Physics I Lab Manual 2011 24

Physics I Lab Manual 2011 24 - f 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: f directly, but rather are written in a way that is easier to remember but requires that you solve the expressions for f 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 rst nding the uncertainty in the individual components, i.e. nd g ( 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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