Measurement and Uncertainties
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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.

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Measurement and Uncertainties

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Measurement and uncertainties

Measurement and Uncertainties

Topic 7.1 Graphical Analysis


Measurement and uncertainties

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


Measurement and uncertainties

  • 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


Measurement and uncertainties

  • 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

Measurement and Uncertainties


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%


Measurement and uncertainties

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


Measurement and uncertainties

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%


Measurement and uncertainties

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


Measurement and uncertainties

  • For other functions, (such as Trigonometrical functions) the mean, the highest and lowest possible answers can be calculated to obtain the uncertainty range


Measurement and uncertainties

  • If one uncertainty is much larger than others, the approximate uncertainty in the calculated answer can be taken as due to that quantity alone


Measurement and uncertainties

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


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