Measurement and Measurement and UncertaintiesUncertainties

Topic 7.Topic 7.11 Graphical Analysis Graphical Analysis

Logarithmic FunctionsLogarithmic Functions

For example A = AFor example A = Aooee--t t

This can be transformed to give This can be transformed to give In A = In AIn A = In Ao o - - t t

This is now in the form y =mx + c This is now in the form y =mx + c Where m = - Where m = - And c = In AAnd c = In Ao o

This can then be plotted as a semi-This can then be plotted as a semi-log graphlog graph

Example 2 Example 2 • y = kxy = kxn n

This can be transformed to give This can be transformed to give In y = In kIn y = In k + n Inx + n Inx

This is now in the form y =mx + c This is now in the form y =mx + c Where m = n Where m = n And c = In kAnd c = In k

This can then be plotted as a log-log This can then be plotted as a log-log graphgraph

The parameters of the original The parameters of the original equation can also be obtained from equation can also be obtained from the slope (the slope (mm) and the intercept () and the intercept (cc) ) of a straight line graph of a straight line graph

Measurement and Measurement and UncertaintiesUncertainties

Absolute, Fractional and Absolute, Fractional and Percentage UncertaintiesPercentage Uncertainties

Absolute uncertainties are in the Absolute uncertainties are in the same units as the value same units as the value

i.e. 5.6 ± 0.05 cm i.e. 5.6 ± 0.05 cm Fractional and percentage Fractional and percentage

uncertainties are this absolute value uncertainties are this absolute value expressed as a fraction or expressed as a fraction or percentage of the value percentage of the value

0.05/5.6 = 0.009 0.05/5.6 = 0.009 5.6cm ± 0.9%5.6cm ± 0.9%

Addition & SubtractionAddition & Subtraction

When adding measurements When adding measurements • add the absolute errors add the absolute errors

When subtracting measurements When subtracting measurements • Add the absolute errors Add the absolute errors

When multiplying or dividing When multiplying or dividing measurements, and powers measurements, and powers • Add the relative or percentage errors of the Add the relative or percentage errors of the

measurements being multiplied or divided measurements being multiplied or divided • then change back to an absolute errorthen change back to an absolute error

ExamplesExamples

What is the product of 2.6 What is the product of 2.6 0.5 0.5 cm and 2.8 cm and 2.8 0.5cm ? 0.5cm ?

First we determine the product of First we determine the product of 2.6 x 2.8 = 7.28 cm2.6 x 2.8 = 7.28 cm22

Then we find the relative errors Then we find the relative errors • i.e. 0.5/2.6 x 100% = 19.2% i.e. 0.5/2.6 x 100% = 19.2% • and 0.5/2.8 x 100% = 17.9%and 0.5/2.8 x 100% = 17.9%

continuedcontinued

Sum of the relative errors Sum of the relative errors • 19.2% + 17.9% = 37.1% 19.2% + 17.9% = 37.1%

Change to absolute error Change to absolute error • 37.1/100 x 7.37.1/100 x 7.2828 = 2.70cm = 2.70cm

Therefore the product is equal to Therefore the product is equal to 7.3 7.3 2.7cm 2.7cm22

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

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

Uncertainties in GraphsUncertainties in Graphs

To determine the uncertainties in To determine the uncertainties in the slope and intercepts of a the slope and intercepts of a straight-line graph you need to straight-line graph you need to draw lines of minimum and draw lines of minimum and maximum fit to the data points, maximum fit to the data points, plus error barsplus error bars