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Measurement and UncertaintiesPowerPoint Presentation

Measurement and Uncertainties

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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

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%

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