ActivityGuide_Appendices

# ActivityGuide_Append - Appendix A Propagation of Errors This document is a compilation of the formulae we use in this laboratory course for

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2003 University of Arkansas Department of Physics. Supported in part by a grant from the National Science Foundation 1 Appendix A: Propagation of Errors This document is a compilation of the formulae we use in this laboratory course for propagating errors. It begins with first-order techniques and moves toward the more sophisticated and general. Don‟t panic. I do not expect you to know how to do all of the calculus you see in equations 5 and 6. This is so you can see where the relationships come from, but we will continue to use the values we “guesstimate” from our measuring devices instead. I. First-Order Formulae When a quantity q is calculated from experimentally determined quantities x, …, z, u, …, w, where each of these have uncertainties x, …, z, u, …, w associated with them, then to first order we may use the following formulae for determining q. Adding and Subtracting: If   w u z x q , then w u z x q . Eq. (1) Products and Quotients: If w u z x q , then w w u u z z x x q q . Eq. (2) Multiplying by a constant: If x C q , where C is a constant such as 2, 4/3, , etc., then x C q . Eq. (3) Raising a variable to a power, n: If n x q , then x x n q q . Eq. (4) All of the above first-order approximations to the propagated uncertainties can be derived from two general first-order formulae. For functions of one variable, q = q(x): x dx dq q , Eq. (5) and for functions of many variables, q = q(x, …, z): z z q x x q q , Eq. (6) where partial derivatives are used here. For students who are not yet familiar with partial differentiation, it is easy once you know how to take the ordinary derivative of a function of one variable. Suppose we have a function, q = q(w, x, …, z), of several variables w, x, …, z and we want to calculate its partial derivative with respect to

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2003 University of Arkansas Department of Physics. Supported in part by a grant from the National Science Foundation 2 one of its variables, say x. To do this, we pretend that all variables other than x are just constants and we take the ordinary derivative of q with respect to x. In other words, think of q as being a function of x only [q = q(x)], which it would be if all other variables were constant. When taking the partial derivative of q with respect to another variable, say w, do the same thing but now let all variables except w be constant and think of q as a function of w only: q = q(w). II.
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## This note was uploaded on 06/04/2009 for the course PHYS 2054 taught by Professor Stewart during the Spring '08 term at Arkansas.

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ActivityGuide_Append - Appendix A Propagation of Errors This document is a compilation of the formulae we use in this laboratory course for

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