containing both 'doubly-reduced' and 'triply-reduced' rulers.

Even the full-sized copies are not perfect. The copier uses lenses, and no lens can be perfect.  There are slight distortions in the copies, and in this experiment we 'map out' these distortions.

The singly-reduced copy looks similar, but the rulers are reduced.  This means that the marks on the page are closer together, and they can therefore measure lengths with more precision than the singly-reduced rulers.  However in order to measure with equivalent accuracy we will need to map out and correct for any distortions arising from the copying process.

The goals of this experiment are as stated above:

If you have a ruler whose smallest division is a millimeter, then the position of a point on the ruler would be measured accurate to a millimeter, and you would also make your best estimate of where that point lies between the marking (e.g., a point between the 3.8 and 3.9 cm markings might lie halfway between those markings, in which case you would estimate the position as 3.85 cm; or it might lie closer to one marking than the other, so you might have an estimate of 3.82 cm or 3.86 cm; you should try to estimate the position between the smallest mark to the nearest tenth of that distance).

Using a piece of typing paper (actually any paper will do as long as its corners are not rounded), cut out a right triangle by trimming one of its corners in the manner indicated by the figure below (cut along the red line, remove the triangle, which will look like the triangle shown in the lower-right-hand corner of the figure).  Make the triangle fairly short.  The longest side should be between 1 and 2 inches long.

Measure the hypotenuse of your triangle, using each level of reduction.  For each level of reduction you have a 'block' consisting of several rulers; for each level, measure in about the center of the middle ruler.  Estimate each measurement to the nearest 1/10 of a division (you won't be accurate at 1/10 division and on the smallest reductions it will be difficult to estimate, but that's no excuse for not doing your best).

We'll make the following conventions for our units of measurement:

Give your results for the hypotenuse below, separated by commas.  A sample format, which gives a brief but complete (though not very accurate) answer, might be '3.14 cm, 5.37 cm_s, 9.48 cm_d, 13.25 cm_t'.  Your numbers of course will differ from those given here.

 

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Describe in words what you did to make your measurements as accurate as possible:

 

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Which of your measurements do you think would be the most accurate, in the sense of having the least uncertainty?

 

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In the same way measure the shorter of the two legs of the triangle and give your results below:

 

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Repeat for the longer of the two legs of the triangle and give your results below:

 

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Consider the two sides whose lengths are closest.  This might be the hypotenuse and the longer leg, or it might be the longer leg and the shorter leg, depending on how you cut your triangle.

According to each ruler, what is the difference between these two sides?  Give you answer in a format similar to that of the first question, as four quantities separated by commas.

 

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For each level of reduction, give the difference between the two sides as a percent of the length of the hypotenuse.  Give your results in the first line as a series of four numbers separated by commas, in order with the result with for the full-sized ruler first, the result for the triply-reduced ruler last.  Use the appropriate number of  significant figures in your results.  Starting in the second line, give your explanation of how you got your results.

 

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According to your results, what would be the length of an object that measures exactly 1 cm on the full-sized copy, if measured using the singly, the doubly, and the triply-reduced copy?  Give your answer in the usual comma-delimited format in the the first line, then starting in a new line explain how you got your results.

 

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What would be the lengths, in units of cm of the full-sized ruler, of three objects, whose respective lengths measure 1 cm_t, 1 cm_d and a cm_s?  Give the three lengths separated by commas in one line, then starting in a new line explain how you got your results.

 

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According to your present results what would be the length, on each of your rulers, of an object whose length on a the doubly-reduced ruler was determined to be 8.34 cm_d?

 

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You made your measurements in the middle of each 'block' of rulers.  We might expect that, due to optical distortions in the copying process, there might be some difference in measurements made at different places on each ruler 'block'.  Investigate this question. 

Are there places on the triply-reduced copies where an object measured at one location gives a different result, due to distortions of the copy, than the same object measured at another location?  If so, at what positions and at what level of reduction do you observe the most distortion?

Give your results and explain how you investigated this question.

 

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If you believe you did detect distortion, how much of the observed difference in measurements do you think you can attribute to actual distortion, and how much to limits on your accuracy and the precision of the markings?

If you did not detect them, this doesn't mean that there aren't distortions.  There almost certainly are, but they might be too slight for you to measure.  In this case, how small would they have to be before you would be unable to detect them?  How big is the largest discrepancy you would be unable to discern?  Give your results and explain your thinking.

 

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Don't actually do this, but if you were to write a 100-word paragraph with a #2 pencil, measuring the pencil before and after, which level of reduction do you think would allow you to determine most accurately the difference in the length of the pencil from eraser to point?

 

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Your instructor is trying to gauge the typical time spent by students on these experiments.  Please answer the following question as accurately as you can, understanding that your answer will be used only for the stated purpose and has no bearing on your grades: 

 

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Revised: 07/26/12