1. Physics and Physical Measurement Topic 1.2 Measurement and Uncertainties

2. The S.I. system of fundamental and derived units

3. Standards of Measurement • SI units are those of the Système International d’Unités adopted in 1960 • Used for general measurement in most countries

4. Fundamental Quantities • Some quantities cannot be measured in a simpler form and for convenience they have been selected as the basic quanitities • They are termed Fundamental Quantities, Units and Symbols

5. The Fundamentals • Length metre m • Mass kilogram kg • Time second s • Electric current ampere A • Thermodynamic temp Kelvin K • Amount of a substance mole mol

6. Derived Quantities • When a quantity involves the measurement of 2 or more fundamental quantities it is called a Derived Quantity • The units of these are called Derived Units

7. The Derived Units • Acceleration ms-2 • Angular acceleration rad s-2 • Momentum kgms-1 or Ns • Others have specific names and symbols • Force kg ms-2 or N

8. Standards of Measurement • Scientists and engineers need to make accurate measurements so that they can exchange information • To be useful a standard of measurement must be • Invariant, Accessible and Reproducible

9. 3 Standards (for information) • The Metre :- the distance traveled by a beam of light in a vacuum over a defined time interval ( 1/299 792 458 seconds) • The Kilogram :- a particular platinum-iridium cylinder kept in Sevres, France • The Second :- the time interval between the vibrations in the caesium atom (1 sec = time for 9 192 631 770 vibrations)

10. Conversions • You will need to be able to convert from one unit to another for the same quanitity • J to kWh • J to eV • Years to seconds • And between other systems and SI

11. KWh to J • 1 kWh = 1kW x 1 h • = 1000W x 60 x 60 s • = 1000 Js-1 x 3600 s • = 3600000 J • = 3.6 x 106 J

12. J to eV • 1 eV = 1.6 x 10-19 J

13. SI Format • The accepted SI format is • ms-1 not m/s • ms-2 not m/s/s • i.e. we use the suffix not dashes

14. Uncertainity and error in measurement

15. Errors • Errors can be divided into 2 main classes • Random errors • Systematic errors

16. Mistakes • Mistakes on the part of an individual such as • misreading scales • poor arithmetic and computational skills • wrongly transferring raw data to the final report • using the wrong theory and equations • These are a source of error but are not considered as an experimental error

17. Systematic Errors • Cause a random set of measurements to be spread about a value rather than being spread about the accepted value • It is a system or instrument value

18. Systematic Errors result from • Badly made instruments • Poorly calibrated instruments • An instrument having a zero error, a form of calibration • Poorly timed actions • Instrument parallax error • Note that systematic errors are not reduced by multiple readings

19. Random Errors • Are due to variations in performance of the instrument and the operator • Even when systematic errors have been allowed for, there exists error.

20. Random Errors result from • Vibrations and air convection • Misreading • Variation in thickness of surface being measured • Using less sensitive instrument when a more sensitive instrument is available • Human parallax error

21. Reducing Random Errors • Random errors can be reduced by • taking multiple readings, and eliminating obviously erroneous result • or by averaging the range of results.

22. Accuracy • Accuracy is an indication of how close a measurement is to the accepted value indicated by the relative or percentage error in the measurement • An accurate experiment has a low systematic error

23. Precision • Precision is an indication of the agreement among a number of measurements made in the same way indicated by the absolute error • A precise experiment has a low random error

24. Limit of Reading and Uncertainty • The Limit of Reading of a measurement is equal to the smallest graduation of the scale of an instrument • The Degree of Uncertainty of a measurement is equal to half the limit of reading • e.g. If the limit of reading is 0.1cm then the uncertainty range is 0.05cm • This is the absolute uncertainty

25. Reducing the Effects of Random Uncertainties • Take multiple readings • When a series of readings are taken for a measurement, then the arithmetic mean of the reading is taken as the most probable answer • The greatest deviation or residual from the mean is taken as the absolute error

