UncertaintyPropagationMethod

UncertaintyPropagationMethod - Uncertainty Propagation...

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Uncertainty Propagation Addition and Subtraction For any combination of addition or subtraction, the absolute error in the result is equal to the sum of the absolute uncertainties of the parts. If D C B A F + + = , then D C B A F δ + + + = . Powers and Roots When a measurement A is raised to a power z , the relative uncertainty in the result is z time the relative uncertainty in A . If z A F = , A A z F F = . Multiplication and Division For any combination of multiplication or division, the relative error in the result is equal to the sum of the absolute uncertainties of the parts. If CD AB F = , then D D C C B B A A F F + + + = . Functions The absolute uncertainty in a function of a measurement A is equal to the absolute uncertainty of the measurement multiplied by the derivative of the function. If ) ( A f F = , then A x dx x df A F = × = ) ( . Table 1 Common functions and their derivatives. Note: Uncertainty in angles must be in radians. f(x) f (x) f(x) f (x) sin(x) cos(x) arcsin(x) 2 1 1 x cos(x) -sin(x) arccos(x) 2 1 1 x tan(x) ) ( cos 1 2 x arctan(x) 2 1 1 x + ln(x) x 1 e x e x
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Examples involving the use of a single rule 1. Find Z = W – X – Y , where W = 0.00123 ± 0.00007 , X = 0.0032 ± 0.0008 , and Y = -0.0061 ± 0.0009 . The principle, or measured part, of Z is Z = 0.00123 – 0.0032 – -0.0061 = 0.00413 . Since finding Z only involves using one operation, addition or subtraction, we know the uncertainty in Z is δ Z = 0.00007 + 0.0008 + 0.0009 = 0.00177 . Thus our result, to the correct number of significant figures is, Z = 0.0041 ± 0.0018 .
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This note was uploaded on 09/02/2010 for the course PHYS MERR1 taught by Professor Carter during the Spring '10 term at UMass (Amherst).

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UncertaintyPropagationMethod - Uncertainty Propagation...

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