physics and physical measurement topic 1.2 measurement and uncertainties
TRANSCRIPT
Physics and Physical Physics and Physical MeasurementMeasurement
Topic 1.2 Measurement and Topic 1.2 Measurement and UncertaintiesUncertainties
The S.I. SystemThe S.I. System
Standards of Standards of MeasurementMeasurement
SI units are those of the Système SI units are those of the Système International d’Unités adopted in International d’Unités adopted in 19601960
Used for general measurement in Used for general measurement in most countries worldwidemost countries worldwide
Fundamental QuantitiesFundamental Quantities
Some quantities cannot be Some quantities cannot be measured in a simpler form and for measured in a simpler form and for convenience they have been convenience they have been selected as the basic quanititiesselected as the basic quanitities
They are termed Fundamental They are termed Fundamental Quantities, Units and SymbolsQuantities, Units and Symbols
The 7 FundamentalsThe 7 Fundamentals
LengthLength metremetre mm MassMass kilogram kilogram kgkg TimeTime secondsecond ss Electric currentElectric current ampereampere AA Thermodynamic tempThermodynamic temp KelvinKelvin
KK Luminous IntensityLuminous Intensity candelacandela cdcd Amount of a substanceAmount of a substance molemole molmol
Derived QuantitiesDerived Quantities
When a quantity involves the When a quantity involves the measurement of 2 or more measurement of 2 or more fundamental quantities it is called fundamental quantities it is called a a Derived QuantityDerived Quantity
The units of these are called The units of these are called Derived UnitsDerived Units
Derived UnitsDerived Units
Examples…Examples… Acceleration Acceleration msms-2-2
MomentumMomentum kgmskgms-1-1 or Ns or Ns
Some derived units have been given their Some derived units have been given their own specific names and symbols…own specific names and symbols…
Force Force N = kg msN = kg ms-2 -2
JouleJoule J = kgmJ = kgm22ss-2-2
Standards of Standards of MeasurementMeasurement
Scientists and engineers need to Scientists and engineers need to make accurate measurements so make accurate measurements so that they can exchange that they can exchange informationinformation
To be useful a standard of To be useful a standard of measurement must be Invariant, measurement must be Invariant, Accessible and Reproducible Accessible and Reproducible
3 Standards 3 Standards (FYI – not tested)(FYI – not tested)
The Meter The Meter :- the distance traveled by a :- the distance traveled by a beam of light in a vacuum over a defined beam of light in a vacuum over a defined time interval ( 1/299 792 458 seconds)time interval ( 1/299 792 458 seconds)
The Kilogram The Kilogram :- a particular platinum-:- a particular platinum-iridium cylinder kept in Sevres, Franceiridium cylinder kept in Sevres, France
The Second The Second :- the time interval between :- the time interval between the vibrations in the caesium atom (1 sec the vibrations in the caesium atom (1 sec = time for 9 192 631 770 vibrations)= time for 9 192 631 770 vibrations)
ConversionsConversions
You will need to be able to convert from You will need to be able to convert from one unit to another for the same one unit to another for the same quanitityquanitity• J to kWh (energy)J to kWh (energy)• J to eV (energy)J to eV (energy)• Years to seconds (time)Years to seconds (time)• And between other systems and SIAnd between other systems and SI
****Note: you should be able to do basic conversions now ****Note: you should be able to do basic conversions now and others will be developed throughout the yearand others will be developed throughout the year
SI FormatSI Format
The accepted SI format isThe accepted SI format is• msms-1-1 not m/s not m/s• msms-2 -2 not m/s/snot m/s/s
The IB will recognize work reported The IB will recognize work reported with “/”, but will only use the SI with “/”, but will only use the SI format when providing info.format when providing info.