26. Absolute/fractional errors and percentage errors • We use ± to show an error in a measurement • (208 ± 1) mm is a fairly accurate measurement • (2 ± 1) mm is highly inaccurate

27. In order to compare uncertainties, use is made of absolute, fractional and percentage uncertainties. • 1 mm is the absolute uncertainty • 1/208 is the fractional uncertainty (0.0048) • 0.48 % is the percentage uncertainity

28. Combining uncertainties • For addition and subtraction, add absolute uncertainities • y = b-c then y ± dy = (b-c) ± (db + dc)

29. Combining uncertainties • For multiplication and division add percentage uncertainities • x = b x c then dx = db + dc x b c

30. Combining uncertainties • When using powers, multiply the percentage uncertainty by the power • z = bn then dz = n db z b

31. Combining uncertainties • If one uncertainty is much larger than others, the approximate uncertainty in the calculated result may be taken as due to that quantity alone

32. Uncertainties in graphs

33. Plotting Uncertainties on Graphs • Points are plotted with a fine pencil cross • Uncertainty or error bars are required • These are short lines drawn from the plotted points parallel to the axes indicating the absolute error of measurement

34. y x Uncertainties on a Graph

35. Significant Figures • The number of significant figures should reflect the precision of the value or of the input data to be calculated • Simple rule: • For multiplication and division, the number of significant figures in a result should not exceed that of the least precise value upon which it depends

36. Estimation • You need to be able to estimate values of everyday objects to one or two significant figures • And/or to the nearest order of magnitude • e.g. • Dimensions of a brick • Mass of an apple • Duration of a heartbeat • Room temperature • Swimming Pool

37. You also need to estimate the result of calculations • e.g. • 6.3 x 7.6/4.9 • = 6 x 8/5 • = 48/5 • =50/5 • =10 • (Actual answer = 9.77)

38. Approaching and Solving Problems • You need to be able to state and explain any simplifying assumptions that you make solving problems • e.g. Reasonable assumptions as to why certain quantities may be neglected or ignored • i.e. Heat loss, internal resistance • Or that behaviour is approximately linear

39. Graphical Techniques • Graphs are very useful for analysing the data that is collected during investigations • Graphing is one of the most valuable tools used because

40. Why Graph • it gives a visual display of the relationship between two or more variables • shows which data points do not obey the relationship • gives an indication at which point a relationship ceases to be true • used to determine the constants in an equation relating two variables

41. You need to be able to give a qualitative physical interpretation of a particular graph • e.g. as the potential difference increases, the ionization current also increases until it reaches a maximum at…..

42. Plotting Graphs • Independent variables are plotted on the x-axis • Dependent variables are plotted on the y-axis • Most graphs occur in the 1st quadrant however some may appear in all 4

43. Plotting Graphs - Choice of Axis • When you are asked to plot a graph of a against b, the first variable mentioned is plotted on the y axis • Graphs should be plotted by hand

44. Plotting Graphs - Scales • Size of graph should be large, to fill as much space as possible • choose a convenient scale that is easily subdivided

45. Plotting Graphs - Labels • Each axis is labeled with the name and symbol, as well as the relevant unit used • The graph should also be given a descriptive title

46. Plotting Graphs - Line of Best Fit • When choosing the line or curve it is best to use a transparent ruler • Position the ruler until it lies along an ideal line • The line or curve does not have to pass through every point • Do not assume that all lines should pass through the origin • Do not do dot to dot!

47. y x

48. Analysing the Graph • Often a relationship between variables will first produce a parabola, hyperbole or an exponential growth or decay. These can be transformed to a straight line relationship • General equation for a straight line is • y = mx + c • y is the dependent variable, x is the independent variable, m is the gradient and c is the y-intercept

49. The parameters of a function can also be obtained from the slope (m) and the intercept (c) of a straight line graph

50. Gradients • Gradient = vertical run / horizontal run • or gradient = y / x • uphill slope is positive and downhill slope is negative • Don´t forget to give the units of the gradient