1 / 23

SI Units and Uncertainties

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

neve-hoover
Download Presentation

SI Units and Uncertainties

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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%

More Related