Uncertainties in Measurements and Calculations

1. Advanced Higher Physics Unit 1 Uncertainties

2. Types of uncertainties There are four ways in which uncertainty from a measurement can arise: • Systematic • Calibration • Scale • Random

3. Systematic Arising from the system being used. For example, a slow running stopclock. These affect every reading and cannot be quantified. They need to be considered in an evaluation for an experiment.

4. Calibration Arising from the accuracy of the instruments used-how well have they been made? Instrument Calibration uncertainties Metres Stick (wood) 0.5 mm Ruler made of steel 0.1 mm 0.5% of reading + 1 in the last digit Digital Meter

5. Calibration-examples For this time measurement, the calibration uncertainty will be: 0:01 35 s Stopclock 0.01 + 0.01 = (0.5% of 1.35s) + 0.01 = ± 0.02 s

6. Scale Reading Arising from the accuracy of how the instruments are read. Scale uncertainty Type of Meter digital ±1 in the least significant digit analogue ±half the smallest unit

7. Scale Reading -example For this time measurement, the scale uncertainty is: 0:01 35 s Stopclock Δt = ±0.01s

8. Random Arising from fluctuations in repeated measurements: Random uncertainty This formula can be found in the Higher part of the data booklet.

9. Random-example Five time intervals have been measured: 1.23s, 1.21s, 1.19s, 1.20s, 1.21s The random uncertainty is given by:

10. Overall Uncertainty in a reading Generally: This formula is NOT included in the data booklet. However, if one of these is three times the others, it dominates. It will be the percentage uncertainty in the final result.

11. Overall uncertainty-example For the earlier time measurements: • Calibration uncertainty = ±0.02s • Scale Reading uncertainty = ±0.01s • Random uncertainty = ±0.01s The overall uncertainty is:

12. Uncertainty in a calculation First, calculate the percentage uncertainty in all the measurements being used. If one of these is three times the others, it dominates. It will be the percentage uncertainty in the final result. If this does not happen then we need to combine the uncertainties in the measurements.

13. Addition and subtraction If then (This formula can be found in the data booklet)

14. Multiplication and division If then (This formula can be found in the data booklet)

15. Powers If then (This formula cannot be found in the data booklet)

16. Graphical Uncertainties 1. Plot points with error bars Y 0 X

17. 2. Centroid is plotted: mean of all the X and Y coordinates Y 0 X

18. 3. Best fit line is drawn through centroid Y 0 X

19. 4. A top parallel line is drawn through point furthest above it. Y 0 X

20. 5. A similar bottom parallel line is drawn below the best fit line. Y 0 X

21. 6. Draw vertical lines through 1st and last plotted points so that it construct a parallelogram round best fit line. Y 0 X

22. 7. Draw the diagonals of this parallelogram Y 0 X

23. 8. Calculate the gradients m1 and m2 of the diagonals. Y 0 X

24. 9. The uncertainty in the gradient is given by: n is the number of data points plotted (This formula can be found in the data booklet)

25. 10. Find y intercepts c1 and c2. Y c1 0 X c2

26. 11. The uncertainty in the intercept is given by: n is the number of data points plotted (This formula can be found in the data booklet)

27. Summary • Measurements including: • Scale Reading • Calibration • Random • Overall Uncertainty

28. Calculations • Start with additions, subtractions and power • Then do formula

29. Activity • Measure the density of a metal cube • Measure the specific heat capacity of block • Measure the average speed of a trolley Think: What formula are you going to use? What quantities will you need to measure?