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# Physics

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a= 1.3mm $$\pm$$ 0.1mm

b= 2.8mm $$\pm$$ 0.2mm

c= 0.8mm $$\pm$$ 0.1mm

Calculate $$Q=\frac{ab^2}{\sqrt{c}}$$ with its uncertainty.

Jan 4, 2018

#1
+103697
+2

a= 1.3mm $$\pm$$ 0.1mm    so a is between 1.2 and 1.4mm

b= 2.8mm $$\pm$$ 0.2mm         b is between 2.6 and 3.0mm     b^2 is between 6.76 and 9.0 mm^2

c= 0.8mm $$\pm$$ 0.1mm         c is between 0.7 and 0.9mm     sqrt(c) is between  0.836 rounded down and 0.987 rounded up

Calculate $$Q=\frac{ab^2}{\sqrt{c}}$$ with its uncertainty.

ab^2 is between 1.2*6.76 and 1.4*9 mm^3 ...       ab^2 is between 8.112   and 12.6 mm^3

ab^2/sqrtc is between      8.112/0.987     and   12.6/0.836    mm^3/mm^0.5.......

ab^2/sqrtc is between 8.218 rounded down  and   15.072 rounded up      mm^2.5

(8.218+15.072)/2  =  11.645

so the average is 11.64, the smallest is 11.645-3.427=8.218    and the biggest is 11.645+3.427=15.072

So that is          11.645 pm 3.427   approx    but this degree of accuracy is not valid and I am not sure what would be valid.

I'd say maybe

11.6 \pm 3.5   OR  12 \pm 4  would be reasonable.

I'm going with 12 pm 4   but I expect there is a more defined way of chosing the degree of uncertainty that is scientifically valid.

Jan 6, 2018
#2
+1705
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The following presentation is adapted from conversational dialogues with Lancelot Link for stochastic measurement errors in the physical sciences.

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To calculate the error range for uncorrelated and random errors, use Gauss’s equations for normal distribution of errors. Using the Gauss equations, gives a 68% confidence interval the measurement error is within this range.

Here are some basic derivatives from the Gauss equations:

$$\small \text{If Q is a combination of sums and/or differences, then the sigma (error) of Q is equal to } \\ \sigma Q = \sqrt{(\sigma a)^2 + (\sigma b)^2 . . . + (\sigma x)^2 . . . } \\ \text{If } Q = b^2 \text{ then } \sigma Q = 2(\sigma (b)) \text{ where } \sigma (b) \text{ is the error of (b)}\\ \text{If } Q = \sqrt{(b)} \text{ then }\sigma Q = \frac{1}{2} (\sigma (b)) \text{ where } \sigma (b) \text{ is the error of (b)}\\ \text{Note how the exponent becomes the multiplier of the error.}\\ \small \text {For this equation, the easiest method is to calculate each error as a }\\ \small \text {decimal and then use the sum of the decimals to calculate the error range. }\\ \text {Uncertainties (In decimal) for a, b, and c. }\\ a_1 = \frac{0.1}{1.3} = 0.0769\\ b_1 = (2)\frac{0.2}{2.8} = 0.1428\\ c_1 = (0.5)\frac{0.1}{0.8} = 0.0625\\ Q= \frac{1.3 \times (2.8)^2}{\sqrt {0.8}} = 11.4 \pm ((0.0769 + 0.1428+ 0.0625) * (11.4))\\ Q= 11.4 \pm 3.2 \\ \small \text { note: pay attention to significant figures when recording the final error range. }\\ \\$$

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Relating to this question and others you’ve posted on here , another important thing you need to do is Stop being a lazy dolt!

I would have answered your chemistry questions, except I was too lazy. I caught this laziness from you—it’s more contagious than dumbness.

GA

Jan 6, 2018
edited by GingerAle  Jan 6, 2018