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### 1.2

1. 1. Physics and PhysicalPhysics and Physical MeasurementMeasurement Topic 1.2 Measurement andTopic 1.2 Measurement and UncertaintiesUncertainties [SI units, uncertainties][SI units, uncertainties]
2. 2. The S.I. system of fundamental andThe S.I. system of fundamental and derived unitsderived units
3. 3. Standards of MeasurementStandards of Measurement SI units are those of the SystèmeSI units are those of the Système International d’Unités adopted in 1960International d’Unités adopted in 1960 Used for general measurement in mostUsed for general measurement in most countriescountries
4. 4. Fundamental QuantitiesFundamental Quantities Some quantities cannot be measured in aSome quantities cannot be measured in a simpler form and for convenience theysimpler form and for convenience they have been selected as the basic quantitieshave been selected as the basic quantities They are termed Fundamental Quantities,They are termed Fundamental Quantities, Units and SymbolsUnits and Symbols
5. 5. The FundamentalsThe Fundamentals LengthLength metremetre mm MassMass kilogramkilogram kgkg TimeTime secondsecond ss Electric currentElectric current AmpereAmpere AA Thermodynamic tempThermodynamic temp KelvinKelvin KK Amount of a substanceAmount of a substance molemole molmol
6. 6. Derived QuantitiesDerived Quantities When a quantity involves the measurementWhen a quantity involves the measurement of 2 or more fundamental quantities it isof 2 or more fundamental quantities it is called a Derived Quantitycalled a Derived Quantity The units of these are called Derived UnitsThe units of these are called Derived Units
7. 7. The Derived UnitsThe Derived Units AccelerationAcceleration msms-2-2 Angular accelerationAngular acceleration rad srad s--22 MomentumMomentum kgmskgms-1-1 or Nsor Ns Others have specific names and symbolsOthers have specific names and symbols ForceForce kg mskg ms-2-2 or Nor N
8. 8. Standards of MeasurementStandards of Measurement Scientists and engineers need to makeScientists and engineers need to make accurate measurements so that they canaccurate measurements so that they can exchange informationexchange information To be useful a standard of measurementTo be useful a standard of measurement must bemust be Invariant, Accessible and ReproducibleInvariant, Accessible and Reproducible
9. 9. 3 Standards (for information)3 Standards (for information) The MetreThe Metre :- the distance traveled by a:- the distance traveled by a beam of light in a vacuum over a definedbeam of light in a vacuum over a defined time interval ( 1/299 792 458 seconds)time interval ( 1/299 792 458 seconds) The KilogramThe Kilogram :- a particular platinum-:- a particular platinum- iridium cylinder kept in Sevres, Franceiridium cylinder kept in Sevres, France The SecondThe Second :- the time interval between the:- the time interval between the vibrations in the caesium atom (1 sec =vibrations in the caesium atom (1 sec = time for 9 192 631 770 vibrations)time for 9 192 631 770 vibrations)
10. 10. ConversionsConversions You will need to be able to convert from oneYou will need to be able to convert from one unit to another for the same quanitityunit to another for the same quanitity • J to kWhJ to kWh • J to eVJ to eV • Years to secondsYears to seconds • And between other systems and SIAnd between other systems and SI
11. 11. KWh to JKWh to J 1 kWh1 kWh = 1kW x 1 h= 1kW x 1 h = 1000W x 60 x 60 s= 1000W x 60 x 60 s = 1000 Js= 1000 Js-1-1 x 3600 sx 3600 s = 3600000 J= 3600000 J = 3.6 x 10= 3.6 x 1066 JJ
12. 12. J to eVJ to eV 1 eV = 1.6 x 101 eV = 1.6 x 10-19-19 JJ 1 J1 J = 1 / 1.6 x 10= 1 / 1.6 x 10-19-19 = 0.625 x= 0.625 x 10101919 = 6.25 x= 6.25 x 10101818 eVeV
13. 13. SI FormatSI Format The accepted SI format isThe accepted SI format is • msms-1-1 not m/snot m/s • msms-2-2 not m/s/snot m/s/s i.e. we use the suffix not dashesi.e. we use the suffix not dashes
14. 14. Uncertainity and error inUncertainity and error in measurementmeasurement
15. 15. ErrorsErrors Errors can be divided into 2 main classesErrors can be divided into 2 main classes Random errorsRandom errors Systematic errorsSystematic errors
16. 16. MistakesMistakes Mistakes on the part of an individual such asMistakes on the part of an individual such as • misreading scalesmisreading scales • poor arithmetic and computational skillspoor arithmetic and computational skills • wrongly transferring raw data to the final reportwrongly transferring raw data to the final report • using the wrong theory and equationsusing the wrong theory and equations These are a source of error but are notThese are a source of error but are not considered as an experimental errorconsidered as an experimental error
17. 17. Systematic ErrorsSystematic Errors Cause a random set of measurements to beCause a random set of measurements to be spread aboutspread about a valuea value rather than beingrather than being spread about the accepted valuespread about the accepted value It is a system or instrument valueIt is a system or instrument value
18. 18. Systematic Errors result fromSystematic Errors result from Badly made instrumentsBadly made instruments Poorly calibrated instrumentsPoorly calibrated instruments An instrument having a zero error, a form ofAn instrument having a zero error, a form of calibrationcalibration Poorly timed actionsPoorly timed actions Instrument parallax errorInstrument parallax error Note that systematic errors are not reducedNote that systematic errors are not reduced by multiple readingsby multiple readings
19. 19. Random ErrorsRandom Errors Are due to variations in performance of theAre due to variations in performance of the instrument and the operatorinstrument and the operator Even when systematic errors have beenEven when systematic errors have been allowed for, there exists error.allowed for, there exists error.
