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Physics I Lab Manual 2011 108

# Physics I Lab Manual 2011 108 - function that involves both...

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APPENDIX B. AN INTRODUCTION TO ERROR ANALYSIS where for g ( a, b ) = ab and dg da b , dg db = a . This gives δg a = bδa and δg b = aδb . When adding these uncorrelated uncertainties using ( δg ) 2 = ( δg a ) 2 + ( δg b ) 2 we get ( δg ) 2 = ( bδa ) 2 + ( aδb ) 2 dividing on the left by g 2 and on the right by ( ab ) 2 (which we can do since g = ab and generalizing for more factors gives the general expression δg g 2 = δa a 2 + δb b 2 + . . . (B.2) Product and quotient formula for uncertainty If formula 2.1 is applied to the N measurements of the same quantity a , each with identical uncertainty δa , the uncertainty in the average of those N results, ¯ a ,is reduced according to δ ¯ a = δa N (B.3) Formula for uncertainty in an average Note that the above expressions do not give the uncertainty δf directly, but rather are written in a way that is easier to remember but requires that you solve the expressions for δf to determine the uncertainty that you will quote in the calculated quantity. These rules can be applied in turn on more complicated functions.
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Unformatted text preview: function that involves both the sum and product of measured quantities h ( a,b,c ) = a × b + c can be determined by ﬁrst ﬁnding the uncertainty in the individual components, i.e. ﬁnd δg ( a,b ) for g ( a,b ) = a × b using formula (2) and then rewrite h as h ( g,c ) = g + c and apply formula (1). Interpretation of results with uncertainty Often times the purpose of lab experiments is to experimentally conﬁrm the value of a “known” quantity (such as the acceleration due to gravity) or to compare a quantity that is measured in two diﬀerent ways (for example the speed a bullet leaves a gun based on a measurement of its trajectory as well as a measurement of its ability to displace a large mass). When comparing two quantities that each have uncertainty (if one of the quantities 98...
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