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SI Units and Uncertainties
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1. SI Units and Uncertainties Unit 1: Measurements

2. SI Units and Uncertainties • SI Unit (Le Système International d’Unités) • Fundamental units • meter (m) • kilogram (kg) • second (s) • ampere (A) • Kelvin (K) • mole (mol) • candela (cd)

3. SI Units and Uncertainties • Derived Units • Any unit made of 2 or more fundamental units • m s-1 • m s-2 • Newton (N) = kg m s-2 • Joule (J) = kg m2 s-2 • Watt (W) = kg m2 s-3 • Coulomb (C) = A s

4. Estimation with SI Units • Fundamental Units • Mass: 1 kg – 2.2lbs / 1 L of H2O / An avg. person is 50 kg • Length: 1 m - Distance between one’s hands with outstretched arms • Time: 1 s - Duration of resting heartbeat • Derived Units • Force: 1 N- weight of an apple • Energy: 1 J- Work lifting an apple off of the ground

5. Scientific Notation and Prefixes • SI prefixes • Table 1 Gm = 1,000,000,000 m = 1,000,000 km 1 GM = 1 x 109 m = 1 x 106 km 0.0000000001 s = 1 ?s = ? ms

6. Can be reduced by repeated readings Cannot be reduced by repeated readings Uncertainties & Errors A. Random Errors • Readability of an instrument • A less than perfect observer • Effects of a change in the surroundings B. Systematic Errors • A wrongly calibrated instrument • An observer is less than perfect for every measurement in the same way

7. x x Random errors x x Uncertainties & Errors (cont.) • An experiment is accurate if…… it has a small systematic error • An experiment is precise if…… it has a small random error Systematic error Perfect

8. Precise but not accurate Accurate but not precise Precise and accurate! Uncertainties & Errors (cont.) Accuracy and Precision: Precision– uniformity Accuracy- conformity to a standard

9. 50 40 30 20 10 Absolute Uncertainty- has units of the measurement Determining the Range of Uncertainty 1) Analogue scales (rulers,thermometers meters with needles) ±half of the smallest division Since the smallest division on the cylinder is 10 ml, the reading would be 32 ± 5 ml 2) Digital scales ±the smallest division on the readout If the digital scale reads 5.052g, then the uncertainty would be ± 0.001g

10. Range of Uncertainty (cont.) 3. Significant Figures If you are given a value without an uncertainty, assume its uncertainty is ±1 of the last significant figure Examples: • The measurement is 14.742 g, the uncertainty of the measurement is 14.742 ± .001 g • The measurement is 50ml, the uncertainty of the measurement is 50 ± 1 ml

11. Range of Uncertainty (cont.) 4. From repeated measurements (an average) Example: A student times a cart going down a ramp 5 times, and gets these numbers: 2.03 s, 1.89 s, 1.92 s, 2.09 s, 1.96 sAverage: 1.98 s Find the deviations between the average value and the largest and smallest values. Largest: 2.09 - 1.98 = 0.11 s Smallest: 1.98 - 1.89 = 0.09 s The average is the best value and the largest deviation is taken as the uncertainty range: 1.98 ± 0.11 s

12. m 15 g Density = = = 3.0 g cm-3 v 5.0 cm3 Mathematical Representation of Uncertainty For calculations, compare the calculated value without uncertainties (the best value) with the max and min values with uncertainties in the calculation. Example 1: Find the density of a block of wood if its mass is 15 g ± 1 g and its volume is 5.0 ± 0.3 cm3 Best value

13. m m 14 g 16 g Density = Density = = = = 3.40 g cm-3 = 2.64 g cm-3 v v 5.3 cm3 4.7 cm3 Mathematical Representation of Uncertainty Example 1 (cont.): Find the density of a block of wood if its mass is 15 g ± 1 g and its volume is 5.0 ± 0.3 cm3 Maximum value: Minimum value:

14. Dy 0.4 = X 100% y 3 Mathematical Representation of Uncertainty (cont.) • The uncertainty range of our calculated value is the largest difference from the best value.. In this case, the density is 3.0 ± 0.4 g cm-3 • The uncertainty in the previous problem could have been written as a percentage = 13% In this case, the density is 3.0 g cm-3 ± 13%

15. Deviates 0.07 Deviates 0.08 Mathematical Representation of Uncertainty (cont.) Example #2: What is the uncertainty of cos q if q = 60o ±5o? • Best value of cos q = cos 60o = 0.50 • Max value of cos q = cos 55o = 0.57 • Min value of cos q = cos 65o = 0.42 The largest deviation is taken as the uncertainty range: In this case, it is 0.50 ± .08 OR 0.50 ± 16%

16. Uncertainty of 2nd quantity Total uncertainty Uncertainty of 1st quantity Mathematical Representation of Uncertainty: Shortcuts! Addition and Subtraction: When 2 or more quantities are added or subtracted, the overall uncertainty is equal to the sum of the individual uncertainties. Dy = Da + Db

17. External radius = 4.0 ±0.1 cm Internal radius = 3.6 ± 0.1 cm Mathematical Representation of Uncertainty: Shortcuts! (cont.) Example for Addition and Subtraction: • Determine the thickness of a pipe wall if the external radius is 4.0 ± 0.1 cm and the internal radius is 3.6 ± 0.1 cm Thickness of pipe: 4.0 cm – 3.6 cm = 0.4 cm Uncertainty = 0.1 cm + 0.1 cm = 0.2 cm Thickness with uncertainty: 0.4 ± 0.2 cm OR 0.4 cm ± 50%

18. Dy = Da + Db + Dc Total percentage/ fractional uncertainty Denominators represent best values y a b c Fractional Uncertainties of each quantity Mathematical Representation of Uncertainty: Shortcuts! (cont.) Multiplication and Division: The overall uncertainty is approximately equal to the sum of the percentage (or fractional) uncertainties of each quantity.

19. Dy = Da + Db 1 + 0.3 = y a b 15 5 Mathematical Representation of Uncertainty: Shortcuts! (cont.) Example for Multiplication and Division: Using the density example from before (where the mass was 15 g ± 1 g and its volume is 5.0 ± 0.3 cm3) = 0.07 + 0.06 = .13 ( this means 13%) 13% of 3 g cm-3 is 0.4 g cm-3 The result of this calculation with uncertainty is: 3.0 ± 0.4 g cm-3 or 3.0 g cm-3 ± 13%

20. Cube- each side is 6.0 ± 0.1 cm = 0.1 x 100 % 6 Mathematical Representation of Uncertainty: Shortcuts! (cont.) For exponential calculations (x2, x3): Just multiply the exponent by the percentage (or fractional) uncertainty of the number. Example: Volume = (6 cm)3 = 216 cm3 Percent uncertainty = 1.7% Uncertainty in value = 3 (1.7%) = ± 5.1% (or 11 cm3) Therefore the volume is 216 ± 11 cm3

21. Problems: • If a cube is measured to be 4.0+_ 0.1 cm in length along each side. Calculate the uncertainty in volume. Answer: 64+_5 Cm

22. Problem ( IB 2010) The length of each side of a sugar cube is measured as 10 mm with an uncertainty of +_2mm. Which of the following is the absolute uncertainty in the volume of the sugar cube? a.+_6 mm c. +_400 mm b. +_8 mm d. +_600 mm

23. Problem: 3. The lengths and width of a rectangular plates are 50+_0.5 mm and 25+_0.5 mm. Calculate the best estimate of the percentage uncertainty in the calculated area. • +_0.02% c. +_3% • +_1 % d. +_5%