20. 20. Random Errors result fromRandom Errors result from Vibrations and air convectionVibrations and air convection MisreadingMisreading Variation in thickness of surface beingVariation in thickness of surface being measuredmeasured Using less sensitive instrument when aUsing less sensitive instrument when a more sensitive instrument is availablemore sensitive instrument is available Human parallax errorHuman parallax error
21. 21. Reducing Random ErrorsReducing Random Errors Random errors can be reduced byRandom errors can be reduced by • taking multiple readings, and eliminatingtaking multiple readings, and eliminating obviously erroneous resultobviously erroneous result • or by averaging the range of results.or by averaging the range of results.
22. 22. AccuracyAccuracy Accuracy is an indication of how close aAccuracy is an indication of how close a measurement is to the accepted valuemeasurement is to the accepted value indicated by the relative or percentageindicated by the relative or percentage error in the measurementerror in the measurement An accurate experiment has a lowAn accurate experiment has a low systematic errorsystematic error
23. 23. PrecisionPrecision Precision is an indication of the agreementPrecision is an indication of the agreement among a number of measurements made inamong a number of measurements made in the same way indicated by the absolutethe same way indicated by the absolute errorerror A precise experiment has a low randomA precise experiment has a low random errorerror
24. 24. Limit of Reading and UncertaintyLimit of Reading and Uncertainty TheThe Limit of ReadingLimit of Reading of a measurement isof a measurement is equal to the smallest graduation of the scale of anequal to the smallest graduation of the scale of an instrumentinstrument TheThe Degree of UncertaintyDegree of Uncertainty of a measurementof a measurement is equal to half the limit of readingis equal to half the limit of reading e.g. If the limit of reading is 0.1cm then thee.g. If the limit of reading is 0.1cm then the uncertainty range isuncertainty range is ±±0.05cm0.05cm This is the absolute uncertaintyThis is the absolute uncertainty
25. 25. Reducing the Effects of RandomReducing the Effects of Random UncertaintiesUncertainties Take multiple readingsTake multiple readings When a series of readings are taken for aWhen a series of readings are taken for a measurement, then the arithmetic mean ofmeasurement, then the arithmetic mean of the reading is taken as the most probablethe reading is taken as the most probable answeranswer The greatest deviation or residual from theThe greatest deviation or residual from the mean is taken as the absolute errormean is taken as the absolute error
26. 26. Absolute/fractional errors andAbsolute/fractional errors and percentage errorspercentage errors We use ± to show an error in aWe use ± to show an error in a measurementmeasurement (208 ± 1) mm is a fairly accurate(208 ± 1) mm is a fairly accurate measurementmeasurement (2 ± 1) mm is highly inaccurate(2 ± 1) mm is highly inaccurate
27. 27. In order to compare uncertainties, use isIn order to compare uncertainties, use is made of absolute, fractional andmade of absolute, fractional and percentage uncertainties.percentage uncertainties. 1 mm is the absolute uncertainty1 mm is the absolute uncertainty 1/208 is the fractional uncertainty (0.0048)1/208 is the fractional uncertainty (0.0048) 0.48 % is the percentage uncertainty0.48 % is the percentage uncertainty
28. 28. Combining uncertaintiesCombining uncertainties For addition and subtraction, add absoluteFor addition and subtraction, add absolute uncertaintiesuncertainties y = b-c then y ±y = b-c then y ± δδy = (b-c) ± (y = (b-c) ± (δδb +b + δδc)c)
29. 29. Combining uncertaintiesCombining uncertainties For multiplication and division addFor multiplication and division add percentage uncertaintiespercentage uncertainties x = b x c thenx = b x c then δδxx == δδbb ++ δδcc x b cx b c
30. 30. Combining uncertaintiesCombining uncertainties When using powers, multiply theWhen using powers, multiply the percentage uncertainty by the powerpercentage uncertainty by the power z = bn thenz = bn then δδzz = n= n δδbb z bz b
