Physics I Lab Manual 2011 24

Physics I Lab Manual 2011 24 - δf directly but rather are...

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EXPERIMENT 2. MEASUREMENT AND PRECISION If the function g ( a,b,. .. ) = a × b × ... or g ( a,b,. .. ) = a/b. .. is the product or quotient of two or more quantities the resulting uncertainty in g is found from δg a = dg da δa δg b = dg db δb 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 2 = ± δa a ² 2 + ± δb b ² 2 + ... (2.2) 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 (2.3) Formula for uncertainty in an average Note that the above expressions do not give the uncertainty
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Unformatted text preview: 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. For example the uncertainty in a 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). Practice Consider measurements of the width w and length l of a piece of paper used to calculated the area of the paper using A = w × l . 14...
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