Physics and Physical Physics and Physical MeasurementMeasurement

Topic 1.2 Measurement and Topic 1.2 Measurement and UncertaintiesUncertainties

The S.I. system of The S.I. system of fundamental and derived fundamental and derived unitsunits

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 countriesmost countries

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 FundamentalsThe Fundamentals

LengthLength metremetre mm MassMass kilogram kilogram kgkg TimeTime secondsecond ss Electric currentElectric current ampereampere AA Thermodynamic tempThermodynamic temp KelvinKelvin

KK 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 Derived Quantitya Derived Quantity

The units of these are called The units of these are called Derived UnitsDerived Units

The Derived UnitsThe Derived Units

Acceleration Acceleration msms-2-2

Angular acceleration Angular acceleration rad srad s--22

MomentumMomentum kgmskgms-1-1 or Ns or Ns Others have specific names and Others have specific names and

symbolssymbols Force Force kg mskg ms-2 -2 or Nor N

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 bemeasurement must be

Invariant, Accessible and Invariant, Accessible and Reproducible Reproducible

3 Standards (for 3 Standards (for information)information)

The Metre :- the distance traveled by a The Metre :- the distance traveled by a beam of light in a vacuum over a beam of light in a vacuum over a defined time interval ( 1/299 792 458 defined time interval ( 1/299 792 458 seconds)seconds)

The Kilogram :- a particular platinum-The Kilogram :- a particular platinum-iridium cylinder kept in Sevres, Franceiridium cylinder kept in Sevres, France

The Second :- the time interval The Second :- the time interval between the vibrations in the caesium between the vibrations in the caesium atom (1 sec = time for 9 192 631 770 atom (1 sec = time for 9 192 631 770 vibrations)vibrations)

ConversionsConversions

You will need to be able to convert You will need to be able to convert from one unit to another for the from one unit to another for the same quanititysame 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

KWh to JKWh to J

1 kWh = 1kW x 1 h1 kWh = 1kW x 1 h = 1000W x 60 x 60 s= 1000W x 60 x 60 s = 1000 Js= 1000 Js-1-1 x 3600 s x 3600 s = 3600000 J= 3600000 J = 3.6 x 10= 3.6 x 1066 J J

J to eVJ to eV

1 eV = 1.6 x 101 eV = 1.6 x 10-19 -19 JJ

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

i.e. we use the suffix not dashesi.e. we use the suffix not dashes

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 spread about measurements to be spread about a value rather than being spread a value rather than being spread about the accepted valueabout the accepted value

It is a system or instrument valueIt is a system or instrument value

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 variations in Are due to variations in performance of the instrument and performance of the instrument and the operatorthe operator

Even when systematic errors have Even when systematic errors have been allowed for, there exists been allowed for, there exists error.error.

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 byRandom errors can be reduced by taking multiple readings, and taking multiple readings, and

eliminating obviously erroneous eliminating obviously erroneous resultresult

or by averaging the range of 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

Limit of Reading and Limit of Reading and UncertaintyUncertainty

The The Limit of ReadingLimit of Reading of a of a measurement is equal to the smallest measurement is equal to the smallest graduation of the scale of an instrumentgraduation of the scale of an instrument

The The Degree of Uncertainty Degree of Uncertainty of a of a measurement is equal to half the limit measurement is equal to half the limit of readingof reading

e.g. If the limit of reading is 0.1cm then e.g. If the limit of reading is 0.1cm then the uncertainty range is the uncertainty range is 0.05cm0.05cm

This is the absolute uncertaintyThis is the absolute uncertainty

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 or residual The greatest deviation or residual from the mean is taken as the from the mean is taken as the absolute errorabsolute error

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

In order to compare uncertainties, In order to compare uncertainties, use is made of absolute, fractional use is made of absolute, fractional and percentage uncertainties. and percentage uncertainties.