Uncertainity and error in Uncertainity and error in measurementmeasurement
ErrorsErrors
Errors can be divided into 2 main Errors can be divided into 2 main classesclasses
Random errorsRandom errors Systematic errorsSystematic errors
MistakesMistakes
Mistakes on the part of an individual Mistakes on the part of an individual such assuch as• misreading scalesmisreading scales• poor arithmetic and computational skillspoor arithmetic and computational skills• wrongly transferring raw data to the final wrongly transferring raw data to the final
reportreport• using the wrong theory and equationsusing the wrong theory and equations
These are a source of error but are not These are a source of error but are not considered as an experimental errorconsidered as an experimental error
Systematic ErrorsSystematic Errors
Cause a random set of Cause a random set of measurements to be affected in measurements to be affected in the same way the same way
It is a system or instrument issueIt is a system or instrument issue
Systematic Errors result Systematic Errors result fromfrom
Badly made instrumentsBadly made instruments Poorly calibrated instrumentsPoorly calibrated instruments An instrument having a zero error, An instrument having a zero error,
a form of calibrationa form of calibration Poorly timed actionsPoorly timed actions Instrument parallax errorInstrument parallax error Note that systematic errors are not Note that systematic errors are not
reduced by multiple readingsreduced by multiple readings
Random ErrorsRandom Errors
Are due to unpredictable variations Are due to unpredictable variations in performance of the instrument in performance of the instrument and the operatorand the operator
Random Errors result fromRandom Errors result from
Vibrations and air convectionVibrations and air convection MisreadingMisreading Variation in thickness of surface Variation in thickness of surface
being measuredbeing measured Using less sensitive instrument Using less sensitive instrument
when a more sensitive instrument is when a more sensitive instrument is availableavailable
Human parallax errorHuman parallax error
Reducing Random ErrorsReducing Random Errors
Random errors can be reduced by Random errors can be reduced by taking multiple readings, and taking multiple readings, and eliminating obviously erroneous eliminating obviously erroneous result or by averaging the range of result or by averaging the range of results.results.
AccuracyAccuracy
Accuracy is an indication of how Accuracy is an indication of how close a measurement is to the close a measurement is to the accepted value indicated by the accepted value indicated by the relative or percentage error in the relative or percentage error in the measurementmeasurement
An accurate experiment has a low An accurate experiment has a low systematic errorsystematic error
PrecisionPrecision
Precision is an indication of the Precision is an indication of the agreement among a number of agreement among a number of measurements made in the same measurements made in the same way indicated by the absolute way indicated by the absolute errorerror
A precise experiment has a low A precise experiment has a low random errorrandom error
Reducing the Effects of Reducing the Effects of Random UncertaintiesRandom Uncertainties
Take multiple readingsTake multiple readings When a series of readings are When a series of readings are
taken for a measurement, then the taken for a measurement, then the arithmetic mean of the reading is arithmetic mean of the reading is taken as the most probable answertaken as the most probable answer
The greatest deviation from the The greatest deviation from the mean is taken as the absolute mean is taken as the absolute errorerror
Absolute/fractional errors Absolute/fractional errors and percentage errorsand percentage errors
We use ± to show an error in a We 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
Absolute, fractional, and Absolute, fractional, and relative uncertaintyrelative uncertainty
Assume we measure something to be Assume we measure something to be 208 ± 1 mm in length...208 ± 1 mm in length...
1 mm is the absolute uncertainty1 mm is the absolute uncertainty 1/208 is the fractional uncertainty 1/208 is the fractional uncertainty
(0.0048)(0.0048) 0.48 % is the relative (percent) 0.48 % is the relative (percent)
uncertaintyuncertainty
Combining uncertainties Combining uncertainties
To determine the uncertainty of a To determine the uncertainty of a calculated value...calculated value...
For addition and subtraction, add For addition and subtraction, add absolute uncertainitiesabsolute uncertainities
For multiplication and division add For multiplication and division add percentage uncertainitiespercentage uncertainities
When using exponents, multiply the When using exponents, multiply the percentage uncertainty by the exponentpercentage uncertainty by the exponent
Combining uncertaintiesCombining uncertainties
If one uncertainty is much larger If one uncertainty is much larger than others, the approximate than others, the approximate uncertainty in the calculated result uncertainty in the calculated result may be taken as due to that may be taken as due to that quantity alonequantity alone
Significant FiguresSignificant Figures
The number of significant figures The number of significant figures should reflect the precision of the should reflect the precision of the values used as input data in a values used as input data in a calculationcalculation
Simple rule: Simple rule: For multiplication and division, the For multiplication and division, the
number of significant figures in a result number of significant figures in a result should not exceed that of the least should not exceed that of the least precise value upon which it dependsprecise value upon which it depends
Uncertainties in graphsUncertainties in graphs
Graphical TechniquesGraphical Techniques
Graphing is one of the most valuable Graphing is one of the most valuable tools in data analysis becausetools in data analysis because• it gives a visual display of the relationship it gives a visual display of the relationship
between two or more variablesbetween two or more variables• shows which data points do not obey the shows which data points do not obey the
relationshiprelationship• gives an indication at which point a gives an indication at which point a
relationship ceases to be truerelationship ceases to be true• used to determine the constants in an used to determine the constants in an
equation relating two variablesequation relating two variables
You need to be able to give a You need to be able to give a qualitative physical interpretation qualitative physical interpretation of a particular graphof a particular graph
Plotting GraphsPlotting Graphs
Independent variables are plotted Independent variables are plotted on the x-axison the x-axis
Dependent variables are plotted on Dependent variables are plotted on the y-axisthe y-axis
Most graphs occur in the 1st Most graphs occur in the 1st quadrant however some may quadrant however some may appear in all 4appear in all 4
Plotting Graphs - Choice of Plotting Graphs - Choice of AxAxiiss
Experimentally speakingExperimentally speaking, the , the dependent variable is plotted on dependent variable is plotted on the y axis and the independent the y axis and the independent variable is plotted on the x axis.variable is plotted on the x axis.
