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# Measurement and Uncertainties PowerPoint PPT Presentation

Measurement and Uncertainties. Topic 7. 1 Graphical Analysis. Logarithmic Functions. For example A = A o e -  t This can be transformed to give In A = In A o -  t This is now in the form y =mx + c Where m = -  And c = In A o This can then be plotted as a semi-log graph.

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

Measurement and Uncertainties

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%

• When subtracting measurements

• 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