31. 31. Combining uncertaintiesCombining uncertainties If one uncertainty is much larger thanIf one uncertainty is much larger than others, the approximate uncertainty in theothers, the approximate uncertainty in the calculated result may be taken as due tocalculated result may be taken as due to that quantity alonethat quantity alone
32. 32. Uncertainties in graphsUncertainties in graphs
33. 33. Plotting Uncertainties on GraphsPlotting Uncertainties on Graphs Points are plotted with a fine pencil crossPoints are plotted with a fine pencil cross Uncertainty or error bars are requiredUncertainty or error bars are required These are short lines drawn from theThese are short lines drawn from the plotted points parallel to the axesplotted points parallel to the axes indicating the absolute error ofindicating the absolute error of measurementmeasurement
34. 34. y x Uncertainties on a GraphUncertainties on a Graph
35. 35. Significant FiguresSignificant Figures The number of significant figures shouldThe number of significant figures should reflect the precision of the value or of thereflect the precision of the value or of the input data to be calculatedinput data to be calculated Simple rule:Simple rule: For multiplication and division, the numberFor multiplication and division, the number of significant figures in a result should notof significant figures in a result should not exceed that of the least precise value uponexceed that of the least precise value upon which it dependswhich it depends
36. 36. EstimationEstimation You need to be able to estimate values ofYou need to be able to estimate values of everyday objects to one or two significant figureseveryday objects to one or two significant figures And/or to the nearest order of magnitudeAnd/or to the nearest order of magnitude e.g.e.g. • Dimensions of a brickDimensions of a brick • Mass of an appleMass of an apple • Duration of a heartbeatDuration of a heartbeat • Room temperatureRoom temperature • Swimming PoolSwimming Pool
37. 37. You also need to estimate the result ofYou also need to estimate the result of calculationscalculations e.g.e.g. • 6.3 x 7.6/4.96.3 x 7.6/4.9 • = 6 x 8/5= 6 x 8/5 • = 48/5= 48/5 • =50/5=50/5 • =10=10 • (Actual answer = 9.77)(Actual answer = 9.77)
38. 38. Approaching and SolvingApproaching and Solving ProblemsProblems You need to be able to state and explain anyYou need to be able to state and explain any simplifying assumptions that you makesimplifying assumptions that you make solving problemssolving problems • e.g. Reasonable assumptions as to why certaine.g. Reasonable assumptions as to why certain quantities may be neglected or ignoredquantities may be neglected or ignored • i.e. Heat loss, internal resistancei.e. Heat loss, internal resistance • Or that behaviour is approximately linearOr that behaviour is approximately linear
39. 39. Graphical TechniquesGraphical Techniques Graphs are very useful for analysing theGraphs are very useful for analysing the data that is collected during investigationsdata that is collected during investigations Graphing is one of the most valuable toolsGraphing is one of the most valuable tools used becauseused because
40. 40. Why GraphWhy Graph • it gives a visual display of the relationshipit gives a visual display of the relationship between two or more variablesbetween two or more variables • shows which data points do not obey theshows which data points do not obey the relationshiprelationship • gives an indication at which point agives an indication at which point a relationship ceases to be truerelationship ceases to be true • used to determine the constants in an equationused to determine the constants in an equation relating two variablesrelating two variables
41. 41. You need to be able to give a qualitativeYou need to be able to give a qualitative physical interpretation of a particularphysical interpretation of a particular graphgraph e.g. as the potential difference increases,e.g. as the potential difference increases, the ionization current also increases untilthe ionization current also increases until it reaches a maximum at…..it reaches a maximum at…..
