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Phy 231
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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Proper relative error ???
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To get the most precise measurement possible you should use a reduced copy of a ruler. To make sure the measurement is also accurate, you should take into account any
tendency toward distortion in the corresponding part of that copy. You can choose whichever level of reduction you think will give you the most accurate and precise measurement.
In the box below, indicate in the first line the ruler markings of both ends of the first rubber band, entering two numbers in comma-delimited format.
In the second line indicate the distance in actual centimeters between the ends, to an estimated precision of .01 cm..
In the third line explain how you obtained the numbers in the second line, and what the meaning of those numbers is. Also indicate how this rubber band is marked, and the limits within which you think your measurement is accurate (e.g., +- .03 cm, indicating that you believe the actual measurement to be between .03 cm less and .03 cm greater than the reported result).
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My question is about and the limits within which you think your measurement is accurate (e.g., +- .03 cm, indicating that you believe the actual measurement to be between .03 cm less and .03
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I chose to use a cm_d ruler to make my measurements. I found that 1 cm full sized was equal to 2.2 cm_d, so I used it as a ratio of (1 cm / 2.2 cm_d)
Am i supposed to take into account both margins of error into my calculations? I know that percent error is (measured - actual) / actual *100%, but i have confusion when there are two margins of error compared to one another.
Here is what i do.
1) I take the measurements on my cm_d ruler as follows.
27.0 cm_d, 10.9 cm_d
'ds = 16.1 cm_d
2) From my measurements I know that i am accurate to 0.05 cm_d for each.
27.0 cm_d +- 0.05 cm_d, 10.9 cm_d +- 0.05 cm_d,
???? do I take both errors into into account????
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In this case you would add the errors, so that the result would be
37.9 cm +- 0.1 cm_d
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3) percent error (27.0) = ((27.05 - 27.0) / 27.0) * 100% = 0.185%
percent error (10.9) = ((10.95 - 10.9) / 10.9) * 100% = 0.459%
and then add these errors together?
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You wouldn't use percent errors with a sum, though you would with products and powers (see below).
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measured 'ds = (27.0 - 10.9) +- (0.185% + 0.459%) = 16.1 cm_d +- .644%
'ds = 16.1 cm_d +-(16.1 * .00644) = 16.1 cm_d +- 0.104 cm_d
4) Do i also take into account my error in my full size ruler?
I found that 1 cm full size = 2.2 cm_d
???? Do i take this as an exact ratio? or do I use the margin of error for both? ????
1.0 cm +- 0.05 cm
((1.05 cm - 1.0 cm) / 1.0 cm) * 100% = 5.0%
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If you obtained this result by, say, measuring 5 cm from a full-sized ruler using a doubly-reduced ruler, and got 11 cm_d, then your result that 1 cm = 2.2 cm_d would have been obtained by dividing the 11 cm_d by the 5 cm to get 2.2 cm_d / cm.
Both the 5 cm and the 11 cm_d would be uncertain. Suppose the 5 cm is uncertain by +-.05 cm, and the 11 cm_d by +-.07 cm_d. Then the percent uncertainties would be .05 cm / (5 cm) = 1%, and .07 cm / (11 cm ) = .6%. The percent uncertainty of the division would therefore be the sum of the percent uncertainties, 1% + .6% = 1.6%.
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Here is what i do.
(16.2 cm_d* (( 1 cm / 2.2 cm_d) = 7.36364 cm
Since the 5% error is larger than the cm_d error of 0.644%, I chose to use that.
So my final answer is:
7.36364 cm +- (7.36364 * 0.05) cm = (7.36 +- 0.37) cm
Is this the proper method to find the final accuracy of +- 0.37 cm?
Is there a lesson online i could take advantage of to properly calculate relative error with two or three different margins of error? I could not find the subject in my Calculus text.
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How To calculate relative error?
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You can search the Web under Propagation of Errors. You'll find a variety of articles.
The treatment in the Wikipedia article is pretty advanced, more than you need at this point. It really requires a good knowledge of statistics, multivariable calculus and linear alegebra.
The article at
http://www.rit.edu/~w-uphysi/uncertainties/Uncertaintiespart2.html
hits the level of rigor about right and should answer your questions.
Bascially the following are sufficient for most purposes:
If the quantity you are calculating is a sum of two quantities x and y with absolute errors | `dx | and | `dy | the absolute error of the sum is | `dx | + | `dy |.
For multiplication or division of two quantities, the percent error of the product is the sum of the percent errors of the two quantities.
For the nth power of a single quantity, the percent error of the power is n times the percent error of the quantity.
These rules come from differentials. For example:
If z = x * y then dz = y * dx + x * dy so that dz / z = (y dx + x dy) / (x y) = dx / x + dy / y.
If z = x^n then dz = n x^(n-1) dx so that dz / z = n x^(n-1) dx / x^n = n * dx / x.
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