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Topic 1: Measurement anduncertainties

1.1 – Measurements in physics

Fundamental and derived unitsFundamental SI units

Quantity SI unit Symbol

Mass Kilogram kg

Distance Meter m

Time Second sElectric current Ampere A

 Amount of substance Mole mol

Temperature Kelvin K

Derived units are combinations of fundamental units Some e!amples are: m"s #$nit for velocit%&

' #kg(m"s)*& #$nit for force&

+ #kg(m)*"s)*& #$nit for energ%&

Scientific notation and metric multipliersIn scientific notation, values are -ritten in t.e form a(1/)n, -.ere a is a number-it.in 1 and 1/ and n is an% integer Some e!amples are: T.e speed of lig.t is 0//////// #m"s& In scientific notation, t.is is e!pressed as

0(1/)  A centimeter #cm& is 1"1// of a meter #m& In scientific notation, one cm is

e!pressed as 1(1/)2* m

Metric multipliers

Prefix Abbreviation Value

peta 3 1/)14

tera T 1/)1*giga 5 1/)6

mega M 1/)7

kilo k 1/)0

.ecto . 1/)*

deca da 1/)1

deci d 1/)21

centi c 1/)2*

milli m 1/)20

micro 8 1/)27

nano n 1/)26

pico p 1/)1*femto f 1/)14

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Significant figuresFor a certain value, all figures are significant, e!cept:1 9eading eros* Trailing eros if t.is value does not .ave a decimal point, for e!ample:

1*0// .as 0 significant figures T.e t-o trailing eros are not significant //1*0// .as 4 significant figures T.e t-o leading eros are not

significant T.e t-o trailing eros are significant;.en multipl%ing or dividing numbers, t.e number of significant figures of t.e resultvalue s.ould not e!ceed t.e least precise value of t.e calculationT.e number of significant figures in an% ans-er s.ould be consistent -it. t.e number of significant figures of t.e given data in t.e <uestion

=rders of magnitude=rders of magnitude are given in po-ers of 1/, like-ise t.ose given in t.e scientificnotation section previousl%

=rders of magnitude are used to compare t.e sie of p.%sical data

Distance Manitu!e "m# $r!er of manitu!e

Diameter of t.eobservable universe

1/)*7 *7

Diameter of t.e Milk%;a% gala!%

1/)*1 *1

Diameter of t.e Solar S%stem

1/)10 10

Distance to t.e Sun 1/)11 11

>adius of t.e Eart. 1/)? ?

Diameter of a .%drogenatom 1/)21/ 1/

Diameter of a nucleus 1/)214 14

Diameter of a proton 1/)214 14

Mass Magnitude #kg& =rder of magnitude

T.e universe 1/)40 40

T.e Milk% ;a% gala!% 1/)@1 @1

T.e Sun 1/)0/ 0/

T.e Eart. 1/)*@ *@

 A .%drogen atom 1/)2*? 2*?

 An electron 1/)20/ 20/

%ime Manitu!e "s# $r!er of manitu!e Age of t.e universe 1/)1? 1?

=ne %ear 1/)? ?

=ne da% 1/)4 4

 An .our 1/)0 0

3eriod of .eart.eart 1/)/ /

Estimation

Estimations are usuall% made to t.e nearest po-er of 1/ Some e!amples are givenin t.e tables in t.e orders of magnitude section

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1.& – 'ncertainties an! errors

>andom and s%stematic errors(an!om error Systematic error  

aused b% fluctuations inmeasurements centered aroundt.e true value

an be reduced b% averaging over

repeated measurements

aused b% fi!ed s.ifts inmeasurements a-a% from t.e truevalue

annot be reduced b% averaging

over repeated measurements

E!amples: Fluctuations in room temperature

T.e noise in circuits

Buman error 

E!amples: E<uipment calibration error suc. as

t.e ero offset error  Incorrect met.od of measurement

 Absolute, fractional and percentage uncertainties3.%sical measurements are sometimes e!pressed in t.e form !C! For e!ample,1/C1 -ould mean a range from 6 to 11 for t.e measurement

Absolute uncertainty  !

)ractional uncertainty  ! "!

Percentae uncertainty  !"!(1//

alculating -it. uncertainties

A!!ition*Subtraction %aCb %aGb #sum ofabsolute uncertainties&

Multiplication*Division %a(b or %a"b %"%a"aGb"b #sum offractional uncertainties&

Po+er  %a)n %"%HnHa"a #HnH timesfractional uncertaint%&

Error barsError bars are bars on grap.s -.ic. indicate uncertainties T.e% can be .oriontal orvertical -it. t.e total lengt. of t-o absolute uncertainties

$ncertaint% of gradient and intercepts,ine of best fit: T.e straig.t line dra-n on a grap. so t.at t.e average distancebet-een t.e data points and t.e line is minimiedMaximum*Minimum line- T.e t-o lines -it. ma!imum possible slope and minimumpossible slope given t.at t.e% bot. pass t.roug. all t.e error bars

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%he uncertainty in the intercepts of a straiht line raph- T.e difference bet-eent.e intercepts of t.e line of best fit and t.e ma!imum"minimum line%he uncertainty in the ra!ient- T.e difference bet-een t.e gradients of t.e line of best fit and t.e ma!imum"minimum line

1. – Vectors an! scalars

ector and scalar <uantitiesScalar Vector  

 A <uantit% -.ic. is defined b% itsmagnitude onl%

 A <uantit% -.ic. is defined b% bot. ismagnitude and direction

E!amples: Distance

Speed

Time

Energ%

E!amples: Displacement

elocit%

 Acceleration

Force

ombination and resolution of vectorsector addition and subtraction can be done b% t.e parallelogram met.od or t.e.ead to tail met.od

;.en resolving vectors in t-o directions, vectors can be resolved into a pair of perpendicular components

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-.ere Av AsinJ and A. AcosJ

FIT.e relations.ip bet-een t-o sets of data can be determined grap.icall%

(elationship %ype of /raph Slope y0intercept

%m!Gc % against ! m c

%k!)n log% against log! n logk

%k!)nGc -it. n given % against !)n k c