When you are asked to plot When you are asked to plot a graph a graph of a against b, the first variable of a against b, the first variable mentioned is plotted on the y axis.mentioned is plotted on the y axis.
Plotting Graphs - ScalesPlotting Graphs - Scales
Size of graph should be large, to fill Size of graph should be large, to fill as much space as possible…3/4 as much space as possible…3/4 rulerule
choose a convenient scale that is choose a convenient scale that is easily subdividedeasily subdivided
Plotting Graphs - LabelsPlotting Graphs - Labels
Each axis is labeled with the name Each axis is labeled with the name of the quantity, as well as the of the quantity, as well as the relevant unit used…relevant unit used…
Temperature/KTemperature/K
speed/msspeed/ms-1-1
The graph should also be given a The graph should also be given a descriptive titledescriptive title
Plotting Uncertainties on Plotting Uncertainties on GraphsGraphs
Error bars showing uncertainty are Error bars showing uncertainty are required - short lines drawn from required - short lines drawn from the plotted points parallel to the the plotted points parallel to the axes indicating the absolute error axes indicating the absolute error of measurementof measurement
Plotting Graphs - Line of Plotting Graphs - Line of Best FitBest Fit
When choosing the best fit line or curve When choosing the best fit line or curve it is easiest to use a transparent rulerit is easiest to use a transparent ruler
Position the ruler until it lies along an Position the ruler until it lies along an ideal lineideal line
The line or curve does not have to pass The line or curve does not have to pass through every pointthrough every point
Do not assume that all lines should pass Do not assume that all lines should pass through the originthrough the origin
Do not do play connect the dots!Do not do play connect the dots!
y
x
Uncertainties on a GraphUncertainties on a Graph
Notice that the best fitting line or curve is one that passes through the error bars of the plotted points. A straight line could not accomplish that with this data set
Analysing the GraphAnalysing the Graph
Often a relationship between variables will Often a relationship between variables will first produce a parabola, hyperbole or an first produce a parabola, hyperbole or an exponential growth or decay. These can be exponential growth or decay. These can be transformed to a straight line relationshiptransformed to a straight line relationship
General equation for a straight line is General equation for a straight line is
y = mx + cy = mx + c– y is the dependent variable, x is the independent y 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
GradientsGradients
Gradient = vertical run / horizontal runGradient = vertical run / horizontal run
gradient = gradient = y / y / xx
Don´t forget to give the units of the Don´t forget to give the units of the gradientgradient
In lab work, always report the In lab work, always report the maximum and minimum gradientmaximum and minimum gradient
Areas under GraphsAreas under Graphs
The area under a graph is a useful tool.The area under a graph is a useful tool.
For example…For example…• on a force vs. on a force vs. displacement graph the displacement graph the
area is work area is work (N x m = J)(N x m = J)• on a speed time graph the area is distance 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 Again, don´t forget the units of the areaarea
Standard Graphs - linear Standard Graphs - linear graphsgraphs
A straight line passing through the A straight line passing through the origin shows proportionalityorigin shows proportionality
y
x
y x
y = k x
Where k is the constantof proportionality
k = rise/run
Standard Graphs - Standard Graphs - parabolaparabola
A parabola shows that y is directly A parabola shows that y is directly proportional to xproportional to x22
y
x2
y
xi.e. y x2 or y = kx2
where k is the constant of proportionality
Standard Graphs - Standard Graphs - hyperbolahyperbola
A hyperbola shows that y is A hyperbola shows that y is inversely proportional to xinversely proportional to x
y
1/x
y
x
i.e. y 1/x or y = k/xwhere k is the constant of proportionality
Standard Graphs - Standard Graphs - hyperbola againhyperbola again
An inverse square law graph is also An inverse square law graph is also a hyperbolaa hyperbola
y
1/x2
y
xi.e. y 1/x2 or y = k/x2
where k is the constant of proportionality