42. 42. Plotting GraphsPlotting Graphs Independent variables are plotted on the x-Independent variables are plotted on the x- axisaxis Dependent variables are plotted on the y-Dependent variables are plotted on the y- axisaxis Most graphs occur in the 1st quadrantMost graphs occur in the 1st quadrant however some may appear in all 4however some may appear in all 4
43. 43. Plotting Graphs - Choice of AxPlotting Graphs - Choice of Axiiss When you are asked to plot a graph of aWhen you are asked to plot a graph of a against b, the first variable mentioned isagainst b, the first variable mentioned is plotted on the y axisplotted on the y axis Graphs should be plotted by handGraphs should be plotted by hand
44. 44. Plotting Graphs - ScalesPlotting Graphs - Scales Size of graph should be large, to fill asSize of graph should be large, to fill as much space as possiblemuch space as possible choose a convenient scale that is easilychoose a convenient scale that is easily subdividedsubdivided
45. 45. Plotting Graphs - LabelsPlotting Graphs - Labels Each axis is labeled with the name andEach axis is labeled with the name and symbol, as well as the relevant unit usedsymbol, as well as the relevant unit used The graph should also be given aThe graph should also be given a descriptive titledescriptive title
46. 46. Plotting Graphs - Line of Best FitPlotting Graphs - Line of Best Fit When choosing the line or curve it is best to use aWhen choosing the line or curve it is best to use a transparent rulertransparent ruler Position the ruler until it lies along an ideal linePosition the ruler until it lies along an ideal line The line or curve does not have to pass throughThe line or curve does not have to pass through every pointevery point Do not assume that all lines should pass throughDo not assume that all lines should pass through the originthe origin DoDo notnot dodo dot to dotdot to dot!!
47. 47. y x
48. 48. Analysing the GraphAnalysing the Graph Often a relationship between variables willOften a relationship between variables will first produce a parabola, hyperbole or anfirst produce a parabola, hyperbole or an exponential growth or decay. These can beexponential growth or decay. These can be transformed to a straight line relationshiptransformed to a straight line relationship General equation for a straight line isGeneral equation for a straight line is y = mx + cy = mx + c – y is the dependent variable, x is the independenty is the dependent variable, x is the independent variable, m is the gradient and c is the y-interceptvariable, m is the gradient and c is the y-intercept
49. 49. The parameters of a function can also beThe parameters of a function can also be obtained from the slope (obtained from the slope (mm) and the) and the intercept (intercept (cc) of a straight line graph) of a straight line graph
50. 50. GradientsGradients Gradient = vertical run / horizontal runGradient = vertical run / horizontal run or gradient =or gradient = ∆∆y /y / ∆∆xx uphill slope is positive and downhill slopeuphill slope is positive and downhill slope is negativeis negative Don´t forget to give the units of the gradientDon´t forget to give the units of the gradient
51. 51. Areas under GraphsAreas under Graphs The area under a graph is a useful toolThe area under a graph is a useful tool • e.g. on a force displacement graph the area ise.g. on a force displacement graph the area is work (N x m = J)work (N x m = J) • e.g. on a speed time graph the area is distancee.g. on a speed time graph the area is distance (ms(ms-1-1 x s = m)x s = m) Again, don´t forget the units of the areaAgain, don´t forget the units of the area
52. 52. Standard Graphs - linear graphsStandard Graphs - linear graphs A straight line passing through the originA straight line passing through the origin shows proportionalityshows proportionality y x y ∝ x y = k x Where k is the constant of proportionality k = rise/run
53. 53. Standard Graphs - parabolaStandard Graphs - parabola A parabola shows that y is directlyA parabola shows that y is directly proportional to xproportional to x22 y x2 y x i.e. y ∝ x2 or y = kx2 where k is the constant of proportionality
54. 54. Standard Graphs - hyperbolaStandard Graphs - hyperbola A hyperbola shows that y is inverselyA hyperbola shows that y is inversely proportional to xproportional to x y 1/x y x i.e. y ∝ 1/x or y = k/x where k is the constant of proportionality
55. 55. Standard Graphs - hyperbolaStandard Graphs - hyperbola againagain An inverse square law graph is also aAn inverse square law graph is also a hyperbolahyperbola y 1/x2 y x i.e. y ∝ 1/x2 or y = k/x2 where k is the constant of proportionality
56. 56. Non-Standard GraphsNon-Standard Graphs You need to make a connection betweenYou need to make a connection between graphs and equationsgraphs and equations y x If this is a graph of r against t2 plotted from data having an expected relationship r = at2 /2 +r0 where a is a constant Then the gradient is a/2 and the y-intercept is r0 - it is not the case that r ∝ t2 , it is a linear relationship The intercept is therefore important tooThe intercept is therefore important too
57. 57. Errors in gradientsErrors in gradients Line of best fit is theLine of best fit is the solid line.solid line. The maximum andThe maximum and minimum slopes gominimum slopes go through the extremesthrough the extremes of the errors barsof the errors bars y x