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 percentage 0.48 % is the percentage

uncertainityuncertainity

Combining uncertainties Combining uncertainties

For addition and subtraction, add For addition and subtraction, add absolute uncertainitiesabsolute uncertainities

y = b-c then y ± y = b-c then y ± y = (b-c) ± (y = (b-c) ± (b + b + c)c)

Combining uncertaintiesCombining uncertainties

For multiplication and division add For multiplication and division add percentage uncertainitiespercentage uncertainities

x = b x c then x = b x c then xx = = bb + + cc

x b cx b c

Combining uncertaintiesCombining uncertainties

When using powers, multiply the When using powers, multiply the percentage uncertainty by the percentage uncertainty by the powerpower

z = bn then z = bn then zz = n = n bb

z bz b

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

Uncertainties in graphsUncertainties in graphs

Plotting Uncertainties on Plotting Uncertainties on GraphsGraphs

Points are plotted with a fine pencil Points are plotted with a fine pencil crosscross

Uncertainty or error bars are Uncertainty or error bars are requiredrequired

These are short lines drawn from These are 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

y

x

Uncertainties on a GraphUncertainties on a Graph

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 value or of the input data to be value or of the input data to be calculatedcalculated

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

EstimationEstimation

You need to be able to estimate values of You need to be able to estimate values of everyday objects to one or two everyday objects to one or two significant figuressignificant 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

You also need to estimate the result of You 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)

Approaching and Solving Approaching and Solving ProblemsProblems

You need to be able to state and You need to be able to state and explain any simplifying assumptions explain any simplifying assumptions that you make solving problemsthat you make solving problems

e.g. Reasonable assumptions as to e.g. Reasonable assumptions as to why certain quantities may be why certain quantities may be neglected or ignoredneglected or ignored

i.e. Heat loss, internal resistancei.e. Heat loss, internal resistance Or that behaviour is approximately Or that behaviour is approximately

linearlinear

Graphical TechniquesGraphical Techniques

Graphs are very useful for Graphs are very useful for analysing the data that is collected analysing the data that is collected during investigationsduring investigations

Graphing is one of the most Graphing is one of the most valuable tools used becausevaluable tools used because

Why GraphWhy Graph

• it gives a visual display of the it gives a visual display of the relationship between two or more relationship between two or more variablesvariables

• shows which data points do not obey shows which data points do not obey the relationshipthe relationship

• 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

e.g. as the potential difference e.g. as the potential difference increases, the ionization current increases, the ionization current also increases until it reaches a also increases until it reaches a maximum at…..maximum at…..

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

When you are asked to plot a When you are asked to plot a graph of a against b, the first graph of a against b, the first variable mentioned is plotted on variable mentioned is plotted on the y axisthe y axis

Graphs should be plotted by handGraphs should be plotted by hand

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 possibleas much space as possible

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 and symbol, as well as the relevant and symbol, as well as the relevant unit usedunit used

The graph should also be given a The graph should also be given a descriptive titledescriptive title

Plotting Graphs - Line of Plotting Graphs - Line of Best FitBest Fit

When choosing the line or curve it is best When choosing the line or curve it is best to use a transparent rulerto 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 dot to dot!Do not do dot to dot!

y

x

Analysing the GraphAnalysing the Graph

Often a relationship between variables Often a relationship between variables will first produce a parabola, hyperbole will first produce a parabola, hyperbole or an exponential growth or decay. or an exponential growth or decay. These can be transformed to a straight These can be transformed to a straight line relationshipline 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-variable, m is the gradient and c is the y-interceptintercept

The parameters of a function can The parameters of a function can also be obtained from the slope also be obtained from the slope ((mm) and the intercept () and the intercept (cc) of a ) of a straight line graphstraight line graph

GradientsGradients

Gradient = vertical run / horizontal runGradient = vertical run / horizontal run

or gradient = or gradient = y / y / xx

uphill slope is positive and downhill uphill slope is positive and downhill slope is negativeslope is negative

Don´t forget to give the units of the Don´t forget to give the units of the gradientgradient

Areas under GraphsAreas under Graphs

The area under a graph is a useful The area under a graph is a useful tooltool

e.g. on a force displacement graph e.g. on a force displacement graph the area is work (N x m = J)the area is work (N x m = J)

e.g. on a speed time graph the area e.g. on a speed time graph the area is distance (msis distance (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

Non-Standard GraphsNon-Standard Graphs

You need to make a connection You need to make a connection between graphs and equationsbetween graphs 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 aconstant

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