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## Measurement and Uncertainties

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**Measurement and Uncertainties**Topic 7.1 Graphical Analysis**Logarithmic Functions**• For example A = Aoe-t • This can be transformed to give In A = In Ao - t • This is now in the form y =mx + c • Where m = - • And c = In Ao • This can then be plotted as a semi-log graph**Example 2**• y = kxn • This can be transformed to give In y = In k+ n Inx • This is now in the form y =mx + c • Where m = n • And c = In k • This can then be plotted as a log-log graph**The parameters of the original equation can also be obtained**from the slope (m) and the intercept (c) of a straight line graph**Absolute, Fractional and Percentage Uncertainties**• Absolute uncertainties are in the same units as the value • i.e. 5.6 ± 0.05 cm • Fractional and percentage uncertainties are this absolute value expressed as a fraction or percentage of the value • 0.05/5.6 = 0.009 • 5.6cm ± 0.9%**Addition & Subtraction**• When adding measurements • add the absolute errors • When subtracting measurements • Add the absolute errors • When multiplying or dividing measurements, and powers • Add the relative or percentage errors of the measurements being multiplied or divided • then change back to an absolute error**Examples**• What is the product of 2.6 0.5 cm and 2.8 0.5cm ? • First we determine the product of 2.6 x 2.8 = 7.28 cm2 • Then we find the relative errors • i.e. 0.5/2.6 x 100% = 19.2% • and 0.5/2.8 x 100% = 17.9%**continued**• Sum of the relative errors • 19.2% + 17.9% = 37.1% • Change to absolute error • 37.1/100 x 7.28 = 2.70cm • Therefore the product is equal to • 7.3 2.7cm2**For other functions, (such as Trigonometrical functions) the**mean, the highest and lowest possible answers can be calculated to obtain the uncertainty range**If one uncertainty is much larger than others, the**approximate uncertainty in the calculated answer can be taken as due to that quantity alone**Uncertainties in Graphs**• To determine the uncertainties in the slope and intercepts of a straight-line graph you need to draw lines of minimum and maximum fit to the data points, plus error bars